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Astronomy & Astrophysics manuscript no. A16˙Dark˙energy˙and˙modified˙gravity c
ESO 2015
February 6, 2015
Planck 2015 results. XIV. Dark energy and modified gravity
Planck Collaboration: P. A. R. Ade90, N. Aghanim63 , M. Arnaud77, M. Ashdown73,7, J. Aumont63 , C. Baccigalupi89, A. J. Banday101,10,
R. B. Barreiro69, N. Bartolo33,70 , E. Battaner103,104, R. Battye72 , K. Benabed64,99, A. Benoˆ
ıt61, A. Benoit-L´
evy25,64,99, J.-P. Bernard101,10,
M. Bersanelli36,52, P. Bielewicz101,10,89, A. Bonaldi72, L. Bonavera69, J. R. Bond9, J. Borrill15,94 , F. R. Bouchet64,92, M. Bucher1, C. Burigana51,34,53,
R. C. Butler51, E. Calabrese97 , J.-F. Cardoso78,1,64, A. Catalano79,76, A. Challinor66,73,13 , A. Chamballu77,17,63, H. C. Chiang29,8,
P. R. Christensen86,40, S. Church96 , D. L. Clements59, S. Colombi64,99 , L. P. L. Colombo24,71, C. Combet79 , F. Couchot74, A. Coulais76,
B. P. Crill71,12, A. Curto7,69 , F. Cuttaia51, L. Danese89, R. D. Davies72, R. J. Davis72 , P. de Bernardis35, A. de Rosa51 , G. de Zotti48,89,
J. Delabrouille1, F.-X. D´
esert57, J. M. Diego69 , H. Dole63,62, S. Donzelli52 , O. Dor´
e71,12, M. Douspis63 , A. Ducout64,59, X. Dupac42 ,
G. Efstathiou66, F. Elsner25,64,99, T. A. Enßlin83, H. K. Eriksen67, J. Fergusson13 , F. Finelli51,53, O. Forni101,10 , M. Frailis50, A. A. Fraisse29 ,
E. Franceschi51, A. Frejsel86 , S. Galeotta50, S. Galli64 , K. Ganga1, M. Giard101,10, Y. Giraud-H´
eraud1, E. Gjerløw67, J. Gonz´
alez-Nuevo69,89,
K. M. G´
orski71,106, S. Gratton73,66 , A. Gregorio37,50,56, A. Gruppuso51 , J. E. Gudmundsson29, F. K. Hansen67, D. Hanson84,71,9, D. L. Harrison66,73,
A. Heavens59, G. Helou12, S. Henrot-Versill´
e74, C. Hern´
andez-Monteagudo14,83, D. Herranz69 , S. R. Hildebrandt71,12, E. Hivon64,99, M. Hobson7,
W. A. Holmes71, A. Hornstrup18 , W. Hovest83, Z. Huang9, K. M. Huffenberger27 , G. Hurier63, A. H. Jaffe59 , T. R. Jaffe101,10, W. C. Jones29,
M. Juvela28, E. Keih¨
anen28, R. Keskitalo15 , T. S. Kisner81, J. Knoche83 , M. Kunz19,63,3, H. Kurki-Suonio28,47, G. Lagache5,63 , A. L¨
ahteenm¨
aki2,47,
J.-M. Lamarre76, A. Lasenby7,73 , M. Lattanzi34, C. R. Lawrence71 , R. Leonardi42, J. Lesgourgues98,88,75, F. Levrier76, A. Lewis26, M. Liguori33,70 ,
P. B. Lilje67, M. Linden-Vørnle18 , M. L´
opez-Caniego42,69, P. M. Lubin31, Y.-Z. Ma23,72, J. F. Mac´
ıas-P´
erez79, G. Maggio50 , D. Maino36,52,
N. Mandolesi51,34, A. Mangilli63,74 , A. Marchini54, P. G. Martin9, M. Martinelli105, E. Mart´
ınez-Gonz´
alez69, S. Masi35 , S. Matarrese33,70,45,
P. Mazzotta38, P. McGehee60, P. R. Meinhold31, A. Melchiorri35,54 , L. Mendes42, A. Mennella36,52 , M. Migliaccio66,73, S. Mitra58,71 ,
M.-A. Miville-Deschˆ
enes63,9, A. Moneti64, L. Montier101,10 , G. Morgante51, D. Mortlock59 , A. Moss91, D. Munshi90 , J. A. Murphy85,
A. Narimani23, P. Naselsky86,40, F. Nati29, P. Natoli34,4,51, C. B. Netterfield21, H. U. Nørgaard-Nielsen18 , F. Noviello72, D. Novikov82,
I. Novikov86,82, C. A. Oxborrow18, F. Paci89, L. Pagano35,54, F. Pajot63, D. Paoletti51,53, F. Pasian50, G. Patanchon1, T. J. Pearson12,60,
O. Perdereau74, L. Perotto79 , F. Perrotta89, V. Pettorino46 ?, F. Piacentini35, M. Piat1, E. Pierpaoli24 , D. Pietrobon71, S. Plaszczynski74 ,
E. Pointecouteau101,10, G. Polenta4,49 , L. Popa65, G. W. Pratt77, G. Pr´
ezeau12,71, S. Prunet64,99 , J.-L. Puget63, J. P. Rachen22,83, W. T. Reach102,
R. Rebolo68,16,41, M. Reinecke83 , M. Remazeilles72,63,1, C. Renault79, A. Renzi39,55 , I. Ristorcelli101,10, G. Rocha71,12 , C. Rosset1, M. Rossetti36,52,
G. Roudier1,76,71, M. Rowan-Robinson59, J. A. Rubi ˜
no-Mart´
ın68,41, B. Rusholme60 , V. Salvatelli35,6, M. Sandri51, D. Santos79, M. Savelainen28,47 ,
G. Savini87, B. M. Schaefer100 , D. Scott23, M. D. Seiffert71,12 , E. P. S. Shellard13, L. D. Spencer90 , V. Stolyarov7,73,95, R. Stompor1, R. Sudiwala90,
R. Sunyaev83,93, D. Sutton66,73, A.-S. Suur-Uski28,47 , J.-F. Sygnet64, J. A. Tauber43, L. Terenzi44,51, L. Toffolatti20,69,51, M. Tomasi36,52,
M. Tristram74, M. Tucci19, J. Tuovinen11, L. Valenziano51, J. Valiviita28,47, B. Van Tent80, M. Viel50,56, P. Vielva69, F. Villa51, L. A. Wade71,
B. D. Wandelt64,99,32, I. K. Wehus71, M. White30 , D. Yvon17, A. Zacchei50 , and A. Zonca31
(Affiliations can be found after the references)
February 5, 2015
ABSTRACT
We study the implications of Planck data for models of dark energy (DE) and modified gravity (MG), beyond the standard cosmological constant
scenario. We start with cases where the DE only directly affects the background evolution, considering Taylor expansions of the equation of
state w(a), as well as principal component analysis and parameterizations related to the potential of a minimally coupled DE scalar field. When
estimating the density of DE at early times, we significantly improve present constraints and find that it has to be below ≈2 % (at 95% confidence)
of the critical density even when forced to play a role for z<50 only. We then move to general parameterizations of the DE or MG perturbations that
encompass both effective field theories and the phenomenology of gravitational potentials in MG models. Lastly, we test a range of specific models,
such as k-essence, f(R) theories and coupled DE. In addition to the latest Planck data, for our main analyses we use background constraints from
baryonic acoustic oscillations, type-Ia supernovae and local measurements of the Hubble constant. We further show the impact of measurements
of the cosmological perturbations, such as redshift-space distortions and weak gravitational lensing. These additional probes are important tools
for testing MG models and for breaking degeneracies that are still present in the combination of Planck and background data sets.
All results that include only background parameterizations (expansion of the equation of state, early DE, general potentials in minimally-coupled
scalar fields or principal component analysis) are in agreement with ΛCDM. When testing models that also change perturbations (even when
the background is fixed to ΛCDM), some tensions appear in a few scenarios: the maximum one found is ∼2σfor Planck TT+lowP when
parameterizing observables related to the gravitational potentials with a chosen time dependence; the tension increases to at most 3σwhen
external data sets are included. It however disappears when including CMB lensing.
Key words. Cosmology: observations – Cosmology: theory – cosmic microwave background – dark energy – gravity
1. Introduction
The cosmic microwave background (CMB) is a key probe of
our cosmological model (Planck Collaboration XIII 2015), pro-
?Corresponding author: Valeria Pettorino, v.pettorino@thphys.
uni-heidelberg.de
viding information on the primordial Universe and its physics,
including inflationary models (Planck Collaboration XX
2015) and constraints on primordial non-Gaussianities
(Planck Collaboration XVII 2015). In this paper we use
1
arXiv:1502.01590v1 [astro-ph.CO] 5 Feb 2015
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
the 2015 data release from Planck1(Planck Collaboration I
2015) to perform a systematic analysis of a large set of dark
energy and modified gravity theories.
Observations have long shown that only a small fraction of
the total energy density in the Universe (around 5%) is in the
form of baryonic matter, with the dark matter needed for struc-
ture formation accounting for about another 26 %. In one sce-
nario the dominant component, generically referred to as dark
energy (hereafter DE), brings the total close to the critical den-
sity and is responsible for the recent phase of accelerated ex-
pansion. In another scenario the accelerated expansion arises,
partly or fully, due to a modification of gravity on cosmologi-
cal scales. Elucidating the nature of this DE and testing General
Relativity (GR) on cosmological scales are major challenges
for contemporary cosmology, both on the theoretical and ex-
perimental sides (e.g., LSST Science Collaboration et al. 2009;
Amendola et al. 2012a;Clifton et al. 2012;Joyce et al. 2014;
Huterer et al. 2015).
In preparation for future experimental investigations of DE
and modified gravity (hereafter MG), it is important to determine
what we already know about these models at different epochs
in redshift and different length scales. CMB anisotropies fix the
cosmology at early times, while additional cosmological data
sets further constrain on how DE or MG evolve at lower red-
shifts. The aim of this paper is to investigate models for dark
energy and modified gravity using Planck data in combination
with other data sets.
The simplest model for DE is a cosmological constant, Λ,
first introduced by Einstein (1917) in order to keep the Universe
static, but soon dismissed when the Universe was found to be ex-
panding (Lemaˆ
ıtre 1927;Hubble 1929). This constant has been
reintroduced several times over the years in attempts to explain
several astrophysical phenomena, including most recently the
flat spatial geometry implied by the CMB and supernova obser-
vations of a recent phase of accelerated expansion (Riess et al.
1998;Perlmutter et al. 1999). A cosmological constant is de-
scribed by a single parameter, the inclusion of which brings the
model (ΛCDM) into excellent agreement with the data. ΛCDM
still represents a good fit to a wide range of observations, more
than 20 years after it was introduced. Nonetheless, theoretical
estimates for the vacuum density are many orders of magnitude
larger than its observed value. In addition, ΩΛand Ωmare of the
same order of magnitude only at present, which marks our epoch
as a special time in the evolution of the Universe (the “coinci-
dence problem”). This lack of a clear theoretical understanding
has motivated the development of a wide variety of alternative
models. Those models which are close to ΛCDM are in broad
agreement with current constraints on the background cosmol-
ogy, but the perturbations may still evolve differently, and hence
it is important to test their predictions against CMB data.
There are at least three difficulties we had to face within this
paper. First, there appears to be a vast array of possibilities in
the literature and no agreement yet in the scientific community
on a comprehensive framework for discussing the landscape of
models. A second complication is that robust constraints on DE
come from a combination of different data sets working in con-
cert. Hence we have to be careful in the choice of the data sets
1Planck (http://www.esa.int/Planck) is a project of the
European Space Agency (ESA) with instruments provided by two sci-
entific consortia funded by ESA member states and led by Principal
Investigators from France and Italy, telescope reflectors provided
through a collaboration between ESA and a scientific consortium led
and funded by Denmark, and additional contributions from NASA
(USA).
so that we do not find apparent hints for non-standard models
that are in fact due to systematic errors. A third area of concern
is the fact that numerical codes available at present for DE and
MG are not as well tested in these scenarios as for ΛCDM, es-
pecially given the accuracy reached by the data. Furthermore,
in some cases, we need to rely on stability routines that deserve
further investigation to assure that they are not excluding more
models than required.
In order to navigate the range of modelling possibilities, we
adopt the following three-part approach.
1. Background parameterizations. Here we consider only pa-
rameterizations of background-level cosmological quanti-
ties. Perturbations are always included, but their evolution
depends only on the background. This set includes models
involving expansions, parameterizations or principal compo-
nent analyses of the equation of state w≡p/ρ of a DE fluid
with pressure pand energy density ρ. Early DE also belongs
to this class.
2. Perturbation parameterizations. Here the perturbations
themselves are parameterized or modified explicitly, not only
as a consequence of a change in background quantities.
There are two main branches we consider: firstly, effective
field theory for DE (EFT, e.g. Gubitosi et al. 2013), which
has a clear theoretical motivation, since it includes all the-
ories derived when accounting for all symmetry operators
in the Lagrangian, written in unitary gauge, i.e. in terms of
metric perturbations only. This is a very general classifica-
tion that has the advantage of providing a broad overview
of (at least) all universally coupled DE models. However, a
clear disadvantage is that the number of free parameters is
large and the constraints are consequently weak. Moreover,
in currently available numerical codes one needs to rely on
stability routines which are not fully tested and may discard
more models than necessary.
As a complementary approach, we include a more phe-
nomenological class of models obtained by directly param-
eterizing two independent functions of the gravitational po-
tentials. This approach can in principle probe all degrees of
freedom at the background and perturbation level (e.g. Kunz
2012) and is easier to handle in numerical codes. While the
connection to physical models is less obvious here than in
EFT, this approach allows us to gain a more intuitive under-
standing of the general constraining power of the data.
3. Examples of particular models. Here we focus on a se-
lection of theories that have already been discussed in the
literature and are better understood theoretically; these can
partly be considered as applications of previous cases for
which the CMB constraints are more informative, because
there is less freedom in any particular theory than in a more
general one.
The CMB is the cleanest probe of large scales, which are
of particular interest for modifications to gravity. We will inves-
tigate the constraints coming from Planck data in combination
with other data sets, addressing strengths and potential weak-
nesses of different analyses. Before describing in detail the mod-
els and data sets that correspond to our requirements, in Sect. 2
we first address the main question that motivates our paper, dis-
cussing why CMB is relevant for DE. We then present the spe-
cific model parameterizations in Sect. 3. The choice of data sets
is discussed in detail in Sec. 4before we present results in Sect. 5
and discuss conclusions in Sect. 6.
2
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
2. Why is the CMB relevant for dark energy?
The CMB anisotropies are largely generated at the last-scattering
epoch, and hence can be used to pin down the theory at early
times. In fact many forecasts of future DE or MG experiments
are for new data plus constraints from Planck. However, there
are also several effects that DE and MG models can have on the
CMB, some of which are to:
1. change the expansion history and hence distance to the last
scattering surface, with a shift in the peaks, sometimes re-
ferred to as a geometrical projection effect (Hu & White
1996);
2. cause the decay of gravitational potentials at late times, af-
fecting the low-multipole CMB anisotropies through the in-
tegrated Sachs-Wolfe (or ISW) effect (Sachs & Wolfe 1967;
Kofman & Starobinskii 1985);
3. enhance the cross-correlation between the CMB and large-
scale structure, through the ISW effect (Giannantonio et al.
2008);
4. change the lensing potential, through additional DE pertur-
bations or modifications of GR (Acquaviva & Baccigalupi
2006;Carbone et al. 2013);
5. change the growth of structure (Peebles 1984;
Barrow & Saich 1993) leading to a mismatch between the
CMB-inferred amplitude of the fluctuations Asand late-time
measurements of σ8(Kunz et al. 2004;Baldi & Pettorino
2011);
6. impact small scales, modifying the damping tail in CTT
`, giv-
ing a measurement of the abundance of DE at different red-
shifts (Calabrese et al. 2011;Reichardt et al. 2012);
7. affect the ratio between odd and even peaks if modifica-
tions of gravity treat baryons and cold dark matter differently
(Amendola et al. 2012b);
8. modify the lensing B-mode contribution, through changes in
the lensing potential (Amendola et al. 2014);
9. modify the primordial B-mode amplitude and scale depen-
dence, by changing the sound speed of gravitational waves
(Amendola et al. 2014;Raveri et al. 2014).
In this paper we restrict our analysis to scalar perturbations.
The dominant effects on the temperature power spectrum are
due to lensing and the ISW effect, as can be seen in Fig. 1,
which shows typical power spectra of temperature anisotropies
and lensing potential for modified gravity models. Different
curves correspond to different choices of the µand ηfunctions,
which change the relation between the metric potentials and the
sources, as well as introducing a gravitational slip; we will de-
fine these functions in Sect. 3.2.2, Eq. (4) and Eq. (6), respec-
tively. Spectra are obtained using a scale-independent evolution
for both µand η. The two parameters in the figure then determine
the change in amplitude of µand ηwith respect to the ΛCDM
case, in which E11 =E22 =0 and µ=η=1.
3. Models and parameterizations
We now provide an overview of the models addressed in this
paper. Details on the specific parameterizations will be discussed
in Sect. 5, where we also present the results for each specific
method.
We start by noticing that one can generally follow two dif-
ferent approaches: (1) given a theoretical set up, one can specify
the action (or Lagrangian) of the theory and derive background
and perturbation equations in that framework; or (2) more phe-
nomenologically, one can construct functions that map closely
101102103
`
0 2000 4000 6000 8000 10000
`(`+ 1)CTT
`/2πµK2
ΛCDM
E11 = 1, E22 = 1
E11 =−1, E22 =−1
E11 = 0.5, E22 = 0.5
E11 = 0, E22 = 1
101102103
`
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
[`(`+ 1)]2Cφφ
`/2πµK2
×10−7
ΛCDM
E11 = 1, E22 = 1
E11 =−1, E22 =−1
E11 = 0.5, E22 = 0.5
E11 = 0, E22 = 1
Fig. 1. Typical effects of modified gravity on theoretical CMB
temperature (top panel) and lensing potential (bottom panel)
power spectra. An increase (or decrease) of E22 with respect
to zero introduces a gravitational slip, higher at present, when
Ωde is higher (see Eq. (4) and Eq. (6)); this in turns changes the
Weyl potential and leads to a higher (or lower) lensing poten-
tial. On the other hand, whenever E11 and E22 are different from
zero (quite independently of their sign) µand ηchange in time:
as the dynamics in the gravitational potential is increased, this
leads to an enhancement in the ISW effect. Note also that even
when the temperature spectrum is very close to ΛCDM (as for
E11 =E22 =0.5) the lensing potential is still different with re-
spect to ΛCDM, shown in black.
onto cosmological observables, probing the geometry of space-
time and the growth of perturbations. Assuming spatial flatness
for simplicity, the geometry is given by the expansion rate Hand
perturbations to the metric. If we consider only scalar-type com-
ponents the metric perturbations can be written in terms of the
gravitational potentials Φand Ψ(or equivalently by any two in-
dependent combinations of these potentials). Cosmological ob-
servations thus constrain one “background” function of time
H(a) and two “perturbation” functions of scale and time Φ(k,a)
and Ψ(k,a) (e.g., Kunz 2012). These functions fix the metric,
and thus the Einstein tensor Gµν. Einstein’s equations link this
tensor to the energy-momentum tensor Tµν, which in turn can be
related to DE or MG properties.
3
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Throughout this paper we will adopt the metric given by the
line element
ds2=a2h−(1 +2Ψ)dτ2+(1 −2Φ)dx2i.(1)
The gauge invariant potentials Φand Ψare related to
the Bardeen (1980) potentials ΦAand ΦHand to the
Kodama & Sasaki (1984) potentials ΨKS and ΦKS in the follow-
ing way: Ψ = ΦA= ΨKS and Φ = −ΦH=−ΦKS. Throughout
the paper we use a metric signature (−,+++) and follow the
notation of Ma & Bertschinger (1995); the speed of light is set
to c=1, except where explicitly stated otherwise.
We define the equation of state ¯p(a)=w(a)¯ρ(a), where ¯p
and ¯ρare the average pressure and energy density. The sound
speed csis defined in the fluid rest frame in terms of pres-
sure and density perturbations as δp(k,a)=c2
s(k,a)δρ(k,a). The
anisotropic stress σ(k,a) (equivalent to πTin the notation of
Kodama & Sasaki 1984) is the scalar part of the off-diagonal
space-space stress energy tensor perturbation. The set of func-
tions {H,Φ,Ψ}describing the metric is formally equivalent to
the set of functions {w,c2
s, σ}(Ballesteros et al. 2012).
Specific theories typically cover only subsets of this function
space and thus make specific predictions for their form. In the
following sections we will discuss the particular theories that we
consider in this paper.
3.1. Background parameterizations
The first main ‘category’ of theories we describe includes pa-
rameterizations of background quantities. Even when we are
only interested in constraints on background parameters, we
are implicitly assuming a prescription for Dark Energy fluctua-
tions. The conventional approach, that we adopt also here, is to
choose a minimally-coupled scalar field model (Wetterich 1988;
Ratra & Peebles 1988), also known as quintessence, which cor-
responds to the choice of a rest-frame sound speed c2
s=1 (i.e.,
equal to the speed of light) and σ=0 (no scalar anisotropic
stress). In this case the relativistic sound speed suppresses the
dark energy perturbations on sub-horizon scales, preventing it
from contributing significantly to clustering.
Background parameterizations discussed in this paper in-
clude:
–(w0,wa) Taylor expansion at first order (and potentially
higher orders);
–Principal Component Analysis of w(a) (Huterer & Starkman
2003), that allows to estimate constraints on win indepen-
dent redshift bins;
–general parameterization of any minimally coupled scalar
field in terms of three parameters s,ζs,∞. This is a
novel way to describe minimally coupled scalar field mod-
els without explicitly specifying the form of the potential
(Huang et al. 2011);
–Dark Energy density as a function of z(including parameter-
izations such as early Dark Energy).
The specific implementation for each of them is discussed
in Sect. 5.1 together with corresponding results. We will con-
clude the background investigation by describing, in Sect. 5.1.6,
a compressed Gaussian likelihood that captures most of the
constraining power of the Planck data applied to smooth Dark
Energy or curved models (following Mukherjee et al. 2008). The
compressed likelihood is useful for example to include more
easily the Planck CMB data in Fisher-forecasts for future large-
scale structure surveys.
3.2. Perturbation parameterizations
Modified gravity models (in which gravity is modified with re-
spect to GR) in general affect both the background and the per-
turbation equations. In this subsection we go beyond background
parameterizations and identify two different approaches to con-
strain MG models, one more theoretically motivated and a sec-
ond more phenomenological one. We will not embark on a full-
scale survey of DE and MG models here, but refer the reader to
e.g. Amendola et al. (2013) for more details.
3.2.1. Modified gravity and effective field theory
The first approach starts from a Lagrangian, derived from an
effective field theory (EFT) expansion (Cheung et al. 2008),
discussed in Gubitosi et al. (2013) in the context of DE.
Specifically, EFT describes the space of (universally coupled)
scalar field theories, with a Lagrangian written in unitary gauge
that preserves isotropy and homogeneity at the background level,
assumes the weak equivalence principle, and has only one extra
dynamical field besides the matter fields conventionally consid-
ered in cosmology. The action reads:
S=Zd4x√−g
m2
0
2[1+ Ω(τ)]R+ Λ(τ)−a2c(τ)δg00
+M4
2(τ)
2a2δg002−¯
M3
1(τ)2a2δg00δKµ
µ
−
¯
M2
2(τ)
2δKµ
µ2−
¯
M2
3(τ)
2δKµ
νδKν
µ+a2ˆ
M2(τ)
2δg00δR(3)
+m2
2(τ)(gµν +nµnν)∂µa2g00∂νa2g00
+Smχi,gµν.(2)
Here Ris the Ricci scalar, δR(3) is its spatial perturbation, Kµ
ν
is the extrinsic curvature, and m0is the bare (reduced) Planck
mass. The matter part of the action, Sm, includes all fluid
components except dark energy, i.e., baryons, cold dark mat-
ter, radiation, and neutrinos. The action in Eq. (2) depends
on nine time-dependent functions (Bloomfield et al. 2013), here
{Ω,c,Λ,¯
M3
1,¯
M4
2,¯
M2
3,M4
2,ˆ
M2,m2
2}, whose choice specifies the
theory. In this way, EFT provides a direct link to any scalar field
theory. A particular subset of EFT theories are the Horndeski
(1974) models, which include (almost) all stable scalar-tensor
theories, universally coupled to gravity, with second-order equa-
tions of motion in the fields and depend on five functions of time
(Bellini & Sawicki 2014;Piazza et al. 2014).
Although the EFT approach has the advantage of being very
versatile, in practice it is necessary to choose suitable parameter-
izations for the free functions listed above, in order to compare
the action with the data. We will describe our specific choices,
together with results for each of them, in Sect. 5.2.
3.2.2. MG and phenomenological parameterizations
The second approach adopted in this paper to test MG is more
phenomenological and starts from the consideration that cosmo-
logical observations probe quantities related to the metric pertur-
bations, in addition to the expansion rate. Given the line element
of Eq. (1), the metric perturbations are determined by the two
potentials Φand Ψ, so that we can model all observationally
relevant degrees of freedom by parameterizing these two poten-
tials (or, equivalently, two independent combinations of them) as
4
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
functions of time and scale. Since a non-vanishing anisotropic
stress (proportional to Φ−Ψ) is a generic signature of modi-
fications of GR (Mukhanov et al. 1992;Saltas et al. 2014), the
parameterized potentials will correspond to predictions of MG
models.
Various parameterizations have been considered in the liter-
ature. Some of the more popular (in longitudinal gauge) are:
1. Q(a,k), which modifies the relativistic Poisson equation
through extra DE clustering according to
−k2Φ≡4πGa2Q(a,k)ρ∆,(3)
where ∆is the comoving density perturbation;
2. µ(a,k) (sometimes also called Y(a,k)), which modifies the
equivalent equation for Ψrather than Φ:
−k2Ψ≡4πGa2µ(a,k)ρ∆; (4)
3. Σ(a,k), which modifies lensing (with the lensing/Weyl po-
tential being Φ+Ψ), such that
−k2(Φ+Ψ)≡8πGa2Σ(a,k)ρ∆; (5)
4. η(a,k), which reflects the presence of a non-zero anisotropic
stress, the difference between Φand Ψbeing equivalently
written as a deviation of the ratio2
η(a,k)≡Φ/Ψ.(6)
In the equations above, ρ∆ = ρm∆m+ρr∆rso that the parameters
Q,µ, or Σquantify the deviation of the gravitational potentials
from the value expected in GR due to perturbations of matter
and relativistic particles. At low redshifts, where most DE mo-
dels become relevant, we can neglect the relativistic contribu-
tion. The same is true for η, where we can neglect the contribu-
tion of relativistic particles to the anisotropic stress at late times.
The four functions above are certainly not independent. It is
sufficient to choose two independent functions of time and scale
to describe all modifications with respect to General Relativity
(e.g. Zhang et al. 2007;Amendola et al. 2008b). Popular choices
include: (µ, η), which have a simple functional form for many
theories; (µ, Σ), which is more closely related to what we actu-
ally observe, given that CMB lensing, weak galaxy lensing and
the ISW effect measure a projection or derivative of the Weyl po-
tential Φ+Ψ. Furthermore, redshift space distortions constrain
the velocity field, which is linked to Ψthrough the Euler equa-
tion of motion.
All four quantities, Q,µ,Σ, and η, are free functions of time
and scale. Their parameterization in terms of the scale factor a
and momentum kwill be specified in Section 5.2.2, together
with results obtained by confronting this class of models with
data.
3.3. Examples of particular models
The last approach is to consider particular models. Even
though these are in principle included in the case described in
Sect. 3.2.1, it is nevertheless still useful to highlight some well
known examples of specific interest, which we list below.
–Minimally-coupled models beyond simple quintessence.
Specifically, we consider “k-essence” models, which are
defined by an arbitrary sound speed c2
sin addition to a
free equation of state parameter w(Armendariz-Picon et al.
2000).
2This parameter is called γin the code MGCAMB, but since γis also
often used for the growth index, we prefer to use the symbol η.
–An example of a generalized scalar field model
(Deffayet et al. 2010) and of Lorentz-violating massive
gravity (Dubovsky 2004;Rubakov & Tinyakov 2008), both
in the ‘equation of state’ formalism of Battye & Pearson
(2012).
–Universal “fifth forces.” We will show results for f(R) the-
ories (Wetterich 1995a;Capozziello 2002;Amendola et al.
2007;De Felice & Tsujikawa 2010), which form a subset of
all models contained in the EFT approach.
–Non-universal fifth forces. We will illustrate results for cou-
pled DE models (Amendola 2000), in which dark matter par-
ticles feel a force mediated by the DE scalar field.
All these particular models are based on specific ac-
tions, ensuring full internal consistency. The reviews by
Amendola et al. (2013), Clifton et al. (2012), Joyce et al. (2014)
and Huterer et al. (2015) contain detailed descriptions of a large
number of models discussed in the literature.
4. Data
We now discuss the data sets we use, both from Planck and in
combination with other experiments. As mentioned earlier, if
we combine many different data sets (not all of which will be
equally reliable) and take them all at face value, we risk attribut-
ing systematic problems between data sets to genuine physical
effects in DE or MG models. On the other hand, we need to avoid
bias in confirming ΛCDM, and remain open to the possibility
that some tensions may be providing hints that point towards DE
or MG models. While discussing results in Sect. 5, we will try to
assess the impact of additional data sets, separating them from
the Planck baseline choice, keeping in mind caveats that might
appear when considering some of them. For a more detailed dis-
cussion of the data sets we refer to Planck Collaboration XIII
(2015).
4.1. Planck data sets
4.1.1. Planck low-`data
The 2013 papers used WMAP polarization measurements
(Bennett et al. 2013) at multipoles `≤23 to constrain the op-
tical depth parameter τ. The corresponding likelihood was de-
noted “WP” in the 2013 papers.
For the present release, we use in its place a Planck
polarization likelihood that is built through low-resolution
maps of Stokes Qand Upolarization measured by LFI at
70 GHz (excluding data from Surveys 2 and 4), foreground-
cleaned with the LFI 30 GHz and HFI 353 GHz maps, used
as polarized synchrotron and dust templates, respectively (see
Planck Collaboration XI (2015)).
The foreground-cleaned LFI 70 GHz polarization maps
are processed, together with the temperature map from the
Commander component separation algorithm over 94 % of the
sky (see Planck Collaboration IX 2015, for further details), us-
ing the low-`Planck temperature-polarization likelihood. This
likelihood is pixel-based, extends up to multipoles `=29 and
masks the polarization maps with a specific polarization mask,
which uses 46 % of the sky. Use of this likelihood is denoted as
“lowP” hereafter.
The Planck lowP likelihood, when combined with the high-
`Planck temperature one, provides a best fit value for the
optical depth τ=0.078 ±0.019, which is about 1 σlower
than the value inferred from the WP polarization likelihood,
5
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
i.e., τ=0.089 ±0.013, in the Planck 2013 papers (see also
Planck Collaboration XIII 2015). However, we find that the LFI
70 GHz and WMAP polarization maps are extremely consistent
when both are cleaned with the HFI 353 GHz polarized dust
template, as discussed in more detail in Planck Collaboration XI
(2015).
4.1.2. Planck high-`data
Following Planck Collaboration XV (2014), the high-`part of
the likelihood (30 < ` < 2500) uses a Gaussian approximation,
−logL(ˆ
C|C(θ)) =1
2(ˆ
C−C(θ))T·C−1·(ˆ
C−C(θ)) +const. , (7)
with ˆ
Cthe data vector, C(θ) the model with parameters θand C
the covariance matrix. The data vector consists of the tempera-
ture power spectra of the best CMB frequencies of the HFI in-
strument. Specifically, as discussed in Planck Collaboration XI
(2015), we use 100 GHz, 143 GHz and 217 GHz half-mission
cross-spectra, measured on the cleanest part of the sky, avoid-
ing the Galactic plane, as well as the brightest point sources
and regions where the CO emission is the strongest. The point
source masks are specific to each frequency. We retain, 66%
of the sky for the 100 GHz map, 57 % for 143 GHz, and 47 %
for 217 GHz. All the spectra are corrected for beam and pixel
window functions. Not all cross-spectra and multipoles are in-
cluded in the data vector; specifically, the T T 100 ×143 and
100 ×217 cross-spectra, which do not bring much extra infor-
mation, are discarded. Similarly, we only use multipoles in the
range 30 < ` < 1200 for 100 ×100 and 30 < ` < 2000 for
143 ×143, discarding modes where the S/N is too low. We do
not co-add the different cross-frequency spectra, since, even af-
ter masking the highest dust-contaminated regions, each cross-
frequency spectrum has a different, frequency-dependent resid-
ual foreground contamination, which we deal with in the model
part of the likelihood function.
The model, C(θ) can be rewritten as
Cµ,ν(θ)=Ccmb +Cfg
µ,ν(θ)
pAµAν
,(8)
where Ccmb is the set of CMB C`s, which is independent of
frequency, Cfg
µ,ν(θ) is the foreground model contribution to the
cross-frequency spectrum µ×ν, and Aµthe calibration factor
for the µ×µspectrum. We retain the following contributions in
our foreground modelling: dust; clustered cosmic infrared back-
ground (CIB); thermal Sunyaev-Zeldovich (tSZ) effect; kinetic
Sunyaev-Zeldovich (kSZ) effect; tSZ-CIB cross-correlations;
and point sources. The dust, CIB and point source contributions
are the dominant contamination. Specifically, dust is the domi-
nant foreground at ` < 500, while the diffuse point source term
(and CIB for the 217 ×217) dominates the small scales. All our
foreground models are based upon smooth C`templates with
free amplitudes. All templates but the dust are based on analyti-
cal models, as described in Planck Collaboration XI (2015). The
dust is based on a mask difference of the 545 GHz map and is
well described by a power law of index n=−2.63, with a wide
bump around `=200. A prior for the dust amplitude is com-
puted from the cross-spectra with the 545 GHz map. We refer
the reader to Planck Collaboration XI (2015) for a complete de-
scription of the foreground model. The overall calibration for the
100×100 and 217×217 power spectra free to vary within a prior
measured on a small fraction of the sky near the Galactic pole.
The covariance matrix Caccounts for the correlations
due to the mask and is computed following the equations in
Planck Collaboration XV (2014). The fiducial model used to
compute the covariance is based on a joint fit of ΛCDM and nui-
sance parameters. The covariance includes the non-Gaussianity
of the noise, but assumes Gaussian statistics for the dust. The
non-whiteness of the noise is estimated from the difference be-
tween the cross- and auto-half mission spectra and accounted
for in an approximate manner in the covariance. Different Monte
Carlo based corrections are applied to the covariance matrix cal-
culation to account for inaccuracies in the analytic formulae at
large scales (` < 50) and when dealing with the point source
mask. Beam-shape uncertainties are folded into the covariance
matrix. A complete description of the computation and its vali-
dation is discussed in Planck Collaboration XI (2015).
The T T unbinned covariance matrix is of size about 8000 ×
8000. When adding the polarization, the matrix has size 23000×
23000, which translates into a significant memory requirement
and slows the likelihood computation considerably. We thus bin
the data and covariance matrix, using a variable bin-size scheme,
to reduce the data vector dimension by about a factor of ten. We
checked that for the ΛCDM model, including single parameter
classical extensions, the cosmological and nuisance parameter
fits are identical with or without binning.
4.1.3. Planck CMB lensing
Gravitational lensing by large-scale structure introduces depen-
dencies in CMB observables on the late-time geometry and clus-
tering, which otherwise would be degenerate in the primary
anisotropies (Hu 2002;Lewis & Challinor 2006). This provides
some sensitivity to dark energy and late-time modifications of
gravity from the CMB alone. The source plane for CMB lensing
is the last-scattering surface, so the peak sensitivity is to lenses
at z≈2 (i.e., half-way to the last-scattering surface) with typi-
cal sizes of order 102Mpc. Although this peak lensing redshift is
rather high for constraining simple late-time dark energy mod-
els, CMB lensing deflections at angular multipoles `<
∼60 have
sources extending to low enough redshift that DE becomes dy-
namically important (e.g., Pan et al. 2014).
The main observable effects of CMB lensing are a smooth-
ing of the acoustic peaks and troughs in the temperature
and polarization power spectra, the generation of significant
non-Gaussianity in the form of a non-zero connected 4-point
function, and the conversion of E-mode to B-mode polar-
ization. The smoothing effect on the power spectra is in-
cluded routinely in all results in this paper. We addition-
ally include measurements of the power spectrum Cφφ
`of the
CMB lensing potential φ, which are extracted from the Planck
temperature and polarization 4-point functions, as presented
in Planck Collaboration XV (2015) and discussed further below.
Lensing also produces 3-point non-Gaussianity, which peaks
in squeezed configurations, due to the correlation between the
lensing potential and the ISW effect in the large-angle tem-
perature anisotropies. This effect has been measured at around
3σwith the full-mission Planck data (Planck Collaboration XV
2015;Planck Collaboration XXI 2015). Although in principle
this is a further probe of DE (Verde & Spergel 2002) and
MG (Acquaviva et al. 2004), we do not include these T–φcorre-
lations in this paper as the likelihood was not readily available.
We plan however to test this effect in future work.
The construction of the CMB lensing likelihood we use
in this paper is described fully in Planck Collaboration XV
6
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
(2015); see also Planck Collaboration XIII (2015). It is a sim-
ple Gaussian approximation in the estimated Cφφ
`bandpow-
ers, covering the multipole range 40 ≤`≤400. The Cφφ
`are
estimated from the full-mission temperature and polariza-
tion 4-point functions, using the SMICA component-separated
maps (Planck Collaboration IX 2015) over approximately 70 %
of the sky. A large number of tests of internal consistency
of the estimated Cφφ
`to different data cuts (e.g., whether
polarization is included, or whether individual frequency
bands are used in place of the SMICA maps) are reported
in Planck Collaboration XV (2015). All such tests are passed
for the conservative multipole range 40 ≤`≤400 that we
adopt in this paper. For multipoles ` > 400, there is marginal
evidence of systematic effects in reconstructions of the lens-
ing deflections from temperature anisotropies alone, based on
curl-mode tests. Reconstructing the lensing deflections on large
angular scales is very challenging because of the large “mean-
field” due to survey anisotropies, which must be carefully sub-
tracted with simulations. We conservatively adopt a minimum
multipole of `=40 here, although the results of the null tests
considered in Planck Collaboration XV (2015) suggest that this
could be extended down to `=8. For Planck, the multipole
range 40 ≤`≤400 captures the majority of the S/N on Cφφ
`
for ΛCDM models, although this restriction may be more lossy
in extended models. The Planck 2014 lensing measurements are
the most significant to date (the amplitude of Cφφ
`is measured at
greater than 40 σ), and we therefore choose not to include lens-
ing results from other CMB experiments in this paper.
4.1.4. Planck CMB polarization
The T E and E E likelihood follows the same principle as the T T
likelihood described in Sect. 4.1.2. The data vector is extended
to contain the T E and EE cross-half-mission power spectra
of the same 100 GHz, 143 GHz, and 217 GHz frequency maps.
Following Planck Collaboration Int. XXX (2014), we mask the
regions where the dust intensity is important, and retain 70 %,
50 %, and 41 % of the sky for our three frequencies. We ig-
nore any other polarized galactic emission and in particular
synchrotron, which has been shown to be negligible, even at
100 GHz. We use all of the cross-frequency spectra, using the
multipole range 30 < ` < 1000 for the 100 GHz cross-spectra
and 500 < ` < 2000 for the 217 GHz cross-spectra. Only the
143 ×143 spectrum covers the full 30 < ` < 2000 range.
We use the same beams as for the T T spectra and do not cor-
rect for leakage due to beam mismatch. A complete descrip-
tion of the beam mismatch effects and correction is described
in Planck Collaboration XI (2015).
The model is similar to the T T one. We retain a single fore-
ground component accounting for the polarized emission of the
dust. Following Planck Collaboration Int. XXX (2014), the dust
C`template is a power law with index n=−2.4. A prior for
the dust amplitude is measured in the cross-correlation with the
353 GHz maps. The calibration parameters are fixed to unity.
The covariance matrix is extended to polarization, as de-
scribed in Planck Collaboration XI (2015), using the correlation
between the T T ,T E, and E E spectra. It is computed similarly
to the T T covariance matrix, as described in Sect. 4.1.2.
In this paper we will only show results that include CMB
high-`polarization data where we find that it has a significant
impact. DE and MG can in principle also affect the B-mode
power spectrum through lensing of B-modes (if the lensing Weyl
potential is modified) or by changing the position and ampli-
tude of the primordial peak (Pettorino & Amendola 2014), in-
cluding modifications of the sound speed of gravitational waves
(Amendola et al. 2014;Raveri et al. 2014). Due to the unavail-
ability of the likelihood, results from B-mode polarization are
left to future work.
4.2. Background data combination
We identify a first basic combination of data sets that we mostly
rely on, for which we have a high confidence that systematics are
under control. Throughout this paper, we indicate for simplicity
with “BSH” the combination BAO +SN-Ia +H0, which we now
discuss in detail.
4.2.1. Baryon acoustic oscillations
Baryon acoustic oscillations (BAO) are the imprint of oscilla-
tions in the baryon-photon plasma on the matter power spec-
trum and can be used as a standard ruler, calibrated to the CMB-
determined sound horizon at the end of the drag epoch. Since
the acoustic scale is so large, BAO are largely unaffected by
nonlinear evolution. As in the cosmological parameter paper,
Planck Collaboration XIII (2015), BAO is considered as the pri-
mary data set to break parameter degeneracies from CMB mea-
surements and offers constraints on the background evolution of
MG and DE models. The BAO data can be used to measure both
the angular diameter distance DA(z), and the expansion rate of
the Universe H(z) either separately or through the combination
DV(z)="(1 +z)2D2
A(z)cz
H(z)#1/3
.(9)
As in Planck Collaboration XIII (2015) we use: the SDSS
Main Galaxy Sample at zeff=0.15 (Ross et al. 2014); the
Baryon Oscillation Spectroscopic Survey (BOSS) “LOWZ”
sample at zeff=0.32 (Anderson et al. 2014); the BOSS CMASS
(i.e. “constant mass” sample) at zeff=0.57 of Anderson et al.
(2014); and the six-degree-Field Galaxy survey (6dFGS) at
zeff=0.106 (Beutler et al. 2011). The first two measurements
are based on peculiar velocity field reconstructions to sharpen
the BAO feature and reduce the errors on the quantity DV/rs; the
analysis in Anderson et al. (2014) provides constraints on both
DA(zeff) and H(zeff). In all cases considered here the BAO obser-
vations are modelled as distance ratios, and therefore provide no
direct measurement of H0. However, they provide a link between
the expansion rate at low redshift and the constraints placed by
Planck at z≈1100.
4.2.2. Supernovae
Type-Ia supernovae (SNe) are among the most important probes
of expansion and historically led to the general acceptance that
a DE component is needed (Riess et al. 1998;Perlmutter et al.
1999). Supernovae are considered as “standardizable candles”
and so provide a measurement of the luminosity distance as a
function of redshift. However, the absolute luminosity of SNe
is considered uncertain and is marginalized out, which also re-
moves any constraints on H0.
Consistently with Planck Collaboration XIII (2015), we use
here the analysis by Betoule et al. (2013) of the “Joint Light-
curve Analysis” (JLA) sample. JLA is constructed from the
SNLS and SDSS SNe data, together with several samples of
low redshift SNe. Cosmological constraints from the JLA sam-
7
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
ple3are discussed by Betoule et al. (2014), and as mentioned in
Planck Collaboration XIII (2015) the constraints are consistent
with the 2013 and 2104 Planck values for standard ΛCDM.
4.2.3. The Hubble constant
The CMB measures mostly physics at the epoch of recombina-
tion, and so provides only weak direct constraints about low-
redshift quantities through the integrated Sachs-Wolfe effect and
CMB lensing. The CMB-inferred constraints on the local ex-
pansion rate H0are model dependent, and this makes the com-
parison to direct measurements interesting, since any mismatch
could be evidence of new physics.
Here, we rely on the re-analysis of the Riess et al. (2011)
(hereafter R11) Cepheid data made by Efstathiou (2014) (here-
after E14). By using a revised geometric maser distance to NGC
4258 from Humphreys et al. (2013), E14 obtains the following
value for the Hubble constant:
H0=(70.6±3.3) km s−1Mpc−1,(10)
which is within 1 σof the Planck TT+lowP estimate. In this pa-
per we use Eq. (10) as a conservative H0prior. We note that
the 2015 Planck TT+lowP value is perfectly consistent with
the 2013 Planck value (Planck Collaboration XVI 2014) and
so the tension with the R11 H0determination is still present
at about 2.4σ. We refer to the cosmological parameter paper
Planck Collaboration XIII (2015) for a more comprehensive dis-
cussion of the different values of H0present in the literature.
4.3. Perturbation data sets
The additional freedom present in MG models can be calibrated
using external data that test perturbations in particular. In the
following we describe other available data sets that we included
in the grid of runs for this paper.
4.3.1. Redshift space distortions
Observations of the anisotropic clustering of galaxies in red-
shift space permit the measurement of their peculiar velocities,
which are related to the Newtonian potential Ψvia the Euler
equation. This, in turn, allows us to break a degeneracy with
gravitational lensing that is sensitive to the combination Φ+Ψ.
Galaxy redshift surveys now provide very precise constraints on
redshift-space clustering. The difficulty in using these data is
that much of the signal currently comes from scales where non-
linear effects and galaxy bias are significant and must be accu-
rately modelled (see, e.g., the discussions in Bianchi et al. 2012;
Gil-Mar´
ın et al. 2012). Moreover, adopting the wrong fiducial
cosmological model to convert angles and redshifts into dis-
tances can bias measurements of the rate-of-growth of structure
(Reid et al. 2012;Howlett et al. 2014). Significant progress in
the modelling has been achieved in the last few years, so we
shall focus here on the most recent (and relatively conservative)
studies. A compilation of earlier measurements can be found in
the references above.
In linear theory, anisotropic clustering along the line of
sight and in the transverse directions measures the combination
f(z)σ8(z), where the growth rate is defined by
f(z)=d ln σ8
d ln a .(11)
3ACosmoMC likelihood module for the JLA sample is available at
http://supernovae.in2p3.fr/sdss_snls_jla/ReadMe.html.
where σ8is calculated including all matter and neutrino den-
sity perturbations. Anisotropic clustering also contains ge-
ometric information from the Alcock-Paczynski (AP) ef-
fect (Alcock & Paczynski 1979), which is sensitive to
FAP(z)=(1 +z)DA(z)H(z).(12)
In addition, fits which constrain RSD frequently also mea-
sure the BAO scale, DV(z)/rs, where rsis the comoving sound
horizon at the drag epoch, and DVis given in Eq. (9). As
in Planck Collaboration XIII (2015) we consider only analyses
which solve simultaneously for the acoustic scale, FAP and fσ8.
The Baryon Oscillation Spectroscopic Survey (BOSS) col-
laboration have measured the power spectrum of their CMASS
galaxy sample (Beutler et al. 2014) in the range k=0.01–
0.20 h Mpc−1.Samushia et al. (2014) have estimated the mul-
tipole moments of the redshift-space correlation function of
CMASS galaxies on scales >25 h−1Mpc. Both papers provide
tight constraints on the quantity fσ8, and the constraints are
consistent. The Samushia et al. (2014) result was shown to be-
have marginally better in terms of small-scale bias compared
to mock simulations, so we choose to adopt this as our base-
line result. Note that when we use the data of Samushia et al.
(2014), we exclude the measurement of the BAO scale, DV/rs,
from Anderson et al. (2013), to avoid double counting.
The Samushia et al. (2014) results are expressed as a 3 ×3
covariance matrix for the three parameters DV/rs,FAP and
fσ8, evaluated at an effective redshift of zeff=0.57. Since
Samushia et al. (2014) do not apply a density field reconstruc-
tion in their analysis, the BAO constraints are slightly weaker
than, though consistent with, those of Anderson et al. (2014).
4.3.2. Galaxy weak lensing
The distortion of the shapes of distant galaxies by large-scale
structure along the line of sight (weak gravitational lensing or
cosmic shear) is particularly important for constraining DE and
MG, due to its dependence on the growth of fluctuations and the
two scalar metric potentials.
Currently the largest weak lensing (WL) survey is the
Canada France Hawaii Telescope Lensing Survey (CFHTLenS),
and we make use of two data sets from this survey:
1. 2D CFHTLenS data (Kilbinger et al. 2013), whose shear
correlation functions ξ±are estimated in the angular range
0.9 to 296.5 arcmin;
2. the tomographic CFHTLenS blue galaxy sample
(Heymans et al. 2013), whose data have an intrinsic
alignment signal consistent with zero, eliminating the need
to marginalize over any additional nuisance parameters, and
where the shear correlation functions are estimated in six
redshift bins, each with an angular range 1.7< θ < 37.9
arcmin.
Since these data are not independent we do not combine them,
but rather check the consistency of our results with each. The
galaxy lensing convergence power spectrum, Pκ
i j(`), can be writ-
ten in terms of the Weyl potential, PΦ+Ψ , by
Pκ
i j(`)≈2π2`Zdχ
χgi(χ)gj(χ)PΦ+Ψ (`/χ, χ),(13)
where we have made use of the Limber approximation in flat
space, and χis the comoving distance. The lensing efficiency is
given by
gi(χ)=Z∞
χ
dχ0niχ0χ0−χ
χ0,(14)
8
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
0.15 0.30 0.45 0.60
Ωm
0.6
0.8
1.0
σ8
Planck TT+lowP
WL + HL1
WL + HL4
WL + BF
WL + IA
WL (linear)
30
40
50
60
70
80
90
100
H0
Fig. 2. Ωm–σ8constraints for tomographic lensing from
Heymans et al. (2013), using a very conservative angular cut, as
described in the text (see Sect. 4.3.2). We show results using lin-
ear theory, nonlinear corrections from Halofit (HL) versions
1, 4, marginalization over baryonic AGN feedback (BF), and
intrinsic alignment (IA) (the latter two using nonlinear correc-
tions and Halofit 4). Coloured points indicate H0values from
WL+HL4.
where ni(χ)is the radial distribution of source galaxies in bin i.
In the case of no anisotropic stress and no additional clustering
from the DE, the convergence power spectrum can be written in
the usual form
Pκ
i j(`)=9
4Ω2
mH4
0Z∞
0
gi(χ)gj(χ)
a2(χ)P(`/χ, χ)dχ . (15)
However, in this paper we always use the full Weyl potential to
compute the theoretical WL predictions. The convergence can
also be written in terms of the correlation functions ξ±via
ξ±
i,j(θ)=1
2πZd` ` Pκ
i j(`)J±(`θ),(16)
where the Bessel functions are J+=J0and J−=J4.
In this paper we need to be particularly careful about the
contribution of nonlinear scales to ξ±, since the behaviour of MG
models in the nonlinear regime is not known very precisely. The
standard approach is to correct the power spectrum on nonlinear
scales using the Halofit fitting function. Since its inception,
there have been several revisions to improve the agreement with
N-body simulations. We use the following convention to label
the particular Halofit model:
1. the original model of Smith et al. (2003);
2. an update from higher resolution N-body simulations, to in-
clude the effect of massive neutrinos (Bird et al. 2012);
3. an update to improve the accuracy on small scales4;
4. an update from higher resolution N-body simulations, in-
cluding DE cosmologies with constant equation of state
(Takahashi et al. 2012).
Given this correction, one can scale the Weyl potential transfer
functions by the ratio of the nonlinear to linear matter power
4http://www.roe.ac.uk/˜jap/haloes/
spectrum
TΦ+Ψ(k,z)→TΦ+Ψ (k,z)sPnonlin
δ(k,z)
Plin
δ(k,z).(17)
Both (Kilbinger et al. 2013) and (Heymans et al. 2013) quote
a “conservative” set of cuts to mitigate uncertainty over the non-
linear modelling scheme. For the 2D analysis of Kilbinger et al.
(2013) angular scales θ < 170are excluded for ξ+, and θ < 540
for ξ−. For the tomographic analysis of Heymans et al. (2013),
angular scales θ < 30are excluded for ξ+for any bin combina-
tion involving the two lowest redshift bins, and no cut is applied
for the highest four redshift bins. For ξ−, angular scales θ < 300
are excluded for any bin combination involving the four lowest
redshift bins, and θ < 160for the highest two bins.
These cuts, however, may be insufficient for our purposes,
since we are interested in extensions to ΛCDM. We therefore
choose a very conservative set of cuts to mitigate the total con-
tribution from nonlinear scales. In order to select these cuts we
choose the baseline Planck TT+lowP ΛCDM cosmology as de-
scribed in Planck Collaboration XIII (2015), for which one can
use Eq. (15). The cuts are then chosen by considering ∆χ2=
|χ2
lin −χ2
nonlin|of the WL likelihood as a function of angular cut.
In order for this to remain ∆χ2<1 for each of the Halofit
versions, we find it necessary to remove ξ−entirely from each
data set, and exclude θ < 170for ξ+for both the 2D and tomo-
graphic bins. We note that a similar approach to Kitching et al.
(2014) could also be followed using 3D CFHTLenS data, where
the choice of cut is more well defined in k-space, however the
likelihood for this was not available at the time of this paper.
On small scales, the effects of intrinsic alignments and bary-
onic feedback can also become significant. In order to check
the robustness of our cuts to these effects we adopt the same
methodology of MacCrann et al. (2014). Using the same base-
line model and choosing Halofit version 4, we scale the mat-
ter power spectrum by an active galactic nuclei (AGN) com-
ponent, derived from numerical simulations (van Daalen et al.
2011), marginalizing over an amplitude αAGN. The AGN bary-
onic feedback model has been shown by Harnois-D´
eraps et al.
(2014) to provide the best fit to small-scale CFHTLens data. For
intrinsic alignment we adopt the model of Bridle & King (2007),
including the additional nonlinear alignment contributions to ξ±,
and again marginalizing over an amplitude αIA. For more details
on this procedure, we refer the reader to MacCrann et al. (2014).
The robustness of our ultra-conservative cuts to nonlinear
modelling, baryonic feedback and intrinsic alignment margina-
lization, is illustrated in Fig. 2for the tomographic data, with
similar constraints obtained from 2D data. Assuming the same
base ΛCDM cosmology, and applying priors of Ωbh2=0.0223±
0.0009, ns=0.96 ±0.02, and 40 km s−1Mpc−1<H0<
100 km s−1Mpc−1to avoid over-fitting the model, we find that
the WL likelihood is insensitive to nonlinear physics. We there-
fore choose to adopt the tomographic data with the ultra-
conservative cuts as our baseline data set.
4.4. Combining data sets
We show for convenience in Table 1the schematic summary
of models. All models have been tested for the combina-
tions: Planck,Planck+BSH, Planck+WL, Planck+BAO/RSD
and Planck+WL+BAO/RSD. Throughout the text, unless other-
wise specified, Planck refers to the baseline Planck TT+lowP
combination. The effects of CMB lensing and Planck TT,TE,EE
9
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Table 1. Table of models tested in this paper. We have tested all models for the combinations: Planck,Planck+BSH, Planck+WL,
Planck+BAO/RSD and Planck+WL+BAO/RSD. Throughout the text, unless otherwise specified, Planck refers to the baseline
Planck TT+lowP combination. The effects of CMB lensing and Planck TT,TE,EE polarization have been tested on all runs above
and are, in particular, used to constrain the amount of DE at early times.
Model Section
ΛCDM . . . . . . . . . . . . . . . . . . . Planck Collaboration XIII (2015)
Background parameterizations:
w....................... Planck Collaboration XIII (2015)
w0,wa. . . . . . . . . . . . . . . . . . . . Sect. 5.1.1: Figs. 3,4,5
whigher order expansion . . . . . . Sect. 5.1.1
1-parameter w(a) . . . . . . . . . . . . Sect. 5.1.2: Fig. 6
wPCA . . . . . . . . . . . . . . . . . . . Sect. 5.1.3: Fig. 7
s,ζs,∞. . . . . . . . . . . . . . . . . . Sect. 5.1.4: Figs. 8,9
Early DE . . . . . . . . . . . . . . . . . . Sect. 5.1.5: Figs. 10,11
Perturbation parameterizations:
EFT exponential . . . . . . . . . . . . Sect. 5.2.1: Fig. 12
EFT linear . . . . . . . . . . . . . . . . . Sect. 5.2.1: Fig. 13
µ, η scale-independent:
DE-related . . . . . . . . . . . . . . . Sect. 5.2.2: Figs. 1,14,15,16,17
time related . . . . . . . . . . . . . . Sect. 5.2.2: Figs. 14,16
µ, η scale-dependent: . . . . . . . . .
DE-related . . . . . . . . . . . . . . . Sect. 5.2.2: Fig. 18
time related . . . . . . . . . . . . . . Sect. 5.2.2
Other particular examples:
DE sound speed and k-essence . . Sect. 5.3.1
Equation of state approach: . . . .
Lorentz-violating massive gravity Sect. 5.3.2
Generalized scalar fields . . . . . Sect. 5.3.2
f(R) . . . . . . . . . . . . . . . . . . . . . Sect. 5.3.3: Figs. 19,20
Coupled DE . . . . . . . . . . . . . . . . Sect. 5.3.4: Figs. 21,22
polarization have been tested on all runs above and are, in parti-
cular, used to constrain the amount of DE at early times. For each
of them we indicate the section in which the model is described
and the corresponding figures. In addition, all combinations in
the table have been tested with and without CMB lensing. The
impact of Planck high-`polarization has been tested on all mo-
dels for the combination Planck+BAO+SNe+H0.
5. Results
We now proceed by illustrating in detail the models and parame-
terizations described in Sect. 3, through presenting results for
each of them. The structure of this section is as follows. We
start in Sect. 5.1 with smooth dark energy models that are ef-
fectively parameterized by the expansion history of the Universe
alone. In Sect. 5.2 we study the constraints on the presence of
non-negligible dark energy perturbations, both in the context
of general modified gravity models described through effective
field theories and with phenomenological parameterizations of
the gravitational potentials and their combinations, as illustrated
in Sect. 3.2.2. The last part, Sect. 5.3, illustrates results for a
range of particular examples often considered in the literature.
5.1. Background parameterizations
In this section, we consider models where DE is a generic
quintessence-like component with equation of state w≡p/ρ,
where pand ρare the spatially averaged (background) DE pres-
sure and density. Although it is important to include, as we do,
DE perturbations, models in this section have a sound speed that
is equal to the speed of light, which means that they are smooth
on sub-horizon scales (see Sect. 3.1 for more details). We start
with Taylor expansions and a principal component analysis of
win a fluid formalism, then consider actual quintessence mo-
dels parameterized through their potentials and finally study the
limits that can be put on the abundance of DE density at early
times. At the end of the sub-section we provide the necessary in-
formation to compress the Planck CMB power spectrum into a
4-parameter Gaussian likelihood for applications where the full
likelihood is too unwieldy.
5.1.1. Taylor expansions of wand w0,waparameterization
If the dark energy is not a cosmological constant with w=−1
then there is no reason why wshould remain constant. In order
to test a time-varying equation of state, we expand w(a) in a
Taylor series. The first order corresponds to the {w0,wa}case,
also discussed in Planck Collaboration XIII (2015):
w(a)=w0+(1 −a)wa.(18)
We use the parameterized post-Friedmann (PPF) model of
Hu & Sawicki (2007) and Fang et al. (2008) to allow for values
w<−1. Marginalized posterior distributions for w0,wa,H0and
σ8are shown in Fig. 3and the corresponding 2D contours can
be found in Fig. 4for wavs w0and for σ8vs Ωm. Results from
Planck TT+lowP+BSH data are shown in blue and corresponds
to the combination we consider the most secure, which in this
case also gives the strongest constraints. This is expected, since
the BAO and SNe data included in the BSH combination provide
the best constraints on the background expansion rate. Results
for weak lensing (WL) and redshift space distortions (RSD) are
also shown, both separately and combined. The constraints from
10
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
−3−2−1 0
w0
0.0
0.2
0.4
0.6
0.8
1.0
P/Pmax
−2−1 0 1
wa
45 60 75 90
H0
0.6 0.75 0.9 1.05
σ8
Planck TT+lowP
Planck TT+lowP+BSH
Planck TT+lowP+WL
Planck TT+lowP+BAO/RSD
Planck TT+lowP+WL+BAO/RSD
Fig. 3. Parameterization {w0,wa}(see Sect. 5.1.1). Marginalized posterior distributions for w0,wa,H0and σ8for various data combi-
nations. The tightest constraints come from the Planck TT+lowP+BSH combination, which indeed tests background observations,
and is compatible with ΛCDM.
−2−1 0 1
w0
−3
−2
−1
0
1
2
wa
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
0.2 0.3 0.4
Ωm
0.75
0.90
1.05
1.20
σ8
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 4. Marginalized posterior distributions of the (w0,wa) parameterization (see Sect. 5.1.1) for various data combinations. The best
constraints come from the priority combination and are compatible with ΛCDM. The dashed lines indicate the point in parameter
space (−1,0) corresponding to the ΛCDM model. CMB lensing and polarization do not significantly change the constraints. Here
Planck indicates Planck TT+lowP.
these probes are weaker, since we are considering a smooth dark
energy model where the perturbations are suppressed on small
scales. While the WL data appear to be in slight tension with
ΛCDM, according to the green contours shown in Fig. 4, the
difference in total χ2between the best-fit in the {w0,wa}model
and in ΛCDM for Planck TT+lowP+WL is ∆χ2=−5.6, which
is not very significant for 2 extra parameters (for normal er-
rors a 2 σdeviation corresponds to a χ2absolute difference of
6.2). The WL contributes a ∆χ2of −2.0 and the ∆χ2
CMB =−3.3
(virtually the same as when using Planck TT+lowP alone, for
which ∆χ2
CMB =−3.2, which seems to indicate that WL is not
in tension with Planck TT+lowP within a (w0,wa) cosmology).
However, as also discussed in Planck Collaboration XIII (2015),
these data combinations prefer very high values of H0, which is
visible also in the third panel of Fig. 3. The combination Planck
TT+lowP+BSH, on the other hand, is closer to ΛCDM, with a
total χ2difference between (w0,wa) and ΛCDM of only −0.8.
We also show in Fig. 5the equation of state reconstructed as a
function of redshift from the linear expansion in the scale factor
afor different combinations of data.
One might wonder whether it is reasonable to stop at first
order in w(a). We have therefore tested a generic expansion in
powers of the scale factor up to order N:
w(a)=w0+
N
X
i=1
(1 −a)iwi.(19)
We find that all parameters are very stable when allowing higher
order polynomials; the wiparameters are weakly constrained and
going from N=1 (the linear case) to N=2 (quadratic case) to
N=3 (cubic expansion) does not improve the goodness of fit
and stays compatible with ΛCDM, which indicates that a linear
parameterization is sufficient.
5.1.2. 1-parameter varying w
A simple example of a varying wmodel that can be written in
terms of one extra parameter only (instead of w0,wa) was pro-
posed in Gott & Slepian (2011), motivated in connection to a
DE minimally-coupled scalar field, slowly rolling down a po-
tential 1
2m2φ2, analogous to the one predicted in chaotic in-
flation (Linde 1983). More generally, one can fully characte-
rize the background by expanding a varying equation of state
11
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
w(z)≡ −1+δw(z)≈ −1+δw0×H2
0/H2(z), where:
H2(z)
H2
0≈Ωm(1 +z)3+ Ωde "(1 +z)3
Ωm(1 +z)3+ Ωde #δw0/Ωde
,(20)
at first order in δw0, which is then the only extra parameter.
Marginalized posterior contours in the plane h–δw0are shown
in Fig. 6. The tightest constraints come from the combination
Planck TT+lowP+lensing+BSH that gives δw0=−0.008 ±
0.068 at 68 % confidence level, which slightly improves con-
straints found by Aubourg et al. (2014).
5.1.3. Principal Component Analysis on w(z)
A complementary way to measure the evolution of the equation
of state, which is better able to model rapid variations, proceeds
by choosing win Nfixed bins in redshift and by performing a
principal component analysis to uncorrelate the constraints. We
consider N=4 different bins in zand assume that whas a con-
stant value piin each of them. We then smooth the transition
from one bin to the other such that:
w(z)=pi−1+ ∆wtanh z−zi
s+1for z<zi,i{1, 4},(21)
with ∆w≡(pi−pi−1)/2, a smoothing scale s=0.02, and a
binning zi=(0,0.2,0.4,0.6,1.8). We have tested also a larger
number of bins (up to N=18) and have found no improvement
in the goodness of fit.
The constraints on the vector pi(i=1,...,N) of values
that w(z) can assume in each bin is difficult to interpret, due to
the correlations between bins. To uncorrelate the bins, we per-
form a principal component analysis (Huterer & Starkman 2003;
Huterer & Cooray 2005;Said et al. 2013). We first run COSMOMC
(Lewis & Bridle 2002) on the original binning values pi; then
extract the covariance matrix that refers to the parameters we
want to constrain:
C≡DppTE− h pihpTi,(22)
012345
z
−1.5
−1.0
−0.5
0.0
w(z)
Planck+BSH
Planck+WL+BAO/RSD
Fig. 5. Reconstructed equation of state w(z) as a function of red-
shift (see Sect. 5.1.1), when assuming a Taylor expansion of w(z)
to first-order (N=1 in Eq. 19), for different combinations of
the data sets. The coloured areas show the regions which con-
tain 95 % of the models. The central blue line is the median
line for Planck TT+lowP+BSH. Here Planck indicates Planck
TT+lowP.
−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6
δw0
60
70
80
90
H0
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 6. Marginalized posterior contours in the h–δw0plane are
shown for 1-parameter varying wmodels (see Sect. 5.1.2)
for different data combinations. Here Planck indicates Planck
TT+lowP.
where pis the vector of parameters piand pTis its transpose.
We calculate the Fisher matrix, F=C−1, and diagonalize it,
F=OTΛO, where Λis diagonal and Ois the orthogonal ma-
trix whose columns are the eigenvectors of the Fisher matrix.
We then define e
W=OTΛ1/2O(e.g., Huterer & Cooray 2005)
and normalize this such that its rows sum up to unity; this ma-
trix can be used to find the new vector q=e
W p of uncorrelated
parameters that describe w(z). This choice of ˜
Whas been shown
to be convenient, since most of the weights (i.e., the rows of
e
W) are found to be positive and fairly well localized in redshift.
In Fig. 7(lower panel) we show the weights for each bin as a
function of redshift. Because they overlap only partially, we can
assume the binning to be the same as the original one and attach
to each of them error bars corresponding to the mean and stan-
dard deviations of the qvalues. The result is shown in Fig. 7, top
panel. The equation of state is compatible with the ΛCDM value
w=−1. Note however that this plot contains more information
than a Taylor expansion to first order.
5.1.4. Parameterization for a weakly-coupled canonical
scalar field.
We continue our investigation of background parameterizations
by considering a slowly rolling scalar field. In this case, as in
inflation, we can avoid writing down an explicit potential V(φ)
and instead parameterize w(a) at late times, in the presence of
matter, as (Huang et al. 2011)
w=−1+2
3sF2 a
ade !,(23)
where the “slope parameter” sis defined as:
s≡V|a=ade ,(24)
with V≡(dln V
dφ)2M2
P/2 being a function of the slope of the po-
tential. Here MP≡1/√8πGis the reduced Planck mass and ade
is the scale factor where the total matter and DE densities are
12
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
0.0 0.5 1.0 1.5 2.0
z
−1.5
−1.0
−0.5
0.0
w(z)
Planck+BSH
0.0 0.5 1.0 1.5 2.0
z
0.0
0.2
0.4
0.6
0.8
1.0
Weight
bin 0
bin 1
bin 2
bin 3
Fig. 7. PCA analysis constraints (described in Sect. 5.1.3). The
top panel shows the reconstructed equation of state w(z) after the
PCA analysis. Vertical error bars correspond to mean and stan-
dard deviations of the qvector parameters, while horizontal error
bars are the amplitude of the original binning. The bottom panel
shows the PCA corresponding weights on w(z) as a function of
redshift for the combination Planck TT+lowP+BSH.
equal. The function F(x) in Eq. (23) is defined as:
F(x)≡√1+x3
x3/2−ln x3/2+√1+x3
x3.(25)
Eq. (23) parameterizes w(a) with one parameter s, while ade
depends on Ωmand sand can be derived using an approxi-
mated fitting formula that facilitates numerical computation
(Huang et al. 2011). Positive (negative) values of scorrespond
to quintessence (phantom) models.
Eq. (23) is only valid for late-Universe slow-roll (V.1
and ηV≡M2
PV00/V1) or the moderate-roll (V.1 and
ηV.1) regime. For quintessence models, where the scalar field
rolls down from a very steep potential, at early times V(a)1,
however the fractional density Ωφ(a)→0 and the combination
V(a)Ωφ(a) aprroaches a constant, defined to be a second param-
eter ∞≡lima→0V(a)Ωφ(a).
One could also add a third parameter ζsto capture the time-
dependence of Vvia corrections to the functional dependence
of w(a) at late time. This parameter is defined as the relative
difference of dpVΩφ/dy at a=ade and at a→0, where y≡
(a/ade)3/2/p1+(a/ade )3. If ∞1, ζsis proportional to the
second derivative of ln V(φ), but for large ∞, the dependence is
more complicated (Huang et al. 2011). In other words, while s
is sensitive to the late time evolution of 1 +w(a), ∞captures
its early time behaviour. Quintessence/phantom models can be
mapped into s–∞space and the classification can be further
refined with ζs. For ΛCDM, all three parameters are zero.
In Fig. 8we show the marginalized posterior distribu-
tions at 68.3 % and 95.4 % confidence levels in the param-
eter space s–Ωm, marginalizing over the other parameters.
In Fig. 9we show the current constraints on quintessence
models projected in s–∞space. The constraints are ob-
tained by marginalizing over all other cosmological parameters.
The models here include exponentials V=V0exp(−λφ/MP)
(Wetterich 1988), cosines from pseudo-Nambu Goldstone
bosons (pnGB) V=V0[1 +cos(λφ/MP)] (Frieman et al.
1995;Kaloper & Sorbo 2006), power laws V=V0(φ/MP)−n
(Ratra & Peebles 1988), and models motivated by supergrav-
ity (SUGRA) V=V0(φ/MP)−αexp [(φ/MP)2] (Brax & Martin
1999). The model projection is done with a fiducial Ωm=0.3
cosmology. We have verified that variations of 1 % compared to
the fiducial Ωmlead to negligible changes in the constraints.
Mean values and uncertainties for a selection of cosmo-
logical parameters are shown in Table 2, for both the 1-
parameter case (i.e., sonly, with ∞=0 and ζs=0, de-
scribing “thawing” quintessence/phantom models, where ˙
φ=
0 in the early Universe) and the 3-parameter case (general
quintessence/phantom models where an early-Universe fast-
rolling phase is allowed). When we vary the data sets and the-
oretical prior (between the 1-parameter and 3-parameter cases),
the results are all compatible with ΛCDM and mutually compat-
ible with each other. Because sand ∞are correlated, caution
has to be taken when looking at the marginalized constraints
in the table. For instance, the constraint on sis tighter for the
3-parameter case, because in this case flatter potentials are pre-
ferred in the late Universe in order to slow-down larger ˙
φfrom
the early Universe. A better view of the mutual consistency can
be obtained from Fig. 9. We find that the addition of polariza-
tion data does not have a large impact on these DE parameters.
Adding polarization data to Planck+BSH shifts the mean of s
by −1/6σand reduces the uncertainty of sby 20 %, while the
95 % upper bound on ∞remains unchanged.
5.1.5. Dark energy density at early times
Quintessence models can be divided into two classes, namely
cosmologies with or without DE at early times. Although the
equation of state and the DE density are related to each other,
it is often convenient to think directly in terms of DE density
rather than the equation of state. In this section we provide a
more direct estimate of how much DE is allowed by the data
as a function of time. A key parameter for this purpose is Ωe,
which measures the amount of DE present at early times (“early
dark energy,” EDE) (Wetterich 2004). Early DE parameteriza-
tions encompass features of a large class of dynamical DE mo-
dels. The amount of early DE influences CMB peaks and can be
strongly constrained when including small-scale measurements
and CMB lensing. Assuming a constant fraction of Ωeuntil re-
cent times (Doran & Robbers 2006), the DE density is parame-
terized as:
Ωde(a)=Ω0
de −Ωe(1 −a−3w0)
Ω0
de + Ω0
ma3w0
+ Ωe(1 −a−3w0).(26)
13
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Parameter Planck+BSH (1-param.) Planck+BSH (3-param.) Planck+WL+BAO/RSD Planck+lensing+BSH
s............... −0.08+0.32
−0.32 −0.11+0.16
−0.12 0.14+0.17
−0.25 −0.03+0.16
−0.17
∞. . . . . . . . . . . . . . . fixed =0≤0.76 (95% CL) ≤0.38 (95 % CL) ≤0.52 (95% CL)
ζs. . . . . . . . . . . . . . . fixed =0 not constrained not constrained not constrained
Table 2. Marginalized mean values and 68 % CL errors for a selection of cosmological parameters for the weakly-coupled scalar
field parameterization described in the text (Sect. 5.1.4). Here “1-param” in the first column refers to the priors ∞=0 and ζs=0
(slow- or moderate-roll “thawing” models).
0.24 0.28 0.32 0.36 0.40
Ωm
−1.2
−0.6
0.0
0.6
s
Planck+WL
Planck+BSH
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 8. Marginalized posterior distributions showing 68 % and
95 % C.L. constraints on Ωmand sfor scalar field models (see
Sect. 5.1.4). The dashed line for s=0 is the ΛCDM model.
Here Planck indicates Planck TT+lowP.
0.00 0.25 0.50 0.75 1.00
∞
−0.4
0.0
0.4
0.8
1.2
s
Quintessence
Phantom
ΛCDM
1 + cos 4φ
e−φ
e−2φ
1 + cos 5φ
φ−11eφ2/2
φ−1/4
φ−1/2φ−1
Planck+WL+BAO/RSD
Planck+lensing+BSH
Planck+BSH
Fig. 9. Marginalized posterior distributions at 68 % C.L. and
95 % C.L. in the parameter space of sand ∞for scalar field
modes (see Sect. 5.1.4). We have computed sand ∞for various
quintessence potentials V(φ), with the functional forms of V(φ)
labelled on the figure. The field φis in reduced Planck mass MP
units. The normalization of V(φ) is computed using Ωm=0.3.
Here Planck indicates Planck TT+lowP.
0.000 0.004 0.008 0.012 0.016
Ωe
0.0
0.2
0.4
0.6
0.8
1.0
P/Pmax
Planck+lensing+BSH
Planck+lensing+WL
Planck+lensing+BAO/RSD
Planck+lensing+WL+BAO/RSD
Planck TT,TE,EE+lowP
Fig. 10. Marginalized posterior distributions for Ωefor the early
DE parameterization in (26) and for different combinations of
data (see Sect. 5.1.5). Here Planck indicates Planck TT+lowP.
This expression requires two parameters in addition to those of
ΛCDM, namely Ωeand w0, while Ω0
m=1−Ω0
de is the present
matter abundance. The strongest constraints to date were dis-
cussed in (Planck Collaboration XVI 2014), finding Ωe<0.010
at 95 % CL using Planck combined with WMAP polarization.
Here we update the analysis using Planck 2015 data. In Fig. 10
we show marginalized posterior distributions for Ωefor different
combination of data sets; the corresponding marginalized lim-
its are shown in Table 3, improving substantially current con-
straints, especially when the Planck TT,TE,EE+lowP polariza-
tion is included, leading to Ωe<0.0036 at 95% confidence level
for Planck TT,TE,EE+lowP+BSH.
As first shown in Pettorino et al. (2013), bounds on Ωecan be
weaker if DE is present only over a limited range of redshifts. In
particular, EDE reduces structure growth in the period after last
scattering, implying a smaller number of clusters as compared
to ΛCDM, and therefore a weaker lensing potential to influence
the anisotropies at high `. It is possible to isolate this effect by
switching on EDE only after last scattering, at a scale factor ae
(or equivalently for redshifts smaller than ze). Here we adopt the
parameterization “EDE3” proposed in Pettorino et al. (2013) to
which we refer for more details:
Ωde(a)=
Ωde0
Ωde0 + Ωm0a−3+ Ωr0 a−4for a≤ae;
Ωefor ae<a<ac:
Ωde0
Ωde0 + Ωm0a−3+ Ωr0 a−4for a>ac.
(27)
14
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Parameter Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT,TE,EE+lowP
+lensing+BSH +lensing+WL +lensing+BAO/RSD +lensing+WL+BAO/RSD +BSH
Ωe. . . . . . . . . . . . <0.0071 <0.0087 <0.0070 <0.0070 <0.0036
w0. . . . . . . . . . . . <−0.93 <−0.76 <−0.90 <−0.90 <−0.94
Table 3. Marginalized 95 % limits on Ωeand w0for the early DE parameterization in Eq. (26) and different combinations of data
(see Sect. 5.1.5). Including high-`polarization significantly tightens the bounds.
10.0 100.0 1000.0
1 + ze
0.0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Ωe
Planck+lensing+BSH
Planck+lensing+WL+BAO/RSD
Planck TT,TE,EE+lowP+BSH
Fig. 11. Amount of DE at early times Ωeas a function of the
redshift zeafter which early DE is non-negligible (see Eq. (27),
Sect. 5.1.5) for different combinations of data sets. The heights
of the columns give the limit at 95 % CL on Ωe, as obtained from
Monte Carlo runs for the values ze=10, 50, 200 and 1000. The
width of the columns has no physical meaning and is just due to
plotting purposes. Here Planck indicates Planck TT+lowP.
In this case, early dark energy is present in the time interval
ae<a<ac, while outside this interval it behaves as in ΛCDM,
including the radiation contribution, unlike in Eq. (26). During
that interval in time, there is a non-negligible EDE contribu-
tion, parameterized by Ωe. The constant acis fixed by continuity,
so that the parameters {Ωe,ae}fully determine how much EDE
there was and how long its presence lasted. We choose four fixed
values of aecorresponding to ze=10, 50, 200 and 1000 and in-
clude Ωeas a free parameter in MCMC runs for each value of
ae. Results are shown in Fig. 11 where we plot Ωeas a function
of the redshift zeat which DE starts to be non-negligible. The
smaller the value of ze, the weaker are the constraints, though
still very tight, with Ωe<
∼2 % (95 % CL) for ze≈50.
5.1.6. Compressed likelihood
Before concluding the set of results on background parame-
terizations, we discuss here how to reduce the full likelihood
information to few parameters. As discussed for example in
Kosowsky et al.(2002) and Wang & Mukherjee (2007), it is pos-
sible to compress a large part of the information contained in
the CMB power spectrum into just a few numbers5: here we
use specifically the CMB shift parameter R(Efstathiou & Bond
1999), the angular scale of the sound horizon at last scattering `A
5There are also alternative approaches that compress the power spec-
tra directly, like e.g. PICO (Fendt & Wandelt 2006).
(or equivalently θ∗), as well the baryon density ωband the scalar
spectral index ns. The first two quantities are defined as
R≡qΩmH2
0DA(z∗)/c, `A≡πDA(z∗)/rs(z∗)=π/θ∗,(28)
where DA(z) is the comoving angular diameter distance to red-
shift z,z∗is the redshift for which the optical depth is unity
and rs(z∗)=r∗is the comoving size of the sound horizon at
z∗. These numbers are effectively observables and they apply to
models with either non-zero curvature or a smooth DE compo-
nent (Mukherjee et al. 2008). It should be noted, however, that
the constraints on these quantities, especially on R, are sensitive
to changes in the growth of perturbations. This can be seen easi-
ly with the help of the “dark degeneracy” (Kunz 2009), i.e., the
possibility to absorb part of the dark matter into the dark energy,
which changes Ωmwithout affecting observables. For this rea-
son the compressed likelihood presented here cannot be used for
models with low sound speed or modifications of gravity (and is
therefore located at the end of this “background” section).
The marginalized mean values and 68 % confidence inter-
vals for the compressed likelihood values are shown in Table 4
for Planck TT+lowP. The posterior distribution of {R, `A, ωb,ns}
is approximately Gaussian, which allows us to specify the like-
lihood easily by giving the mean values and the covariance ma-
trix, as derived from a Monte Carlo Markov chain (MCMC) ap-
proach, in this case from the grid chains for the wCDM model.
Since these quantities are very close to observables directly
derivable from the data, and since smoothly parameterized DE
models are all compatible with the Planck observations to a com-
parable degree, they lead to very similar central values and es-
sentially the same covariance matrix. The Gaussian likelihood
in {R, `A, ωb,ns}given by Table 4is thus useful for combin-
ing Planck temperature and low-`polarization data with other
data sets and for inclusion in Fisher matrix forecasts for future
surveys. This is especially useful when interested in parameters
such as {w0,wa}, for which the posterior is very non-Gaussian
and cannot be accurately represented by a direct covariance ma-
trix (as can be seen in Fig. 4).
The quantities that make up the compressed likelihood are
supposed to be “early universe observables” that describe the ob-
served power spectrum and are insensitive to late time physics.
However, lensing by large-scale structure has an important
smoothing effect on the C`and is detected at over 10 σin the
power spectrum (see section 5.2 of Planck Collaboration XIII
(2015)). We checked by comparing different MCMC chains that
the compressed likelihood is stable for ΛCDM, wCDM and the
{w0,wa}model. However, the “geometric degeneracy” in curved
models is broken significantly by the impact of CMB lensing on
the power spectrum (see Fig. 25 of Planck Collaboration XIII
2015) and for non-flat models one needs to be more careful. For
this reason we also provide the ingredients for the compressed
likelihood marginalized over the amplitude ALof the lensing
power spectrum in the lower part of Table 4. Marginalizing
15
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Smooth DE models Planck TT+lowP R`AΩbh2ns
R. . . . . . . . . . . . . . . . . . . 1.7488 ±0.0074 1.0 0.54 −0.63 −0.86
`A. . . . . . . . . . . . . . . . . . . 301.76 ±0.14 0.54 1.0−0.43 −0.48
Ωbh2. . . . . . . . . . . . . . . . 0.02228 ±0.00023 −0.63 −0.43 1.0 0.58
ns. . . . . . . . . . . . . . . . . . . 0.9660 ±0.0061 −0.86 −0.48 0.58 1.0
Marginalized over ALPlanck TT+lowP R`AΩbh2ns
R. . . . . . . . . . . . . . . . . . . 1.7382 ±0.0088 1.0 0.64 −0.75 −0.89
`A. . . . . . . . . . . . . . . . . . . 301.63 ±0.15 0.64 1.0−0.55 −0.57
Ωbh2. . . . . . . . . . . . . . . . . 0.02262 ±0.00029 −0.75 −0.55 1.0 0.71
ns. . . . . . . . . . . . . . . . . . . 0.9741 ±0.0072 −0.89 −0.57 0.71 1.0
Table 4. Compressed likelihood discussed in Sect. 5.1.6. The left columns give the marginalized mean values and standard deviation
for the parameters of the compressed likelihood for Planck TT+lowP, while the right columns present the normalized covariance or
correlation matrix Dfor the parameters of the compressed likelihood for Planck TT+lowP. The covariance matrix Cis then given
by Ci j =σiσjDi j (without summation), where σiis the standard deviation of parameter i. While the upper values were derived for
wCDM and are consistent with those of ΛCDM and the {w0,wa}model, we marginalized over the amplitude of the lensing power
spectrum for the lower values, which leads to a more conservative compressed likelihood.
over ALincreases the errors in some variables by over 20 %
and slightly shifts the mean values, giving a more conserva-
tive choice for models where the impact of CMB lensing on the
power spectrum is non-negligible.
We notice that the constraints on {R, `A, ωb,ns}given in
Table 4for Planck TT+lowP data are significantly weaker
than those predicted by table II of Mukherjee et al. (2008),
which were based on the “Planck Blue Book” specifications
Planck Collaboration (2005). This is because these forecasts also
used high-`polarization. If we derive the actual Planck covari-
ance matrix for the Planck TT,TE,EE+lowP likelihood then we
find constraints that are about 50 % smaller than those given
above, and are comparable and even somewhat stronger than
those quoted in Mukherjee et al. (2008). The mean values have
of course shifted to represent what Planck has actually mea-
sured.
5.2. Perturbation parameterizations
Up to now we have discussed in detail the ensemble of back-
ground parameterizations, in which DE is assumed to be a
smooth fluid, minimally interacting with gravity. General modi-
fications of gravity, however, change both the background and
the perturbation equations, allowing for contribution to cluste-
ring (via a sound speed different than unity) and anisotropic
stress different from zero. Here we illustrate results for pertur-
bation degrees of freedom, approaching MG from two differ-
ent perspectives, as discussed in Sect. 3. First we discuss results
for EFT cosmologies, with a “top-down” approach that starts
from the most general action allowed by symmetry and selects
from there interesting classes belonging to so-called “Horndeski
models”, which, as mentioned in Sect. 3.2.1, include almost all
stable scalar-tensor theories, universally coupled, with second-
order equations of motion in the fields. We then proceed by pa-
rameterizing directly the gravitational potentials and their com-
binations, as illustrated in Sect. 3.2.2. In this way we can test
more phenomenologically their effect on lensing and clustering,
in a “bottom-up” approach from observations to theoretical mo-
dels.
5.2.1. Modified gravity: EFT and Horndeski models
The first of the two approaches described in Sect. 3.2.1 adopts
effective field theory (EFT) to investigate DE (Gubitosi et al.
2013), based on the action of Eq. (2). The parameters that
appear in the action, when choosing the nine time-dependent
functions {Ω,c,Λ,¯
M3
1,¯
M4
2,¯
M2
3,M4
2,ˆ
M2,m2
2}, describe the effec-
tive DE. The full background and perturbation equations for
this action have been implemented in the publicly available
Boltzmann code EFTCAMB (Hu et al. 2014b)6. Given an expan-
sion history (which we fix to be ΛCDM, i.e., effectively w=−1)
and an EFT function Ω(a), EFTCAMB computes cand Λfrom
the Friedmann equations and the assumption of spatial flatness
(Hu et al. 2014a). As we have seen in Sect. 5.1, for smooth DE
models the constraints on the DE equation of state are compat-
ible with w=−1; hence this choice is not a limitation for the
following analysis. In addition, EFTCAMB uses a set of stabil-
ity criteria in order to specify whether a given model is stable
and ghost-free, i.e. without negative energy density for the new
degrees of freedom. This will automatically place a theoretical
prior on the parameter space while performing the MCMC anal-
ysis.
The remaining six functions, ¯
M3
1,¯
M4
2,¯
M2
3,M4
2,ˆ
M2,m2
2, are
internally redefined in terms of the dimensionless parameters αi
with irunning from 1 to 6:
α4
1=M4
2
m2
0H2
0
, α3
2=
¯
M3
1
m2
0H0
, α2
3=
¯
M2
2
m2
0
,
α2
4=
¯
M2
3
m2
0
, α2
5=ˆ
M2
m2
0
, α2
6=m2
2
m2
0
.
We will always demand that
m2
2=0 (or equivalently α2
6=0),(29)
¯
M2
3=−¯
M2
2(or equivalently α2
4=−α2
3),(30)
which eliminates models containing higher-order spatial deriva-
tives (Bellini & Sawicki 2014). In this case the nine functions of
time discussed above reduce to a minimal set of five functions
of time that can be labelled {αM, αK, αB, αT, αH}, in addition to
the Planck mass M2
∗(the evolution of which is determined by H
6http://www.lorentz.leidenuniv.nl/˜hu/codes/, version
1.1, Oct. 2014.
16
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
and αM), and an additional function of time describing the back-
ground evolution, e.g., H(a). The former are related to the EFT
functions via the following relations (Bellini & Sawicki 2014):
M2
∗=m2
0Ω + ¯
M2
2; (31)
M2
∗HαM=m2
0˙
Ω + ˙
¯
M2
2; (32)
M2
∗H2αK=2c+4M4
2; (33)
M2
∗HαB=−m2
0˙
Ω−¯
M3
1; (34)
M2
∗αT=−¯
M2
2; (35)
M2
∗αH=2ˆ
M2−¯
M2
2.(36)
These five αfunctions are closer to a physical description
of the theories under investigation. For example: αTenters in
the equation for gravitational waves, affecting their speed and
the position of the primordial peak in B-mode polarization; αM
affects the lensing potential, but also the amplitude of the pri-
mordial polarization peak in B-modes (Amendola et al. 2014;
Raveri et al. 2014;Pettorino & Amendola 2014). It is then pos-
sible to relate the desired choice for the Horndeski variables to
an appropriate choice of the EFT functions,
∂τ(M2
∗)=HM2
∗αM,(37)
m2
0(Ω + 1) =(1 +αT)M2
∗,(38)
¯
M2
2=−αTM2
∗,(39)
4M4
2=M2
∗H2αK−2c,(40)
¯
M3
1=−M2
∗HαB+m2
0˙
Ω,(41)
2ˆ
M2=M2
∗(αH−αT),(42)
where His the conformal Hubble function, m0the bare Planck
mass and M∗the effective Plank mass. Fixing αMcorresponds
to fixing M∗through Eq. (37). Once αThas been chosen, Ωis
obtained from Eq. (38). Finally, αBdetermines ¯
M3
1via Eq. (41),
while the choice of αHfixes ˆ
M2via Eq. (42). In this way, our
choice of the EFT functions can be guided by the selection of
different “physical” scenarios, corresponding to turning on dif-
ferent Horndeski functions.
To avoid possible consistency issues with higher derivatives,
we set7¯
M2
3=¯
M2
2=0 in order to satisfy Eq. (30). From Eq. (39)
and Eq. (31) this implies αT=0, so that tensor waves move with
the speed of light. In addition, we set αH=0 so as to remain
in the original class of Horndeski theories, avoiding operators
that may give rise to higher-order time derivatives (Gleyzes et al.
2014). As a consequence, ˆ
M2=0 from Eq. (42) and M2
∗=
m2
0(1 + Ω) from Eq. (31). For simplicity we also turn offall other
higher-order EFT operators and set ¯
M3
1=M4
2=0. Comparing
Eq. (32) and Eq. (41), this implies αB=−αM.
In summary, in the following we consider Horndeski models
in which αM=−αB,αKis fixed by Eq. (34), with M2=0 as a
function of cand αT=αH=0. We are thus considering non-
minimally coupled “K-essence” type models, similar to the ones
discussed in Sawicki et al. (2013).
The only free function in this case is αM, which is linked to
Ωthrough:
αM=a
Ω + 1
dΩ
da .(43)
By choosing a non-zero αM(and therefore a time evolving Ω)
we introduce a non-minimal coupling in the action (see Eq. 2),
7Because of the way EFTCAMB currently implements these equations
internally, it is not possible to satisfy Eq. (30) otherwise.
which will lead to non-zero anisotropic stress and to modifica-
tions of the lensing potential, typical signatures of MG models.
Here we will use a scaling ansatz, αM=αM0aβ, where αM0 is
the value of αMtoday, and β > 0 determines how quickly the
modification of gravity decreases in the past.
Integrating Eq. (43) we obtain
Ω(a)=exp (αM0
βaβ)−1,(44)
which coincides with the built-in exponential model of EFTCAMB
for Ω0=αM0/β. The marginalized posterior distributions for
the two parameters Ω0and βare plotted in Fig. 12 for different
combinations of data. For αM0 =0 we recover ΛCDM. For small
values of Ω0and for β=1, the exponential reduces to the built-in
linear evolution in EFTCAMB,
Ω(a)= Ω0a.(45)
The results of the MCMC analysis are shown in Table 5. For
both the exponential and the linear model we use a flat prior
Ω0∈[0,1]. For the scaling exponent βof the exponential model
we use a flat prior β∈(0,3]. For β→0 the MG parameter αM
remains constant and does not go to zero in the early Universe,
while for β=3 the scaling would correspond to M functions in
the action (2) which are of the same order as the relative energy
density between DE and the dark matter background, similar to
the suggestion in (Bellini & Sawicki 2014). An important fea-
ture visible in Fig. 12 is the sharp cutoffat β≈1.5. This cutoff
is due to “viability conditions” that are enforced by EFTCAMB
and that reject models due to a set of theoretical criteria (see
Hu et al. (2014a) for a full list of theoretical priors implemented
in EFTCAMB). Disabling some of these conditions allows to ex-
tend the acceptable model space to larger β, and we find that the
constraints on αM0continue to weaken as βgrows further, ex-
tending Fig. 12 in the obvious way. We prefer however to use
here the current public EFTCAMB version without modifications.
A better understanding of whether all stability conditions imple-
mented in the code are really necessary or exclude a larger region
than necessary in parameter space will have to be addressed in
the future. The posterior distribution of the linear evolution for Ω
is shown in Fig. 13 and is compatible with ΛCDM. Finally, it is
interesting to note that in both the exponential and the linear ex-
pansion, the inclusion of WL data set weakens constraints with
respect to Planck TT+lowP alone. This is due to the fact that
in these EFT theories, WL and Planck TT+lowP are in tension
with each other, WL preferring higher values of the expansion
rate with respect to Planck.
5.2.2. Modified gravity and the gravitational potentials
The second approach used in this paper to address MG is more
phenomenological and, as described in Sect. 3.2.2, starts from
directly parameterizing the functions of the gravitational poten-
tials listed in Eqs. (3)–(6). Any choice of two of those functions
will fully parameterize the deviations of the perturbations from
a smooth DE model and describe the cosmological observables
of an MG model.
In Simpson et al. (2013) the amplitude of the deviation with
respect to ΛCDM was parameterized similarly to the DE-related
case that we will define as case 1below, but using µ(a) and
Σ(a) instead of µ(a,k) and η(a,k)8. They found the constraints
8The parameterization of µand Σin Simpson et al. (2013) uses
ΩDE(a)/ΩDE instead of ΩDE (a); their µ0and Σ0correspond to our µ0−1
and Σ0−1 respectively.
17
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Parameter TT+lowP+BSH TT+lowP+WL TT+lowP+BAO/RSD TT+lowP+BAO/RSD+WL TT,TE,EE TT,TE,EE+BSH
Linear EFT . . . . . . . . .
αM0 .............. <0.052(95 %CL) <0.072(95 %CL) <0.057(95 %CL) <0.074(95 %CL) <0.050(95 %CL) <0.043(95 %CL)
Exponential EFT . . . . .
αM0 .............. <0.063(95 %CL) <0.092(95 %CL) <0.066(95 %CL) <0.097(95 %CL) <0.054(95 %CL) <0.062(95 %CL)
β................ 0.87+0.57
−0.27 0.91+0.54
−0.26 0.88+0.56
−0.28 0.92+0.53
−0.25 0.90+0.55
−0.26 0.92+0.53
−0.24
Table 5. Marginalized mean values and 68 % CL intervals for the EFT parameters, both in the linear model, αM0, and in the
exponential one, {αM0, β}(see Sect. 5.2.1) Adding CMB lensing does not improve the constraints, while small-scale polarization
can more strongly constraint αM0.
0.00 0.04 0.08 0.12 0.16
αM0
0.0
0.4
0.8
1.2
1.6
β
Planck
Planck+ BSH
Planck+ WL
Planck+ BAO/RSD
Planck+ BAO/RSD + WL
Fig. 12. Marginalized posterior distributions at 68% and 95 %
C.L. for the two parameters αM0 and βof the exponential evo-
lution, Ω(a)=exp(Ω0aβ)−1.0, see Sect. 5.2.1. Here αM0 is
defined as Ω0βand the background is fixed to ΛCDM. ΩM0 =0
corresponds to the ΛCDM model also at perturbation level. Note
that Planck means Planck TT+lowP. Adding WL to the data sets
results in broader contours, as a consequence of the slight ten-
sion between the Planck and WL data sets.
µ0−1=0.05 ±0.25 and Σ0−1=0.00 ±0.14 using RSD data
from the WiggleZ Dark Energy Survey (Blake et al. 2011) and
6dF Galaxy Survey (6dFGS) (Beutler et al. 2012), together with
CFHTLenS WL. Baker et al. (2014) provided forecasts on µ0−1
and Σ0−1 for a future experiment that combines galaxy cluster-
ing and tomographic weak lensing measurements. The ampli-
tude of departures from the standard values was parameterized
as in Simpson et al. (2013), but a possible scale dependence was
introduced. In Zhao et al. (2010), the authors constrained µ0and
η0and derived from those the limits on Σ0, using WMAP-5 data
along with CFHTLenS and ISW data. Together with a princi-
pal component analysis, they also constrained µand ηassuming
a time evolution of the two functions, introducing a transition
redshift zswhere the functions move smoothly from an early
time value to a late time one; they obtained µ0=1.1+0.62
−0.34, η0=
0.98+0.73
−1.0for zs=1 and µ0=0.87 ±0.12, η0=1.3±0.35 for
zs=2. A similar parametrization was also used in Daniel et al.
(2010) in terms of µ0and $(equivalent to µ0−1 and η0−1 in our
convention) using WMAP5, Union2, COSMOS and CFHTLenS
data, both binning these functions in redshift and assuming a
time evolution (different from the one we will assume in the fol-
0.00 0.04 0.08 0.12
αM0
0.0
0.2
0.4
0.6
0.8
1.0
P/Pmax
Planck
Planck + BSH
Planck + WL
Planck + BAO/RSD
Planck + BAO/RSD + WL
Fig. 13. Marginalized posterior distribution of the linear EFT
model background parameter, Ω, with Ωparameterized as a lin-
ear function of the scale factor, i.e., Ω(a)=αM0 a, see Sect.
5.2.1. The equation of state parameter wde is fixed to −1, and
therefore, Ω0=0 will correspond to the ΛCDM model. Here
Planck means Planck TT+lowP. Adding CMB lensing to the
data sets does not change the results significantly; high-`po-
larization tightens the constraints by a few percent, as shown in
Tab. 5.
lowing), obtaining −0.83 < µ0<2.1 and −1.6< $ < 2.7 at
95% confidence level for their present values. In Macaulay et al.
(2013) the authors instead parameterized Ψ/Φ(the inverse of
η) as (1 −ζ) and use RSD data from 6dFGS, BOSS, LRG,
WiggleZ and VIPERS galaxy redshift surveys to constrain de-
partures from ΛCDM; they did not assume a functional form for
the time evolution of ζ, but rather constrained its value at two
different redshifts (z=0 and z=1), finding a 2 σtension with
the ΛCDM limit (ζ=0) at z=1.
In this paper, we choose the pair of functions µ(a,k) (related
to the Poisson equation for Ψ) and η(a,k) (related to the gravita-
tional slip), as defined in Eqs. (4) and (6), since these are the
functions directly implemented in the publicly available code
MGCAMB9(Zhao et al. 2009;Hojjati et al. 2011) integrated in the
latest version of CosmoMC.
9Available at http://www.sfu.ca/˜aha25/MGCAMB.html (Feb.
2014 version), see appendix A of (Zhao et al. 2009) for a detailed de-
scription of the implementation.
18
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Parameter Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT,TE,EE+lowP
+BSH +WL +BAO/RSD +WL+BAO/RSD +BSH
E11 . . . . . . . . . . 0.099+0.34
−0.73 0.06+0.32
−0.69 −0.20+0.19
−0.47 −0.24+0.19
−0.33 −0.30+0.18
−0.30 0.08+0.33
−0.69
E22 . . . . . . . . . . 0.99 ±1.3 1.03 ±1.3 1.92+1.4
−0.96 1.77 ±0.88 2.07 ±0.85 0.9±1.2
µ0−1 . . . . . . . . 0.07+0.24
−0.51 0.04+0.22
−0.48 −0.14+0.13
−0.34 −0.17+0.14
−0.23 −0.21+0.12
−0.21 0.06+0.23
−0.48
η0−1 . . . . . . . . 0.70 ±0.94 0.72 ±0.90 1.36+1.0
−0.69 1.23 ±0.62 1.45 ±0.60 0.60 ±0.86
Σ0−1 . . . . . . . . 0.28 ±0.15 0.27 ±0.14 0.34+0.17
−0.14 0.29 ±0.13 0.31 ±0.13 0.23 ±0.13
τ. . . . . . . . . . . 0.065 ±0.021 0.063 ±0.020 0.061+0.020
−0.022 0.062 ±0.019 0.057 ±0.019 0.060 ±0.019
H0(km/s/Mpc) . 68.5±1.1 68.17 ±0.58 69.2±1.1 68.26 ±0.69 68.55 ±0.66 67.90 ±0.48
Table 6. Marginalized mean values and 68 % C.L. errors on cosmological parameters and the parameterizations of Eqs. (46) and
(47) in the DE-related case (see Sect. 5.2.2), for the scale-independent case.
Max. degeneracy Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP
+BSH +WL +BAO/RSD +WL+BAO/RSD
DE-related . . . . 0.84+0.30
−0.40 (2.1σ) 0.80+0.28
−0.39 (2.1σ) 1.08+0.35
−0.42 (2.6σ) 0.90+0.33
−0.37 (2.4σ) 1.03 ±0.34 (3.0σ)
+CMB lensing 0.42+0.18
−0.34 (1.2σ) 0.38+0.18
−0.28 (1.4σ) 0.58+0.24
−0.37 (1.6σ) 0.40+0.18
−0.28 (1.4σ) 0.51+0.21
−0.30 (1.7σ)
Time-related . . . 0.67+0.26
−0.66 (1.0σ) 0.69+0.25
−0.67 (1.0σ) 1.12+0.40
−0.64 (1.8σ) 0.55+0.25
−0.32 (1.7σ) 0.70+0.27
−0.33 (2.1σ)
Table 7. Marginalized mean values and 68 % C.L. errors on the present day value of the function 2[µ(z,k)−1] +[η(z,k)−1],
which corresponds to the (approximate) maximum degeneracy line identified within the 2 dimensional posterior distributions. This
function gives a quick idea of the maximum possible tension found for each data set combination in these classes of models, for the
scale-independent case. The upper part of the table refers to the DE-related parametrisation, with and without CMB lensing, while
the lower part refers to the time-related one (see Sect. 5.2.2). For convenience, we write explicitly in brackets for each case the
tension in units of σwith respect to the standard ΛCDM zero value. The DE-related case is more in tension than the time-related
parameterization, with a maximum tension that ranges between 2.1 σand 3σ, depending on the data sets. When CMB lensing is
included, also the DE-related parameterization becomes compatible with ΛCDM, with a maximum possible ‘tension’ of at most 1.7
σwhen WL and BAO/RSD are included.
Other functional choices can be easily derived from them
(Baker et al. 2014). We then parameterize µand ηas follows.
Since the Planck CMB data span three orders of magnitude in `,
it seems sensible to allow for two scales to be present:
µ(a,k)=1+f1(a)1+c1(λH/k)2
1+(λH/k)2; (46)
η(a,k)=1+f2(a)1+c2(λH/k)2
1+(λH/k)2.(47)
For large length scales (small k), the two functions reduce to
µ→1+f1(a)c1and η→1+f2(a)c2; for small length scales
(large k), one has µ→1+f1(a) and η→1+f2(a). In other words,
we implement scale dependence in a minimal way, allowing µ
and ηto go to two different limits for small and large scales. Here
the fiare functions of time only, while the ciand λparameters
are constants. The cigive us information on the scale dependence
of µand η, but the fimeasure the amplitude of the deviation from
standard GR, corresponding to µ=η=1.
We choose to parameterize the time dependence of the fi(a)
functions as
1. coefficients related to the DE density, fi(a)=EiiΩDE(a),
2. time-related evolution, fi(a)=Ei1 +Ei2(1 −a).
The first choice is motivated by the expectation that the contri-
bution of MG to clustering and to the anisotropic stress is pro-
portional to its effective energy density, as is the case for matter
and relativistic particles. The second parameterization provides
a complementary approach to the first: Ei1 describes the MG
contribution at late times, while Ei2 is relevant at early times.
Therefore the adoption of the time-related evolution allows, in
principle, for deviations from the standard behaviour also at high
redshift, while the parameterization connected to the DE density
leads by definition to (µ, η)→1 at high redshift, since the red-
shift evolution is tied to that of ΩDE(z).
For case 1 (referred to as “DE-related” parameterization) we
then have five free parameters, E11,c1,E22,c2, and λ, while
for case 2 (the “time-related” parameterization) we have two
additional parameters, E12 and E21. The choice above looks
very similar to the BZ parameterization (Bertschinger & Zukin
2008) for the quasi-static limit of f(R) and scalar-tensor theo-
ries. However, we emphasise that Eqs. (46)–(47) should not be
seen as a quasi-static limit of any specific theory, but rather as
a (minimal) way to allow for (arbitrary) scale dependence, since
the data cover a sufficiently wide range of scales. Analogously
to the EFT approach discussed in the previous section, we set
the background evolution to be the same as in ΛCDM, so that
w=−1. In this way the additional parameters purely probe the
perturbations.
The effect of the Eii parameters on the CMB temperature and
lensing potential power spectra has been shown in Fig. 1for the
“DE-related” choice. In the temperature spectrum the amplitude
of the ISW effect is modified; the lensing potential changes more
than the temperature spectrum for the same amplitude of the Eii
parameter.
We ran Monte Carlo simulations to compare the theoretical
predictions with different combinations of the data for both cases
1and 2. For both choices we tested whether scale dependence
19
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
plays a role (via the parameters cand λ) with respect to the
scale-independent case in which we fix c1=c2=1. Results
show that a scale dependence of µand ηdoes not lead to a
significantly smaller χ2with respect to the scale-independent
case, both for the DE-related and time-related parameterizations.
Therefore there is no gain in adding ciand λas extra degrees of
freedom. For this reason, in the following we will mainly show
results obtained for the scale-independent parameterization.
Table 6shows results for the DE-related case for different
combinations of the data. Adding the BSH data sets to the Planck
TT+lowP data does not significantly increase the constraining
power on MG parameters; Planck polarization also has little im-
pact. On the contrary, the addition of RSD data tightens the con-
straints significantly. The WL contours, including the ultracon-
servative cut that removes dependence on nonlinear physics, re-
sult in weaker constraints. In the table, µ0−1 and η0−1 are
obtained by reconstructing Eqs. (46) and (47) from E11 and E22
at the present time. In addition, the present value of the Σpa-
rameter, defined in Eq. (5), can be obtained from µand ηas
Σ = (µ/2)(1 +η) using Eqs. (4) and (6).
0.00 0.25 0.50 0.75 1.00
Σ0−1
−0.8
0.0
0.8
1.6
µ0−1
DE-related
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 15. Marginalized posterior distributions for 68% and 95%
C.L. for the two parameters {µ0−1,Σ0−1}obtained by evalu-
ating Eqs. (46) and (47) at the present time in the DE-related
parametrization when no scale dependence is considered (see
Sect. 5.2.2). Σis obtained as Σ = (µ/2)(1 +η). The time-related
evolution would give similar contours. In the labels, Planck
stands for Planck TT+lowP.
Some tension appears, in particular, when plotting the
marginalized posterior distributions in the planes (µ0−1, η0−1)
and (µ0−1, Σ0−1), as shown in Figs. 14 and 15. Here the con-
straints on the two parameters that describe the perturbations in
MG are simultaneously taken into account. In Fig. 14, left and
right panels refer to the DE-related and time-related parameteri-
zations defined in 5.2.2, respectively, while the dashed lines in-
dicate the values predicted in ΛCDM. Interestingly, results ap-
pear similar in both parameterizations. In the DE-related case
(left panel), the ΛCDM point lies at the border of the 2σcon-
tour, already when considering Planck TT+lowP alone. More
precisely, when looking at the goodness of fit, with respect to the
standard ΛCDM assumption, the MG scenario (which includes
two extra parameters E11 and E22) leads to an improvement of
∆χ2=−6.3 when using Planck TT+lowP (similarly divided be-
tween lowP and TT) and of ∆χ2=−6.4 when including BSH
(with a ∆χ2
CMB ∼ −5.6 equally divided between TT and lowP).
When Planck data (TT+lowP) are combined also with WL data,
the tension increases to ∆χ2=−10.6 (with the CMB still con-
tributing about the same amount, ∆χ2
CMB =−6.0). When con-
sidering Planck TT+lowP+BAO/RSD, ∆χ2=−8.1 with respect
to ΛCDM while, when combining both WL and BAO/RSD, the
tension is maximal, with ∆χ2=−10.8 and χ2
CMB =−6.9. There
is instead less tension for the time-related parameterization, as is
visible in the right panel of Fig. 14.
Once the behaviour of the coefficients in the two param-
eterizations is known, we can use Eq. (46) to reconstruct the
evolution of µand ηwith scale factor (or redshift, equiva-
lently). In Fig. 16 we choose to show the linear combination
2[µ(z,k)−1] +[η(z,k)−1], which corresponds approximately to
the maximum degeneracy line in the 2 dimensional µ−1, η −1
parameter space, which allows to better visualize the joint con-
straints on µand ηand their maximal allowed departure from
ΛCDM. As expected, the DE-related dependence forces the
combination to be compatible with ΛCDM in the past, when the
DE density is negligible; the time-related parameterization, in-
stead allows for a larger variation in the past.
The tension can be understood by noticing that the best fit
power spectrum corresponds to a value of µand η(E11 =−0.3,
E22 =2.2 for Planck TT+lowP) close to the thick dot dashed
line shown in Fig. 1for demonstration. This model leads to
less power in the CMB at large scales and a higher lensing
potential, which is slightly preferred by the data points with
respect to ΛCDM. This explains also why the MG parame-
ters are somewhat degenerate with the lensing amplitude AL
(which is an ‘unphysical’ parameter redefining the lensing am-
plitude that affects the CMB power spectrum). As discussed
in Planck Collaboration XIII (2015) (see for ex. Sect. 5.1.2),
ΛCDM would lead to a value of AL(Calabrese et al. 2008)
somewhat larger than 1. When varying it in MG, we find a
mean value of AL=1.116+0.095
−0.13 which is compatible with
AL=1 at 1 σ. The price to pay is the tension with ΛCDM in
MG parameter space, which compensates the need for a higher
ALthat one would have in ΛCDM. The CMB lensing likeli-
hood extracted from the 4- point function of the Planck maps
(Planck Collaboration XV 2015) on the other hand does not pre-
fer a higher lensing potential and agrees well with ΛCDM. For
this reason the tension is reduced when we add CMB lensing,
as shown in Fig. 17. We also note that constraints for this class
of model are sensitive to the estimation of the optical depth τ.
Smaller values of τtend to shift the results further away from
ΛCDM.
In order to have a quick overall estimate of the tension
for all cases discussed above, we then show in Table 7the
marginalized mean and 68% CL errors for the linear combina-
tion 2[µ(z,k)−1] +[η(z,k)−1] that . In the table, we indicate
in brackets, for convenience, the ‘tension’ with ΛCDM for each
case. This is the maximum allowed tension, since it is calcu-
lated along the maximum degeneracy direction. The DE-related
parameterization is more in tension with ΛCDM than the time-
related one. The maximum tension reaches 3 σwhen including
WL and BAO/RSD, being therefore mainly driven by external
data sets. The inclusion of CMB lensing shifts results towards
ΛCDM, as discussed.
Finally, in general, µand ηdepend not only on redshift but
also on scale, via the parameters (ci,λ). When marginalizing
over them, constraints become weaker, as expected. The com-
parison with the scale-independent case is shown in Fig. 18 for
Planck TT+lowP+BSH and different values of k. When allow-
20
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
−10123
η0−1
−1.0
−0.5
0.0
0.5
1.0
µ0−1
DE-related
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
−10123
η0−1
−1.0
−0.5
0.0
0.5
1.0
µ0−1
Time-related
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 14. 68 % and 95 % contour plots for the two parameters {µ0−1, η0−1}obtained by evaluating Eqs. (46) and (47) at the present
time when no scale dependence is considered (see Sect. 5.2.2). We consider both the DE-related (left panel) and time-related
evolution cases (right panel). Results are shown for the scale-independent case (c1=c2=1). In the labels, Planck stands for Planck
TT+lowP.
012345
z
−0.5
0.0
0.5
1.0
1.5
2.0
2[µ(z)−1] + [η(z)−1]
DE-related
Planck+BSH
Planck+WL+BAO/RSD
012345
z
−0.5
0.0
0.5
1.0
1.5
2.0
2[µ(z)−1] + [η(z)−1]
Time-related
Planck+BSH
Planck+WL+BAO/RSD
Fig. 16. Redshift dependence of the function 2[µ(z,k)−1] +[η(z,k)−1], defined in Eqs. (46,47), which corresponds to the maximum
degeneracy line identified within the 2 dimensional posterior distributions. This combination shows the strongest allowed tension
with ΛCDM. The left panel refers to the DE-related case while the right panel refers to the time-related evolution (see Sect. 5.2.2).
In both panels, no scale dependence is considered. The coloured areas show the regions containing 68 % and 95 % of the models. In
the labels, Planck stands for Planck TT+lowP.
ing for scale dependence, the tension with ΛCDM is washed out
by the weakening of the constraints and the goodness of fit does
not improve with respect to the scale independent case.
5.3. Further examples of particular models
Quite generally, DE and MG theories deal with at least one extra
degree of freedom that can usually be associated with a scalar
field. For ‘standard’ DE theories the scalar field couples mini-
mally to gravity, while in MG theories the field can be seen as the
mediator of a fifth force in addition to standard interactions. This
happens in scalar-tensor theories (including f(R) cosmologies),
massive gravity, and all coupled DE models, both when matter
is involved or when neutrino evolution is affected. Interactions
and fifth forces are therefore a common characteristic of many
proposed models, the difference being whether the interaction is
universal (i.e., affecting all species with the same coupling, as in
scalar-tensor theories) or is different for each species (as in cou-
pled DE, Wetterich 1995b;Amendola 2000 or growing neutrino
models, Fardon et al. 2004;Amendola et al. 2008a). In the fol-
lowing we will test well known examples of particular models
within all these classes.
5.3.1. Minimally coupled DE: sound speed and k-essence
In minimally coupled quintessence models, the sound speed
is c2
s=1 and DE does not contribute significantly to clus-
tering. However, in so-called “k-essence” models, the kinetic
21
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
−10123
η0−1
−1.0
−0.5
0.0
0.5
1.0
µ0−1
DE-related
Planck+BSH
Planck+lensing+BSH
−10123
η0−1
−1.0
−0.5
0.0
0.5
1.0
µ0−1
DE-related
Planck+WL+BAO/RSD
Planck+lensing+WL+BAO/RSD
Fig. 17. 68 % and 95 % marginalised posterior distributions for the two parameters {µ0−1, η0−1}obtained by evaluating Eqs. (46)
and (47) at the present time when no scale dependence is considered (see Sect. 5.2.2). Here we show the effect of CMB lensing,
which shifts the contours towards ΛCDM. In the labels, Planck stands for Planck TT+lowP.
term in the action is generalised to an arbitrary function of
(∇φ)2(Armendariz-Picon et al. 2000): the sound speed can then
be different from the speed of light and if cs1, the DE per-
turbations can become non-negligible on sub-horizon scales and
impact structure formation. To test this scenario we have per-
formed a series of analyses where we allow for a constant equa-
tion of state parameter wand a constant speed of sound c2
s(with
a uniform prior in log cs). We find that the limits on wdo not
change from the quintessence case and that there is no signifi-
cant constraint on the DE speed of sound using current data. This
can be understood as follows: on scales larger than the sound
horizon and for wclose to −1, DE perturbations are related to
−4 0 4 8
η(z= 0, k)−1
−2
−1
0
1
2
3
µ(z= 0, k)−1
k= 10−10 Mpc−1
k= 102Mpc−1
scale independent
Fig. 18. 68 % and 95 % contour plots for the two parameters
{µ0(k)−1, η0(k)−1}obtained by evaluating Eqs. (46) and
(47) at the present time for the DE-related parameterization
(see Sect. 5.2.2). We consider both the scale-independent and
scale-dependent cases, choosing kvalues of 10−10Mpc−1and
102Mpc−1.
dark matter perturbations through ∆DE '(1+w)∆m/4 and inside
the sound horizon they stop growing because of pressure sup-
port (see e.g., Sapone & Kunz 2009). In addition, at early times
the DE density is much smaller than the matter density, with
ρDE/ρm=[(1 −Ωm)/Ωm]a−3w. Since the relative DE contribu-
tion to the perturbation variable Q(a,k) defined in Eq. (3) scales
like ρDE∆DE /(ρm∆m), in k-essence type models the impact of the
DE perturbations on the total clustering is small when 1 +w≈0.
For the DE perturbations in k-essence to be detectable, the sound
speed would have had to be very small, and |1+w|relatively
large.
5.3.2. Massive gravity and generalized scalar field models
We now give two examples of subclasses of Horndeski models,
written in terms of an alternative pair of DE perturbation func-
tions (with respect to µand ηused before, for example), given
by the anisotropic stress σand the entropy perturbation Γ:
wΓ = δp
ρ−dp
dρδ . (48)
When Γ = 0 the perturbations are adiabatic, that is δp=dp
dρδρ.
For this purpose, it is convenient to adopt the ‘equation
of state’ approach described in Battye & Pearson (2012). The
gauge-invariant quantities Γand σcan be specified in terms of
the other perturbation variables, namely δρ,θ,hand ηin the
scalar sector, and their derivatives.
We then show results for two limiting cases in this for-
malism, corresponding to Lorentz-violating massive gravity
(LVMG) for which (σ,0,Γ = 0) and generalized scalar field
models (GSF) in which the anisotropic stress is zero (σ=0,Γ,
0).
Lorentz-violating massive gravity (LVMG) If the Lagrangian is
L≡L(gµν ) (i.e. only written in terms of metric perturbations,
as in the EFT action) and one imposes time translation invari-
ance (but not spatial translational invariance), one finds that this
22
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
corresponds to an extra degree of freedom, ξi, that has a physi-
cal interpretation as an elastic medium, or as Lorentz-violating
massive gravity (Dubovsky 2004;Rubakov & Tinyakov 2008;
Battye & Pearson 2013). In this case, the scalar equations are
characterized by Γ = 0 (the model is adiabatic) and a non-
vanishing anisotropic stress:
σ=(w−c2
s)δ
1+w−3η,(49)
including one degree of freedom, the sound speed c2
s, which
can be related equivalently to the shear modulus of the elas-
tic medium or the Lorentz violating mass. Tensor (gravitational
wave) equations will also include a mass term. The low sound
speed may lead to clustering of the DE fluid, which allows the
data to place constraints on c2
s. But as wapproaches −1, the DE
perturbations are suppressed and the limits on the sound speed
weaken. We can take this degeneracy between 1 +wand c2
sinto
account by using the combination λc=|1+w|αlog10 c2
sin the
MCMC analysis, where α=0.35 was chosen to decorrelate
wand λc. With this, we find Planck TT+lowP+lensing gives
lower limit of λc>−1.6 at 2σand a tighter one when includ-
ing BAO/RSD and WL, with λc>−1.3 at 2σ. For any w,−1
these limits can be translated into limits on log10 c2
sby comput-
ing λc/|1+w|α. The ΛCDM limit is however fully compatible
with the data, i.e. there is no detection of any deviation from
w=−1 (and in this limit c2
sis unconstrained).
Generalized scalar field models (GSF) One can allow for
generalized scalar fields by considering a Lagrangian L ≡
L(φ, ∂µφ, ∂µ∂νφ, gµν, ∂αgµν ), in which the dependence on the
scalar fields is made explicit, imposing full reparameteriza-
tion invariance (xµ→xν+ξµ), allowing for only linear cou-
plings in ∂αgµν and second-order field equations. In this case the
anisotropic stresses are zero and
wΓ = (α−w)(δ−3β1H(1 +w)θ−3β2H(1 +w)
2k2−6( ˙
H − H2)
˙
h
+3(1 −β2−β1)H(1 +w)
6¨
H − 18H˙
H+6H3+2k2H
¨
h).(50)
This has three extra parameters (α, β1, β2), in addition to w. If
β1=1 and β2=0 this becomes the generalized k-essence
model. An example of this class of models is “kinetic gravity
braiding” (Deffayet et al. 2010) and similar to the non-minimally
coupled k-essence discussed via EFT in Sect. 5.2.1. The αpa-
rameter in Eq. (50) can be now interpreted as a sound speed,
unconstrained as in results above. There are however two addi-
tional degrees of freedom, β1and β2. RSD data are able to con-
strain them, with the addition of Planck lensing and WL making
only a minor change to the joint constraints. As in the LVMG
case, we use a new basis γi=|1+w|αiβiin the MCMC analysis,
where α1=0.2 and α2=1 were chosen to decorrelate wand
γi. The resulting 2σupper limits are γ1<0.67 and γ2<0.61
(for w>−1), γ2<2.4 (for w<−1) for the combination of
Planck TT+lowP+lensing+BAO/RSD+WL. As for the LVMG
case, there is no detection of a deviation from ΛCDM and for
w=−1 there are no constraints on β1and β2.
5.3.3. Universal couplings: f(R)cosmologies
A well-investigated class of MG models is constituted by the
f(R) theories that modify the Einstein-Hilbert action by substi-
tuting the Ricci scalar with a more general function of itself:
S=1
2κ2Zd4x√−g(R+f(R)) +Zd4xLM(χi,gµν),(51)
where κ2=8πG.f(R) cosmologies can be mapped to a subclass
of scalar-tensor theories, where the coupling of the scalar field
to the matter fields is universal.
For a fixed background, the Friedmann equation provides
a second-order differential equation for f(R(a)) (see e.g.,
Song et al. 2007a;Pogosian & Silvestri 2008). One of the initial
conditions is usually set by requiring
lim
R→∞
f(R)
R=0,(52)
and the other initial (or boundary condition), usually called B0,
is the present value of
B(z)=fRR
1+fR
H˙
R
˙
H−H2.(53)
Here, fRand fRR are the first and second derivatives of f(R),
and a dot means a derivative with respect to conformal time.
Higher values of B0suppress power at large scales in the
CMB power spectrum, due to a change in the ISW effect. This
also changes the CMB lensing potential, resulting in slightly
smoother peaks at higher `s (Song et al. 2007b;Schmidt 2008;
Bertschinger & Zukin 2008;Marchini et al. 2013).
It is possible to restrict EFTcamb to describe f(R)-
cosmologies. Given an evolution history for the scale factor and
the value of B0,EFTcamb effectively solves the Friedmann equa-
tion for f(R). It then uses this function at the perturbation level
to evolve the metric potentials and matter fields. The merit of
EFTcamb over the other available similar codes is that it checks
the model against some stability criteria and does not assume the
quasi-static regime, where the scales of interest are still linear but
smaller than the horizon and the time derivatives are ignored.
As shown in Fig. 19, there is a degeneracy between the op-
tical depth, τ, and the f(R) parameter, B0. Adding any structure
formation probe, such as WL, RSD or CMB lensing, breaks the
degeneracy. Figure 20 shows the likelihood of the B0param-
eter using EFTcamb, where a ΛCDM background evolution is
assumed, i.e., wDE =−1.
As the different data sets provide constraints on B0that
vary by more than four orders of magnitude, we show plots for
log10 B0; to make these figures we use a uniform prior in log10 B0
to avoid distorting the posterior due to prior effects. However,
for the limits quoted in the tables we use B0(without log) as the
fundamental quantity and quote 95 % limits based on B0. In this
way the upper limit on B0is effectively given by the location of
the drop in probability visible in the figures, but not influenced
by the choice of a lower limit of log10 B0. Overall this appears
to be the best compromise to present the constraints on the B0
parameter. In the plots, the GR value (B0=0) is reached by a
plateau stretching towards minus infinity.
Finally, we note that f(R) models can be studied also with
the MGcamb parametrization, assuming the quasi-static limit. We
find that for the allowed range of the B0parameter, the results
with and without the quasi-static approximation are the same
within the uncertainties. The 95 % confidence intervals are re-
ported in Table 8. These values show an improvement over the
WMAP analysis made with MGcamb (B0<1 (95 % C.L.) in
Song et al. (2007a)) and are similar to the limits obtained in
Marchini & Salvatelli (2013) with MGcamb.
23
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
f(R) models Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP
+BSH +WL +BAO/RSD +WL+BAO/RSD
B0............ <0.79 (95 % CL) <0.69 (95 % CL) <0.10 (95 % CL) <0.90 ×10−4(95 % CL) <0.86 ×10−4(95 % CL)
B0(+lensing) . . . . <0.12 (95 % CL) <0.07 (95% CL) <0.04 (95 % CL) <0.97 ×10−4(95 % CL) <0.79 ×10−4(95 % CL)
Table 8. 95 % CL intervals for the f(R) parameter, B0(see Sect. 5.3.3). While the plots are produced for log10 B0, the numbers in
this table are produced via an analysis on B0since the GR best fit value (B0=0) lies out of the bounds in a log10 B0analysis and its
estimate would be prior dependent.
−7.5 −6.0 −4.5 −3.0 −1.5 0.0
log(B0)
0.02
0.04
0.06
0.08
0.10
0.12
τ
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+BAO/RSD+WL
Fig. 19. 68 % and 95 % contour plots for the two parameters,
{Log10(B0), τ}(see Sect. 5.3.3). There is a degeneracy between
the two parameters for Planck TT+lowP+BSH. Adding lensing
will break the degeneracy between the two. Here Planck indi-
cates Planck TT+lowP.
5.3.4. Non-universal couplings: coupled Dark Energy
Universal couplings discussed in the previous subsection gener-
ally require screening mechanisms to protect baryonic interac-
tions in high density environments, where local measurements
are tightly constraining (see e.g. Khoury 2010). An alternative
way to protect baryons is to allow for non-universal couplings,
in which different species can interact with different strengths:
baryons are assumed to be minimally coupled to gravity while
other species (e.g., dark matter or neutrinos) may feel a “fifth
force,” with a range at cosmological scales.
A fifth force between dark matter particles, mediated by the
DE scalar field, is the key ingredient for the coupled DE sce-
nario Amendola (2000). In the Einstein frame, the interaction is
described by the Lagrangian
L=−1
2∂µφ∂µφ−V(φ)−m(φ)¯
ψψ +Lkin[ψ],(54)
in which the mass of matter fields ψis not a constant (as
in the standard cosmological model), but rather a function of
the DE scalar field φ. A coupling between matter and DE can
be reformulated in terms of scalar-tensor theories or f(R) mo-
dels (Wetterich 1995b;Pettorino & Baccigalupi 2008;Wetterich
−7.5 −6.0 −4.5 −3.0 −1.5 0.0
log(B0)
0.0
0.2
0.4
0.6
0.8
1.0
P/Pmax
Planck+lensing
Planck+lensing+BSH
Planck+lensing+WL
Planck+lensing+BAO/RSD
Planck+lensing+BAO/RSD+WL
Fig. 20. Likelihood plots of the f(R) theory parameter, B0(see
Sect. 5.3.3). CMB lensing breaks the degeneracy between B0and
the optical depth, τ, resulting in lower upper bounds.
2014) via a Weyl scaling from the Einstein frame (where matter
is coupled and gravity is standard) to the Jordan frame (where the
gravitational coupling to the Ricci scalar is modified and mat-
ter is uncoupled). This is exactly true when the contribution of
baryons is neglected.
Dark matter (indicated with the subscript c) and DE densities
are then not conserved separately, but coupled to each other:
ρ0
φ=−3Hρφ(1 +wφ)+βρcφ0,(55)
ρ0
c=−3Hρc−βρcφ0.
Here each component is treated as a fluid with stress energy ten-
sor Tν(α)µ=(ρα+pα)uµuν+pαδν
µ, where uµ=(−a,0,0,0) is the
fluid 4-velocity and wα≡pα/ραis the equation of state. Primes
denote derivatives with respect to conformal time and βis as-
sumed, for simplicity, to be a constant. This choice corresponds
to a Lagrangian in which dark matter fields have an exponen-
tial mass dependence m(φ)=m0exp−βφ (originally motivated
by Weyl scaling scalar-tensor theories), where m0is a constant.
The DE scalar field (expressed in units of the reduced Planck
mass M=(8πGN)−1/2) evolves according to the Klein-Gordon
equation, which now includes an extra term that depends on the
density of cold dark matter:
φ00 +2Hφ0+a2dV
dφ=a2βρc.(56)
24
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
Following Pettorino & Baccigalupi (2008), we choose an in-
verse power-law potential defined as V=V0φ−α, with αand
V0being constants. The amplitude V0is fixed thanks to an
iterative routine, as implemented by Amendola et al. (2012b);
Pettorino et al. (2012). To a first approximation αonly affects
late-time cosmology. For numerical reasons, the iterative routine
finds the initial value of the scalar field, in the range α≥0.03,
which is close to the ΛCDM value α=0 and extends the range
of validity with respect to past attempts; the equation of state w
is approximately related to αvia the expression (Amendola et al.
2012b): w=−2/(α+2) so that a value of α=0.03 corresponds
approximately to w(α=0.03) =−0.99. The equation of state
w≡p/ρ is not an independent parameter within coupled DE
theories, being degenerate with the flatness of the potential. Dark
matter particles then feel a fifth force with an effective gravita-
tional constant Geffthat is stronger than the Newtonian one by a
factor of β2, i.e.
Geff=G(1 +β2),(57)
so that a value of β=0 recovers the standard gravitational in-
teraction. The coupling affects the dynamics of the gravitational
potential (and therefore the late ISW effect), hence the shape
and amplitude of perturbation growth, and shifts the position
of the acoustic peaks to larger multipoles, due to an increase
in the distance to the last-scattering surface; furthermore, it re-
duces the ratio of baryons to dark matter at early times with
respect to its present value, since coupled dark matter dilutes
faster than in an uncoupled model. The strength of the coupling
is known to be degenerate with a combination of Ωc,nsand H(z)
(Amendola & Quercellini 2003;Pettorino & Baccigalupi 2008;
Bean et al. 2008;Amendola et al. 2012b). Several analyses have
previously been carried out, with hints of coupling different from
zero, e.g., by Pettorino (2013), who found β=0.036±0.016 (us-
ing Planck 2013 +WMAP polarization +BAO) different from
zero at 2.2 σ(the significance increasing to 3.6 σwhen data from
HST were included).
The marginalized posterior distribution, using Planck 2015
data, for the coupling parameter βis shown in Fig. 21, while
the corresponding mean values are shown in Table 9.Planck TT
data alone gives constraints compatible with zero coupling and
the slope of the potential is consistent with a cosmological con-
stant value of α=0 at 1.3 σ. When other data sets are added,
however, both the value of the coupling and the slope of the po-
tential are pushed to non-zero values, i.e., further from ΛCDM.
In particular, Planck+BSH gives a value which is ∼2.5σin
tension with ΛCDM, while, separately, Planck+WL+BAO/RSD
gives a value of the coupling βcompatible with the one from
Planck+BSH and about 2.3 σaway from ΛCDM.
When comparing with ΛCDM, however, the goodness of fit
does not improve, despite the additional parameters. Only the
χ2
BAO/RS D improves by ≈1 in CDE with respect to ΛCDM, the
difference not being significant enough to justify the additional
parameters. The fact that the marginalized likelihood does not
improve, despite the apparent 2 σtension, may hint at some de-
pendence on priors: for example, the first panel in Fig. 22 shows
that there is some degeneracy between the coupling βand the
potential slope α; while contours are almost compatible with
ΛCDM in the 2 dimensional plot, the marginalization over α
takes more contributions from higher values of β, due to the
degeneracy, and seems to give a slight more significant peak
in the one dimensional posterior distribution shown in Fig. 21.
Whether other priors also contribute to the peak remains to be
understood. In any case, the goodness of fit does not point to-
wards a preference for non-zero coupling. Degeneracy between
the coupling and other cosmological parameters is shown in
the other panels of the same figure, with results compatible
with those discussed in Amendola et al. (2012b) and Pettorino
(2013). Looking at the conservation equations (i.e., Eqs. 55 and
56), larger positive values of βcorrespond to a larger transfer
of energy from dark matter to DE (effectively adding more DE
in the recent past, with roughly Ωφ∝β2for an inverse power-
law potential) and therefore lead to a smaller Ωmtoday; as a
consequence, the distance to the last-scattering surface and the
expansion rate are modified, with H0/H=−3/2(1 +weff), where
weffis the effective equation of state given by the ratio of the to-
tal pressure over total (weighted) energy density of the coupled
fluid; a larger coupling prefers larger H0and higher σ8.
The addition of polarization tightens the bounds on the cou-
pling, increasing the tension with ΛCDM, reaching 2.8σand
2.7σfor Planck+BSH and Planck+WL+BAO/RSD, respecti-
vely. Also in this case the overall χ2does not improve between
coupled DE and ΛCDM.
0.00 0.03 0.06 0.09 0.12 0.15
β
0.0
0.2
0.4
0.6
0.8
1.0
P/Pmax
Planck
Planck+BSH
Planck+WL
Planck+BAO/RSD
Planck+WL+BAO/RSD
Fig. 21. Marginalized posterior distribution for the coupling β
(see Sect. 5.3.4). The value corresponding to standard gravity is
zero. Results and goodness of fit are discussed in the text.
6. Conclusions
The quest for Dark Energy and Modified Gravity is far from
over. A variety of different theoretical scenarios have been pro-
posed in literature and need to be carefully compared with the
data. This effort is still in its early stages, given the variety of the-
ories and parameterizations that have been suggested, together
with a lack of well tested numerical codes that allow us to make
detailed predictions for the desired range of parameters. In this
paper, we have provided a systematic analysis covering a gene-
ral survey of a variety of theoretical models, including the use
of different numerical codes and observational data sets. Even
though most of the weight in the Planck data lies at high redshift,
Planck can still provide tight constraints on DE and MG, espe-
cially when used in combination with other probes. Our focus
has been on the scales where linear theory is applicable, since
these are the most theoretically robust. Overall, the constraints
that we find are consistent with the simplest scenario, ΛCDM,
with constraints on DE models (including minimally-coupled
scalar field models or evolving equation of state models) and
25
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
CDE models Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP Planck TT+lowP
+BSH +WL +BAO/RSD +WL+BAO/RSD
β............. <0.066 (95 %) 0.037+0.018
−0.015 0.043+0.026
−0.022 0.034+0.019
−0.016 0.037+0.020
−0.016
α............. 0.43+0.15
−0.33 0.29+0.077
−0.26 0.44+0.18
−0.29 0.40+0.15
−0.29 0.45+0.17
−0.33
H0(km/s/Mpc) . . . . 65.4+3.2
−2.667.47+0.88
−0.79 67.6±2.8 66.7±1.1 66.9±1.1
.............. TT,TE,EE+lowP TT,TE,EE+lowP TT,TE,EE+lowP TT,TE,EE+lowP TT,TE,EE+lowP
.............. +BSH +WL +BAO/RSD +WL+BAO/RSD
β............. <0.062 (95 %) 0.036+0.016
−0.013 0.036+0.019
−0.026 0.034+0.018
−0.015 0.038+0.018
−0.014
α............. 0.42+0.14
−0.32 <0.58 (95 %) 0.42+0.13
−0.33 0.37+0.13
−0.28 0.42+0.16
−0.31
Table 9. Marginalized mean values and 68 % C.L. intervals for coupled DE (see Sect. 5.3.4). Planck here refers to Planck TT+lowP.
Results and goodness of fits are discussed in the text. CMB lensing does not improve the constraints significantly.
0.2 0.4 0.6 0.8
α
0.00
0.02
0.04
0.06
0.08
0.10
β
0.25 0.30 0.35 0.40
Ωm
0.76 0.80 0.84 0.88
σ8
63 66 69 72
H0
Planck+BSH Planck +WL Planck+BAO/RSD Planck+WL+BAO/RSD
Fig. 22. Marginalized posterior distribution for coupled DE and different combinations of the data sets (see Sect. 5.3.4). Here Planck
refers to Planck TT+lowP. We show the degeneracy of the coupling βwith α,Ωm,σ8and H0.
MG models (including effective field theory, phenomenological
parameterizations, f(R) and coupled DE models) that are signifi-
cantly improved with respect to past analyses. We discuss here
our main results, drawing our conclusions for each of them and
summarizing the story-line we have followed in this paper to
discuss DE and MG.
Our journey started from distinguishing background and per-
turbation parameterizations. In the first case, the background is
modified (which in turn affects the perturbations), leading to the
following results.
1. The equation of state w(z) as a function of redshift has been
tested for a variety of parameterizations.
(a) In (w0,wa), Planck TT+lowP+BSH is compatible with
ΛCDM, as well as BAO/RSD. When adding WL to
Planck TT+lowP, both WL and CMB prefer the (w0,wa)
model with respect to ΛCDM at about 2σ, although with
a preference for high values of H0(third panel of Fig. 3)
that are excluded when including BSH.
(b) We have reconstructed the equation of state in red-
shift, testing a Taylor expansion up to the third order
in the scale factor and by doing a Principal Component
Analysis of w(z) in different redshifts bins. In addition,
we have tested an alternative parametrization, that allows
to have a varying w(z) that depends on one parameter
only. All tests on time varying w(z) are compatible with
ΛCDM for all data sets tested.
2. ‘Background’ Dark Energy models are generally of
quintessence type where a scalar field rolls down a poten-
tial. We have shown via the (s, ∞) parameterization, re-
lated respectively to late and early time evolution, that the
quintessence/phantom potential at low redshift must be rel-
atively flat: dln V/dφ < 0.35/MPfor quintessence; and
dln V/dφ < 0.68/MPfor phantom models. A zero slope
(ΛCDM) remains consistent with the data and compared to
previous studies, the uncertainty has been reduced by about
10 %. We have produced a new plot (Fig. 9) that helps to vi-
sualize minimally coupled scalar field models, similarly to
analogous plots often used in inflationary theories.
3. Information on DE, complementary to (w0,wa), comes from
asking whether there can be any DE at early times. First,
we have obtained constraints on early DE parametrizations,
assuming a constant DE relative density at all epochs until it
matches ΛCDM in recent times: we have improved previous
constraints by a factor ∼3−4, leading to Ωe<0.0071 at 95%
C.L. from Planck TT+lowP+lensing+BSH and Ωe<0.0036
for Planck TT,TE,EE+lowP+BSH. In addition, we have also
asked how much such tight constraints are weakened when
the fraction of early DE is only present in a limited range in
redshift and presented a plot of Ωe(z) as a function of ze, the
redshift starting from which a fraction Ωeis present. Also
in this case constraints are very tight, with Ωe<
∼2 % (95 %
C.L.) even for zeas late as ≈50.
The background is then forced to be very close to ΛCDM, unless
the tight constraints on early DE can somehow be evaded in a
realistic model by counter balancing effects.
In the second part of the paper, we then moved on to un-
derstanding what Planck can say when the evolution of the per-
turbations is modified independently of the background, as is the
case in most MG models. For that, we followed two complemen-
tary approaches: one (top-down) that starts from a very general
26
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
EFT action for DE (Sect. 5.2.1); the other (bottom-up) that starts
from parameterizing directly observables (Sect. 5.2.2). In both
cases we have assumed that the background is exactly ΛCDM,
in order to disentangle the effect of perturbations. We summarize
here our results.
1. Starting from EFT theories for DE, which include (almost)
all universally coupled models in MG via nine generic fun-
ctions of time, we have discussed how to restrict them to
Horndeski theories, described in terms of five free functions
of time. Using the publicly available code EFTcamb, we have
then varied three of these functions (in the limits allowed
by the code) which correspond to a non-minimally coupled
K-essence model (i.e. αB,αM,αKare varying functions of
the scale factor). We have found limits on the present value
αM0<0.052 at 95% C.L. (in the linear EFT approximation),
in agreement with ΛCDM. Constraints depend on the stabi-
lity routines included in the code, which will need to be fur-
ther tested in the future, together with allowing for a larger
set of choices for the Horndeski functions, not available in
the present version of the numerical code.
2. When starting from observables, two functions of time and
scale are required to describe perturbations completely, in
any model. Among the choices available (summarized in
Sect. 3.2.2), we choose µ(a,k) and η(a,k) (other observ-
ables can be derived from them). In principle, constraints
on these functions are dependent on the chosen parame-
terization, which needs to be fixed. We have tested two
different time dependent parameterizations (DE-related and
time-related) and both lead to similar results, although the
first is slightly more in tension than the other with ΛCDM
(Fig. 14). In this framework, ΛCDM lies at the 2σlimit when
Planck TT+lowP+BSH is considered, the tension increa-
sing to about 3σwhen adding WL and BAO/RSD to Planck
TT+lowP. As discussed in the text, the mild tension with
Planck TT+lowP is related to lower power in the TT spec-
trum and a larger lensing potential in the MG model, with
respect to ΛCDM. The inclusion of CMB lensing shifts all
contours back to ΛCDM. We have reconstructed the two ob-
servables in redshift for both parameterizations, along the
maximum degeneracy line (Fig. 16). When scale dependence
is also included, constraints become much weaker and the
goodness of fit does not improve, indicating that the data do
not seem to need the addition of additional scale dependent
parameters.
The last part of the paper discusses a selection of particular MG
models of interest in literature.
1. We first commented on the simple case of a minimally cou-
pled scalar field in which not only the equation of state is
allowed to vary but the sound speed of the DE fluid is not
forced to be 1, as it would be in the case of quintessence.
Such a scenario corresponds to k-essence type models. As
expected, given that the equation of state is very close to the
ΛCDM value, the total impact of DE perturbations on the
clustering is small.
2. We adopt an alternative way to parameterize observables
(the equation of state approach) in terms of gauge invari-
ant quantities Γand σ. We have used this approach to in-
vestigate Lorentz-breaking massive gravity and generalised
scalar fields models, updating previous bounds.
3. As a concrete example of universally coupled theories, we
have considered f(R) models, written in terms of B(z), con-
ventionally related to the first and second derivatives of f(R)
with respect to R. Results are compatible with ΛCDM. Such
theories assume that some screening mechanism is in place,
in order to satisfy current bounds on baryonic physics at so-
lar system scales.
4. Alternatively to screening mechanisms, one can assume that
the coupling is not universal, such that baryons are still feel-
ing standard gravitational attraction. As an example of this
scenario, we have considered the case in which the dark
matter evolution is coupled to the DE scalar field, feel-
ing an effective fifth force stronger than gravity by a fac-
tor β2. Constraints on coupled dark energy show a tension
with ΛCDM at the level of about 2.5 σ, slightly increasing
when including polarization. The apparent tension, however,
seems to hint at a dependence on priors, partly related to the
degeneracy between the coupling and the slope of the back-
ground potential (and possibly others not identified here).
Future studies will need to identify the source of tension and
possibly disentangle background from perturbation effects.
We have thus presented the first systematic analysis of DE
and MG models, for a variety of models, data sets and numeri-
cal codes. This sets the state of the art for future tests. There
are several ways in which the analysis can be extended. We
have made an effort to (at least start) to put some order in the
variety of theoretical frameworks discussed in literature. There
are of course scenarios not included in this picture that de-
serve future attention, such as additional cosmologies within the
EFT (and Horndeski) framework, other massive gravity models
(see de Rham (2014) for a recent review), general violations of
Lorentz invariance as a way to modify GR (Audren et al. 2014),
non-local gravity (which, for some choices of the action, ap-
pears to fit Planck 2013 data sets (Dirian et al. 2014) as well as
ΛCDM, although there is no connection to a fundamental the-
ory available at present); models of bigravity (Hassan & Rosen
2012) appear to be affected by instabilities in the gravitational
wave sector (Cusin et al. 2014) and are not considered in this
paper. In addition to extending the range of theories, which re-
quires new numerical codes, future tests should verify whether
all the assumptions (such as stability constraints, as pointed out
in the text) in the currently available codes are justified. Further
promising input may come from data sets such as WL and
BAO/RSD, that allow to tighten considerably constraints on MG
models in which perturbations are modified. We also anticipate
that these constraints will strengthen with future releases of the
Planck data, including improved likelihoods for polarization and
new likelihoods, not available at the time of this paper, such as
ISW, ISW-lensing and B-mode polarization, all of which can be
used to further test MG scenarios.
Acknowledgements. It is a pleasure to thank Luca Amendola, Emilio Bellini,
Diego Blas, Sarah Bridle, Noemi Frusciante, Catherine Heymans, Alireza
Hojjati, Bin Hu, Thomas Kitching, Niall MacCrann, Marco Raveri, Ignacy
Sawicki, Alessandra Silvestri, Fergus Simpson, Christof Wetterich and Gongbo
Zhao for interesting discussions on theories, external data sets and numerical
codes. Part of the analysis for this paper was run on the Andromeda and Perseus
clusters of the University of Geneva and on WestGrid computers in Canada.
This research was partly funded by the DFG TransRegio TRR33 grant on ‘The
Dark Universe’ and by the Swiss NSF. The Planck Collaboration acknowl-
edges the support of: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI,
CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC,
MINECO, JA, and RES (Spain); Tekes, AoF, and CSC (Finland); DLR and MPG
(Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland);
RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU).
A description of the Planck Collaboration and a list of its members, indicating
which technical or scientific activities they have been involved in, can be found
at http://www.cosmos.esa.int/web/planck/planck-collaboration.
27
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
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1APC, AstroParticule et Cosmologie, Universit´
e Paris Diderot,
CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris
Cit´
e, 10, rue Alice Domon et L´
eonie Duquet, 75205 Paris Cedex
13, France
2Aalto University Mets¨
ahovi Radio Observatory and Dept of Radio
Science and Engineering, P.O. Box 13000, FI-00076 AALTO,
Finland
3African Institute for Mathematical Sciences, 6-8 Melrose Road,
Muizenberg, Cape Town, South Africa
4Agenzia Spaziale Italiana Science Data Center, Via del Politecnico
snc, 00133, Roma, Italy
5Aix Marseille Universit´
e, CNRS, LAM (Laboratoire
d’Astrophysique de Marseille) UMR 7326, 13388, Marseille,
France
6Aix Marseille Universit´
e, Centre de Physique Th´
eorique, 163
Avenue de Luminy, 13288, Marseille, France
7Astrophysics Group, Cavendish Laboratory, University of
Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, U.K.
8Astrophysics & Cosmology Research Unit, School of Mathematics,
Statistics & Computer Science, University of KwaZulu-Natal,
Westville Campus, Private Bag X54001, Durban 4000, South
Africa
9CITA, University of Toronto, 60 St. George St., Toronto, ON M5S
3H8, Canada
10 CNRS, IRAP, 9 Av. colonel Roche, BP 44346, F-31028 Toulouse
cedex 4, France
11 CRANN, Trinity College, Dublin, Ireland
12 California Institute of Technology, Pasadena, California, U.S.A.
13 Centre for Theoretical Cosmology, DAMTP, University of
Cambridge, Wilberforce Road, Cambridge CB3 0WA, U.K.
14 Centro de Estudios de F´
ısica del Cosmos de Arag´
on (CEFCA),
Plaza San Juan, 1, planta 2, E-44001, Teruel, Spain
15 Computational Cosmology Center, Lawrence Berkeley National
Laboratory, Berkeley, California, U.S.A.
16 Consejo Superior de Investigaciones Cient´
ıficas (CSIC), Madrid,
Spain
17 DSM/Irfu/SPP, CEA-Saclay, F-91191 Gif-sur-Yvette Cedex,
France
18 DTU Space, National Space Institute, Technical University of
Denmark, Elektrovej 327, DK-2800 Kgs. Lyngby, Denmark
19 D´
epartement de Physique Th´
eorique, Universit´
e de Gen`
eve, 24,
Quai E. Ansermet,1211 Gen`
eve 4, Switzerland
20 Departamento de F´
ısica, Universidad de Oviedo, Avda. Calvo
Sotelo s/n, Oviedo, Spain
21 Department of Astronomy and Astrophysics, University of
Toronto, 50 Saint George Street, Toronto, Ontario, Canada
22 Department of Astrophysics/IMAPP, Radboud University
Nijmegen, P.O. Box 9010, 6500 GL Nijmegen, The Netherlands
23 Department of Physics & Astronomy, University of British
Columbia, 6224 Agricultural Road, Vancouver, British Columbia,
Canada
24 Department of Physics and Astronomy, Dana and David Dornsife
College of Letter, Arts and Sciences, University of Southern
California, Los Angeles, CA 90089, U.S.A.
25 Department of Physics and Astronomy, University College
London, London WC1E 6BT, U.K.
26 Department of Physics and Astronomy, University of Sussex,
Brighton BN1 9QH, U.K.
27 Department of Physics, Florida State University, Keen Physics
Building, 77 Chieftan Way, Tallahassee, Florida, U.S.A.
28 Department of Physics, Gustaf H¨
allstr¨
omin katu 2a, University of
Helsinki, Helsinki, Finland
29 Department of Physics, Princeton University, Princeton, New
Jersey, U.S.A.
30 Department of Physics, University of California, Berkeley,
California, U.S.A.
31 Department of Physics, University of California, Santa Barbara,
California, U.S.A.
32 Department of Physics, University of Illinois at
Urbana-Champaign, 1110 West Green Street, Urbana, Illinois,
U.S.A.
33 Dipartimento di Fisica e Astronomia G. Galilei, Universit`
a degli
Studi di Padova, via Marzolo 8, 35131 Padova, Italy
34 Dipartimento di Fisica e Scienze della Terra, Universit`
a di Ferrara,
Via Saragat 1, 44122 Ferrara, Italy
35 Dipartimento di Fisica, Universit`
a La Sapienza, P. le A. Moro 2,
Roma, Italy
36 Dipartimento di Fisica, Universit`
a degli Studi di Milano, Via
Celoria, 16, Milano, Italy
37 Dipartimento di Fisica, Universit`
a degli Studi di Trieste, via A.
Valerio 2, Trieste, Italy
38 Dipartimento di Fisica, Universit`
a di Roma Tor Vergata, Via della
Ricerca Scientifica, 1, Roma, Italy
39 Dipartimento di Matematica, Universit`
a di Roma Tor Vergata, Via
della Ricerca Scientifica, 1, Roma, Italy
40 Discovery Center, Niels Bohr Institute, Blegdamsvej 17,
Copenhagen, Denmark
41 Dpto. Astrof´
ısica, Universidad de La Laguna (ULL), E-38206 La
Laguna, Tenerife, Spain
42 European Space Agency, ESAC, Planck Science Office, Camino
bajo del Castillo, s/n, Urbanizaci´
on Villafranca del Castillo,
Villanueva de la Ca˜
nada, Madrid, Spain
43 European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ
Noordwijk, The Netherlands
44 Facolt`
a di Ingegneria, Universit`
a degli Studi e-Campus, Via
Isimbardi 10, Novedrate (CO), 22060, Italy
31
Planck Collaboration: Planck 2015 results. XIV. Dark energy and modified gravity
45 Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100
L’Aquila, Italy
46 HGSFP and University of Heidelberg, Theoretical Physics
Department, Philosophenweg 16, 69120, Heidelberg, Germany
47 Helsinki Institute of Physics, Gustaf H¨
allstr¨
omin katu 2, University
of Helsinki, Helsinki, Finland
48 INAF - Osservatorio Astronomico di Padova, Vicolo
dell’Osservatorio 5, Padova, Italy
49 INAF - Osservatorio Astronomico di Roma, via di Frascati 33,
Monte Porzio Catone, Italy
50 INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11,
Trieste, Italy
51 INAF/IASF Bologna, Via Gobetti 101, Bologna, Italy
52 INAF/IASF Milano, Via E. Bassini 15, Milano, Italy
53 INFN, Sezione di Bologna, Via Irnerio 46, I-40126, Bologna, Italy
54 INFN, Sezione di Roma 1, Universit`
a di Roma Sapienza, Piazzale
Aldo Moro 2, 00185, Roma, Italy
55 INFN, Sezione di Roma 2, Universit`
a di Roma Tor Vergata, Via
della Ricerca Scientifica, 1, Roma, Italy
56 INFN/National Institute for Nuclear Physics, Via Valerio 2,
I-34127 Trieste, Italy
57 IPAG: Institut de Plan´
etologie et d’Astrophysique de Grenoble,
Universit´
e Grenoble Alpes, IPAG, F-38000 Grenoble, France,
CNRS, IPAG, F-38000 Grenoble, France
58 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, Pune
411 007, India
59 Imperial College London, Astrophysics group, Blackett
Laboratory, Prince Consort Road, London, SW7 2AZ, U.K.
60 Infrared Processing and Analysis Center, California Institute of
Technology, Pasadena, CA 91125, U.S.A.
61 Institut N´
eel, CNRS, Universit´
e Joseph Fourier Grenoble I, 25 rue
des Martyrs, Grenoble, France
62 Institut Universitaire de France, 103, bd Saint-Michel, 75005,
Paris, France
63 Institut d’Astrophysique Spatiale, CNRS (UMR8617) Universit´
e
Paris-Sud 11, Bˆ
atiment 121, Orsay, France
64 Institut d’Astrophysique de Paris, CNRS (UMR7095), 98 bis
Boulevard Arago, F-75014, Paris, France
65 Institute for Space Sciences, Bucharest-Magurale, Romania
66 Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, U.K.
67 Institute of Theoretical Astrophysics, University of Oslo, Blindern,
Oslo, Norway
68 Instituto de Astrof´
ısica de Canarias, C/V´
ıa L´
actea s/n, La Laguna,
Tenerife, Spain
69 Instituto de F´
ısica de Cantabria (CSIC-Universidad de Cantabria),
Avda. de los Castros s/n, Santander, Spain
70 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via
Marzolo 8, I-35131 Padova, Italy
71 Jet Propulsion Laboratory, California Institute of Technology, 4800
Oak Grove Drive, Pasadena, California, U.S.A.
72 Jodrell Bank Centre for Astrophysics, Alan Turing Building,
School of Physics and Astronomy, The University of Manchester,
Oxford Road, Manchester, M13 9PL, U.K.
73 Kavli Institute for Cosmology Cambridge, Madingley Road,
Cambridge, CB3 0HA, U.K.
74 LAL, Universit´
e Paris-Sud, CNRS/IN2P3, Orsay, France
75 LAPTh, Univ. de Savoie, CNRS, B.P.110, Annecy-le-Vieux
F-74941, France
76 LERMA, CNRS, Observatoire de Paris, 61 Avenue de
l’Observatoire, Paris, France
77 Laboratoire AIM, IRFU/Service d’Astrophysique - CEA/DSM -
CNRS - Universit´
e Paris Diderot, Bˆ
at. 709, CEA-Saclay, F-91191
Gif-sur-Yvette Cedex, France
78 Laboratoire Traitement et Communication de l’Information, CNRS
(UMR 5141) and T´
el´
ecom ParisTech, 46 rue Barrault F-75634
Paris Cedex 13, France
79 Laboratoire de Physique Subatomique et Cosmologie, Universit´
e
Grenoble-Alpes, CNRS/IN2P3, 53, rue des Martyrs, 38026
Grenoble Cedex, France
80 Laboratoire de Physique Th´
eorique, Universit´
e Paris-Sud 11 &
CNRS, Bˆ
atiment 210, 91405 Orsay, France
81 Lawrence Berkeley National Laboratory, Berkeley, California,
U.S.A.
82 Lebedev Physical Institute of the Russian Academy of Sciences,
Astro Space Centre, 84/32 Profsoyuznaya st., Moscow, GSP-7,
117997, Russia
83 Max-Planck-Institut f¨
ur Astrophysik, Karl-Schwarzschild-Str. 1,
85741 Garching, Germany
84 McGill Physics, Ernest Rutherford Physics Building, McGill
University, 3600 rue University, Montr´
eal, QC, H3A 2T8, Canada
85 National University of Ireland, Department of Experimental
Physics, Maynooth, Co. Kildare, Ireland
86 Niels Bohr Institute, Blegdamsvej 17, Copenhagen, Denmark
87 Optical Science Laboratory, University College London, Gower
Street, London, U.K.
88 SB-ITP-LPPC, EPFL, CH-1015, Lausanne, Switzerland
89 SISSA, Astrophysics Sector, via Bonomea 265, 34136, Trieste,
Italy
90 School of Physics and Astronomy, CardiffUniversity, Queens
Buildings, The Parade, Cardiff, CF24 3AA, U.K.
91 School of Physics and Astronomy, University of Nottingham,
Nottingham NG7 2RD, U.K.
92 Sorbonne Universit´
e-UPMC, UMR7095, Institut d’Astrophysique
de Paris, 98 bis Boulevard Arago, F-75014, Paris, France
93 Space Research Institute (IKI), Russian Academy of Sciences,
Profsoyuznaya Str, 84/32, Moscow, 117997, Russia
94 Space Sciences Laboratory, University of California, Berkeley,
California, U.S.A.
95 Special Astrophysical Observatory, Russian Academy of Sciences,
Nizhnij Arkhyz, Zelenchukskiy region, Karachai-Cherkessian
Republic, 369167, Russia
96 Stanford University, Dept of Physics, Varian Physics Bldg, 382 Via
Pueblo Mall, Stanford, California, U.S.A.
97 Sub-Department of Astrophysics, University of Oxford, Keble
Road, Oxford OX1 3RH, U.K.
98 Theory Division, PH-TH, CERN, CH-1211, Geneva 23,
Switzerland
99 UPMC Univ Paris 06, UMR7095, 98 bis Boulevard Arago,
F-75014, Paris, France
100 Universit¨
at Heidelberg, Institut f¨
ur Theoretische Astrophysik,
Philosophenweg 12, 69120 Heidelberg, Germany
101 Universit´
e de Toulouse, UPS-OMP, IRAP, F-31028 Toulouse cedex
4, France
102 Universities Space Research Association, Stratospheric
Observatory for Infrared Astronomy, MS 232-11, Moffett Field,
CA 94035, U.S.A.
103 University of Granada, Departamento de F´
ısica Te´
orica y del
Cosmos, Facultad de Ciencias, Granada, Spain
104 University of Granada, Instituto Carlos I de F´
ısica Te´
orica y
Computacional, Granada, Spain
105 University of Heidelberg, Institute for Theoretical Physics,
Philosophenweg 16, 69120, Heidelberg, Germany
106 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478
Warszawa, Poland
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