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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,