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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,