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Future CMB ISW-Lensing bispectrum constraints on modified gravity in the

Parameterized Post-Friedmann formalism

Bin Hu1, Michele Liguori1,2, Nicola Bartolo1,2, and Sabino Matarrese1,21INFN, Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy

2Dipartimento di Fisica e Astronomia “G. Galilei”,

Universita degli Studi di Padova , Via Marzolo 8, 35131 Padova, Italy

(Dated: September 24, 2018)

We forecast the constraints on both Hu-Sawicki model and Bertschinger-Zukin model of f(R)modified gravity within the Parameterized Post-Friedmann (PPF) formalism for the Planck satelliteexperiment by performing the joint analysis of ISW-Lensing bispectrum and CMB power spectrum.We find that, even considering the temperature-temperature mode of CMB power spectrum only,Planck data are expected to reduce the error bars on the modified gravity parameter B0 (related tothe present value of Compton wavelength of the extra scalar degree of freedom) at least one ordermagnitude compared with WMAP. The spectrum-bispectrum joint analysis can further improve theresults by a factor ranging from 1.14 to 5.32 depending on the specific modified gravity model. One ofour main results is that the cross-correlation between ISW-Lensing bispectrum and power spectrumcan be safely neglected when performing the joint analysis. For simplicity, we only investigate thelikelihood of one parameter (B0) and fix all other cosmological parameters to their best-fit valuesin WMAP7yr results.

I. INTRODUCTION

Cosmic acceleration can arise from either an exoticform of energy with negative pressure, namely “dark en-ergy”, or a modification of gravity that appears on largescales. As shown in [1], every expansion history that canbe parameterized by a dark energy model with ρDE(ln a)can be reproduced by a one parameter family of f(R)models, which approaches the Einstein-Hilbert action inthe high curvature limit. Hence, the observations of thecosmic background expansion cannot help us to distin-guish the viable modified gravity from dark energy mod-els. However, the dynamical evolution of fluctuations canprovide us a tool to test gravity theory (e.g. [2–14] andreferences therein). The main difference between mod-ified gravity and dark energy models is the anisotropicstress on large scales. In most of the dark energy mod-els the large scale anisotropic stress vanishes, while, inmodified gravity models usually it does not. The deeperphysical reason for this phenomenon is that in modifiedgravity usually the evolution of the Newtonian potentialΨ deviates from that of the spatial curvature potentialΦ.

A viable modified gravity model should at least satisfythe following three conditions: first, it should pass thestringent tests in the local solar-system; second, it shouldrespect the standard expansion history and accelerate thelate-time expansion; third, it should avoid modifying thephysics at recombination, which has been investigated byvarious CMB experiments with a high precision.

The studies of modified gravity models, in principle,can be classified into two different frameworks (see, e.g.,the discussion in Ref. [19]). The first is a model depen-dent method. One can start from a specific Lagrangian,investigating its dynamical behavior to finally give itspredictions. Various viable modified gravity models havebeen proposed (see the review article [15]). In this paper

we mainly focus on f(R) models (see [18] for a review),such as Starobinsky model [16], Hu-Sawicki (HS) model[17]. The other method is inspired by the ParameterizedPost-Newtonian (PPN) approach for the solar-systemtest. In this case one can try to build a more or less modelindependent framework, in which many modified gravitymodels can be parameterized in an unified way. The sim-plest idea is to directly generalize Eddington parameter(γ ≡ Φ/Ψ) [20] in static solar-system to a one unknownfunction of space and time γ(t,x) in Friedmann universe.Many studies (e.g., [19, 21–25]) based on this frameworkshow that this parameterization works quite well withthe large scale structure data. However, as pointed in[26], the horizon-scale and superhorizon evolution needto be consistent with the background expansion of theuniverse. This is one of the reason why the simple γ pa-rameter is not sufficient when dealing with, such as ISWeffect. Furthermore, as stated in [27] using a parame-terization with insufficient freedom significantly tightensthe apparent theoretical constraints. Still in the samespirit of the standard solar-system PPN approach, an-other viable parameterization has been proposed, the socalled Parameterized Post-Friedmann (PPF) framework.The idea of PPF framework is to parameterize the back-ground/linear fluctuation gravitational equations, whichis firstly proposed in [79]1.After that different formal-ism in the domain of PPF have been proposed, suchas Hu-Sawicki PPF formalism [28–30] that bridges be-tween different scale regimes. In details, PPF allows todescribe (and match) three different regimes of gravitymodifications to GR, namely the super-horizon regime,the (sub-horizon) quasi-static regime (with an intermedi-ate regime between the two) and a non-linear regime onsmaller scales. Aside that Barker-Ferreira-Skordis-Zuntz

1 We are grateful to Tessa Baker for pointing this out

2

also give their algorithm in [36, 80] which try to keep thisparameterized framework as general as possible. Since inthis work we focus on the Hu-Sawicki PPF formalism, inwhat follows we will call it PPF for short.The current constraints on general f(R) models within

the PPF formalism are B0 < 0.42(95%C.L.) by usingCMB and ISW-galaxy correlation, and B0 < 1.1×10−3 at95% C.L. [32] using a larger set of data, such as WMAP5[63], Arcminute Cosmology Bolometer Array Receiver(ACBAR) [64], Cosmic Background Imager (CBI) [65],Very Small Array (VSA) [66], Supernova CosmologyProject (SCP) Union [67], Supernovae and H0 for theEquation of States (SHOES) [68], BAO data [69], ISW-galaxy correlation ISWWLL likelihood code [70, 71], theEG measurement, probing the relation between weakgravitational lensing and galaxy flows [72], cluster abun-dance (CA) from the likelihood code of [73], as well asthe σ8 measurement of the Chandra Cluster CosmologyProject (CCCP) [74]. The main constraint comes fromcluster abundance. The first constraint on f(R) mod-els from local cluster abundance from the several Gpcscale to the tens of Mpc scale is derived in [77], and thenew constraint |fR0| < 3.5 × 10−3 at 1D marginalized95% confidence level through cluster-galaxy lensing ofmaxBCG clusters at (0.220) Mpc scale from the SloanDigital Sky Survey (SDSS) [75, 76], is obtain in [78].Besides what mentioned above, there exist many otherparameterizations [33–36].On the other hand, from the theoretical point of view,

it has become clear that the cross-correlation between theIntegrated Sachs-Wolfe (ISW) effect [37] and the gravi-tational Weak Lensing (WL) in CMB anisotropies canprovide a wealth of cosmological information, includingthe dynamics of cosmic acceleration [38–42]. Recently ithas been shown that this effect will provide the most im-portant contamination to the local-type primordial bis-pectrum [43–49]. In order to detect the primordial non-Gaussian signals, one needs to accurately subtract thissecondary anisotropy in the Planck data. In fact vari-ous forecasts indicate that the Planck mission can mea-sure this secondary effect with a high precision [43–49].From the observational point of view, determining thecosmological parameters will require not only additionalmeasurements, but also characterizations of the data be-yond the power spectrum. ISW-Lensing bispectrum canrepresent one of these novel statistical variables. In thispaper we focus on constraining modified gravity modelsvia future Planck ISW-Lensing bispectrum data.

II. PARAMETERIZATION OF MODIFIED

GRAVITY

Compared with the PPN formalism, PPF is a param-eterization of a series of differential equations, ratherthan a specific static/stationary solution. This is be-cause what we are interested in cosmology is not onlyinhomogeneous but also time-varying quantities. For-

tunately, in some dominant regime for measures of therate of structure growth, such as the weak lensing andpeculiar velocity surveys, the time derivatives of pertur-bations can be neglected relative to spatial derivatives.This approximation leads to the so called quasi-staticapproximation, which can reduce the time differentialequations into simple algebra relations. However, onthe super-horizon scales the long wavelength perturba-tions have enough time to feel the changes of the gravita-tional potentials, consequently, the time derivative termsin the differential equations cannot be neglected. Oneexplicit example is the ISW effect. In this section, wefirst recall one of the conventional PPF parametrizationwhich is mainly inspired by the quasi-static dynamics,the Bertschinger-Zukin parameterization [19]; and thenwe introduce a more suitable parameterization for ourpurposes, the Hu-Sawicki PPF formalism, which prop-erly takes the super-horizon perturbation evolution intoaccount, even for large value of B0 (fR0).

A. Bertschinger-Zukin parameterization

As a straightforward extension of the PPN formal-ism, Bertschinger-Zukin formalism (BZ) [19] proposeda quite efficient parameterization of modified gravity inthe quasi-static regime, where the dynamical behaviorof the temporal and spatial gravitational potentials aregoverned by two modified Poisson-like equations

−k2Φ = 4πGΦ(t,k)a2ρmδ , (1)

−k2Ψ = 4πGΨ(t,k)a2ρmδ , (2)

with GΨ(t,k) = γ(t,k)GΦ(t,k)2. Given the above ob-

servation, Bertschinger and Zukin proposed the followingparameterization for f(R) model

GΦ

G0

=1 + α1k

2as

1 + α2k2as, γ =

1 + β1k2as

1 + β2k2as, (3)

with

Ψ(t,k) = γ(t,k)Φ(t,k) . (4)

where G0 is the Newton constant at the present epoch.We can easily see that in the quasi-static approxima-tion time derivative terms are neglected so that long-wavelength perturbations cannot be properly taken intoaccount (e.g., γ → 1 for k → 0, while in general, mod-ified gravity theories can show distinctive features alsoon the longwavelength regime)3 However, authors of [62]

2 Notice that the metric convention of Bertschinger-Zukin is differ-ent w.r.t. Hu-Sawicki, we have: ΦBZ = ΨHS and ΨBZ = −ΦHS.In this paper, except in this subsection, we follow Hu-Sawickiconventions.

3 In [19] authors also discussed the super-horizon perturbation dy-namics which, however, is not captured by parameterized equa-tion (3).

3

recently demonstrated that for small values of B0 (fR0),the parameterization of [19] is practically good enoughfor the current data analysis purpose. In our followingstudies, we also investigate this kind of parameteriza-tion. Constraining modified gravity models via the ISW-Lensing bispectrum forces to take the ISW effect intoaccount properly. Given this request, in what follows weturn to Hu-Sawicki PPF formalism in that it allows tojoin the superhorizon to the subhorizon regime.

B. Hu-Sawicki model in PPF formula

The PPF parameterization is defined by 3 functionsg(ln a, kH), fζ(ln a), fG(ln a) and 1 parameter cΓ. Theycorrespond to the metric ratio, the super-horizon rela-tionship between the metric and density, the deviationof Newton constant on super-horizon scale from thaton quasi-static regime, and the relationship between thetransition scale and the Hubble scale [28]. The 4th or-der nature of f(R) theories provides enough freedom toreproduce any cosmological background history by an ap-propriate choice of the f(R) function [1]. For simplicity,in this paper we concentrate on one specific parameter-ization form, namely Hu-Sawicki model [17], which cansatisfy the local solar system tests

f(R) = −m2 c1(R/m2)n

c2(R/m2)n + 1, (5)

where the mass scale reads

m2 ≡ H20Ωm = (8315Mpc)−2

(

Ωmh2

0.13

)

. (6)

The non-linear terms in f(R) introduce higher-orderderivatives into this theory. However, we are more famil-iar with a second-order derivative theory. Fortunately,we can reduce the derivatives to second order by definingan extra scalar field χ ≡ df/dR, namely the “scalaron”,which absorbs the higher derivatives. The square of theCompton wavelength of the scalaron in units of the Hub-ble length is defined as

B =fRR

1 + fRR′

H

H ′, (7)

with fR = df/dR, fRR = d2f/dR2 and ′ ≡ d/d ln a.At the high curvature regime, (6) can be expanded

w.r.t. m2/R as

limm2/R→0

f(R) ≈ −c1c2m2 +

c1c22m2

(

m2

R

)n

+ · · · . (8)

From Eq. (8) we can see that, the first and second termsrepresent a cosmological term and a deviation from it,respectively. In order to mimic a ΛCDM evolution onthe background, the vale of c1/c2 can be fixed as [17]

c1c2

≈ 6ΩΛ

Ωm. (9)

By using the relation (9) the number of free parame-ters can be reduced to two. From the above analysis,we can see that, strictly speaking, due to the appear-ances of correction terms to the cosmological constant,Hu-Sawicki model cannot exactly mimic ΛCDM. Sincem2/R increase very fast with time, the largest value (atthe present epoch) is m2/R ∼ 0.03, the largest deviationto ΛCDM background happens in the case n = 1 with1% errors, corresponding to m2/Rc2 ∼ 0.01 in Eq.(8).For large n values, such as n = 4, 6 we can safely neglectthis kind of theoretical errors. We have checked that forn = 1 case, this 1% deviation from ΛCDM will bring usa 10% errors in the variance of the parameter B0, whilefor n = 4, 6 our results are not affected.Without loss of generality we can choose the two free

parameters to be (n, c2). However, as shown in [1], formore general f(R) models the ΛCDM evolution on thebackground can be reproduced exactly by only introduc-ing one free parameter. This means that there existssome degeneracy between the two parameters, as shownin Fig. 1. Usually General Relativity (GR) is recoveredwhen

B0 = 0 . (10)

Then, from Fig. 1 we can see no matter what valuen takes, we are always allowed to set B0 = 0 by ad-justing c2. Furthermore, in order to mimic ΛCDM onthe background, c2 and n need to satisfy one constraint:the first term in the denominator of (5) should be muchlarger than the second. The blank area in Fig. 1 cor-responds those parameter regions that are ruled out byc2(R/m2)n > 1. Actually, this condition gives

Bmax0 =

0.1 , (n = 1) ,

1.2 , (n = 4) , (11a)

4.0 , (n = 6) .

In the approximation (8), we assume m2/R → 0, how-ever, with the expansion of the universe this approxima-tion starts to be not so accurate for our purposes. Inorder to obtain our results as accurate as possible, in thefollowing calculation we use the exact expression (5) inplace of the approximated one.We have decided to study the Hu-Sawicki model within

the PPF formalism for the particular relevance of thismodel, and to exploit the generality of the PPF for-malism for the study of the perturbations, using for thecomputations the publicy available PPF module [30] ofCAMB [31]. In Ref. [50] we apply the same formalism tomore general f(R) models and other models of modifiedgravity. As stated previously, the idea of the Hu-SawickiPPF formalism is to parameterize gravity theory in orderto join different scale regimes. In what follows, we willbriefly review this parameterization. Firstly, we definethe metric ratio as

g =Φ+Ψ

Φ−Ψ. (12)

4

0.0

0.5

1.0B0

2

4

6

n

-15

-10

-5

0

logHc2L

FIG. 1. The degeneracy between parameters c2 and n.

In principle, g is completely equivalent to the functionγ(t, k) introduced by Bertschinger and Zukin in [19].

1. Super-Horizon Regime

On super-horizon scales, the dynamical equation ofmotion for the spatial curvature potential has the fol-lowing form [28]

Φ′′ +

(

1− H ′′

H ′+

B′

1−B+B

H ′

H

)

Φ′

+

(

H ′

H− H ′′

H ′+

B′

1−B

)

Φ = 0 , (kH → 0) . (13)

Furthermore, the comoving curvature conservation rela-tion (ζ′ = 0) gives

Ψ =−Φ−BΦ′

1−B, (kH → 0). (14)

On super-horizon scales, because the spatial gradients ofthe perturbations can be neglected compared with theirtemporal evolution, the metric ratio reduces to a functiongSH(ln a) dependent only on time

g(ln a, kH = 0) = gSH(ln a) =Φ +Ψ

Φ−Ψ. (15)

The relevant phenomenon about the decay/growth of thegravitational potentials in the CMB photons is the ISWeffect. In Fig. 2 we compute the ISW effect in the mod-ified gravity models we consider in this paper. Our cal-culations demonstrate that ISW effect is quite sensitiveto the value of the parameter n in the Hu-Sawicki model.The left plot shows that, for the Hu-Sawicki model withn = 1, the ISW effect is always decreasing with increas-ing of B0. However, for n = 4 case, the ISW effect firstlydecreases monotonically then bounce again after reach-ing some critical values. This behaviors are also reflectedon our likelihood analysis (see later, Fig. 5).

1000

10 100

l

Cosmic VarianceB0=0.0n=4, B0=0.1n=4, B0=0.35n=4, B0=0.75BZ, B0=0.42BZ, B0=2.0

10 100

l(l+

1)C

lTT/2

π[µK

2 ]

l

Cosmic VarianceB0=0.0n=1, B0=0.01n=1, B0=0.05n=1, B0=0.1n=6, B0=0.85

FIG. 2. ISW effect in modified gravity models: Bertschinger-Zukin (BZ) model, Eq.(3) and f(R) Hu-Sawicki model,Eq.(5).

2. Quasi-Static Regime

In the quasi-static regime, all the time derivative termsare neglected w.r.t. the spatial gradients. At the linearperturbation order, this brings a lot of simplifications,because we can reduce the ordinary differential equationsinto algebraic relations. As an example, the modifiedPoisson equation takes the following form [28]

k2Φ− =4πG

1 + fGa2ρm∆m , (16)

where fG represents the deviation from the standardgravitational constant (here Φ− = (Φ− Ψ)/2). Further-more, for f(R) models, we have

fG = fR , g → gQS = −1/3 , (17)

below the Compton wavelength.

3. Matching the superhorizon and the subhorizon regimes

In order to match the super-horizon scale behavior, anadditional term Γ must be introduced to the modifiedPoisson equation [28]

k2[

Φ− + Γ]

= 4πGa2ρm∆m , (18)

where Γ satisfy the following equation

(1 + c2Γk2H)[

Γ′ + Γ+ c2Γk2H(Γ− fGΦ−)

]

= S , (19)

with the source term

S = −[

1

g + 1

H ′

H+

3

2

H2m

H2a3(1 + fζ)

]

Vm

kH

+

[

g′ − 2g

g + 1

]

Φ− . (20)

In Eq.(19), the coefficient cΓ represents the relationshipbetween the transition scale and the Hubble scale, and

5

the function fζ the relationship between the metric andthe density perturbation. For f(R) models, we have

cΓ = 1 , fζ = cζg , cζ ≈ 1/3 . (21)

Up to now, the only unknown function in the intermeadi-ate regime is the metric ratio. An interpolating functionis proposed in [28] as

g(ln a, k) =gSH + gQS(cgkH)ng

1 + (cgkH)ng

, (22)

with

gQS = −1/3 , ng = 2 , cg = 0.71√

B(t) . (23)

III. ISW-LENSING BISPECTRUM

The primordial local non-Gaussianity is generatedwhen the short-wavelength fluctuations are modulatedby the long-wavelength modes which cross the horizonmuch earlier than the former. This kind of interactionswill induce some correlations between the random fluctu-ations on large scales and those on small scales. In princi-ple, any mechanisms through which small and large scalefluctuations couple together will generate similar featuresas those in the primordial local-type non-Gaussianity.The late-time cross-correlation between ISW andWeak

Lensing provides one such mechanism. On one hand,when the universe starts to accelerate at late-time, thedecay of the gravitational potential generates the sec-ondary anisotropies in the CMB anisotropies, the ISWeffect

δT

T(n)|ISW =

∫

dχ(Φ−Ψ),τ (n, χ) , (24)

where τ and χ are the conformal time and comoving dis-tance, respectively. On the other hand, the gravitationallensing induced by the fluctuations of matter density de-flects the paths of CMB photons as they travel from thelast-scattering surface to the observer

δT (n) = δT (n+ ∂φ)

≃ δT (n) +[

(∂φ) · (∂δT )]

(n) , (25)

where the lensing potential is defined as

φ(n) = −∫ χ∗

0

dχχ∗ − χ

χ∗χ

(

Φ−Ψ)

(n, χ) . (26)

From Eq.(24) and (25), we can see that Weyl poten-tial (Φ − Ψ) sources both ISW and WL, so the long-wavelength mode from ISW couples with the short-wavelength mode from WL. Consequently, this kindof correlation will induce the secondary ISW-Lensingbispectrum, which has been recognized as the mostdominant contamination of local-type primordial bispec-trum [43–48]

Going to harmonic space and computing the reducedbispectrum, one can obtain [38, 39, 44, 60]

bISW−Ll1l2l3

=

[−l1(l1 + 1) + l2(l2 + 1) + l3(l3 + 1)

2CT

l2CφTl3

+5perm.]

. (27)

where CφTl = 〈φ∗

lmaISWlm 〉. Since the gravitational lensingis just remapping the primordial CMB temperature andpolarization fields via the gradient of a foreground po-tential, from the above expression we can see that ISW-Lensing bispectrum includes information from both theprimordial fluctuations and the secondary ones. Thisfact is important for the understanding of the spectrum-bispectrum joint analysis, because the bispectrum cancorrelate with the power spectrum through both thetemperature-temperature mode of power spectrum (CT

l )and the secondary lensing-temperature cross power spec-

trum (CφTl ), as detailed in Sec.IVC.

-5

-4

-3

-2

-1

0

1

2

3

4

5

10 100 1000

l3 ClΦ

Τ/1

04 µK2

l

GRn=1, B0=0.05n=1, B0=0.1

n=4, B0=0.35n=4, B0=0.75n=6, B0=0.85BZ, B0=0.42

BZ, B0=2.0

FIG. 3. The cross spectrum between lensing potential andtemperature l3CΦT

l .

-4

-3

-2

-1

0

1

2

3

4

0 500 1000 1500 2000 2500 3000

l4 b 4,l,

l+4/

108 µK

3

l

GRn=1, B0=0.05n=1, B0=0.1

n=4, B0=0.35n=4, B0=0.75n=6, B0=0.85BZ, B0=0.42

BZ, B0=2.0

FIG. 4. The squeezed slice of reduced bispectrum l4b4,l,l+4.

6

In Fig.3, we plot the lensing-temperature cross powerspectrum for several different gravity models. From thisfigure, we can explicitly see the non-linear dependence

of CφTl on B0: firstly, in the GR limit B0 → 0 (red

solid curve), the cross-correlation is large and positive;secondly, with increasing of B0, the cross-correlation de-creases and oscillates around zero (green dotted curvefor n = 1, pink dashed curve for n = 4 Hu-Sawickimodels, and black dotted-dashed curve for Bertschinger-Zukin model); finally, with further increasing of B0, thecross-correlation amplitude increase again but negatively(dotted deep blue curve for n = 1, dashed-dotted lightblue curve for n = 4 Hu-Sawicki models, and dashed-dotted orange curve for the Bertschinger-Zukin model).The reasons of these phenomena lie in the non-linear de-pendence of ISW effect on B0, i.e. the first few multi-poles firstly decrease and then bounce with the increas-ing of B0, see Fig.2. The other important feature isthat the cross-correlations are suppressed greatly afterthe first hundred multipoles, it almost vanishes in theregime l ≥ 1000, because the cross-correlation is drivenby the late time ISW effect.

We also investigate the squeezed profile of reducedISW-Lensing bispectrum, see Fig.4. From this figure, wecan easily see the modulation of short-wavelength modeby the long-wavelength mode.

IV. FORECASTS AND RESULTS

In this section, we will present the joint likelihood anal-ysis of the CMB power spectrum and ISW-Lensing bis-pectrum for several modified gravity models by using thespecifications of WMAP and Planck experiments. In thispaper, we implement the Hu-Sawicki model by using thePPF module [30] of CAMB [31] and the Bertschinger-Zukin model by using MGCAMB [24]. Our results showthat the constraint on B0 is quite model dependent.However, generally speaking, the ISW-Lensing bispec-trum can constrain cosmological parameters at the samelevel as the power spectrum does. Consequently, the jointanalysis of the power spectrum and the ISW-lensing bis-pectrum can improve the B0 error bars by a factor 1.14to 5.32.

A. Power spectrum

In this subsection, we forecast constraints on B0 fromobservations of the CMB temperature power spectrum.To this aim we firstly consider the simple likelihood ex-tracted from the chi-squared variable:

χ2P =

3000∑

l=2

[

Cthl − Cfid

l

]2

σ2P,l

, (28)

TABLE I. B0 error bars (68% CL) from power spectrum χ2

for the Hu-Sawicki model

n WMAP PLANCK

1 4.20 × 10−2 6.68 × 10−5

4 ——– 2.06 × 10−3

6 ——– 1.51 × 10−2

a The blank elements in the table represent that WMAP datacannot efficiently constrain B0 parameter in Hu-Sawicki modelwith n = 4, 6, because the likelihood does not vanish in the tailswhich can be read from Fig.5 and 6.

TABLE II. B0 error bars (68% CL) from power spectrum χ2

for the Bertschinger-Zukin model

Ideal (lmax = 1000) PLANCK

1.63 × 10−1 1.10 × 10−2

a In order to compare with [42], we forecast B0 error bars in theideal experiment with lmax = 1000 instead of WMAP.

where Cl is the lensed ”TT” power spectrum, and thevariance is

σ2P,l =

2

(2l + 1)

(

b2lCTl +Nl

)2

, (29)

Nl and bl denoting the experimental noise and beam re-spectively (listed in Tab.VII). We allow only B0 to varywhile fixing all other cosmological parameters to theirWMAP7yr best-fit values, as in [42]. The B0-likelihoodobtained in this way is shown in figure Fig.5 and 6. Inagreement with previous studies [32, 42], we find thatits behaviour is highly non-Gaussian and for this reasonwe cannot rely on a standard Fisher matrix approach inorder to forecast error bars on B0. Despite this issuewe still computed a full 7-parameter Fisher matrix as amean to get an order of magnitude estimate of the cor-relation between B0 and standard ΛCDM parameters.Our results are summarized in detail in Appendix A andshow that there are no significant degeneracies betweenB0 and other parameters (the largest cross-correlationcoefficient we find is Ωbh

2-B0 = −0.18). This in turnjustifies the simple 1-parameter likelihood approach usedin this section in order to study B0 error bars.We investigate the likelihood function for both WMAP

and Planck experiments in three Hu-Sawicki models withdifferent values of n = 1, 4, 6 in Fig.5 and 6. Further-more, in order to compare with the results in [42] we alsoinclude the Bersthinger-Zukin model in Fig.6. The corre-sponding 68% CL’s are summarized in Tab.I and II. No-tice that the likelihood of Hu-Sawicki model for n = 4, 6in Fig. 5 and Fig. 6 shows that WMAP data does nothave enough constraining power on parameter B0. Thisis because with WMAP resolution we are insensitive tothe lensing signal which carries most of late-time evolu-tion information. From the above results, we firstly seethat Planck is able to reduce the error bar at least one

7

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

likel

ihoo

d

B0

n=4, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015

likel

ihoo

d

B0

n=4, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1

likel

ihoo

d

B0

n=1, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.0001 0.0002 0.0003

likel

ihoo

d

B0

n=1, PLANCK

FIG. 5. WMAP/PLANCK experiments power spectrum like-lihood of B0, for n = 1 (top) and n = 4 (bottom).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

likel

ihoo

d

B0

BZ model, ideal

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

likel

ihoo

d

B0

BZ model, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

likel

ihoo

d

B0

n=6, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

likel

ihoo

d

B0

n=6, PLANCK

FIG. 6. Hu-Sawicki model (n=6) and Bertschinger-Zukinmodel power spectrum likelihood.

order of magnitude. In Fig.7, we demonstrate that theseimprovements mainly come from lensing signal. The χ2

for lensed TT spectrum (pink curve) increases very faston small scales and saturate around multipoles l ∼ 2000,while the unlenesd one (blue curve) only receives the con-tributions from late time ISW effect on large scales. InFig.7 we also plot the ratio ∆Cl/

√2σl for the Hu-Sawicki

model with parameter values n = 1, B0 = 10−4, where∆Cl is defined as

∆Cl ≡ CTl (B0 = 10−4)− CT

l (B0 = 0) , (30)

and the variance σl can be found in Eq. (29). FromFig.7 we can see that the ISW effect contribution to thefinal signal to noise ratio is not negligible in amplitudecompared with the lensing part. However, the numberof multipoles which ISW could change is much less thanthose of lensing effect. Furthermore, we can also see thatthe χ2 for the lensed case (pink curve) saturates at mul-tipoles (l ∼ 2000) around O(1) for B0 = 10−4, whichagrees with our global analysis. We have checked that

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

1 10 100 1000 10000

l

|∆Cl|/√2σl, n=1, B0=10-4

|∆C~

l|/√2σl, n=1, B0=10-4

Χ2/2, n=1, B0=10-4

Χ~2/2, n=1, B0=10-4

FIG. 7. . The demonstration of main signal of constrain-ing B0 in Hu-Sawicki model with n = 1, B0 = 10−4. Thered (unlensed) and green (lensed) curves shows the fractionaldifference in TT spectrum between f(R) model and generalrelativity. And the blue (unlensed) and pink (lensed) curvesshows the increase of χ2/2 with respect to the maximum ofmultipoles.

the other models considered in this paper share a similarbehavior.

Notice that for the Bertschinger-Zukin model, our er-ror bars for an ideal experiment (no noise, lmax = 1000)compared with those in [42] (σ = 0.61) are improved bya factor 3.74. This is due to the fact that in our analysiswe account for the weak lensing in the power spectrum.

B. Bispectrum

As already mentioned in the previous section, evenfor perfectly Gaussian primordial perturbations ISW andWL effects generate some non-Gaussian features in theCMB. In this subsection, we explore the possibility toconstrain cosmological parameters via the ISW-lensingbispectrum. For a rotation-invariant sky, all the cos-mological information is encoded in the angular-averagedbispectrum (e.g., see [59]), defined as:

Bl1l2l3 =

√

(2l1 + 1)(2l2 + 1)(2l3 + 1)

4π

(

l1 l2 l30 0 0

)

bl1l2l3 ,

(31)where the reduced bispectrum bl1l2l3 produced by theISW-Lensing correlation is given by Eq. (27). The aver-aged bispctrum is different from 0 only for l1, l2, l3 triplesthat satisfy triangle inequality and parity conservationrelationships:

l1 ≤ l2 ≤ l3 ,

l1 + l2 + l3 = even , (32a)

|lj − lk| ≤ li ≤ lj + lk , (32b)

8

TABLE III. B0 error bars (68% CL) from bispectrum χ2 forthe Hu-Sawicki model

n WMAP PLANCK

1 ——– 2.90 × 10−4

4 ——– 1.89 × 10−3

6 ——– 6.29 × 10−3

a The blank elements in the table represent that WMAP datacannot efficiently constrain B0 parameter in Hu-Sawicki model,because the likelihood does not vanish in the tails which can beread from Fig. 8 and 9.

TABLE IV. B0 error bars (68% CL) from bispectrum χ2 forthe Bertschinger-Zukin model

Ideal (lmax = 1000) PLANCK

1.92 × 10−1 3.85 × 10−2

a We can see that the results from ideal experiment withlmax = 1000 is in agreement with those obtained in[42].

Following the approach of the previous section, we nowconsider the chi-squared variable [38, 39, 59, 60]:

χ2B =

∑

l1,l2,l3

[

Bthl1l2l3

−Bfidl1l2l3

]2

σ2B,l1l2l3

, (33)

where the bispectrum variance reads

σ2B,l1l2l3 = ∆l1l2l3C

Tl1C

Tl2C

Tl3 , (CTl = CT

l +Nl) , (34)

with the weights

∆l1l2l3 =

6 , (l1 = l2 = l3) ,

2 , (two identical) , (35a)

1 , (all different) .

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

likel

ihoo

d

B0

n=4, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.002 0.004 0.006

likel

ihoo

d

B0

n=4, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.02 0.04 0.06 0.08 0.1

likel

ihoo

d

B0

n=1, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.0002 0.0004 0.0006 0.0008 0.001

likel

ihoo

d

B0

n=1, PLANCK

FIG. 8. WMAP/PLANCK experiments bispectrum likeli-hood of B0, for n = 1 (top) and n = 4 (bottom).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

likel

ihoo

d

B0

BZ model, ideal

0

0.2

0.4

0.6

0.8

1

0 0.04 0.08 0.12 0.16 0.2

likel

ihoo

d

B0

BZ model, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

likel

ihoo

d

B0

n=6, WMAP

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

likel

ihoo

d

B0

n=6, PLANCK

FIG. 9. Hu-Sawicki model (n=6) and Bertschinger-Zukinmodel bispectrum likelihood.

The CMB bispectrum likelihood for the Hu-Sawickimodels with n = 1, 4, 6 and the Bertschinger-Zukinmodel are plotted in Fig.8 and Fig.9, respectively, andthe corresponding B0 error bars are listed in Tab.III andTab.IV. Similar as power spectrum case, WMAP dataonly cannot give any constraint on B0. Compared withthe B0 error bars obtained from the power spectrum anal-ysis, we do not find any significant improvement from thebispectrum alone. In details, for the Hu-Sawicki modelswith n = 4, 6 the ratio between the error bars from thepower spectrum and those from the bispectrum are 1.09and 2.40, respectively. For the Hu-Sawicki model withn = 1 and the Bertschinger-Zukin model, the error barsfrom the bispectrum are even larger than those from thepower spectrum. In the following section we are howevergoing to explore the possibility of improving our con-straint through a joint analysis of power spectrum andbispectrum.

C. Joint Analysis

As already specified in the last part of Sec.III, thepower spectrum can correlate with the ISW-Lensingbispectrum via both the primordial and the secondaryanisotropies (ISW and weak lensing). However, compar-ing with the primordial and ISW channels, the cross-correlation through weak lensing is sub-dominant. Thereason is that cross-correlator of the power spectrum andbispectrum via weak lensing channel is an 8-point func-tion of a Gaussian random fluctuations (alm, φlm), whileboth the primordial and ISW are a 6-point function.Therefore, we will focus on the unlensed cross-correlationchannel in what follows, i.e. the power spectrum cross-correlation with the bispectrum through unlensed fluctu-ations.

If we compose the power spectrum and the bispectruminto a vector ∆ ≡ (∆Cl,∆Bl1l2l3), the joint likelihood

9

function reads

L ∝ exp(

−∆ · CoV−1 ·∆T)

, (36)

where the inverse of the covariance matrix can be writtenas a 2× 2 blocked form

CoV−1 ≡ 1

σ2Pσ

2B − (σ2

PB)2

[

σ2B,l1l2l3

−σ2PB,l1l2l3l

−σ2PB,ll1l2l3

σ2P,l

]

.

(37)When the power spectrum and the bispectrum are weaklycorrelated (σ2

PB ≪ σ2P,B), the above formula can be re-

duced into[61]

L ∝ exp(−χ2/2) , χ2 = χ2P + χ2

B + χ2PB . (38)

For the computation of cross likelihood we have

χ2PB =

∑

l,l1,l2,l3

−2σ2PB,ll1l2l3

×∆CTl ×∆BISW−L

l1l2l3

σ2P,lσ

2B,l1l2l3

,(39)

where the variance reads

σ2PB,ll1l2l3 ≡

⟨

∆CTl ∆BISW−L

l1l2l3

⟩

=√

(2l1 + 1)(2l2 + 1)(2l3 + 1)

4π

(

l1 l2 l30 0 0

)

×

1

2

[

− l1(l1 + 1) + l2(l2 + 1) + l3(l3 + 1)]

⟨

CTl C

φTl2

CTl3

⟩

+ 5perm.

− CTl B

ISW−Ll1l2l3

=

√

(2l1 + 1)(2l2 + 1)(2l3 + 1)

4π

(

l1 l2 l30 0 0

)

×

1

2

[

− l1(l1 + 1) + l2(l2 + 1) + l3(l3 + 1)]

×[

2δll3(2l3 + 1)

CTl C

φTl2

CTl3 +

2δll2(2l2 + 1)

CTl C

φTl2

CTl3

+8δll2δll3(2l + 1)2

CTl C

φTl2

CTl3

]

+ 5perm.

, (40)

with

CφTl =

∑

m,m′

δmm′φ∗

lmaTlm′

2l+ 1, (41)

CTl =

∑

m,m′

δmm′aT∗

lmaTlm′

2l+ 1. (42)

We emphasize again that here we cross-correlate ISW-Lensing bispectrum with unlensed power spectrum.From the above result, we can see that the first two termsin Eq.(40), which peak at the squeezed configurations,give the main contributions to the cross-correlation, whilethe last term only affect the equilateral configurations.Since our ISW-Lensing bispectrum signal peaks at thesqueezed limit, the contributions from the third term is

0.001

0.01

0.1

1e-05 0.0001 0.001 0.01 0.1

χ2 Cro

ss/χ

2 Bis

p

B0

HS,n=1HS,n=4HS,n=6

BZ

FIG. 10. Likelihood ratio between cross-correlation and bis-pectrum χ2

PB/χ2B.

negligible. In Appendix.B, we show a computation ex-ample about σ2

PB.In Fig.10, we show the likelihood ratio between cross-

correlation and bispectrum for several B0 values. Thisresult indicates that the maximum correction from cross-correlations are 3%, so we can safely neglect the cross-correlations between the power spectrum and the bispec-trum in the joint analysis. In Fig.11, we plot the jointlikelihoods of B0 for the Planck experiment, and the cor-responding 68% CL results are listed in Tab.V. Compar-ing with the error bars from the power spectrum alone,there is an improvement of a factor ranging from 1.14 to5.32 (depending on the specific modified gravity model).

0

0.2

0.4

0.6

0.8

1

0 5e-05 0.0001 0.00015 0.0002

likel

ihoo

d

B0

n=1, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.001 0.002 0.003

likel

ihoo

d

B0

n=4, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.002 0.004 0.006 0.008 0.01

likel

ihoo

d

B0

n=6, PLANCK

0

0.2

0.4

0.6

0.8

1

0 0.01 0.02 0.03 0.04 0.05

likel

ihoo

d

B0

BZ model, PLANCK

FIG. 11. Joint analysis results: spectrum and bispectrumproduct, no cross-correlation included.

V. CONCLUSIONS

The correlation between the Integrated Sachs Wolfe(ISW) effect and gravitational lensing is expected to

10

TABLE V. B0 error bars (68% CL) from the power spectrum-bispectrum joint analysis

n/BZ Joint Power/Joint

1 5.85 × 10−5 1.14

4 6.90 × 10−4 2.99

6 2.84 × 10−3 5.32

BZ 7.87 × 10−3 1.40

generate a bispectrum signature in the CMB . In factthe Planck satellite can detect this secondary non-Gaussianity [43–48]. The high precision measurements ofCMB anisotropies expected from Planck data will enableus to detect such signature at a high level of significance[43–48]. This will open the interesting possibility of ac-tually using ISW-lensing bispectrum measurements as atool to constrain cosmological parameters (see, e.g., [38–42]). Since the ISW-Lensing bispectrum is composed bythe product of primordial power spectrum and the ISW-Lensing cross-spectrum, it carries information on bothearly (recombination) and late-time epochs. In this paperwe investigated the possibility to use forthcoming Planck

data to constrain some modified gravity models and theirdeviations from GR as a candidate to explain the late-time acceleration of the universe, focusing in particularon the possibility to perform a joint analysis of CMBpower spectrum and ISW-Lensing bispectrum.Firstly we compared the ability to constrain cosmologi-

cal parameters via CMB power spectrum only in WMAPand Planck missions. Due to the high resolution and sen-sitivity of the Planck dataset, Planck data will allow toimprove the constraint on the modified gravity param-eter B0 at least by one order magnitude compared towhat one can do with WMAP. Secondly we performed ajoint power spectrum/bispectrum analysis. Our resultsshow that the joint analysis could reduce the B0 1-σ er-ror bar by a factor ranging from 1.14 to 5.32 w.r.t. towhat can be achieved with the power spectrum alone.The improvement factor depends on the model consid-ered: for the Hu-Sawicki models with large value of nthe improvements are quite significant unlike those withsmall value of n and the Bertschinger-Zukin model. Fur-themore, we have provided the full formulae to computethe cross-correlation between the power spectrum andthe bispectrum (that are necessary in the joint-analysisapproach). One of our main results is that this cross-correlation can be safely neglected and this conclusionseems quite model independent.

ACKNOWLEDGMENTS

We we are greatly indebted to Wayne Hu, Alan Heav-ens, Daniele Bertacca for their helpful discussions andcorrespondences. This research has been partially sup-ported by the ASI/INAF Agreement I/072/09/0 for the

TABLE VI. B0 error bars (68% CL) summary for Planckexperiment

Model Pow Bisp Joint

1 6.68 × 10−5 2.90 × 10−4 5.85 × 10−5

4 2.06 × 10−3 1.89 × 10−3 6.90 × 10−4

6 1.51 × 10−2 6.29 × 10−3 2.84 × 10−3

BZ 1.10 × 10−2 3.85 × 10−2 7.87 × 10−3

Planck LFI Activity of Phase E2 and by the PRIN 2009project ”La Ricerca di non-Gaussianita‘ Primordiale”

Appendix A: CMB power spectrum Fisher matrix

analysis

In this Appendix, we present the results of our Cl

Fisher matrix analysis. The Fisher matrix for CMBanisotropies and polarization is [51–56]

Fij =∑

l

∑

X,Y

∂CXl

∂pi

(

Covl

)

−1

XY

∂CY l

∂pj, (A1)

where the covariance matrix is defined as(

Covl

)

TT=

2

(2l + 1)fsky

[

CTl + w−1T B−2

l

]2

, (A2)

(

Covl

)

EE=

2

(2l + 1)fsky

[

CEl + w−1P B−2

l

]2

, (A3)

(

Covl

)

TE=

2

(2l + 1)fskyC2

Cl , (A4)

with

wX,c =(

σX,cθbeam

)

−2

, wX =∑

c

wX,c , (A5)

Bl =1

wX

∑

c

wX,cBc,l , X ∈ (T, P ) (A6)

Bc,l ≃ exp

−l(l+ 1)θ2beam8 ln 2

. (A7)

In the above expressions, B2l is the Gaussian beam win-

dow function ,θbeam is the beam’s full-width at half-maximum (FWHM), and σX are the sensitivities. In thispaper we choose beams and sensitivities representative ofWMAP and Planck experiments, following specificationsfrom [56, 58]. For convenience we list channels, beamsand noise levels we used in Tab.VII.As shown in the main text, the likelihood of B0 is

highly non-Gaussian, which makes forecasting B0 errorbars through a Fisher matrix analysis unfeasible. On theother hand we still use the Fisher matrix as a way to get areasonable order of magnitude estimate of the correlationbetween different parameters.We jointly analyze 7 cosmological parameters within

the Hu-Sawicki model (As, ns, τ,Ωbh2,Ωmh2, h, B0). The

11

TABLE VII. WMAP and Planck experiment

Experiment Frequency θbeam σT σP

WMAP: 94 12.6 49.9 70.7

60 21.0 30.0 42.6

40 28.2 17.2 24.4

Planck: 217 5.0 13.1 26.8

143 7.1 6.0 11.5

Junk:2012qt 100 10.0 6.8 10.9

70 14.0 12.8 18.3

a Frequencies in GHz. Beam size θbeam is the FWHM inarcminutes. Sensitivities σT and σP are in µK per FWHMbeam.

TABLE VIII. Fisher matrix joint analysis results (68%CL) onlensed power spectrum for the Hu-Sawicki model

CP Fiducial valuea WMAP PLANCK

As 2.43× 10−9 9.61× 10−11 3.46× 10−11

ns 0.963 1.15 × 10−2 3.38 × 10−3

τ 0.088 1.58 × 10−2 3.92 × 10−3

Ωbh2 0.0226 5.53 × 10−4 1.47 × 10−4

Ωmh2 0.1109 5.31 × 10−3 1.23 × 10−3

h 0.70 1.72 × 10−2 6.20 × 10−3

B0(n = 4)b 0 1.81 × 10−3 8.33 × 10−5

B0(n = 1) 0 4.41 × 10−4 1.92 × 10−5

a Fiducial values of the first six parameters are taken asWMAP7yr best-fit values.

b Results of the first six parameters include polarizationcontributions, while those of B0 are only from CTT

lmode.

first 6 are the standard ΛCDM parameters, while B0 isthe new modified gravity parameter. Actually, as statedin Sec.II B the Hu-Sawicki model is charaterized by 2parameters (B0, n), however, in our calculations we fixn for a few specific values. The are various reasons forthis choice: firstly, there exists a parameter degeneracybetween n and B0; secondly, the current constraints onthe parameter n is rather weak [9] and the results arestrongly dependent on the value of n. Given the aboveconsiderations, we fix n for a few specific values in thefollowing numerical analysis, i.e. we can deal with theHu-Sawicki model with different n’s as different models.

Our Fisher matrix results are summarized in Ta-ble VIII. The first six rows are obtained for the Hu-Sawicki model with n = 4. Actually, we find that thedependence of these standard parameters on n is ratherweak, i.e. we can obtain similar results for other values of

n. As a numerical check, we verified that our results forthe 6 “standard” parameters match well both WMAP-7yr [57] and Planck blue book results [58]. We find nosignificant degeneracies between B0 and other parame-ter, thus justifying the simple 1-parameter likelihood ap-proach we use in the main text to forecast B0 error bars

TABLE IX. Cross-correlation coefficients of the standard cos-mological parameters with B0 in Hu-Sawicki model (n = 4)

B0-CP cross-correlation coefficients

As −7.9× 10−2

ns −0.12

τ −8.0× 10−2

Ωbh2

−0.18

Ωmh2−4.7× 10−2

h 8.2× 10−2

Furthermore, in Tab. IX we list the cross-correlationcoefficients of several cosmological parameters with B0 inHu-Sawicki model with n = 4. Our results show that allthe cross-correlation coefficients are small.

Appendix B: Computation example of σ2PB

As an example of the computation for the cross-likelihood, Eq. 40 we show explicitly the following term:

⟨

CTl C

φTl2

CTl3

⟩

− CTl B

ISW−Ll1l2l3

=∑

m′,m′

2,m′

3,m,m2,m3

δmm′δm2m′

2δm3m′

3

(2l + 1)(2l2 + 1)(2l3 + 1)

⟨

aT∗

lmaTlmφ∗

l2m2aTl2m2

aT∗

l3m3aTl3m3

⟩

− CTl B

ISW−Ll1l2l3

=∑

m′,m′

2,m′

3,m,m2,m3

δmm′δm2m′

2δm3m′

3

(2l + 1)(2l2 + 1)(2l3 + 1)

×

4⟨

aT∗

lmaTlmφ∗

l2m2aTl2m2

aT∗

l3m3aTl3m3

⟩

+2⟨

aT∗

lmaTlmφ∗

l2m2aTl2m2

aT∗

l3m3aTl3m3

⟩

+4⟨

aT∗

lmaTlmφ∗

l2m2aTl2m2

aT∗

l3m3aTl3m3

⟩

+2⟨

aT∗

lmaTlmφ∗

l2m2aTl2m2

aT∗

l3m3aTl3m3

⟩

, (B1)

where we have subtracted 3 disconnected terms.

[1] Y. -S. Song, W. Hu and I. Sawicki, “The Large ScaleStructure of f(R) Gravity,” Phys. Rev. D 75, 044004

(2007) [astro-ph/0610532].

12

[2] P. Zhang, M. Liguori, R. Bean and S. Dodelson, “ProbingGravity at Cosmological Scales by Measurements whichTest the Relationship between Gravitational Lensing andMatter Overdensity,” Phys. Rev. Lett. 99, 141302 (2007)[arXiv:0704.1932 [astro-ph]].

[3] S. Wang, L. Hui, M. May and Z. Haiman, “Is ModifiedGravity Required by Observations? An Empirical Con-sistency Test of Dark Energy Models,” Phys. Rev. D 76,063503 (2007) [arXiv:0705.0165 [astro-ph]].

[4] B. Jain and P. Zhang, “Observational Tests of Mod-ified Gravity,” Phys. Rev. D 78, 063503 (2008)[arXiv:0709.2375 [astro-ph]].

[5] J. -P. Uzan, “The acceleration of the universe and thephysics behind it,” Gen. Rel. Grav. 39, 307 (2007) [astro-ph/0605313].

[6] P. Zhang, “Testing f(R) gravity against the large scalestructure of the universe.,” Phys. Rev. D 73, 123504(2006) [astro-ph/0511218].

[7] S. Camera, A. Diaferio and V. F. Cardone, “Tomogra-phy from the Next Generation of Cosmic Shear Experi-ments for Viable f(R) Models,” JCAP 1107, 016 (2011)[arXiv:1104.2740 [astro-ph.CO]].

[8] F. Schmidt, “Weak Lensing Probes of Modified Gravity,”Phys. Rev. D 78, 043002 (2008) [arXiv:0805.4812 [astro-ph]].

[9] M. Martinelli, A. Melchiorri, O. Mena, V. Salvatelli andZ. Girones, “Future constraints on the Hu-Sawicki mod-ified gravity scenario,” Phys. Rev. D 85, 024006 (2012)[arXiv:1109.4736 [astro-ph.CO]].

[10] E. Calabrese, A. Cooray, M. Martinelli, A. Mel-chiorri, L. Pagano, A. Slosar and G. F. Smoot, “CMBLensing Constraints on Dark Energy and ModifiedGravity Scenarios,” Phys. Rev. D 80, 103516 (2009)[arXiv:0908.1585 [astro-ph.CO]].

[11] E. Calabrese, A. Slosar, A. Melchiorri, G. F. Smoot andO. Zahn, “Cosmic Microwave Weak lensing data as a testfor the dark universe,” Phys. Rev. D 77, 123531 (2008)[arXiv:0803.2309 [astro-ph]].

[12] S. Camera, D. Bertacca, A. Diaferio, N. Bartolo andS. Matarrese, “Weak lensing signal in Unified Dark Mat-ter models,” Mon. Not. Roy. Astron. Soc. 399, 1995(2009) [arXiv:0902.4204 [astro-ph.CO]].

[13] X. Wang, X. Chen and C. Park, “Topology of large scalestructure as test of modified gravity,” Astrophys. J. 747,48 (2012) [arXiv:1010.3035 [astro-ph.CO]].

[14] H. Gil-Marin, F. Schmidt, W. Hu, R. Jimenez andL. Verde, “The Bispectrum of f(R) Cosmologies,” JCAP1111, 019 (2011) [arXiv:1109.2115 [astro-ph.CO]].

[15] T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis,“Modified Gravity and Cosmology,” Phys. Rept. 513, 1(2012) [arXiv:1106.2476 [astro-ph.CO]].

[16] A. A. Starobinsky, “A New Type of Isotropic Cosmolog-ical Models Without Singularity,” Phys. Lett. B 91, 99(1980).

[17] W. Hu and I. Sawicki, “Models of f(R) Cosmic Acceler-ation that Evade Solar-System Tests,” Phys. Rev. D 76,064004 (2007) [arXiv:0705.1158 [astro-ph]].

[18] A. De Felice and S. Tsujikawa, “f(R) theories,” LivingRev. Rel. 13, 3 (2010) [arXiv:1002.4928 [gr-qc]].

[19] E. Bertschinger and P. Zukin, “Distinguishing ModifiedGravity from Dark Energy,” Phys. Rev. D 78, 024015(2008) [arXiv:0801.2431 [astro-ph]].

[20] A. S. Eddington, “The Mathematical Theory of Relativ-ity,” (Cambridge University Press,1922).

[21] L. Amendola, M. Kunz and D. Sapone, JCAP 0804, 013(2008) [arXiv:0704.2421 [astro-ph]].

[22] G. -B. Zhao, L. Pogosian, A. Silvestri and J. Zylber-berg, “Searching for modified growth patterns with to-mographic surveys,” Phys. Rev. D 79, 083513 (2009)[arXiv:0809.3791 [astro-ph]].

[23] G. -B. Zhao, L. Pogosian, A. Silvestri and J. Zylberberg,“Cosmological Tests of General Relativity with FutureTomographic Surveys,” Phys. Rev. Lett. 103, 241301(2009) [arXiv:0905.1326 [astro-ph.CO]].

[24] A. Hojjati, L. Pogosian and G. -B. Zhao, “Testing gravitywith CAMB and CosmoMC,” JCAP 1108, 005 (2011)[arXiv:1106.4543 [astro-ph.CO]].

[25] T. Giannantonio, M. Martinelli, A. Silvestri and A. Mel-chiorri, “New constraints on parametrised modified grav-ity from correlations of the CMB with large scale struc-ture,” JCAP 1004, 030 (2010) [arXiv:0909.2045 [astro-ph.CO]].

[26] E. Bertschinger, Astrophys. J. 648, 797 (2006) [astro-ph/0604485].

[27] J. Zuntz, T. Baker, P. Ferreira and C. Skordis, “Ambigu-ous Tests of General Relativity on Cosmological Scales,”arXiv:1110.3830 [astro-ph.CO].

[28] W. Hu and I. Sawicki, “A Parameterized Post-FriedmannFramework for Modified Gravity,” Phys. Rev. D 76,104043 (2007) [arXiv:0708.1190 [astro-ph]].

[29] W. Hu, Phys. Rev. D 77, 103524 (2008) [arXiv:0801.2433[astro-ph]].

[30] W. Fang, W. Hu and A. Lewis, “Crossing the PhantomDivide with Parameterized Post-Friedmann Dark En-ergy,” Phys. Rev. D 78, 087303 (2008) [arXiv:0808.3125[astro-ph]].

[31] A. Lewis, A. Challinor and A. Lasenby, “Efficient com-putation of CMB anisotropies in closed FRW models,”Astrophys. J. 538, 473 (2000) [astro-ph/9911177].

[32] L. Lombriser, A. Slosar, U. Seljak and W. Hu, “Con-straints on f(R) gravity from probing the large-scalestructure,” arXiv:1003.3009 [astro-ph.CO].

[33] R. Bean and M. Tangmatitham, “Current constraints onthe cosmic growth history,” Phys. Rev. D 81, 083534(2010) [arXiv:1002.4197 [astro-ph.CO]].

[34] D. Bertacca, N. Bartolo and S. Matarrese, “A new ap-proach to cosmological perturbations in f(R) models,”arXiv:1109.2082 [astro-ph.CO].

[35] E. V. Linder, “Cosmic growth history and expan-sion history,” Phys. Rev. D 72, 043529 (2005) [astro-ph/0507263].

[36] T. Baker, P. G. Ferreira, C. Skordis and J. Zuntz,“Towards a fully consistent parameterization of mod-ified gravity,” Phys. Rev. D 84, 124018 (2011)[arXiv:1107.0491 [astro-ph.CO]].

[37] R. K. Sachs and A. M. Wolfe, “Perturbations of a cos-mological model and angular variations of the microwavebackground,” Astrophys. J. 147, 73 (1967) [Gen. Rel.Grav. 39, 1929 (2007)].

[38] D. M. Goldberg and D. N. Spergel, Phys. Rev. D 59,103002 (1999) [astro-ph/9811251].

[39] L. Verde and D. N. Spergel, “Dark energy and cosmicmicrowave background bispectrum,” Phys. Rev. D 65,043007 (2002) [astro-ph/0108179].

[40] F. Giovi, C. Baccigalupi and F. Perrotta, Phys. Rev. D68, 123002 (2003) [astro-ph/0308118].

[41] V. Acquaviva, C. Baccigalupi and F. Perrotta, “Weaklensing in generalized gravity theories,” Phys. Rev. D 70,

13

023515 (2004) [astro-ph/0403654].[42] E. Di Valentino, A. Melchiorri, V. Salvatelli and

A. Silvestri, “Parametrised modified gravity and theCMB Bispectrum,” Phys. Rev. D 86 (2012) 063517[arXiv:1204.5352 [astro-ph.CO]].

[43] K. M. Smith and M. Zaldarriaga, “Algorithms for bispec-tra: Forecasting, optimal analysis, and simulation,” Mon.Not. Roy. Astron. Soc. 417, 2 (2011) [astro-ph/0612571].

[44] D. Hanson, K. M. Smith, A. Challinor and M. Liguori,“CMB lensing and primordial non-Gaussianity,” Phys.Rev. D 80, 083004 (2009) [arXiv:0905.4732 [astro-ph.CO]].

[45] A. Mangilli and L. Verde, Phys. Rev. D 80, 123007 (2009)[arXiv:0906.2317 [astro-ph.CO]].

[46] A. Lewis, A. Challinor and D. Hanson, “The shape ofthe CMB lensing bispectrum,” JCAP 1103, 018 (2011)[arXiv:1101.2234 [astro-ph.CO]].

[47] A. Lewis, “The full squeezed CMB bispectrum from in-flation,” arXiv:1204.5018 [astro-ph.CO].

[48] R. Pearson, A. Lewis and D. Regan, “CMB lensing andprimordial squeezed non-Gaussianity,” JCAP 1203, 011(2012) [arXiv:1201.1010 [astro-ph.CO]].

[49] V. Junk and E. Komatsu, “Cosmic Microwave Back-ground Bispectrum from the Lensing–Rees-Sciama Cor-relation Reexamined: Effects of Non-linear Matter Clus-tering,” Phys. Rev. D 85, 123524 (2012) [arXiv:1204.3789[astro-ph.CO]].

[50] B. Hu et al., in preparation.[51] U. Seljak, “Measuring polarization in cosmic microwave

background,” [astro-ph/9608131].[52] M. Zaldarriaga and U. Seljak, “An all sky analysis of

polarization in the microwave background,” Phys. Rev.D 55, 1830 (1997) [astro-ph/9609170].

[53] M. Kamionkowski, A. Kosowsky and A. Stebbins, “AProbe of primordial gravity waves and vorticity,” Phys.Rev. Lett. 78, 2058 (1997) [astro-ph/9609132].

[54] J. R. Bond, G. Efstathiou and M. Tegmark, “Forecast-ing cosmic parameter errors from microwave backgroundanisotropy experiments,” Mon. Not. Roy. Astron. Soc.291, L33 (1997) [astro-ph/9702100].

[55] M. Zaldarriaga, D. N. Spergel and U. Seljak, “Microwavebackground constraints on cosmological parameters,” As-trophys. J. 488, 1 (1997) [astro-ph/9702157].

[56] D. J. Eisenstein, W. Hu and M. Tegmark, “Cosmic com-plementarity: Joint parameter estimation from CMB ex-periments and redshift surveys,” Astrophys. J. 518, 2(1999) [astro-ph/9807130].

[57] [WMAP team], “http://lambda.gsfc.nasa.gov/product/map/current/params/lcdm sz lens wmap7.cfm,”

[58] [Planck Collaboration], “The Scientific programme ofplanck,” astro-ph/0604069.

[59] E. Komatsu and D. N. Spergel, “Acoustic signatures inthe primary microwave background bispectrum,” Phys.Rev. D 63, 063002 (2001) [astro-ph/0005036].

[60] E. Komatsu, “Hunting for Primordial Non-Gaussianity inthe Cosmic Microwave Background,” Class. Quant. Grav.27, 124010 (2010) [arXiv:1003.6097 [astro-ph.CO]].

[61] E. Sefusatti, M. Crocce, S. Pueblas and R. Scoccimarro,“Cosmology and the Bispectrum,” Phys. Rev. D 74,023522 (2006) [astro-ph/0604505].

[62] A. Hojjati, L. Pogosian, A. Silvestri and S. Tal-bot, “Practical solutions for perturbed f(R) gravity,”arXiv:1210.6880 [astro-ph.CO].

[63] J. Dunkley et al. [WMAP Collaboration], Astrophys. J.Suppl. 180, 306 (2009) [arXiv:0803.0586 [astro-ph]].

[64] C. -L. Kuo, P. A. R. Ade, J. J. Bock, J. R. Bond,C. R. Contaldi, M. D. Daub, J. H. Goldstein andW. L. Holzapfel et al., Astrophys. J. 664, 687 (2007)[astro-ph/0611198].

[65] A. C. S. Readhead, B. S. Mason, C. R. Contaldi,T. J. Pearson, J. R. Bond, S. T. Myers, S. Padin andJ. L. Sievers et al., Astrophys. J. 609, 498 (2004) [astro-ph/0402359].

[66] K. Grainge, P. Carreira, K. Cleary, R. D. Davies,R. J. Davis, C. Dickinson, R. Genova-Santos andC. M. Gutierrez et al., Mon. Not. Roy. Astron. Soc. 341,L23 (2003) [astro-ph/0212495].

[67] M. Kowalski et al. [Supernova Cosmology Project Collab-oration], Astrophys. J. 686, 749 (2008) [arXiv:0804.4142[astro-ph]].

[68] A. G. Riess, L. Macri, S. Casertano, M. Sosey, H. Lam-peitl, H. C. Ferguson, A. V. Filippenko and S. W. Jhaet al., Astrophys. J. 699, 539 (2009) [arXiv:0905.0695[astro-ph.CO]].

[69] W. J. Percival et al. [SDSS Collaboration], Mon. Not.Roy. Astron. Soc. 401, 2148 (2010) [arXiv:0907.1660[astro-ph.CO]].

[70] S. Ho, C. Hirata, N. Padmanabhan, U. Seljakand N. Bahcall, Phys. Rev. D 78, 043519 (2008)[arXiv:0801.0642 [astro-ph]].

[71] C. M. Hirata, S. Ho, N. Padmanabhan, U. Seljakand N. A. Bahcall, Phys. Rev. D 78, 043520 (2008)[arXiv:0801.0644 [astro-ph]].

[72] R. Reyes, R. Mandelbaum, U. Seljak, T. Baldauf,J. E. Gunn, L. Lombriser and R. E. Smith, Nature 464,256 (2010) [arXiv:1003.2185 [astro-ph.CO]].

[73] R. Mandelbaum, A. Slosar, T. Baldauf, U. Seljak,C. M. Hirata, R. Nakajima, R. Reyes and R. E. Smith,arXiv:1207.1120 [astro-ph.CO].

[74] A. Vikhlinin, A. V. Kravtsov, R. A. Burenin, H. Ebeling,W. R. Forman, A. Hornstrup, C. Jones and S. S. Murrayet al., Astrophys. J. 692, 1060 (2009) [arXiv:0812.2720[astro-ph]].

[75] B. Koester et al. [SDSS Collaboration], Astrophys. J.660, 239 (2007) [astro-ph/0701265].

[76] K. N. Abazajian et al. [SDSS Collaboration], Astrophys.J. Suppl. 182, 543 (2009) [arXiv:0812.0649 [astro-ph]].

[77] F. Schmidt, A. Vikhlinin and W. Hu, Phys. Rev. D 80,083505 (2009) [arXiv:0908.2457 [astro-ph.CO]].

[78] L. Lombriser, F. Schmidt, T. Baldauf, R. Mandelbaum,U. Seljak and R. E. Smith, Phys. Rev. D 85, 102001(2012) [arXiv:1111.2020 [astro-ph.CO]].

[79] M. Tegmark, Phys. Rev. D 66, 103507 (2002) [astro-ph/0101354].

[80] T. Baker, P. G. Ferreira and C. Skordis, Phys. Rev. D87, 024015 (2013) [arXiv:1209.2117 [astro-ph.CO]].