roman scoccimarro- large-scale structure in brane-induced gravity i: perturbation theory

8/3/2019 Roman Scoccimarro- Large-Scale Structure in Brane-Induced Gravity I: Perturbation Theory

1/36

arXiv:0906.4545v2

[astro-ph.CO

]19Jul2009

Large-Scale Structure in Brane-Induced Gravity

I. Perturbation Theory

Roman ScoccimarroCenter for Cosmology and Particle Physics, Department of Physics,

New York University, NY 10003, New York, USA

We study the growth of subhorizon perturbations in brane-induced gravity using perturbationtheory. We solve for the linear evolution of perturbations taking advantage of the symmetry undergauge transformations along the extra-dimension to decouple the bulk equations in the quasistaticapproximation, which we argue may be a better approximation at large scales than thought before.We then study the nonlinearities in the bulk and brane equations, concentrating on the workings ofthe Vainshtein mechanism by which the theory becomes general relativity (GR) at small scales. Weshow that at the level of the power spectrum, to a good approximation, the effect of nonlinearitiesin the modified gravity sector may be absorbed into a renormalization of the gravitational constant.

Since the relation between the lensing potential and density perturbations is entirely unaffected bythe extra physics in these theories, the modified gravity can be described in this approximationby a single function, an effective gravitational constant for nonrelativistic motion that depends onspace and time. We develop a resummation scheme to calculate it, and provide predictions for thenonlinear power spectrum. At the level of the large-scale bispectrum, the leading order correctionsare obtained by standard perturbation theory techniques, and show that the suppression of thebrane-bending mode leads to characteristic signatures in the non-Gaussianity generated by gravity,generic to models that become GR at small scales through second-derivative interactions. Wecompare the predictions in this work to numerical simulations in a companion paper.

I. INTRODUCTION

The understanding of the current acceleration of

cosmic expansion is one of the most pressing chal-lenges in cosmology. While the cause of accelerationis presently not well understood, a substantial num-ber of observations are being planned in the nearfuture that will try to solve this puzzle.

Cosmic acceleration can be explained in at leasttwo different ways. Since in general relativity (GR)gravitational effects can be generated by pressurein addition to energy density (being, respectively,space-space and time-time components of the stress-energy tensor), the so-called dark energy solution tocosmic acceleration postulates a new contributionto the stress-energy tensor that dominates at late

times and effectively has an equation of state wherepressure is comparable to minus energy density (innatural units), leading to a repulsive force that fu-els acceleration. The simplest, best motivated, suchmodel in agreement with current observations is thecosmological constant.

Another possible explanation is that cosmic accel-eration signals a breakdown of GR at present cos-

mological scales, rather than a new contribution tostress-energy. In this case, gravity is modified inthe infrared (large-distance) regime to provide ac-

celeration even though the universe is dominated bynon-relativistic matter at late times.Mapping the expansion history of the universe

from deceleration to present acceleration typicallycannot distinguish between these two possibilities,as e.g. the Friedmann equation in a theory of mod-ified gravity can always be well approximated by aGR Friedmann equation with a dark energy com-ponent with suitable equation of state. The key isto add information on the growth of structure [1],which results from a competition between gravita-tional attraction and the expansion history: differenttheories of gravity will result in different growth ofstructure for a fixed (observed) expansion history.

By now, a large number of tests along these lineshave been proposed in the literature [114, 1426],and will be pursued in upcoming surveys.

Modifying gravity at large scales is a difficult task,and so far there has been limited success in findingconsistent theories. GR is a very constrained theory,since it follows from requirements of locality in 3+1dimensions, massless spin two graviton and general
http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2http://arxiv.org/abs/0906.4545v2


2/36

covariance [2731]. Any modification to it entails

new degrees of freedom. When allowing for com-plex models of dark energy, however, the meaningof what constitutes a modification of gravity canbecome nontrivial to define. For example, manytheories usually considered in the cosmology liter-ature as modification of gravity can be written asGR plus a (possibly non-minimally coupled) scalarfield, i.e. some form of dark energy (a notable ex-ample in this class are f(R) theories [32, 33]). Insuch scalar-tensor theories, where only a scalar de-gree of freedom is added to GR, it is a matter ofchoice how to describe the physics (since both areequivalent). A more restrictive definition of grav-

ity modification is to define departures from GR assuch theories in which tensor perturbations propa-gate differently than in GR even in the absence ofany contributions to the stress energy of the back-ground, i.e. gravitons behave differently than in GReven in flat space [34]. In this case one considers asgravity modification the addition of scalars only ifthey change the properties of the spin-2 sector, e.g.altering the speed of propagation of gravitons.

Finally, the most restrictive definition of modi-fied gravity (and the one we adopt in this paper) in-volves modification in the number of degrees of free-dom in the spin-2 sector as a requisite, e.g. extragraviton polarizations, or many gravitons. Exam-ples include theories that realize the idea of massivegravity, e.g. large-scale extra-dimensions [3538],degravitation [39] and bigravity [40, 41]. These the-ories generically have a scalar-tensor regime, wherethe predictions can be approximated by the additionof one or more effective scalars (e.g. the zero-helicitycomponent of the graviton/s), but the theory in fullis not equivalent to an arbitrary dark energy model.The best-known example is the DGP model [35], inwhich the large distance behavior involves from the4D point of view an infinite number of massive spin-2 states that describe the fact that the theory isfundamentally five-dimensional. This class of theo-

ries also have more practical advantages. As alreadymentioned, general covariance imposes very power-ful constraints on the nonlinear couplings of spin-2fields. Requiring consistency leads to strongly con-strained theories that are easiest to test (or rule out),since they contain only a small number of free pa-rameters (instead of free functions). See e.g. [42] formore discussion along these lines.

From the point of view of cosmology, the extra de-

grees of freedom in the metric that behave as scalars(in the sense of the 3+1-SVT decomposition, seebelow) are the ones that lead to readily observableconsequences. To be a viable theory of gravity, how-ever, any extra scalar degree of freedom must besuppressed in the solar system, where classical tests(light deflection in combination with non-relativisticmotion) have shown GR to be a very good approxi-mation.

To date perhaps the most promising mechanismfor such a suppression is a generalization of the Vain-shtein effect found in massive gravity [43], where ex-tra degrees of freedom become nonlinear through

their self-interactions and decouple from matter,which then leads to the recovery of GR [123]. TheVainshtein mechanism happens under rather genericconditions and plays a role in various theories ofmodified gravity [34, 4446]. Being a nonlinear ef-fect, the scale at which it happens is related to theamplitude of the sources; for stars such as the sunthe Vainshtein scale is tiny by cosmological stan-dards, but for cosmological sources it is expected tobe in the interesting few Mpc range [47]. Therefore,at scales smaller than the Vainshtein scale these the-ories will be consistent with GR, and thus difficult todistinguish from dark energy by cosmological obser-vations, i.e. the expansion history plus growth testsmentioned above would fail [124].

Such considerations show that probes of cosmicacceleration that are sensitive to small scale pertur-bations may yield no (or weak) tests on the causeof the expansion history, and could be mistakenlyinterpreted in favor of dark energy if the possibil-ity of nonlinear effects in the modified gravity sec-tor is ignored. It is therefore important to look forsignatures of such nonlinearities in modified gravitytheories that can be used as a diagnostic in futureobservations. Thats in part the motivation for thispaper and its companion paper II [48].

In this paper we study the growth of structure in

brane-induced gravity in five dimensions [35], knownas the DGP model, which presents all the ingre-dients discussed above. This theory belongs to abroad class of modified gravity theories due to large-scale extra-dimensions [3638], and presents simi-larities with other realizations of the idea of mas-sive gravity such as degravitation [39] and bigrav-ity [40, 41]. For theories that implement massive

2


3/36

gravity through something akin to the Higgs mech-

anism see e.g. [49, 50].The DGP model has two branches of cosmological

solutions, the so-called normal and self-acceleratingbranches. The latter, in the presence of non-relativistic matter, makes a transition from earlytime deceleration to acceleration [51, 52] with de Sit-ter expansion in the long-time limit. In this paper,we concentrate on the self-acelerating branch as anexample of a theory of modified gravity that leadsto acceleration, even though several lines of workhave shown that this branch has significant prob-lems related to the appearance of a ghost mode atthe linear level [34, 5358], and thus it may not be

regarded as a viable description of our universe atall scales, particularly when quantum effects are in-cluded [125]. Its worth pointing out, however, thata ghost is not a necessary ingredient to obtain self-accelerated solutions [34].

On the other hand, as we discuss below, such is-sues do not seem to impact the cosmological solu-tions we study here, in the sense that cosmologicalperturbations that respond to a standard fluctua-tion spectrum source do not show any pathologi-cal behavior. In addition, and most importantly,it is expected that the techniques developed herewill be widely applicable to the study of cosmologi-cal perturbations in other (consistent) large-distancemodifications of gravity. We thus take the DGPmodel in the spirit of the simplest example of the-ories (parametrized by a single free parameter, asCDM dark energy) where observational signaturesthat can help determine the cause of cosmic acceler-ation in the near future can be addressed most eas-ily. The DGP model itself is strongly constrained bycurrent observations, see e.g. [15, 5962].

Previous work in the literature discussed gravita-tional instability in this model in different approx-imations: [47] obtained the quasistatic linear andnonlinear evolution in the spherical approximation,while [63] developed the linear quasistatic approxi-

mation further by going beyond spherical symmetryand including an explicit treatment of the bulk per-turbations, which justified the assumptions in [47].To follow perturbations of wavelength larger thanthe Hubble radius one must go beyond the qua-sistatic approximation; this was first done in [64]using a scaling ansatz and later corroborated by [65]using an exact numerical solution of the linear equa-

tions. Except for [47] all these treatments work in

linear perturbation theory using the master variableformalism developed in [66, 67] to solve for the bulkperturbations.

In this paper we extend previous work in severaldirections. First, we derive the linear solutions with-out relying on the master variable formalism, follow-ing the standard treatment of cosmological pertur-bations in GR after choosing suitable perturbationvariables based on the properties of gauge transfor-mations along the extra dimension [126]. Next wetackle the nonlinear case, focussing on the recov-ery of GR by the Vainshtein mechanism using a re-summation technique to obtain the modified Poisson

equation propagator. Finally, we apply these resultsto the calculation of the nonlinear power spectrumand the corrections induced in the bispectrum dueto the nonlinear dynamics in the modified gravitysector. We then comment on the impact of theseresults on looking for modified gravity in observa-tions. In a companion paper [48], we develop anN-body code and test some of the analytic predic-tions developed here. For similar work on N-bodysimulations see [68].

II. FIVE-DIMENSIONAL METRIC

PERTURBATIONS

We start by reviewing metric perturbations in thecontext of braneworlds, in which the observable uni-verse is a 3+1 brane embedded in a 4+1 dimensionalspacetime. Most of this section contains well knownresults (see e.g. [69, 70] for reviews), but it serves tofix our notation, explain our unusual gauge choice,and motivate our strategy to solve the bulk equa-tions described in section III D below.

A. SVT Decomposition

In this paper we are interested in the scalar per-turbations to the metric as seen by an observer inthe brane. Following the standard SVT (Scalar, Vec-tor, Tensor) decomposition (see e.g. [71, 72]) slightlygeneralized for one extra dimension, we can write ingeneral for the scalar part of the metric,

3


4/36

ds2 = e2(1 + 2) dt2 + 2 iB dxidt

e2(1 2)ij + (2ij ij3

2)A

dxidxj

(1 + g44) dw2 + 2iC dxidw + 2F dtdw,(1)

where xA = (t , x, y, z, w) with A = 0, . . . , 4, w isthe coordinate on the extra (4th spatial) dimensionwith w = 0 denoting the unperturbed brane posi-tion, and (t, w) and (t, w) characterize the back-ground solution

e a + |w| a, e 1 + |w| aa

, (2)

where a(t) is the scale factor, and dots denote timederivatives. We assume Z2 symmetry, all metric per-turbations except C, F are functions of |w|, C, F in-stead are odd (henceforth we work with w 0). Inthe absence of perturbations the metric in Eq. (1) re-duces to the background solution found in [51], with = +1 corresponding to the self-accelerated branchand = 1 to the normal branch. From now onwe will concentrate on the self-accelerated branch,although occasionally we will discuss how some re-sults change in the normal branch. Note that, al-though not obvious from Eq. (1), the backgroundbulk spacetime (w > 0) is Minkowski, by making achange of coordinates this can be made explicit butthe brane will have a complicated trajectory in thatcoordinate system, see e.g. [67]. Finally, we assumea spatially flat universe throughout this paper.

A few comments are in order regarding the formof Eq. (1). The SVT decomposition implied there isbased on the behavior of metric perturbations withrespect to transformations of the 3 spatial dimen-sions of the brane. A bulk observer, could performa similar SVT decomposition but with respect to 4

spatial dimensions, which leads to 4S, 6V and 5Tmodes that would correspond respectively to spin0,1,2 in a local inertial frame. The 5T modes arethe true physical degrees of freedom (only ones thatpropagate in the absence of sources) for a spin 2massless graviton in 5D, the rest are induced bystress energy fluctuations. Out of the total of 15modes, however, 2S and 3V modes can be set to zero

by appropriate choice of gauge, leaving 10 physical

modes (2S, 3V and 5T). However, because the back-ground spacetime does not have symmetry under full4D spatial rotations, these SVT modes do not evolveindependently. Instead, it is easier to work with the3D SVT decomposition modes which do evolve inde-pendently thanks to the fact that w = const. hyper-surfaces in the background space are homogeneousand isotropic. The price to pay is that some V andT modes in 4D project into S modes in 3D. Thatswhy apart from the usual 4 scalar modes encoded in,,A,B, we have to deal with 3 new scalar modesdescribed by C,F,g44.

The 3D SVT decomposition divides metric pertur-

bations into 7S (shown in Eq. 1), 6V and 2T modes.Gauge transformations can be used to set 3S and2V modes to zero, leaving 4S, 4V and 2T physicalmodes (again, a total of 10 physical modes). Forexample, the scalar modes B and F in Eq. (1) cor-respond to linear combination of 4D S and V modes(spin 0 and 1). The scalar mode C corresponds toa linear combination of 4D S,V and T modes (spin0,1,2). We now discuss briefly how to use suitablegauge transformations to eliminate 3 out of 7 scalarmodes.

B. Gauge Transformations and Gauge Choices

Gauge transformations for scalar perturbationscan be written as

xA xA xA, xA (x0, ixs, x4). (3)

These three functions can be chosen to eliminatesome of the scalar potentials in Eq. (1), keepingin mind that there is an additional degree of free-dom we have not yet discussed, corresponding to thebrane position [73] which will not stay fixed at w = 0in the presence of stress energy fluctuations. To seewhich scalar metric perturbations can be set to zero

we look at the transformation properties of metricperturbations under a scalar gauge transformation,

+ x0 + x0 + x4, (4)

x0 x4, (5)

4


5/36

A

A + 2 e2 xs (6)

B B + e2 x0 e2 xs (7)

C C x4 e2 (xs) (8)

F F + e2 (x0) x4 (9)

g44 g44 + 2 (x4) (10)

where dots and primes denote time and w deriva-

tives, respectively.From these equations one can deduce a few useful

gauges. The simplest to see, is a generalization ofthe longitudinal gauge, in this case one sets

A = B = C = 0. (11)

This can always be obtained, since these conditionsuniquely fix xs, x0, x4, respectively. An impor-tant property of this gauge is that the brane is bent,i.e. the position of the brane is not at w = 0 in thepresence of fluctuations, but rather at some function

w = wb(x

).The gauge that we use in this paper can be easilyobtained from the longitudinal gauge, Eq. (11). Wemake a gauge transformation w wx4 that onlyinvolves the fourth spatial coordinate (thus keepingA = B = 0), such that (x4) is fixed by the condi-tion g44 = 0 and the w-independent part of x4 isfixed by the condition that the brane is straight inthis gauge, i.e. wb = 0. This can always be done be-cause it is possible to set to zero the brane positionwith a w-independent shift of coordinates. There-fore, our gauge reads

A = B = g44 = wb = 0 (12)

Note that in going from the longitudinal gauge toour gauge we generate a nonzero C which is givenby,

C =1

2

w0

g44(x, |w|) dw |wb(x)| |w|, (13)

where quantities with hats are in the longitudinal

gauge, and the form of the second term respectsthat C must me odd in w while bringing back thebrane to the unperturbed position. Close to thebrane, say w = 0+, the first term in Eq. (13) doesnot contribute and C is directly given by the branebending wb(x

). Thus we may call C the brane-bending mode. In geometric terms, 2C determinesthe brane extrinsic curvature at subhorizon scales.

Since our gauge can be imposed at any w, allderivatives of A,B,g44 with respect to w will van-ish as well. In the following we present the equationsof motion always in this gauge. Note also that theinduced metric at the brane is given in terms of the

five-dimensional metric by g(t, xi

) = [g(5)

]w=0.The most popular choice in the literature is the

so-called Gaussian Normal (GN) gauge, where onefixes x4 as in Eq. (12), but instead fixes (xs)

fromthe condition C = 0 and (x0) from the conditionF = 0. To fully specify the gauge one fixes the w-independent part of xs by setting A = 0 at thebrane and that ofx0 by setting B = 0 at the brane.Since these two choices can only be taken at w = 0,the w-derivatives of A and B in this gauge will notbe zero at the brane (and elsewhere), and one mustkeep track of them. In this sense the GN gauge seemsa bit cumbersome, although it is perfectly fine as agauge choice [127].

C. ADM form

To understand the role of the brane-bending moreC in geometric terms, see e.g. [53, 54], it is useful tocast the metric in ADM form [74],

ds2 = g (dx + N dw)(dx + N dw) (N dw)2,

(14)where g is the 4D metric of a hypersurface at fixedw and from Eq. (1) the lapse N and shift N func-

tions are given by,

N = 1 +g44

2, N = (F, C), (15)

where we have used linear perturbation theory forN. It is also useful to write the extrinsic curvaturetensor of a hypersurface at fixed w,

5


6/36

K =1

2N

N; + N; wg

, (16)

where the covariant derivatives are with respect tothe 4D metric g . From this we have, for the back-ground metric (bars denote background values)

K00 = 12

(e2), Kij =1

2(e2) ij , (17)

where primes denote derivatives in the extra spatialdimension. Therefore, the extrinsic curvature scalarreads

K = (3 + ), (18)whereas for perturbations we have to linear order

K = e2

F(3 ) + F

+ (3 ) e22C.(19)

A similar (though longer) expression holds for thefull extrinsic curvature tensor. This will imply (af-ter we solve the bulk equations and develop the qua-sistatic solution) that the extrinsic curvature willbe dominated by second derivatives of the brane-

bending mode at subhorizon scales, and by normalderivatives at larger (but still smaller than Hub-ble) scales. At scales larger than the Hubble radiusthe contributions from F become important, but wewont deal with such wavelengths in this paper, thusF will play essentially no role. Since extrinsic curva-ture is the new degree of freedom that makes gravitydifferent from GR, this shows which metric pertur-bations are going be relevant for structure formation.

III. FIELD EQUATIONS AND LINEAR

THEORY

A. Basics

The field equations corresponding to brane-induced gravity in five dimensions are [75],

g(5)

2rcG(5)AB + GAB [D] = 8G TAB [D], (20)

where G(5)AB is the 5D Einstein tensor, and GAB, TAB

the standard 3 + 1 Einstein and stress-energy ten-sors, which vanish when any index is larger than3; this condition for TAB enforces there is no fluxof energy-momentum to the extra dimension (recallthe position of the brane is fixed at w = 0 even in thepresence of fluctuations). In Eq. (20), g(5) denotesthe absolute value of the 5D metric determinant, and

[D] D(w)

g(4). The parameter rc denotes thecross-over scale from 4D to 5D behavior, indeed onecan rewrite by dimensional analysis Eq. (20) for apoint source at the brane as

1

rc

r2 +

r21

r 1

r31

r , (21)

where denotes a generic component of the metric.Therefore, for r rc one has the standard 4D be-havior r1, whereas for r rc the 5D behavior r2 is obtained. More precisely, small pertur-bations about Minkowski g = +h (h 1)the metric at the brane can be readily obtained inmomentum space [76]

h = G(p) (T 13

T)

= G(p) (T 12 T) + 16 G(p) T ,(22)

where a tilde denotes objects in Fourier space (de-pending on four-momentum p), T T , and

G(p) =8G

p2 +p/rc, (23)

from which it follows Eq. (21) for a static pointsource. We will come back to Eq. (23) when wediscuss linear cosmological perturbations as the un-usual p1 dependence will lead to interesting scale

dependence of the growth factor at large scales. Inthe second line of Eq. (22), we have decomposedthe tensor structure of the full theory (previous line)in terms of the standard massless spin-2 part (firstterm) and the helicity-zero part (second term) thateffectively acts as a scalar. The fact that the fulltheory in the massless limit prc 1 does not re-cover the massless case (GR) is the well-known van

6


7/36

Dam-Veltman-Zakharov (vDVZ) discontinuity [77

79]. As conjectured by Vainshtein in massive grav-ity [43], the discontinuity should disappear in the fullnonlinear theory (in which the helicity-zero modegets suppressed), otherwise one could rule out a mas-sive graviton (m r1c here) from local observationsno matter how tiny its mass. Understanding thisVainshtein mechanism for realistic cosmological per-turbations is one of the main goals of this paper. Weshall see that the behavior of the graviton propaga-tor just discussed has a close analogy to the modifiedPoisson propagator for cosmological perturbations,as expected.

It will be useful for future reference to write the

gravitational part of the action for Eq. (20) in ADMform,

Sgrav =1

16G

d4x dw

g(5)

R D(w)

+1

2rcN(R + K2 KK)

, (24)

where R is the Ricci scalar for the 4D metric. Thefirst term is the usual Einstein-Hilbert term, whereasthe second contribution proportional to the lapsefunction N is the 5D Einstein-Hilbert term decom-posed in ADM form into intrinsic (R) and extrinsic

(K) curvature of hypersurfaces at constant w. Forthe Ricci scalar of the background we have,

R = 6e2(2 2 + ) (25)

whereas for perturbations,

R = 2e22( 2) 2R +6e2( + 4 + ) (26)

The stress-energy tensor we assume throughout

this paper is that corresponding to a dark matterfluid,

T = (1 + ) uu , (27)

where is the mean density, describes density fluc-tuations and the four-velocity,

u =v

gvv, (28)

where v = dx/d = (a, vi) with v the standardvelocity fluctuations about the Hubble flow, and is the conformal time. The only non-vanishing com-ponent of the stress tensor fluctuations are (in linearperturbation theory),

T00 = , (29)

T0

i = a vi. (30)From this, conservation of energy and momentum

gives respectively (always to first order),

3 + 1a

= 0, (31)

+ H +1

a2 = 0. (32)

The first is the standard continuity equation withthe extra term representing the change in volume

as the universe expands due to the fact that a(1 ) is the perturbed scale factor. The second is thegeodesic equation for the fluid, in the approximationof weak fields and small velocities. These equationsare independent of the theory of gravity (as long asit is a metric theory and stress energy is conserved),and must be supplemented by the field equations ofthe gravity theory under consideration, which relatesstress energy fluctuations to metric fluctuations.

B. Background Evolution

Lets briefly recall the evolution of the backgroundin the self-accelerated branch, this will also be usefulto set the time variables we are going to use whendiscussing perturbations. From the 00 field equationwe obtain the modified Friedmann equation,

H2 =H

rc+

8

3G, (33)

7


8/36

Since > 0, the theory makes sense only when

rcH > 1, thus probing distances larger than thecrossover scale rc necessarily implies considering per-turbations of wavelength larger than the Hubble ra-dius. As time goes on, one immediately sees fromEq. (33) that since 0, one enters a de Sit-ter phase where the universe expands exponentiallywith H = 1/rc. The acceleration follows from the iifield equation, which gives

H = 32

H2rcH 1

rcH 1/2 , (34)

which implies that the acceleration of the Universeis given by

a

a= H+ H2 = H2 2

2 1 , (35)

where we defined,

rcH, (36)thus the Universe starts to accelerate when H dropsbelow the critical value crit = 2. In the early uni-verse 1, in fact can be thought of as a timevariable which decreases with cosmic time ( > 1,achieving = 1 in the infinite future when the dark

matter density asymptotes to zero). In terms of thistime variable, the scale factor reads,

a() =0(0 1)

( 1)1/3

, (37)

and 0 = H0rc can also be written in terms of thez = 0 matter density, 0 = (1 0m)1. One can goback and forth between a and using Eq. (37) and

its inverse relation, 2 = 1 +

1 + 40(0 1)a3.We can define the acceleration parameter,

q aa

a2 =

2

2 1 , (38)

which goes from

1/2 in the early universe to 1

as the universe gets into the purely de Sitter phase.We can then write simply,

e = a (1 + |w|H), e = 1 + |w|Hq. (39)Finally, note that before the universe enters into

the acceleration phase ( > 2), q < 0 and fromEqs. (1) and (39) it follows that there is a Rindlerhorizon in the bulk where located at

wh =

(qH)1, (40)

which becomes 2H1 in the early universe, and goesto infinity when the universe approaches the accel-erated phase, and disappears afterwards.

In what follows we will use and q to character-ize the time dependences of the scale factor in thismodel.

C. Linear Perturbations: at the Brane

We now consider linear metric perturbations atthe brane. Since the dark matter has negligibleanisotropic stress, Tij = 0, and thus from the ij-equations (i = j) at w = 0 one obtains the follow-ing simple boundary condition for the brane-bendingmode C,

C0 = rc (0 0), (41)

where a 0 subscript denotes quantities evaluated atthe brane. The 00-equation at the brane gives the

modified Poisson equation in linear theory,

20 12rc

2C0 + 3a2

2rc

0 + HF0

= 3a2H

0 + H0

+ 4Ga2 . (42)

8


9/36

At this point it is worth making a few observations.

First, note that we can obtain the GR limit by tak-ing 0 = 0 in Eq. (41) which sets to zero C0in Eq. (42), and then ignoring the contribution inparenthesis (effectively setting rc to infinity) givesthe Poisson equation in GR. The terms in braces onthe right hand side of Eq. (42) correspond to theusual contributions in GR from retardation effectsthat dominate at superhorizon scales, while at sub-horizon scales they are negligible compared to 20.The reason that normal derivatives only appear atmost to first order in Eq. (42) is that at least an-other first normal derivative of the background met-ric (that has a jump at w = 0) is needed to get

a nonzero contribution localized in an infinitesimalbrane.We are interested in finding the growth of struc-

ture at subhorizon scales where time derivatives aresmall compared to spatial gradients, and thus onecan assume a quasistatic evolution. In this case,Eq. (42) becomes,

1

22(0 + 0) = 4Ga2 , (43)

where we have used the constraint in Eq. (41) to re-place the brane-bending mode C0 in terms of poten-

tials 0, 0, and have neglected the terms in paren-thesis and braces as they involve spatial derivativesof order less than second supressed by the Hubbleradius or rc H10 . Then, all that remains to findis the relation between 0 and 0. This can be ob-tained directly from the 44 bulk equation close to thebrane (w = 0+), which in the same approximationreads,

2(0 0) = 1(q + 2)

2(20 0), (44)

and thus the system is closed. This is essentiallythe argument given in [47] to derive the growth fac-

tor at subhorizon scales. Note that Eq. (44) relatesthe extrinsic curvature, Eq. (19), to the Ricci scalar,Eq. (26), at subhorizon scales; we shall come backto this below. At high redshift, = Hrc 1 whileq 1/2, and Eq. (44) sets 0 = 0 and GR isrecovered in Eq. (43).

The modified Poisson equation that results fromEqs.(43)-(44) is,

20 = 4Geffa2 , Geff G 2(q + 2) 42(q + 2) 3 ,

(45)thus modification of gravity in this approximationcan be understood as a time-dependent (but scale-independent) modification of the gravitational con-stant. Note that at late times, in the de Sitter limit,, q 1 and Geff (2/3)G. Thus gravity becomesweaker leading to a slower growth of structure com-pared to a dark energy model with the same expan-sion history.

These arguments, while certainly reasonable, arenot rigorous enough to understand the limit of ap-

plicability of the approximations made. For exam-ple, that Geff is only a function of time results fromkeeping only the highest (second) spatial derivatives.We would like to understand the limitations of theseresults at both large and small scales. The limita-tions at large scales come from neglecting the nor-mal derivatives to the brane, 0, for which we needto solve the equations in the bulk [63]. The limita-tions at small scales come from the fact that brane-induced gravity becomes GR at small scales throughthe Vainshtein mechanism, where nonlinear effectssuppress the brane-bending mode C0. For this weneed to understand the nonlinearities responsible for

this mechanism and study at what scale they do op-erate. We first consider the linear bulk equations,postponing until section IV the discussion of nonlin-ear effects.

D. Linear Perturbations: in the Bulk

Before we discuss the solution of the bulk equa-tions, it will be useful to split perturbations intotwo classes according to their behavior under gaugetransformations that shift the brane position, w w + x4. We define two new metric variables,

+ + ( ) C, f F C, (46)which, as can be seen from Eqs. (4-10), are in-

variant under shifts of the brane position. We alsodefine a new metric variable,

2 , (47)

9


10/36

which gives the leading contribution to the per-

turbed Ricci scalar, Eq. (26), at subhorizon scales.We keep the brane bending mode as our fourth met-ric variable, thus we are looking for solutions inthe bulk for , f, , C, where the first two variablesare invariant under gauge transformations involvingonly w, and the other two are not. As we shall see,this split between , f and , C will play a key rolein solving the bulk equations.

The equations that determine the shift vector Nin the bulk are very simple. The ij field equationreads,

3C

+ 2(2

+

) C = 2 , (48)which determines C once and are known. Afterusing this result, the 0j field equations give

f + ( + ) f = 2( + ). (49)

This determines f from the evolution of . An equa-tion for follows from subtracting one third of theii field equations from the 00 equations,

5 = 4e2+4() +{, , f , C }, (50)

where the 5D dAlembert operator in the back-ground metric reads

5 (3 + ) w 2w, (51)

and the 4D one,

e2 2t + (3 ) e2 t e22, (52)

and the term in braces in Eq. (50) denotes termsthat involve the four metric variables (but none oftheir derivatives). All the terms that do not involve have the property that their coefficients vanish inthe de Sitter limit (q = 1), as can be seen explicitly

for the term, i.e. = = H/(1 + wH) inthe de Sitter phase. In this limit, {} = 22 e2 ,and Eq. (50) becomes a closed equation for . Thisshows that during the accelerated phase we are veryclose to having decoupled the bulk equations, evenbeyond the quasistatic approximation.

So far we have written exact bulk equations inlinear theory. The remaining bulk equations are

in some cases rather complicated but we are in-

terested in sub-horizon perturbations where timederivatives are suppressed compared to spatial gra-dients. Therefore, in the following we make the ap-proximation that we consider modes inside the Hub-ble radius, i.e. (k/aH)2 1, where k is the comov-ing wavenumber. This is a very good approximationsince, e.g. at z = 0

k

aH= 300

k0.1 h Mpc1

. (53)

In this approximation, Eq. (50) becomes

+ (3

+

)

+ e

22 0, (54)where we have also neglected , , f and C terms(all of which disappear as the universe approachesthe de Sitter phase). This will be further justi-fied shortly. Equation (54) can be readily solvedin Fourier space as

= 0 (1 + s)k/aH s wH (55)

where we have dropped the solution that diverges atinfinity. Strictly speaking, before the universe accel-erates one must impose vanishing boundary condi-

tions at the Rindler horizon given by Eq. (40), butthe quasistatic approximation breaks down at suchlarge distances away from the brane, and in any caseEq. (55) decays so fast that any mixing with the sec-ond solution is going to be negligibly small. Equa-tion (55) is analogous to the quasistatic bulk solutionobtained in [63] for the master variable (however, itis not the same as the master variable is a gauge in-variant quantity, and is only invariant under gaugetransformations involving only the extra-dimensioncoordinate).

We can check the validity of the quasistatic solu-tion by comparing the time derivatives to the other

terms in the Eq. (50). We have

k2

a2

(1 q) s

1 + s+ q ln(1 + s)

2, (56)

and compared to spatial derivatives in the left handside of Eq. (50) we see that time derivatives becomeimportant only far from the brane, i.e. when s 1

10


11/36

or w

H1; at this point the quasistatic solution

in Eq. (55) has decayed exponentially to zero for thewavenumbers of interest, see Eq. (53).

Having a solution for we can now solve for ffrom Eq. (49). Here one must be careful to includetime derivatives that act as a source of f in Eq. (49),leading to

f 2

(1q)s+q(1+s)ln(1+s)

= 2e22,(57)

where we have set the homogeneous solution fH (1+s)1(1+qs)1 to zero since it violates the i4 bulkequations, as we shall see in Eq. (65) below. Note

that close to the brane a subleading term of order (aH/k) becomes dominant since Eq. (57) vanishesas s 0. At the brane (s = 0+) we then have

f0 20aH

k

2 +

d ln 0d ln a

. (58)

In what follows we neglect this subleading contri-

bution, and use Eq. (57) everywhere in the bulk.Then f is at most of order . Note that neglectingf altogether results in a shift vector that is a purefour-gradient, i.e. N = C.

It remains to find the dynamics of and of thebrane-bending mode C which determine, respec-tively, the Ricci scalar and the extrinsic curvature inconstant w hypersurfaces at subhorizon scales. As itis clear from the ADM formalism, Eqs. (15) and (24),the 44 field equation relates the Ricci scalar to theextrinsic curvature (the Gauss-Codazzi equation),

R = KK K2, (59)

which to linear order gives, in the subhorizon ap-proximation (using the same notation as in Eq. 50)

(2 + )C + ( 3) = e2 + {f,C, C, , , }. (60)

Again, working in the quasistatic approximation,and neglecting compared to 2, which will be

justified shortly, we obtain a relationship between and C that can be written,

= (2+)C+e2()2e22. (61)In principle one could add to this the solution of

= ( 3), which is = (1 + s)k2/3(1 +2qs/(3 q))k2/6, with k = k/aH, but such termwould violate the i4 field equation below. This isimportant because otherwise the solution for the lin-ear growth factor would be different at subhorizon

scales. Replacing Eq. (61) in Eq. (48) and ignoringsubleading terms gives a very simple equation for Cin terms of alone,

3C = , (62)which leads to

C = C0 +03

ak

(1 + s)k/aH 1

, (63)

and this in turn gives a solution for from Eq. (61).

We are now in a position to check the validity ofthe approximations made. First, to go from Eq. (50)to Eq. (54) we have dropped the terms involving f,, and C, apart from assuming the quasistaticapproximation for . Since f is at most of order ,Eq. (57), it is justified to drop it compared to 2.The remaining terms involving , and C are po-tentially dangerous, as C does not decay into thebulk and only decays slowly, thus they can becomelarger than 2 as it becomes strongly damped inthe bulk. However, using Eqs. (61)-(63) it follows

that all these terms in Eq. (50) cancel leaving onlycontributions proportional to . This is as expectedfrom the properties of fields under gauge transforma-tions that shift the brane position. Finally, we seefrom Eqs. (61)-(63) that it is also justified to drop compared to 2 in the quasistatic regime.

Other checks can be done from the remaining bulkequations. The i4 field equation reads,

11


12/36

f + ( )f = 23

e2

3 + ( )(2 )

+(82 + 22) C

(64)

which in view of Eqs. (61-63) can be written as

f + ( )f = 2e2 2e22, (65)which is solved by Eq. (57), and does not support

a slow decay into the bulk, i.e. the homogenous so-lution of Eq. (49), fH (1 + s)1(1 + qs)1, doesnot solve Eq. (65) with zero right hand side, andthus must be discarded. Finally, the 04 bulk equa-

tion can be shown to hold by using Eq. (57) for f aswell.These results are summarized in Fig. 1, where

we show the four bulk solutions as a function ofthe coordinate into the bulk in units of the Hub-ble radius, s wH, for two different wavectorsk = 0.005 h Mpc1 (solid) and k = 0.02 h Mpc1

(dashed). Note that fields invariant under shifts ofthe brane position (, f) are concentrated at thebrane and leak only partially into the bulk, whereasthose who are not (C, ) approach a constant farfrom the brane. However, one should keep in mindthat this neat separation of bulk behavior accordingto symmetry breaks down at small scales (high-k)when nonlinearities are included, as we shall discussin section IV.

E. The Linear Quasistatic Solution

Now we can go back to the brane equationsand finalize the linear quasistatic solution. FromEqs. (46)-(47) we can map back to the .,F vari-abls relevant to the Poisson equation, Eq. (42),

=1

32(

)C+ 2

, (66)

=1

3

( )C+ +

, (67)

from which we can obtain the normal derivatives atthe brane,

0 20 2

30 =

2

3

k

a0, (68)

FIG. 1: Bulk solutions, for k = 0.005h Mpc1 (solid)and k = 0.02h Mpc1 (dashed), as a function ofs = wHfor redshift z = 0, when the matter density is given by0m

= 0.2. The brane is located at s = 0. All metricvariables are normalized by the value of at the brane,0. Fields invariant under shifts of the brane position(, f) decay strongly into the bulk, whereas C, do not.

plus subleading (order H0) contributions. We thussee that normal derivatives are suppressed comparedto the leading contributions in the Poisson equationEq. (42) by one inverse power of krc/a. These termsare genuinely of quasistatic nature (i.e. they giverise to the 5D decay of the gravitational potentialfor static sources), and thus we must keep them aspart of the quasistatic solution. They are dominantcompared to retardation effects standard from GR(which are also modified here at late times by 5D dy-

namics), which are suppressed by two inverse powersof k/aH. Clearly, krc/a compares a physical scalea/k against the spatial scale rc, whereas k/aH com-pares the light crossing time across a perturbationof size a/k to the Hubble time H1, a measure ofretardation effects.

An important feature of brane-induced gravityis that from the point of view of standard 4D

12


13/36

physics, the bulk dynamics introduces through nor-

mal derivatives such as those in Eq. (42), unusualnon-local operators. From Eq. (68) we have thatnormal derivatives of 0 can be written in configu-ration space as

0 = 1

a

2 0. (69)

This is a consequence of the 5D nature of the gravitytheory.

We can then summarize the linear quasistaticequations as follows. To make equations compact,we will first make use of different (though linearly

related) metric variables. We later write down every-thing in terms of the Newtonian potential 0. UsingEq. (61) at w = 0+ we have,

H(q + 2) 2C0 + Hqa

2 0 = 20, (70)

which is a more accurate version of Eq. (44). Towrite the Poisson equation, Eq. (42), in the qua-sistatic approximation, we can drop the terms inbraces in Eq. (42) (as usual in GR) since they aredown from the leading terms by the factor (aH/k)2,and also we can drop F0 as from the bulk solutionF0 C0, and thus this term is subdominant by thesame factor to the term with

2

C0. Then we have,

20 12rc

2C0 a2rc

2 0 = 4Ga2 . (71)

Note, as argued above, that the nonlocal termsin these equations belong to the quasistatic approxi-mation. For a point source of mass m, 1/r3,and the right hand side of the Poisson equationscales as rg/r

3, where rg Gm is the gravitational(Schwarzschild) radius of the source. The first twoterms in Eq. (71) scale as /r2, where is the or-der of magnitude of metric perturbations, while the

third goes as /(rrc), then we can symbolically write

the Poisson equation as (compare to Eq. 21),

r2+

rrc rg

r3, (72)

and thus the normal derivative term governs thetransition of the static on-brane potential from theusual 4D behavior rg/r for r rc. Eq. (71) thusreflects precisely the behavior of the propagator inEq. (23) for static sources, with the nonlocality inthe Poisson equation a consequence of the p1 be-havior of the on-brane propagator. This is a feature

of extra-dimensional theories of gravity that cannottake place in a local model of dark energy (no matterhow complicated), and it is interesting to note thatit is expected to happen in theories with more thanone extra dimension as well, as long as there is a 5Dregime before hitting the truly large distance regime(as in cascading models, see [3638]).

As we shall see, these non-local corrections be-come interesting well before scales of order rc andimprove the validity of the quasistatic approxima-tion at large scales compared to the quasistatic so-lution of [47, 63] which sets the normal derivativesto zero. Therefore, it is expected that at least some

of the deviations reported at large scales in numeri-cal solutions that include normal derivatives [64, 65]can be explained by this simple result.

Note that in Eqs. (70)-(71), the brane metric vari-ables are all related through Eqs. (41), (46) and (47).Thus there is only one independent brane potentialthat obeys a Poisson-like equation. We want to com-pute the growth of density perturbations, and sincethe Newtonian potential 0 is responsible for accel-erations, Eq. (32), we write everything in terms ofthe Newtonian potential. Thus we have the effectivePoisson equation in Fourier space

k202(2 + q) 3

2(2 + q) 4 + 3a

krc

(1 + q) 1(2 + q) 2

2= 4Ga2 , or k20 4Geff(k, )a2 , (73)

where we have defined an effective gravitational constant Geff that depends on scale and time [128]. Forcompleteness, we note that one can also derive the following Poisson-like equation for the brane-bendingmode,

13


14/36

k2 C0rc

[2(2 + q) 3]

2akrc

(2q 1)[(2 + q) 2]

= 8Ga2 . (74)

FIG. 2: The effective gravitational constant Geff(k, z)that governs the linearized Poisson equation for the New-tonian potential, Eq. (73), as a function ofk for differentredshifts z = 0, 0.5, 1, 2, 3, 10 (from bottom to top). Non-linear effects modify significantly this behavior at smallscales, see Fig. 6 below.

Figure 2 shows Geff(k, z) as a function of k fordifferent redshifts. We see that at all times Geff isless than G, more so at late times (when the modi-fications away from GR become more important, as

the universe accelerates), and at large scales wherethe transition to the 5D regime of even weaker grav-ity takes place, inducing scale dependence. For thepurpose of this plot we have assumed 0m = 0.2 forwhich 0 = 1.25 and the universe starts acceleratingat z = 0.86.

The two features seen in Fig. 2 are easy to under-stand from the basic physics of brane-induced grav-

ity. The scale-indepedent suppression of Geff arisesfrom the first term within braces in Eq. (73), whichis clearly always larger than unity, while the scale de-pendence arises through the second term that is al-ways positive, enhancing the previous effect at largescales.

The physical origin of these terms is very simple.They arise because the gravitational effect of sources

at the brane is not only to create 4D intrinsic cur-vature (encoded by the 4D Ricci scalar), but alsoto create extrinsic curvature, which at small scalesis dominated by the brane-bending mode C0 (firstterm within braces in Eq. 73) and at larger scalesby normal derivatives (second contribution withinbraces in Eq. 73), see Eq. (19).

The precise fraction that goes into the usual 4DRicci scalar (available to accelerate dark matter par-ticles and thus grow structure) versus extrinsic cur-vature is controled by the Gauss-Codazzi equation,Eq. (59), that relates intrinsic to extrinsic curva-ture. Therefore, as long as the extrinsic curvature

remains important, density perturbations will havesuppressed clustering because part of their ability toinduce intrinsic curvature (the only type available inGR) is being used to create extrinsic curvature of thebrane. We will see in the next section that nonlin-ear effects suppress the extrinsic curvature at smallscales leading to the recovery of GR.

From the modified Poisson equation, Eq. (73), andthe conservation equations, Eqs. (31-32), at subhori-zon scales we have the linear density perturbationgrowth equation,

d2

d2 + Hd

d = 2

0 = 4Geff(k, )a2

(75)

where is conformal time (ad = dt) and H = aH isthe conformal expansion rate. Changing from con-formal time to our time variable, d = H(q 1) d,we can find the growth factor as a function of timeand scale. The results are shown in Fig. 3. The over-all behavior of the growth factor is expected given

14


15/36

FIG. 3: The growth factor in the quasistatic approxima-tion as a function ofk for different redshifts z = 0, 0.5, 1(from top to bottom).

the properties of Geff shown in Fig. 2, the scale de-pendence is much smaller due to the fact that thegrowth results from the integrated history of thestrength of gravity and for most of the evolution thescale dependence in Geff is small. Our results are ingood agreement with those in [64, 65] for the scaledependence at large scales, suggesting that most ofthe effect is due to the normal derivatives in the qua-sistatic approximation.

IV. BEYOND LINEAR THEORY

A. Nonlinearities

We now discuss how the linear results derived inthe previous sections change when nonlinear effects

are included. The first step is to identify the sources

of new nonlinearities not present in GR. Looking atthe action in ADM form, Eq. (24), the new con-tributions are in the second term proportional tothe lapse function N. We see right away that newrelevant nonlinearities appear through the 44 fieldequation, Eq. (59), where the extrinsic curvature isK 2C, see Eq. (19). Since 2C0 in lin-ear theory, clearly the quadratic terms in Eq. (59)will become important as becomes of order unity.Neglecting subleading terms the nonlinear version ofEq. (60) becomes

(2 + ) 2C+ e2

2[(2C)2 (ijC)2] = 2,

(76)which agrees with the equation written down in [80]for the brane-bending mode; see Eq. (87) below fora version of this equation at the brane includingnormal derivatives. Therefore the dynamics of thebrane-bending mode is nonlinear [44, 47, 53, 54, 8082]; this is crucial as it determines the relationshipbetween the Newtonian potential 0 and the spatialcurvature potential 0, through Eq. (41), which itis easy to check does not receive significant correc-tions even in the nonlinear regime. These two factstogether determine how the theory reduces to GR atsmall scales, well explain this in more detail in thenext sections.

There are other, similar, nonlinear correctionsthat affect the bulk solutions derived in the pre-vious section. The main reason is that while obeys an approximately decoupled equation from Cat the linear level due to its properties under gaugetransformations involving the extra dimension, whennonlinear effects are included this result no longerholds, as fields of different symmetry are coupled.Thus there is a coupling between fields being ap-proximately constant in the bulk (, C) and thosethat decay strongly (, f). For example, Eq. (48)becomes (k = i, j)

ij [3C + 2(2 + ) C 2 + ] + 3e2 [ijCkkC ikCjkC] = 0, (77)

15


16/36

therefore there are corrections of order (k/aH)2HC

to Eq. (48) and Eq. (62), with the consequence thatthe details of how C behaves in the bulk will bemodified somewhat. In second-order perturbationtheory, is easy to see that a new contribution devel-ops that makes C decay slowly into the bulk as e,this is expected to kick in at high k.

Similarly, the bulk equation for , Eq. (54), de-velops in second order a source term going as e2 (2C)2, which again becomes important athigh-k, meaning that will not decay as stronglyinto the bulk as given by linear theory, Eq. (55).This means that the normal derivatives at high-kwill not be as large as given by Eq. (68), which in

turn implies that the nonlocal terms in the equa-tions of motion will go away at small scales fasterthan expected based on linear theory. Summariz-ing, these nonlinear effects will make small nonlo-cal terms even more subdominant in the nonlinearregime. Therefore, we conclude it is safe to neglectthese nonlinearities in the bulk equations.

Finally, there are other possible nonlinear termsthat involve the interaction ofC with the other met-ric perturbations. Since it is precisely the brane-bending mode C that is a ghost (at least in the deSitter limit), one concern is that interaction of Cwith normal fields can develop a catastrophic insta-

bility of cosmological solutions (see e.g. [83, 84] fortoy models and discussion of quantum effects).

The most dangerous of such interaction termsthat correct e.g. Eq. (60) have the form 2C toquadratic order and 2C to cubic order. Giventhat does not grow with k at high-k due to thenonlinear effects discussed above and that its powerspectrum decreases rapidly with k (P k5), theseterms do not give significant corrections on any cos-mological scales, since 1. In regions wherethe weak field approximation breaks down ( 1),e.g. near a black hole, the situation can of coursebe very different. Although in such high density re-

gions, one may have difficulties exciting C due to theVainshtein suppression, and since C does not obey asuperposition principle, one may not add free so-lutions to cosmological ones, thus any analysis mustbe done self-consistently. See Fig. 7 in paper II [48]for a self-consistent solution showing how a cosmo-logical background is enough to suppress C at verylarge scales compared to astrophysical objects.

B. Vainshtein Mechanism for Cosmological

Perturbations

Having identified the main new sources of nonlin-earities, we are ready to discuss their effect. Beforewe do that in detail, it is useful to give a simple pic-ture in geometrical terms of the role of nonlineari-ties, which lead to the recovery of GR at small scales,by the Vainshtein mechanism [43]. For detailed dis-cussion of this in the context of the Schwarzschildsolution see [44, 46, 81, 8588], here we are ratherinterested in how the mechanism works in a cosmo-logical setting, for previous discussion along theselines see [47, 80].

We start from the field equations at the brane,which for 4D components can be written as

(R 12

gR) 1rc

(K gK) = 8GT, (78)

where the first term in parenthesis is the usual Ein-stein tensor characterizing intrinsic curvature, andthe second term denotes the contribution comingfrom extrinsic curvature of the brane embedded inthe bulk. The relationship between intrinsic and ex-trinsic curvatures is given by Eq. (59), and valideverywhere in the bulk (including arbitrarily closeto the brane, which is what we are interested inthis section). Roughly speaking, we can interpretthe self-accelerated solution by saying that for thisbranch the extrinsic curvature is of the same sign asintrinsic curvature, leading to the Friedmann equa-tion Eq. (33) and a self-accelerated universe whereintrinsic curvature ( H2) is balanced against ex-trinsic curvature of the brane ( H) when densityof matter has dropped sufficiently. The thresholdextrinsic curvature at which this transition happensis given by the inverse curvature radius r1c . There-fore, the universe accelerates when extrinsic curva-ture starts to play a role.

We can understand the behavior of perturbationsand the recovery of GR at small scales in similar, ge-

ometric terms. For this purpose lets rewrite Eq. (59)in the following symbolic form (hereafter we alsodrop numerical factors of order unity)

R = R + R K2 = (K+ K)2 (79)where quantities with bars refer to background val-ues. Since R = K2 H2, for perturbations we have

16


17/36

R KK+ (K)2, (80)with R 2/a2 and K 2C/a2, assumingwe are at scales well within the Hubble radius. Wecan then write

R

R=

K

K+K

K

2(81)

where

R

R

k

aH2

, (82)

K

K k

aH

ka

C (83)

Then at large scales, the solution of Eq. (81) is

R

R K

K 1 (84)

and fluctuations in intrinsic and extrinsic curvatureare comparable, thus gravity is modified by the con-tribution of extrinsic curvature fluctuations encodedin the brane bending mode C. The same is true atscales approaching the Hubble radius when normal

derivatives dominate the contribution to the extrin-sic curvature, leading to 5D behavior.

When becomes of order unity, however, the sec-ond term in Eq. (81) becomes important. Therefore,at small scales we have instead,

R

RK

K

2 1 (85)

and fluctuations in intrinsic curvature ( ) aremuch larger than in extrinsic curvature ( 1/2).Thus fluctuations in extrinsic curvature of the braneare suppressed at small scales and GR is restored.

Note that although fluctuations in extrinsic cur-

vature become large in the nonlinear regime, thereis no metric perturbation that is of order unity [86].One can see that from Eq. (83), while K/K maybe large, gi4 = iC is always small, since k/aH isvery large in the nonlinear regime. We will explicitlycheck this with N-body simulations in paper II.

We can put the condition for transition to GR,K/K 1 in a more familiar form by considering

spherical isolated perturbations, for which

rg/r

where rg is the gravitational radius correspondingto the mass enclosed by the perturbation [47]. SinceC rc, using Eq. (83) we can translate the re-quirement K/K 1 to find the scale r away fromthe perturbation where the transition to GR takesplace

1

H

1

r2

rcrgr

1, (86)

which gives the Vainshtein scale r3

rcH1rg r2crg familiar from the Schwarzschild solution [44, 46,81, 8588]. The difference here is that in the cosmo-

logical context rg is the gravitational radius corre-sponding to the mass enclosed by the perturbation,not the total mass including the background, whosegravitational effects are already taken into accountin the evolution of the Hubble constant [47]. Indeed,although at small scales the relation between metricand stress-energy fluctuations approaches GR, per-turbations still evolve in a background that has amodified expansion history. The difference betweentotal mass and perturbation mass becomes totallynegligible for astrophysical objects within the rangeof their influence as the mass contributed by thebackground is completely negligible. For cosmolog-ical perturbations, the opposite is the case for longwavelength perturbations and thus while countingtotal mass one would expect no corrections to GRbelow scales H1 [45, 89] (and nonlinearities at thatscale), properly taking into account the backgroundone expects that for long-wavelength perturbations > r and thus modifications to their growth.

To obtain the precise value for r in a givencosmology one has to solve the nonlinear equationEq. (81) (which is meant to represent Eq. 59). Theanswer for the Vainshtein scale r can only be givenin a statistical sense, that is, the solution of thatequation will depend on the spectrum of perturba-tions and cosmological parameters. In the context

of the Schwarzschild solution, one can think of it ashaving to estimate r from a collection of objects(how many and of what masses depending on thespectrum of perturbations) in a situation where thesuperposition principle does not apply (hence it isnot possible to use the known answer for a singleobject). How different objects interact is controlledby the nonlinearities. We will tackle the calcula-

17


18/36

tion of the Vainshtein scale using perturbation the-

ory below, and N-body simulations in paper II [48].See [90] for a recent discussion of the impact of Vain-shteins mechanism for macroscopic violations of theequivalence principle.

C. The Effective Poisson Equation

Let us now discuss in detail the behavior of pertur-bations including nonlinearities. We can incorporate

the nonlinear dynamics of the brane-bending mode

into the Poisson equation and thus find an effectivePoisson equation that relates the Newtonian poten-tial to the density perturbation. Equation (59) atthe brane becomes (generalizing Eq. 70),

H(q + 2) 2C0 + Hqa

2 0 20 = 12a2

(ijC0)2 (2C0)2

(87)

which we can write in Fourier space as,

(q + 2) + 1rc

k2C0 + Hqak 0 + k2 0 =

1

2a2

(k1 k2)2 k21k22

C0(k1)C0(k2) [D] d

3k1d3k2, (88)

where [D] D(k k1 k2). This, together with the Poisson equation, Eq. (71), which in view of Eq. (41)we can rewrite in terms of 0 as

k20 12rc

k2C0 ak2rc

0 = 4Ga2 , (89)

determines how gravity responds to density perturbations. After expressing all metric variables interms of0 and C0, using that 0 = 20 + [(q 1) + 1]C0/rc, we can use Eq. (89) to eliminate the brane-bendingmode C0 in terms of0 and and thus Eq. (88) results in an effective nonlinear Poisson equation that relatesthe gravitational potential 0 to the density contrast ,

+ k20

=G

G

2

rca

2 1 (k1 k2)2

+ k210

+ k220

[D] d

3k1d3k2

, (90)

where

G Geff G, 4Ga2, (91)and we have used the fact that when the nonlinearterm becomes important the normal derivative termscan be neglected, i.e. a/(krc) 1 at small scales.

Its easy to see that this equation simplifies con-siderably in the spherical approximation. Indeed, inthat case the kernel [1(k1 k2)2] = 2/3 and Fourier

transforming back to real space, Eq. (90) becomes a

local quadratic equation for (2

0) with source, whose solution is

20 = + 34g

GG

1 +

8

3g 1

, (92)

18


19/36

where g is a dimensionless nonlinearity parameter

given by

g 2G

G

2rca

2 =

(2 1)2 ( 1)3(22 2 + 1)2 , (93)

and decreases as time goes on (due to the dilutionin matter density) from g = 1/3 at early times tog 0.1 at z = 0. As a result, the transition tostandard gravity happens at relatively large den-sity contrasts > 10. One can also interpret thespherical modified Poisson equation, by defining a-dependent Newtons constant by

GG

1

20

= 1 +(G/G)

1 + 83g , (94)

which also emphasizes that for voids, < 0, G isfurther suppressed (recall G/G < 0 in our case)compared to > 0, thats another way of sayingthat new non-Gaussianities will be induced.

Equation (92) is the same as the modified Pois-son equation given by Eq. (3.1) in [47], after usingthat their variables and can be written in ournotation as = (8/3)g and = 3 (G/G). Note,however, that our treatment includes the scale de-

pendence ofG/G induced by the normal derivativesto the brane at the linear level, neglected in [47].

In paper II we also perform numerical simulationsunder the spherical approximation (used in [91] tosimulate DGP and degravitation theories) and com-pare them to the full solution to check their validity.We find that the spherical approximation is accurateto about 10% in the density power spectrum.

Equation (90) gives, beyond the spherical approx-imation, a nonlocal and nonlinear relationship be-tween 20 and the density perturbations . We willexplore two consequences of this equation. First, wediscuss how to calculate the relationship between 0

and , i.e. the propagator for Eq. (90), using an ap-proximate resummation technique that captures thenon-local corrections beyond the spherical approxi-mation. We will use this to calculate the nonlinearpower spectrum and evaluate the transition scale toGR. Finally, we use perturbation theory in Eq. (90)to calculate the newly induced non-Gaussianities inthe bispectrum.

D. The Modified Poisson Propagator

The right hand side of Eq. (90) describes the cor-rections to the standard Poisson equation. At largescales, density perturbations are small and the first,linear, term dominates over the second, nonlinear,contribution. This is what leads to the modifiedgrowth of perturbations described by an effectivescale and time dependent gravitational constant. Assmall scales are approached, however, the nonlinearterm becomes important, and as we discussed aboveit makes the right hand side of Eq. (90) subdomi-nant compared to the left hand side, recovering GRat small scales.

To follow the nonlinear growth of structure we areinterested in how 20 responds to density fluctu-ations . Since we are interested in calculating sta-tistical averages (e.g. the power spectrum) we candescribe the response in Fourier space as

0 =D0(k)

D(k)

(k) +1

2!

D20(k)D1D2

12

+1

3!

D30(k)D1D2D3

(123 12 3 cyc.)

+ . . . (95)

where

Ddenotes a functional derivative, integration

over repeated Fourier arguments is understood, andi (ki). In the language of renormalized pertur-bation theory [92, 93] the coefficient of the linearterm can be regarded as the two-point propagatorfor the modified Poisson equation,

(k) D(k k) D0(k)

D(k)

, (96)

the coefficient of the quadratic term as its three-point propagator

(2) (k1,k2) D(k

k12)

1

2!D20(k)

D(k

1)D

(k2

),(97)

and so on. The two-point propagator describesthe renormalized linear response, with asymptotics

(kr 1) k2

GeffG

, (98)

(kr 1) k2

, (99)

19


20/36

FIG. 4: Top panel: The two-point propagator for themodified Poisson equation in the naive spherical approx-imation, Eq. (100), for different redshifts correspondingto z = 0 (solid), z = 0.75 (long dashed), and z = 1.5(short dashed). Bottom panel: Same for the three-pointpropagator, Eq. (101). Note that the time dependenceis not monotonic in this case.

where r is the Vainshtein scale. Recall that at largescales the effective Newtons constant Geff dependson k, with nonlocal terms giving (k 0) rck1,in precise correspondence with the graviton propa-gator. On the other hand, at small scales where ex-trinsic curvature fluctuations are subdominant com-pared to density (intrinsic curvature) fluctuations returns to its GR (unmodified Poisson) value. Thetransition happens through nonlinear effects in thedynamics of the brane bending mode, which is the

nontrivial part of evaluating the expectation valuein Eq. (96).

The three-point propagator (2) is important at

large scales where it gives rise to additional non-Gaussianity. However, as we shall see below it is arather small correction (few percent) at large scales,whereas at small scales we know from the discus-sion on the Vainshtein mechanism above that it

must be highly suppressed since corrections to lin-

ear response at small scales go roughly as extrin-sic curvature fluctuations which scale as 1/2,which from Eq. (95) suggest

(2) 3/2, where

2(k) 4k3P(k) is the amplitude of density per-turbations. Indeed, it is easy to see this in the spher-ically symmetric case. From Eq. (92) we can roughlyestimate

(k) k2

1 +

(G/G)1 + 83g(k)

, (100)

where we naively replaced (k) when takingexpectation values in Eq. (94). This is of course nota valid calculation of the spherical propagator (thedependence on the power spectrum is simply a guess,and higher-order statistics than the power spectrumshould enter into the correct spherical answer), butit is enough for estimation purposes. To make thisdistinction clear we refer to this calculation as thenaive spherical approximation. Similarly, we can es-timate the three-point propagator as

(2) (k)

k2 2

3gG

G

1 +

8

3g (k)

3/2

. (101)

Thus we see that when 1, (2)

3/2

andthat

(2) at all scales. The overall scale of

(2) compared to is set by the nonlinear cou-

pling g, which is always much smaller than unity(see Eq. 93). Higher-order propagators will be sup-pressed by correspondingly more powers of g, mak-ing the description of the modified Poisson equationby the two-point propagator alone a very good ap-proximation.

Figure 4 shows these results as a function of k forredshifts z = 0, 0.75, 1, where for the power spec-trum we have used the HaloFit fitting formula [94]for standard gravity corresponding to a linear nor-

malization given by the DGP expansion history. Wewill compare these simple results with the calcula-tion including effects beyond the spherical approxi-mation that follows below.

All these arguments suggest that to obtain an ef-fective resummation of the two-point propagator is very reasonable to use linear response in Eq. (90).For convenience in writing the equations that follow

20


21/36

below we write the propagator in terms of a new

function M(k),

(k) + k20(k) G

G

(k) (k) M(k), (102)

thus

(k) = k2

1 +

G

GM(k)

, (103)

where M describes on average the Vainshtein non-linear dynamics, with asymptotics

M(k 0) 1, M(k ) 0. (104)

We must note that the resummation is essential torecover the large-k limit, and that M(k) in Eq. (102)denotes effective (in mean field sense) corrections tothe Poisson equation, which is what enters when cal-culating correlation functions; i.e. it is not meant to

describe a particular realization in the ensemble (for

that it would have to be a random field itself). Wemay regard the function M(k) as a form-factor thatdescribes the suppression of the brane-bending modeat small scales; or, in terms of the graviton propa-gator, it corresponds to including such momentumdependent form-factor in the second term on the sec-ond line of Eq. (22), corresponding to the helicity-zero mode, thus leading to the disappearance of thevDVZ discontinuity at high momentum.

The nonlinearity of the effective Poisson equationis encoded in the fact that M depends on the spec-trum and bispectrum of density perturbations, whilethe non-locality essentially comes from the fact that

the bispectrum has a nontrivial dependence on tri-angle shape. Although we only explicitly used k asan argument, M also depends on time, as do all vari-ables in Eqs. (102-103). In order to find an equationfor M we rewrite Eq. (90) using Eq. (102),

(M 1) (k) = g

1 (k1 k2)2

1M1 2M2 [D] d3k1d

3k2, (105)

and multiplying by (k), taking expectation values, and Fourier transforming back to real space, we find

(r) (r) = g

1 (k1 k2)2

B(k1,k2) M1M2 eik12r d3k1d

3k2, (106)

where the correlation functions are,

(r) =

eikr P(k) d3k, (r) =

eikr P(k) M(k) d3k, (107)

and P and B are the power spectrum and bispectrum of density perturbations,

(k)(k) = P(k) D(k + k), (k1)(k2)(k3) = B(k1,k2) D(k1 + k2 + k3). (108)As mentioned above, the non-Gaussian corrections induced by the nonlinearities in the modified gravity

sector are small, since they are suppressed by powers of g (see Section V C below for an explicit calculation ofthe bispectrum). Therefore, for the purpose of calculating M(k) we shall use the standard gravity bispectrum.Since the Vainshtein scale is in the nonlinear regime [47], we need a description of the bispectrum at nonlinearscales. Thus we use the bispectrum fitting formula obtained in [95], which reads

B(k1,k2) = 2F2(k1,k2)P1P2 + 2F2(k2,k3)P2P3 + 2F2(k3,k1)P3P1, (109)

where k3 = k12 and the kernel F2 is given by

F2(k1,k2) =5

7a(k1)a(k2) +

1

2(k1 k2)

k1k2

+k2k1

b(k1)b(k2) +

2

7(k1 k2)2 c(k1)c(k2), (110)

21


22/36

where the functions a, b and c are given by [95]

a =1 + 0.28

0.7 Q3(n) (q/4)

n+3.5

1 + (q/4)n+3.5, b =

1 + 0.4 (n + 3) qn+3

1 + qn+3.5, c =

1 + 4.5/[1.5 + (n + 3)4] (2q)n+3

1 + (2q)n+3.5,

(111)where q = k/knl with knl the scale at which the dimensionless linear power is unity, n is the effective spectralindex at scale k, and Q3(n) = (4 2n)/(1 + 2n+1). At large scales, a,b,c 1 and tree-level perturbationtheory is recovered, at small scales b, c 0 and a hierarchical bispectrum with saturation value Q3(n)follows [96]. Possible deviations from the hierarchical ansatz in the highly nonlinear regime will only impactM(k) at very large wavenumbers, where it is nearly zero.

In order to solve for M(k) from Eq. (106) it is convenient to define an effective amplitude Qeff(r),

Qeff(r) [(r)]2

1 (k1 k2)2

B(k1,k2) M1M2 e

ik12r d3k1d3k2, (112)

which converts Eq. (106) into a local equation for(r),

(r) (r) = g Qeff(r) [(r)]2, (113)and the solution for M is obtained from the result-ing quadratic equation for (r) by inverse Fouriertransform,

M(k) =

d3r

(2)3eikr

1 + 4 g Qeff(r) 1

2g QeffP(k).

(114)Its easy to see that this M(k) satisfies the asymp-totics in Eq. (104) and represents an effective re-summation of the Vainshtein mechanism. TheVainshtein scale for cosmological perturbation thusdepends on the characteristic nonlinearity g (seeEq. 93), the bispectrum of perturbations (since thenonlinearities in the field equations are quadratic)weighted by the nonlocal kernel of second deriva-tives 1 (k1 k2)2 represented here by Qeff, and thepower spectrum of perturbations.

To carry out the calculation ofM, we then proceedas follows:

i) We start assuming the naive spherical solutionfor the modified Poisson propagator (and thusM), Eq. (100).

ii) Given this M, we calculate Qeff from Eq. (112)at the desired redshift. For the power spec-trum, we use the nonlinear power spectrumgiven by HaloFit [94] corresponding to a linear

spectrum in standard gravity with the growthfactor that follows the modified expansion his-tory, let us call this GRH. For example, for them = 0.2 model we use, we assume 8 = 0.9 atz = 0 for GR, then 8 = 0.754 for GRH, and8 = 0.696 for linearized DGP. Since the Qeffratio involves equal number of power spectrain numerator and denominator, the assumedpower spectrum should not affect the answerby much. As we shall see, the GRH powerspectrum never differs from the fully nonlin-

ear DGP model by more than 30%.

iii) Evaluate the new M(k) from Eq. (114).

iv) With this new solution for M(k) we go back ofstep i) and iterate these steps until convergenceis achieved.

Figure 5 shows the results of this iterating proce-dure for z = 0. Short dashed lines show the initialguess that corresponds to the naive spherical solu-tion for M (bottom panel) and that implied for Qeff(top panel) after using Eq. (112), long dashed linesshows the solutions after 3 iterations, and solid lines

after 10 iterations. We see that convergence is quitefast, typically a few iterations (at high redshift con-vergence is much faster). Some noise develops in Qeffat small scales (r


23/36

FIG. 5: Convergence of the iteration scheme to solve theintegral equation for the modified Poisson propagator atz = 0. Short dashed lines show the initial guess thatcorresponds to the naive spherical solution, long dashedlines shows the solutions after 3 iterations, and solid linesafter 10 iterations. The top panel corresponds to Qeff,see Eqs. (112-113), while the bottom panel shows the

function M that defines the modified Poisson propaga-tor, see Eqs. (103) and (114).

modified Poisson propagator is included. We seethat nonlinearities in the modified gravity sectordrive Geff/G to unity at small scales, as expectedfrom the Vainshtein mechanism. We now discusswhat this renormalization of the gravitational con-stant implies for the nonlinear power spectrum.

V. STATISTICS

A. Nonlinear Equations of motion

We now consider the equations of motion for den-sity and velocity fields at the nonlinear level, andinclude the nonlinearities in the modified gravity sec-tor, first in terms of the running gravitational con-

FIG. 6: Effective gravitational constant (solid line) as afunction of scale at z = 0, 1, 3 after resummation of non-linearities in the brane-bending mode (Vainshtein mech-anism). Dashed lines show the linear theory calculation,as in Fig. 2.

stant (two-point modified Poisson propagator) andwhat this implies for the nonlinear power spectrum,and second by including the three-point propaga-tor and what it implies for the bispectrum. Let uschange to a time variable in terms of the scale factora, to

x ln a, (115)

then we can rewrite conservation of stress-energy as

x

v = (v), (116)

where we have written the velocity field asv Hv,and using v,

x+ (1 + q) 20 = [(v )v], (117)

23


24/36

FIG. 7: Ratio of renormalized growth factor to standardlinear growth factor as a function of scale for z = 0, 1.The enhancement towards small scales reflects the onsetof the Vainshtein mechanism, by which gravity becomesGR.

where

is the dimensionless gradient,

aH

, k kaH

(118)

and we can write the modified Poisson equation as,

0(k) = (k) (k)+

(2) (k1,k2) 12 [D]d

3k1d3k2.

(119)

Equations (116), (117) and (119) form the closedsystem of equations to be solved, and constitutes thestarting point to compute nonlinear corrections tothe power spectrum and bispectrum. We can writea second-order equation for the evolution of densityperturbations (putting linear terms in the left handside and quadratic nonlinearities in the right handside),

2

x2+ (1 + q)

x+ k2 =

(1 + q) +

x

(v) + [(v )v] k2

(2) 12 [D] d

3k1d3k2 (120)

where we have slightly abused notation by writingsome terms in real space (the standard gravity non-linearities) and others in Fourier space (the modifiedgravity nonlinearities, which are resummed into thePoisson propagators s). This form of the equa-

tions of motion is useful to derive linear and second-order perturbative solutions.

At this point it is worth mentioning that had wecarried out the full perturbative expansion of theDGP equations (see [97]), we would end up with aninfinite series of vertices corresponding to the per-turbative solution of Eq. (90) for 0 as a functionof . These terms will have two effects when cal-

culating correlation functions: they will renormalizethe growth factor (essentially because one can dressany linear propagator inside a diagram by includ-ing such vertices: this is described by ), and theycan change the mode-coupling properties (described

by (2) and higher-order ones). While the first ef-

fect leads to the Vainshtein mechanism through adressing of the gravitational constant, the second issubdominant to the mode-coupling effects in GR,because the vertices are suppressed by powers of gas opposed to those in GR which are always of orderunity. Thus, while we will need the resummation ofthe two-point Poisson propagator to describe nonlin-

24


25/36


26/36

FIG. 9: The ratio ofPGRH, the power spectrum for GRwith a modified expansion history, to that in the DGPmodel for two values of m, highlighting the dependenceon the somewhat different Vainshtein scales.

our renormalized growth will bring the linear powerspectrum to the amplitude it would have in GR witha modified expansion history. We shall compare ourpredictions to numerical simulations in detail in pa-per II.

Figure 8 shows the results for the nonlinear powerspectrum at z = 0, 1. The dashed lines correspondto the GRH nonlinear spectrum, i.e. a model withthe same expansion history (or Hubble constant) butwith standard gravity [129]. The difference in large-scale normalization (8 = 0.696 versus 8 = 0.754at z = 0) reflects the difference in gravitationalforce law alone, and is stronger at lower redshift

where the modified gravity has had more time to

act. This difference gets enhanced at intermediate

scales, but this is an expected outcome of the dif-ference in normalizations, i.e. one-loop correctionswould be stronger for PGRH than for PDGP; the ef-fect of the running G in PDGP is subdominant. Atsmall scales the running of G in PDGP becomes im-portant and PDGP approaches PGRH. Some of thesefeatures are also seen in f(R) models, see [99].

Figure 9 shows more clearly these effects by tak-ing the ratio PGRH/PDGP at z = 0 for two differentvalues of m, the standard m = 0.2 that we usedthroughout this paper and m = 0.27 (which we usein paper II to compare to simulations). Both modelswould have 8 = 0.9 at z = 0 in CDM. The lower

m is, the larger rc is, and thus the larger the Vain-shtein scale is, with stronger nonlinearities in themodified gravity sector (i.e. larger g). Therefore,although the difference between these two models issignificant (up to 5% for k


27/36

contributions do not add new multipoles (only changing the = 0, 2 amplitudes), we follow the standard

approach from PT (see e.g. [3]). We write the second-order solution as,

(2)(k) =

D0 + 1

2D1

k1k2

+k2k1

L1(k1 k2) + D2 L2(k1 k2)

12 [D] d

3k1d3k2, (122)

where the Di are arbitrary functions of time, and L are Legendre polynomials. The terms inside the squarebrackets reflect the structure of the right-hand side of Eq. (120) and correspond to the kernel F2 in Eq. (110)in the large-scale limit (a = b = c = 1). From the constraint that (2) = 0 it follows that D1 = D0 + D2,and the two independent multipole amplitudes obey the following equations of motion (a dot denotes d/dx),

D0 + (1 + q) D0 32

1

GeffG

D0 =

(1 + q) +d

dxD+D+

+

1

3(D+)

2 + gG

G 1

D2+, (123)

D2 + (1 + q) D2 32

1

GeffG

D2 = 23

(D+)2 g G

G

1

D2+, (124)

where D+ is the linear growth factor. These equa-tions are straightforward to solve numerically giveninitial conditions as determined by Eq. (110) in thelarge-scale limit, i.e. D0 = 17/21 and D2 = 4/21with Di (D+)2. From Eq. (122) one can com-pute the bispectrum using Eq. (109), and from itthe reduced bispectrum Q123

B123/(P1P2+P2P3+

P3P1). Figure 10 shows the result of the ratio ofQ tothat in standard gravity for m = 0.2 and triangleswith k1 = 0.1 h Mpc

1 and k2 = 2k1 as a function ofangle between the wavevectors, showing that thereis a few percent enhancement for isosceles configura-tions, and no correction at all for squeezed triangles( = 0, ) where the kernel in Eq. (110) vanishes.The Q ratio is roughly independent of scale, and itis most sensitive to m, increasing with decreasingm, again as a result that lower m leads to a largerrc and thus larger g in Eq. (121). We compare theseresults against measurements in N-body simulationsin paper II.

Although the amplitude of this correction is smallcompared to the current statistical errors in redshiftsurveys, the amplitude is model dependent and maybe significantly (though only parametrically) differ-ent in other modifications of gravity. The character-istic shape dependence (no correction for squeezedtriangles, and maximal difference for isosceles tri-angles) should be robust for the class of models of

massive gravity where GR is restored by derivativeinteractions as in the Vainshtein mechanism. To theextent that such a mechanism is universal [45], weare after a generic feature of modified gravity modelsthat can be tested through the study of cosmologi-cal perturbations. In f

roman scoccimarro- large-scale structure in brane-induced gravity i: perturbation theory

Documents