Observational Tests of Modified Gravity

University of Pennsylvania ScholarlyCommons

Department of Physics Papers Department of Physics

9-2-2008

Observational Tests of Modified Gravity

Bhuvnesh Jain University of Pennsylvania, [email protected]

Pengjie Zhang Shanghai Astronomical Observatory; Joint Institute for Galaxy and Cosmology

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Recommended Citation Jain, B., & Zhang, P. (2008). Observational Tests of Modified Gravity. Retrieved from https://repository.upenn.edu/physics_papers/85

Suggested Citation: B. Jain and P. Zhang. (2008). "Observational tests of modified gravity." Physical Review D. 78, 063503.

This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/85 For more information, please contact [email protected]. Observational Tests of Modified Gravity

Abstract Modifications of general relativity provide an alternative explanation to dark energy for the observed acceleration of the Universe. Modified gravity theories have richer observational consequences for large scale structures than conventional dark energy models, in that different observables are not described by a single growth factor even in the linear regime. We examine the relationships between perturbations in the metric potentials, density and velocity fields, and discuss strategies for measuring them using gravitational lensing, galaxy cluster abundances, galaxy clustering/dynamics, and the integrated Sachs- Wolfe effect. We show how a broad class of gravity theories can be tested by combining these probes. A robust way to interpret observations is by constraining two key functions: the ratio of the two metric potentials, and the ratio of the gravitational ‘‘constant’’ in the Poisson equation to Newton’s constant. We also discuss quasilinear effects that carry signatures of gravity, such as through induced three-point correlations. Clustering of dark energy can mimic features of modified gravity theories and thus confuse the search for distinct signatures of such theories. It can produce pressure perturbations and anisotropic stresses, which break the equality between the two metric potentials even in general relativity. With these two extra degrees of freedom, can a clustered dark energy model mimic modified gravity models in all observational tests? We show with specific examples that observational constraints on both the metric potentials and density perturbations can in principle distinguish modifications of gravity from dark energy models. We compare our result with other recent studies that have slightly different assumptions (and apparently contradictory conclusions).

Disciplines Physical Sciences and Mathematics | Physics

Comments Suggested Citation: B. Jain and P. Zhang. (2008). "Observational tests of modified gravity." Physical Review D. 78, 063503.

This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/85 PHYSICAL REVIEW D 78, 063503 (2008) Observational tests of modiﬁed gravity

Bhuvnesh Jain1 and Pengjie Zhang2,3,* 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA 2Shanghai Astronomical Observatory, Shanghai, China 200030 3Joint Institute for Galaxy and Cosmology (JOINGC) of Shanghai Astronomical Observatory (SHAO) and University of Science and Technology of China (USTC), 80 Nandan Road, Shanghai, China 200030 (Received 8 October 2007; published 2 September 2008) Modifications of general relativity provide an alternative explanation to dark energy for the observed acceleration of the Universe. Modified gravity theories have richer observational consequences for large- scale structures than conventional dark energy models, in that different observables are not described by a single growth factor even in the linear regime. We examine the relationships between perturbations in the metric potentials, density and velocity fields, and discuss strategies for measuring them using gravitational lensing, galaxy cluster abundances, galaxy clustering/dynamics, and the integrated Sachs-Wolfe effect. We show how a broad class of gravity theories can be tested by combining these probes. A robust way to interpret observations is by constraining two key functions: the ratio of the two metric potentials, and the ratio of the gravitational ‘‘constant’’ in the Poisson equation to Newton’s constant. We also discuss quasilinear effects that carry signatures of gravity, such as through induced three-point correlations. Clustering of dark energy can mimic features of modified gravity theories and thus confuse the search for distinct signatures of such theories. It can produce pressure perturbations and anisotropic stresses, which break the equality between the two metric potentials even in general relativity. With these two extra degrees of freedom, can a clustered dark energy model mimic modified gravity models in all observational tests? We show with specific examples that observational constraints on both the metric potentials and density perturbations can in principle distinguish modifications of gravity from dark energy models. We compare our result with other recent studies that have slightly different assumptions (and apparently contradictory conclusions).

DOI: 10.1103/PhysRevD.78.063503 PACS numbers: 98.65.Dx, 04.50.Kd, 95.36.+x

ics and its relativistic version (tensor-vector-scalar, TeVeS) I. INTRODUCTION [2] are able to replace dark matter at galaxy scales to The energy contents of the Universe pose an interesting reproduce the galaxy rotation curves, which provided the puzzle, in that general relativity (GR) plus the standard earliest and most direct evidences for the existence of dark model of particle physics can only account for about 4% of matter. The DGP model [3], in which gravity lives in a 5D the energy density inferred from observations. By intro- brane world, naturally leads to late time acceleration of the ducing dark matter (DM) and dark energy (DE), which Universe. Adding a correction term fðRÞ to the Einstein- account for the remaining 96% of the total energy budget Hilbert action [4] also allows late time acceleration of the of the Universe, cosmologists have been able to account for Universe to be realized. a wide range of observations, from the overall expansion of In this paper we will focus on modified gravity (MG) the Universe to the large-scale structure of the early and theories that are designed as an alternative to dark energy late Universe [1]. to produce the present day acceleration of the Universe. In The dark matter/dark energy scenario assumes the va- these models, such as DGP and fðRÞ models, gravity at late lidity of GR at galactic and cosmological scales and in- cosmic times and on large scales departs from the predic- troduces exotic components of matter and energy to tions of GR. We will consider the prospects of distinguish- account for observations. Since GR has not been tested ing MG models containing dark matter but no dark energy independently on these scales, a natural alternative is that from GR models with dark matter and dark energy. By the failures of GR plus the standard model of particle design, successful MG models will be indistinguishable physics imply a failure of GR. This possibility, that mod- from viable DE models against observations of the expan- ifications in GR at galactic and cosmological scales can sion history of the Universe. To break this degeneracy, replace dark matter and/or dark energy, has become an area observations of large-scale structure (LSS) must be used of active research in recent years. to test the growth of perturbations. Attempts have been made to modify GR at galactic [2] LSS in MG theories can be more complicated to predict, or cosmological scales [3–5]. Modified Newtonian dynam- but is also richer because different observables such as lensing and galaxy clustering probe independent perturbed *[email protected] variables. This differs from conventional DE scenarios

1550-7998=2008=78(6)=063503(19) 063503-1 Ó 2008 The American Physical Society BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) where the linear growth factor of the density field fixes all 22]. This entity has total mean matter density GR and observables on sufficiently large scales. One of the goals of equation of state parameter w ¼ pGR= GR. However, this study is to examine carefully what various LSS ob- when discussing perturbations in this entity, we may sepa- servables measure once the assumption of GR (with rate it into a matter component (dissipationless particles smooth DE) is dropped. which can be described as a pressureless fluid free of Structure formation in modified gravity in general dif- anisotropic stress) and a dark energy component. fers [6–18] from that in GR. Theories of LSS in these Throughout this paper, when we refer to ‘‘smooth’’ or modified gravity models are still in their infancy. ‘‘clustered’’ dark energy, we refer to this dark energy However, perturbative calculations at large scales have subset of the overall dark sector. shown that it is promising to connect predictions in these We may consider the Hubble parameter HðzÞ to be fixed theories with observations of LSS. Most studies have fo- by observations. In a dark energy model, GR is given by 2 cused on probes of a single growth factor with one or a few the Friedmann equation of GR: GR ¼ 3H =8G. The observables. In this paper we will consider a variety of LSS equation of state parameter is w ¼1 2H=_ 3H2. observables that can be measured with high precision with The corresponding modified gravity model has matter current or planned surveys. Our emphasis will be on density MG to be determined from its Friedmann-like model-independent constraints of MG enabled by combin- equation. We will consider MG models dominated by ing different observables. dark matter and baryons at late times and denote fluid Carrying out robust tests of MG in practice is challeng- variables such as the density with subscript MG. ing as in the absence of a fundamental theory, the modifications to gravity are often parametrized by free A. Metric and fluid perturbations functions, to be fine-tuned and fixed by observations. With the smooth variables fixed, we will consider per- Given the parameter space available to both DE and MG turbations as a way of testing the models. In the Newtonian theories, it is unclear how the two classes of theories can be gauge, scalar perturbations to the metric are fully specified distinguished. Kunz and Sapone [19] presented a rather by two scalar potentials and : pessimistic example. They found that one can tune a 2 2 2 2 clustered dark energy model to reproduce observations of ds ¼ð1 þ 2 Þdt þð1 2Þa ðtÞdx~ ; (1) gravitational lensing and matter fluctuations in the DGP where aðtÞ is the expansion scale factor. This form for the model. It is not clear if this conclusion applies to all perturbed metric is fully general for any metric theory of modified gravity models and if adding more LSS observ- gravity, aside from having excluded vector and tensor ables helps to break this severe degeneracy. perturbations (see [23] and references therein for justifica- In this paper, we first discuss ways of parametrizing tions). Note that corresponds to the Newtonian potential modified gravity models and dark energy models. for the acceleration of particles, and that in general rela- Section II presents the definitions and evolution equations tivity ¼ in the absence of anisotropic stresses. for perturbations in the metric and the energy-momentum A metric theory of gravity relates the two potentials tensor. We then classify independent LSS observables above to the perturbed energy-momentum tensor. We in- based on the perturbations that are probed by them. troduce variables to characterize the density and velocity Section III is devoted to the use of observational probes perturbations for a fluid, which we will use to describe of LSS for testing MG. We consider the four fundamental matter and dark energy (we will also consider pressure and perturbation variables and the observations that can be used anisotropic stress below). The density fluctuation is to probe them. The additional information available in the given by quasilinear regime is discussed in the Appendix. In Sec. IV we consider the question of distinguishing MG from DE ðx;~ tÞðtÞ ðx;~ tÞ ; (2) scenarios. The specific question we want to answer is: ðtÞ given a set of LSS observations, can a general MG model ð Þ ð Þ be mimicked by a DE model? If not, what LSS observables where x;~ t is the density and t is the cosmic mean are required to break the degeneracy? We conclude in density. The second fluid variable is the divergence of the Sec. V. peculiar velocity j ~ rjT0=ðp þ Þ¼rv;~ (3) II. PERTURBATION FORMALISM where v~ is the (proper) peculiar velocity. Choosing By definition, the dark sector (dark matter and dark instead of the vector v implies that we have assumed v to energy) can only be inferred from their gravitational con- be irrotational. This approximation is sufficiently accurate sequence. In general relativity, gravity is determined by the in the linear regime, even for unconventional dark energy ¼ total stress-energy tensor of all matter and energy (G models and minimally coupled modified gravity models. 8GT). Thus we can treat dark matter and dark energy In principle, observations of large-scale structure can as a single entity, without loss of physical generality [20– directly measure the four perturbed variables introduced

063503-2 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) above: the two scalar potentials and , and the density _ w_ GR ¼Hð1 3wÞGR GR and velocity perturbations specified by and . As shown 1 þ w below, these variables are the key to distinguishing modi- p= k2 fied gravity models from dark energy. Each has a scale and þ þ : (6) 1 þ w a redshift dependence, so it is worth noting which variables and at what scale and redshift are probed by different We have allowed for anisotropic stress sources in the observations. It is convenient to work with the Fourier energy-momentum tensor, parametrized by the scalar , transforms, such as which enters the Euler equation. Z Note that the above equations describe the multicompo- 3 ~ nent fluid of baryons, dark matter, and dark energy; the ^ðk;~ tÞ¼ d xðx;~ tÞe ik x~: (4) density and velocity variables for this fluid are subscripted GR above (these variables will represent a fluid with no When we refer to length scale , it corresponds to a dark energy for MG theories below). The metric potential statistic such as the power spectrum on wave number k ¼ variables are and in either case. Further, we do not 2=. We will henceforth work exclusively with the subscript p and as these sources occur only in the DE Fourier space quantities and drop the ^ symbol for plus GR scenario. convenience. The linearized constraint equation gives the Poisson equation for weak field gravity: B. Evolution and constraint equations 2 2 GR k ¼4Ga GR GR þ 3ð1 þ wÞHa ; We consider here the fluid equations for DE and MG k2 scenarios. We work in the Newtonian gauge and follow the 2 ’4Ga GRGR; (7) formalism and notation of [20], except that we use physical 2 time t instead of conformal time. We are interested in the where in the second line we have dropped the HGR=k evolution of perturbations after decoupling, so we will term as it is negligible for nonrelativistic motions on scales neglect radiation and neutrinos as sources of perturbations. well below the horizon. We will make the approximation of nonrelativistic motions Nonzero anisotropic stress leads to inequality between and restrict ourselves to subhorizon length scales. One can the two potentials: also self-consistently neglect time derivatives of the metric 2ð Þ¼12 2ð1 þ Þ potentials in comparison to spatial gradients. These ap- k Ga w : (8) proximations will be referred to as the quasistatic, It is common to take ¼ for ordinary matter and dark Newtonian regime. We will not consider the evolution of matter; however clustered dark energy can have a non- perturbations on superhorizon length scales; [24] shows negligible anisotropic stress. that differences in their evolution may have observable Equations (5)–(8) fully describe the evolution of pertur- consequences for some MG models [discussed further bations in DE scenarios in the quasistatic, Newtonian under the cosmic microwave background (CMB) below]. regime. Next we consider the analogous relations for modified gravity scenarios. 1. Dark energy with GR scenario We first consider the DE scenario, assuming GR. Using 2. Modified gravity scenario the perturbed field equations of GR to first order gives a set For minimally coupled gravity models with baryons and of constraint and evolution equations. The evolution of the cold dark matter, but without dark energy, we can neglect density and velocity perturbations includes gravity and pressure and anisotropic stress terms in the evolution equa- pressure perturbations p as sources. In the Newtonian tions to get the continuity equation: limit they give the familiar continuity and Euler equations for a perfect fluid. Keeping all first order terms and using _ MG _ MG MG ¼ 3 ’ ; (9) the notation _ d=dt,gives a a where the second equality follows from the quasistatic _ GR _ p GR ¼ð1 þ wÞ 3 3H þ 3HwGR approximation as for GR. The Euler equation is a k2 GR p _ ¼ þ ’ð1 þ wÞ 3H þ 3HwGR: (5) MG HMG : (10) a a For a generic MG theory, the analog of the constraint In the second line we have dropped the _ term as it is equations (7) and (8) can take different forms. We will negligible compared to the other terms in the quasistatic attempt to characterize the general behavior in the weak regime. The Euler equation is given by field limit for small perturbations (small ) and nonrela-

063503-3 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) tivistic motions. On subhorizon scales the field equations as it includes scalar and vector fields that are coupled to the in MG theories can then be significantly simplified. We growth of scalar perturbations. parametrize modifications in gravity by two functions For a generic metric theory of MG, one would expect ~ Geffðk; tÞ and ðk; tÞ to get the analog of the Poisson that a Poisson-like equation is valid to leading order in the equation and a second equation connecting and [25]. potentials and the density perturbation, at least on large We first write the generalization of the Poisson equation in scales in the linear regime where Fourier modes are un- terms of an effective gravitational constant Geff: coupled. In this regime, we expect that since the left-hand side of the field equations involves curvature, it must have 2 2 k ¼4Geffðk; tÞ MGa MG: (11) second derivatives of the metric perturbations, while the right-hand side is simply given by the energy-momentum Note that the potential in the Poisson equation comes tensor. On smaller scales, in general a MG theory may not from the spatial part of the metric, whereas it is the obey superposition and require higher order terms and ‘‘Newtonian’’ potential that appears in the Euler equa- higher derivatives of the potentials. Similarly a generic tion (it is called the Newtonian potential as its gradient relation between and is likely to have a linearized gives the acceleration of material particles). Thus in MG, relation of the form in Eq. (13). While it is not necessary one cannot directly use the Poisson equation to eliminate that the leading term be linear in both the potentials, the potential in the Euler equation. A more useful version observational constraints require that it be very close to of the Poisson equation would relate the sum of the poten- linear with ’ 1 on small scales where tests of gravity tials, which determines lensing, with the mass density. We ~ exist (see [28] for a review). therefore introduce Geff and write the constraint equations With the linearized equations above, the evolution of for MG as either the density or velocity perturbations can be de- 2 ~ 2 scribed by a single second order differential equation. In k ð þ Þ¼8Geffðk; tÞ MGa MG; (12) the case of MG theories, this equation is simpler as the only source is provided by the Newtonian potential . From ¼ ð Þ k; t ; (13) Eqs. (9) and (10) we get, for the linear solution, ðk;~ tÞ’ ~ 1 ð ~Þ ð Þ where Geff ¼ Geffð1 þ Þ=2. Note that if one starts in initial k D k; t , real space then the corresponding parameters would not be 2 ~ € _ k Fourier transforms of and Geff. Thus the Fourier trans- þ 2H þ ¼ 0: (14) form of the parametrized post-Newtonian parameter a = , the ratio of the metric potentials in real space con- For a given theory, Eqs. (12) and (13) then allow us to strained by solar system tests, is given by a convolution of substitute for in terms of to determine Dðk; tÞ, the and [26]. Only if is scale independent would it be the linear growth factor for the density: ~ Fourier transform of . A similar reasoning applies to Geff 8 ~ in using the Poisson equation. We prefer to work in Fourier € _ Geff 2 D þ 2HD MGa D ¼ 0: (15) space because of the ease of describing perturbations: each ð1 þ Þ Fourier mode evolves independently in the large-scale, linear regime. Furthermore, the equations describing cos- We can also use the relations given above to obtain the mological perturbations in MG theories such as fðRÞ grav- linear growth factors for and the potentials from D. Note ity and DGP are generally expressed in Fourier space. that in general the growth factors for the potentials have a ~ The parameter Geff characterizes deviations in the ð þ different k dependence than D. In the Appendix we give Þ relation from that in GR. Since the combination details on the linear and second order solutions and sum- ~ þ is directly responsible for gravitational lensing, Geff marize quasilinear signatures of MG theories. has a specific physical meaning: it determines the power of matter inhomogeneities to distort light. This is the reason C. Power spectra we prefer it over working with more direct generalization Before we turn to large-scale structure observables, we of Newton’s constant, Geff. define the power spectra of the perturbed variables. The ~ The Geff parametrization is equivalent to the Q three-dimensional power spectrum of ðk; zÞ for instance is parametrization independently proposed by [18] (see also defined as [27]), where Q parametrizes deviations in Poisson equation (7) from GR. For minimally coupled gravity models, ~ ~0 3 ~ 0 hðk; zÞðk ;zÞi ¼ ð2Þ Dðk þ k ÞPðk; zÞ; (16) with no dark energy fluctuations, it is also equivalent to that proposed by [16]. And is also equivalent to the parameter where we have switched the time variable to redshift z. The $ proposed by [17]. DGP and fðRÞ gravity can be de- power spectra of perturbations in other quantities are described by our parametrization. So is the widely adopted fined analogously. We will denote the cross spectra of two Yukawa potential. An exception to our approach is TeVeS different variables with appropriate subscripts, for ex-

063503-4 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) 4 ample, P denotes the cross spectrum of the density and must modify gravity on horizon scales of order 10 Mpc;it the potential . is an open question how they transition to GR on very small We write down next the relation between the power scales to satisfy experimental constraints from solar system spectra of the two potentials and the density in DE and tests. We will assume that the MG theories of interest differ MG scenarios. From the Poisson equation (7) for GR we from GR over the observationally accessible scales. have The most stringent current tests of gravity come from ð Þ laboratory and solar system tests and from binary pulsar GR ð Þ¼ð4 Þ2 4 2 P;GR k; z : P k; z G a GR 4 ; (17) observations; see [28] for a review. Interesting probes of k gravity on sub-Mpc scales also exist: galaxy rotation where P is the power spectrum of the potential . Using curves, satellite dynamics, strong lensing observations of the Friedmann equation for GR the above equation is often galaxies and clusters, and x-ray plus lensing observations written as of clusters (e.g., [29]). Modifications in gravity can affect the propagation of gravitational wave. Future gravitational 9 ð Þ 2 2 P;GR k; z wave experiments such as LISA can detect gravitational GR : P ðk; zÞ¼ H0 ; (18) 4 a2k4 wave from distant supermassive black hole pairs in the where H0 is the present day value of the Hubble parameter, coalescence phase and thus test this effect [30]. We will not and is the dimensionless density parameter. consider these tests in this paper. We will restrict our The Poisson equation (12) for MG gives the following attention to large-scale structure on scales where theoreti- equations for the power spectra of the metric potentials: cal predictions can be made using linear or quasilinear perturbation theory. ~ 2 4 2 4 MG: P þðk; zÞ¼½8Geffðk; zÞ a MGP;MGðk; zÞ=k ~ 2 ½8Geffðk; zÞ P MGðk; zÞ A. Connection of observables to perturbation variables or P ¼ a4 2 ; ; ½1 þ 1ðk; zÞ2 MG k4 In principle, observations of large-scale structure can (19) directly measure four fundamental variables that describe the perturbed metric and (fluid) energy-momentum tensor: where we have used Eq. (13) to get the equation for P. the two scalar potentials and that characterize the For LSS observables, we will need the power spectra of metric, and the density and velocity perturbations specified ( þ ) for lensing, of for dynamics, and of for tracers by and . Next we discuss the prospects for different of LSS. We will use Eqs. (17)–(19) to connect them, along probes of these variables. with the relations between the two potentials [Eq. (8) for Sum of potentials þ : Gravitational lensing in either GR and Eq. (13) for MG]. With these relations we can the weak or strong lensing regime probes the sum of the express different observable power spectra in terms of a metric potentials. We will consider the weak lensing shear single density power spectrum—for MG this will involve (or equivalently the lensing convergence) power spectrum ~ the functions Geffðk; zÞ and ðk; zÞ. as the primary statistical discriminator of MG via lensing. The spatial components of the geodesic equation for a photon trajectory xðÞ (where parametrizes the path) III. LARGE-SCALE STRUCTURE OBSERVATIONS are We will assume that the background expansion rate is 2 d x dx dx determined by a set of observations: type Ia supernovae, þ ¼ 0: (21) d2 d d baryon acoustic oscillation (BAO), and other probes at low redshift and the CMB and nucleosynthesis at high redshift. For the metric of Eq. (1), this gives the following relation These observations measure the luminosity or angular for the first order perturbation to the photon trajectory diameter distance at a given redshift. The distance mea- [generalizing, for example, from Eq. (7.72) of [31]]: sures in a spatially flat universe are, within factors of 1 þ z, 2 ð1Þ d x 2 simply the comoving coordinate distance: ¼q r~ ?ð þ Þ; (22) d2 Z 0 z dz ðzÞ¼ 0 : (20) where q is the norm of the tangent vector of the unper- 0 Hðz Þ turbed path and r~ ? is the gradient transverse to the un- Furthermore, BAO can directly measure the Hubble con- perturbed path. This gives the deflection angle formula stant at the redshift of galaxies. Z We are interested in the constraints available on per- i ¼ @ið þ Þds; (23) turbed quantities. Hence we will consider observational probes of large-scale structure to constrain modified grav- where s ¼ q is the path length and i is the ith compo- ity scenarios. In nearly all cases we will be interested in nent of the deflection angle (a two-component vector on scales in the range 1–103 Mpc. The MG theories of interest the sky). Since all lensing observables are obtained by

063503-5 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) taking derivatives of the deflection angle, they necessarily bution: depend only on the combination þ (to first order in the Z 2 WgiðÞWjðÞk l potentials). Cg ðlÞ¼ d Pgð þÞ k ¼ ; ; For weak lensing tomography we use the shear power i j spectrum for two sets of source galaxies with redshift (26) distributions centered at z and z . Following standard i j where W is the normalized (foreground) galaxy redshift treatments of weak lensing, this may be derived from the gi distribution (e.g., [40]). Galaxy–galaxy lensing has been deflection angle formula to get the shear power spectrum well measured from the Sloan Digital Sky Survey (SDSS). on angular wave number l ([32]): It is a very useful check on galaxy bias; hence it aids the Z 4 l interpretation of galaxy clustering measurements ([41]) as C ðlÞ¼ dW ðÞW ðÞk P þ k ¼ ; ; (24) ij i j well. Assumptions: In using weak lensing observations with where the weight function Wi is simply the above formalism, one assumes that intrinsic correlations are negligible or removable (in general these can i W / (25) differ for different gravity theories), that the weak lensing i i approximation is valid, and that galaxy properties that for source galaxies at a single comoving distance i affect photometric redshift determination are not affected ðziÞ (it can be easily generalized for sources specified by a by the gravity theory. redshift distribution). We have assumed a flat background Newtonian potential : This can be measured by dy- geometry for simplicity; our results throughout this paper namical probes, typically involving galaxy or cluster ve- can be generalized to a curved spatial geometry by replac- locity measurements. If gravity is the only force ing in the argument of W by the angular diameter determining galaxy accelerations at large scales (as ex- distance. pected), we have from Eq. (10): Note that in the literature the lensing power spectra for dðagÞ GR are expressed in terms of the density power spectrum k2 ¼ ; (27) dt PðkÞ assuming the standard Poisson and Friedmann equations. Usually anisotropic stress is neglected so that one where g rvg. On sub-Mpc scales this relation can be can substitute into the above equation the relation between used to constrain using galaxy satellite dynamics and 4 4 2 2 the power spectra: P þ ¼ 9k H0 P=a from rotation curves (e.g., [42]). Redshift distortion effects in Eq. (18). For MG, this substitution breaks down due to the galaxy power spectrum probe larger scales, which we the modifications of the Poisson equation and the address in more detail here. Friedmann equation. However the correct substitution The redshift space power spectrum of galaxies is a well- ~ can be made in terms of Geffðk; zÞ using Eq. (19) and the measured quantity. It can be expressed in the large-scale, modified Friedmann equation (which depends on the spe- small angle limit as (e.g., [43]) cific theory). 2u2 u4 k2u22 Since lensing probes the sum of the metric potentials, Ps ðkÞ¼ P ðkÞþ P ðkÞþ P ðkÞ F v ; g g gg 2 g 2ð Þ with the deflection angle formula following from the geo- H H H z desic equation (which simply describes how curvature (28) affects trajectories), it may not by itself test the field where u ¼ kk=k is the cosine of the angle of the k vector equations of the gravity theory. However lensing measure- with respect to radial direction; Pg, Pg , and P are the ments at multiple source redshifts are sensitive to the g g real space galaxy power spectra of galaxies, galaxy- and growth of the lensing potential, which does offer a test of g , respectively; is the 1D velocity dispersion; and FðxÞ the MG theory. And by combining lensing with other g v is a smoothing function, normalized to unity at x ¼ 0, observables, the relation of P þ to P can be tested. determined by the velocity probability distribution. The Recent studies that have examined constraints on MG dependence on u enables separate measurements of all theories with weak lensing include [18,33–39]. three power spectra, though P is the hardest to measure Another important observable in lensing is galaxy–gal- g axy lensing, the mean tangential shear around foreground with high precision [44,45]. Furthermore, measurements of s (lens) galaxies. Its Fourier transform, the galaxy-lensing Pg at smaller scales provide information on pairwise ve- cross spectrum, depends on þ and on the galaxy locity dispersion v [46]. number density. It is given by an equation similar to In the linear regime, we can rewrite Eq. (27)as Eq. (24), with the power spectrum of the lensing potential ð Þ 2 ¼ d aD =dt in the integrand replaced by the three-dimensional cross- k g: (29) D power spectrum, and with one of the weight functions replaced by one representing the foreground galaxy distri- Here D is the growth factor of g. For MG models, D has

063503-6 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) a simple relation to D, the linear density growth factor: differ due to galaxy bias. Further the galaxy-density rela- _ D / aD ¼ aHD, where d lnD=d lna. In the linear tion may be nonlocal and vary slightly in different gravity regime we have gðk;tÞ¼gðk;tiÞDðk;tÞ. Note that the theories due to differences in the tidal field that influence above equation does not require D to be scale indepen- collapsed objects such as galaxy halos. We will restrict dent, so it is applicable to modified gravity models and ourselves to large scales (k knl, the nonlinear wave clustered dark energy models. Note also that we do not number) where bias is scale independent in simple models distinguish the growth factor of g from that of because of galaxy formation. This allows us to infer the mass power we only use its time (redshift) derivative, which is expected spectrum from the galaxy power spectrum without detailed to be very similar. Velocity measurements at multiple red- modeling of their relation, because it is possible to fit for shifts are required to measure from the above equation, the bias directly from the data. We discuss below the as described in [47]. caveats to this assumption for clustered dark energy. The galaxy density in three-dimensional space may be For clustered DE models, the galaxy vg is not necessarily equal to v of the total fluid. From the Euler equa- expressed in terms of the density and bias parameters b1 tion (6) applied separately to different components of the and b2 as fluid, we can see that the DM and DE velocities evolve ng b2 2 ¼ b1 þ : (30) differently since only the latter is affected by pressure g n 2 perturbations in the DE. As a first order approximation, g galaxies and baryonic gas velocities trace that of the DM. This expansion is useful for small values of ; it can be So what one actually measures is g ’ DM DE . used in a perturbative expansion to explore what measure- This distinction can be relevant for DE models with large ments are sufficient to measure the bias parameters b1, b2 perturbations on subhorizon scales if these are not corre- as well as (see [49] for details on the bias formalism). lated with the matter fluctuations (i.e., if the DE power Equation (28) shows how the three-dimensional galaxy spectrum has a different shape from the matter power power spectrum Pg can be obtained from redshift space spectrum). measurements. A second way of measuring Pg is from Assumptions/caveats: The galaxy peculiar velocity only imaging data with photometric redshifts. This provides probes where there are galaxies. So potentially there is a measurements of the angular power spectrum of galaxies, bias related to the environment of galaxies. However, since which is a projection of the three-dimensional galaxy gravity is a long range force, the potential where galaxies power spectrum reside is determined by matter over a much larger region Z W2ðÞ l and thus should be unbiased with respect to the overall . C ðlÞ¼ d g P k ¼ ; ; (31) Galaxies themselves are not sufficiently massive to con- g 2 g tribute to this long range potential. However, to obtain v_ g where W is the normalized redshift distribution of gal- from limited redshift bins, one does need to parametrize g the redshift dependence of v . axies included in the sample. With good photo-z’s it is a g narrow range with width of order 0.1 in redshift, so that The accuracy of the velocity information inferred from many such angular spectra can be measured at different the redshift space galaxy power spectrum relies on the mean redshifts from a survey (e.g., [50]). modeling of the redshift distortion. The derivation of Eq. (28) is quite general—it can be applied to general DE or MG models. However, Eq. (28) does not describe 1. Galaxy bias with clustered dark energy redshift distortions to percent-level accuracy [43]. In clustered dark energy models it is not a priori clear Nonetheless, with improved modeling of the correlation whether the galaxy overdensity is related to the matter function in redshift space [48] the associated systematic overdensity m or to the total fluid overdensity GR.We errors in velocity (and ) measurements can be reduced. argue below that at least for some galaxy populations, g is Density contrast : The clustering of galaxies is one of directly related to m, even though the evolution of the the earliest measures of large-scale structure, and its mea- matter density responds to the full gravitational potential surements have advanced over the last three decades. The (which receives contributions from dark energy clustering galaxy power spectrum Pg is the simplest statistical mea- as well). sure of correlations in the galaxy number density. Several One way to see this is to consider the centers of mass of other probes of large-scale structure also probe the density galaxy halos at sufficiently high redshift zi where the dark field: clustering of the Lyman-alpha forest, clustering of energy density is negligible. The clustering of these halo quasars and galaxy clusters, the abundance of galaxy clus- centers is then simply a biased version of the mass distri- ters, and (in the future) 21-cm emission measurements of bution. Hence at zi one can write gðziÞ¼bðziÞM, with the high-redshift universe. bðziÞ independent of scale for large enough scales. As they However, given a measured galaxy power spectrum, the evolve to redshifts below unity, their motions are given by power spectrum P of the underlying mass density may the potential , just as for the matter field. Hence their

063503-7 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) _ ’ evolution obeys the continuity and Euler equations: g ¼ Q þ b2 _ 2 Qg 2 : (32) g=a and g ’Hg þ k =a. The matter density b1 b1 obeys the same equations with M and M as the density and velocity perturbations. This means that the bias factor By using Pg and measurements of Qg for different tri- preserves its scale independence: at low redshift, it relates angles, both bias parameters and P can be determined. the galaxy power spectrum to the matter power spectrum (A similar analysis can be done in real space, e.g., using and is not directly sensitive to the clustering of dark energy. counts in cells. The skewness S3 is given by the shape of For example the halo-model expression [51,52] for the bias the power spectrum and bias parameters.) While this is a evolution is bðzÞ’1 þð 1Þ=scðzÞ, where scðzÞ/ simplified model, it helps us address what changes for MG: DðzÞ is the density required for spherical collapse at z, the predictions for P and Q both change, with the former and scðzÞ= with the smoothed rms mass fluctua- given by the new linear growth factor on large scales and tion. The expression for ellipsoidal collapse has two addi- the latter by next order terms in perturbation theory (see the tional parameters but still has no scale dependence. Appendix for more details). For well-specified gravity Clustered dark energy follows Eq. (6) with w 0 and scenarios, these calculations can be done and thus the 0, so it has a different time and spatial dependence bias factors determined from measurements. b1 from m. If the dark energy clusters significantly, it is A second approach to measuring is to use the galaxy- therefore possible that galaxies have a scale dependent mass cross correlation measured by galaxy–galaxy lensing bias relative to it and therefore to the total density field. in combination with the galaxy power spectrum (e.g., The above argument is very general but relies on some [41]). This has the advantage that one uses only two-point approximations. These are well justified for massive halos, statistics that can be measured with high accuracy. for which the evolution at low redshift is very simple: However, as discussed below and by [25], for MG theories consider galaxy halos of mass M M, where M is the there is a complication because the Poisson equation is standard halo-model nonlinear mass. The centers of mass needed as well since lensing measures the potentials rather of these halos can be mapped to high- peaks in the nearly than . So for MG theories, the extraction of the bias Gaussian mass distribution at high redshift. Moreover, they parameter in this approach is more complicated, but never- ~ do not move significantly, so it is evident that their power theless feasible by jointly fitting for bias and Geff. spectrum at large scales evolves simply by the growth of its amplitude. Such massive halos correspond to galaxy clus- 3. Galaxy cluster mass function ters and luminous red galaxies (LRGs) at moderate to high- A different probe of is provided by the mass function redshift. For galaxies in lower mass halos, halo motions of galaxy clusters. Given Gaussian initial conditions and a and mergers change their clustering at low redshift, so one spherical/ellipsoidal collapse model, the number density of has to be careful in modeling their bias factors. galaxy clusters can be related to the linear density contrast. Another route to in any GR scenarios is through the In the spherical collapse scenario, a region containing mass metric potentials. Given lensing measurements of þ M will collapse if the overall density fluctuation exceeds a and dynamical measurements of , one can obtain the threshold . The number of such regions can be predicted potential . Using this, the Poisson equation (7) then gives c from the Gaussian statistics and this fixes the halo mass , since the gravitational constant is known in GR. Thus function dn=dM, the number of halos with mass M. is not independent of the metric potentials even for clus- In the standard CDM cosmology, gravitational dynam- tered DE models. ics is determined by GR. The mass function of galaxy clusters is sensitive to the smoothed mass density variance 2. Empirical determination of bias 2 R on scale R, which is dependent on the cluster mass and To leading order then, knowledge of b1 allows us to is typically of order 10 Mpc (e.g., [55]). This is related to relate Pg to P. Barring extreme scenarios of clustered the density power spectrum as dark energy, we take to be the full density field. Z d3k Provided a halo-model description applies reasonably 2 ¼ P ðkÞW2 ðkRÞ; (33) well to our universe, bias can be determined by combining R ð2Þ3 top hat observations and using two- and three-point statistics. For concreteness we consider the bias parameters b1, b2 that where Wtop hat is the window function for averaging with a can be determined from the data using the power spectrum spherical top hat. and bispectrum (denoted B) measurements. In a determi- For clustered DE models, the cluster formation picture nistic bias model, one can then get the density power becomes complicated. The presence of the anisotropic 2 spectrum. With Pg ¼ b1P and the reduced three-point stress invalidates the spherical collapse model and more parameter Q B=P2 (see the Appendix and [49] for full complicated models such as ellipsoidal collapse with tidal expressions), one has a relationship between the Q parame- fields need to be used. Furthermore, the fate of an over- ter of galaxies and mass [53,54]: dense region is no longer determined by the matter fluc-

063503-8 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) tuation m alone. DE fluctuations DE and affect Velocity divergence : Many existing velocity measure- through Eqs. (7) and (8). And p affects the evolution of ments are based on distance indicators: the difference of DE through Eqs. (5) and (6). Thus a combination of m, the true distance from what is inferred from the recession DE, p, and acts in determining the evolution of a given velocity gives an estimate of the peculiar velocity of a region of matter—the resulting collapse condition has yet sample of galaxies or clusters [57]. The pairwise velocity to be worked out. Since many galaxy clusters form recently at small separation can be measured through anisotropic at z & 1, where DE is non-negligible, DE fluctuations galaxy clustering in redshift space at cosmological dis- could leave some detectable signatures in cluster tances [46]. While challenging, there are ongoing attempts abundance. to improve measurements of bulk flow measurements, For probing the dark universe, this is a valuable feature. based on SNe Ia [58]. An independent method is the kinetic It implies that galaxy cluster abundances contain informa- SZ effect [59] of clusters which is directly proportional to tion on not only fluctuations in matter but also fluctuations the cluster peculiar velocity and enables a rather model- in dark energy, and thus is a promising probe of the total . independent measurement method [60]. These measure- To extract such information requires further modeling of ments are likely to have lower signal-to-noise ratio than the the DE model, but a simplified model can be obtained as redshift space distortions discussed above. Further it is follows. The collapse condition based on energy conserva- unclear whether they estimate of the total fluid in a tion should be linear in the DM and DE perturbation clustered DE scenario, for the reason discussed above. variables, since they are all first order variables of the CMB: The CMB power spectrum is given by Z Z energy-momentum tensor. At high redshift, the dark en- 0 0 0 CTTðlÞ¼ dk d FCMBðk; l; Þjl½kðz Þ; (34) ergy contribution should vanish (assuming DE m). Thus we may assume that matter fluctuations are the where the spherical Bessel function j is the geometric only source of growth for the late time m as well as l term through which the CMB power spectrum depends DE, p, and responsible for the LSS. In this picture, all perturbation variables are correlated and have determi- on the distance to the last scattering surface. The function nistic relations [56]. The collapse condition can be sim- FCMB combines several terms describing the primordial power spectrum and the growth of the potential. We will plified into a modified condition on m alone. An effective eff regard FCMB as identical to the GR prediction since we do c can be defined for specific DE models, such that when eff not invoke MG in the early universe (up to the redshift at a region reaches m c , it will collapse. The usual collapse model deals with isolated objects and last scattering). The CMB anisotropy does receive contributions at red- thus Birkhoff’s theorem is implicitly required. shifts below last scattering, in particular, due to the inte- Modifications in GR result in a generic breakdown of grated Sachs-Wolfe (ISW) effect [61]. In the presence of Birkhoff’s theorem. This significantly complicates the dark energy or due to modifications in gravity, gravita- modeling of cluster abundance in MG models, since the tional potentials are in general time varying and thus fate of a given region is determined not only by matter and produce a net change in the energy of CMB photons: energy inside this region, but also matter and energy out- Z ð þ Þ side. However, given a MG model, one can still predict the T ¼ d ð Þ a t d: (35) probability for a given region with overdensity m to T ISW dt collapse and thus predict cluster abundances. Assumption/caveats: Unlike the use of gravitational The ISW effect, like gravitational lensing, depends on and þ lensing to probe þ , it is model dependent to probe probes the combination . The ISW signal is over- from cluster abundance. (1) The cluster abundance requires whelmed by the primary CMB at all scales (although it careful modeling, even in the simplest case of smooth DE does produce a bump at the largest scales in the CMB models. For example, the tidal field makes the spherical power spectrum). For this reason, it has to be measured collapse model only a rough approximation. (2) The indirectly, through cross correlation with other tracers of large-scale structure. The resulting cross-correlation signal observable-mass relation is needed to connect observable is then [e.g., x-ray flux, Sunyaev Zel’dovich (SZ) flux, or cluster Z richness] to the mass of clusters. These cluster properties l 2 d CISWðlÞ¼ P _ _ k ¼ ; a : (36) often involve complicated gastrophysical processes and gð þÞ 2 cannot be predicted with sufficiently high precision from ð Þ _ þ first principles. As a consequence, using cluster abundance Here, Pgð _ þ_ Þ k; is the cross-power spectrum of ( to probe often requires model-dependent calibrations. _ ) and galaxies or other tracers of the LSS such as quasars In spite of these caveats, it is hoped that the well-posed or clusters. By cross correlating the CMB temperature with problem of the evolution of a region in an initially galaxy overdensity g, the ISW effect has been detected at Gaussian random field will be calculable and related to & 5 confidence level [62] and provides independent the linear density field in generic MG or DE models. evidence for dark energy, given the prior of a spatially

063503-9 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) flat universe and GR. This cross-correlation signal depends redshift relation will be measured to 1% accuracy by the on galaxy bias, which has to be marginalized to infer next-generation SNIa and BAO surveys at low-z and by the cosmology. With the aid of gravitational lensing, uncer- CMB at high-z. The next-generation BAO surveys can tainties of galaxy bias can be avoided [63,64]. further measure HðzÞ at low redshift. With the expansion Furthermore, since the ISW amplitude peaks on the largest rate of DE and MG models tightly constrained, measure- scales, it also has a strong correlation with large-scale bulk ments of perturbed variables become powerful flows and produces a cross-correlation signal with poten- discriminators. tially better signal-to-noise ratio than that of the density- The distance-redshift relation at redshift z is given by an ISW cross correlation [65]. integral over the expansion rate, and therefore the energy The primary CMB is Gaussian and statistically isotropic. densities, from redshift 0 to z. This measurement at z & 1 However, gravitational lensing distorts the CMB sky and has provided evidence of acceleration, consistent with induces anisotropy and Fourier mode coupling in the CDM. On the other hand, CMB measurements at CMB, which should not exist otherwise. This feature al- high-z for both distances and perturbations are consistent lows reconstruction of the lensing potential from future with a universe governed by GR, with its energy density high resolution CMB maps [66]. The CMB sky is the dominated by matter and radiation [40]. Thus either dark furthest lensing source and thus can probe þ at red- energy or modification of gravity must produce effects that shifts well above unity. This will be useful to constrain are significant at z & 1 and negligible at z 1000.In those MG and DE models in which deviations from Fig. 1 we show as examples the deviation (from CDM CDM persist at these redshifts. with DE ¼ 0:7) of a model with DE ¼ 0:75 and a flat ¼ 0 3 ISW measurements and future measurements of lensing DGP model with m : . It is clear that for both dis- and galaxy clustering can probe scales approaching the horizon scale. This provides an additional test of MG models in which the growth of perturbations is altered at relatively high redshift on superhorizon scales. Bertschinger [23] showed that growth on superhorizon scales is constrained to be universal for MG models with ¼ . Hu and Sawicki [24] showed how it differs for fðRÞ models which do not obey this constraint, and describe the transition from superhorizon to subhorizon scales. If measurements achieve high accuracy on these large scales, they can be combined with information on subhorizon scales to provide additional constraints on the ratio of potentials for such MG models. Summary: The quantity that can be measured most robustly is the sum of potentials þ , through gravitational lensing and the ISW effect. With a bit more modeling, the Newtonian potential can be inferred from galaxy velocity measurements (i.e., redshift space distortions). To obtain model-independent constraints on the total density perturbation is challenging if one allows for dark energy clustering in the GR scenario. Galaxy clustering is likely to be an effective measure of the matter fluctuation m, while cluster abundance is a promising probe of as it is sensitive to DE fluctuations as well. Although the galaxy pecu- FIG. 1 (color online). Upper panel: We plot the normalized liar velocity is likely to be well measured in the future, the ð Þ DE peculiar velocity (and therefore the overall v and )is distance d z (solid curves, almost coincident) and the linear growth rate DðzÞ=aðzÞ (dotted curves) for 0

063503-10 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) tances and perturbations, significant deviations occur at (a) PLANCK and ground based missions (2008–). low-z in such models [67]. We have indicated the approximate redshift range over The most promising scale/redshift range in the near which these surveys will provide accurate measurements. future is 10–100 s Mpc at redshifts 0:3–1. Imaging It would be most useful to have different observables and spectroscopic observations are likely to be made on overlap in redshift and length scale in the range z ’ these scales and will be robust to many sources of error and 0:3–1 and at scale ’ 10 to several 100 Mpc. This range dependence on specific models. We list below several of scales covers the linear and quasilinear regimes of categories of surveys that will test MG and DE models. structure formation (we are assuming that MG effects are Two sets of surveys are indicated: surveys planned for the present on these scales). Next we consider two promising near future (significant data within 5 years), and surveys combinations of observables that on the 5–7 yr time scale ~ planned to start in about a decade. (The list is not complete will enable measurements of the MG functions Geff and as several projects have been formulated or modified re- on these scales. cently.) Lensing and galaxy power spectra: Planned next- (1) Multicolor imaging survey: With photometric red- generation imaging surveys (see above) will have area shifts for millions of galaxies, these surveys provide coverage in excess of 1000 sq. degrees, enabling a few measurements of weak lensing, galaxy cluster abun- percent-level measurements of lensing power spectra. The dances, and the angular clustering of galaxies, clus- same imaging surveys will also measure the angular clusters, and quasars. These measurements probe tering of the galaxies Cg (at z 0:3–0:6, the redshift of the þ , g and their cross correlation. Upcoming lensing mass) to percent-level accuracy; cluster abundan- surveys include the following: ces will also be measured through optical and SZ surveys: (a) DES, KIDS, PS1, HSC (2008–). 0

063503-11 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008)

IV. MODIFIED GRAVITY VS DARK ENERGY This degeneracy certainly deserves further investigation. In this section, we consider in more detail the question: can Specific models of MG and DE can be tested by com- one always succeed in tuning DE models to produce ob- bining observations of the expansion rate and large-scale servational consequences identical to a given MG model? structure. For example, in the CDM model (and well- If the answer is yes, then one can never unambiguously test defined scalar field models), the growth of the large-scale for deviations from GR. structure is completely determined by the expansion his- The answer to the above question is incomplete in fully tory: there exists a fixed relation between the expansion describing the dark degeneracy. The complementary ques- rate and the growth of LSS. This consistency check has tion, which needs to be answered is: can one always tune been carried out in the literature and is indeed able to MG models to produce observational consequence identi- distinguish specific models investigated [10]. cal to a given DE model? If the answer is yes, then one can Furthermore, this consistency check can be performed in never unambiguously justify the existence of DE. a model-independent way to search for signatures of vio- However, this question is more difficult since it requires lation of GR, with the prior of smooth DE [74–76]. a general parametrization of the relation between the ex- DE models that depart from scalar field models can be pansion rate HðzÞ and the nature of a general gravity much more complicated, with a breakdown of the corre- theory—such a parametrization is not yet available. This spondence between the expansion rate and large-scale limit in theoretical understanding of MG forces us to structure. The expansion rate is determined by GR and investigate only the first question, since we know the w. However, two extra DE properties, the anisotropic stress most general way of parametrizing the influence of DE and the response of pressure to perturbations (p), can on the expansion history of the Universe. Furthermore, we affect the growth of the LSS. These two properties are have constrained our study to a special class of MG mod- determined by the microphysics of the DE model and are els, in which gravity is minimally coupled to matter. The independent of GR and w. As a consequence, the growth study of both questions for the most general MG models is of LSS is no longer fixed by the expansion rate and the beyond the scope of this paper. above consistency check cannot be applied to search for The relationship between the four perturbation variables signatures of the violation of GR. Current observational , , , and is fixed for a complete DE or MG theory. 2 constraints on the sound speed (cs p=)[77] and the These consistency relations are the key to probing the anisotropic stress [78] are weak. Furthermore, these studies nature of DE and MG. With just two variables being use a particular form of and p and assume that one can observable, one can only test against one consistency be switched off when studying the other. Thus a potentially relation and, as we see below, by tuning the 2 extra degrees wide range of DE models with non-negligible anisotropic of freedom in clustered DE models, any MG model can be stress and pressure fluctuations are still viable against mimicked. However, with more observed variables, one observations. To investigate the feasibility of distinguish- can test other consistency relations and hope to break the ing between DE and MG, we will allow for arbitrary degeneracy between DE and MG models. In this section, anisotropic stress and pressure perturbations. we explore the feasibility of distinguishing DE and MG Modifications of gravity (at least the class of theories we models. In this section we consider only the question of have considered) involve two extra quantities which gov- distinguishability in principle, without regard to the accu- ~ ern LSS, namely, modification of Newton’s constant, Geff, racy of observations in the foreseeable future. and the ratio of potentials = . Although these quantities determine the gravitational interaction of perturba- A. Two perturbation observables tions, they do in general affect the expansion rate HðzÞ— unlike for GR and its Newtonian limit. This is in part due to First we assume that both potentials are observables, i.e., the fact that for MG, Birkhoff’s theorem no longer holds we require and to be identical in the two models. From and thus the usual exercise of calculating HðzÞ from the the discussions in previous sections, these two quantities Newtonian dynamics of a spherical matter distribution no are the most likely to be measured to high precision. So we ~ set them identical in the constraint equations for GR and longer applies. Thus Geff and represent real extra degrees of freedom in MG theories. MG to get relations between the remaining variables. The extra degrees of freedom in MG and clustered DE Comparing Eq. (8) for in GR with the constraint models can produce similar observational consequences. Eqs. (12) and (13) for MG gives For example, the anisotropic stress breaks the equality 2 1 1 ~ between and , mimicking the role of in MG models. ¼ Geff MG 1 MG: (37) Thus one might expect that by tuning the 2 extra degrees of 3 þ 1 G GR freedom in DE models, one can mimic a given MG model to fit observations. Indeed, Kunz and Sapone [19] explic- In addition by combining the Poisson equation (7) for itly construct a DE model which reproduces degenerate , GR with Eqs. (12) and (13) for MG, we obtain a second , and m with the flat DGP modified gravity model [79]. constraint

063503-12 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) 2 ~ B. Three or more observables þ 3ð1 þ Þ GR ¼ Geff MG GR w Ha 2 1 MG: (38) k þ 1 G GR If both potentials and are observable then the theory is constrained much more tightly, especially if they are mea- The question then is whether Eqs. (5), (6), and (38), have sured multiple redshifts. solutions for GR, GR, and p, in terms of MG variables For a DE model to mimic the given MG model, , , [recall that is now fixed by Eq. (37)]. Without a funda- and must satisfy the three equations (7), (12), and (13). ~ mental theory, p can take any form [21]: hence there is This imposes the following consistency relation for Geff always a form of p satisfying all three equations. Namely, and : there is always a DE model which can mimic the given MG ~ model to produce identical and . Geff MG 1 ¼ 2 1: (41) The degeneracy persists for other combinations of two G GR perturbation variables. We have discussed above that in a clustered DE model, it is difficult to establish whether So if the given MG model does not obey the above relation, galaxies, other tracers, or cluster abundances probe or no DE model can produce , , and identical to the m (or neither). If we assume that a subset of LSS obser- given MG model, no matter how the DE properties are fine- vations will provide measurements of , then combining tuned. that with measurements of þ from lensing, we have Equation (41) represents a strong constraint on MG ~ models as it shrinks the 2-parameter Geff space in ~ 2 Geff MG MG models into a straight line. We show as an example ¼ 1 : (39) 3ð1 þ wÞ G GR that the DGP model does not satisfy this condition. ~ 1 In a flat DGP model, Geff ¼ G and ¼ð1 þ Thus if is free, it can be chosen to match the above 1=3DGPÞ=ð1 1=3DGPÞ [11]. Here DGP ¼ 1 _ 2 equation for any set of theories. 2rcHð1 þ H=3H Þ¼1 2rcH=ð1 2rcHÞ < 0, with 2 3 Extra information can break this degeneracy. The re- H ¼ H=rc þ ma and rc ¼ 1=ð1 mÞ. We have sponse of pressure to density perturbations and the aniso- normalized Hðz ¼ 0Þ¼1. For the DGP model, MG ¼ 3 tropic stress are determined by the microphysics of the DE ma (up to a normalization, which is irrelevant for model. It requires a theory to provide such closure relations this discussion). By requiring the DE model to reproduce (see [24] for more detailed discussions). For example, the the expansion history of the given MG model, we have 2 quintessence model predicts vanishing and negligible GR ¼ H . Figure 2 shows that the consistency condition pressure perturbation on subhorizon scale. Even if advan- is significantly violated by all flat DGP models. This means ces in the understanding of general DE theory do not that no DE model can produce , , and identical to a provide such specific information, some general con- flat DGP model that satisfies observational constraints on straints can still break the degeneracy. For example, if the expansion history. 2 2 2 p takes the form p ¼ cs and cs ¼ csðtÞ, as is true This conclusion seems to contradict [19]. However, [19] for the adiabatic case, solutions do not exist in general for requires the dark matter density fluctuation m to be iden- Eqs. (5), (6), and (38). In this case, one cannot find a DE tical in the GR and MG scenarios. We require the total to model to mimic the given MG model. be identical instead. In GR, the Poisson equation specifies Another physically well-motived example is for the the relation, not the m relation, so with as anisotropic stress . A natural source of is the velocity an observable, one directly constructs consistency relations perturbations in the fluid. By the requirement on gauge and thus distinguishes between DE and MG. invariance, the evolution in may be parametrized in the There is also a constraint on the anisotropic stress, from following form in the Newtonian gauge [78,80]: Eqs. (8), (12), (13), and (38),

8 c2 1 1 þ 3H_ ¼ vis ; (40) ¼ : (42) 3 1 þ w 3ð1 þ wÞ where cvis is the viscous parameter. This equation in gen- Comparing Eq. (6) with Eq. (10), we obtain eral contradicts Eqs. (37) and (39) above and thus no DE w_ c2 k2 model that satisﬁed Eq. (40) can mimic the given MG 3Hw þ s ¼ 0: (43) model. 1 þ w 1 þ w a Extra information can also come from additional ob- Since H aH2 k2=a, we have servables. The equations above show that if we have just one additional observable, such as or , there will in ð1 þ Þ 1 1 2 ’ w ¼ general be no solution for the remaining two variables that cs : (44) 3 satisﬁes three equations [e.g., (5), (6), and (38)]. We consider this next. Comparing Eq. (5) with Eq. (9), we obtain

063503-13 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) wð=a 3_ Þ c2 ¼ þ w ’ w þ 1 : (45) s 3H 3

2 Combining both constraints on cs, we obtain

1 ’ w þ 1 þ 3w: (46)

2 w is fixed by the condition GR ¼ H . is calculated from the given MG theory. So the above equation can be checked from the viewpoint of MG models unambiguously. Again, Fig. 3 shows that this condition is severely violated for the DGP model: thus no DE model can mimic a flat DGP model to reproduce identical , , and .We have verified that this is true for fðRÞ models as well. These relations present general constraints, without re- sort to real observation data. Observations show that at the present epoch, w<1=3 since the universe is accelerat- ing, while >0 since the structure is growing. From 2 Eq. (45), we have cs < 1=3, if the related DE model FIG. 2. First consistency condition for at least one DE model can reproduce , , and . Furthermore, Eq. (46) tells that to mimic , , and in a flat DGP model. The dashed line given <0 today. by Eq. (41) represents the required condition, while the solid curve is the actual relation in flat DGP. When a ! 0, ! 1 and for a !1, ! 1=2. The points on the curve with a ¼ 0:5 and V. DISCUSSION ¼ 1 a are indicated. (For flat DGP lines with different m lie on We have described the role of perturbations in testing top of each other.) The disagreement between the two curves shows that DGP is a modified gravity model that cannot be theories of MG against large-scale structure observations. mimicked by any dark energy model. We have chosen the class of MG theories that are described by scalar perturbations, so that two metric potentials suf- fice to describe the perturbed space-time. We then consider the quasistatic, Newtonian limit of the perturbed field equations and compare DE and MG theories. Our main focus is on the relationship of different observables—lensing, large-scale dynamics of galaxies, galaxy clustering, cluster abundances, and various cross correlations—to the four perturbation variables of MG theories including the two metric potentials, the density field, and the divergence of the peculiar velocity. In Sec. III we give the relationship of measured power spectra in real space, redshift space, and on the sky with theoretical predictions for these perturbation variables. We highlight the use of two effective functions to test MG theories: the ratio of the metric potentials = and the effective gravi- ~ tational constant Geff. We also consider in the Appendix quasilinear signatures of MG and show how the MG functions affect second order corrections to the power spectrum and the bispectrum. We discuss in detail what is actually measured by various large-scale structure observations once the assumptions of smooth dark energy and GR are dropped. While lensing and dynamical probes have a direct connection to FIG. 3. The second necessary condition for at least one DE different potential variables, tracers of the density field and model to mimic , , and in the given DGP model. The cluster abundances must be treated carefully in extracting condition 1 ’ w þ 1 þ 3w, given by Eq. (46) implies the information about the density field from them. The most variable on the y axis should be zero for all a. For flat DGP with robust tests of MG effects can be made by combining m ¼ 0:2, this condition is also severely violated for a>0. different observables from planned multicolor imaging

063503-14 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) surveys and redshift surveys: see Sec. III B for two ex- couplings to matter can mimic any modification of grav- amples that will be feasible in the near future. ity.) Our result, that DE models—with no coupling to Observables that may not be useful in constraining matter—can be distinguished from MG, appears to be smooth DE can be crucial for testing MG because there consistent with the analysis of [24,83]. In Sec. IV we are more variables to be measured and different observ- showed how consistency relations, obtained from the evo- ables are sensitive to them. For instance, the redshift space lution and constraint equations obeyed by perturbation power spectrum Pg is not considered a valuable probe of variables, help to distinguish between DE and MG. The structure formation in DE studies as other methods produce violation of the consistency conditions that can occur with lower statistical errors on dark energy parameters. But in a sufficient observables (see Sec. IV B) would thus imply MG scenario, Pg is useful because it probes the either the modification of gravity or a coupling of the ‘‘dark Newtonian potential that other probes are not sensitive energy’’ with matter in a GR scenario. to. It can be combined with galaxy–galaxy lensing, which There are several assumptions and caveats in this study. We study MG theories with scalar perturbations to the probes Pgð þÞ, to constrain the ratio of potentials = (Sec. III B). This is a case where observables from multi- metric; our formalism does not apply to theories with color imaging surveys and spectroscopic surveys must be additional vector or tensor degrees of freedom. While combined to test MG theories. More generally, once one describing the quasilinear regime of large-scale structure allows for MG scenarios, multiple observables are needed (where planned observations have the best signal-to-noise to test theories. Thus the diversity of LSS observations, ratio), one needs to be aware that it is not clear how to which has lost some of its appeal in the recent trend of obtain nonlinear predictions of some MG theories. We going after a single dark energy figure of merit, becomes have chosen to work in Fourier space, where the descrip- vital (see [81] for a broader criticism of dark energy driven tion of clustering is simpler, but this means that there is not research, and [82] for a rebuttal). always a direct relationship of our effective functions for MG theories with the real space description of these theo- Finally we consider a question posed recently in the ~ literature: can a DE model be constructed to mimic any ries (e.g., Geff is related to its real space counterpart in the MG theory? We show that with observations of multiple Poisson equation by a convolution with the density field). perturbed variables (three are sufficient in general), unique And finally, we have left for future work a detailed study of signatures of MG theories can be established. We show the accuracy with which MG tests can be performed by the with the example of the flat DGP model how, given suffi- next generation of surveys. ciently accurate measurements of the lensing and Newtonian potentials and the density, no DE model can ACKNOWLEDGMENTS mimic the DGP model. We are grateful to Jacek Guzik, Eric Linder, and Wayne Our results may be compared with other recent studies in Hu for helpful discussions and comments on an early draft. the literature, some of which appeared while this work was We thank Francis Bernardeau, Raul Jimenez, Matt in progress [19,24,83]. These studies tackle the question of Martino, Roman Scoccimarro, Ravi Sheth, Fritz how dark energy and modified gravity can be distin- Stabenau, Masahiro Takada, Jean-Philippe Uzan, and guished. We have clarified the apparent conflicts between Licia Verde for stimulating discussions. PJ. Z. is supported this paper and [19] in Sec. IV B. References [24,83,84] by the one-hundred talents program of the Chinese present a formal argument that any deviation from GR can Academy of Science (CAS), the National Science be absorbed into the dark sector as an effective dark Foundation of China Grant No. 10533030, and CAS component. The effective stress-energy tensor of this com- Grant No. KJCX3-SYW-N2. B. J. is supported in part by eff ponent, T, is defined as the deviation of the given MG NSF Grant No. AST-0607667, the Department of Energy, eff theory from Einstein’s field equations. T is conserved, as and the Research Corporation. expected for the usual DE model. From this argument, one might conclude that MG cannot be distinguished from DE APPENDIX: PERTURBATION THEORY IN gravitationally. However, Hu and Sawicki and Bashinsky MODIFIED GRAVITY [24,83] also pointed out that the effective dark component has a generic, though implicit, coupling to matter, despite The fluid equations in the Newtonian regime are given eff by the continuity, Euler, and Poisson equations. Keeping the conservation of T. This coupling hides in the closure relations for this dark component, which depend on exter- the nonlinear terms that have been discarded in the study of nal matter and the metric, instead of just its internal mi- linear perturbations in the rest of the paper, the continuity crophysics. A theory with such a dark component is equation gives different from conventional DE models of the kind we Z 3 ~ ~ _ d k1 k k1 þ ¼ ðk~1Þðk~ k~1Þ; (A1) have considered, even ones with strong clustering of the ð2Þ3 k2 DE. (Indeed, it should not be surprising that allowing for a 1 dark component with an arbitrary stress-energy tensor and where the term on the right shows the nonlinear coupling of

063503-15 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) modes. Note that the time derivatives are with respect to order terms. These rely either on features in k and t in conformal time in this Appendix. The Euler equation is measurements of P þ and P, or on the three-point Z 3 2 functions, which even at a single redshift can have distinct d k1 k k~1 ðk~ k~1Þ _ þ H k2 ¼ signatures of MG [86]. Quasilinear signatures due to ð2 Þ3 2 2 2 2k1jk~1 k~2j ðk; zÞ can also be detected via second order terms in the redshift distortion relations for the power spectrum and ðk~1Þðk~ k~1Þ: (A2) bispectrum. Our discussion generalizes that of [6] who We neglect pressure and anisotropic stress as the energy examined a Yukawa-like modification of the Newtonian density is taken to be dominated by nonrelativistic matter potential. Note that inclusion of higher order terms in the [85]. The Poisson equation is given by Eq. (12) and sup- density field in the quasistatic Newtonian equations is plemented by the relation between and given by consistent even if higher order terms in the field equations Eq. (13). Using these equations we can substitute for of modified gravity theories are neglected: the latter in- in the Euler equation to get volve potentials which are taken to be very small in the weak field limit, unlike the density. ~ Z 3 2 8Geff d k1 k k~1 ðk~ k~1Þ _ þ H þ a2 ¼ ð1 þ Þ MG ð2 Þ3 2 2 2 2k1jk~1 k~2j 1. Second order solution From a perturbative treatment of Eqs. (A1) and (A3) the ðk~1Þðk~ k~1Þ: (A3) second order term for the growth of the density field is Equations (A1) and (A3) are two equations for the two given by variables and . They constitute a fully nonlinear de- € _ 2 _ _ _ ~ 2 þ H2 MGa MG2 ¼ HI1½1;1þI2½1; 1 scription of MG theories and can be solved once and Geff _ _ are specified. An important caveat is that they may never- þ I1½1;1; (A6) theless be invalid for particular theories, for example, if the I1 I2 superposition principle is violated. where and denote convolutionlike integrals of the two Next we consider perturbative expansions for the density arguments shown, given by the right-hand side of Eqs. (A1) field and the resulting behavior of the power spectrum and and (A3) as follows: 2 bispectrum. Let ¼ 1 þ 2 þ ..., where 2 Oð1Þ. Z 3 ~ ~ ½ _ ð ~Þ¼ d k1 k k1 _ ð ~ Þ ð ~ ~ Þ Higher order effects due to gravitational dynamics become I1 1;1 k 3 2 1 k1 1 k k1 ; (A7) ð2Þ k detectable on 10 s of Mpc at low redshift. While this is 1 strictly true only for general relativity, any MG theory that and is close enough to GR to fit observations can also be Z 3 2 ~ ~ ~ _ _ d k1 k k1 ðk k1Þ _ _ expected to have this feature. In the quasilinear regime, I2½1; 1ðk~Þ¼ 1ðk~1Þ1ðk~ k~1Þ: 10 100 Mpc ð2 Þ3 2 2 2 i.e., on length scales between – , mode- 2k1jk~1 k~2j coupling effects can be calculated using perturbation the- (A8) ory. For MG, let us simplify the notation by introducing the _ _ function: Finally, the last term in Eq. (A6) is simply I1½1;1¼ ½ € þ ½ _ _ ~ I1 1;1 I1 1; 1 . Note that by continuing the itera- 8Geff MGðk; tÞ¼ ; (A4) tion higher order solutions can be obtained. ð1 þ Þ From the above equations it follows that if MG MGðtÞ then 2 may be specified by k integrals over 1, which is simply 4G in GR but can vary with time and so that one may express the functional relationship 2 scale in MG theories. The evolution of the linear growth 2½1 (where it is understood that 2 at a given wave factor is given by substituting for in Eq. (A5) to get number k~ depends on 1 at all other wave numbers). But € _ 2 1 þ H1 MG MGa 1 ¼ 0: (A5) for the general case of a MG theory with scale dependent MG, the second order solution has additional scale and In GR, the relation of to is given by the Poisson time dependence behavior that is not determined by the equation with constant G. In MG, this relation involves linear solution [owing to the third term on the left-hand ~ both and Geff. If either of these functions have a depen- side in Eq. (A6)]. So the functional relationship must be dence on k or z, then the solution for the growth factor modified to 2 2½1; MG. This means that quasilinear changes. The linear solutions for and þ are then evolution provides an additional signature of MG. That is, simply obtained using Eqs. (12) and (13). even if the initial power spectrum is not fully specified We show below that in addition the second order solu- (e.g., if the running of the spectral index is not well con- ~ tion has a functional dependence on Geff and that can strained), the comparison of linear and quasilinear growth differ. Thus potentially distinct signatures of the scale and rates can reveal the signature of MG. In practice, whether ~ time dependence of Geffðk; zÞ can be inferred from higher the quasilinear signature is significant must be determined

063503-16 OBSERVATIONAL TESTS OF MODIFIED GRAVITY PHYSICAL REVIEW D 78, 063503 (2008) by computations for specific models (see [87] for a specific linear power spectrum [both numerator and denominator 4 model for which it is not). Note also that the second order are Oð1Þ, see [86]] and is nearly constant with triangle correction to the density power spectrum also involves the size in GR. It is however sensitive to the shape of the 2 third order density field as it is given by P2 h2iþ triangle. The dependence on size and shape changes for h13i. The qualitative features we highlight for 2 will MG theories and is in principle a probe of MG. It is beyond also be found in 3, which is also given by iterations of the the scope of this paper to elaborate on the measurement of nonlinear Eqs. (A1) and (A2). the bispectrum from galaxy surveys; we will instead focus We summarize the comparison of linear and second on the prospects for lensing measurements. Shirata et al. order solutions for the density for GR versus MG. We [71] have tested Yukawa-like modifications of gravity us- have identified the function MGðk; tÞ as containing all the ing Q for the galaxy density measured in real and redshift information about MG that affects density and velocity space. fields. For the density field the first and second order The lensing bispectrum contains perhaps the clearest solutions can be compared to GR as follows: signature of MG. It is a projection of the three-dimensional (i) Linear growth in GR: In smooth dark energy GR bispectrum models, 1ðk;~ tÞ is a separable function of scale and 6 ~ 2 3 3 k B þ ð8Geffa Þ h i time. MG ~ 2 3 2 (ii) Linear growth in MG: In MG theories, 1ðk;~ tÞ is a ’ð8Geffa MGÞ h12i: (A11) separable function of k and t if and only if the MG Since both 1 and 2 are functions of MG, measurements function MG is independent of scale. ~ (iii) Second order solution in GR: In smooth dark energy of B þ are sensitive to Geff and MG separately. The reduced lensing bispectrum in a MG theory can be GR models, the second order solution 2ðk;~ tÞ is not separable. It is however determined by integrals over expressed in terms of the density power spectrum and bispectrum as 1. (iv) Second order solution in MG: In MG models with ~ ð Þ ~ ð Þ ~ ð Þ ð ~ ~ ~ Þ / Geff k1 Geff k2 Geff k3 B k1; k2; k3 =MG MGðk; tÞ, 2 is no longer determined solely by 1 Q þ 2 ~2 ~2 2 2 : k Geffðk1ÞP ðk1ÞGeffðk2ÞP ðk2Þ=k k þ sym . . . and contains additional signatures of MG. 3 1 2 Note that for weak lensing measurements, quasilinear cor- (A12) ~ rections are given by the density times Geff [by substituting For equilateral triangles, Q in MG theories is simpler higher order terms into Eq. (19)]. So the resulting signa- since the Geff factors in all the terms are the same. One then tures can be straightforwardly computed using the higher has 2 order solutions for the density field. k Q ðMGÞ / Q þðMGÞ ~ : (A13) Geff MG 2. Three-point correlations The ratio of Q for MG versus GR for equilateral triangles is Distinct quasilinear effects are found in three-point cor- given by relations (we will use the Fourier space bispectrum), as it is Q þðMGÞ Q ðMGÞ G GR the lowest order probe of gravitationally induced non- / : (A14) ~ Gaussianity. The bispectrum for the density field B is Q þðGRÞ QðGRÞ GeffMG defined by Note that Q itself depends on MG. Bernardeau [86] shows ~ ~ ~ 3 ~ ~ ~ ~ ~ ~ ¼ 1 ~ hðk1Þðk2Þðk3Þi ¼ ð2Þ Dðk1 þ k2 þ k3ÞBðk1; k2; k3Þ: that with but a scale dependent Geff, Q for given (A9) initial power spectrum is relatively insensitive to MG.If that holds for generic initial power spectra and gravity 3 2 3 B h ih 2i h i¼0 ~ Since 1 (using 1 for an initially models, it would imply that Q þðMGÞ probes Geff for Gaussian density field), the second order solution enters at models with ¼ 1. leading order in the bispectrum. Note also that the wave In general a measurement of the lensing power spectrum vector arguments of the bispectrum form a triangle due to and reduced bispectrum (roughly speaking, of P þ and the Dirac delta function on the right-hand side above. In Q þ) is sufficient to measure departures from GR. There practice, a very useful measure of non-Gaussianity is the ~ are three underlying functions (P , MG, and Geff)tobe reduced bispectrum function Q, which for the density field determined. For given k and source redshift, we have is given by measurements of P and of Q as a function of triangle B ðk~1; k~2; k~3Þ shape. Thus while the equilateral triangles may be regarded ~ Q : as sensitive primarily to Geff, elongated triangles will be Pðk1ÞPðk2ÞþPðk2ÞPðk3ÞþPðk1ÞPðk3Þ sensitive to MG, and therefore to . In practice one must (A10) take into account the fact that the bispectrum has lower To leading order Q is independent of the amplitude of the signal-to-noise ratio than the power spectrum on quasi-

063503-17 BHUVNESH JAIN AND PENGJIE ZHANG PHYSICAL REVIEW D 78, 063503 (2008) linear scales [88], so one must fit for the desired informa- lations of the density and potential fields. tion from all triangle configurations and sizes to constrain Independent of the shape and amplitude of the power the MG functions. spectrum, the dependence of the reduced bispectrum To summarize this section, quasilinear effects thus offer Q on triangle size and shape is a useful test of MG. two signatures of MG. The reduced lensing bispectrum, for example, has a ~ (i) A scale and time dependent feature on quasilinear strong dependence on Geff. scales in the power spectrum that depends on and We have not considered here whether a clustered DE model ~ Geff. This enters through the second order contribu- can mimic both these signatures. It would be of interest to tion to the power spectrum. carry out the second order calculations for a set of MG (ii) Signatures in the bispectrum: additional signatures models and compare predicted deviations with observatio- of modified gravity are present in three-point corre- nal error bars.

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