A Explicit expression for the kernels, K_{f(R)}^{(n)} and K_{\rm DGP}^{(n)}

Regularized cosmological power spectrum and correlation function in modified gravity models


Based on the multi-point propagator expansion, we present resummed perturbative calculations for cosmological power spectra and correlation functions in the context of modified gravity. In a wide class of modified gravity models that have a screening mechanism to recover general relativity (GR) on small scales, we apply the eikonal approximation to derive the governing equation for resummed propagator that partly includes the non-perturbative effect in the high- limit. The resultant propagator in the high- limit contains the new corrections arising from the screening mechanism as well as the standard exponential damping. We explicitly derive the expression for new high- contributions in specific modified gravity models, and find that in the case of gravity for a currently constrained model parameter, the corrections are basically of the sub-leading order and can be neglected. Thus, in gravity, similarly to the GR case, we can analytically construct the regularized propagator that reproduces both the resummed high- behavior and the low- results computed with standard perturbation theory, consistently taking account of the nonlinear modification of gravity valid at large scales. With the regularized multi-point propagators, we give predictions for power spectrum and correlation function at one-loop order, and compare those with -body simulations in gravity model. As an important application, we also discuss the redshift-space distortions and compute the anisotropic power spectra and correlation functions.


I Introduction

The precision observation of large-scale structure of the Universe now plays a very crucial role in scrutinizing the standard cosmological model that has emerged recently based on the multiple cosmological observations. Amongst various cosmological issues, one important subject is to clarify the origin and nature of cosmic acceleration, first discovered by the distant supernova observations Perlmutter et al. (1999); Riess et al. (1998). The cosmic acceleration may be originated from the dark energy, or rather it may indicate the breakdown of general relativity on very large scales. To observationally explore this, the measurements of both the cosmic expansion and the growth of structure are thought to be essential, giving us a chance to test gravity on cosmological scales or to constrain dark energy equation of state (e.g., Weinberg et al. (2013) for review). The large-scale structure observations with galaxy redshift surveys indeed offer an opportunity to measure these two quantities simultaneously.

The key measurements are the baryon acoustic oscillations (BAO) and redshift-space distortions (RSD), imprinted on the large-scale clustering pattern of galaxy distribution. With BAO as a standard ruler, we can simultaneously measure the angular diameter distance and Hubble parameter at the distant galaxies through the Alcock-Paczynski effect (e.g., Alcock and Paczynski (1979); Seo and Eisenstein (2003); Blake and Glazebrook (2003); Glazebrook and Blake (2005); Shoji et al. (2009); Padmanabhan and White (2008)). On the other hand, RSD caused by the peculiar velocity of galaxies induces apparent clustering anisotropies, whose strength is related to the growth rate of structure formation (e.g., Kaiser (1987); Hamilton (1997); Peebles (Princeton University Press, 1980); Linder (2008)). Since both BAO and RSD are now reliably and simultaneously measured through the clustering statistics of galaxy distribution (e.g., Eisenstein et al. (2005); Percival et al. (2007); Reid et al. (2012); Anderson et al. (2014); Beutler et al. (2013) for recent measurements) typically on scales close to the linear regime of gravitational evolution, the precision estimation of power spectrum and/or correlation function is a major priority of the ongoing and upcoming galaxy surveys.

With increasing interests in precision measurements, accurate theoretical modelings of power spectrum and/or correlation function is crucial and is essential to correctly estimate the geometric distances and structure growth, taking full account of the nonlinear systematics including gravitational clustering and RSD. Development of theoretical templates is thus an important research subject, and there have been numerous numerical and analytical studies along this line Jeong and Komatsu (2006); Crocce and Scoccimarro (2008); Matsubara (2008, 2011); McDonald (2007); Taruya and Hiramatsu (2008); Taruya et al. (2009); Bernardeau et al. (2008); Pietroni (2008); Valageas (2007); Taruya et al. (2012); Wang et al. (2013); Taruya et al. (2010); Reid and White (2009); Carlson et al. (2012); Seljak and McDonald (2011); Vlah et al. (2012); Okumura et al. (2012); Heitmann et al. (2009); Lawrence et al. (2010). Thanks to these efforts, we are now able to discuss the accuracy of theoretical template at a percent level. However, one important remark is that the calculation of such templates, especially for the prediction of gravitational clustering, heavily relies on the underlying theory of gravity. So far, general relativity (GR) has been implicitly assumed as the underlying theory of gravity in most studies. As a consequence, although such templates can be employed for consistency tests of GR, their use for characterizing or detecting deviation from GR gravity can be limited.

Further theoretical developments are therefore required in a wide context of modified gravity models. While a model-independent approach, in which we do not assume gravity but rather parametrize it in a fairly generic way (e.g., Hu and Sawicki (2007a); Baker et al. (2013); Brax et al. (2012); Daniel et al. (2008); Clifton et al. (2012)) is very helpful and should be exploited, most of the approaches proposed so far have been restricted to the linear regime. Since the applicable range of linear theory calculation is known to be rather narrower at lower redshifts, our ability to constrain or test such models is expected to be significantly reduced Taruya et al. (2014).

In this paper, we attempt to extend the framework of theoretical templates to deal with modified theories of gravity. Here, we specifically examine this issue based on the analytical approach with perturbation theory calculations, relevant for the measurement of BAO and RSD on large scales. Previously, we have presented the basic formalism to treat general modified gravity models Koyama et al. (2009), and in specific gravity models, we have computed power spectra in both real and redshift spaces based on the standard perturbation theory (PT) Bernardeau et al. (2002). The standard PT is, however, known to produce a poorly convergent series expansion, and because of the bad high- behavior (e.g., Crocce and Scoccimarro (2006a); Carlson et al. (2009); Taruya et al. (2009)), difficulty arises in computing the correlation function through a direct integration of power spectrum.

In the present paper, we shall apply the specific resummed PT scheme referred to as the multipoint propagator expansion or expansion Bernardeau et al. (2008). The building blocks of this PT scheme are the multipoint propagators, with which the non-perturbative properties at high- can be efficiently resummed, giving us an improved convergence of the PT expansion. In the case of GR, making full use of the analytical properties, the regularized propagators, which consistently reproduces both the standard PT results at low- and the expected resummed behaviors at high-, have been successfully constructed Bernardeau et al. (2012a); Bernardeau et al. (2014), and the expansion has been applied to the predictions of real- and redshift-space power spectra and correlation functions, showing a very good agreement with -body simulations Crocce et al. (2012); Taruya et al. (2012, 2013). Clearly, a crucial point for applying this approach to the modified gravity models is whether we can systematically construct regularized propagators in a semianalytic manner. Here, we specifically show that while there appear non-trivial corrections originating from the screening mechanism in modified gravity model, in the case of gravity model for a currently constrained model parameter, these corrections are basically small. Thus, in gravity, the propagator can be constructed in a similar manner to the GR case. Then the analytically computed propagators are compared with -body simulations, and a good agreement is found. With these propagators as building blocks, we will proceed to the calculation of the power spectrum and correlation function in both real and redshift spaces.

The paper is organized as follows. In Sec. II, we briefly review the basic formalism to treat perturbations in general modified gravity models, and introduce a resummed PT scheme based on the multipoint propagator expansion. Sec. III discusses the non-perturbative high- behavior of the propagators based on the eikonal approximation, and Sec. IV presents an explicit expression for matter power spectrum in terms of the regularized propagators, which satisfy both the expected high- and low- behaviors. Then, the comparison of PT results with -body simulations is made in Sec. V, and the applications to the redshift-space observables are discussed in Sec. VI. Finally, the newly developed PT calculation is compared with standard PT prediction in Sec. VII, and we summarize our findings in Sec. VIII.

Ii Basic equations for perturbations

In this section, we begin by reviewing the framework to treat the evolution of matter fluctuations in modified gravity models Koyama et al. (2009), and present a set of basic equations relevant for the perturbation theory (PT) treatment. Then, a resummed PT scheme with multipoint propagator expansion Bernardeau et al. (2008) is briefly reviewed, and the properties of these multipoint propagators are mentioned.

ii.1 Dynamics of matter fluctuations in modified theories of gravity

In this paper, we are particularly interested in the evolution of matter fluctuations, ignoring a tiny contribution of massive neutrinos. Inside the Hubble horizon, the so-called quasi-static approximation may be applied, and the time derivatives of the perturbed quantities can be neglected compared to the spatial derivatives. In GR, based on this approximation, we can find the Newtonian correspondence, and the standard Poisson equation is recovered. On the other hand, in modified theory of gravity, the Poisson equation is generically modified due to a new scalar degree of freedom, referred to as the scalaron. On large scales, the scalaron mediates the scalar force, and behaves like the Brans-Dicke scalar field without potential and self-interactions, while it should acquire some interaction terms on small scales, which play an important role to recover GR and to evade the solar-system constraints. Indeed, there are several known mechanism such as chameleon and Vainshtein mechanisms (e.g., Khoury and Weltman (2004); Deffayet et al. (2002)), in which the nonlinear interaction terms naturally arise and eventually become dominant, leading to a recovery of GR. As a result, the Poisson equation is coupled to the field equation for scalaron with self-interaction term. Under the quasi-static approximation, we have Koyama et al. (2009)


with and being the Brans-Dicke parameter. The quantities is the Newton potential, and the function represents the nonlinear self-interaction, which may be generally expanded as


On the other hand, for the matter sector, the evolution of matter fluctuations is governed by the conservation of energy momentum tensor, which would remain unchanged even if the gravity sector is modified. Under the single-stream approximation, which is relevant for the scale of our interest, the matter fluctuations are treated as a pressureless fluid flow, whose evolution equations are given by Bernardeau et al. (2002)


Eqs. (2)–(4) are the basic equations for perturbations in a general framework of modified gravity models. In Fourier space, they can be reduced to a more compact form. Assuming the irrotationality of fluid quantities, the velocity field is expressed in terms of the velocity divergence, . Then, we introduce the two-component multiplet (e.g.,Crocce and Scoccimarro (2006a)):


where the subscript selects the density and the velocity components of CDM plus baryons. The governing equations for become Koyama et al. (2009)


where the time variable is defined by , and is the Kronecker delta. Here, we introduced the shortcut notations, and . The matrix is given by


with the function defined by


From the component of , we can define the effective Newton constant as


Note that in the cases with =0, the effective Newton constant is given by


For a positive , the effective gravitational constant is larger than GR and the gravitational force is enhanced. On the other hand, if , becomes .

In Eq. (7), there appear two types of vertex functions. One is the standard vertex function arising from the nonlinearity of the fluid flow, :


Note the symmetric properties of the vertex function, . Another vertex function is characterized by the kernel , which represents the mode coupling of the density fields with velocity-divergence field. This coupling comes from the non-linear interaction terms of the scalaron [i.e., Eqs. (1) and (2) through (3)]. The explicit form of the higher-order vertex functions is given by (see Appendix B for derivation):


Note that the expression of vertex functions and is not yet symmetrized under the permutation of wave vectors, and it has to be symmetrized.

So far the framework to treat perturbations is general, and can be applied to any gravity model that satisfies the conservation law of the matter sector. As representative examples of modified gravity models that can explain the late-time cosmic acceleration, we shall below consider the gravity Hu and Sawicki (2007b); Starobinsky (2007) and Dvali-Gabadadze-Poratti (DGP) model Dvali et al. (2000), and present the explicit expressions for model-dependent parameters and coupling functions and . While these models are rather specific and have been tightly constrained recently by observations, the mechanisms to recover GR on small scales are typical and a broad class of modified gravity models can fall into either of two models. We thus expect that even the PT calculations in these specific modified gravity models can give a fairly generic view on the deviation of gravity from GR.


The gravity is a representative modified gravity model for which the Einstein-Hilbert action is generalized to include an arbitrary function of the scalar curvature :


where is the Lagrangian for matter sector. This theory is known to be equivalent to the Brans-Dicke theory with parameter , but due to the nonlinear functional form of , the Brans-Dicke scalar can acquire a nontrivial potential. This can be seen from the trace of the modified Einstein equations. In the universe dominated by ordinary matter, we have


where and . The field is identified with the scalaron, i.e., the extra scalar field, and its perturbations are defined as


where the bar indicates that the quantity is evaluated on the background universe. Imposing the conditions and , the background expansion can be close to CDM cosmology, and the quasi-static approximation leads to


The above equation indeed corresponds to Eq. (2) with . Then, expanding in terms of , we obtain the explicit functional form of the coupling functions:


which only depends on time. Then, this gives


where we define .

In this paper, we will present the results of PT calculations in gravity, and the predictions of propagator, power spectrum, and correlation function are compared with -body simulations. For this purpose, in this paper, we will below consider the specific function of the form:


where is a dimensional constant of length squared. In particular, we are interested in the high curvature limit , and can be expanded as


Here, is the constant energy density related to . The quantity is the background curvature at present time, and we defined . With the current observational constraint (e.g., Marchini and Salvatelli (2013); Lombriser et al. (2012); Yamamoto et al. (2010); Okada et al. (2013); Schmidt et al. (2009), see also Wang et al. (2012) for a strong constraint from small-scales), the background cosmology becomes indistinguishable with CDM model, but the extra term is still non-negligible for the evolution of matter fluctuations, giving rise to a different growth history of structure.

DGP model

The DGP braneworld model is another modified gravity model that has a screening mechanism. The DGP model is the 5D gravity theory with the induced 4D gravity on a brane in which we are living. Thus, on large scales larger than the characteristic scale , the gravity becomes , while on small scales, gravity becomes , but it is not described by GR. As a result, the Friedman equation is modified on the brane Dvali et al. (2000):


where represents two distinct branches of the solutions ( is the self-accelerating branch, and is called the normal branch).

Notable point in the DGP model is that the GR is recovered via the Vainshtein mechanism, by which the scalaron becomes massless, but acquires a large second-order derivative interaction. The resultant coupling functions become Koyama et al. (2009)


The quasi-static perturbations on 4D brane are described by the Brans-Dicke theory, where the Brans-Dicke parameter is given by


with being the cosmic time derivative of the Hubble parameter. Then, the function becomes


ii.2 Multipoint propagator expansion

Provided the basic equations for matter fluctuations, a straightforward approach to deal with the nonlinear evolution perturbatively is to just expand the perturbed quantities like , and to solve the equations order by order, regarding the initial field as a small expansion parameter. This is the so-called standard PT treatment Bernardeau et al. (2002). As we mentioned in Sec. I, the standard PT is known to produce a poorly convergent series expansion, and is difficult to compute the correlation function because of the bad UV behavior.

Alternatively, we may first introduce the non-perturbative statistical quantities, and expand the statistical quantities for our interest in terms of these. The multi-point propagator expansion or the expansion is one such PT expansion, and is regarded as a resummed PT treatment, in which the standard PT expansion is reorganized by the non-perturbative quantities Bernardeau et al. (2008). A key property is that all the statistical quantities such as the power spectra and bispectra can be reconstructed by an expansion series written solely in terms of the multipoint propagators. The multi-point propagator is a fully non-perturbative quantity, and with this object, a good convergence of the PT expansion is guaranteed. This is in marked contrast to the standard PT expansion. Although these have been confirmed and checked in the case of GR, we expect them to hold even in modified gravity models as long as the deviation from GR is small.

The -point propagator is defined by


with being the ensemble average. Here, is the initial density field given at an early time . With the multi-point propagator, the power spectra of cosmic fields are systematically constructed as follows. Defining the power spectra as


we have Bernardeau et al. (2008)


where the quantity is the initial power spectrum defined as


As it is clear from Eq. (30), the nonlinear effects in the power spectrum are wholly encapsulated in the multi-point propagators, and thus the construction of the propagators keeping their non-perturbative properties is quite essential in the analytic treatment of PT. Therefore, subsequent sections are devoted to the discussion on how to analytically construct the propagators in the context of modified gravity. Section III discusses the non-perturbative high- behavior of the propagators based on the eikonal approximation, and Section IV presents a consistent construction of the regularized propagators which satisfies both the expected high- and low- behaviors.

Iii Resummed linear propagator with eikonal approximation

In this section, we derive the resummed linear propagator, in which the behaviors of the high- limit is reproduced at the tree-level calculation as a result of resummation. This resummed linear propagator will be used to systematically construct the multi-point propagator in next section. Here, following Ref. Bernardeau et al. (2012b), we apply the eikonal approximation to the perturbation equations (7). The eikonal approximation enables us to derive the effective evolution equation for short-wave fluctuations under the influence of long-wave modes, which are regarded as external random background. With this treatment, if we neglect the non-linear mode couplings, the fluid equations can be rewritten as linear equations embedded in an external random medium.

iii.1 Eikonal approximation

To be more explicit, consider first the non-linear mode coupling in Eq. (7) associated with standard vertex function, . Through the relation , the contribution of the non-linear coupling can be split into two different cases: the one coming from coupling two modes of very different amplitudes, or , and the one coming from coupling two modes of comparable amplitudes. In the first case, the small wave modes ought to be much smaller than . Let us denote these small modes by , and divide the domain of integral into soft and hard domains. Then, the coupling term may be rewritten as Bernardeau et al. (2012b)


where the first term at the right-hand side represents the contribution from the soft domain, taking the limit, . The expression for the function becomes


where the subscript implies that the integral is restricted to the soft domain.

Similarly, the mode coupling arising from the non-linear interaction of the scalaron [i.e., the second term in RHS of Eq. (7)] can be split into two domains: the soft domain in which one of the modes is much larger than others, and the hard domain in which there is no particularly larger mode than others. We obtain


Here, the matrix includes the contribution from the soft domain, and the non-vanishing contribution appears only in . Let us rewrite it with


Here, the function represents the sum of all possible combinations of the soft/hard domains of the integral. We find that the non-vanishing contribution of leads to the modification of the effective Newton constant given in Eq. (10), as a result of the screening mechanism in modified gravity:


The explicit expression for is given by


for the gravity model, and


for the DGP model. Note that the factor comes from the number of possible combinations of the soft/hard domains of the integral. The functions and are the dimensionless kernels, whose explicit expressions are presented in Appendix A. In the high- limit, these contributions are supposed to be subdominant compared to that coming from the standard vertex function [see Eq. (33)], and may be treated perturbatively as a higher-order contribution. This point will be discussed in Sec. III.3.

iii.2 Resummed propagator

Based on the eikonal approximation in Sec. III.1, we can now reabsorb the effect of the non-linear coupling with long-wavelength modes in the linear terms, and . As a result, the evolution equation for perturbation, Eq. (7), can be recast as


The solution to this equation can be given in terms of the resummed propagator, , and it reads


Here, the resummed propagator satisfies


with the boundary condition, .

In the absence of the term , the solution of this resummed propagator is expressed in terms of the standard linear propagator of Eq. (7), , which satisfies :


In the presence of the asymmetric matrix , no tractable analytic expression is obtained, however, assuming that the term just gives a sub-dominant contribution compared to the function , we obtain the approximate expression:


The resummed propagator given above can be used to systematically compute the multi-point propagators defined in Eq. (28), where the non-perturbative high- behaviors have been already encapsulated (see Sec. IV). In GR, the correction vanishes, and all the multi-point propagators are shown to have the exponential damping behaviors Bernardeau et al. (2008). Thus, the non-vanishing contribution of is a non-trivial result in modified gravity models. The influence of this on the multi-point propagators will be quantitatively estimated in next subsection.

iii.3 Impact of screening effect on resummed propagator

Let us discuss the impact of screening effect on the resummed propagator, focusing on the new contribution, . To start with, we define


In the cases with the negligible effect of modified gravity at an early time , contracting with vector gives the two-point propagator in the high- limit, . To evaluate the impact of the new correction term, we adopt Eq. (43) and substitute it into the above. We then write


Using Eq. (35), we recast as


Based on the leading-order calculation in which the field is treated as linear-order quantity, we can explicitly evaluate under the Gaussian initial condition. We then find that the contribution from in Eq. (46) vanishes in both gravity and DGP models, and the correction from can give the leading-order non-vanishing contribution. Up to this contribution, the propagator can be recast as


The correction represents the first non-vanishing contributions from of . The explicit expression for is presented in Appendix C for the gravity and DGP models [Eqs. (87) (88)].

Figure 1: Resummed linear propagator in gravity model. Left panel shows the redshift evolution of the propagator at Mpc (left) and Mpc (right), while right panel plots the scale dependence of the propagator at . Top panels plot the ratio of propagator in gravity to that in GR for density (red) or velocity-divergence fields (blue). Dashed and solid lines respectively represent the result with and without new correction arising from the screening effect, i.e., and . On the other hand, to see the size of the correction, bottom panels show the fractional difference between the propagators with and without the correction, i.e., . For all panels, we assume gravity of the functional form in Eq. (23), and the model parameter is set to . In computing the propagators, we set the initial redshift to , and adopt the cosmological parameters;