Lagrangian perturbation theory for modified gravity

Abstract

We present a formalism to compute Lagrangian displacement fields for a wide range of cosmologies in the context of perturbation theory up to third order. We emphasize the case of theories with scale dependent gravitational strengths, such as chameleons, but our formalism can be accommodated to other modified gravity theories. In the non-linear regime two qualitative features arise. One, as is well known, is that nonlinearities lead to a screening of the force mediated by the scalar field. The second is a consequence of the transformation of the Klein-Gordon equation from Eulerian to Lagrangian coordinates, producing frame-lagging terms that are important especially at large scales, and if not considered, the theory does not reduce to the $\Lambda$CDM model in that limit. We apply our formalism to compute the 1-loop power spectrum and the correlation function in $f(R)$ gravity by using different resummation schemes. We further discuss the IR divergences of these formalisms.Comment: 32 pages, 10 figures. v2: CLPT scheme added, IR divergences discussed. v3: Matches published versio

Marked correlation functions in perturbation theory

“…, (A.6) and the second order growth functions, D (2) , are the solutions to equations [31][32][33] (T − A(k))D…”

“…These are constructed by 2-and 3-point correlations of Lagrangian displacements, and are well known in the literature; we refer the reader to the above references for their expressions in ΛCDM, and to refs. [31,33] for MG. In this section we display those functions that are not presented in the above references but are necessary for the marked correlation functions.…”

An accurate perturbative approach to redshift space clustering of biased tracers in modified gravity

“…While halo bias and RSD have been studied in tandem for modified gravity in the context of N-body simulations, for example [69][70][71], they have only been studied separately, to date, for perturbative approaches to the clustering statistics [72][73][74][75][76][77][78][79][80][81][82][83]. In [81,83], CLPT was extended to predict the two-point statistics for biased tracers in MG, based on the LPT framework for MG developed in [84] and also an analytical model for the prediction of the Lagrangian bias factors in MG [83], extending the Peak-Background Split formalism (PBS) [43,[85][86][87]. Applied on the f(R) Hu-Sawicki [88] and the nDGP [89] models, it was shown to perform very well against results obtained by N-body simulations across a wide variety of screening levels and cosmological redshifts.…”

COLA with massive neutrinos

“…We can show that the corresponding Fourier transform with respect to q at second order for the specific case of scale-dependent modified gravity without massive neutrinos is [52,58,59]…”

“…When taking the Fourier transform of Eq. (3.1) with respect to q we will need an expression for F q ∇ x 2 Φ N ( x, τ ) which will contain frame-lagging terms above linear order [58]. For the general case where the scale-dependence at n th order is encapsulated in the effective Newton's constant µ (n) (k, τ ) the Fourier transform with respect to x up to second order is…”