Relativistic effects and primordial non-Gaussianity in the matter density fluctuation

Abstract

We present the third-order analytic solution of the matter density fluctuation in the proper-time hypersurface of nonrelativistic matter flows by solving the nonlinear general relativistic equations. The proper-time hypersurface provides a coordinate system that a local observer can set up without knowledge beyond its neighborhood, along with physical connections to the local Newtonian descriptions in the relativistic context. The initial condition of our analytic solution is set up by the curvature perturbation in the comoving gauge, clarifying its impact on the nonlinear evolution. We compute the effective non-Gaussian parameters due to the nonlinearity in the relativistic equations. With proper coordinate rescaling, we show that the equivalence principle is respected and the relativistic effect vanishes in the large-scale limit. 98.80.Jk,98.62.Py Rapid developments in large-scale galaxy surveys over the past decades have enabled the precision measurements of galaxy clustering, which can be used to probe the nature of dark energy and the perturbation generation mechanism in the early Universe [1]. In parallel, the recent theoretical development ([2-6]; see [7] for review) has revealed that the subtle relativistic effects are present in galaxy clustering, providing new opportunities to extract additional and critical information about the gravity on large scales and the initial conditions for structure formation. In particular, the relativistic formalism has been extended [8-10] to the second-order in perturbations for the computation of higher-order statistics such as the bispectrum.One of the critical elements in the relativistic formalism is galaxy bias, which relates the galaxy number density to the underlying matter distribution. Beyond the linear order in perturbations, however, galaxy bias poses a nontrivial problem due to the gauge issues in general relativity. It was shown [11] that the proper-time hypersurface of nonrelativistic matter flows provides a physical description of the local observer, moving with dark matter and baryons that will collapse to form galaxies. This physical justification has led us to study the nonlinear relativistic effects of the matter density fluctuation in the proper-time hypersurface of nonrelativistic matter flows. This is timely and interesting, for there has been recent conflict in literature -It is argued in [12,13] by performing the second-order relativistic calculations that the nonlinear evolution of gravity generates the local-type non-Gaussianity, which would in turn exhibit a prominent signature in galaxy clustering on large scales. On the other hands, in [14-16] the opposite claim is asserted that the observable quantities are not affected by these nonlinear relativistic effects of gravity, while the calculations are in general based on studying the special case (or the squeezed limit), where only the linear- * Electronic address: jyoo@physik.uzh.ch † Electronic address: jinn-ouk.gong@apctp.org order calculations are required. In this article, we present the ...

Galaxy two-point correlation function in general relativity

“…Drawing upon these studies, we demonstrate that our relativistic derivation of galaxy clustering is not affected by such long modes either from scalar or tensor perturbations. As described in [37,42], this also implies that our expression is devoid of infrared divergences. It is known that most relativistic expressions for galaxy clustering in literature have variances that diverge in the infrared [28].…”

Exact analytic solution for non-linear density fluctuation in a ΛCDM universe

“…For the first time, we have derived the exact analytic solutions of the matter density and the velocity fluctuations in a ΛCDM universe, accounting for the nonlinear relativistic effects and greatly extending the results of [16] in the EdS universe. Our general approach to solving the nonlinear dynamical equations allows us to derive the solutions in a ΛCDM universe, in which the time-dependence of the solutions is more complicated than that in the EdS universe.…”

“…In this work, we derive the exact third-order analytic solution of the matter density and the velocity fluctuations in a ΛCDM universe, substantially extending the works in the Einstein-de Sitter (EdS) universe [15,16]. The matter density fluctuation is the dominant contribution to galaxy clustering on all scales, and there is no relativistic correction to it at the linear order in perturbation.…”

“…and compare to our analytic solutions in Section 3. Using the full ADM energy constraint equation (2.11) and taking the limit t → 0, we derive the nonlinear relation 15) and the solutions up to third order in perturbation in previous work are readily obtained as 16) identical to our analytic solution in Section 3. Using the relation to the second order in perturbations, we can repeat this exercise to recover the relativistic corrections in δ (3) 2 and κ…”

“…To implement this change in the perturbative approach in (5.6), we introduce a time-dependent third-order SPT kernel F δ2 16) such that the density fluctuation is 17) and similarly so for κ and G κ2 3 . Therefore, the additional terms in the algebraic equations (5.12) are…”