Consequences of symmetries and consistency relations in the large-scale structure of the universe for non-local bias and modified gravity

Abstract

Consistency relations involving the soft limit of the (n+1)correlation functions of dark matter and galaxy overdensities can be obtained, both in real and redshift space, thanks to the symmetries enjoyed by the Newtonian equations of motion describing the dark matter and galaxy fluids coupled through gravity. We study the implications of such symmetries for the theory of galaxy bias and for the theories of modified gravity. We find that the invariance of the fluid equations under a coordinate transformation that induces a long-wavelength velocity constrains the bias to depend only on a set of invariants, while the symmetry of such equations under Lifshitz scalings in the case of matter domination allows one to compute the time-dependence of the coefficients in the bias expansion. We also find that theories of modified gravity which violate the equivalence principle induce a violation of the consistency relation which may be a signature for their observation. Thus, given adiabatic Gaussian initial conditions, the observation of a deviation from the consistency relation for galaxies would signal a breakdown of the so-called non-local Eulerian bias model or the violation of the equivalence principle in the underlying theory of gravity

Proper-time hypersurface of nonrelativistic matter flows: Galaxy bias in general relativity

“…Beyond the linear order in Newtonian dynamics, local biasing models (e.g., [47,48]) are frequently used, in which the galaxy number density fluctuation is a nonlinear function of the matter density fluctuation to be expanded in a Taylor series. At higher order, however, additional non-local terms can be included in biasing [49], and it was shown (e.g., [50][51][52][53]) that consistent renormalization of galaxy bias requires the presence of non-local derivative terms in addition to the local terms. At the second order, the additional non-local term is the contraction s 2 = s ij s ij of the gravitational tidal tensor [49], and its evidence was measured [54] in simulations, where…”

Angular-averaged consistency relations of large-scale structures

“…Some exact results have recently been obtained [24][25][26][27][28][29][30] in the form of "kinematic consistency relations." They relate the (l þ n)-density correlation, with l large-scale wave numbers and n small-scale wave numbers, to the n-point small-scale density correlation, with l prefactors that involve the linear power spectrum at the large-scale wave numbers.…”

Consistency relations for large-scale structures: Applications for the integrated Sachs-Wolfe effect and the kinematic Sunyaev-Zeldovich effect

“…As described in recent works (Kehagias & Riotto 2013;Peloso & Pietroni 2013;Creminelli et al 2013;Kehagias et al 2014a;Peloso & Pietroni 2014;Creminelli et al 2014;Valageas 2014b;Horn et al 2014Horn et al , 2015, it is possible to obtain exact relations between density correlations of different orders in the squeezed limit, where some of the wavenumbers are in the linear regime and far below the other modes that may be strongly nonlinear. These "kinematic consistency relations", obtained at the leading order over the large-scale wavenumbers, arise from the equivalence principle and the assumption of Gaussian primordial perturbations.…”

“…The reader may then note that redshift-space distortions (RSD) also involve velocities, but previous works that studied the galaxy density field in redshift space (Creminelli et al 2014;Kehagias et al 2014a) found that there is no equal-time effect, as in the real-space case. Indeed, in both real space and redshift space, the long mode only generates a uniform change of coordinate (in the redshift-space case, this shift involves the radial velocity).…”

“…Recently, some exact results have been obtained (Kehagias & Riotto 2013;Peloso & Pietroni 2013;Creminelli et al 2013;Kehagias et al 2014a;Peloso & Pietroni 2014;Creminelli et al 2014;Valageas 2014b;Horn et al 2014Horn et al , 2015 in the form of "kinematic consistency relations". They relate the ( + n)-density correlation, with large-scale wave numbers and n small-scale wave numbers, to the n-point small-scale density correlation.…”