Exact analytical solution of the linear structure growth rate inΛCDMcosmology and its cosmological applications

Abstract

We derive the exact analytical solution of the linear structure growth rate in $\ensuremath{\Lambda}\mathrm{CDM}$ cosmology with flat or curved geometry, under the Newtonian gauge. Unlike the well known solution under the Newtonian limit [D. J. Heath, Mon. Not. R. Astron. Soc. 179, 351 (1977)], our solution takes all general relativistic corrections into account and is hence valid at both the sub- and superhorizon scales. With this exact solution, we evaluate cosmological impacts induced by these relativistic corrections. (1) General relativistic corrections alter the density growth from $z=100$ to $z=0$ by 10% at $k=0.01\text{ }\text{ }h/\mathrm{Mpc}$ and the impact becomes stronger toward larger scales. We caution the readers that the overdensity is not gauge invariant and the above statement is restrained to the Newtonian gauge. (2) Relativistic corrections introduce a ${k}^{\ensuremath{-}2}$ scale dependence in the density fluctuation. It mimics a primordial non-Gaussianity of the local type with ${f}_{\mathrm{NL}}^{\mathrm{local}}\ensuremath{\sim}1$. This systematical error may become non-negligible for future all sky deep galaxy surveys. (3) Cosmological simulations with box size greater than 1 Gpc are also affected by these relativistic corrections. We provide a postprocessing recipe to correct for these effects. (4) These relativistic corrections affect the redshift distortion. However, at redshifts and scales relevant to redshift distortion measurements, such effect is negligible.