Third-order cosmological perturbations of zero-pressure multi-component fluids: puregeneral relativistic non-linear effects

Abstract

The present expansion stage of the universe is believed to be mainly governed bythe cosmological constant, collisionless dark matter and baryonic matter. Thelatter two components are often modeled as zero-pressure fluids. In our previouswork we have shown that to second order in the cosmological perturbations, therelativistic equations for the density and velocity perturbations of the zero-pressure,irrotational, multi-component fluids in a spatially near flat background withoutgravitational waves effectively coincide with the Newtonian equations. As the Newtonianequations only have quadratic order non-linearity, it is of practical interest toderive the third-order perturbation terms in a general relativistic treatment, whichcorrespond to pure general relativistic corrections. In our previous work we haveshown that even in a single-component fluid there exist a substantial numberof pure relativistic third-order correction terms. We have, however, shown thatthose correction terms are independent of the horizon scale, and are quite small(∼5 × 10−5 smaller compared with the relativistic/Newtonian second-order terms) near thehorizon scale due to the weak level anisotropy of the cosmic microwave backgroundradiation. Here, we present pure general relativistic correction terms appearing inthe third-order perturbations of the multi-component zero-pressure fluids. Theforms of the pure general relativistic correction terms are quite similar to theones in a single-component situation. The third-order correction terms involveonly the ‘linear order spatial curvature perturbation in the comoving gauge’φv which has the order of the ‘perturbed Newtonian gravitational potential divided byc2 ’, and thus is small on nearly all scales. Consequently, we show that, as ina single-component situation, the third-order correction terms are quite small(∼5 × 10−5 smaller) near the horizon, and independent of the horizon scale. We emphasize that theseresults are based on our proper choice of perturbation variables and gauge conditions fordescribing the relativistic perturbations. Still, there do exist a substantial number of puregeneral relativistic correction terms in third-order perturbations which couldpotentially become important in future development of precision cosmology. Althoughφv is small on nearly all scales, our third-order corrections are applicable only in weaklynon-linear regimes where perturbation analysis is viable. We include the cosmologicalconstant in all our analyses.