Newtonian versus relativistic nonlinear cosmology

Abstract

Both for the background world model and its linear perturbations Newtonian cosmology coincides with the zero-pressure limits of relativistic cosmology. However, such successes in Newtonian cosmology are not purely based on Newton's gravity, but are rather guided ones by previously known results in Einstein's theory. The action-at-a-distance nature of Newton's gravity requires further verification from Einstein's theory for its use in the large-scale nonlinear regimes. We study the domain of validity of the Newtonian cosmology by investigating weakly nonlinear regimes in relativistic cosmology assuming a zero-pressure and irrotational fluid. We show that, first, if we ignore the coupling with gravitational waves the Newtonian cosmology is exactly valid even to the second order in perturbation. Second, the pure relativistic correction terms start appearing from the third order.Third, the correction terms are independent of the horizon scale and are quite small in the large-scale near the horizon. These conclusions are based on our special (and proper) choice of variables and gauge conditions. In a complementary situation where the system is weakly relativistic but fully nonlinear (thus, far inside the horizon) we can employ the post-Newtonian approximation. We also show that in the large-scale structures the postNewtonian effects are quite small. As a consequence, now we can rely on the Newtonian gravity in analyzing the evolution of nonlinear large-scale structures even near the horizon volume.

Newtonian versus relativistic cosmology

“…We stress that our results do not contradict the claims about the exact Newtonian-relativistic correspondence of equations of motions written in terms of Bardeen variables in different foliations found in literature, e.g. by Hwang and Noh [11,12]. We have calculated the relativistic corrections as they appear in one specific and fixed foliation of space-time, i.e.…”

“…The relativistic-Newtonian correspondence up to second order found in [11] holds in the following way: The Newtonian density contrast is identified with a relativistic gauge-invariant expression that reduces to the density contrast on comoving hypersurfaces. However, the Newtonian gravitational and velocity-potentials are identified with gauge-invariant expressions which reduce to the relativistic potentials on zero-shear hypersurfaces.…”

“…The other strategy is based on the use of gaugeinvariant variables. Gauge-invariant quantities were first introduced by Bardeen in [6] (see also [5] for a more recent review) and are the basis for the work of Hwang and Noh, who found an exact correspondence up to second order between Newtonian and relativistic cosmology [11,12].…”