Second-order cosmological perturbations. II. Produced by scalar-tensor and tensor-tensor couplings

Abstract

We study the second-order perturbations in the Einstein-de Sitter Universe in synchronous coordinates. We solve the second-order perturbed Einstein equation with scalar-tensor, and tensortensor couplings between 1st-order perturbations, and obtain, for each coupling, the solutions of scalar, vector, and tensor metric perturbations, including both the growing and decaying modes for general initial conditions. We perform general synchronous-to-synchronous gauge transformations up to 2nd order, which are generated by a 1st-order vector field and a 2nd-order vector field, and obtain all the residual gauge modes of the 2nd-order metric perturbations in synchronous coordinates. We show that only the 2nd-order vector field is effective for the 2nd-order transformations that we consider because the 1st-order vector field was already fixed in obtaining the 1st-order perturbations. In particular, the 2nd-order tensor is invariant under 2nd-order gauge transformations using ξ (2)µ only, just like the 1st-order tensor is invariant under 1st-order transformations.

Second-order cosmological perturbations. IV. Produced by scalar-tensor and tensor-tensor couplings during the radiation dominated stage

“…Here we give the basic setups of this paper, using the same notations as in Refs. [60][61][62][63]. A flat Robertson-Walker (RW) metric in synchronous coordinates is given by ds 2 = g µν dx µ dx ν = a 2 (τ )[−dτ 2 + γ ij dx i dx j ],…”

“…On the other hand, the tensor modes (2.28) are waves propagating at the speed of light. In the MD stage [60][61][62], the scalar, and density contrast are not waves and do not propagate; only the tensor modes still propagate at the speed of light. Therefore,whether or not the scalar modes propagate actually depends on the background matter; nevertheless, the tensor modes always propagate at speed of light, regardless of the background matter.…”

“…So far, the solutions for the 2nd-order perturbations have all been given. The expressions of the 2nd-order solutions for the RD stage generally contain many more terms than those for the MD stage [61,62].…”

“…In our previous works, we extended the study of Refs. [59,60], including the scalar-scalar [61], scalartensor, and tensor-tensor couplings [62]; obtained all the 2nd-order solutions of the scalar, vector, tensor, and density perturbations during the MD stage; identified the residual gauge modes under synchronous-to-synchronous transformations, and also presented the 2nd-order transformation from synchronous to Poisson coordinates. In Ref.…”

Adiabatic regularization of power spectrum and stress tensor of relic gravitational wave without low-frequency distortion

“…The second order RGW [33,34] is not considered here. We emphasize that, at the level of first order metric perturbations, the RGW equation is homogeneous, and these k-modes of RGW are independent, as they do not couple to each other, and no energy exchanges between different k-modes.…”