In the conventional adiabatic regularization the vacuum ultraviolet divergences of a quantum field in curved spacetime are removed by subtracting the k-mode of the stress tensor to the 4th-order. For a scalar field in de Sitter space, we find that the 4th-order regularized spectral energy density is negative. Moreover, the 2nd-order regularization for minimal coupling (ξ = 0) and the 0th-order regularization for conformal coupling (ξ = 1 6 ) yield a positive and UV-convergent spectral energy density and power spectrum. The regularized stress tensor in the vacuum is maximally symmetric and can drive inflation, while its k-modes representing the primordial fluctuations are nonuniformly distributed. Conventional regularization of a Green's function in position space is generally plagued by a log IR divergence. Only in the massless case with ξ = 0 or 1 6 , we can directly regularize the Green's functions and obtain vanishing results that agree with the adiabatic regularization results. In this case, the regularized power spectrum and stress tensor are both zero, and no trace anomaly exists. To overcome the log IR divergence problem in the massive cases with ξ = 0 and 1 6 , we perform Fourier transformation of the regularized power spectra and obtain the regularized analytical Green's functions which are IR-and UV-convergent. inflationary universe, mathematical and relativistic aspects of cosmology, quantum fields in curved spacetimes PACS numbers: 98.80.Cq , 98.80.Jk , 04.62.

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.

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With the continuous upgrade of detectors, greater numbers of gravitational wave (GW) events have been captured by the LIGO Scientific Collaboration and Virgo Collaboration (LVC), which offer a new avenue to test general relativity and explore the nature of gravity. Although various model-independent tests have been performed by LVC in previous works, it is still interesting to ask what constraints can be placed on specific models by current GW observations. In this work, we focus on three models of scalar-tensor theories, the Brans–Dicke theory (BD), the theory with scalarization phenomena proposed by Damour and Esposito-Farèse (DEF), and screened modified gravity (SMG). Of the four possible neutron star–black hole events that have occurred so far, we use two of them to place constraints. The other two are excluded in this work because of possible unphysical deviations. We consider the inspiral range with the cutoff frequency at the innermost stable circular orbit and add a modification of dipole radiation into the waveform template. The scalar charges of neutron stars in the dipole term are derived by solving the Tolman–Oppenheimer–Volkoff equations for different equations of state. The constraints are obtained by performing the full Bayesian inference with the help of the open source software Bilby. The results show that the constraints given by GWs are comparable to those given by pulsar timing experiments for DEF theory, but are not competitive with the current solar system constraints for BD and SMG theories.