Analytical Solutions For Second-Order Cosmological Perturbations And Study Of Relic Gravitational Waves

In this dissertation,we study cosmological perturbations and cosmological relic gravi-tational waves.We perform detailed studies on three aspects:1.analytical solutions of 2nd-order cosmological perturbations;2.estimation of spectrum and parameters of relic gravita-tional wave(RGW)using space-borne interferometers;3.adiabatic regularization of power spectrum and stress tensor of RGW.These three aspects belong to three different fields of research respectively:perturbation theory of general relativity,statistical analysis of weak signals,quantum field theory in curved spacetime.Since they are based on different theoret-ical frameworks,we apply different methods of study on them.1.Within the framework of General Relativity,in the flat Robertson-Walker spacetime,we present a systematic study of the 2nd-order scalar,vector,tensor metric perturbations and the 2nd-order energy,pressure,velocity perturbations of matter for matter-dominated and radiation-dominated stages respectively.By omitting the 1st-order vector modes,there are three kinds of 1st-order coupling terms:scalar-scalar,scalar-tensor and tensor-tensor,which serve as effective sources of the 2nd-order perturbations.We obtain the complete analytical solutions of the 2nd-order perturbations for each kind of the coupling types.Note that the 2nd-order solutions by scalar-scalar couplings in the MD stage can be expressed as powers of time,yet the 2nd-order solutions in other cases cannot and contain integrations which require numerical method.We perform the residual gauge transformation from synchronous to syn-chronous coordinates,and identify the residual gauge modes in the analytical solutions.We also transform the solutions in synchronous coordinate to Poisson coordinate,and obtain the 1st-and 2nd-order analytical solutions in the Poisson coordinate.2.RGW is a stochastic background of fluctuations in the Universe.We study the de-tection of RGW and the estimation of its spectrum and parameters by using space-borne in-terferometers,such as LISA.RGW can be described by three parameters:tensor-scalar ratio,spectral index and running spectral index.Since the Michelson interferometer has higher sen-sitivity,we adopt a single and a pair of it respectively to analyze the detection capability.For one-year detection under the observed-inferred RGW model,the sensitivity of a pair is higher than a single by around 2 orders of magnitude,and higher than Advanced LIGO by 4-5 orders of magnitude.We find that when noise is dominant,a single cannot properly estimate RGW spectrum.With a pair of interferometers,we use cross-correlated integrated signals to esti-mate one RGW parameter and give the signal-to-noise ratio.However,we cannot estimate the spectrum by this method either.Regarding of this,we propose the sample averaging method to estimate the spectrum.We also use correlation for un-integrated signals as a third method to estimate RGW spectrum and three parameters.For all three approaches,we adopt maxi-mum likelihood estimation to give the estimation formulae for the spectrum and parameters.We also provide numerical simulations to illustrate the feasibility.3.Quantum fields in curved spacetime like RGW generated during inflation contain UV divergences.To remove these divergences,we propose the inside-horizon regularization on RGW spectra during inflation.This scheme does not distort the low-frequency region nor bring IR divergences which happen in usual regularization.This scheme also does not violate the energy conservation due to the independence between different k-modes of RGW in linear order.We obtain the regularized spectrum,energy density and pressure of RGW at the end of inflation.Then we take them as initial conditions and evolve them into the present acceleration stage.We find that the results at present contain no UV divergences and free of low-frequency distortion.We also try regularization schemes at the present stage.However,we find that their outcomes contain abrupt drops.We also find that the RGW spectra generally exhibit quick oscillations in frequency domain,even if the initial spectra during inflation have no oscillations.This pattern is due to the interference between the positive and negative frequency modes developed during cosmic expansion,and may be probed by future RGW detections.