Second-Order Cosmological Perturbation

The cosmological perturbation,which can be divided into three kinds and known as the scalar perturbation and the vector perturbation as well as the tensor perturbation,can cause influence on the gravitational waves in our universe.Knowing the physics nature of these influence will be very helpful for the scientists to detect the gravitational waves.So,it is very important for us to find the evolution equations of these influence in order to understand their nature.Here,we only consider the irrotational case,in which the vector perturbation rep-resents the remaining gauge mode in the synchronous gauge and can be set to zero.So,our research is only to find the second order gravitational waves equations to describe the scalar perturbation ’ s influence and on the tensor perturbation ’ s influence on the gravitational waves.Our methods is to rewrite the Einstein Equation into the extrinsic curvature form in the co-moving coordinates to obtain the energy constraint and the momentum constraint as well as the evolution equation,and then expend the metric into series form and plug-ging these series form of metric into the energy constraint,the momentum constraint and the evolution equation to obtain the solution of the first order metric perturbation.After that,we will based on the first order metric perturbation to obtain the second order gravitational waves equations by plugging these series form of metric,which is up to the second order,into the energy constraint,the momentum constraint and the evolution equation.In our research,we have found that the cosmological perturbation’ s influence on the gravitational waves are caused by three factors,i.e.the scalar perturbation,the tensor perturbation and the coupling between scalar perturbation and tensor perturbation.