Abstract
The future detection projects of gravitational waves are expected to have the sensitivity of detecting the primordial gravitational waves, which may be useful to get new insight into the very early universe. It is essential to analyze the evolution equation of the gravitational waves to estimate the present field strength of the primordial gravitational waves. In this paper, we obtain analytic solutions of the gravitational wave equation in the presence of non-relativistic matters as well as the cosmological constant. Although it is difficult to obtain the solution directly, we find that the equation for the square of the amplitudes has a simple polynomial solution. This quantity, which is directly related to the energy density of the gravitational waves, turns out to be useful to construct analytic solutions for the amplitudes by using Weierstrass's elliptic functions.
Highlights
The direct detection of gravitational waves [1] opened an era of gravitational wave astronomy
We will show that for the transition from the matter-dominant era to the cosmological-constant-dominant era, the evolution of the square of the amplitude of gravitational waves can be expressed as a polynomial of the reciprocal of the scale factor a, which is used to construct exact solutions to the gravitational wave equation in terms of Weierstrass elliptic function ℘
We have shown that gravitational wave equation can be solved exactly in the presence of matters and the cosmological constant
Summary
The direct detection of gravitational waves [1] opened an era of gravitational wave astronomy. In the case of an equation with four definite singular points, which is called Heun’s equation (see for example [15]), we cannot usually get any analytical expression, so that it is difficult to expect the matching beyond the region of the convergence in the power series solutions. As we will see later, the generic evolution equation of the gravitational waves has six singular points, so that we do not expect to have any analytic solutions. We will show that for the transition from the matter-dominant era to the cosmological-constant-dominant era, the evolution of the square of the amplitude of gravitational waves can be expressed as a polynomial of the reciprocal of the scale factor a, which is used to construct exact solutions to the gravitational wave equation in terms of Weierstrass elliptic function ℘.
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