Propagation of gravitational waves in an expanding background in the presence of a point mass

We solve the Laplace equation $\square h_{ij}=0$ describing the propagation of gravitational waves in an expanding background metric with a power law scale factor in the presence of a point mass in the weak-field approximation (Newtonian McVittie background). We use boundary conditions at large distances from the mass corresponding to a standing spherical gravitational wave in an expanding background which is equivalent to a linear combination of an incoming and an outgoing propagating gravitational wave. We compare the solution with the corresponding solution in the absence of the point mass and show that the point mass increases the amplitude of the wave and also decreases its frequency (as observed by an observer at infinity) in accordance with gravitational time delay.

Figure 1

(a) A superposition of the analytic solution with the numerical simulation initial condition taken at $\tau =1$. The spherical wave with $l=6$ and scale factor $a(\tau )=\tau$ is shown. (b) The numerically evolved solution (dashed red line) for the gravitational wave $Q(\tau ,\rho )$ at $\tau =16.5$ with $R_s=0$ is in excellent agreement with the corresponding analytic evolved solution (blue line). This is a test of the quality of the numerical solution.

Figure 2

The evolution of the profile of a partial spherical gravitational wave with $l=6$, $R_s=5$ (red dashed line) in comparison with the corresponding free solution ($R_s=0$, blue continuous line). The free wave reaches its maximum first (b), while the wave in the presence of the point mass shows a delay in reaching its maximum (c) due to gravitational redshift. The wave in the presence of the mass has a higher amplitude [compare (b) with (c)] in the vicinity of the mass, and the phase difference increases with time in the vicinity of the mass (d).

Figure 3

The time evolution of the first spatial maximum (at $\rho =7.9$) of the partial spherical wave with $l=6$, $R_s=5$ (red line) in comparison with the corresponding free solution ($R_s=0$, blue continuous line) at the same spatial point. The free wave reaches its maximum first [Fig. 2], while the wave in the presence of the point mass shows a delay in reaching its maximum [Fig. 2(c)] due to gravitational redshift. The wave in the presence of the mass has an amplitude that increases with time, as indicated with the dashed red line that is tangent to the gravitational wave maxima. As expected, the product $a(\tau )Q(\tau )$ is constant for the free wave in an expanding background.

Figure 4

The time power spectra of the gravitational wave in the presence (red line) and in the absence (blue line) of the mass. Notice that lower frequencies have a higher amplitude for the wave in the presence of the mass, as expected due to the gravitational time delay.

Figure 5

The relative difference of the wave periods $\mathrm{\Delta }T/T_0$, where $T$ is the period in the presence of mass, and $T_0$ is the period in the absence of mass, as a function of $R_s$. It is clear that as the value of the variable $\rho$ increases, the statistical slope of the curve decreases. This is an anticipated result due to theoretical slope of the curve $\frac{1}{2a\rho }$.

Figure 6

The ratio of the amplitudes of the waves $A/A_0$ in the presence of a mass ($A$) and in the absence of the mass ($A_0$) as a function of the parameter $R_s$, when $\rho =7.9$ and $\rho =15.89$.