Amplitude of gravitational waves at emission and how they decay

Question

I am a bit confused with some aspects of gravitational waves (GW). GW we observe here on Earth with LIGO/Virgo have an amplitude of $\approx 10^{-21}$ with peak frequency $\nu = 150\, Hz$. If we assume that the strain from two black holes with same mass merging goes as \begin{align} h &\approx 2\frac{GM}{c^2}*\frac{1}{r}*\left(\frac{v}{c}\right)^2 \\ &\approx 3 \times 10^{4} m * 10^{-25} m^{-1} * (0.25)^2 \\ &\approx 2 \times 10^{-22} \;, \end{align} we obtain a pretty good approximation. What I want to know is:

1. since GW decay as 1/r, can we assume that the amplitude at emission $h_{em}$ is roughly obtained assuming $r=2GM/c^2$ (i.e. produced near the Schwarzschild radius of the final BH), so that \begin{align} h_{em} &\approx \left(\frac{v}{c}\right)^2 \\ &\approx 10^{-2} \;? \end{align}
2. GW decay should also depend on time (as hinted, for example, here). But after spending a few hours looking in the literature, I haven't found any derivation showing both time and distance decays of the strain. Does anyone have references on the subject?
3. the physical process leading to astrophysical GW is much less energetic than, say, inflation. So, when we say primordial GW (PGW) are harder to detect, is it only due to the decay with distance (and/or time) that is much greater?
4. The period of an astrophysical GW depends on the radius R of the orbit as $T_{GW}=2\pi R/\nu$. This formula does not make sense for PGW. Is there a way to obtain the period of a PGW?

Cheers!

Answer 1

1. Sort of, although the approximation you are using has been derived from gravitational waves in the "weak field limit" and would not be accurate close to the source of the gravitational waves. Only difficult numerical simulations can follow the non-linear, General Relativistic distortion of spacetime close to a merging black hole system.

2. GWs interact extremely weakly with matter and are expected to propagate right across the observable universe with negligible dissipation. That is why they are hard to detect. I think the answer you refer to is suggesting that GWs are subject to the same phenomena as light in terms of gravitational interactions like lensing and Shapiro delay and of course redshift caused by the expansion of the universe.

3. The GWs from merging binary systems have a very distinct pattern of rising amplitude and frequency versus time, culminating in a "chirp". That means you can pick them out more easily (though easily is a comparative term) from the morass of noise that totally swamps all genuine signals in a gravitational wave detector. The GWs also come from a particular direction and with a particular polarisation, which means that multiple detectors can make source identification much more secure. In contrast, primordial GWs are expected over a broad continuum of frequencies with low amplitudes. They basically have the appearance of another source of noise in a GW detector and that is going to make them very hard to separate.

An audio analogy is appropriate. It is much easier to pick out a chirping bird amongst a general hum of say traffic noise than it would be to pick out the pseudo-white noise contributed by a breeze rustling the leaves in the tree the bird is in.

4. See 3. No periodicity is expected.