The most likely signal to be detected using a pulsar timing array is a stochastic background formed

Question:

The most likely signal to be detected using a pulsar timing array is a stochastic background formed by perhaps billions of binary black holes in the nuclei of galaxies.

(a) For simplicity, suppose that these binaries are all on circular orbits and that they lose energy at a rate given by Eqs. (27.73). Show that for each binary the gravitational wave energy radiated per unit log wave frequency is where M is the chirp mass given by Eq. (27.76c) and E is its orbital energy given by −1/2μM/a.

(b) Suppose that, on astrophysical grounds, binaries only radiate gravitational radiation efficiently if they merge in less than t_merge ∼ 300Myr. Show that this requires

This evaluates to ∼1million solar masses for f ∼ 30 nHz. By contrast, measured masses of the large black holes in the nuclei of galaxies range from about 4million solar masses as in our own modest spiral galaxy to perhaps 20 billion solar masses in the largest galaxies observed.

(c) Now suppose that these black holes grew mostly through mergers of holes with very different masses. Show that the total energy radiated per unit log frequency over the course of the many mergers that led to a black hole with mass M is

(d) The number density per unit mass dn/dM today of these merging black holes has been estimated to be

where ρ_BH ∼ 2 × 10⁻¹⁵ J m⁻³ is the contribution of black holes (i.e., of their masses) to the average energy density in the local universe and Mt = 5 × 10⁷ solar masses ∼10³⁸ kg. Show that

This is an overestimate, because most of the black holes were likely assembled in the past, when the universe was about three times smaller than it is today. If one thinks of the gravitational waves as gravitons that lose energy as the universe expands, then this energy density should be reduced by a factor of three. Furthermore, as the black holes are thought to grow through accretion of gas and not by mergers, this estimate should be considered an upper bound.

(e) Making the same assumptions as in the previous exercise, determine whether it will be possible to detect this background in 5 years of observation.

Equations.