Pre-Phase A Report - Lisa - Nasa

34 Chapter 1 Scientific Objectives
test the predictions of general relativity very accurately in extremely strong fields. The
orbital velocity near periapsis is roughly 0.5 c, and the period for the relativistic precession
of periapsis is similar to the period for radial motion. In addition, if the MBH is rapidly
rotating, the orbital plane will rapidly precess. In view of the complexity of the orbits, the
number of parameter values to be searched for, and the expected evolution of the orbit
parameters, the SNR needed to detect the signals reliably probably will be about 10.
If these events are observed, then each one will tell us the mass and spin of the central
MBH, as well as its distance and position. The ensemble of events will give us some
indication of the numbers of such black holes out to z ∼ 1, and they will give us useful
information about the MBH population, particularly the distribution of masses and spins.
1.2.3 Primordial gravitational waves
Just as the cosmic microwave background is left over from the Big Bang, so too should
there be a background of gravitational waves. If, just after the Big Bang, gravitational
radiation were in thermal equilibrium with the other fields, then today its temperature
would have been redshifted to about 0.9 K. This radiation peaks, as does the microwave
radiation, at frequencies above 10 10 Hz. At frequencies accessible to LISA, or indeed even
to ground-based detectors, this radiation has negligible amplitude. So if LISA sees a
primordial background, it will be non-thermal.
Unlike electromagnetic waves, gravitational waves do not interact with matter after a few
Planck times (10−45 s) after the Big Bang, so they do not thermalize. Their spectrum
today, therefore, is simply a redshifted version of the spectrum they formed with, and
non-thermal spectra are probably the rule rather than the exception for processes that
produce gravitational waves in the early universe.
The conventional dimensionless measure of the spectrum of primordial gravitational waves
is the energy density per unit logarithmic frequency, as a fraction of the critical density
to close the Universe, ρc :
ωGW(f) = f dρGW . (1.17)
ρc df
The background radiation consists of a huge number of incoherent waves arriving from
all directions and with all frequencies; it can only be described statistically. The rms
amplitude of the fluctuating gravitational wave in a bandwidth f about a frequency f is
hrms(f, ∆f =f) = 10 −15 [ΩGW(f)] 1/2
1mHz H0
f 75 km s−1 Mpc−1
, (1.18)
where H0 is the present value of Hubble’s constant. That this seems to be large in LISA’s
band is deceptive: we really need to compare this with LISA’s instrumental noise, and
this is best done over the much narrower bandwidth of the frequency resolution of a 1 yr
observation, 3×10−8 Hz. Since the noise, being stochastic, scales as the square root of
the bandwidth, this gives us the relation
1/2 3/2 ΩGW(f) 1mHz
hrms(f, ∆f =3×10 −8 Hz) = 5.5×10 −22
×
10 −8
H0
75 km s −1 Mpc −1
f
, (1.19)
3-3-1999 9:33 Corrected version 2.08