2.6 Luminosity in gravitational waves

The general formula for the local stress-energy of a gravitational wave field propagating through flat spacetime, using the TT-gauge, is given by the Isaacson expression [262Jump To The Next Citation Point335]
1 ⟨ TT TTjk⟩ Tαβ = ---- hjk,αh ,β , (14 ) 32π
where the angle brackets denote averages over regions of the size of a wavelength and times of the length of a period of the wave. The energy flux of a wave in the xi direction is the T 0i component.

The gravitational wave luminosity in the quadrupole approximation is obtained by integrating the energy flux from Equation (14View Equation) over a distant sphere. When one correctly takes into account the projection factors mentioned after Equation (2View Equation), one obtains [262Jump To The Next Citation Point]

( ) 1- ∑ ... ... 1-...2 Lgw = 5 QjkQjk − 3 Q , (15 ) j,k
where Q is the trace of the matrix Qjk. This equation can be used to estimate the backreaction effect on a system that emits gravitational radiation.

Notice that the expression for Lgw is dimensionless when c = G = 1. It can be converted to normal luminosity units by multiplying by the scale factor

5 52 L0 = c ∕G = 3.6 × 10 W. (16 )
This is an enormous luminosity. By comparison, the luminosity of the sun is only 3.8 × 1026 W, and that of a typical galaxy would be 1037 W. All the galaxies in the visible universe emit, in visible light, on the order of 1049 W. We will see that gravitational wave systems always emit at a fraction of L0, but that the gravitational wave luminosity can come close to L0 and can greatly exceed typical electromagnetic luminosities. Close binary systems normally radiate much more energy in gravitational waves than in light. Black hole mergers can, during their peak few cycles, compete in luminosity with the steady luminosity of the entire universe!

Combining Equations (2View Equation) and (15View Equation) one can derive a simple expression for the apparent luminosity of radiation ℱ, at great distances from the source, in terms of the gravitational wave amplitude [335]:

|˙h|2 ℱ ∼ 16 π. (17 )
The above relation can be used to make an order-of-magnitude estimate of the gravitational wave amplitude from a knowledge of the rate at which energy is emitted by a source in the form of gravitational waves. If a source at a distance r radiates away energy E in a time T, predominantly at a frequency f, then writing ˙h = 2πf h and noting that ℱ ∼ E∕(4πr2T ), the amplitude of gravitational waves is
∘ --- h ∼ -1-- E-. (18 ) πfr T
When the time development of a signal is known, one can filter the detector output through a copy of the expected signal (see Section 5 on matched filtering). This leads to an enhancement in the SNR, as compared to its narrow-band value, by roughly the square root of the number of cycles the signal spends in the detector band. It is useful, therefore, to define an effective amplitude of a signal, which is a better measure of its detectability than its raw amplitude:
√-- heff ≡ nh. (19 )
Now, a signal lasting for a time T around a frequency f would produce n ≃ f T cycles. Using this we can eliminate T from Equation (18View Equation) and get the effective amplitude of the signal in terms of the energy, the emitted frequency and the distance to the source:
∘ --- heff ∼ -1- E-. (20 ) πr f
Notice that this depends on the energy only through the total fluence, or time-integrated flux E ∕4πr2 of the wave. As in many other branches of astronomy, the detectability of a source is ultimately a function of its apparent luminosity and the observing time. However, one should not ignore the dependence on frequency in this formula. Two sources with the same fluence are not equally easy to detect if they are at different frequencies: higher frequency signals have smaller amplitudes.

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