Gravitational Wave Astronomy

Daniel Williams

Contents

Part I
Gravitational Physics

Chapter 1
Overview of Gravitational Waves

 1.1 Polarisation
 1.2 Directionality
 1.3 Amplitude
 1.4 Frequency
 1.5 Luminosity

A gravitational wave is a propagating oscillation of a gravitational field, but because mass cannot be negative a gravitational wave cannot be the sole component of a gravitational field. Gravitational waves couple weakly to detectors, so the sources we attempt to observe must be highly luminous.

1.1 Polarisation

Through the equivalence principle we know that a single particle cannot probe a gravitational wave, and instead we measure inhomogeneities in the field by comparing the positions of two (or more) particles. Due to the tensoric nature of a gravitational wave the angle separating polarisation states is π ∕4  rather than π∕2  .



Figure 1.1: The + [Left] and × [Right] polarisation states of a gravitational wave.

The two polarisation states of a gravitational wave are denoted h+  and h× , and these are the primary time-dependent observables of the wave.

1.2 Directionality

Gravitational wave antennas have poor directional sensitivity, as they are quadrupolar detectors, and as a result multiple detectors are required to localise a detection, by measuring the differences in times-of-arrival. The current generation of ground-based detectors have baselines of around L ∼ 3 × 106m  . This corresponds to a resolution of 0.1rad  across the network at a frequency of 1Hz  .

For long-lasting sources we can attain much better resolution via annual parallax, with sub-arcsecond resolution, around 10− 6rad  .

Space-based detectors are hampered in this respect by the very low frequencies they work at; LISA would be limited to around 1rad  , but very high-SNR sources could be localised to around 1 arc-minute.

1.3 Amplitude

The lowest-order post-Newtonian approximation for gravitational radiation is the quadrupole order, depending only upon the density, ρ  , and the velocity fields of the Newtonian system. Defining a spatial tensor, Q
  jk  , which is the second moment of the mass distribution, as

      ∫
Q   =   ρx x d3x ,
  jk       j k
(1.1)

then the emitted amplitude of the gravitational wave is

      2d2Qjk-
hjk = r dt2 ,
(1.2)

which we may interpret as the amplitude of the wave at an infinite distance from the source, in (otehrwise) flat space, in the Lorentz gauge.

A typical component of the d2Q  ∕dt2
    jk  tensor will have a magnitude (M v2)
      nonsph  , which is twice the non-spherical part of the source’s kinetic energy, and so

    2(M v2)nonsph    2
h ≤ -----r------= 2vnonsphϕext

1.4 Frequency

In most cases the frequency of a gravitational wave will be related to the natural frequency of the self-gravitating body which produced it, defined as

ω = ∘ πG-¯ρ
 0
(1.3)

For ¯ρ  the mean mass-energy density of the source. This translates to a characteristic expression

       (    )1       (      )
     1--3M-- 2         10M⊙-
f0 = 4π   R3    ≈ 1kHz   M
(1.4)

1.5 Luminosity

The general formula for the stress-energy of a gravitational wave in the TT-gauge is given by the Isaacson expression,

          ⟨         ⟩
Tαβ = -1-- thtjk,αtth jk
      32π        ,β
(1.5)

In the quadrupole approximation the energy flux is integrated over a distant sphere to obtain

        (                )
L   = 1 ( ∑  .Q.. Q... − 1..Q.2)
  gw   5   j,k  jk jk   3
(1.6)

This equation is in natural units, so a conversion factor,       5           52
L0 = c ∕G = 3.6 × 10  W  is required. From this and equation (1.2) we can find the luminosity of radiation, ℱ ,

     || ||2
     |h˙|-
ℱ  ∼ 16π.
(1.7)

Part II
Detection

Part III
Astrophysics

Chapter 2
Sources of Gravitational Waves

2.1 Bursts from collapses

Both neutron stars and black holes form through gravitational collapse of a progenitor star or white dwarf. If the collapse is non-spherical some of the binding energy will be carried away as gravitational waves. Type II supernovae are believed to occur at a rate of between 0.1 and 0.01 per year in the Galaxy. There are so many unknown quantities in any given supernova collapse that numerical simulations are likely to provide poor templates for this type of event.

We can make a rough estimate of the amplitude for a supernova with energy E  , at a distance r  , producing a waveform at frequency f  for a time T  as

           (        )1∕2(    )1∕2
h ∼ 6 × 10− 21 ---E----     1ms-   .
             10− 7M ⊙      T
(2.1)

2.2 Pulsars