Astrophysical sources of
gravitational waves
Gravitational waves
Linearized general relativity: Einstein equations become a wave equation
Gravitational radiation is primarily generated by time-dependent mass
quadrupole moment:
Nearby point particles in free fall, small separation
 Tidal effect:
h
"plus" and "cross" polarizations:
+
h
x
Gravitational wave detectors
Ground-based
- LIGO
- Virgo
(- Einstein Telescope)
Space-based
- LISA
Sources for ground-based detectors
Coalescing binaries
Neutron stars in (wide) binary systems
Young neutron stars
Supernovae
Sources for LISA
Galactic white dwarfs
Supermassive binary black holes
Primordial backgrounds
Capture orbits
Isolated neutron stars
Mostly composed of neutrons in superfluid state
Crust composed of dense but ordinary matter
Magnetic field
 Precessing magnetic dipole
- Emission of EM radiation
- Carries away rotational energy
- Star becomes less oblate
- Cracking of the crust
"Mountains" (~0.1 mm)
 Asymmetry
 Varying mass quadrupole moment
Gravitational waves from neutron stars
Model neutron stars as rigid bodies
Characterize by inertia tensor
Angular momentum and rotational kinetic energy:
"Body frame": diagonalizes inertia tensor
Principal moments of inertia:
Gravitational waves from neutron stars
Neutron stars are almost oblate:
Assume rotation around
Body frame rotates along with star
Inertia tensor in body frame:
Inertia tensor in stationary frame:
Components will be time-dependent
linear combinations of
Compare inertia tensor with quadrupole tensor:
... hence
... and
Gravitational waves from neutron stars
Result:
Want gravitational radiation in
arbitrary direction
Rotate around x axis over angle  :
Gravitational waves from neutron stars
Substitute into quadrupole expressions:
... to arrive at:
Define ellipticity by
Define "gravitational wave frequency"
Gravitational waves from neutron stars
Write polarizations as
... where
Typical neutron star:
- Mass
- Radius

- Frequency ~ 1 kHz (higher end)
- Located near center of galaxy, r ~ 10 kpc
- Ellipticity unknown;
possible
Gravitational waves from neutron stars
Energy emitted in gravitational waves:
Effect on
if no other braking mechanisms:
... hence
Observations:
with n = 2 - 3 , hence GW not dominant
LIGO result for the Crab pulsar:
Not more than 4% of energy emitted in GW
Gravitational waves from neutron stars
What can be learned?
Neutron star structure
poorly understood:
Structure of the crust?
Equation of state?
Superfluid interior
"Pinning of fluid vortices to crust
Origin of magnetic field?
More exotic objects?
(Strange quark stars?)
Gravitational waves from neutron stars
Crustal structure:
- Spin-down causes cracking of the
crust; "mountains"
- Strength of continuouis GW signal can
shed light
Pulsar "glitches":
- Fluid modes get excited; causes
transient GW signal
- Learn about equation of state of
dense nuclear matter
NS in low-mass X-ray binaries:
- Why NS spin frequencies
clustered around 350 Hz?
Accretion will cause
inhomogeneities; GW carry away
angular momentum?
- From X-ray and GW: study fluid
instabilities which only exist in GR
Gravitational waves from coalescing binaries
2 point particles at large distance R orbiting center of mass:
... where
... and reduced mass
Quadrupole moment, non-relativistic limit:
... with density
Gravitational waves from coalescing binaries
Quadrupole moment:
Insert into expressions for polarizations:
Result:
Equating centripetal and gravitational force:

Gravitational waves from coalescing binaries
Define chirp mass:
... then
Power emitted:
Orbital energy:
... and if this is to become more negative then R must decrease
Gravitational waves from coalescing binaries
R must decrease
One has
... so that
Hence if
, inwards radial motion small compared to tangential motion
Quasi-circular, adiabatic inspiral
Gravitational waves from coalescing binaries
Use Kepler's law again:
Energy balance:
... then
In the limit where
,
: innermost stable circular orbit
Rule of thumb for termination frequency also for comparable masses:
Gravitational waves from coalescing binaries
Orbital evolution with energy loss:
Previous expressions for polarizations:
Need to re-calculate derivatives of quadrupole moment
- Neglect
- Frequency won't change much over single orbit, so neglect
- In the amplitudes, replace
by
- Inside the cosine and sine, replace
by
Gravitational waves from coalescing binaries
Waveforms with orbital energy loss taken into account:
Frequency evolution:
... can be substituted into
... to find the GW phase:
What kinds of sources suitable for studying inspiral signal?
Gravitational waves from coalescing binaries
Coalescing binaries are short duration sources:
... can be inverted to obtain
Binary neutron stars:
... hence in band for just under 3 minutes
binary black hole:
Most massive system that can be seen has
Number of cycles in band:
; in band for 19 sec
,
Gravitational waves from coalescing binaries
Detector response:
... and we know
... so
For matched filtering we need Fourier transform:
Stationary phase approximation:
Keep only the dominant contributions
Gravitational waves from coalescing binaries
Fourier transform:
Largest contribution from t = ts where
Expand exponent:
... and define
Integral is approximately equal to
... and
Gravitational waves from coalescing binaries
Result:
We have expressions for
and hence
where
In particular,
Putting everything together (and recalling angle-dependent prefactor):
Gravitational waves from coalescing binaries
Signal-to-noise ratio:
With the stationary phase approximation:
Largest if source "face-on" (
) and at zenith or nadir (
):
For a given system, useful to know average over sky position and orientation:
... so that
Given a minimum
needed for detection, angle-averaged distance reach:
Gravitational waves from coalescing binaries
Angle-averaged distance reach (minimum S/N = 5):
Black: Advanced LIGO
Red: Initial LIGO
Green: Initial Virgo
Gravitational waves from coalescing binaries
LISA:
- Sensitivity band [10-4 , 0.1 ] Hz
- Will see binary supermassive black holes, masses 106 - 109 Msun
- Signals in band for months!
- Any such event in LISA's past light cone will be picked up
- Expected 20 - 80 yr-1
Gravitational waves from coalescing binaries
What happens after inspiral?
Well-understood:
Numerical
Perturbations
Perturbation theory,
Expansions in (v/c)
simulations
of Kerr black hole
Inspiraling binary black holes
Spin-orbit interactions:
Precession of the orbital plane
 Modulation of the waveform
Going beyond quadrupole: higher
signal harmonics
- Sub-dominant by powers of v/c
- Can be picked up if sufficient S/N
Amplitude
- Important for asymmetric, high mass
systems
Binary black holes are
"laboratories" for testing GR
First-ever tests of strong-field
dynamics of gravity
Time
Merging neutron stars
Many possible equations of
state (EOS)
Extremes:
- "Soft" EOS: prompt collapse to a
black hole
- "Hard" EOS: unstable bar mode;
eventually BH
Merging neutron stars
Many possible equations of
state (EOS)
Extremes:
- "Soft" EOS: prompt collapse to a
black hole
- "Hard" EOS: unstable bar mode;
eventually BH
Merging neutron stars
Many possible equations of
state (EOS)
Extremes:
- "Soft" EOS: prompt collapse to a
black hole
- "Hard" EOS: unstable bar mode;
eventually BH
Black hole "kicks"
Asymmetric radiation, but
averages out over many cycles
No averaging during last cycle
Kicks enhanced by black hole
spins
- Anti-aligned spins:
Frame dragging effects cause
Vmax ~ 450 km/s
Compare with escape velocities from:
- Globular clusters: ~ 30 km/s
- Galaxies: ~ 1000 km/s
Super-kicks:
Black hole super-kicks
Due to frame dragging: orbital plane "bobs" up and down
During final cycle of inspiral: effect in one direction
Vmax ~ 4000 km/s
Black hole "kicks"
Supermassive black hole can be
expelled from galactic center
Will drag matter along; expect
bright X-ray source
Possible kick observation?
To see intricate binary BH
dynamics:
Need GW observations!
Capture orbits and the
No Hair Theorem
Extreme mass ratio inspirals:
- Small black hole captured by
very large one
- No time for circularization of
orbit
Complicated orbits:
- Extreme periastron precession
- Extreme precession of orbital
plane
- "Zoom-whirl" orbits
Powerful tool for mapping
spacetime geometry around
central black hole
Precision tests of No Hair
Theorem:
Black hole only determined by
M, J?
Probing the non-linear
structure of GR
GW "tails":
Waves interacting with
themselves and background
spacetime
Einstein Telescope: measure 1%
deviation from GR