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Stochastic Backgrounds of Gravitational Waves
L P Grishchuk
Cardiff University, UK
Fujihara Seminar, Japan, May 28, 2009
Contents
 Basic definitions, cosmological and astrophysical gravitational
wave backgrounds, classical and quantum-mechanical generation
mechanisms
 Correct and incorrect formulae in the literature
 Progress during the last 20 years
 Current situation (as a function of wavelength): suspected
detection (CMB), serious limits (pulsar timing), interesting limits
(LIGO et al), disappointing limits (HFGW)…
 Near future: expecting a discovery of relic gravitational waves in the
Planck CMB polarization data. Extrapolation to higher frequencies
 Astrophysical backgrounds – a blessing and a burden
 Enthusiastic conclusion
Qualitative definitions:
Stochastic signal – something random, noisy, unpredictable.
Background signal – something happening almost everywhere, in all
directions, at all times.
It is difficult to distinguish a useful stochastic gw signal from ordinary, nongravitational-wave noise, and one useful stochastic background from another.
A background signal can appear to be a random process, while being intrinsically a
deterministic, but very complicated function. For example, the gravitationalwave field consisting of many overlapping periodic signals with arbitrary, but fixed,
amplitudes and phases appears random. In principle, it is resolvable in components.
A stochastic background signal can be intrinsically random, like processes in
quantum mechanics. The field is characterized by a quantum state and by quantummechanical averages over that state. If the field is defined as a classical Fourier
expansion, the complex Fourier coefficients are taken from some probability
distributions. In cosmology, one normally has access to only one realization of this
random process. A background of quantum-mechanical origin is represented by
primordial (relic) gravitational waves – our window to the birth of the Universe.
More definitions and characteristics:
Cosmological gw background – generated before era of reionization at redshifts
z= 10 – 20. Astrophysical gw background – generated after that time.
Cosmological backgrounds : relic gravitational waves, ‘pre-Big-Bang’ models,
phase transitions, string networks, non-linear generation by density structures….
Astrophysical backgrounds: coalescing super-massive and ordinary black holes,
supernovae, neutron stars, binary white dwarf population,…
Broad spectrum (many decades of frequency, like in some cosmological
backgrounds) or relatively narrow spectrum (e.g. population of pulsars)
Isotropic background (even relic gravitational waves are not quite isotropic)
or strongly anisotropic background (e.g. binaries in the Galactic plane or a
‘stochastic boiling’ of an individual supernova)
Stationary versus non-stationary (relic gravitational waves are non-stationary)
“Not-quite-stochastic” backgrounds (few overlapping signals in a frequency
bin, ‘pop-corn’ noise, small duty cycle …)
Gravitational waves in cosmology and astrophysics
Spatial Fourier expansion of metric perturbations over
Polarization tensors for gravitational waves, ‘plus’ and ‘cross’ (or circular) polarizations,
For a classical random field, the Fourier coefficients
For a quantum field,
commutation relations
D
are random complex numbers.
are the annihilation and creation operators satisfying the
and acting on the
quantum states of quantized gravitational waves. Initial vacuum state:
Rigorous definitions for relic gravitational waves are based on quantum mechanics :
Mean-square amplitude of the field in the initial (Heisenberg) vacuum state:
Gravitational wave power spectrum is a function of wave-numbers (and time):
Statistical properties are determined by the statistics of squeezed vacuum states
In classical approximation, one works with random (Gaussian) Fourier coefficients:
Today’s mean-square amplitude is given by
is an rms amplitude per logarithmic frequency interval. It
depends on frequency but is dimensionless. Very convenient for
comparisons with dimensionless amplitudes of all other signals.
20 years ago…
rms field amplitude
in log interval
another important quantity,
(energy density in log interval)
Something has happened after 1988, the definition used these days is
different (and incorrect).
See arXiv:0707.3319 :
One may introduce a new cosmological parameter defined by this formula,
but this is not the Omega-parameter accepted in astronomy. Special care
is needed when comparing the results.
Understanding of 1986-1988
What has changed in 20 years ?
What has changed in 20 years ?
1988 - 2009
Suspected detection,
arXiv:0810.0756
Serious limits,
arXiv astro-ph/0609013
Interesting limits,
arXiv astro-ph/0608606
Current predictions
Spectrum of relic gravitational waves normalized to CMB anisotropies
arXiv:0707.3319
Energy density of relic gravitational waves
arXiv:0707.3319
Suspected detection. Analysis of the 5-year WMAP TE and TT data
The likelihood function for R, where
The maximum likelihood value:
20% of temperature quadrupole is produced by relic gravitational waves
Zhao, Baskaran, Grishchuk, arXiv:0810.0756
There are indications of the presence of relic gravitational waves in the
WMAP5 data.
And this is what we believe (Zhao, Baskaran,Grishchuk, arXiv:0810.0756)
will be seen by the Planck mission:
We expect 3sigma detection in TE channel
and 2sigma detection in ‘realistic’ BB channel
Big picture:
Relic gravitational waves as a signature of the ‘birth of the Universe’
(arXiv:0903.4395)
Using the predicted relic g.w. background as a benchmark signal, we can now
discuss other cosmological and astrophysical backgrounds from various sources.
[Note that the pre-Big-Bang, quintessential, cyclic and other scenarios rely on the
same mechanism of superadiabatic amplification of zero-point quantum oscillations
of gravitational waves. They differ from the theory of relic g.w. only in the assumptions
about the evolution of the early Universe cosmological scale factor.]
There is no gravitational wave competitors to relic gravitational
waves in the range of wavelengths relevant to CMB measurements,
but there are many competitors in the range of shorter wavelengths
One man’s signal is another man’s noise !
It appears that realistic astrophysical backgrounds do not exceed
the expected (optimistic) relic gw background
in the LIGO-VIRGO frequency window
Astrophysical backgrounds in different frequency bands
Important for the study of cosmic objects formation rates, but also as
foreground noises for relic gravitational waves
CMB - no competitors to relic gravitational waves
Pulsar timing – massive black hole collisions, string networks, etc.
Very optimistically, the signal level can be approaching the current
upper limit
Space-based interferometers – white dwarf binaries in our Galaxy, as
well as hypothetical strong electroweak phase transitions, bubble
collisions, turbulence, etc. White dwarf binaries will certainly appear in
the LISA window. Will need resolving, modeling and subtraction.
Ground-based interferometers – various emission mechanisms in
core collapse supernovae, neutron stars, compact binaries, etc.
Most optimistic predictions approach
, More realistically,
May swamp relic signal, discrimination is necessary.
Some conclusions:
Stochastic backgrounds of gravitational waves are difficult to
detect, but the current work on theoretical modelling and
comparison of primordial, cosmological, astrophysical
backgrounds, as well as data analysis techniques, must continue.
The relic gravitational waves are a unique probe of the birth and
dynamics of the very early Universe. They should be explored
in all frequency windows. First detection is likely to come from the
ongoing CMB observations. This will provide crucial information on
the shape of the spectrum and its likely level at higher frequencies.
The chances of advanced space-based and ground-based
interferometers to see the relic signal are reasonable. In the higherfrequency windows, the astrophysical backgrounds will be
encountered first. They should be properly dealt with.