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

Stochastic signal – something random, noisy, unpredictable. Background signal – something happening almost everywhere, in alldirections, at all times.

It is difficult to distinguish a useful stochastic gw signal from ordinary, non-gravitational-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 gravitational-wave 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 inquantum mechanics. The field is characterized by a quantum state and by quantum-mechanical 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.

Qualitative definitions:

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,

D

For a quantum field, are the annihilation and creation operators satisfying the

commutation relations and acting on the

quantum states of quantized gravitational waves. Initial vacuum state:

For a classical random field, the Fourier coefficients are random complex numbers.

Mean-square amplitude of the field in the initial (Heisenberg) vacuum state:

Gravitational wave power spectrum is a function of wave-numbers (and time):

In classical approximation, one works with random (Gaussian) Fourier coefficients:

Rigorous definitions for relic gravitational waves are based on quantum mechanics :

Statistical properties are determined by the statistics of squeezed vacuum states

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.

another important quantity,(energy density in log interval)

20 years ago…

rms field amplitude 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.

What has changed in 20 years ?

Understanding of 1986-1988

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

Spectrum of relic gravitational waves normalized to CMB anisotropies

arXiv:0707.3319

Current predictions

Energy density of relic gravitational waves

arXiv:0707.3319

The likelihood function for R, where

The maximum likelihood value:

Zhao, Baskaran, Grishchuk, arXiv:0810.0756

Suspected detection. Analysis of the 5-year WMAP TE and TT data

20% of temperature quadrupole is produced by relic gravitational waves

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

Relic gravitational waves as a signature of the ‘birth of the Universe’ (arXiv:0903.4395)

Big picture:

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 thesame 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.]

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

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

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, aswell as hypothetical strong electroweak phase transitions, bubble collisions, turbulence, etc. White dwarf binaries will certainly appear inthe 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 anddynamics 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 higher-frequency windows, the astrophysical backgrounds will be encountered first. They should be properly dealt with.