JPL, 12 April 2007

Discovering Relic Gravitational Waves in Cosmic Microwave Background

Radiation

L. P. Grishchuk Cardiff University and Moscow State University

[CMB conclusions are based on paper with D. Baskaran and A.G. Polnarev, Phys. Rev. D74,

083008 (2006), gr-qc/0605100]

Content •Inevitability of the existence of relic gravitational waves

•Superadiabatic (parametric) amplification. Quantization

•Gravitational wave mode functions, power spectra, statistics

•Comparison of gravitational waves and density perturbations. Initial conditions, quantization

•Radiative transfer in a perturbed universe

•Metric power spectra and power spectra of CMB polarization anisotropies

•TE correlation functions and signatures of relic gravitational waves

• Evidence for relic gravitational waves in the WMAP data

• What the upcoming CMB observational missions are expected to see ?

Superadiabatic (parametric) amplification: frequency of the oscillator

can be changed by the variation of length of the pendulum (or strength of the

gravitational field) ,

lg /2 l g

Cosmological oscillators

Spatial Fourier expansion of metric perturbations:

Polarization tensors for gravitational waves:

Polarization tensors for density perturbations:

DFrom the position of Einstein’s equations, cosmological density perturbations are

“scalar gravitational waves” that exist only when supported by matter perturbations

Gravitational-wave equation; parametrically excited oscillator:

where

Relic gravitational waves allow us to make direct inferences about the early universe Hubble parameter and scale factor (`birth’ of the Universe and its early dynamical evolution). But it is only through extra assumptions that we can make inferences about, say, inflationary scalar field potential (if it had anything to do at all with the “engine” that drove our early cosmological evolution).

and

[Interesting comments: J. A. Wheeler - “engine-driven cosmology”, E. Schrodinger

“alarming phenomenon” (he was right to worry about electromagnetic waves but

not about gravitational waves; for some details see PRD 48 , 5581, 1993) ]

Conclusion: inevitable generation of relic gravitational waves (Grishchuk, JETP,1974)

Quantization of gravitational waves

Gravitational wave Lagrangian for a given mode n:

where

Position and momentum of the oscillator:

Creation and annihilation operators:

Commutation relationships:

The Hamiltonian:

Initial conditions: ground state (vacuum state) of the Hamiltonian:

Quantum-mechanical Schrodinger evolution transforms initial vacuum state into a

strongly squeezed multiparticle vacuum state (standing waves). This determines

today’s amplitudes, power spectra, statistics, etc. of relic gravitational waves and

primordial density perturbations; PRD 42, 3413 (1990)

where the coupling function

Engine-driven cosmology with some knowledge of some parts of the evolution

Wavelength in comparison with the Hubble radius

(Gravitational wave equation: )

The shape of the `superadiabatic barrier’ fully determines the shape of today’s spectrum of relic gravitational waves:

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

Power spectrum is a function of wavenumbers (and time):

To simplify calculations, one can work with a `classical’ version of the theory in which the field is characterized by classical random Fourier coefficients:

Now the averaging is taken over realizations. For example, the power spectrum:

Rigorous definitions are based on quantum mechanics :

Statistical properties are determined by the statistics of squeezed vacuum states

and no other statistical assumptions are being made.

Evolution of mode functions (standing waves)

Today’s spectrum of relic gravitational waves and techniques of detection in various frequency bands

Comparison with density perturbations

“Spin-0 gravitational waves” in presence of matter (scalar field) perturbations:

The only equation to be solved:

or equivalent equation:

where

,

- `gauge-invariant’ metric perturbation

and - sometimes denoted

Quantization of density perturbations

Lagrangian:

where

Exactly the same theory as theory of gravitational waves, under the replacements:

With proper quantization and initial conditions, the final (at the end of the

amplification regime) amplitudes of gravitational waves and

density perturbations must be roughly equal to each other

(Physics-Uspekhi 48, 1235 (2005) (gr-qc/0504018))

Stokes parameters and polarization tensor for CMB radiation

Metric tensor and antisymmetric tensor

are used to build invariants of the radiation field. First two invariants:

Characterization of the electromagnetic radiation field

Metric tensor on a 2-sphere:

Symmetric trace-free part of the polarization tensor:

Two remaining invariants are called the E- and B- modes of CMB polarization:

(Pioneering papers: Newman and Penrose 1966, Thorne 1980, Zaldariaga

and Seljak 1997, Kamionkowski, Kosowsky and Stebbins 1997)

Multipole decomposition of the radiation field in ordinary spherical harmonics:

Randomness and statistics of the multipole coefficients of the radiation field are fully determined by the randomness and statistics of the coefficients in the Fourier expansion of the perturbed metric field

Radiative transfer equation (Chandrasekhar):

Having solved the radiation transfer equation one can find the multipoles of the correlation functions, where

Symbolic vector and the Stokes parameters:

Power spectra of metric perturbations (b) and their time-derivatives (d) at recombination, and CMB temperature (a) and polarization (c) anisotropies today in terms of

Cross-power spectrum of the metric times its time-derivative (lower panel) at recombination, and TE cross-correlation spectrum today (upper panel)

BB- and EE- correlations in the model that includes the epoch of reionization.

Broken lines show the results for a model without reionization.

Our theoretical result: TE correlation at lower l’s must be negative (red line) if caused by gravitational waves and positive (green line) if caused by density perturbations. Especially interesting interval:

WMAP observational result (Page et al. 2006): “The detection of the TE anticorrelation near l=30 is a fundamental measurement of the physicsof the formation of cosmological perturbations…”

Effects of inclusion of relic gravitational waves with spectral index n=1.2 on TT and TE correlations of CMB. Contributions of gravitational waves and density perturbations are equal at the quadrupole, l=2.

Conclusion: there is evidence for relic gravitational waves in the already available CMB data, and more accurate observations of the TE

correlation at lower multipoles have the potential of a firm positive answer

A cleaner, but more difficult, probe: BB correlations (Clover, BICEP , etc). In the more distant future: direct detection with LISA, advanced LIGO, etc.

Conclusions

Relic gravitational waves are inevitably generated in the early Universe. Their

existence relies only on the validity of general relativity and quantum mechanics

If the observed large-scale CMB anisotropies are caused by perturbations of

quantum-mechanical origin, relic gravitational waves contribute to lower l’s at

least at the level of the contribution from primordial density perturbations

The TE correlation due to relic gravitational waves must be negative at lower l’s

and appears to be present in the already available WMAP data

The expected (negative) TE-signal is about 100 times stronger than the BB-signal

and (presumably) is easier to measure. Must be seen by the Planck mission.

The BB signature is a clean probe of relic gravitational waves and should be

searched for at relatively high l’s (around l=90) by ground-based experiments