Professor, Department of Astronomy and Astrophysics
University of Chicago

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Gravitational Waves

A time-variable tensor metric perturbation similarly leaves an imprint in the temperature anisotropy [Sachs & Wolfe, 1967]. A tensor metric perturbation can be viewed as a standing gravitational wave and produces a quadrupolar distortion in the spatial metric. If its amplitude changes, it leaves a quadrupolar distortion in the CMB temperature distribution [Polnarev, 1985]. Inflation predicts a nearly scale-invariant spectrum of gravitational waves. Their amplitude depends strongly on the energy scale of inflation,[*](power $\propto E_i^4$ [Rubakov et al, 1982,Fabbri & Pollock, 1983]) and its relationship to the curvature fluctuations discriminates between particular models for inflation. Detection of gravitational waves in the CMB therefore provides our best hope to study the particle physics of inflation.

Figure: Gravitational waves and the energy scale of inflation $E_i$. Left: temperature and polarization spectra from an initial scale invariant gravitational wave spectrum with power $\propto E_i^4=(4 \times 10^{16} {\rm GeV})^4$. Right: 95% confidence upper limits statistically achievable on $E_i$ and the scalar tilt $n$ by the MAP and Planck satellites as well as an ideal experiment out to $\ell =3000$ in the presence of gravitational lensing $B$-modes.

Gravitational waves, like scalar fields, obey the Klein-Gordon equation in a flat universe and their amplitudes begin oscillating and decaying once the perturbation crosses the horizon. While this process occurs even before recombination, rapid Thomson scattering destroys any quadrupole anisotropy that develops (see §3.6). This fact dicates the general structure of the contributions to the power spectrum (see Figure 4, left panel): they are enhanced at $\ell=2$ the present quadrupole and sharply suppressed at multipole larger than that of the first peak [Abbott & Wise, 1984,Starobinskii, 1985,Crittenden et al, 1993]. As is the case for the ISW effect, confinement to the low multipoles means that the isolation of gravitational waves is severely limited by cosmic variance.

The signature of gravitational waves in the polarization is more distinct. Because gravitational waves cause a quadrupole temperature anisotropy at the end of recombination, they also generate a polarization. The quadrupole generated by a gravitational wave has its main angular variation transverse to the wavevector itself [Hu & White, 1997a]. The resulting polarization that results has components directed both along or orthogonal to the wavevector and at 45$^\circ$ degree angles to it. Gravitational waves therefore generate a nearly equal amount of $E$ and $B$ mode polarization when viewed at a distance that is much greater than a wavelength of the fluctuation [Kamionkowski et al, 1997,Zaldarriaga & Seljak, 1997]. The $B$-component presents a promising means of measuring the gravitational waves from inflation and hence the energy scale of inflation (see Figure 4, right panel). Models of inflation correspond to points in the $n,E_i$ plane [Dodelson et al, 1997]. Therefore, the anticipated constraints will discriminate among different models of inflation, probing fundamental physics at scales well beyond those accessible in accelerators.

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Next: Gravitational Lensing Up: Gravitational Secondaries Previous: Rees-Sciama and Moving Halo
Wayne Hu 2001-10-15