Tensor Anisotropies in an Open Universe

We calculate the anisotropies in the cosmic microwave background induced by long-wavelength primordial gravitational waves in a universe with negative spatial curvature, such as are produced in the “open inflation” scenario. The impact of these results on the COBE normalization of open models is discussed. Subject headings: cosmology:theory – cosmic microwave background There is considerable observational prejudice suggesting that we live in a universe with negative spatial curvature, an open universe. Recently Bucher, Goldhaber and Turok (1995) have devised an open inflationary cosmogony. This has allowed one, for the first time, to calculate the spectrum of primordial fluctuations in an open model. The scalar field modes, which give rise to density perturbations, come in 3 types and have been extensively discussed in the literature (for a recent review see Cohn 1996). As with all inflationary models, a nearly scale-invariant spectrum of gravitational waves (tensor modes) is also produced, and the spectrum of such modes has recently been computed (Tanaka & Sasaki 1997, Bucher & Cohn 1997). If the energy density during inflation is high enough, these modes induce a measurable cosmic microwave background (CMB) anisotropy (Abbott & Wise 1984). In this Letter we calculate these CMB anisotropies, and their implications for the COBE normalization of open models. The gravitational waves represent tensor perturbations to the metric ds = a(η) [ −dη + (γij + hij) dx dx ] , (1) where a(η) is the scale factor, dη = dt/a(t) is the conformal time and γij is the 3-metric for a space of constant (negative) curvature K = −H2 0 (1 − Ω0). The perturbations decompose as hij = 2hQij, where the harmonic modes are the transverse-traceless tensor eigenfunctions of the Laplacian ∇Qij = −kQij (Kodama & Sasaki 1984, Abbott & Schaefer 1986). The photon temperature distribution function Θ, can likewise be expanded in mode functions Θ(η, ~x, n̂) = ∑ ~k ∑ l Θl( ~k)Gl(~x,~k, n̂), which form a complete basis constructed out of covariant derivatives of Qij .