Tensor modes and the cosmic microwave background Jonathan Pritchard and Marc Kamionkowski Introduction Scalar and Tensor Power Spectra Inflationary theories predict the existence of a stocastic background of TAPIR, California Institute of Technology, Pasadena, CA 91125 Fig. 10. CMB Power gravitational waves (tensor perturbations to the metric). Detection of spectra describe the these waves would constrain inflationary models and increase our statistics of temperature knowledge of the earliest moments of the Universe. While direct and polarisation detection by interferometers such as LISA is a future possibility, Formalism fluctuations. Features indirect detection via the cosmic microwave background (CMB) is may be understood in actively being pursued by groups like BICEP and QUAD. terms of the horizon sizes Gravitational waves produce temperature perturbations via the at recombination lR and integrated Sachs-Wolfe (ISW) effect. A photon propagating past a matter-radiation leq, and gravitational wave which is oscillating with changing amplitude the width of the surface acquires a net change in energy. Consequently an initially isotropic of last scattering lΔ. temperature distribution becomes anisotropic in the presence of Baryonic oscillation gravitational waves. Compton scattering of this anisotropic generates the features of temperature distribution generates polarisation. the scalar spectrum. The Detailed numerical calculations of the statistical power spectra of the oscillation of inflationary fluctuations exist. E.g. CMBFAST. Here we develop complementary gravitational waves analytic expressions that reproduce the main features of these determines the tensor calculations and help illustrate the underlying physics. spectrum.

Fig. 1. The Cosmic Microwave Background Growth of anisotropy (CMB) originates at the time of when protons and Before recombination the optical depth for photons is large leading to a strong electrons combine to form coupling between photons and baryons. During this tightly-coupled period the neutral Hydrogen. short distance between scatterings suppresses the growth of anisotropy. As Photons no longer scatter recombination proceeds the photon mean free path increases and anisotropy grows. from charged particles and This growth is described by the two Boltzmann equations free-stream to the present day. Fluctuations at recombination are frozen in. These may be approximated by taking the optical depth to be large and expanding Fig. 2. Unpolarised light Fig. 3. Analytic approximations for the power spectrum reproduce in inverse powers of κ. The resulting evolution equation for the source is from an anisotropic the main features of numerical calculation using CMBFAST. temperature distribution Compton scatters from a charged particle leading The evolution of Ψ is driven by the oscillation of the tensor mode. In solving this to polarised outgoing light. Projection Gravitational Wave Evolution The functions for projecting from Inflation predicts a nearly scale-invariant primordial gravitational wave we exploit the narrowness of the visibility function which allows us to treat the The angular dependence value of at the peak as representative. This works provided that we incorporate of Compton scattering Fourier space to the unit sphere may power spectrum. This is processed by waves entering the cosmological be approximated by exploiting the horizon and evolving to generate the spectrum at recombination. Simple an exponential damping term due to phase damping on small scales. We arrive at picks out the quadrupole an expression for the integrated source part of the incident narrowness of the visibility function scaling arguments for the gravitational wave amplitude h then allow us to photon distribution. to write determine the scaling of the CMB power spectra We then use Debye’s asymptotic approximation for the Bessel function Recombination History and average to simplify the form for Fig. 5. A single k mode induces These regimes correspond to pre-horizon entry, matter dominated, Fig. 4. Most relevant the projection factors which involve a correlations on a range of angular radiation dominated and phase damped epochs. thermal history is sum of spherical Bessel functions. scales. Peak contribution at l~k(τ0- Fig. 7. Gravitational waves contained in the visibility τR). are frozen until horizon entry function g( ) describing the τ (kτ=1) after which they decay probability that an Fig. 6. The exact projection and oscillate. Evolution observed photon last factors are highly oscillatory. depends on the expansion scattered at a conformal Sharp peaks in P and P rate and the presence of time τ. It is sharply peaked T E faithfully reproduce features anisotropic stresses, which at the surface of last imprinted in the fluctuations at damp the amplitude. Exact scattering and can be recombination. The broad peak analytic expressions for h readily approximated by a Fig. 9. On scales smaller than in P leads to smoothing of exist in the matter and narrow Gaussian. This B the width of the SLS coherent features in the observed power radiation dominated limit. In Fig. 8. Integrated source shows approximation also scattering of photons from spectrum. Period averaged the mixed case the WKB oscillation with wavenumber k. reproduces the optical different phase regions leads to approximations are easier to approximation may be used. These features are projected into the depth κ in the region of cancellation and exponential calculate with. Plots are for l=50. angular power spectrum. the SLS. suppression of the anisotropy.