Tensor cosmological perturbation ($h_{ij}$)

A tensor cosmological perturbation is the transverse, trace-free part of a metric perturbation about a Friedmann-Lemaître-Robertson-Walker metric. In conformal coordinates, its spatial metric has the form $g_{ij}=a^2({\delta }_{ij}+h_{ij})$, with ${\partial }_i{h}_{ij}=0$ and $h_{ii}=0$. Its two independent polarizations describe a gravitational wave. Neutrino tensor anisotropic stress can damp its evolution through neutrino free streaming.

If the tensor metric perturbation is $h_{ij}=2E_{ij}$ and the polarization tensors have contraction $M_{ij}^{(p)}{M}^{(q)\ast ij}={\delta }_{pq}$, the quadratic action for each Fourier amplitude is $(M_{\mathrm{Pl}}^2/2)\int a^2(|{\psi }_{(p)}^{\prime }{|}^2-k^2|{\psi }_{(p)}{|}^2)$. Thus each helicity is normalized like a real massless scalar by the displayed rescaling. Its power per polarization is $(H_k/(2\pi M_{\mathrm{Pl}}){)}^2$. Factors of two in other tensor spectra depend on the definitions of the metric amplitude, the polarization norm and whether both helicities are summed.

A complex helicity-$+2$ plane wave along the $z$ axis has polarization contraction ${\sin }^2\theta e^{2i\phi }$. Its remaining tensor CMB line-of-sight source depends only on $\cos \theta$. Orthogonality of the azimuthal factors of spherical harmonics therefore restricts its temperature coefficients to $m=2$. A real perturbation also includes the conjugate mode.

For a matter-era superhorizon tensor with $h=h_{\mathrm{prim}}[1-(k\eta {)}^2/10+\cdots ]$, the tensor CMB line-of-sight source gives ${\Theta }_{22}\propto h_{\mathrm{prim}}(k{\eta }_0{)}^2$ and ${\Theta }_{32}/{\Theta }_{22}=-ik{\eta }_0/(3\sqrt{7})$ when ${\eta }_{\ast }\ll {\eta }_0$. The extra spatial phase makes the octupole one order smaller.

For a tensor cosmological perturbation $h_{ij}$, the time geodesic equation in a local orthonormal tetrad gives $d\log ϵ/d\eta =-{\dot{h}}_{ij}{e}^i{e}^j/2$. The photon comoving energy removes the background redshift; the dot is a partial time derivative, whereas the energy derivative follows the ray.

For massless neutrinos with a temperature perturbation $\Theta$ and background energy density ${\bar{\rho }}_{\nu }$, the physical trace-free spatial stress-energy tensor isSome conventions define ${\Pi }^{ij}=-{\pi }_{\nu }^{ij}$ and correspondingly reverse the sign of the stress source in the gravitational-wave equation.

Projecting free-streaming neutrinos onto the tensor spherical harmonic givesHere $j_2$ is a Spherical Bessel function. The continuous value at zero follows by integrating $(1-{\mu }^2{)}^2$, whose integral is $16/15$.

For either tensor helicity, neutrino free streaming produces the memory equationThe neutrino tensor free-streaming kernel encodes the angular propagation of the perturbation. Energy transferred to neutrino directional anisotropy damps the oscillation amplitude of a gravitational wave after horizon entry. This effect was calculated by Weinberg.