{title2=$\gamma_{ij}^{L}$}

An adiabatic tensor mode is a long-wavelength transverse-traceless metric perturbation whose leading local effect can be produced by a <large spatial diffeomorphism>. Its effect on short modes is therefore constrained by a Ward identity.

Adiabatic tensor mode

The background spatial metric is $g_{ij}=a^2\delta_{ij}$. For the <large spatial diffeomorphism>
$$
\epsilon^0=0,
\qquad \epsilon^i=\omega^i{}_jx^j,
\qquad \omega_{ij}=\omega_{ji},
\qquad \omega_i{}^i=0,
$$
homogeneity and isotropy imply that the purely spatial background <Christoffel symbols> vanish. Hence
$$
\nabla_i\epsilon_j=a^2\omega_{ji},
\qquad
\boxed{\Delta\gamma_{ij}=-2\omega_{ij}}.
$$
The shift is constant, transverse and traceless, and is therefore the zero-momentum <adiabatic tensor mode>.

A scalar transforms by its <Lie derivative>:
$$
\Delta\varphi(\mathbf x)
=-\omega_{ij}x^j\partial_i\varphi(\mathbf x).
$$
Fourier transformation and <integration by parts> in momentum space give
$$
\boxed{\Delta\varphi(\mathbf k)
=\omega_{ij}k_i\frac{\partial}{\partial k_j}\varphi(\mathbf k)}.
$$
The term proportional to $\omega_i{}^i$ vanishes because the deformation is traceless.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 312 / 2 / i / Solution

A large spatial diffeomorphism changes spatial coordinates by a transformation that remains nonzero at large distance. In cosmology, suitable large spatial diffeomorphisms generate zero-momentum adiabatic modes.


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