Expansion of cosmological perturbations in helicities

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When dealing with perturbations in cosmology one usually doesn't distinguish between original functions and their Fourier transforms. Coordinate and momenetum representations are connected by Fourier transform: \begin{equation*} h_{\mu\nu}(\eta,\, \vec{x})= \int d^3ke^{i\vec{k}\vec{x}} h_{\mu\nu}(\vec{k}), \end{equation*} which is reduced to the replacement $\partial_i\longleftrightarrow ik_i$ (see problem \ref{per29nnn}), and $\vec{k}$ has a meaning of conformal momentum.

Due to isotropy, metrics is invariant under spatial rotations, while at fixed conformal momentum $\vec{k}$ it is invariant under the rotations around the direction of $\vec{k}$, i.e. posses posses $SO(2)$ symmetry. Arbitrary three--dimensional tensor can be expanded in irreducible representations of $SO(2)$, which have certain helicity (eigenvalues of a rotation operator $L_\alpha = -i\frac{\partial}{d \alpha}$ at angle $\alpha$ ). \\ For example, three--dimensional scalar has zero helicity, since it doesn't transform under rotations around $\vec{k}$. Since for three--dimensional vector $v_i \propto k_i$, it has unit helicity, or more precisely a superposition of helicities +1 and -1. Three--dimensional tensor of the form $h_{ij}\propto v_i v_j$ or $h_{ij}\propto \delta_{ij}$ has the same helicity. Symmetric transverse traceless tensor $h_{ij}^{TT}$ has double helicity. Mathematically this can be expressed as \begin{equation}\label{hijTT} h_{ii}^{TT}=0,~ k_ih_{ij}^{TT}=k_jh_{ij}^{TT}=0. \end{equation} In linear theory, the Einstein's equations and equations of energy--momentum tensor covariance split into independent equations for helical components of cosmological perturbations. Thus, the perturbations are divided into tensor (double helicity), vector (unit helicity) and scalar (zero helicity). Expansion of perturbed metrics in helical components in general case has the form (see, for example \cite{Rubakov2}): \begin{equation}\label{h00_gen} h_{00} = 2\Phi, \end{equation} \begin{equation}\label{h0i_gen} h_{0i} = i k_i Z+ Z_i^T, \end{equation} \begin{equation}\label{hij_gen} h_{ij} = -2\Psi \delta_{ij}-2k_i k_j E+i(k_i W_j^T+k_j W_i^T)+h_{ij}^{TT}, \end{equation} where $\Phi, Z, \Psi, E$ are scalar functions of coordinates,$Z_i^T, W_i^T$ are transverse vectors ($k_i Z_i^T=k_i W_i^T =0$) and $h_{ij}^{TT}$ is a transverse traceless tensor.

In the subsequent problems the all types of perturbations are considered: scalar, vecotr and tensor. Also, we will use the gauge $h_{0i}=0.$