Scalar cosmological perturbation

A scalar cosmological perturbation transforms as a scalar under spatial rotations and can be represented using scalar metric potentials, density and pressure perturbations, and the longitudinal part of the fluid velocity.

Synchronous gauge sets the lapse and shift perturbations to zero, $h_{0\mu }=0$. Its coordinates follow freely falling observers, but residual coordinate freedom produces gauge modes that must be distinguished from physical perturbations.

For scalar perturbations of a generic fluid in cosmic time, momentum conservation in synchronous gauge in cosmology gives

Newtonian gauge removes scalar shear and writes scalar metric perturbations in terms of gravitational potentials. The fluid Euler equation then contains an explicit gradient of the Newtonian potential.

The longitudinal velocity perturbation of a cosmological fluid can be represented by a scalar velocity potential through $\delta T^0{}_i=(\bar{\rho }+\bar{P}){\partial }_i\delta u$.

Scalar anisotropic stress is the trace-free scalar part of the spatial stress perturbation. Under a common convention it enters as $\delta T^i{}_j=\delta P{\delta }_j^i+{\partial }^i{\partial }_j{\pi }^S$.