# What is the equation for the amplitude of scalar perturbations ?

1. Oct 26, 2013

### phypar

What is the "equation for the amplitude of scalar perturbations"?

I am studying inflation now, and in a book I read "equation for the amplitude of scalar perturbations", in the paper the author does not explain what is it, could any one give some detail on this equation or any reference? Thanks a lot in advance.

2. Oct 26, 2013

### Chronos

3. Oct 28, 2013

### phypar

4. Oct 28, 2013

### bapowell

Yes, that is the equation of the gauge invariant quantity $v = a \delta \phi + \frac{\phi'}{H}\mathcal{H}_L$ where $a$ is the scale factor, $\delta \phi$ is the scalar field fluctuation, $H$ is the Hubble parameter, $\phi'$ is the derivative of the field with respect to conformal time, and $\mathcal{H}_L$ is longitudinal part of the space-space metric perturbation, $\delta g_{ij} = 2(\mathcal{H}_L \delta_{ij} + \partial_i \partial_j \mathcal{H}_T)$. Now, the intrinsic curvature perturbation on spacelike hypersurfaces is $\mathcal{R} = \mathcal{H}_L + \mathcal{H}_T/3$. In most calculations, perturbations are evaluated on comoving hypersurfaces: here the transverse component $\mathcal{H}_T = 0$ and the scalar field fluctuation $\delta \phi = 0$. This leaves the well-known result that $\mathcal{R} = \mathcal{H}_L$ and the gauge invariant "potential" reduces to $v = \frac{\phi'}{H}\mathcal{R}$, which is Eq. 3.3. The expression Eq. 3.5 is the equation of the motion for $z$, whether or not a gauge has been chosen. I realize this is probably coming across as needlessly convoluted, but perturbations are strongly dependent on the choice of gauge (the way that spacetime is cut up into hypersurfaces and threaded with worldlines).

Last edited: Oct 28, 2013