What is the equation for the amplitude of scalar perturbations ?

[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 anyone give some detail on this equation or any reference? Thanks a lot in advance.

 

[Chronos]

This may be helpful: http://www.mpi-hd.mpg.de/lin/events/group_seminar/inflation/schmidt.pdf

 

[phypar]

This may be helpful: http://www.mpi-hd.mpg.de/lin/events/group_seminar/inflation/schmidt.pdf

thank you for give the reference, is Eq.(3.5) the equation for the scalar perturbation?

 

[bapowell]

thank you for give the reference, is Eq.(3.5) the equation for the scalar perturbation?

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).

 

[sardar]

The equation for the amplitude of scalar perturbations is a mathematical expression that describes the magnitude of fluctuations in the density of matter in the early universe. It is a key component in the theory of inflation, which suggests that the universe underwent a period of rapid expansion in its early stages.

The specific equation for the amplitude of scalar perturbations depends on the specific model of inflation being studied. In general, it takes into account factors such as the initial conditions of the universe, the rate of expansion, and the energy density of matter and radiation.

As for references, there are many books and articles on the topic of inflation and the equations involved. Some recommended sources include "Inflation and Cosmology" by Alan Guth, "The Early Universe" by Edward Kolb and Michael Turner, and "Introduction to Cosmology" by Barbara Ryden. Additionally, there are numerous scientific papers available online that discuss the equations for the amplitude of scalar perturbations in more detail.