## Wednesday, May 1, 2013

### inflationary perturbations

guest post by Shaun Hotchkiss (from the blog trenches of discovery):

if, through the CMB and large scale structure, we can study the primordial density perturbation over a range of scales corresponding to five orders of magnitude (i.e. $k_\mathrm{max} = 10^5 k_\mathrm{min}$), what fraction $\Delta \phi/\phi$ of the inflationary potential does this probe (if inflation is true and the inflationary potential is $V(\phi)=m^2\phi^2$)?

bonus question: where's the first minimum of $x^{\sinh(x)}$, $x>0$?

1. Spirou asked me to provide an answer... so here it is:

If there have been $10^5$ orders of magnitude of $k$ observed, then this corresponds to $N=\log(10^5)$ e-folds of inflation.

In $m^2\phi^2$ inflation, inflation ends when $\phi=1/\sqrt{2}$ (i.e when the slow-roll paramter $\epsilon=1$). In this model $dN/d\phi=\phi$ and so tracing back $\sim 60$ e-folds corresponds to $\phi\simeq 11$.

$log(10^5)=11.5$, and evolving forward 11.5 e-folds from $\phi=11$, corresponds to $\phi\simeq10$.

Therefore, over the entire 5 orders of magnitude of observable scales from CMB to smallest large-scale structure, the inflaton field $\phi$ has changed from $\phi \simeq 11$ to $\phi \simeq 10$.

And the answer to the question is therefore $\simeq 10\%$.

(which is a little sobering... and note that the same order of magnitude values holds for most inflationary potentials simply because $\epsilon$ tends to be so much smaller earlier in inflation when our observable scales are crossing the horizon, compared to the end of inflation.

Though, any model where $\epsilon$ is not monotonic, or undergoes a sudden increase at the end of inflation will be different.

2. hi Shaun, thanks a lot for this CQW and for providing the answer!