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 $x^{\sinh(x)}$, $x>0$?

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

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

ReplyDeleteIf 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.

)

Deletehi Shaun, thanks a lot for this CQW and for providing the answer!

ReplyDelete