I am trying to figure out how to solve the Mukhanov-Sasaki equation to compute the power spectrum of an inflation potential that exhibits an ultra slow-roll phase that gives rise to an enhancement in the power spectrum, suitable for primordial black hole formation. In terms of the efold variable, the MS equation is

$$ \frac{d^{2}\zeta_{k}}{dN^{2}}+\left( 3-\epsilon_{1}+\epsilon_{2} \right)\frac{d\zeta_{k}}{dN}+\left( \frac{k}{aH} \right)^{2}\zeta_{k}=0 $$ with the Bunch-Davies vacuum as the initial condition

$$ u_{k}=\lim_{k>>aH}\frac{e^{-ik\tau}}{\sqrt{2k}} $$ I have computed the Hubble flow parameters but now I am unsure how to solve this ODE for all the modes of interest. I know that I need to consider them deep inside the horizon and evolve them to horizon exit where they freeze over, however, I am unsure how to proceed.

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  • $\begingroup$ What specifically are you unsure about: how to initialize the equation for each $k$ in the Bunch-Davies limit? How to solve it once you've done that? $\endgroup$ – bapowell Nov 26 '18 at 15:46
  • $\begingroup$ Yes, that's what I want to know. I know that I have to apply the BD vacuum to all the modes some efolds before horizon exit and let it evolve sufficiently long until it reaches a constant value. $\endgroup$ – Judas503 Nov 26 '18 at 16:35

You need to solve the MS equation separately for each $k$, initialized far enough in the past so that each mode is approximately in the Bunch-Davies limit. I've found that choosing $N_i$ so that $k = \mathcal{O}(100)a(N_i)H(N_i)$ is sufficient for this. Note that there is a different $N_i$ for each $k$. Here, $a(N) = a_0 \exp(N_0-N)$, where you need to pick a value for $a_0$. For example, you could set $a_0$ such that a particular scale, like the quadrupole, is at horizon crossing at $N_0$, i.e. $k_{\ell = 2} = a_0 H_0$.

Initializing further back in time will help with accuracy, but there are diminishing returns. Also, if you go back too far, you'll begin to see trans-Planckian modulations in your power spectra.

Then, you just evolve each mode forward in time until inflation ends. The largest-scale modes will have frozen-out long before the end of inflation, but this way you can compute $P(k)$ by evaluating each individual mode $u_k$ on an equal-time slice. Alternatively, you could evolve each mode until they are well-outside the horizon, say, when $k < aH/100$ or something. The modes freeze-out quite soon after horizon exit so applying a generous cut-off like this should work well.

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  • $\begingroup$ Thanks. I have one last question. In the paper I'm studying, $k=0.05Mpc^{-1}$ has been chosen to leave the horizon at N=0. So if I want to evolve sufficiently far enough, then N will take negative values (I have only solved the inflation background for N 0 to 56). $\endgroup$ – Judas503 Nov 26 '18 at 18:27
  • $\begingroup$ So what about smaller momentum modes, e.g. $k = 0.002 \, h{\rm Mpc}^{-1}$? These cross the horizon at $N<0$ in your model as well. $\endgroup$ – bapowell Nov 26 '18 at 19:09
  • $\begingroup$ I know. I was concerned whether the answers might turn out funny because I haven't solved that far back. $\endgroup$ – Judas503 Nov 26 '18 at 21:32

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