A Calculating the power spectra of scalar perturbation

[shinobi20]

Well, you originally asked for the spectrum as a function of $k$, remember?

No, I just want $P_S$ AT the horizon crossing (which now we say at $N=60$), because my main goal is to get $r$ at the horizon crossing.

 

[bapowell]

Yes. Refer to posts #4 and #5 where you confirm that you desire to find $P(k)$. And to be clear: your main goal here is to find $r$ on a particular observable scale (not simply "at horizon crossing"), which you've selected to be N=60. But, as is standard, we quote observables on a given scale $k$, and so you need the $k=aH$ mapping to go from $k$ to $N$.

 

[shinobi20]

Yes. Refer to posts #4 and #5 where you confirm that you desire to find $P(k)$. And to be clear: your main goal here is to find $r$ on a particular observable scale (not simply "at horizon crossing"), which you've selected to be N=60. But, as is standard, we quote observables on a given scale $k$, and so you need the $k=aH$ mapping to go from $k$ to $N$.

So you mean, choosing $N=60$ is enough to calculate $r$ at that point though I should state it as you've mentioned in the latter part? Or you mean I should still show $k=aH$ vs $N$ even though I've decided already to compute $r$ at $N=60$?

 

[bapowell]

If you wish to find $r$ at N=60, you're done.

 

[shinobi20]

If you wish to find $r$ at N=60, you're done.

Ok, so that is settled. What is the corresponding formula for the spectral index NOT in terms of $k$? I mean, like the one I posted for $P_S$, it is the lowest order in slow roll.

 

[bapowell]

Check out that Stewart and Lyth references I posted earlier.