Calculating the power spectra of scalar perturbation

[bapowell]

Yeah, $r$ is just another way of reporting the tensor spectrum amplitude at the scale of interest.

 

[shinobi20]

Yeah, $r$ is just another way of reporting the tensor spectrum amplitude at the scale of interest.

I don't understand why papers got different values for $r$ for different models knowing that the observational value is already established. But that aside, I have plotted $aH$ vs. $N$, so how do I know at which point I should get $\dot\phi_*$ and $H_*$ and solve $P_S$ at the horizon crossing?

Another thing, $aH$ is just $\dot a$ right? And $a$ is dimensionless, $H$ is measured in GeV, so my plot for $aH$ is in the units of GeV.

 

[bapowell]

1) I don't understand why this graph isn't monotonic: what's with the sharp rise as N descends from 140?
2) You read off the value of $aH$ associated with the $N$ of interest.
3) Why do you say that the observational value of $r$ is established?

 

[shinobi20]

1) I don't understand why this graph isn't monotonic: what's with the sharp rise as N descends from 140?
2) You read off the value of $aH$ associated with the $N$ of interest.
3) Why do you say that the observational value of $r$ is established?

1) I also don't know how to interpret the plot of $aH$ vs $N$, maybe you know of some resources that plots $aH$ vs $N$? Even in the cold regime would ok.
2) In warm inflation, I could plot out the temperature and the Hubble parameter as a function of the number of e-folding, and tell from the plot what is the duration $N$ of warm inflation. So in warm inflation I know the value of $N$ for a given constraint/parameter. Should that imply that whatever $N$ I get would be the point where I calculate $r$?
3) What I mean by that is observationaly, we know that $P_S \approx 10^{-9}$, right?

 

[bapowell]

1) aH must be monotonically increasing as a function of N during inflation, since it is equal to 1 over the comoving horizon size (which is decreasing during inflation). I.e. there is something wrong with that plot.
2) Sounds reasonable; but the N should not necessarily be the duration of inflation, but the time when observable scales leave the horizon (around N = 60).
3) Right, but $r$ then tells you about the tensor amplitude.

 

[shinobi20]

1) aH must be monotonically increasing as a function of N during inflation, since it is equal to 1 over the comoving horizon size (which is decreasing during inflation). I.e. there is something wrong with that plot.
2) Sounds reasonable; but the N should not necessarily be the duration of inflation, but the time when observable scales leave the horizon (around N = 60).
3) Right, but $r$ then tells you about the tensor amplitude.

1) In this model that I have plotted, the duration $N \approx 137$, then at $N \approx 137$ is where warm inflation ends and that is where the peak is so I think that is where aH should be equal to 1 (if I scale my y-axis properly). What do you think?
2)I'm still confused by the vagueness of the terminologies used. Can you explain more? For example, in my plot $N$ starts at 0 and changes until ~137. So $N \approx 137$, that is the number of e-folding during inflation but of course slow roll inflation doesn't take place right away, slow roll starts around a few e-folds after $N=0$, say $N=5$. When do the observable scales leave the horizon? $N=0$ or $N=5$ or $N=137$?
3) Yes, but the tensor amplitude is approximately constant, so $r$ changes value for different models/parameters only because of the scalar amplitude $P_S$ (assuming the value of $P_S$ deviates from $10^{-9}$)

 

[bapowell]

Observable scales leave the horizon 60 e-folds before the end of inflation in cold inflation. Is that no longer the case in warm inflation? I never studied it.

Yes, the tensor spectrum is nearly constant, but it's the amplitude that $r$ gives you.

 

[shinobi20]

Observable scales leave the horizon 60 e-folds before the end of inflation in cold inflation. Is that no longer the case in warm inflation? I never studied it.

Yes, the tensor spectrum is nearly constant, but it's the amplitude that $r$ gives you.

As I know there is no consensus on this since authors sometimes write that they will take $N=60$ but they'll add that there are still debates on this since the region is still unknown. The dilemma here is that, as you can see from my op, as $Q$ changes, $N$ also changes, i.e. increasing $Q$ prolongs $N$ (as in my plots, $N \approx 137$), so should that imply that whatever $N$ I got that would be the horizon crossing? or should I take $N=60$ as the horizon crossing? What will happen to the 77 e-folds before 60?

 

[bapowell]

I don't know what you mean by "whatever $N$ I got..". Got from where? You have a range of N, and for each N there is a mode leaving the horizon at that time. The inflation that happens before N = 60 generates fluctuations that today exist on scales well outside the cosmological horizon. That's the point of the N=60: it's the farthest back we can probe observationally, since our observations are limited by structures (really, correlations) within today's horizon.

 

[shinobi20]

I don't know what you mean by "whatever $N$ I got..". Got from where? You have a range of N, and for each N there is a mode leaving the horizon at that time. The inflation that happens before N = 60 generates fluctuations that today exist on scales well outside the cosmological horizon. That's the point of the N=60: it's the farthest back we can probe observationally, since our observations are limited by structures (really, correlations) within today's horizon.

What I mean is that, as I solve the dynamical equations for different $Q$ and the initial conditions $\phi(0)$, $\dot\phi(0)$ for all of the case are the same, the duration is different for each case of them. An example would be for $Q=10^{-2}$, $N \approx 137$ as you can see from the plots in the previous post.

 

[bapowell]

The duration is not relevant to the observables as long as it's sufficient to solve the horizon/flatness problems. Whether I've got an inflation model that lasts for N=1000 or N=100, I want observables at N=60 for each.

 

[shinobi20]

The duration is not relevant to the observables as long as it's sufficient to solve the horizon/flatness problems. Whether I've got an inflation model that lasts for N=1000 or N=100, I want observables at N=60 for each.

Then in that case, I should evaluate everything at 60 e-folds before the end of inflation. So suppose I got $N=200$, I should evaluate the observables at $N=140$ right?

 

[bapowell]

If N = 200 is the end, then yes. FYI, conventionally the variable $N$ is defined as the number of e-folds before the end of inflation (which is why I've been saying N=60 this whole time).

 

[shinobi20]

If N = 200 is the end, then yes. FYI, conventionally the variable $N$ is defined as the number of e-folds before the end of inflation (which is why I've been saying N=60 this whole time).

Yeah, what you mean by $N=60$ is where we start counting towards the end of inflation. If that is the case, I can just find the quantity in the plot at $N=60$ before the end of inflation, so what's the point of plotting $k=aH$ right? So I can just get $\dot\phi_*$, and $H_*$ at $N=60$ before the end of inflation in the plot.

 

[bapowell]

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

 

[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.