# Calculating the power spectra of scalar perturbation

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bapowell
Oh, that is why I should specify at which ##k=aH## I want to measure ##r## and that would give me the corresponding ##N##. Typical back in the days are ##aH=0.05/0.002 h^{-1} Mpc##. But why should I take the derivative of ##k=aH## with respect to ##N##? I could just plot ##k=aH## vs ##N## and find the ##k=aH## based on Planck data and get the corresponding ##N##?
Sure, that works. I only mentioned the explicit mapping so that you can see how to go from k <-> N in general. But, yes, if you have k = aH as a function of N, you just pick the k of interest, read off the N, then solve for r at that N.

As a sidenote, is ##k## defined as ##k=aH##? I thought ##k## is an independent quantity and we just set a condition ##k=aH## to specify the Hubble crossing.
That's right.

Sure, that works. I only mentioned the explicit mapping so that you can see how to go from k <-> N in general. But, yes, if you have k = aH as a function of N, you just pick the k of interest, read off the N, then solve for r at that N.

That's right.
Oh, that is very informative. Last question though, I have set my equations to be the KG equation, Friedmann equation, and the number of e-folding equation (##N=\int Hdt## or ##\frac{dN}{dt} = H##). So I don't have any equation involving the scale factor ##a## that I need to plot ##aH##. So Suppose I use the acceleration equation ##\frac{\ddot a}{a} = \frac{1}{3M_p^2} (V-\dot\phi^2)##, I would need two initial conditions ##a(0)## and ##\dot a(0)##, what do you think I should set for those two? As I remember ##a(0)## is usually set as 1? I'm not sure.

bapowell
But you do have an equation for the scale factor! Can you write ##a## as a function of ##N##?

But you do have an equation for the scale factor! Can you write ##a## as a function of ##N##?
Oh, I'm sorry. I could write,

(1) ##\ddot \phi + 3H (1+Q) \dot\phi + V_{,\phi} = 0 \quad ## KG equation

(2) ##H = \frac{\dot a}{a} = \Bigg(\frac{1}{3 M_p^2} ( \frac{1}{2}\dot\phi^2 + V + \rho_r) \Bigg)^\frac{1}{2} \quad## Friedmann equation

(3) ##\frac{dN}{dt} = \frac{\dot a}{a} \quad## Number of e-folding

But due to (2) and (3) being equal, I can only explicitly use either (1) (2) or (1) (3), in my case I choose the latter so that I can also use ##N## for plotting. So how would I specify the initial conditions for ##a(0)##? Since before, I'm using ##H## instead so I just specify ##H(0) ∝ \phi(0), \dot \phi(0)##

bapowell
Pick whatever you want for a(0): it's the proportionate change in ##a## that matters. Also, you've got a misplaced power of 2 in (2) (should be ##H^2##)

Pick whatever you want for a(0): it's the proportionate change in ##a## that matters. Also, you've got a misplaced power of 2 in (2) (should be ##H^2##)
Typo, I've edited it. Why do a lot of papers say that for example, we calculate ##P_S## at the horizon crossing (which is typically taken to be around 50~60 e-folds, but we can take ##N = 60##). So I have a notion that what they mean is just to get at least ##N=60## for the duration of slow roll inflation then just pick the value of the quantities at ##N=60## (start of slow-roll as we count down from 60 although the plot is counting up) and that's it, plug into ##P_S## and I got ##r##.

bapowell
##N=60## is just a rule-of-thumb. Remember, as I've been saying: the mapping between k (the scale of interest observationally) and N is model dependent. In one model, the quadrupole might map to N=60; in another, it might map to N = 55. Whichever N we pick, that's the duration of the full inflationary period, including non-slow roll. The amount of slow roll inflation also varies by model, but surely lasts for several dozens of efolds around the time that observable scales leave the horizon.

EDIT: The range of N, usually between 50 and 60, is dictated by the amount of inflation needed to solve the flatness/horizon problems. There is some leeway in the amount of inflation because the reheat temperature can generally be tweaked to compensate.

##N=60## is just a rule-of-thumb. Remember, as I've been saying: the mapping between k (the scale of interest observationally) and N is model dependent. In one model, the quadrupole might map to N=60; in another, it might map to N = 55. Whichever N we pick, that's the duration of the full inflationary period, including non-slow roll. The amount of slow roll inflation also varies by model, but surely lasts for several dozens of efolds around the time that observable scales leave the horizon.
Ok, I'll work on it first and just update this thread. Do you know of any references/books/papers that I can read that is more focused on the details of this topic? I mean, the methods on how to do things. like what we have discussed. Books usually are too general and some papers are too concise to be useful. Even just solving the simplest model (cold inflation scenario is fine) is fine as long as I get the vital points and specially how to calculate ##r##.

bapowell
I learned the basics of the inflationary perturbations calculation from Section 2.3 of Will Kinney's PhD thesis (he was my PhD advisor), available online here: http://www.acsu.buffalo.edu/~whkinney/cv/thesis/thesis.ps. He solves the power spectrum in the slow roll limit for a minimally coupled massless scalar in order to demonstrate the main ideas and important steps. (This kind of spectrum is actually the tensor spectrum; the scalar spectrum is more complicated because the field fluctuations couple to the metric, but the main ideas are the same). I'd recommend starting there if you have the appropriate background: some QFT, GR, FRW cosmology. Then, I can point you to some references that cover the full scalar spectrum calculation. Enjoy!

I learned the basics of the inflationary perturbations calculation from Section 2.3 of Will Kinney's PhD thesis (he was my PhD advisor), available online here: http://www.acsu.buffalo.edu/~whkinney/cv/thesis/thesis.ps. He solves the power spectrum in the slow roll limit for a minimally coupled massless scalar in order to demonstrate the main ideas and important steps. (This kind of spectrum is actually the tensor spectrum; the scalar spectrum is more complicated because the field fluctuations couple to the metric, but the main ideas are the same). I'd recommend starting there if you have the appropriate background: some QFT, GR, FRW cosmology. Then, I can point you to some references that cover the full scalar spectrum calculation. Enjoy!
I've seen in one paper that the scalar spectra amplitude ##P_S## value at the pivot scale ##k_*## is set by the CMB data at ~##10^{-9}## with ##k_* = 0.05 Mpc^{-1}##. But if that is the case, what is the point of calculating the tensor to scalar ratio ##r## if that should always be the value of ##P_S##?

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

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?

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.

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.

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.

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.

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