Calculating the power spectra of scalar perturbation

[shinobi20]

I'd like to numerically calculate the power spectra of the scalar perturbation at the Hubble crossing in warm inflation, my problem is that I don't know how to do it. As I know, the Hubble crossing happens at the onset of warm inflation where the different modes become larger than the Hubble length. Now suppose I have solved the dynamical equations of warm inflation with respect to time. So given the scalar power spectra at the Hubble crossing,

$$P_S = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \delta\phi_*^2 = \Bigg( \frac{H_*}{\dot\phi_*} \Bigg)^2 \Bigg(\frac{\sqrt{3(1+Q)} H_*T_*}{2\pi^2}\Bigg)$$

where ##H## is the Hubble parameter, ##\phi## is the inflaton field, ##T## is the temperature, and ##Q = \frac{\Gamma}{3H}## is the ratio of the effectiveness of the dissipation ##\Gamma##. The "##_*##" denotes the quantities are evaluated at the horizon crossing.

How do I solve for the quantities AT the horizon crossing? Can I plot out an evolution of some quantity and identify that at some point on the plot, that is the horizon crossing? Or how should I proceed in solving this? Does anyone know of any resources/ material that I can look into to be able to know how to do this?

 

[bapowell]

What is the condition for horizon crossing in warm inflation? In standard inflation it's [itex]k = aH[/itex].

 

[shinobi20]

What is the condition for horizon crossing in warm inflation? In standard inflation it's [itex]k = aH[/itex].

As I know, it is the same thing. But let's assume that we are in the cold inflation case. I know that we should evaluate those quantities at the horizon crossing, but these are just all statements, how do I actually do it i.e. plotting it out or determining when is the Hubble crossing from the plot of something.

The power spectrum for the scalar perturbations in the cold inflation case at the Hubble crossing is given by,

$$P_S = \Bigg(\frac{H_*}{2\pi}\Bigg)^2 \Bigg(\frac{H_*}{\dot\phi_*}\Bigg)^2$$

 

[bapowell]

What are you trying to do, determine [itex]P(k)[/itex]?

 

[shinobi20]

What are you trying to do, determine [itex]P(k)[/itex]?

Yes, in order to get the tensor to scalar ratio.

 

[bapowell]

You've got two relations: the power spectrum and the horizon crossing condition. The expression for P can be solved in terms of some time variable, pick one (a, t, N, ...), as can the expression k = aH. The idea is to solve for the power spectrum at horizon crossing parametrically.

 

[shinobi20]

You've got two relations: the power spectrum and the horizon crossing condition. The expression for P can be solved in terms of some time variable, pick one (a, t, N, ...), as can the expression k = aH. The idea is to solve for the power spectrum at horizon crossing parametrically.

I understand what you mean by solving P and k in terms of say, N; so I need to plot k=aH with respect to N and tell from the plot where is the horizon crossing?

 

[bapowell]

Once you have k = aH as a function of N, you can obtain P(k) parametrically.

 

[shinobi20]

Once you have k = aH as a function of N, you can obtain P(k) parametrically.

So I first need to know when k =aH from the plot of k vs. N right? But what is k? There isn't an explicit expression for k.

 

[bapowell]

No, you're plotting k = aH, say, as a function of N. This function tells you the value of k that is crossing the horizon at the corresponding value of N.

 

[shinobi20]

No, you're plotting k = aH, say, as a function of N. This function tells you the value of k that is crossing the horizon at the corresponding value of N.

But that is my question originally, suppose I plot k=aH as a function of N, how would I know when IT crossed the horizon?

 

[bapowell]

What is IT? You have k vs. N. Pick an N. The associated k is crossing the horizon at that N. I think we're talking past each other...

 

[shinobi20]

What is IT? You have k vs. N. Pick an N. The associated k is crossing the horizon at that N. I think we're talking past each other...

Oh, what I mean by IT is ##k##. So if I plot ##aH## as a function of ##N## and from observation we need at least ##N=60## of e-folding for the duration of inflation, then the corresponding ##k## crosses the horizon? But that would just mean if say, I plot ##H## vs. ##N## and just choose whatever ##H## is at ##N=60## that would be the corresponding ##H_*## at the horizon crossing, as well as for ##\dot\phi_*##.

 

[bapowell]

Yep. There is just one k that crossed the horizon at N = 60. But you said you wanted P(k), so presumably you are interested in the spectrum at more than just N = 60, right?

 

[shinobi20]

Yep. There is just one k that crossed the horizon at N = 60. But you said you wanted P(k), so presumably you are interested in the spectrum at more than just N = 60, right?

Wait wait, maybe what I'm thinking needs some patching. Basically I want to numerically compute the tensor to scalar ratio without using the slow roll parameters. So, the tensor to scalar ratio is given by ##r = \frac{P_T}{P_S}## where ##P_S## is the power spectrum for the scalar perturbations, while ##P_T## is the power spectrum for the tensor perturbations. I just showed ##P_S## in the op since if I know how to do it in ##P_S##, ##P_T## would follow. So, based on what I know ##r## is calculated at the instant ##k## starts leaving the horizon (indicating that the slow roll inflation started, not inflation); this is usually denoted by ##N=60## before the end of inflation. So I think what I need is just ##P_S## at the onset of slow roll inflation not the whole spectrum even during inflation? Is this correct?

 

[bapowell]

A few things. If you wish to compute r without reference to slow roll, you can't use the expression you posted for [itex]P_S[/itex], since it is lowest-order in slow roll (specifically, the quantity [itex]H/2\pi[/itex]). The only way to do this in general is to compute [itex]P_S[/itex] and [itex]P_T[/itex] by solving the mode equations numerically, across a range of k. But, assuming you wish to use the slow roll expressions for [itex]P_S[/itex] and [itex]P_T[/itex], then to get r at, say, the CMB quadrupole, you would evaluate [itex] r = P_T/P_S[/itex] at N = 60.

 

[shinobi20]

A few things. If you wish to compute r without reference to slow roll, you can't use the expression you posted for [itex]P_S[/itex], since it is lowest-order in slow roll (specifically, the quantity [itex]H/2\pi[/itex]). The only way to do this in general is to compute [itex]P_S[/itex] and [itex]P_T[/itex] by solving the mode equations numerically, across a range of k. But, assuming you wish to use the slow roll expressions for [itex]P_S[/itex] and [itex]P_T[/itex], then to get r at, say, the CMB quadrupole, you would evaluate [itex] r = P_T/P_S[/itex] at N = 60.

Oh... So what I usually see in cosmology books and some in the literature are approximations for ##P_S## and ##P_T##. So how would I go on to numerically solve the mode equation to get ##P_S## and ##P_T##? Do you know of any reference/papers that detail this method?

In the literature, is it often the case that physicists just use the lowest order approximation for ##P_S## and ##P_T##? What justification did they consider?

 

[bapowell]

Yes. In fact, I did this for my thesis ;). See this paper for a discussion of the numerical solution: https://arxiv.org/pdf/0706.1982.pdf. In that work we had a weird way of including the background cosmology in the mode equation (basically, we did Monte Carlo over inflationary solutions described in terms of Taylor expansions of the Hubble parameter, the so-called flow method). You probably won't set your problem up this way, opting instead to solve the mode equation along with the Klein Gordon equation for a specific potential and initial condition on [itex]\dot{\phi}[/itex]. I've done it this way too, but haven't published the code anywhere. That said, the above reference might still be useful for its discussion of how to intialize the mode functions and how to build P(k) from individual solutions. That stuff is necessary regardless of how you are thinking about the background, but I'm happy to provide any insights on how to do the solution with the KG for specific potentials.

As for the literature, the lowest-order expressions are often used because authors are often working in the slow roll limit, or close to it. There are higher-order approximations to these results (papers by Stewart and Lyth come to mind, e.g. https://arxiv.org/pdf/gr-qc/9302019.pdf) that are useful if you need to go beyond basic slow roll, but there's no substitute for numerical computation if you want the full-fledged spectrum regardless of assumptions about the background expansion (you'll see that in our paper referenced above, we used the numerical computation to find power spectra for strongly non-slow roll inflation models).

 

[shinobi20]

Yes. In fact, I did this for my thesis ;). See this paper for a discussion of the numerical solution: https://arxiv.org/pdf/0706.1982.pdf. In that work we had a weird way of including the background cosmology in the mode equation (basically, we did Monte Carlo over inflationary solutions described in terms of Taylor expansions of the Hubble parameter, the so-called flow method). You probably won't set your problem up this way, opting instead to solve the mode equation along with the Klein Gordon equation for a specific potential and initial condition on [itex]\dot{\phi}[/itex]. I've done it this way too, but haven't published the code anywhere. That said, the above reference might still be useful for its discussion of how to intialize the mode functions and how to build P(k) from individual solutions. That stuff is necessary regardless of how you are thinking about the background, but I'm happy to provide any insights on how to do the solution with the KG for specific potentials.

As for the literature, the lowest-order expressions are often used because authors are often working in the slow roll limit, or close to it. There are higher-order approximations to these results (papers by Stewart and Lyth come to mind, e.g. https://arxiv.org/pdf/gr-qc/9302019.pdf) that are useful if you need to go beyond basic slow roll, but there's no substitute for numerical computation if you want the full-fledged spectrum regardless of assumptions about the background expansion (you'll see that in our paper referenced above, we used the numerical computation to find power spectra for strongly non-slow roll inflation models).

Thanks for that resource! And you published a paper with Kinney, I've followed some of his notes on inflation. I have solved the dynamical equations of inflation (KG equation, Friedmann equations, etc) simultaneously in Mathematica but in the context of warm inflation, then plot out say, ##\phi##, ##\dot\phi##, ##H##, etc. So I think I'd just use the lowest order in slow roll approximation for ##P_S## and ##P_T##, I think solving it from full numerical calculation would be hard for me now (just starting in this field). But at what point in the plot of say ##\dot\phi## vs ##N## should I choose ##\dot\phi_*## to plug into ##P_S##? I think ##H## is not a problem since it looks approximately constant so I can just measure the value of ##H## anywhere in the plot and set it as ##H_*##.

We say that we countdown the number of e-foldings (N=60, 59, 58, ..., 0) from the start of slow-roll inflation and when we say that we calculate a quantity at the horizon crossing we mean say at N=60 (start of slow roll inflation, while in the plot it is N=0) so does that mean ##\dot\phi_*## is the initial condition itself?

 

[bapowell]

Which N you choose depends on what length scale you wish to measure r on. The mapping between and N and k is model dependent, but for a given model, once you pick the physical length scale of interest (the scale [itex]k = 0.002 h^{-1}{\rm Mpc}[/itex] was a standard one back in the day, as was [itex]k = 0.05 h^{-1}{\rm Mpc}[/itex]), you can determine the corresponding N. Do you know how to map k to N?

 

[shinobi20]

Which N you choose depends on what length scale you wish to measure r on. The mapping between and N and k is model dependent, but for a given model, once you pick the physical length scale of interest (the scale [itex]k = 0.002 h^{-1}{\rm Mpc}[/itex] was a standard one back in the day, as was [itex]k = 0.05 h^{-1}{\rm Mpc}[/itex]), you can determine the corresponding N. Do you know how to map k to N?

It is N, it starts from N=0, 1, ..., 60. Should I set N= -60, -59, -58,... ,0? I don't know yet the relationship between k and N.

 

[bapowell]

You can find the relationship by taking the derivative of k = aH wrt N...

 

[shinobi20]

Which N you choose depends on what length scale you wish to measure r on. The mapping between and N and k is model dependent, but for a given model, once you pick the physical length scale of interest (the scale [itex]k = 0.002 h^{-1}{\rm Mpc}[/itex] was a standard one back in the day, as was [itex]k = 0.05 h^{-1}{\rm Mpc}[/itex]), you can determine the corresponding N. Do you know how to map k to N?

Wait, ##r## is usually measured at the onset of inflation right? What's the significance of the length scale to choosing N? N should just be at least 60 e-folding right?

 

[bapowell]

It's measured on CMB scales, which generally do not correspond to the onset of inflation. Remember, inflation can last arbitrarily long, for 1000s of e-folds or more. The time that we can probe with CMB and LSS is the dozen or so e-folds around N=60, corresponding to length scales from the CMB quadrupole down to around [itex]k \approx 0.1 {\rm Mpc}^{-1}[/itex]. You can measure r anywhere in this observational window, but generally a standard "pivot" point is chosen. Check out the latest Planck results etc to see where people are currently constraining these observables (you need to be careful: since the scalar and tensor spectra generally have different shapes, r will vary by scale and so comparison with other's results will require that you evaluate it at the same scale!) The two pivots I quoted above were standards back when I was writing papers, 5-10 years ago.

 

[shinobi20]

It's measured on CMB scales, which generally do not correspond to the onset of inflation. Remember, inflation can last arbitrarily long, for 1000s of e-folds or more. The time that we can probe with CMB and LSS is the dozen or so e-folds around N=60, corresponding to length scales from the CMB quadrupole down to around [itex]k \approx 0.1 {\rm Mpc}^{-1}[/itex]. You can measure r anywhere in this observational window, but generally a standard "pivot" point is chosen. Check out the latest Planck results etc to see where people are currently constraining these observables (you need to be careful: since the scalar and tensor spectra generally have different shapes, r will vary by scale and so comparison with other's results will require that you evaluate it at the same scale!) The two pivots I quoted above were standards back when I was writing papers, 5-10 years ago.

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##?

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.

 

[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.

 

[shinobi20]

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##?

 

[shinobi20]

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##)

 

[shinobi20]

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.

 

[shinobi20]

##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!

 

[shinobi20]

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##?