# A Calculating the power spectra of scalar perturbation

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1. May 16, 2017

### 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?

2. May 18, 2017

### bapowell

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

3. May 18, 2017

### shinobi20

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,

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

4. May 18, 2017

### bapowell

What are you trying to do, determine $P(k)$?

5. May 18, 2017

### shinobi20

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

6. May 18, 2017

### 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.

7. May 18, 2017

### shinobi20

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?

8. May 18, 2017

### bapowell

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

9. May 18, 2017

### shinobi20

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.

10. May 18, 2017

### 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.

11. May 18, 2017

### shinobi20

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?

12. May 18, 2017

### 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...

13. May 18, 2017

### shinobi20

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 $\dot\phi_*$.

14. May 18, 2017

### 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?

15. May 18, 2017

### shinobi20

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?

16. May 18, 2017

### bapowell

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

17. May 18, 2017

### shinobi20

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?

18. May 18, 2017

### 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 $\dot{\phi}$. 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).

19. May 18, 2017

### shinobi20

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?

20. May 18, 2017

### 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 $k = 0.002 h^{-1}{\rm Mpc}$ was a standard one back in the day, as was $k = 0.05 h^{-1}{\rm Mpc}$), you can determine the corresponding N. Do you know how to map k to N?