Initial conditions for tensor perturbations

[WannabeNewton]

Hi all. Say we have a background inflaton field $\varphi$ and that we've integrated the background equation for $\varphi$, $H(\eta)$, and $a(\eta)$ up to the number of e-folds of inflation corresponding to $\epsilon = 1$ in the slow-roll parameter. We then wish to solve for the $k$ modes of the tensor perturbation in the transverse-traceless gauge using $\ddot{h}+ 2\frac{\dot{a}}{a}\dot{h} + k^2 h = 0$ by plugging in the $a(\eta)$ solved from the background (the tensor modes don't couple to scalar perturbations $\delta \varphi$) where a dot means derivative with respect to conformal time. To do this we would need to know the initial values of $h$ and $\dot{h}$ at conformal time $\eta(N = 0)$ where $N$ is e-folds of inflation. In Dodelson, it is noted that at early enough times all $k$ modes are subhorizon which reduces the equation $\ddot{v} + (k^2 - \frac{\ddot{a}}{a})v = 0$ for the coefficient $v$ of the creation operator $\hat{a}_k$ associated with a given mode to the equation for a simple harmonic oscillator. This gives us boundary conditions on $h$ and $\dot{h}$ at early times, including the initial time of inflation.

But is there a more rigorous, possibly model-dependent, but experimentally based method of obtaining the initial conditions for the $k$ modes of the tensor perturbations? For example when solving for $\varphi$ from the Klein-Gordon equation, one can use the approximately constant energy scale of inflation to set the initial value for $\varphi$ and use the slow-roll condition to set the initial value for the background inflaton $\dot{\varphi}$ and tune these initial values based on agreement with experiment. However for $h$ the choices of initial $h$ and $\dot{h}$ are based solely on the harmonic oscillator boundary conditions so I was wondering if, in practice, one chooses these initial conditions in a way more akin to the way one would choose the initial conditions for $\varphi$.

I would also appreciate any references on the general framework of choosing initial conditions for background fields and perturbations in inflation.

As an aside, any references discussing the range of physical modes $\frac{\Delta k}{a(\eta)}$ at the present time that one would be interested in for tensor perturbations would be quite helpful as well. Thanks in advance.

 

[Chalnoth]

The k-modes are produced during inflation. They stem from random quantum vacuum fluctuations in the inflating universe.

 

[bapowell]

A couple points. The variable N is the number of efolds before the end of inflation, so that N=0 marks the end of the inflationary expansion, not the beginning. Generally, you wish to take N_{init} large enough so that all k-modes of observational interest are in the vacuum at the initial time.

Regarding the vacuum state -- the initial conditions on the modes -- this is very much model-dependent. The standard computation takes the modes to be in the so-called Bunch-Davies vacuum at early times (this is the closest thing to the no-particle state during de Sitter expansion -- it is the vacuum perceived by an inertial observer and is equivalent to the 0th-order adiabatic vacuum). The arbitrariness of selecting this state is emphasized in the transplanckian problem of inflationary perturbations: see http://arxiv.org/abs/hep-th/0203198 for a quick and easy introduction to these ideas. Essentially, the idea is that quantum gravitational physics must ultimately inform the nature of the quantum vacuum, i.e. it is ultimately an assumption that the modes reduce to plain wave solutions as small wavelength.

There has also been much work considering non-vacuum initial states as a source of non-Gaussian large-scale perturbations.

Regarding the range of scales, I don't have the numbers with me right now, but they run from the present-day horizon (the CMB quadrupole) down to the horizon size at decoupling (essentially the first acoustic peak). Tensor perturbations on smaller scales have redshifted away by the present epoch.

 

[WannabeNewton]

Hi bapowell, thanks for the reply!

A couple points. The variable N is the number of efolds before the end of inflation, so that N=0 marks the end of the inflationary expansion, not the beginning.

I should have mentioned that my normalization is $a_i = 1$ at the start of inflation so that $N = 0$ initially since with this normalization $a = e^{N}$.

Generally, you wish to take N_{init} large enough so that all k-modes of observational interest are in the vacuum at the initial time.

I'm not sure what you mean by "in the vacuum at the initial time". Could you explain that? Do you mean that for any given $k$ of interest, the associated tensor perturbation mode $h_k$ is in its vacuum state initially, with the vacuum chosen to be the minimum energy vacuum in quasi de-Sitter? My understanding is that we take $N$ large enough so as to have all $k$ modes of interest on the superhorizon scale so that they freeze and ten we can evaluate the power spectrum at this $N$ for the range of $k$.

The arbitrariness of selecting this state is emphasized in the transplanckian problem of inflationary perturbations: see http://arxiv.org/abs/hep-th/0203198 for a quick and easy introduction to these ideas. Essentially, the idea is that quantum gravitational physics must ultimately inform the nature of the quantum vacuum, i.e. it is ultimately an assumption that the modes reduce to plain wave solutions as small wavelength.

Thank you for the information! However I can't seem to access the paper itself that you linked, it says access denied. But as it turns out, there is also quite a lucid discussion of the Bunch-Davies vacuum in Baumann's TASI lectures.

As a bit of an unrelated question, as of now I have a very ad-hoc method of getting the initial conditions for the background inflaton $\varphi$. Basically $\varphi_i$ is chosen so that $\rho_i \propto H_i^2$ is the known energy scale of inflation and $\dot{\varphi}_i$ is chosen so that the initial value of the slow-roll parameter satisfies $\epsilon_i \ll 1$ i.e. initially the kinetic energy is much less than the potential. I think the choice of $\varphi_i$ based on the energy scale of inflation is rather standard but I'm not sure really. Then I tune the initial conditions to fit observational data through e.g. the power spectrum for a range of $k$ modes of the tensor perturbation $h$. Regardless, in practice how are $\varphi_i$ and $\dot{\varphi}_i$ usually chosen?

Regarding the range of scales, I don't have the numbers with me right now, but they run from the present-day horizon (the CMB quadrupole) down to the horizon size at decoupling (essentially the first acoustic peak).

Yeah it turns out that for now I'm just working with the $k$ modes in the NANOgrav range.

If you don't mind could you explain what you mean by the CMB quadrupole being the present-day horizon? Does that mean that the length scale associated with the $l = 2$ part of the CMB temperature perturbation $\Theta$ is on the order of the present-day Hubble radius $(aH)^{-1}$? Or on the order of the comoving horizon $\eta$?

 

[bapowell]

I'm not sure what you mean by "in the vacuum at the initial time". Could you explain that? Do you mean that for any given $k$ of interest, the associated tensor perturbation mode $h_k$ is in its vacuum state initially, with the vacuum chosen to be the minimum energy vacuum in quasi de-Sitter? My understanding is that we take $N$ large enough so as to have all $k$ modes of interest on the superhorizon scale so that they freeze and ten we can evaluate the power spectrum at this $N$ for the range of $k$.

So this only makes sense with the conventional definition of N signifying the number of efolds before the end of inflation, so that modes are initialized in the vacuum at very large N (the distant past) and evolved forwards in time, exiting the horizon at N for which $k = aH$, through to the end of inflation at $N=0$.

Thank you for the information! However I can't seem to access the paper itself that you linked, it says access denied.

Too bad. It is a paper by Ulf Danielsson called "A note on inflation and transplanckian physics". Should be available on the arxiv.

A I think the choice of $\varphi_i$ based on the energy scale of inflation is rather standard but I'm not sure really. Then I tune the initial conditions to fit observational data through e.g. the power spectrum for a range of $k$ modes of the tensor perturbation $h$. Regardless, in practice how are $\varphi_i$ and $\dot{\varphi}_i$ usually chosen?

The initial values of $\varphi$ and $\dot{\varphi}$ are arbitrary, chosen to satisfy the conditions for inflation.

If you don't mind could you explain what you mean by the CMB quadrupole being the present-day horizon? Does that mean that the length scale associated with the $l = 2$ part of the CMB temperature perturbation $\Theta$ is on the order of the present-day Hubble radius $(aH)^{-1}$?

Yes, exactly.

 

[Mordred]

it is avail on arxiv
A note on inflation and transplanckian physics
http://arxiv.org/abs/hep-th/0203198

 

[WannabeNewton]

Thank you! I had one more question, this one quite unrelated but short enough not to warrant a whole other thread. Do you know of any pedagogical references on neutrino damping of tensor perturbations? All I could find was a paper by Weinberg but it was extremely concise.

 

[bapowell]

What's the Weinberg paper so that I have a better idea of the subject matter?

 

[George Jones]

What's the Weinberg paper so that I have a better idea of the subject matter?

I think WannabeNewton means

http://arxiv.org/abs/astro-ph/0306304

WannabeNewton, does Weinberg explain it any better in his 2008 text "Cosmology"?

 

[WannabeNewton]

http://arxiv.org/abs/astro-ph/0306304

Yes that's the one.

WannabeNewton, does Weinberg explain it any better in his 2008 text "Cosmology"?

Unfortunately not, at least in my opinion; it's extremely dense. His paper is actually better.

Something at the level of Dodelson, Liddle, or Mukhanov would be ideal :)