Can we produce the CMB power spectrum from a given model of inflation, theoretically? If yes, how?

Question

To the best of my knowledge, the CMB power spectrum is obtained from a statistical analysis of the observed temperature anisotropies of the CMB sky. Is there a way of getting the power spectrum theoretically from the quantum aspects of a given model of inflation? In that case, one can compare that with the experimentally obtained power spectrum. If yes, I would love to know how is this accomplished.

Answer 1

Absolutely! This is how we learn about the physics of inflation. Given an inflaton potential, we can work out the spectrum of density perturbations it generates. With this power spectrum (plus several so-called "late-time" parameters), we can compute the expected CMB temperature and polarization anisotropies. This is a very involved, multi-step analysis so I will only highlight the main parts here.

Given the scalar inflaton field with potential $V(\phi)$, we write the field as a perturbation about a homogeneous value, $\phi({\bf x},t) = \phi_0(t) + \delta \phi({\bf x},t)$, drop it into the Klein Gordon Equation, and obtain, $$\ddot{\delta \phi} + 3H\dot{\delta \phi} -\left(\frac{\nabla^2}{a^2} - \left.\frac{{\rm d}^2V(\phi)}{{\rm d}\phi^2}\right|_{\phi = \phi_0}\right)\delta \phi = 0.$$ Moving to Fourier space gives $$\ddot{\delta \phi_k} + 3H\dot{\delta \phi_k} + \left(\frac{k}{a}\right)^2\delta \phi_k = 0.$$ This equation has no general analytic solution, but we can solve it exactly for de Sitter expansion ($H = const$) and then add perturbations to the solution to study the quasi-de Sitter expansion observed during real inflation (so-called slow roll inflation, in which time derivatives of $H$ are increasingly suppressed at higher-order). The quantity of interest is the variance (or mean squared amplitude) of the fluctuation, $$\langle | \delta \phi_k|^2\rangle = \frac{H^2}{2k^3}.$$ This is the variance of the field on horizon scales, where the quantum fluctuation is becoming classical. The variance of the full fluctuation also includes a piece from the classical evolution of the field, which is $H^2/\dot{\phi}^2$, giving $$\langle | \delta_k|^2\rangle = \frac{H^2}{\dot{\phi}^2}\langle |\delta \phi_k|^2\rangle = \frac{H^4}{2\dot{\phi}^2 k^3}.$$

From the total variance, we can form the power spectrum, defined $$P(k) = \frac{k^3}{2\pi^2} \langle | \delta_k|^2\rangle = \frac{1}{4\pi}\frac{H^4}{\dot{\phi}^2}.$$ Notice that the $k$-dependence is not explicit in this expression: it actually arises from the time-dependence of the cosmological quantities $H$ and $\dot{\phi}$. As modes evolve in time, their amplitudes freeze once they cross the horizon, $k = aH$, and so this expression is to be evaluated at that time. Once we have the power spectrum across the scales of interest, we use it to "seed" the density perturbations in the baryon-photon plasma. These perturbations are what are observed as anisotropies in the temperature and polarization of the CMB.

Since the observed CMB is influenced by more than simply the initial spectrum (the growth of the perturbations from their birth up until last scattering depends on things like the baryon and CDM densities, for example), we must specify several more quantities to obtain the final CMB spectrum. Thankfully, there is publicly-available software that does this (e.g. CAMB). So, for a given initial spectrum $P(k)$ and values for these other quantities, we obtain a set of CMB spectra we can use to compare our model with observations.

Slow roll inflation gives rise to a power law spectrum, $$P(k) = P(k_0) \left(\frac{k}{k_0}\right)^{n-1},$$ where the various parameters can be written in terms of the inflaton potential and field. For example, the spectral index, which governs the tilt of the spectrum, is determined by the first and second derivatives of the potential, $$n -1 = \frac{m_{\rm Pl}^2}{4\pi}\left[\frac{3}{2}\left(\frac{V'}{V}\right)^2 - \frac{V''}{V}\right].$$
This is an example of how a determination of the spectral index (inferred through the CMB anisotropies) can be used to learn about the shape of the inflaton potential. Other spectral parameters, like the overall amplitude and higher-order $k$-dependence of the spectrum, tell us about the height of the potential and the higher-order $\phi$-terms in its Taylor expansion, respectively.

This has been a very high-level and terse answer, since easily an entire textbook could be written that addresses just this topic.