I want a good article that will help me generate the CMBR spectrum from first principles of basic cosmological perturbation theory. In other words, I want to start with the cosmological perturbation theory, and want to derive by hand (i) the origin of the different peaks, (ii) way of inferring things like curvature of universe or baryonic/dark matter content by looking at the CMBR spectrum, (iii) reason why the spectrum is plotted only as a function of $\ell$, not $\ell \ \& \ m$, as it should be in the case of spherical harmonics.
1 Answer
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Here are some comments on the physics.
(i) the origin of the different peaks,
The peaks are remnants of sound waves in the primordial at the time of recombination. The size of the peak at a given angular scale is predominantly determined by modes of a corresponding wavenumber. The phase is determined by the initial conditions of the Universe; some modes are at their peak at the time of recombination (leading to peaks), and others are close to zero (leading to troughs).
Meanwhile, dark matter tends to enhance even peaks of the power spectrum. These peaks arise from overdensities in the cosmic fluid. What happens is that dark matter forms clumps, gravitationally, and does not oscillate with the fluid. When the amplitude of the fluid density wave increases, it means there is an overdensity; this overdensity is increased by the dark matter clump. When the sound waves go the other way and form a density minimum, the fluid elements have to climb out of the gravitational field produced by the dark matter, and so these density minima have a lower amplitude than the pressure maxima. The alternating pressure maxima and minima translate into peaks in the power spectrum, with even peaks (density maxima) enhanced by the effect of the dark matter, and odd peaks (density minima) reduced.
(ii) way of inferring things like curvature of universe or baryonic/dark matter content by looking at the CMBR spectrum,
The way this is done in practice is to do a giant Bayesian fit by computing the spectrum associated with a set of cosmological parameters and comparing to data.
You can get intuition for what aspects of the spectrum are affected by which parameters by studying various limits and special cases. The curvature of the Universe, for example, arises because the wavelength associated with the largest peak in the CMB angular power spectrum forms a standard ruler. Since we now the distance between adjacent peaks at recombination (it corresponds to a wavenumber, which corresponds to an angular size), and since we know the redshift of the CMB and hence its distance (particularly the angular diameter distance), we can form a triangle with us at one corner and two peaks on the other corners. The geometry of this triangle lets us determine to very good precision whether the triangle has been drawn on a curved or flat surface.
(iii) reason why the spectrum is plotted only as a function of ℓ, not ℓ & 𝑚, as it should be in the case of spherical harmonics.
The CMB is statistically isotropic, meaning that \begin{equation} \langle a_{\ell m}^\star a_{\ell' m'} \rangle = P_\ell \delta_{\ell \ell'} \delta_{m m'} \end{equation} What is plotted (up to some normalization) is the $P_\ell$, which does not depend on $m$ because of the assumption of statistical isotropy. It's quite a good thing that there is no $m$ dependence in $P_\ell$ -- this means that for each $\ell$, the different values of $m$ give us $2\ell + 1$ independent measurements of $P_\ell$. Therefore, even though we have only one Universe, we get many measurements of $P_\ell$ for $\ell$ large enough, reducing cosmic variance.
There are many, many references.
- Wayne Hu's tutorial pages are an excellent start. He also has a good review
- I fond Mukhanov's textbook to be excellent.
- Other books include Dodelson, Kolb and Turner, and Weinberg. At a lower level which won't cover all of this detail is Ryden.