1 - Introduction

As we have seen before, when we expand a real function in terms of the Fourier series, each of the terms in the series accounts for changes in the function at different length scales. Similarly, in an expansion of the temperature field on the sphere in terms of the spherical harmonics, different terms (that is, different values of $\ell$ and $m$) capture changes of the temperature across different angular scales.

Nevertheless, the information contained in the list of coefficients (either in the Fourier or the spherical harmonics case) is too much information for what we care about. If you think about the CMB, you can imagine what it would look like from a different observation point. Imagine a different civilization is looking at the CMB just like us, but from a galaxy several gigaparsecs away. We both exist in the same universe with the same laws of physics. We are both even looking at the same parts of the Universe, and receiving photons that came from the same region of the Universe during recombination. Nevertheless, the distribution of temperatures across the sky would look completely different, just as a result of observing the Universe from a different location.

This means that the temperature field $T(\theta,\phi)$ contains some information that is subjective to our point of view, and that are accidental rather than fundamental. This is similar to what one encounters if one is observing the distribution of planets around us. We'll find some planets close to us, some others far from us. Some big, some small, some heavy, some light. We know that the mass of Jupiter, or the exact distance between Saturn and the Sun are accidents, not fundamental. Similarly, the fact that the temperature at $(\theta,\phi)=(23^\circ,261^\circ)$ is $10\%$ larger than at $(\theta,\phi)=(78^\circ,13^\circ)$ is just an accident. We should not give too much attention, then, to the specific value of the temperature at a specific location. Since the temperature field is fully captured by the $a_{\ell m}$ coefficients, this means also that we shouldn't care that much about the specific value of each of the $a_{\ell m}$ coefficients. What should we then pay attention to?

2 - Cosmological Perturbation theory

In a first introduction to cosmology, one usually studies the universe as if it was perfectly homogeneous and isotropic. This is done by a series of reasons:

On theoretical principles (the cosmological principle), it is reasonable to expect empty space to be perfectly isotropic and homogeneous: every point of space should be equivalent to any other point if there's nothing in it. By extension, we could also assume that if there is anything in space, it should also be spread in a homogeneous and isotropic way.
Experimentally, the Universe is observed to be homogeneous and isotropic to a good approximation when looked at large enough scales.
In practical terms, cosmology can be studied using perturbation theory in terms of the inhomogeneities and anisotropies. For example, if we consider the density of matter across space, it is in general a function of spacetime $\rho(t,\mathbf x)$. We can write this as $\rho(t,\mathbf x)=\rho_0+\delta\rho(t,\mathbf x)$. If we assume that $|\delta\rho(t,\mathbf x)|\ll \rho_0$, we can solve equations a different orders of $\delta\rho(t,\mathbf x)$, first considering only the constant term, then only terms up to first order in $\delta\rho(t,\mathbf x)$, etc. This is only possible to the extent that $|\delta\rho(t,\mathbf x)|\ll \rho_0$, which is only true at large scales, at which the Universe is empirically observed to be homogeneous and isotropic.

Under the assumptions of homogeneity and isotropy, anything that comes as a product whatever occurred during the early Universe should also be homogeneous and isotropic. This applies to the photons that escaped after recombination. Therefore, under this version of Cosmology, the temperature field of the CMB should be a constant across the sky. In such case, only the spherical harmonic with $(\ell,m)=(0,0$ would contribute to the spherical harmonic expansion. This version of cosmology doesn't lack interest, but it only allows us to study the evolution of the Universe as a whole, considering that it looks exactly the same everywhere.

CMB physics becomes only interesting when we go beyond the zero order approximation described above, and we actually consider densities (as well as many other quantities) to be different at different locations of space. Very often, one considers only perturbations at first order. That is, terms(\delta\rho(t,\mathbf x)\) are kept in all equations, but terms $\delta\rho(t,\mathbf x)^n$ with $n>1$ are not kept. This is called linear perturbation theory.

An advanced course in cosmology usually consists on solving Einstein's equations at first order in perturbation theory, which is a rather arduous task. Nevertheless, Einstein's equations are differential equations, and they are conceptually not hard to understand. The main point is this: the value of any quantity (like the density, pressure, curvature, etc) at a given time can be obtained using the differential equations if their value is known at a previous time (what is usually known as an initial condition). The relation between the initial conditions and the value at a later time (like today, the observation time) is very complicated, and it requires an understanding of all physical processes involved. But the following is the key message: the value of the CMB temperature at each direction of the sky as seen from Earth is a direct consequence of the initial value of all relevant physical quantities at all positions in space.

If we want to produce theoretical predictions of what the CMB looks like today we then need to understand (i) the initial conditions of the Universe at some time at or after the Big Bang, and all the physical processes that may take place since then. This seems quite ambiguous (and it is), but it is at the same time what makes cosmology a beautiful field of study. Understanding all physical processes is obviously important, but not relevant for the point that I want to make in this section of the notes. Here, I am more interested in the initial conditions, of which I talk more about in the following section.

3 - The initial conditions

The Universe is quite complicated, and a full description of its state at any time requires the specification of a large number of quantities some related to the values of the spacetime metric, and some to the concentration of all different types of matter. For illustration, I will talk here just about the energy density $\rho(t,\mathbf x)$ in general, although this can be applied to all matter species. With this, we can summarize the previous section as follows: if we know the value of $\rho(t,\mathbf x)$ at an initial time $t=t_0$ for all points of space, we can calculate its value at any later time $t\gt t_0$ everywhere in space.

The question then is, how do we describe the initial conditions $\rho(t_0,\mathbf x)$ and what is the initial time $t_0$. If we want to have the most predictive power, we would like the initial time $t_0$ to be as close as possible to the Big Bang. There is not much value (to Cosmology) in explaining todays observations based on observations of one year ago. We would like, if possible, explain today's observations based on theoretical principles about what was going on right after the Big Bang. That is the current state of Cosmology.

The earliest state of the Universe of which we have a mathematical model is the period of inflation. If inflation occurred, it did so in the period between $10^{-33}$ and $10^{-32}$ seconds after the Big Bang. The actual physical stuff that existed during inflation is understood to be a quantum field; physicists play around with the properties of this quantum field and explore different possibilities. This is other of those highly complicated parts of cosmology; quantum field theory a very complex subject, much more so than general relativity, and definitely one of which we don't have a very clean understanding). Still, there are some fundamental concepts about quantum field theory and quantum mechanics that can allow us to conceptually understand what the initial conditions for the universe may and may not be.

Quantum fields are fields (quantities defined everywhere in space) that are subject to the laws of quantum mechanics. Among many (many) other things, this means that quantities are described by wavefunctions, which have intrinsic uncertainties. You can think of a quantum field as a field that each location of space is described by a wavefunction, so that the actual value of the field is not perfectly defined. This means that the field can't be quiet, it can't just stay still at a fixed value. Realize how important this is: it doesn't matter that empty space is fully homogeneous and isotropic; as soon as a quantum field is put in it, it will necessarily oscillate at random, breaking down the perfect isotropy and homogeneity.

In the current state of cosmology, there is an inflationary period very soon after the Big Bang. After that is over, the inflation quantum field somehow decays into other types of quantum fields (the ones that we can actually study here on Earth, as described by the standard model of particle physics, like photons, electrons, quarks, etc). Then, we have a pretty much full account of everything that happens to those fields from the early Universe until today. So the question about the initial conditions, if they are generated by the inflation field, is the following: how do we describe the fluctuations of the inflation field, and how do they get transferred into the other quantum fields that we know of?

Due to the nature of quantum mechanics, the fluctuations of the inflation field are random, truly random. Just like any other random thing, it is described by a probability distribution. And in quantum mechanics, the probability distribution is pretty much given by the square of the wavefunction. Then, all we need to understand the fluctuations of the inflation field is a wavefunction. The wavefunction is obtained solving Schrodingers equation, which can be solved once we specify what is the hamiltonian that describes the system. So once again, we reduce the understanding of the initial conditions to a new question: what is the hamiltonian that describes the inflation field? Well, we just don't know, but we can try. That is, we can choose different types of hamiltonian, study the fluctuations that they produce, and treat those as the initial conditions for the rest of the evolution of the universe.