# Table of contents

1. Introduction to theoretical cosmology 2. Background cosmology

- 2.1 The Cosmological Principle
- 2.2 Mass-Energy content of the Universe
- 2.3 Derivation of the Friedman equations
- 2.4 Cosmological parameters and the standard model of cosmology
- 2.5 Measures of time in cosmology
- 2.6 Measures of distances in cosmology
- 2.7 Summary

# Introduction to theoretical cosmology

Here are some slides (or download the PDF if it you can't see it below).

# Background Cosmology

Here are some notes on background cosmology in addition to the slides in the previous section which also covers this. For more details see Chapter 1 and Chapter 3 in Baumann.

## The Cosmological Principle

Figure: The possible geometries of the Universe.

The Cosmological principle says that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale. Isotropic simply means the Universe looks the same (in a statistical sense) in every direction and homogenous means it looks the same no matter where you are. In modern cosmology this is not a principle per se, that the Universe has to obey, but its more of a convenient assumption whos validity can be tested with observations. The main evidence for isotropy comes from the cosmic microwave background whose temperature is found to be the same in every direction on the sky and the main evidence for homogenity comes from looking at the large scale distribution on galaxies. Why is it such a convenient assumption? Because it places strong constraints on the possible geometries of the Universe. The metric we use for describing the geometry of the Universe must take a very particular (and simple form): it must be the Friedmann–Lemaître–Robertson–Walker metric (FRLW) which in polar coordinates reads $$ds^2 = -dt^2 + a(t)^2(\frac{dr^2}{1-kr^2} + r^2d\Omega^2)$$ where $a(t)$ is the scale-factor (the "size" of the Universe) and $k$ denotes the curvature of the Universe. For this we have only three different options: $k\gt 0$ for which we have a closed Universe (like the surface of a sphere; in this case $1/\sqrt{k}$ would then be the "radius"), $k\lt 0$ for which we have an open Universe and $k=0$ for which the Universe is flat. Another way of describing these options is how triangles looks like: in a flat universe the angles sum to $180$ degrees, in a closed Universe to $\gt 180$ degrees and in an open Universe to $\lt 180 degrees$. Thus an object will look larger in a closed Universe and smaller in an open Universe than in a flat Universe. For us this is all we need to know and will be very relevant when we talk about how we can constrain the curvature of the Universe from how the CMB looks. Observations so far point to the Universe being flat so in this course we will for simplicity assume $k=0$ for which the metric can be written in usual Cartesian coordinates as $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ This only has one free function $a(t)$ which tells us all we need to know about how fast the Universe expands and how stuff moves in our Universe at large. This is the thing we want to measure in cosmology and the key equation that determines how it evolves is called the Friedmann equation and follows from plugging the metric above into the Einstein equations. Before we can evaluate these for the given metric we need to know how to model the right hand side of the Einstein equations: the energy momentum tensor for the Universe as a whole.

## Mass-Energy content of the Universe

For studying background cosmology all energy forms are modelled as *perfect fluids*. A perfect fluid is a fluid that can be completely characterized by its rest frame mass density $\rho$ and isotropic pressure $p$. The energy momentum tensor for such a fluid takes the simple form
$$T^{\mu\nu} = U^\mu U^\nu(\rho + p) + pg^{\mu\nu}$$
where $U^\mu$ is the 4-velocity of the fluid. In the rest frame $U^\mu = (1,0,0,0)$ and it takes the simple form
$$T^\mu_\nu = \pmatrix{-\rho & 0 & 0 & 0\\ 0 & p & 0 & 0\\ 0 & 0 & p & 0\\ 0 & 0 & 0 & p\\}$$
i.e. $T^0_0 = -\rho$ and $T^i_j = p\delta_{ij}$. An important quantity is the *equation of state*
$$w = \frac{p}{\rho}$$
which tells us what the pressure is given the energy density. For example normal nonrelativistic matter (dust) or cold dark matter has $w\approx 0$, a relativistic fluid has $w = \frac{1}{3}$ and dark energy in the form of a cosmological constant has $w = -1$ ($p=-\rho$ and $T^{\mu\nu} = -\frac{\Lambda}{8\pi G} g^{\mu\nu}$). Curvature (in the Friedman equations) enters as if it was a form of energy with equation of state $w = -\frac{1}{3}$.

The conservation of energy-momentum in general relativity (in the absence of interactions between different fluiods) is expressed by $\nabla_\mu T^{\mu\nu} = 0$ for each fluid individually. For a FRLW Universe this becomes $$\frac{d\rho}{dt} + 3H(\rho+p) = 0$$ or using the equation of state $$\frac{d\rho}{dt} + 3H\rho(1+w) = 0$$ For a constant equation of state the solution is $$\rho \propto \frac{1}{a^{3(1+w)}}$$ and tells us how the energy density is dilluted as the Universe expands. For normal matter $\rho \propto 1/a^3$ - normal volume dilution in an expanding Universe. For radiation $\rho \propto 1/a^4 = 1/a \cdot 1/a^3$. The number of photons are diluted just as for normal matter, but we also have that the wavelength of the photons is streched by the expansion so the energy ($E = h/\lambda$) is reduced. For dark energy in the form of a cosmological constant $\rho = $ constant so the energy density stays constant as the Universe expands! These are the main forms of energy we have in our Universe and the only ones we will need in this course.

- Normal matter ("baryons"): $w = 0$, $\rho_b \propto \frac{1}{a^3}$
- Cold dark matter: $w = 0$, $\rho_{\rm CDM} \propto \frac{1}{a^3}$
- Radiation (photons): $w = \frac{1}{3}$, $\rho_\gamma \propto \frac{1}{a^4}$
- Massless neutrinos: $w = \frac{1}{3}$, $\rho_\nu \propto \frac{1}{a^4}$
- Dark energy (cosmological constant): $w=-1$, $\rho_\Lambda = $ constant
- Curvature: $w = -\frac{1}{3}$, $\rho_k \propto \frac{1}{a^2}$

Even though all these cases have a constant equation of state, this does not have to hold in general as we will see when we discuss inflation later in this course.

## Derivation of the Friedman equations

The brute-force algorithm for getting to the Einstein equations are as follows. Start with the metric $g_{\mu\nu}$. We assume the curvature is zero in this derivation for which doing the derivation in Cartesian coordinates $(t,x,y,z)$ is simplest and the metric (whose form follows from the assumption about homogenity and isotropy) can be written: $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ We have $g_{00} = -1$ and $g_{ij} = a^2(t)\delta_{ij}$. First compute the inverse metric $g^{\mu\nu}$. Since the metric is diagonal this is easy $g^{00} = -1$ and $g^{ij} = a^{-2}(t) \delta^{ij}$. Next compute the Christoffel symbols $$\Gamma^\mu_{\alpha\beta} = \frac{g^{\mu\delta}}{2}(g_{\delta\beta,\alpha} + g_{\alpha\delta,\beta} - g_{\alpha\beta,\delta})$$ To do this systematically divide it into cases where $\mu = 0$ (time index) and $\mu = i$ (a space index). Then notice that since $g$ is diagonal only the term with $\mu = \delta$ will be non-zero. This simplifies the expression to $$\Gamma^0_{\alpha\beta} = \frac{g^{00}}{2}(g_{0\beta,\alpha} + g_{\alpha 0,\beta} - g_{\alpha\beta,0})$$ $$\Gamma^i_{\alpha\beta} = \frac{g^{ii}}{2}(g_{i\beta,\alpha} + g_{\alpha i,\beta} - g_{\alpha\beta,i})$$ and note that there is no implicit summation over $i$ in the last expression. Since the Christoffel symbol is symmetric in the two lower indices we have three cases for each of the two terms: $\alpha\beta=00$ (time-time), $\alpha\beta = 0j$ (one time-one space) and $\alpha\beta = jk$ (space-space). What further simplifies the derivation is that the metric only depends on $t$ so any space derivative ($,i$) vanishes. For example $$\Gamma^0_{00} = \frac{g^{00}}{2}(g_{0j,i} + g_{i0,j} - g_{ij,0}) = \frac{-1}{2}(0 + 0 - g_{ij,0}) = \frac{1}{2}(a^2\delta_{ij})_{,0} = a\dot{a}\delta_{ij}$$ This gives you six different symbols to compute and only two of these turn out to be non-zero namely $$\Gamma^0_{ij} = a\dot{a}\delta_{ij}$$ $$\Gamma^i_{0j} = \Gamma^i_{j0} = \frac{\dot{a}}{a}\delta^i_j$$ Having computed these we can go on to evaluate the Ricci tensor which is given by $$R_{\mu\nu} = \Gamma^\alpha_{\mu\nu,\alpha} - \Gamma^\alpha_{\mu\alpha,\nu} + \Gamma^\alpha_{\mu\nu}\Gamma^\beta_{\alpha\beta} - \Gamma^\beta_{\mu\alpha}\Gamma^\alpha_{\beta\nu}$$ To evaluate this we again split into two cases $R_{00}$ and $R_{ij}$ (there is also $R_{0j}$, but since the metric is diagonal this is forced to be zero). This gives us $$R_{00} = -3\frac{\ddot{a}}{a}$$ $$R_{ij} = (a\ddot{a} + 2\dot{a}^2)\delta_{ij}$$ From this we can compute the Ricci scalar $$R = g^{\mu\nu}R_{\mu\nu} = g^{00}R_{00} + g^{ij}R_{ij} = 6\left(\frac{\ddot{a}}{a} + \frac{\dot{a}^2}{a^2}\right)$$ which finally gives us the Einstein tensor $G_{\mu \nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$. We can go ahead and compute all the different component of this. The $G_{0i}$ component will be zero, the $00$ component gives $$G_{00} = 3\frac{\dot{a}^2}{a^2}$$ and the $ij$ component becomes $$G_{ij} = \left(-2\ddot{a}a - \dot{a}^2\right)\delta_{ij}$$ We are now in the positon to evaluate the Einstein equation $G_{\mu\nu} = 8\pi G T_{\mu\nu}$. Taking the energy content of the Universe to consist of the components $(n)$ (denoting baryons, cold dark matter, radiation, etc.) then $T_{\mu\nu} = \sum_n T_{\mu\nu}^{(n)}$ where each component $(n)$ is assumed to be a perfect fluid $$T_{\mu\nu}^{(n)} = (\rho_n + p_n)u_\mu u_\nu + p_ng_{\mu\nu}$$ where $u_\mu$ is the 4-velocity of the fluid which in our co-moving frame is simply $u^\mu = (1,0,0,0)$ thus $T_{00}^{(n)} = \rho_n$ and $T_{jk}^{(n)} = p_n a^2\delta_{ij}$. The $00$ component of the Einstein equations then gives us the Hubble equation $$H^2 = \frac{8\pi G}{3} \sum_n \rho_n$$ where $H \equiv \frac{\dot{a}}{a}$. The $11$, $22$ and $33$ components gives us 3 more equations (because of isotropy all of these are equal) which reads $$\frac{\ddot{a}}{a} = -\frac{4\pi G}{3}\sum_n (\rho_n + 3p_n)$$ In addition to this we have the conservation equation $\nabla_\mu T^{\mu\nu} = 0$, which in the absence of any interactions between the different species simply reduces to one equation for each component $\nabla_\mu T^{\mu\nu}_{(n)} = 0$ whose $0$ component gives us $$\nabla_\mu T^{\mu 0}_{(n)} = \partial_\mu T^{\mu 0}_{(n)} + \Gamma^\mu_{\mu\alpha} T^{\alpha 0}_{(n)} + \Gamma^0_{\mu\alpha} T^{\mu \alpha}_{(n)} \implies \dot{\rho}_n + 3H(\rho_n + p_n) = 0$$

## Cosmological parameters and the standard model of cosmology

From the Friedmann equation $$H^2 = \frac{8\pi G}{3}\sum_i\rho_i$$ we can define a critical density $\rho_c = \frac{3H^2}{8\pi G}$ so that it reads $$1 = \sum_i \frac{\rho_i}{\rho_c}$$ or by introducing the density parameters $\Omega_i \equiv \frac{\rho_i}{\rho_c}$ $$1 = \sum_i \Omega_i$$ thus $\Omega_i = \Omega_i(a)$ tells us how much of the mass-energy content of the Universe is in the form of component $i$ at any given time. The value of the density parameters today $\Omega_{i0}$ are the cosmological parameters that we aim at constraining with observations. In terms of these the Friedmann equation can be written $$H^2 = H_0^2\sum_i \frac{\Omega_{i0}}{a^{3(1+w_i)}}$$ where we have used that the energy density of a component with equation of state $w_i$ decays as $\rho_i \propto a^{-3(1+w_i)}$.

### The $\Lambda$CDM model

What components do we have in our Universe? We have normal matter ("baryons"), radiation, neutrinos (treated as massless in this course) and we also have to include two additional components: cold dark matter and dark energy in the form of a cosmological constant. The cosmological constant was added after the discovery of the accelerated expansion of the Universe ($\ddot{a} \gt 0$) and we will see some important reasons for why we need dark matter in this course, but if you want to know more about it right now then I reccomend the lecture What We Know and Don't Know about Dark Matter by Neal Weiner on Youtube (though this will make much more sense after we have gone through the thermal history of the Universe). This gives us the so-called $\Lambda$-Cold-Dark-Matter ($\Lambda$CDM) model which is the standard model of cosmology today. In addition to this mass-energy content the model also assumes that gravity is described by general relativity and that the cosmological principle applies. It is fully determined by giving the value of a few parameters: the pressent Hubble rate $h \equiv H_0 / (100$km/s/Mpc) $\,\approx 0.7$, the baryon density $\Omega_{b0} \approx 0.05$, the cold dark matter density $\Omega_{\rm CDM 0} \approx 0.25$, the dark energy density $\Omega_{\Lambda 0} \approx 0.7$, the curvature $\Omega_{k 0} \approx 0$ and the present temperature of the CMB which determines $\Omega_{\gamma 0}$ and $\Omega_{\nu 0}$. The Friedmann equation for this model is $$H(a) = H_0 \sqrt{\Omega_{\Lambda 0} + \frac{\Omega_{k 0}}{a^2} + \frac{\Omega_{b0} + \Omega_{\rm CDM 0}}{a^3} + \frac{\Omega_{\gamma 0} +\Omega_{\nu 0}}{a^4}}$$ where $\Omega_{\Lambda 0} + \Omega_{k 0} + \Omega_{b0} + \Omega_{\rm CDM 0} + \Omega_{\gamma 0} +\Omega_{\nu 0} \equiv 1$. The present time radiation density parameter is (we will derive this later on) $$\Omega_{\gamma 0} = 2\cdot \frac{\pi^2}{30} \frac{(k_bT_{\rm CMB 0})^4}{\hbar^3 c^5} \frac{8\pi G}{3H_0^2}$$ where $T_{\rm CMB 0} \simeq 2.7255$ K is the temperature of the CMB today and since we know this to great precision this is sometimes not considered a free parameter though it really is. Likewise the neutrino energy density is $$\Omega_{\nu 0} = N_{\rm eff}\cdot \frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Omega_{\gamma 0}$$ where $N_{\rm eff} = 3.046$ is the effective number of neutrinos (we'll get back to explaining the $0.046$ part later).

In the minimal setup (with $\Omega_{k0} = 0$ and considering $T_{\rm CMB 0} = 2.7255$ K as fixed) we only have $3$ free parameters: $\Omega_{b0}, \Omega_{\rm CDM0}$ and $h$ (remember $\Omega_{\Lambda 0} = 1 - \sum\Omega_{i0}$ so this is fixed by the other parameters). We will later introduce $3$ more parameters: two related to the initial conditions of the Universe as set up by inflation and one related to the reionization of the Universe at late times (gives us free electrons that will scatter the CMB photons), but that is it. With only $6$ free parameters we are able to fully account for a wide range of observations which is what has made $\Lambda$CDM the standard model of cosmology today.

### Common extensions of the $\Lambda$CDM model

We have a successfull model, but its always a good idea to also test alternatives and people do this all the time. In some extensions of the standard model $N_{\rm eff}$ is treated as a free parameter to allow us to test the possibillity of having extra relativisic species in our Universe, but so far observations is perfectly consistent with $3.046$ and we will use this value in this course. To test if dark energy is really a cosmological constant people often consider extensions where the dark energy density evolves with a constant or time-varying equation of state $w$. One can either do this with conrete models or to look at parameterisations of this effect. For example a commonly used parametrisation for this is the Chevallier-Polarski-Linder (CPL) parametrisation $$w(a) = w_0 + w_a(1-a) \implies \rho_\Lambda \propto \frac{1}{a^{3(1+w_0-w_a)}} e^{3w_a (a-1)}$$ which adds two free parameters (and where a cosmological constant corresponds to $w_0 = -1$ and $w_a=0$) that can be constrained with data. In this course we will only consider a pure cosmological constant so this is not relevant for us, but I just wanted to mention it in case you read about this elsewhere. In this course we only consider the flat case $\Omega_{k0} = 0$, but in general one can add this as another free parameter and let the data constrain it. To date observations are consistent with us living in a flat Universe so its fine putting this to zero. Including this as a free parameter in the Friedmann equation is simple enough, however the mathematics of the CMB is a bit more involved if we allow curvature which is the main reason for ignoring it.

### Physical and derived parameters

A final note on a slightly different set of parameters that are commonly used. One often consider the parameter combination $\Omega_{i0} h^2$ - called "physical parameters" - as the cosmological parameters. Why? Look at the Friedmann equation: $H^2$ is a sum of terms on the form $H_0^2 \Omega_{i0}$. If we increase all the $\Omega_{i0}$'s by a factor $4$ and reduce $h$ by a factor $2$ then $H$ stays the same! Thus for an observable that only depends on the Hubble rate we can only really measure the combination $\Omega_{i0}h^2$. This is called a *parameter degeneracy* and will be important later in this course when we discuss how we can use features in the CMB and matter power-spectrum to constrain the cosmological parameters. Luckily different types of observations will often have different parameter degeneracies and by combining different observations we are able to constrain the individual parameters.
Related to this we see that a measurement of the Hubble function alone can also only tell us what the sums $\Omega_{b0} + \Omega_{\rm CDM 0} \equiv \Omega_{m0}$ and $\Omega_{\gamma 0} +\Omega_{\nu 0} \equiv \Omega_{R0}$ is. These *derived parameters* are called the total matter and total radiation density parameters.

## Measures of time in cosmology

There are many different ways of telling time in our Universe other than simply cosmic time. There is a one-to-one relation between time and scale-factor so we can also use this as a measure of time. A closely related concept is that of redshift: how much the wavelength of a photon released at time $t$ has been stretched when it reaches us. This is also directly related to what we actually observe. Finally in the early phases of our Universe everything is a plasma with temperature $T$ that goes down as the Universe expands. It's much more convenient to talk about a temperature of $100$ GeV rather than $z\sim 10^{15}$ or $t \sim 10^{-12}$ sec - in particular since this temperature can be directly related to particle physics energy scales (which determine when stuff "happens" in the plasma) so we often use this in the early Universe. Here is a list of the different measures of time we have in our Universe and how they are related:

- Scale factor $a$
- Cosmic time $t = \int_0^a \frac{da}{aH}$
- Conformal time $a d\eta = dt$ so $\eta = \int_0^a \frac{da}{a^2 H}$. And if we multiply this by $c$ we get a distance - and this is what we usually will refer to as the conformal time - sorry!
- Redshift $z$ where $1+z = \frac{1}{a}$.
- Temperature of the primordial plasma $T = \frac{T_{\rm CMB 0}}{a}$ with $T_{\rm CMB 0} = 2.7255$ Kelvin.

## Measures of distances in cosmology

Like with time we also have many different ways of measuring distance in our Universe. In a flat non-expanding Universe this is trivial, but in a non-flat and/or expanding Universe there are complications coming in due to the geometry is not the same as we know and love from Euclid and the fact that distances change in time with the expansion of the Universe. The different measures we have are closely related to different ways of actually measuring the distance. Below all formulas will be those that only apply in a flat Universe.

Comoving distance is the distance between two points measured along a path defined at the present cosmological time. The comoving distance to an object at redshift $z$ is: $$\chi = \int_{1}^a \frac{cda}{a^2H} = \int_{0}^z \frac{cdz}{H} = \eta - \eta_0$$ This is one of the reasons why its convenient to view the conformal time as a distance rather than a time.

Luminosity distance: if the intrinsic luminosity $L$ of a distant object is known, we can calculate its luminosity distance by measuring the flux $S$ and determine $d_L = \sqrt{\frac{L}{4\pi S}}$. This is related to the comoving distance via: $$d_L = \frac{\chi}{a} = (1+z)\chi$$

Angular diameter distance: An object of size $d$ at redshift $z$ that appears to us to have an angular size $\theta$ has the angular diameter distance of $d_A = d / \theta$. This is related to the comoving distance via: $$d_A = a\chi = \frac{\chi}{1+z}$$

For small redshifts $z \ll 1$ all distance measures gives the same result $$d \simeq \frac{cz}{H_0}$$ and if we interpret the redshift as a Doppler effect, for which $v=cz$, then the formula above says that $$v = H_0d$$ which is Hubble's law: galaxies are moving away from the Earth at speeds proportional to their distance. In other words, the farther they are the faster they are moving away from Earth.

See Section 1.2.3 in Baumann or Chapter 2.2 in Dodelson for a more thorough discussion about this. I also recommend the article Distance measures in cosmology by David Hogg.

## Summary

For a quick summary look through the slides Introduction to theoretical cosmology (shown at the top of this page). The standard model of cosmology is built on the following assumptions:

- Space and time is a 4D manifold equipped with a metric. Particles and light travel on geodesics of this metric. The evolution of this metric is described by General Relativity
- The cosmological principle applies: the universe is statistically isotropic and homogenous on large enough scales. This again implies the geometry of the Universe is described by the Friedmann–Lemaître–Robertson–Walker metric (here for a flat Universe): $$ds^2 = -dt^2 + a^2(t)(dx^2 + dy^2 + dz^2)$$ where $a$ is the scale-factor.
- The mass-energy content (relevant for describing the evolution of) the Universe is: normal matter ("baryons" $w=0$), cold dark matter ($w=0$), radiation ($w=1/3$), neutrinos (we assume massless ones in this course; $w=1/3$) and dark energy (a cosmological constant; $w=-1$). All of which can be described as perfect fluids. This gives us the Friedmann equation: $$H(a) = H_0 \sqrt{\Omega_{\Lambda 0} + \frac{\Omega_{k 0}}{a^2} + \frac{\Omega_{b0} + \Omega_{\rm CDM 0}}{a^3} + \frac{\Omega_{\gamma 0} +\Omega_{\nu 0}}{a^4}}$$ where the Hubble function $H = \frac{1}{a}\frac{da}{dt}$. This tells us how the scale-factor evolves given the cosmological parameters.
- The free parameters of the model is the present Hubble rate $H_0$ (commonly described by giving the value of "little $h$": $h = H_0 / (100$ km/s/Mpc$)$), the temperature of the CMB today $T_{\rm CMB 0} \approx 2.7255$ K and the density parameters $\Omega_{b0}, \Omega_{\rm CDM 0}, \Omega_{\Lambda 0}, \Omega_{k 0},\Omega_{\gamma 0},\Omega_{\nu 0}$ where $$\Omega_{\gamma 0} = 2\cdot \frac{\pi^2}{30} \frac{(k_bT_{\rm CMB 0})^4}{\hbar^3 c^5} \frac{8\pi G}{3H_0^2} \approx 2.47\cdot 10^{-5} / h^2$$ $$\Omega_{\nu 0} = N_{\rm eff}\cdot \frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Omega_{\gamma 0} \approx 1.71\cdot 10^{-5} / h^2$$ with $N_{\rm eff} = 3.046$. In addition to this we have two parameters related to the initial conditions of the Universe and one related to reionization at late times which we will get back to.