# Cosmology from Einstein's General Relativity

By

Dr Lim Zhi Han

●

Published 2022-10-09

Einstein's theory of general relativity (GR). GR is a theory that relates the energy content of a system and its spacetime geometry. The theory hinges on the idea that spacetime can be curved, and the curvature dependent on its mass-energy content. Particles move in (free-falling) motion by following paths of least action governed by the geometry of curved spacetime.

The dynamics is encoded in a set of differential equations known as the Einstein field equation:

R_{ab}-\frac{1}{2}Rg_{ab}=\frac{8\pi G}{c^{4}}T_{ab}

The LHS of the Einstein field equation consists of geometrical terms (R_{ab},R and g_{ab}) that can be found with the metric. The RHS of the field equation (T_{ab}) contain information about the mass-energy content in the system. In essense, the equation relates mass-energy with spacetime. As aptly summarised by John Wheeler with 12 words,

The theory of general relativity is applicable for isolated systems such as stars and black holes, as well as for the Universe as a whole. While a full discussion of the theory is out of the scope of this course, we will examine the results of the theory in the context expanding Universe.

The FLRW metric

ds^{2}=-c^{2}dt^{2}+S(t)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}\right)

describes the spacetime geometry of an expanding homogeneous and isotropic Universe. Fitting it into the Einstein field equations, one obtain the following equations¹:

\begin{equation}
\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}}=\frac{8}{3}\pi G\rho(t)
\end{equation}

and

\begin{equation}
-2\frac{\ddot{S}(t)}{S(t)}-\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}}=\frac{8\pi G}{c^{2}}p(t)
\end{equation}

Eqs (1) and (2) are known as the first and second Friedmann equations respectively. They are differential equations of the scale factor S(t) that models the evolution of the expanding Universe. The equations are first derived by Alexander Friedmann in 1922, six years after Einstein published the theory of general relativity.

We find that the GR-derived Friedmann equations, are similar to Eqs (1) and (2) which were derived using Newtonian gravity. The difference, as we shall see below, is in the energy content.

Recall that in the Newtonian derivation of the Friedmann equations, the Universe contain matter where

\begin{equation}
\rho_{\text{matter}}\propto S^{-3}
\end{equation}

This is rather intuitive as energy dilutes as the universe expands, and in our 3D world, volume \propto S^{-3}.

In the context of general relativity, the energy content of the Universe may not come only from matter. Other forms of energy such as radiation and vacuum energy can be considered in the GR-derived Friedmann equations. In the following section, we will examine how the different energy content affects the expansion of the Universe.

In the Friedmann equations, there are three time dependent functions S(t), \rho(t) and p(t). The ultimate goal is to solve the Friedmann equations to obtain the scale factor S(t). However we cannot solve for S(t) without knowing the energy density \rho and pressure p. To do so, some assumptions and simplifications needs to be made.

We begin with the first Friedmann equation Eq.(1)

\dot{S}^{2}+kc^{2}=\frac{8}{3}\pi G\rho S

Differentiating this with respect to time, we have

\begin{align*}
2\dot{S}\ddot{S} & =\frac{8\pi G}{3}\left(\dot{\rho}S^{2}+\rho2S\dot{S}\right)
\end{align*}

Putting this into the second Friedmann equation Eq.(2), we have

\begin{align*}
-2\frac{\ddot{S}(t)}{S(t)}-\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}} & =\frac{8\pi G}{c^{2}}p(t)\\
-\frac{1}{S}\frac{8\pi G}{3\dot{S}}\left(\dot{\rho}S^{2}+\rho2S\dot{S}\right)-\frac{8}{3}\pi G\rho & =\frac{8\pi G}{c^{2}}p\\
-\frac{\dot{\rho}S}{3\dot{S}}-\frac{2\rho}{3}-\frac{\rho}{3} & =\frac{p}{c^{2}}\\
-\frac{\dot{\rho}S}{3\dot{S}} & =\rho+\frac{p}{c^{2}}\\
\dot{\rho} & =-\frac{3\dot{S}}{S}\left(\rho+\frac{p}{c^{2}}\right)\\
\frac{\dot{\rho}}{\rho} & =-\frac{3\dot{S}}{S}\left(1+\frac{p}{\rho c^{2}}\right)\\
\frac{\dot{\rho}}{\rho} & =-\frac{3\dot{S}}{S}\left(1+w\right)
\end{align*}

where

\begin{equation}
w=\frac{p}{\rho c^{2}}
\end{equation}

w is known as the "equation of state". We now assume for our cosmological models that density and pressure are proportional, meaning that while p and \rho are functions of time, their ratio w is a constant. Continuing from the above working,

\begin{align}
\frac{1}{\rho}\frac{d\rho}{dt} & =-3(1+w)\frac{1}{S}\frac{dS}{dt}\nonumber \\
\frac{d}{dt}\ln\rho & =-3(1+w)\frac{d}{dt}\ln S\nonumber \\
\ln\rho & =\ln S^{-3(1+w)}+\text{constant }\nonumber \\
\rho & \propto S^{-3(1+w)}
\end{align}

Eq. (5) relates the energy content (characterized by w) to the scale factor. Putting this into Eq. (1), we have

\dot{S}^{2}+kc^{2}\propto\frac{8}{3}\pi GS^{-3(1+w)}S

It is now (almost) possible to solve this differential equation!

We still need to know the value for w. Different types of energy content have different values of w. Common energy contents that are discussed in modern cosmology are

• Matter w_{\text{m}}=0
• Radiation w_{\text{r}} = \displaystyle \frac{1}{3}
• Vacuum energy (aka cosmological constant) w_{\Lambda}=-1
• Dark Energy w_{de}<-{\displaystyle \frac{1}{3}}

In a matter-dominated Universe, w=0. From Eq.(5),

\rho\propto S^{-3}

The first Friedmann equation becomes

\dot{S}^{2}+kc^{2}\propto\frac{8}{3}\pi GS^{-2}

The early Universe is often modeled as a radiation dominated Universe. Matter in the early Universe will be highly energetic, moving so fast at near light speed that they can be approximated to be radiation-like.

Putting w_{\text{r}}=\frac{1}{3} into Eq.(5), we find that the energy density of radiation relates to the scale factor as:

\begin{equation}
\rho_{\text{r}}\propto S^{-4}
\end{equation}

"Dark-energy" sounds too evil...

In the 1990s, two competing groups of astronomers were hunting down supernovae. Their goal was to extend Hubble's plot to very far-away objects. The research question is "How does the Hubble parameter change over time?" Based on cosmological models known at the time, the researchers were pretty sure that the expanding universe is decelerating (see Fig.(EX 1)).

In 1998 and 1999, both groups published their results: The expansion of the Universe is not slowing down. It is in fact accelerating! This shook the scientific community because no one was expecting that, yet both (competing) groups got the same results.

On one hand, if we think about it, the expansion of the Universe is in itself an astonishing "fact" (backed up by observational evidences from Hubble and after). But we can always attribute the expansion to an initial condition, that the Universe started off expanding. We could religiously accept this initial condition, or apply the anthropic principle and reason that if it is otherwise, the Universe would not have evolved to this stage for us to discuss about this.

Acceleration, on the other hand is a different game. The cosmological models that we know of, the energy contents that we are familiar with, all point towards a Universe whose rate of expansion decreases over time. We cannot accept a deccelerating Universe by putting on an initial condition. It needs us to either tweak in the (cosmological) model, or accept a new kind of energy content. While it seems neccessary for to believe in the existence of a new energy content in the Universe to account for the new observational evidence for an accelerating Universe, no one has a concrete idea of what this energy is. It has never been directly or even indirectly observed. All we could do is to "reason" that it must be there, in dominating amounts! Astronomers call this mysterious energy "dark energy"

To create an accelerating Universe (on paper), one can choose some value for w and put it into the Friedmann equations to obtain \ddot{S}(t)>0. The choice for w is not unique (we shall see that later), but a popular choice that astronomers like to use is w=-1, corresponding to what we call the vacuum energy.

Historically the vacuum energy was first introduced by Einstein for another purpose. He called it the cosmological constant \Lambda. Nowadays these terms are used interchangably. Let us put w_{\Lambda}=-1 into Eq.(5), (4), (1) and (2):

\begin{align*}
w_{\Lambda} & =-1\\
\rho_{\Lambda} & \propto S^{-3(1-1)}=1\\
\Rightarrow\rho_{\Lambda} & =\text{constant}\\
\text{Let }\rho_{\Lambda} & =\frac{\Lambda}{8\pi G}
\end{align*}

where \Lambda is a positive constant. The energy density associated with vacuum energy is a constant.

The pressure associated with \Lambda can be found using the equation of state Eq.(4):

\begin{align*}
-1 & =\frac{p_{\Lambda}}{\rho_{\Lambda}c^{2}}\\
\Rightarrow p_{\Lambda} & =-\frac{\Lambda}{8\pi G}c^{2}
\end{align*}

The first Friedmann equation Eq.(1) becomes

\frac{\dot{S}(t)^{2}+kc^{2}}{S(t)^{2}}=\frac{8}{3}\pi G\frac{\Lambda}{8\pi G}=\frac{\Lambda}{3}

Solving this equation gives the evolution of a expanding vacuum energy dominated Universe.

The second Friedmann equation Eq.(2) becomes

\begin{align*}
-2\frac{\ddot{S}}{S}-\frac{\Lambda}{3} & =\frac{8\pi G}{c^{2}}\frac{-\Lambda}{8\pi G}c^{2}\\
-2\frac{\ddot{S}}{S} & =\frac{\Lambda}{3}-\Lambda\\
\ddot{S} & =\frac{\Lambda}{3}S\,>0
\end{align*}

Hence a vacuum energy dominated Universe follows an accelerated expansion.

w<-\frac{1}{3}

The 2011 Physics Nobel Prize which was awarded to astronomers who discovered that our Universe is accelerating. For more information on the accelerating Universe and dark energy, read this writeup.