The concept of an expanding universe is one of the most profound discoveries in cosmology. It has reshaped our understanding of the cosmos and our place within it. This chapter will delve into the history and mathematics behind this concept, providing a comprehensive understanding of how we describe an expanding universe.
1 How Hubble Discovered That the Universe is Expanding
The discovery that the universe is expanding is often attributed to the American astronomer Edwin Hubble. In the 1920s, he made observations of galaxies beyond the Milky Way and discovered a pattern: galaxies were moving away from us, and the farther the galaxy, the faster it was moving away. This was contrary to the prevailing belief at the time, which was that the universe was static and unchanging.
This relationship between the velocity of a galaxy ($v$) and its distance ($d$) from the observer is expressed as Hubble’s Law:
\begin{equation} v = H_0 \cdot d \end{equation}
where $H_0$ is Hubble’s constant, representing the rate of the universe’s expansion. The current value of $H_0$ is around $70 \pm 7 \ \mathrm{km/s/Mpc}$ (kilometers per second per megaparsec). This means that for every additional megaparsec (a distance unit equal to about 3.26 million light years) an object is away from us, it appears to be moving away from us an additional $70 \pm 7 \ \mathrm{km/s}$ due to the expansion of the universe.
The implications of Hubble’s discovery were profound. It suggested that if galaxies are moving away from us now, they must have been closer together in the past. This ultimately led to the development of the Big Bang theory, the idea that the universe began as an incredibly dense, hot state and has been expanding and cooling ever since.
2 Friedman Equation
2.1 Gravitational Forces and Conservation of Energy
To understand the expansion of the universe, let’s start with a Newtonian description. Consider a large spherical region of space containing a large number of evenly spaced galaxies of mass $M$ and radius $R$. We’re interested in understanding the force acting on a single satellite galaxy of mass $m$ located at the edge of the sphere.
Using Newton’s Law of Gravitation, the gravitational force acting on the satellite galaxy is $F = - \frac{GmM}{R^2}$.
According to Newton’s second law of motion, this force is also equal to $ma$, where $a$ is the acceleration of the galaxy.
Therefore, we have $ma = - \frac{GmM}{R^2}$, which simplifies to $a = - \frac{GM}{R^2}$.
If we express this acceleration as the rate of change of the radius $R$ (as in circular motion), we get $\frac{\mathrm{d}^2 R}{\mathrm{d} t^2} = - \frac{GM}{R^2}$.
If we multiply both sides by $\frac{\mathrm{d} R}{\mathrm{d} t}$, we get $\frac{\mathrm{d} R}{\mathrm{d} t} \frac{\mathrm{d}^2 R}{\mathrm{d} t^2} = - \frac{GM}{R^2} \frac{\mathrm{d} R}{\mathrm{d} t}$.
Integrating both sides with respect to time, we get
\begin{equation} \frac{1}{2} \left( \frac{\mathrm{d} R}{\mathrm{d} t} \right)^2 - \frac{GM}{R} = k \tag{2} \end{equation}
where $k$ is a constant of integration.
This equation represents the conservation of energy, with the $\frac{1}{2} \left( \frac{\mathrm{d} R}{\mathrm{d} t} \right)^2$ term representing the kinetic energy per unit mass and the $\frac{GM}{R}$ term representing potential energy per unit mass.
2.2 Friedman Equation in Newtonian Form
Now, let’s consider the case where the universe is expanding. Since the sphere of galaxies is part of the universe, the radius of the sphere of mass $M$ is also expanding. We represent the change in the radius as $R = a (t) \cdot r$, where $a (t)$ is the time-dependent scale factor that describes the expansion of the universe, and $r$ is the co-moving distance (which is constant).
The co-moving distance between any two objects in the universe remains constant with time, provided there is no other motion than the expansion of the universe acting on them. In simpler terms, co-moving distance does not change as the universe expands because it’s measured in a coordinate grid that expands with the universe. To illustrate, consider two galaxies. As the universe expands, the “actual” (proper) distance between the galaxies increases, but their co-moving coordinates stay the same because the grid on which they are being measured is also stretching.
We want to account for satellite galaxy’s acceleration (due to the expanding sphere) in equation (2), which is rewritten below for convenience.
\[ \frac{1}{2} \left( \frac{\mathrm{d} R}{\mathrm{d} t} \right)^2 - \frac{GM}{R} = k \]
If the universe is homogeneous and isotropic (the cosmological principle), then the mass within any volume is given by $M = \frac{4}{3} \pi R^3 \cdot \rho$.
Recall Hubble’s law: $v = H \cdot R = H \cdot a \cdot r$ (we drop the dependence on $t$ for convenience of writing).
Recall also that $v = \frac{\mathrm{d} R}{\mathrm{d} t} = \frac{\mathrm{d} a}{\mathrm{d} t} r = \dot{a} r$.
Substituting the above $v$ into Hubble’s law, we get $H = \frac{\dot{a}}{a}$, which is known as the Hubble parameter. The Hubble constant $H_{0}$ is the Hubble parameter $H$ at the present moment.
Subsituting the above results into equation (2), we get the Friedman equation in Newtonian form.
\begin{equation} \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \rho}{3} + \frac{2 k}{r^2} \frac{1}{a^2} \tag{3} \end{equation}
The Friedman equation in Newtonian form does not account for the fact that the shape of the universe is influended by the presence of matter and energy.
2.3 Friedman Equation Using General Relativity Formulation
To make the Friedmann equation compatible with General Relativity, we need to introduce the concept of energy density $\varepsilon$ (instead of the mass density $\rho$) and curvature $\kappa$.
The energy density is defined as the total energy per unit volume.
\begin{equation} \varepsilon = \frac{E}{V} = \frac{mc^2}{V} = \rho c^2 \tag{4} \end{equation}
The reason we’re doing this is because the curvature of space-time is related to mass/energy.
We need to replace the Newtonian energy (the $k$ in equation (3)) with Einsteinian curvature. This is because the presence of energy affects the curvature of space.
\begin{equation} \frac{2 k}{r^2} \rightarrow - \frac{\kappa c^2}{R_{0}} \tag{5} \end{equation}
where $\kappa$ is the curvature constant, $R_{0}$ is the radius of curvature.
The Friedmann equation in the context of General Relativity becomes:
\begin{equation} \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \varepsilon}{3 c^2} - \frac{\kappa c^2}{R_{0}^2} \frac{1}{a^2} \end{equation}
The curvature constant $\kappa$ can take on the values $- 1$, $0$, or $1$.
- If $\kappa = - 1$, space is said to be negatively-curved, the RHS of equation (6) is always positive, and this describes an open universe that will keep on expanding forever.
- If $\kappa = + 1$, space is said to be positively-curved, the RHS of equation (6) is will eventually become negative, causing the contraction of the universe. This type of a universe is said to be closed.
- If $\kappa = 0$, space is said to be zero curvature. The universe will continue to expand forever, but will slow down in its expansion with $\dot{a} \rightarrow 0$, after an infinite amount of time. This type of a universe is said to be flat.
The question now is, how to determine the value of $\kappa$ to know the fate of the universe.
2.4 Determining the Fate of the Universe
Let’s rewrite the Friedmann equation here for conveninece.
$$ \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \varepsilon}{3 c^2} - \frac{\kappa c^2}{R_{0}^2} \frac{1}{a^2} $$
The special case of $\kappa = 0$ represents the crossover point between an open and closed universe. If we set $\kappa = 0$ in equation (6), we get:
$$\frac{8 \pi G \varepsilon _{c}}{3 c^2} = \left( \frac{\dot{a}}{a} \right)^2 = H^2$$
\begin{equation} \varepsilon _{c} = \frac{3 c^2 H^2}{8 \pi G} \tag{7} \end{equation}
This gives us an expression of the critical density of the universe in terms of the Hubble parameter, whose value at the present is $70 \pm 7 \ \mathrm{km/s/Mpc}$.
- If the energy density $\varepsilon$ is greater than $\varepsilon _{c}$, then the universe is positively-curved ($\kappa = + 1$).
- If the energy density $\varepsilon$ is less than $\varepsilon _{c}$, then the universe is negatively-curved ($\kappa = - 1$).
The current estimate of the critical energy density is about $7.8 \times 10^{- 10} \ \mathrm{J/m^3}$ and the current estimate of the critical mass density is about $8.7 \times 10^{-27} \ \mathrm{kg/m^3}$ (about four hydrogen atoms per cubic meter). This energy density is very small compared to everyday energy densities. For instance, the energy density of a AA battery is about $10^8 \ \mathrm{J/m^3}$, which is expected because cosmic energy densities are averaged over vast volumes of mostly empty space.
It turns out the mean density of the observable universe is close to the critical density, suggesting that we live in a nearly flat universe.
3 The Time Evolution of the Scale Factor
The evolution of the scale factor $a (t)$ with time is an important piece of the puzzle, as it allows us to understand how the universe has been expanding. However, the Friedman equation remains an equation of two unknowns: $a$ and $\varepsilon$, both of which are functions of time.
3.1 The Fluid Equation
Recall that the Friedman Equation in Newtonian form is a statement about the conservation of energy. We can get another equation that relates $a$ and $\varepsilon$ by considering another energy conservation law. With two equations and two unknowns, we can solve for one of these terms.
Recall the first law of Thermodynamics:
\begin{equation} \mathrm{d} Q = \mathrm{d} U + P \mathrm{d} V \tag{8} \end{equation}
where $\mathrm{d} Q$ is the heat transfer, $\mathrm{d} U$ is the change in internal energy, $P$ is the pressure and $\mathrm{d} V$ is the change in volume. The pressure can be thought of as the result of the thermal motion of atoms in a gas cloud.
Assuming that the universe is isolated and does not exchange heat with anything, we can set $\mathrm{d} Q$ to 0.
The internal energy within a volume of space can be expressed as energy density $\varepsilon$ times the volume $V$ of that space.
\begin{equation} U (t) = \varepsilon (t) \cdot V (t) = \varepsilon (t) \cdot \frac{4}{3} \pi R (t)^3 = \varepsilon (t) \cdot \frac{4}{3} \pi \cdot (a (t) \cdot r)^3 \tag{9} \end{equation}
Taking the derivative of $U$ with respect to time $t$, we get:
\begin{equation} \dot{U} = \frac{4}{3} \pi r^3 a^3 \left( 3 \varepsilon \frac{\dot{a}}{a} + \dot{\varepsilon} \right) \tag{10} \end{equation}
Taking the derivative of $V$ with respect to time $t$, we get:
\begin{equation} \dot{V} = \frac{4}{3} \pi r^3 a^2 \dot{a} \tag{11} \end{equation}
Substituting the above terms into the equation for the first law of Thermodynamics, we get:
\begin{equation} \frac{4}{3} \pi r^3 a^3 \left( \dot{\varepsilon} + 3 \frac{\dot{a}}{a} (\varepsilon + P) \right) = 0 \tag{12} \end{equation}
Since $R = a (t) \cdot r \neq 0$, it follows that:
\begin{equation} \dot{\varepsilon} + 3 \frac{\dot{a}}{a} (\varepsilon + P) = 0 \tag{13} \end{equation}
This equation, known as the Fluid equation, is of great importance as it allows us to describe the evolution of the energy density of the universe.
Recall that $H = \frac{\dot{a}}{a}$, we can rewrite the above equation as:
\begin{equation} H = - \frac{1}{3 (\varepsilon + P)} \cdot \dot{\varepsilon} \tag{14} \end{equation}
which shows that the expansion of the region of space is proportional to the negative rate of change of the energy density, implying that as the universe expands, its energy density decreases.
3.2 The Acceleration Equation
Combining the Friedman equation and the Fluid equation, we can derive the acceleration equation that describes how the expansion of the universe changes with time.
The fluid equation contains the term $\dot{\varepsilon}$, whereas the Friedman equation contains only $\varepsilon$. To combine the two equations, we first differentiate the Friedman equation with respect to time. This gives us:
\begin{equation} \frac{\ddot{a}}{a} = \frac{8 \pi G}{3 c^2} \left( \dot{\varepsilon} \frac{a}{\dot{a}} + 2 \varepsilon \right) \tag{15} \end{equation}
Recall the Fluid equation:
\[ \dot{\varepsilon} + 3 \frac{\dot{a}}{a} (\varepsilon + P) = 0 \]
Rearranging the terms:
\begin{equation} \dot{\varepsilon} \frac{a}{\dot{a}} = - 3 (\varepsilon + P) \tag{16} \end{equation}
Substituting this into the Friedman equation:
\begin{equation} \frac{\ddot{a}}{a} = - \frac{4 \pi G}{3 c^2} (\varepsilon + 3 P) \tag{17} \end{equation}
This equation, known as the acceleration equation, tells us that the rate at which the universe is expanding (or contracting) is driven by the total energy density and pressure within the universe.
If both the energy density and pressure are positive (which they are), then the RHS of the acceleration equation is negative, telling us that the relative velocity between any two points in the universe will be reducing with time, causing the universe to slow down and eventually contract.
This led to a puzzle for Einstein: his equations implied that the universe should be either contracting, not static as he initially thought. To maintain a static universe, Einstein introduced a positive term $\Lambda$, later called the cosmological constant ($\Lambda$), to the RHS of the equation.
4 Vacuum Catastrophe
4.1 The Energy Density of the Vacuum
When the cosmological constant term is added to the Friedman equation (now expressed in terms of General Relativity), we get:
\begin{equation} \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G \varepsilon}{3 c^2} - \frac{\kappa c^2}{R_{0}^2} \frac{1}{a^2} + \frac{\Lambda}{3} \tag{18} \end{equation}
If we rewrite the above equation so that the cosmological constant $\Lambda$ is included in a term that has the same units as the energy density, we get:
\begin{equation} \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3 c^2} \left( \underbrace{\varepsilon}_{\varepsilon _{m}} + \underbrace{\frac{\Lambda c^2}{8 \pi G}}_{\varepsilon _{\Lambda}} - \underbrace{\frac{3 \kappa c^4}{8 \pi \operatorname{GR}_{0}^2 a^2}}_{\varepsilon _{\kappa}} \right) \tag{19} \end{equation}
The total energy density $\varepsilon$ can be divided into three components: mass density $\varepsilon _{m}$, vacuum energy density $\varepsilon _{\Lambda}$, and curvature energy density $\varepsilon _{k}$.
From this, we see that the cosmological constant represents a component of the universe, $\varepsilon _{\Lambda} = \frac{\Lambda c^2}{8 \pi G}$, in which a constant energy density causes the universe to accelerate in its expansion. This mysterious energy driving the expansion of the universe, coined “dark energy”, is the energy associated with the vacuum of space. If we can determine this vacuum energy, then we should be able to calculate the value of the cosmological constant.
4.2 Vacuum Catastrophe
To delve into why the vacuum might have energy, we have to step into the realm of quantum field theory.
4.2.1 Quantum Fields and Particles
The universe, according to quantum field theory, is filled with a variety of quantum fields that pervade all of space. Particles correspond to ripples in the their corresponding quantum fields. For instance, an electron is an excitation of the electron field, a photon is an excitation of the electromagnetic field, and so on.
4.2.2 Heisenberg Uncertainty Principle
The Heisenberg Uncertainty Principle is a fundamental concept in quantum mechanics. It states that it is impossible to simultaneously measure the exact position and momentum of a particle. This is not due to any technological limitations, but rather a fundamental aspect of quantum systems. The principle can be mathematically represented as:
\begin{equation} \Delta p \cdot \Delta x \geqslant \frac{\hslash}{2} \tag{20} \end{equation}
where $\Delta p$ is the uncertainty in momentum, $\Delta p$ is the uncertainty in position, and $\hslash$ is the reduced Planck constant, equal to the Planck constant $h$ divided by $2 \pi$.
The Heisenberg Uncertainty Principle also applies to energy and time. This is often less discussed but is equally fundamental. It can be mathematically represented as:
\begin{equation} \Delta E \cdot \Delta t \geq \frac{\hslash}{2} \tag{21} \end{equation}
where $\Delta E$ is the uncertainty in energy and $\Delta t$ is the uncertainty in time.
4.2.3 Implications for Vacuum Energy
The Heisenberg Uncertainty Principle implies that even empty space, or a vacuum, has energy. This is because the principle forbids any quantum system from having zero uncertainty in both position and momentum, or in both energy and time. This means that a quantum field, even in its ground state, must have fluctuations. These fluctuations give rise to what is known as zero-point energy, or vacuum energy.
The vacuum energy has been experimentally confirmed through phenomena such as the Lamb shift and the Casimir effect, which are small effects that can only be explained by the existence of vacuum fluctuations. These phenomena provide evidence that even empty space is full of activity on the quantum scale, with particles and antiparticles constantly popping in and out of existence.
4.2.4 Planck Length and Planck Time
The Planck length and Planck time are fundamental units in the system of natural units known as Planck units. They are derived from three fundamental physical constants: the speed of light $c$, the gravitational constant $G$, and the reduced Planck constant $\hslash$.
The Planck length, denoted by $l_{P}$, is defined as:
\begin{equation} l_{P} = \sqrt{\frac{\hslash G}{c^3}} \approx 1.6 \times 10^{- 35} \mathrm{m} \tag{22} \end{equation}
The Planck time, denoted by $t_{P}$, is defined as the time it takes light to travel one Planck length in a vacuum, and is given by:
\begin{equation} t_{P} = \sqrt{\frac{\hslash G}{c^5}} \approx 5.4 \times 10^{- 44} \mathrm{s} \tag{23} \end{equation}
The Planck length and Planck time are believed to be the smallest meaningful length and time, respectively. Below these scales, it is thought that the very concepts of space and time may break down.
In the context of vacuum energy, the Planck length and Planck time are used to regularize the calculation. Without these scales, the calculation of vacuum energy would yield an infinite result, which is clearly unphysical. The Planck scales provide a natural cutoff, beyond which we do not include contributions to the vacuum energy.
4.2.5 Calculating the Vacuum Energy Density
To calculate the vacuum energy density, we first consider a small volume of space with sides of length equal to the Planck length. This gives us a volume of $l_{P}^3$.
Next, we consider a quantum harmonic oscillator in this volume. A quantum harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. It is a good model for many physical systems, including quantum fields.
The energy of a quantum harmonic oscillator is given by:
\begin{equation} E = \left( n + \frac{1}{2} \right) \hslash \omega \tag{24} \end{equation}
where $n$ is the quantum number (which can be any non-negative integer), $\hslash$ is the reduced Planck constant, and $\omega$ is the angular frequency of the oscillator.
For the vacuum energy, we are interested in the ground state of the oscillator, which corresponds to $n = 0$. This gives us the zero-point energy:
\[ E_{0} = \frac{1}{2} \hslash \omega \]
The frequency $\omega$ can be related to the Planck time $t_{P}$ as $\omega = \frac{1}{t_{P}}$. Substituting this into the equation for $E_{0}$, we get:
\[ E_{0} = \frac{1}{2} \frac{\hslash}{t_{P}} \]
This is the energy in one Planck volume. To get the energy density, we divide by the volume:
\begin{equation} \varepsilon _{\Lambda} = \frac{E_{0}}{l_{P}^3} = \frac{\hslash / (2 t_{P})}{l_{P}^3} \approx 10^{114} \ \mathrm{J/m^3} \tag{25} \end{equation}
The predicted energy of the vacuum is $10^{114} \ \mathrm{J/m^3}$. The equivalent predicted mass density, using Einstein’s mass-energy equivalence $E = mc^2$, is approximately $10^{97} \ \mathrm{kg/m^3}$.
4.2.6 Comparison with Measurements
Let’s recall the Friedman equation here for convenience.
$$\left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3 c^2} \left( \underbrace{\varepsilon}_{\varepsilon _{m}} + \underbrace{\frac{\Lambda c^2}{8 \pi G}}_{\varepsilon _{\Lambda}} - \underbrace{\frac{3 \kappa c^4}{8 \pi \operatorname{GR}_{0}^2 a^2}}_{\varepsilon _{\kappa}} \right)$$
The total energy density $\varepsilon$ can be divided into three components: mass density $\varepsilon _{m}$, vacuum energy density $\varepsilon _{\Lambda}$, and curvature energy density $\varepsilon _{\kappa}$.
- The mass density $\varepsilon _{m}$ includes both visible matter and dark matter. Visible matter includes stars, galaxies, and gas clouds, while dark matter is a hypothetical form of matter that does not interact with light and is only detectable through its gravitational effects. The mass density is often expressed as a fraction of the critical density $\varepsilon _{c}$, which is the density required for the universe to be flat. Current measurements indicate that $\varepsilon _{m} \approx 0.3 \varepsilon _{c}$, with about $0.05$ in visible matter and $0.25$ in dark matter.
- The vacuum energy density $\varepsilon _{\Lambda}$ is associated with the energy of empty space, or the cosmological constant. Current measurements indicate that $\varepsilon _{\Lambda} \approx 0.7 \varepsilon _{c}$, meaning that about 70% of the energy content of the universe is in the form of dark energy.
- The curvature energy density $\varepsilon _{\kappa}$ is associated with the curvature of space. Current measurements indicate that $\varepsilon _{\kappa} \approx 0$, meaning that the universe appears to be flat.
4.2.7 Vacuum Energy Problem
The predicted energy of the vacuum is about $10^{114} \ \mathrm{J/m^3}$. If we substitute the predicted value of the vacuum energy density into the Friedmann equation ($H = \sqrt{\frac{8 \pi G \varepsilon _{\Lambda}}{3 c^2} }$), we find that the Hubble parameter $H$ would be extremely large, implying that the universe should be doubling in size every $10^{- 43} \mathrm{s}$.
However, the observed vacuum energy density is much smaller. It is measured using observations of the large scale structure of the universe and the cosmic microwave background radiation, and is approximately $10^{- 10} \ \mathrm{J/m^3}$. This discrepancy between the predicted and observed vacuum energy density is known as the vacuum catastrophe and is one of the unsolved problems in theoretical physics.