Chapter 5

This chapter is all about solving our 3 equations for different types of substances. Our equations are:

We also have 3 potential substances to fill our universe with (these are the 3 we think exist, but we'll consider them individually and in various combinations to build up our understanding). They are defined by their EoS parameter $w$:

Here, I've expressed the dark energy component as either a cosmological constant ($w=-1$) or more generally as a linear function of $a$. Current observations favor (or don't disagree with) values of $w_0 \approx -1$ and $w_a \approx 0$, consistent with a cosmological constant explanation for dark energy. Due to its simplicity, we'll focus more on $\Lambda$ alone, but the latter form is more general.

Because energy densities and pressures are additive, i.e.,

and assuming the components don't interact, the fluid equation should hold for each substance individually. Technically, as long as any increase (in the equation) in one substance is compensated for by an exact decrease in the other substances, the fluid equation $Eqn_{\rm Fluid}$ would hold. Mathematically:

If the substances don't interact, however, there's no way for them to conspire to satisfy the fluid equation, so we can require that each substance individually satisfies the equation. For a given substance $i$, the differential fluid equation can be solved as a function of $a$:

Again, 0 refers to the current time, so $\varepsilon_{i,0} = \varepsilon_i(a=1)$. Now we know how each component evolves as the universe expands by substituting the value of $w$ into the above equation:

For non-relativistic matter, the energy is wrapped up in the particle rest masses, and as the universe expands, the number of particles in a given volume will decrease by the amount the volume of the universe expanded. For relativistic matter or radiation, the energy is ALSO affected by their wavelength getting redshifted (remember, $1+z = a^{-1}$), adding an extra factor of $a$. The cosmological constant is, well, a constant, so doesn't change with the expansion of space.

Inserting this expression for energy density into the Friedmann equation, we now have a single equation that depends only on $a$ and constants that is solvable:

Ugly, but solvable. Let's add components bit by bit before presenting the general solution (especially since there isn't an analytic solution in that case).

Empty Universes

The equation simplifies considerably when $\varepsilon = 0$, depending only on the curvature $\kappa$. Such a universe can't be positively curved due to $\dot{a}^2$, but it can have flat or negative curvature. The flat case is static ($\dot{a} = 0$, or $a$ is const.). The negatively curved version is slightly more interesting, with

The universe can expand or contract at a constant rate. If expanding ($+$), $a(t)$ can be directly solved for:

We also know that $a(t=0) = 0$, so the equation above tells us that $1 = c t_0 / R_0$, or $t_0 = R_0/c$. So we can more simply write $a(t) = t/t_0$.

In an empty universe, the universe expands at a constant rate, so the Hubble parameter ($\dot{a}/{a}$) is constant, or always equal to the Hubble constant $H_0 = \dot{a}/a = (1/t_0)/(1) = t_0^{-1}$.

What about observations in this galaxy? (Ignore the fact that there's nothing to observe.) Light from a "galaxy" gets redshifted following the always true relation $1+z = a(t_e)^{-1}$. Substituting in above relations, we can find the age of the universe when that photon was emitted, $t_e = [H_0(1+z)]^{-1}$. This is useful to know for the next thing.

This is the next thing! What about the proper distance between two galaxies in an empty universe? One way to calculate the proper distance today, if you know $a(t)$, is via

The above equation is true for any universe, empty, full of ponies, whatever. Solving this equation for an empty universe and substituting in expressions for $t_0$ and $t_e$:

We've replaced $t_0$ and $t_e$ with $H_0$ and $z$ because those are observables, while various times are not. Now, what about the separation of our two ghost galaxies when the photon was first emitted? We know the distance b/t them now, when $a(t_0) = 1$, so the distance back then is smaller by the same amount that the universe was smaller, i.e., $d_p(t_e) = d_p(t_0) \frac{a(t_e)}{a(t_0)} = d_p(t_0) (1+z)^{-1}$ (using the relation above indicated to be always true).

For universes with actual stuff in them, solving these equations for $a(t)$ and $d_p(t_0)$ are harder, but the procedure is the same.

Single Component, Flat Universe

Here we set $\kappa = 0$ and consider the universe contains only one substance with EOS parameter $w$ (unspecified). Doing math, we find

These expressions are valid for a pure non-relativistic, relativistic, or dark energy (except cosmological constant-like) substance filled universes. The proper distance between two galaxies in this type of universe is

The proper distance to a galaxy (assuming one existed) at $t=0$ is called the horizon distance, because that is the farthest away an object can be seen — light would have had to travel longer than the age of the universe for us to see it, which is not possible. Instead of integrating from $t_e$ to $t_0$ light usual, we can replace $t_e$ with 0. As $t \rightarrow 0$, the redshift $z \rightarrow \infty$, which makes sense since $a(t \rightarrow 0) \rightarrow 0$ and $1+z = 1/a(t)$, so $z$ must become infinite. Making that substitution in the equation for proper distance yields the horizon distance in this universe:

Remember, these equations (and the 3 following cases) are for flat ($\kappa=0$) universes, so by definition, $\varepsilon(t) = \varepsilon_{\rm crit}(t) = \varepsilon_{\rm c,0} a^{-3(1+w)}$, where $\varepsilon_{\rm c,0} = 3 c^2 H_0^2 / (8 \pi G)$.

If that component is non-relativistic matter

Setting $w=0$ for non-relativistic matter, the above equations become

And, of course, $d_p(t_e) = d_p(t_0) (1+z)^{-1}$.

If that component is relativistic (matter or radiation)

Setting $w=0$ for a relativistic component, the above equations become

If that component is a cosmological constant ($\Lambda$)

The above functional form doesn't work for $w=-1$, so that needs to be solved for explicitly, which yields

More than one component at a time

If we have all 3 of the above components AND curvature, we can write down an integral that can always be solved numerically, and occasionally analytically. The form of the equation is more tractable if put in terms of density parameters ($\Omega = \varepsilon / \varepsilon_{\rm crit}$), where the sum of density parameters $\Omega = \sum_i \Omega_i$ reveals the curvature ($\kappa = 0$ implies $\Omega = 1$; $\kappa = +1$ implies $\Omega > 1$; $\kappa = -1$ implies $\Omega < 1$).

Again, 0 denotes the value of the parameter at the current time, and $1-\Omega_0$ is a proxy for the curvature parameter (if $\kappa=0$, then $\Omega_0 = 1$ and that term goes to 0, just like the curvature term). We can come up with analytic solutions by setting some of these parameters to 0. Even though our universe contains all 3 components, not all components "matter" at all times; e.g., while radiation dominates the evolution of the universe at early times, its energy density is now less than 0.01% of the total energy density of the universe, so we can ignore radiation when considering the future evolution of the universe.

Non-relativistic Matter & Curvature

Setting $\Omega_{\rm r,0}=0$ and $\Omega_{\Lambda,0}=0$:

($\Omega_0 = \Omega_{\rm m,0}$, since it's the only non-zero parameter). If $\Omega_0 < 1$, both terms on the right will always be positive and the universe will expand forever ($H > 0$ always). If $\Omega_0 > 1$, however, the second of the two right terms is negative and as $a$ increases as the universe expands, that term will eventually dominate and $H<0$, meaning the universe reaches a maximum size and starts collapsing back in on itself. This maximum size $a_{\rm max} = \Omega_0 / (\Omega_0-1)$ — you should derive it for yourself!

Solving the integral to get $a(t)$ isn't trivial, but you can get an analytic solution in parametric form, which is given in the textbook. Historically, these solutions were considered the only possible options (since radiation is and will remain negligible, and $\Lambda$ was ignored after the discovery of Hubble's law). The crucial parameter that determined whether the universe would expand forever (and die in a "Big Chill") or eventually collapse in a "Big Crunch."

Non-relativistic Matter & $\Lambda$ with Flat Curvature

Spoiler alert: we think this combination is most likely to describe the future evolution of our universe (with $\Omega_\Lambda > 0$ that is).

If $\kappa=0$, $\Omega_0 = 1$ and thus $\Omega_{\Lambda,0} = 1 - \Omega_{\rm m,0}$. If $\Omega_\Lambda < 0$, then it has positive energy density (like the other substances), contributing to slow the expansion of space. The universe will reach a maximum size and collapse. Solving the integral:

Our universe has $\Omega_\Lambda > 0$, however. The solution from the integral in this case is

Non-relativistic Matter & $\Lambda$ with non-Flat Curvature

As you'll learn, the evidence suggests (but doesn't require) that our universe is flat, but it may not be. In that case, we have

Finally, we've gotten to a form of the equation without an analytic solution. By carefully choosing parameters, you can get all kinds of funky behavior for $a(t)$. Measurements don't support any of these funky options, unfortunately, so don't worry about them.

Non-relativistic Matter & Radiation

At high redshift, because $\Omega_{\rm r}(a) = \Omega_{\rm r,0} a^{-4}$ is a steeper function wrt $a$ than other other component, we have to consider its contribution to understand the evolution of the early universe. As long as $a$ isn't super small, non-relativistic matter also matters (pun intended).

BTW, $a_{\rm m\Lambda}$ and $a_{\rm rm}$ are not just convenient variables in these equations. They give the scale factor when the energy densities of those parameters at that time are equal to each other.

"Our Universe" or something close to it: Benchmark Model

The ultimate goal of the field of cosmology is to measure the variables in the above equations, namely $H_0$, $\Omega_{\rm m,0}$, $\Omega_{\rm r,0}$, $\Omega_{\Lambda,0}$ (as well as various other numbers, but these are the ones that govern the overall expansion of space). The textbook defines a "benchmark model", sometimes called "concordance cosmology" because there is generally concordance or consistency between various independent constraints on these parameters that all agree (for the most part). This model may not be right, but unless our current understanding is incorrect (which it may be!) the true values of the parameters will be close to the values given in the textbook for this model.