Matter Content and Exact Solutions
How different forms of matter and energy shape the expansion history through their equations of state
The Friedmann equations determine the evolution of the scale factor a(t) once the matter content of the universe is specified through an equation of state. Different types of matter -- radiation, dust, and dark energy -- produce qualitatively different expansion histories. In this section we solve the Friedmann equations exactly for each case and introduce the critical density and cosmological distance measures.
The Equation of State Parameter
A cosmological fluid is characterized by its equation of state, the relationship between its pressure p and energy density \(\rho\). For a barotropic fluid, this takes the simple linear form:
$$p = w\rho c^2$$
The equation of state parameter w is constant for each fundamental matter type
With a constant equation of state parameter w, the continuity equation\(\dot{\rho} + 3H(\rho + p/c^2) = 0\) can be integrated directly:
$$\dot{\rho} + 3\frac{\dot{a}}{a}\rho(1+w) = 0$$
$$\Rightarrow \quad \rho(a) = \rho_0\, a^{-3(1+w)}$$
The energy density dilutes as the universe expands, with the rate depending on w
The Three Fundamental Components
1. Radiation (w = 1/3)
Radiation includes photons and any relativistic species. The pressure equals one-third the energy density, a result that follows from the tracelessness of the stress-energy tensor for massless particles:
$$p_r = \frac{1}{3}\rho_r c^2 \quad \Longrightarrow \quad \rho_r \propto a^{-4}$$
$$a(t) \propto t^{1/2} \quad \text{(radiation-dominated era)}$$
The extra factor of \(a^{-1}\) beyond the volume dilution \(a^{-3}\) comes from cosmological redshift
The energy density of radiation decreases as \(a^{-4}\): three powers of a come from the volume dilution (the number density of photons decreases as \(a^{-3}\)), and one additional power comes from the redshift of each photon's energy (\(E = h\nu \propto a^{-1}\)). The Hubble parameter during radiation domination is \(H = 1/(2t)\).
2. Pressureless Dust (w = 0)
Non-relativistic matter (dust) has negligible pressure compared to its energy density. This includes baryonic matter and cold dark matter at late times:
$$p_m = 0 \quad \Longrightarrow \quad \rho_m \propto a^{-3}$$
$$a(t) \propto t^{2/3} \quad \text{(matter-dominated era)}$$
Pure volume dilution: the mass of each particle is constant, only the number density decreases
For a flat (\(k = 0\)), matter-dominated universe with no cosmological constant, the solution is the Einstein-de Sitter model. This was the standard cosmological model for much of the 20th century. The Hubble parameter is \(H = 2/(3t)\), and the age of the universe is \(t_0 = 2/(3H_0)\).
3. Cosmological Constant / Dark Energy (w = -1)
A cosmological constant (or vacuum energy) has the remarkable equation of state \(p = -\rho c^2\). The negative pressure means that dark energy acts as a repulsive gravitational effect:
$$p_\Lambda = -\rho_\Lambda c^2 \quad \Longrightarrow \quad \rho_\Lambda = \text{const}$$
$$a(t) \propto e^{Ht} \quad \text{where} \quad H = \sqrt{\frac{\Lambda c^2}{3}} = \text{const}$$
The energy density remains constant as space expands -- the energy of the vacuum is a property of space itself
This is the de Sitter solution: a universe undergoing eternal exponential expansion. The Hubble parameter is constant, and the scale factor doubles every\(t_{\text{double}} = \ln 2 / H \approx 0.693/H\). In our universe, dark energy began dominating at \(z \approx 0.4\) (about 5 billion years ago), and we are now entering an approximately de Sitter phase.
Summary: Equation of State and Scaling
| Component | w | \(\rho(a)\) | \(a(t)\) (flat, single component) | H(t) |
|---|---|---|---|---|
| Radiation | 1/3 | \(\propto a^{-4}\) | \(\propto t^{1/2}\) | \(1/(2t)\) |
| Matter (dust) | 0 | \(\propto a^{-3}\) | \(\propto t^{2/3}\) | \(2/(3t)\) |
| Cosmological constant | -1 | const | \(\propto e^{Ht}\) | const |
| General | w | \(\propto a^{-3(1+w)}\) | \(\propto t^{2/[3(1+w)]}\) | \(2/[3(1+w)t]\) |
Conformal Time and Comoving Distance
It is often convenient to use conformal time \(\eta\)instead of cosmic time t, defined by:
$$d\eta = \frac{c\,dt}{a(t)} \quad \Longrightarrow \quad \eta = c\int_0^t \frac{dt'}{a(t')}$$
In conformal time, the FLRW metric takes the form \(ds^2 = a(\eta)^2\left[-d\eta^2 + d\chi^2 + S_k^2(\chi)\,d\Omega^2\right]\)
The beauty of conformal time is that the metric becomes conformally flat -- light rays travel on 45-degree lines in a conformal spacetime diagram, just as in Minkowski space. This makes causal structure transparent.
The comoving distance to an object at redshift z is the distance measured along a spatial slice at the present time:
$$\chi(z) = c\int_0^z \frac{dz'}{H(z')} = c\int_{t_{\text{emit}}}^{t_0}\frac{dt'}{a(t')}$$
The comoving distance is the conformal time elapsed since the light was emitted
Cosmological Horizons
The finite age of the universe and the finite speed of light create fundamental limits on what we can observe and what can influence us. These limits define the cosmological horizons.
Particle Horizon
The particle horizon is the maximum comoving distance from which light could have reached us since the Big Bang:
$$d_{\text{particle}}(t) = a(t)\int_0^t \frac{c\,dt'}{a(t')} = a(t)\,\eta(t)$$
The proper distance to the particle horizon. Objects beyond this distance have never been in causal contact with us.
For a matter-dominated universe, \(d_{\text{particle}} = 3ct\) (three times the naive estimate\(ct\), because the universe was smaller in the past, so light had "less distance to cover"). For our universe today, the particle horizon corresponds to a comoving distance of about 46.3 billion light-years, which is the radius of the observable universe.
Event Horizon
The event horizon is the maximum comoving distance from which light emitted now can ever reach us in the infinite future:
$$d_{\text{event}}(t) = a(t)\int_t^\infty \frac{c\,dt'}{a(t')}$$
Events beyond the event horizon will never be observable, no matter how long we wait
An event horizon exists only if the integral converges, which occurs when the expansion accelerates sufficiently (as in a universe with a positive cosmological constant). In our universe, the event horizon is at a comoving distance of about 16.7 billion light-years. Galaxies currently beyond this distance will eventually disappear from view as their light is stretched to infinite wavelength.
Hubble Radius vs. Particle Horizon
An important distinction that often causes confusion:
Hubble Radius
$$d_H = \frac{c}{H(t)}$$
The distance at which the recession velocity equals c. Objects beyond the Hubble radius are receding superluminally now, but this does not violate special relativity (it is the expansion of space, not motion through space).
Particle Horizon
$$d_p = a(t)\int_0^t \frac{c\,dt'}{a(t')}$$
The maximum distance from which we have received light. In general,\(d_p > d_H\) because light from the early universe has had time to reach us even though those regions are now beyond the Hubble radius. We can see objects receding faster than light.
Critical Density and Density Parameters
The first Friedmann equation can be rewritten in a revealing form by defining the critical density:
$$\rho_{\text{crit}} = \frac{3H^2}{8\pi G} \approx 9.47 \times 10^{-27} \text{ kg/m}^3$$
The energy density required for a spatially flat universe (evaluated today with \(H = H_0\))
This corresponds to about 5.7 protons per cubic meter -- an astonishingly low density that nevertheless determines whether the universe is open, flat, or closed. We define dimensionless density parameters for each component:
$$\Omega_r = \frac{\rho_r}{\rho_{\text{crit}}}, \quad \Omega_m = \frac{\rho_m}{\rho_{\text{crit}}}, \quad \Omega_\Lambda = \frac{\rho_\Lambda}{\rho_{\text{crit}}}, \quad \Omega_k = -\frac{kc^2}{a^2H^2}$$
$$\Omega_r + \Omega_m + \Omega_\Lambda + \Omega_k = 1$$
The cosmic sum rule: the total density parameter always equals unity
In terms of the density parameters, the Friedmann equation becomes:
$$H^2(z) = H_0^2\left[\Omega_{r,0}(1+z)^4 + \Omega_{m,0}(1+z)^3 + \Omega_{k,0}(1+z)^2 + \Omega_{\Lambda,0}\right]$$
Hubble parameter as a function of redshift z, using \(1 + z = 1/a\)
This form makes it clear which component dominates at each epoch:
- Radiation domination at very high redshift (\(z \gtrsim 3400\)): the \((1+z)^4\) term wins
- Matter domination at intermediate redshifts (\(3400 \gtrsim z \gtrsim 0.4\)): the \((1+z)^3\) term wins
- Dark energy domination at low redshift (\(z \lesssim 0.4\)): the constant \(\Omega_\Lambda\) term wins
Current Values (Planck 2018)
\(\Omega_{r,0} \approx 9.15 \times 10^{-5}\)
Radiation (photons + neutrinos)
\(\Omega_{m,0} \approx 0.315\)
Matter (baryons + dark matter)
\(\Omega_{\Lambda,0} \approx 0.685\)
Dark energy
\(|\Omega_{k,0}| < 0.002\)
Curvature (consistent with zero)
Physical Insight: The Cosmic Budget
Our universe's energy budget is dominated by dark energy (~69%) and dark matter (~26%), with ordinary baryonic matter comprising only about 5% and radiation a negligible fraction today. Yet in the early universe, radiation dominated everything. The crossover between eras is one of the most important features of cosmological evolution, and the precise values of the density parameters determine the entire thermal and structural history of the cosmos.