The Expanding Universe III — Dynamics

Part 3 of five. So far (a(t)) has been an unknown we carried around. Now we make it obey a law — Einstein's equation, fed the contents of the universe — and derive the Friedmann equations and their exact solutions.

In Part 1 symmetry fixed the shape of spacetime; in Part 2 we watched light move on it. Both left \(a(t)\) undetermined. The missing law is the Einstein equation

\[ G_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu} ,\]

with \(G_{\mu\nu}\) the curvature and \(T_{\mu\nu}\) the matter content. Plan: constrain \(T_{\mu\nu}\) by symmetry, compute \(G_{\mu\nu}\) for RW, set them equal, solve.

What matter can be: perfect fluids

Homogeneity and isotropy constrain the matter as tightly as they did the geometry. Decompose \(T_{\mu\nu}\) into a scalar \(T_{00}\), vectors \(T_{0i}\), and a tensor \(T_{ij}\). Isotropy forbids a preferred direction, so the mean of any 3-vector vanishes: \(T_{0i}=0\). Isotropy also forces the 3-tensor \(T_{ij}\) to be proportional to the spatial metric, \(T_{ij}\propto g_{ij}\), and homogeneity makes the coefficient a function of time only. Hence

\[ T_{00}=\rho(t)c^2 , \quad T_{0i}=0 , \quad T_{ij}=P(t)\,g_{ij} \;\Longrightarrow\; T^\mu{}_\nu=\mathrm{diag}(-\rho c^2, P, P, P) .\]

This is a perfect fluid, and in covariant form

\[ T_{\mu\nu} = \left(\rho+\frac{P}{c^2}\right)U_\mu U_\nu + P\,g_{\mu\nu} .\]

Two functions of time: energy density \(\rho c^2\) and pressure \(P\).

Continuity — deriving how \(\rho\) evolves

Energy–momentum conservation is \(\nabla_\mu T^\mu{}_\nu=0\). Unpacking the covariant derivative with the FRW Christoffels (Part 2), the \(\nu=0\) component gives

\[ \partial_\mu T^\mu{}_0 + \Gamma^\mu_{\mu\lambda}T^\lambda{}_0 - \Gamma^\lambda_{\mu 0}T^\mu{}_\lambda = 0 . \]

With \(T^0{}_0=-\rho c^2\), \(T^i{}_j=P\delta^i_j\), and \(\Gamma^i_{0j}=c^{-1}(\dot a/a)\delta^i_j\) (three of them), this becomes the continuity equation:

\[ \boxed{\;\dot\rho + 3\frac{\dot a}{a}\left(\rho+\frac{P}{c^2}\right)=0.\;}\]

The \(3(\dot a/a)\) is dilution by the growing volume (\(V\propto a^3\)); the pressure term is the work the fluid does as space expands. (The same logic applied to a conserved number current \(\nabla_\mu N^\mu=0\) gives \(n\propto a^{-3}\) — pure volume dilution.)

Equation of state and the density scalings

Close the system with an equation of state \(w\equiv P/\rho c^2\). For constant \(w\), the continuity equation is separable:

\[ \frac{\dot\rho}{\rho} = -3(1+w)\frac{\dot a}{a} \;\Longrightarrow\; \boxed{\;\rho\propto a^{-3(1+w)}.\;}\]

🧠 Defn

The three players.

• Matter (dust, CDM, baryons): \(w=0\Rightarrow\rho_m\propto a^{-3}\) — volume dilution only.

• Radiation (photons, relativistic \(\nu\)): \(w=\tfrac13\Rightarrow\rho_r\propto a^{-4}\) — one extra power of \(a\) from the \(E\propto a^{-1}\) redshift of Part 2.

• Dark energy (\(\Lambda\)): \(w=-1\Rightarrow\rho_\Lambda=\,\)const — it does not dilute, so it inevitably wins.

The Friedmann equations

Computing \(G_{\mu\nu}\) for the RW metric is a Christoffel grind (Baumann does it once "to appreciate computer algebra"). The nonzero Ricci pieces are \(R_{00}=-3\ddot a/(c^2a)\) and \(R_{ij}=c^{-2}[\ddot a/a+2(\dot a/a)^2+2kc^2/(a^2R_0^2)]g_{ij}\), giving the Einstein tensor components

\[ G^0{}_0 = -\frac{3}{c^2}\!\left[\left(\frac{\dot a}{a}\right)^2+\frac{kc^2}{a^2R_0^2}\right], \qquad G^i{}_j = -\frac{1}{c^2}\!\left[2\frac{\ddot a}{a}+\left(\frac{\dot a}{a}\right)^2+\frac{kc^2}{a^2R_0^2}\right]\delta^i_j .\]

Setting \(G^0{}_0=(8\pi G/c^4)T^0{}_0=-(8\pi G/c^2)\rho\) gives the (first) Friedmann equation:

\[ \boxed{\;\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{a^2R_0^2}.\;}\]

The spatial component \(G^i{}_j=(8\pi G/c^4)T^i{}_j\) gives the second, the Raychaudhuri equation:

\[ \boxed{\;\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right).\;}\]

The first says the expansion rate is sourced by the total density, offset by curvature; the second says gravity decelerates the expansion whenever \(\rho+3P/c^2>0\) — which is why ordinary matter demands a Big Bang, and why acceleration needs something with \(w<-\tfrac13\).

📝 Note

A Newtonian shadow. For a test mass on an expanding ball of dust of radius \(R=aR_0\), energy conservation \(\tfrac12\dot R^2-GM/R=E\) with \(M=\tfrac{4\pi}{3}R^3\rho\) is exactly the first Friedmann equation, with \(-kc^2/R_0^2\) playing the role of \(2E\). A flat universe (\(k=0\)) is one whose kinetic and potential energies cancel — the coincidence that becomes the flatness problem.

The master equation

Measure densities against the critical density \(\rho_{c,0}=3H_0^2/8\pi G\), and define \(\Omega_i\equiv\rho_{i,0}/\rho_{c,0}\). Dividing the Friedmann equation by \(H_0^2\) and inserting the scalings turns it into the equation Cosmic.jl solves for every calculation in this course:

\[ \boxed{\;\frac{H^2}{H_0^2} = \Omega_r a^{-4} + \Omega_m a^{-3} + \Omega_k a^{-2} + \Omega_\Lambda \equiv E(a)^2,\;}\]

with curvature "density" \(\Omega_k\equiv-kc^2/(R_0H_0)^2\). At \(a=1\) this is the closure relation \(\Omega_r+\Omega_m+\Omega_\Lambda+\Omega_k=1\). Measuring these numbers is the central task of observational cosmology — and it is exactly the species list you hand cosmology().

Exact solutions

Single component (flat). With one fluid, \(\dot a/a=H_0\sqrt{\Omega_i}\,a^{-3(1+w)/2}\). Separating and integrating \(\int a^{3(1+w)/2-1}\mathrm d a\propto t\),

\[ a(t)\propto\begin{cases} t^{1/2} & \text{radiation }(w=\tfrac13)\\ t^{2/3} & \text{matter }(w=0)\\ e^{H_0\sqrt{\Omega_\Lambda}\,t} & \Lambda\ (w=-1)\end{cases}\]

These three limiting cases have names, and they come up again and again, so it is worth knowing what each one is in plain terms:

🧠 Defn

Three model universes.

• Einstein–de Sitter — a flat universe made of nothing but matter (\(\Omega_m=1\)), expanding as \(a\propto t^{2/3}\). It was the textbook cosmology for decades. Its trouble: with \(a=(t/t_0)^{2/3}\) one finds \(t_0=\tfrac23 H_0^{-1}\approx9.6\,\mathrm{Gyr}\) — younger than the oldest known stars. This "age problem" was one of the clues that something (dark energy) was missing.

• de Sitter — a universe with nothing but a cosmological constant (vacuum energy), expanding exponentially, \(a\propto e^{Ht}\). Empty of matter but never diluting, it accelerates forever. It describes both the very early universe (inflation) and the far future of ours, once \(\Lambda\) has won.

• Milne — an empty universe with only spatial curvature, coasting as \(a\propto t\). A useful "no-gravity" reference point.

Because the densities scale so differently, our universe is dominated by one component at a time — radiation, then matter, then dark energy:

Feeding the full species list to Cosmic.jl and integrating the Friedmann equation reproduces these power laws as the limiting slopes of one smooth curve:

Matter + curvature: the fate of the universe. Before dark energy, the destiny of the cosmos hinged on curvature. Working in conformal time, the matter+curvature Friedmann equation \(2\,\mathrm d\widetilde\mathcal H/\mathrm d\theta+\widetilde\mathcal H^2+k=0\) (with \(\theta\equiv c\eta/R_0\)) integrates to the exact parametric solution

\[ a(\theta)=A\begin{cases}\sin^2(\theta/2) & k=+1\\ \theta^2 & k=0\\ \sinh^2(\theta/2) & k=-1\end{cases}, \qquad t(\theta)=\frac{R_0A}{2c}\begin{cases}\theta-\sin\theta\\ \theta^3/3\\ \sinh\theta-\theta\end{cases}\]

A closed (\(\Omega_0>1\)) universe reaches a maximum \(a_{\rm max}=\Omega_0/(\Omega_0-1)\) where \(H=0\) and then recollapses to a Big Crunch; an open (\(\Omega_0<1\)) universe expands forever; the flat case is the knife-edge Einstein–de Sitter. Cosmic.jl integrates all three directly from \(\Omega_k\) (the closed branch stops at turnaround, where \(E(a)\to0\)):

julia> mk(Ωk) = cosmology(Ω_c = 1 - Ω_b(c) - Ω_γ(c) - Ω_ν(c) - Ωk, Ω_k = Ωk);

julia> c_flat, c_open, c_close = mk(0.0), mk(0.7), mk(-0.3);   # k = 0, -1, +1

julia> Ω_m(c_close), Ω_k(c_close)     # a closed universe has Ω_m > 1 and Ω_k < 0
(1.2985, -0.3)

julia> a_max = Ω_m(c_close) / abs(Ω_k(c_close))   # it turns around here, where H = 0
4.328

julia> E(c_close, a_max)              # the expansion rate does vanish at turnaround
2.1e-8

Matter + \(\Lambda\): our universe. A flat universe with matter and a cosmological constant has \(H^2/H_0^2=\Omega_m a^{-3}+\Omega_\Lambda\). Unlike the single-fluid and curvature cases above, this one is not solved by a direct power-law ansatz — it needs a substitution, so let's actually do it. Write \(\dot a/a=H_0\sqrt{\Omega_m a^{-3}+\Omega_\Lambda}\) and multiply through by \(a^{3/2}\):

\[ a^{1/2}\dot a = H_0\sqrt{\Omega_m+\Omega_\Lambda a^3} = H_0\sqrt{\Omega_m}\,\sqrt{1+(a/a_\Lambda)^3}, \qquad a_\Lambda\equiv\left(\frac{\Omega_m}{\Omega_\Lambda}\right)^{1/3} .\]

The left side is \(\tfrac23\mathrm d(a^{3/2})/\mathrm d t\), which is the tell that \(x\equiv(a/a_\Lambda)^{3/2}\) is the right variable: \(a^{3/2}=a_\Lambda^{3/2}x\), so \(\tfrac23a_\Lambda^{3/2}\dot x=H_0\sqrt{\Omega_m}\sqrt{1+x^2}\). Since \(a_\Lambda^{-3/2}=\sqrt{\Omega_\Lambda/\Omega_m}\), the \(\Omega_m\)'s cancel and this separates cleanly,

\[ \frac{\mathrm d x}{\sqrt{1+x^2}} = \tfrac32\sqrt{\Omega_\Lambda}\,H_0\,\mathrm d t \;\Longrightarrow\; \sinh^{-1}x = \tfrac32\sqrt{\Omega_\Lambda}H_0 t ,\]

fixing the integration constant by \(a=0\) (so \(x=0\)) at \(t=0\), the Big Bang. Undoing \(x=(a/a_\Lambda)^{3/2}\) gives the closed form, and setting \(a(t_0)=1\) gives the age:

\[ \boxed{\;a(t)=\left(\frac{\Omega_m}{\Omega_\Lambda}\right)^{1/3}\sinh^{2/3}\!\left(\tfrac32\sqrt{\Omega_\Lambda}\,H_0 t\right)\;}, \qquad t_0=\frac{2}{3\sqrt{\Omega_\Lambda}H_0}\sinh^{-1}\!\sqrt{\frac{\Omega_\Lambda}{\Omega_m}} .\]

The two limits confirm it stitches together the two universes we already know. Early on \(a\ll a_\Lambda\) means \(x\ll1\), where \(\sinh^{-1}x\approx x\), so \(a(t)\propto t^{2/3}\) — plain Einstein–de Sitter, matter not yet noticing \(\Lambda\). Late on \(a\gg a_\Lambda\) means \(x\gg1\), where \(\sinh x\approx\tfrac12e^x\), so \(a(t)\propto e^{H_0\sqrt{\Omega_\Lambda}\,t}\) — de Sitter, matter diluted away and only the constant left. With \(\Omega_m=0.31,\Omega_\Lambda=0.69\) the age comes out \(t_0\approx0.96\,H_0^{-1}\approx14\,\mathrm{Gyr}\) — a universe old enough for its oldest stars, unlike EdS — and matches what Cosmic returns below.

A universe of matter obeying \(\rho+3P/c^2>0\) must have had a singularity — because \(\ddot a<0\) everywhere means \(a(t)\), run back from today's expansion, hit zero. That is the FRW baby version of the Hawking–Penrose theorems.

Verifying with Cosmic.jl

The tightest check of the course: \(E(a)\) is literally the species sum, and the deceleration parameter follows from its slope.

First, that \(E(a)^2\) really is the sum over species — compare it against the textbook two-power form at, say, \(a=0.1\):

julia> a = 0.1;

julia> E(c, a)^2                              # the exact Friedmann sum, Σ ρ_s(a)/ρ_c0
313.3

julia> Ω_r(c)*a^-4 + Ω_m(c)*a^-3 + Ω_de(c)    # the textbook Ω_r a⁻⁴ + Ω_m a⁻³ + Ω_Λ
311.9

They agree to about half a percent, and the small gap is physics, not error: at \(a=0.1\) the massive neutrino is partway between behaving like radiation and like matter, and the clean two-power split can't capture it — a reminder of why Cosmic tracks each species' real \(\rho(a)\).

Next, the deceleration parameter \(q_0=-\ddot a a/\dot a^2\big|_0\), which is negative if the universe accelerates. We can read it straight off the slope of \(E(a)\):

julia> δ = 1e-4;

julia> q0 = -1 - (log(E(c, exp(δ))) - log(E(c, exp(-δ)))) / (2δ)   # from the slope of E(a)
-0.544

julia> 0.5*Ω_m(c) + Ω_r(c) - Ω_de(c)          # the textbook formula ½Ω_m + Ω_r − Ω_Λ
-0.534

Both are negative — our universe is accelerating. Finally, the age problem and its resolution, side by side:

julia> 2/3 * hubble_time(c)      # the Einstein–de Sitter (matter-only) age, in Gyr
9.634

julia> age(c)                    # the real, Λ-included age
13.787

The matter-only universe would be only 9.6 Gyr old — younger than the oldest stars. Dark energy stretches the real age to 13.8 Gyr, comfortably older, exactly as promised.

Where we are

We have the law of motion: the Friedmann equations, the density scalings, and the exact solutions for each era and geometry. Everything is ready for the measured numbers. Part 4 — Our Universe puts them in — the cosmic budget, the age, equality, and the thermal history — Cosmic.jl against the observed values.

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