The Friedmann Equations from Einstein's Equations

Derivation

The Friedmann Equations from Einstein's Equations

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Statement

Substituting the spatially homogeneous and isotropic Friedmann–Robertson–Walker (FRW) metric together with a perfect-fluid stress–energy tensor \(T^{\mu}{}_{\nu}=\mathrm{diag}(-\rho,\,p,\,p,\,p)\) into the Einstein field equations \(G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}\) yields, from the \(tt\) component, the Friedmann equation \(\left(\dfrac{\dot a}{a}\right)^{2}=\dfrac{8\pi G}{3}\rho-\dfrac{kc^{2}}{a^{2}}+\dfrac{\Lambda c^{2}}{3}\); from the spatial components the acceleration equation \(\dfrac{\ddot a}{a}=-\dfrac{4\pi G}{3}\left(\rho+\dfrac{3p}{c^{2}}\right)+\dfrac{\Lambda c^{2}}{3}\); and, from their combination (equivalently \(\nabla_{\mu}T^{\mu}{}_{\nu}=0\)), the cosmic fluid equation \(\dot\rho+3\dfrac{\dot a}{a}\left(\rho+\dfrac{p}{c^{2}}\right)=0\).

Why it matters

These are the master equations of physical cosmology. From a single scale factor \(a(t)\) and an equation of state \(p=p(\rho)\) they determine the entire expansion history of the universe: the Hubble rate, the deceleration or acceleration, the age, the critical density, and the fate of the cosmos. Every quantitative statement in the \(\Lambda\mathrm{CDM}\) concordance model — the microwave-background acoustic scale, big-bang nucleosynthesis abundances, the supernova evidence for dark energy — is read off solutions of these equations.

The derivation is also the cleanest non-trivial application of general relativity: a maximally symmetric spatial slice collapses the ten coupled Einstein equations to just two independent ordinary differential equations, showing exactly how geometry (\(G_{\mu\nu}\)) and matter (\(T_{\mu\nu}\)) talk to each other on cosmological scales.

Assumptions

The cosmological principle: space is homogeneous and isotropic about every point.If dropped, the metric acquires position- and direction-dependent functions, the off-diagonal and anisotropic Einstein components no longer vanish, and one is forced into Bianchi or Lemaître–Tolman–Bondi models with many more degrees of freedom.

Matter is a perfect fluid: \(T_{\mu\nu}=\left(\rho+p/c^{2}\right)u_{\mu}u_{\nu}+p\,g_{\mu\nu}\), with no shear, viscosity, or heat flux.If dropped, anisotropic stress \(\pi_{\mu\nu}\) sources the spatial Einstein equations, the two Friedmann equations decouple from a single \(\rho,p\) pair, and dissipative terms enter the fluid equation.

The fluid is comoving: its four-velocity is \(u^{\mu}=(c,0,0,0)/\sqrt{-g_{tt}}\) in FRW coordinates, i.e. \(u^{\mu}=(1,0,0,0)\) with \(c=1\).If dropped, peculiar velocities generate momentum density \(T^{t}{}_{i}\neq0\), breaking the diagonal form and the isotropy the metric assumes.

General relativity holds with the Einstein–Hilbert action plus at most a cosmological constant \(\Lambda\); the connection is the metric-compatible torsion-free Levi-Civita connection.If dropped for a modified-gravity theory (\(f(R)\), scalar–tensor, extra dimensions), \(G_{\mu\nu}\) gains extra curvature terms and the Friedmann equations acquire new source terms mimicking or replacing dark energy.

Spatial curvature is one of the three maximally symmetric constants \(k\in\{-1,0,+1\}\) (or a continuous \(k\) with units of inverse length squared).If dropped, the spatial metric is not a constant-curvature space and the Ricci tensor of the slice is not proportional to its metric, invalidating the simple \(k/a^2\) term.

Derivation

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

Adopt the FRW line element (from frw-metric-cosmological-principle); homogeneity forces all metric functions to depend on \(t\) alone and isotropy fixes the constant-curvature spatial part. Comoving coordinates \((t,r,\theta,\phi)\). A

\[ g_{tt}=-c^{2},\quad g_{rr}=\frac{a^{2}}{1-kr^{2}},\quad g_{\theta\theta}=a^{2}r^{2},\quad g_{\phi\phi}=a^{2}r^{2}\sin^{2}\theta \]

Read the diagonal metric components straight off the line element; the inverse metric \(g^{\mu\nu}\) is the reciprocal of each diagonal entry. This is the raw input to the connection. A

\[ \Gamma^{\lambda}{}_{\mu\nu}=\tfrac12 g^{\lambda\sigma}\left(\partial_{\mu}g_{\sigma\nu}+\partial_{\nu}g_{\sigma\mu}-\partial_{\sigma}g_{\mu\nu}\right) \]

Compute the Levi-Civita connection. Only \(t\)-derivatives of \(a\) and \(r,\theta\)-derivatives of the spatial part survive. The metric-compatibility and torsion-free conditions make this connection unique. B

\[ \Gamma^{t}{}_{rr}=\frac{a\dot a}{c^{2}\left(1-kr^{2}\right)},\quad \Gamma^{t}{}_{\theta\theta}=\frac{a\dot a r^{2}}{c^{2}},\quad \Gamma^{t}{}_{\phi\phi}=\frac{a\dot a r^{2}\sin^{2}\theta}{c^{2}} \]

The time–space Christoffels carry the expansion. Each spatial \(\Gamma^{t}{}_{ii}\) equals \(g_{ii}\dot a /(a c^{2})\); these feed the \(tt\) Ricci component. B

\[ \Gamma^{r}{}_{tr}=\Gamma^{\theta}{}_{t\theta}=\Gamma^{\phi}{}_{t\phi}=\frac{\dot a}{a},\qquad \Gamma^{r}{}_{rr}=\frac{kr}{1-kr^{2}} \]

The \(\dot a/a\) Christoffels are the Hubble factor \(H\); the purely spatial \(\Gamma^{r}{}_{rr}\), \(\Gamma^{r}{}_{\theta\theta}=-r(1-kr^{2})\), etc. encode the spatial curvature \(k\). B

\[ R_{\mu\nu}=\partial_{\lambda}\Gamma^{\lambda}{}_{\mu\nu}-\partial_{\nu}\Gamma^{\lambda}{}_{\mu\lambda}+\Gamma^{\lambda}{}_{\lambda\sigma}\Gamma^{\sigma}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\sigma}\Gamma^{\sigma}{}_{\mu\lambda} \]

Assemble the Ricci tensor by contracting the Riemann tensor. Homogeneity and isotropy guarantee \(R_{\mu\nu}\) is diagonal with only two independent entries (a time entry and one spatial entry, the latter proportional to \(g_{ii}\)). C

\[ R_{tt}=-\frac{3\ddot a}{a},\qquad R_{ij}=\left[\frac{\ddot a}{a}+\frac{2\dot a^{2}}{a^{2}}+\frac{2kc^{2}}{a^{2}}\right]\frac{g_{ij}}{c^{2}} \]

Carry out the contraction. The single time component and the isotropic spatial component are the only non-zero pieces; every off-diagonal Ricci component cancels by symmetry. C

\[ R=g^{\mu\nu}R_{\mu\nu}=g^{tt}R_{tt}+g^{ij}R_{ij}=\frac{6}{c^{2}}\left[\frac{\ddot a}{a}+\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right] \]

Contract Ricci with the inverse metric to get the Ricci scalar. Using \(g^{tt}=-1/c^{2}\) and \(g^{ij}R_{ij}=3\) times the bracket, the two pieces add. B

\[ G_{\mu\nu}=R_{\mu\nu}-\tfrac12 R\,g_{\mu\nu}\ \Longrightarrow\ G_{tt}=3\left[\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right] \]

Form the Einstein tensor. For the \(tt\) component, \(-\tfrac12 R\,g_{tt}=+\tfrac12 R\,c^{2}\); the \(\ddot a\) terms cancel between \(R_{tt}\) and \(\tfrac12 R c^2\), leaving only first-derivative and curvature terms. B

\[ G_{ij}=-\left[\frac{2\ddot a}{a}+\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right]\frac{g_{ij}}{c^{2}} \]

The spatial Einstein components, again isotropic (proportional to \(g_{ij}\)). This will pair with the pressure \(p\) on the matter side. B

\[ T_{\mu\nu}=\left(\rho+\frac{p}{c^{2}}\right)u_{\mu}u_{\nu}+p\,g_{\mu\nu},\quad u^{\mu}=(1,0,0,0)\ \Rightarrow\ T_{tt}=\rho c^{2},\ \ T_{ij}=p\,g_{ij} \]

Insert the comoving perfect fluid (from stress-energy-perfect-fluid). With \(u_{t}=-c^{2}/(c)= -c\) normalized so \(u^\mu u_\mu=-c^2\), the energy density sits in \(tt\) and isotropic pressure in the spatial block. A

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

Write the Einstein field equations with cosmological constant (from einstein-field-equations), in SI form with the \(1/c^{4}\) coupling so both sides carry units of inverse length squared. A

\[ 3\left[\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right]+\Lambda\left(-c^{2}\right)=\frac{8\pi G}{c^{4}}\,\rho c^{2} \]

Take the \(tt\) component: substitute \(G_{tt}\) from step 9, \(g_{tt}=-c^{2}\), and \(T_{tt}=\rho c^{2}\). This is the raw \(tt\) Einstein equation before isolating \(H^2\). B

\[ \boxed{\ \left(\frac{\dot a}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^{2}}+\frac{\Lambda c^{2}}{3}\ } \]

Divide step 13 by 3 and rearrange: move the curvature term to the right, absorb \(\Lambda\). This is the Friedmann equation. Symbols rearranged, no numbers introduced. A

\[ -\left[\frac{2\ddot a}{a}+\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right]\frac{g_{ij}}{c^{2}}+\Lambda g_{ij}=\frac{8\pi G}{c^{4}}\,p\,g_{ij} \]

Take any spatial component: substitute \(G_{ij}\) from step 10 and \(T_{ij}=p\,g_{ij}\). Every term is proportional to \(g_{ij}\), so divide it out. B

\[ -\frac{1}{c^{2}}\left[\frac{2\ddot a}{a}+\frac{\dot a^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}\right]+\Lambda=\frac{8\pi G}{c^{4}}\,p \]

Cancel the common \(g_{ij}\). Now use the Friedmann equation (step 14) to eliminate the combination \(\dot a^{2}/a^{2}+kc^{2}/a^{2}=8\pi G\rho/3+\Lambda c^{2}/3\). B

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

Substitute and solve step 16 for \(\ddot a/a\): the curvature term cancels, leaving the acceleration (second Friedmann) equation. Pressure and energy density both decelerate; \(\Lambda\) accelerates. A

\[ \nabla_{\mu}T^{\mu}{}_{\nu}=0\ \xrightarrow{\ \nu=t\ }\ \dot\rho+3\frac{\dot a}{a}\left(\rho+\frac{p}{c^{2}}\right)=0 \]

The contracted Bianchi identity \(\nabla_{\mu}G^{\mu}{}_{\nu}=0\) forces energy–momentum conservation. The \(\nu=t\) component gives the cosmic fluid equation; it also follows by differentiating step 14 and using step 17, so only two of the three equations are independent. C

Result

\[ \left(\frac{\dot a}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2}}{a^{2}}+\frac{\Lambda c^{2}}{3},\qquad \frac{\ddot a}{a}=-\frac{4\pi G}{3}\!\left(\rho+\frac{3p}{c^{2}}\right)+\frac{\Lambda c^{2}}{3},\qquad \dot\rho+3H\!\left(\rho+\frac{p}{c^{2}}\right)=0 \]

Reading. The first equation says the square of the expansion rate \(H=\dot a/a\) is set by the total energy density, reduced by positive spatial curvature and boosted by the cosmological constant — geometry balances against content. The second says pressure gravitates: a fluid with \(\rho+3p/c^{2}>0\) decelerates the expansion, while a sufficiently negative pressure (dark energy, \(w<-1/3\)) or a positive \(\Lambda\) drives acceleration. The third is local energy conservation carried by the expansion: as space stretches, density dilutes at a rate set by \(H\) and the equation of state, so \(\rho\propto a^{-3(1+w)}\) for \(p=w\rho c^{2}\).

Units check. In SI, \([\dot a/a]=\mathrm{s^{-1}}\), so the left side of Friedmann is \(\mathrm{s^{-2}}\). Right side: \([G\rho]=(\mathrm{m^{3}kg^{-1}s^{-2}})(\mathrm{kg\,m^{-3}})=\mathrm{s^{-2}}\ \checkmark\); \([kc^{2}/a^{2}]=(\mathrm{m^{2}s^{-2}})/\mathrm{m^{2}}=\mathrm{s^{-2}}\ \checkmark\) (with \(a\) carrying length when \(k=\pm1\)); \([\Lambda c^{2}]=(\mathrm{m^{-2}})(\mathrm{m^{2}s^{-2}})=\mathrm{s^{-2}}\ \checkmark\). Acceleration equation: \([\ddot a/a]=\mathrm{s^{-2}}\) and \([p/c^{2}]=(\mathrm{kg\,m^{-1}s^{-2}})/(\mathrm{m^{2}s^{-2}})=\mathrm{kg\,m^{-3}}=[\rho]\ \checkmark\).

Limiting cases

Flat, matter-dominated (\(k=0,\ \Lambda=0,\ p=0\)): \(\rho\propto a^{-3}\) gives the Einstein–de Sitter solution \(a\propto t^{2/3}\), \(H=2/(3t)\).
Flat, radiation-dominated (\(p=\rho c^{2}/3\)): \(\rho\propto a^{-4}\) gives \(a\propto t^{1/2}\), the early-universe scaling that fixes nucleosynthesis timing.
Vacuum / de Sitter (\(\rho=p=0,\ \Lambda>0,\ k=0\)): \(H=\sqrt{\Lambda c^{2}/3}=\text{const}\), so \(a\propto e^{Ht}\) — exponential expansion, the inflationary and late-time attractor.
Static Einstein universe (\(\ddot a=\dot a=0\)): requires \(\Lambda=4\pi G\rho/c^{2}\) and \(k=+1\); unstable, historically Einstein's motive for \(\Lambda\).
Empty curved (\(\rho=p=\Lambda=0,\ k=-1\)): Milne universe, \(a\propto t\), \(H=1/t\) — coasting expansion with zero deceleration.

Breaks when

At the initial singularity \(a\to0\): \(\rho\) and curvature invariants diverge, quantum-gravity effects become order unity at the Planck density, and the classical Einstein equations — hence these Friedmann equations — cease to be valid.
When homogeneity or isotropy fails (structure formation on small scales, strong anisotropies, large peculiar velocities): off-diagonal and anisotropic-stress components of \(G_{\mu\nu}\) and \(T_{\mu\nu}\) revive, and the single scale factor \(a(t)\) is no longer an adequate description; one needs perturbation theory or full numerical relativity.
Under modified gravity or extra fields (\(f(R)\), scalar–tensor, braneworlds): additional geometric terms enter \(G_{\mu\nu}\), changing the functional form of the Friedmann equation and mimicking a nonstandard dark-energy component.
For an imperfect fluid with shear viscosity or free-streaming (e.g. neutrinos near decoupling): anisotropic stress \(\pi_{ij}\neq0\) breaks the clean \(T_{ij}=p\,g_{ij}\) form used in step 11.

Failure modes

Dropping the \(3p/c^{2}\) in the acceleration equation — treating pressure as a passive spectator. In GR pressure gravitates; forgetting it removes the very term that lets dark energy accelerate the universe.
Sign error on curvature: writing \(+kc^{2}/a^{2}\) in the Friedmann equation. Positive \(k\) (closed) reduces \(H^{2}\); the term must be subtracted.
Confusing \(\Lambda\) placement: putting \(\Lambda\) on the matter side as a density \(\rho_\Lambda=\Lambda c^{2}/8\pi G\) and also keeping the explicit geometric \(\Lambda\) term — double counting the vacuum energy.
Using coordinate (non-comoving) velocity for \(u^{\mu}\), producing spurious \(T^{t}{}_{i}\) momentum flux that contradicts the isotropy already assumed in the metric.
Treating \(a(t)\) as dimensionless while \(k=\pm1\): then \(kc^{2}/a^{2}\) has wrong units. Either \(a\) carries length (with \(k=\pm1\)) or \(k\) carries inverse length squared (with dimensionless \(a\)) — be consistent.
Assuming the three equations are independent: the fluid equation follows from the other two via the Bianchi identity, so imposing all three plus an equation of state over-determines the system.

Discussion

The structure of the derivation is a lesson in how symmetry organizes general relativity. The maximal symmetry of the spatial slices forces \(G_{\mu\nu}\) and \(T_{\mu\nu}\) to be simultaneously diagonal with only two independent components, so ten coupled second-order PDEs collapse to two ODEs. The \(tt\) equation is a constraint (first order in \(\dot a\), no \(\ddot a\)) — a Hamiltonian constraint reflecting time-reparametrization invariance — while the spatial equation is the genuine evolution equation. This constraint/evolution split is the cosmological face of the general \(3+1\) ADM structure of GR.

Physically, the two equations partition gravity into an "energy budget" statement and a "force" statement. Friedmann's equation is a first integral: multiplying by \(a^{2}\) it reads like Newtonian energy conservation for a test shell, \(\tfrac12\dot a^{2}-\tfrac{4\pi G}{3}\rho a^{2}=-\tfrac12 kc^{2}\), with \(-kc^{2}/2\) playing the role of total mechanical energy. The acceleration equation is the corresponding "\(F=ma\)", and it is here that relativity departs decisively from Newton: the active gravitational mass density is \(\rho+3p/c^{2}\), not \(\rho\) alone, so pressure both weighs and, when negative enough, antigravitates.

The three equations connect directly across the physics2u threads. Through fields, they are the FRW reduction of the Einstein field equations. Through energy, the fluid equation is \(\nabla_\mu T^{\mu}{}_\nu=0\) and yields the scaling laws \(\rho_m\propto a^{-3}\), \(\rho_r\propto a^{-4}\), \(\rho_\Lambda=\text{const}\). Through symmetry, the whole reduction is a consequence of the homogeneity/isotropy Killing vectors. Through force, the acceleration equation is the gravitational equation of motion for the cosmos as a whole.

At a deeper level the redundancy among the equations is not an accident but the contracted Bianchi identity \(\nabla_\mu G^{\mu}{}_\nu\equiv0\), a geometric identity that holds off-shell. Imposing the field equations then forces \(\nabla_\mu T^{\mu}{}_\nu=0\): energy–momentum conservation is a consequence of the field equations, not an independent postulate. In the Hamiltonian (ADM) formulation the Friedmann equation is precisely the Hamiltonian constraint \(\mathcal{H}=0\) generated by the lapse, and its preservation under evolution is guaranteed by the same identity — which is why one may derive the fluid equation either from \(\nabla_\mu T^{\mu}{}_\nu=0\) directly or by differentiating Friedmann and substituting the acceleration equation.

Common misconceptions. (i) The expansion is not galaxies flying through space from a central explosion; \(a(t)\) rescales the comoving metric itself and there is no preferred center. (ii) "Dark energy accelerates because it has energy" is incomplete — it accelerates because its pressure is negative (\(w<-1/3\)); ordinary energy density decelerates. (iii) A closed (\(k=+1\)) universe is not automatically destined to recollapse once \(\Lambda>0\): the fate depends on the full density mix, not curvature alone.

Worked examples

\[ \text{Critical density today: } \rho_{c}=\frac{3H_{0}^{2}}{8\pi G},\qquad H_{0}=67.4\ \mathrm{km\,s^{-1}Mpc^{-1}} \]

Set \(k=0,\ \Lambda\) absorbed into \(\rho\); the flat-universe Friedmann equation defines the critical density. Convert \(H_0\) to SI first, then substitute numbers. A

\[ H_{0}=\frac{67.4\times10^{3}\ \mathrm{m\,s^{-1}}}{3.086\times10^{22}\ \mathrm{m}}=2.18\times10^{-18}\ \mathrm{s^{-1}} \]

Use \(1\ \mathrm{Mpc}=3.086\times10^{22}\ \mathrm{m}\). This is the Hubble rate in inverse seconds. A

\[ \rho_{c}=\frac{3\,(2.18\times10^{-18})^{2}}{8\pi\,(6.674\times10^{-11})}\ \mathrm{kg\,m^{-3}} \]

Substitute \(H_0\) and \(G=6.674\times10^{-11}\ \mathrm{m^{3}kg^{-1}s^{-2}}\). Numerator \(=3\times4.75\times10^{-36}=1.43\times10^{-35}\); denominator \(=1.677\times10^{-9}\). A

\[ \rho_{c}=8.5\times10^{-27}\ \mathrm{kg\,m^{-3}}\approx5.1\ \frac{\text{GeV}/c^{2}}{\mathrm{m^{3}}}\approx5\ \text{protons per m}^{3} \]

Reading. The mean density that makes space flat is astonishingly small — a few hydrogen atoms per cubic metre. Measured total density sits within a percent of this, so the observable universe is spatially flat to high precision.

Units check. \([H^{2}/G]=\mathrm{s^{-2}}/(\mathrm{m^{3}kg^{-1}s^{-2}})=\mathrm{kg\,m^{-3}}\ \checkmark\).

\[ \text{Age of a flat matter-dominated universe: } H^{2}=\frac{8\pi G}{3}\rho,\quad \rho=\rho_{0}\left(\frac{a_{0}}{a}\right)^{3} \]

Take \(k=\Lambda=0,\ p=0\). The fluid equation gives \(\rho\propto a^{-3}\); insert into Friedmann to get a solvable ODE for \(a(t)\). B

\[ \frac{\dot a}{a}=\sqrt{\frac{8\pi G\rho_{0}}{3}}\left(\frac{a_{0}}{a}\right)^{3/2}\ \Rightarrow\ a^{1/2}\,da=C\,dt,\quad C\equiv H_{0}a_{0}^{3/2} \]

Separate variables. Here \(H_0^{2}=8\pi G\rho_0/3\). Integrating the left side from \(0\) to \(a_0\) gives the age. B

\[ \tfrac23 a_{0}^{3/2}=C\,t_{0}=H_{0}a_{0}^{3/2}t_{0}\ \Rightarrow\ t_{0}=\frac{2}{3H_{0}} \]

Evaluate the definite integral; \(a_0^{3/2}\) cancels. Now put in \(H_0=2.18\times10^{-18}\ \mathrm{s^{-1}}\). A

\[ t_{0}=\frac{2}{3\,(2.18\times10^{-18})}=3.06\times10^{17}\ \mathrm{s}\approx9.7\ \text{Gyr} \]

Reading. A pure matter universe with this \(H_0\) would be only \(\sim9.7\) billion years old — younger than the oldest stars. The observed \(13.8\) Gyr age requires the extra push of a cosmological constant, which slows early expansion less and lengthens the inferred age. This tension is a textbook argument for dark energy.

Units check. \([1/H_0]=\mathrm{s}\); dividing by \(3.156\times10^{16}\ \mathrm{s\,Gyr^{-1}}\) gives Gyr \(\checkmark\).

Problems

(A) For a flat radiation-dominated universe (\(p=\rho c^{2}/3\)), use the fluid equation to show \(\rho\propto a^{-4}\) and hence find \(a(t)\).
Solution
Fluid equation: \(\dot\rho+3H(\rho+p/c^2)=\dot\rho+3H(\rho+\rho/3)=\dot\rho+4H\rho=0\). Thus \(\dot\rho/\rho=-4\dot a/a\Rightarrow\rho\propto a^{-4}\). Friedmann: \(\dot a/a=\sqrt{8\pi G\rho_0/3}\,(a_0/a)^{2}\Rightarrow a\,da\propto dt\Rightarrow a^2\propto t\Rightarrow a\propto t^{1/2}\). Then \(H=1/(2t)\).
(B) Derive the equation of state scaling \(\rho\propto a^{-3(1+w)}\) for \(p=w\rho c^{2}\) with constant \(w\).
Solution
Insert \(p=w\rho c^2\) into the fluid equation: \(\dot\rho+3H\rho(1+w)=0\), so \(\dfrac{d\rho}{\rho}=-3(1+w)\dfrac{da}{a}\). Integrating, \(\ln\rho=-3(1+w)\ln a+\text{const}\), giving \(\rho=\rho_0\,(a/a_0)^{-3(1+w)}\). Checks: \(w=0\to a^{-3}\) (matter), \(w=1/3\to a^{-4}\) (radiation), \(w=-1\to\) const (\(\Lambda\)).
(B) Given \(H_0=70\ \mathrm{km\,s^{-1}Mpc^{-1}}\), compute the Hubble time \(1/H_0\) in Gyr and the flat critical density.
Solution
\(H_0=70\times10^3/3.086\times10^{22}=2.27\times10^{-18}\ \mathrm{s^{-1}}\). Hubble time \(1/H_0=4.41\times10^{17}\ \mathrm{s}=4.41\times10^{17}/3.156\times10^{16}=14.0\ \text{Gyr}\). Critical density \(\rho_c=3H_0^2/8\pi G=3(2.27\times10^{-18})^2/(1.677\times10^{-9})=9.2\times10^{-27}\ \mathrm{kg\,m^{-3}}\).
(C) Starting from the Friedmann and acceleration equations, verify by differentiation that the fluid equation is not independent (for \(k,\Lambda\) constant).
Solution
Differentiate \(H^2=\dot a^2/a^2=\tfrac{8\pi G}{3}\rho-kc^2/a^2+\Lambda c^2/3\): LHS \(\dfrac{d}{dt}\left(\dfrac{\dot a^2}{a^2}\right)=2\dfrac{\dot a}{a}\left(\dfrac{\ddot a}{a}-\dfrac{\dot a^2}{a^2}\right)\). RHS \(=\tfrac{8\pi G}{3}\dot\rho+2kc^2\dot a/a^3\). Substitute \(\ddot a/a=-\tfrac{4\pi G}{3}(\rho+3p/c^2)+\Lambda c^2/3\) on the left and use the Friedmann equation to replace \(\dot a^2/a^2\). After cancelling the \(k\) and \(\Lambda\) terms, one is left with \(\dfrac{8\pi G}{3}\dot\rho=-\dfrac{8\pi G}{3}\cdot 3H(\rho+p/c^2)\), i.e. \(\dot\rho+3H(\rho+p/c^2)=0\). Hence it is a consequence, as guaranteed by the Bianchi identity.
(C) A flat universe contains matter (\(\Omega_{m,0}=0.31\)) and a cosmological constant (\(\Omega_{\Lambda,0}=0.69\)). Find the redshift \(z_{eq}\) at which the deceleration switches to acceleration (\(\ddot a=0\)).
Solution
Write densities as \(\rho_m=\rho_{m,0}(1+z)^3\), \(\rho_\Lambda=\text{const}\), with \(p_m=0\), \(p_\Lambda=-\rho_\Lambda c^2\). Acceleration equation \(\ddot a=0\) requires \(\rho_{\text{tot}}+3p_{\text{tot}}/c^2=0\), i.e. \(\rho_m+\rho_\Lambda+3(-\rho_\Lambda)=\rho_m-2\rho_\Lambda=0\). So \(\rho_{m,0}(1+z)^3=2\rho_{\Lambda,0}\Rightarrow(1+z)^3=2\Omega_{\Lambda,0}/\Omega_{m,0}=2(0.69)/0.31=4.45\). Thus \(1+z=4.45^{1/3}=1.645\Rightarrow z_{eq}\approx0.65\), matching the observed onset of cosmic acceleration.