--- title: The Friedmann Equations and Cosmic Dynamics draft: false module: Cosmic Expansion and Dynamics moduleNumber: 11 lessonNumber: 3 order: 1103 summary: > The scale factor obeys the Friedmann equation, the acceleration equation, and the fluid equation, only two of which are independent. An equation of state fixes how each component behaves under expansion, so radiation dilutes as the inverse fourth power of the scale factor, matter as the inverse cube, and vacuum energy not at all. The critical density defines the density parameters, and the deceleration parameter encodes whether gravity or dark energy is winning. topics: [Cosmology] sources: - book: Ryden ref: "Ch. 4 — Cosmic Dynamics; Ch. 5 — Model Universes" - book: Carroll & Ostlie ref: "Ch. 29 — Cosmology; §29.3 The Friedmann Equations" - book: Maoz ref: "Ch. 9 — The Big Bang and the Expansion of the Universe" --- The metric of the previous lesson contains an undetermined function, the scale factor $a(t)$. What fixes it is the content of the universe acting through gravity. In general relativity the connection between geometry and content is the Einstein field equation; specialized to the Robertson-Walker metric it collapses to two ordinary differential equations for $a(t)$, the **Friedmann equations**, supplemented by a thermodynamic **fluid equation** that describes how each component's energy density changes as the universe expands. This lesson assembles these equations, introduces the equation of state that distinguishes radiation, matter, and vacuum energy, works out how each component's density scales with $a$, and defines the critical density, the density parameters $\Omega_i$, and the deceleration parameter that together summarize the dynamical state of the cosmos. ## The Friedmann equation The Newtonian energy argument of the first lesson already produced the correct form. Retaining the relativistic result, in which the source of gravity is the total energy density $\varepsilon = \rho c^{2}$ rather than the mass density alone, the **Friedmann equation** is $$ \left(\frac{\dot a}{a}\right)^{2} = \frac{8\pi G}{3 c^{2}}\,\varepsilon - \frac{k c^{2}}{a^{2}} , $$ or equivalently, writing it with mass density $\rho = \varepsilon/c^{2}$, $$ H^{2} = \left(\frac{\dot a}{a}\right)^{2} = \frac{8\pi G}{3}\,\rho - \frac{k c^{2}}{a^{2}} . $$ This is a first-order equation relating the expansion rate $H = \dot a/a$ to the density and the curvature. It expresses energy conservation for the expansion: the "kinetic" term $H^{2}$ is balanced against the "potential" term set by the density and a constant of integration fixed by the curvature $k$. Given the density as a function of $a$, this equation determines the entire expansion history.[^ry-friedmann] $$ % caption: The Friedmann equation as an energy balance: the expansion rate squared is % the gravitating density term minus the curvature term. Their competition sets the fate. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % central balance \node[anchor=center] at (0,1.0) {H squared}; \node[black, anchor=center] at (0,0.5) {expansion}; % left: density drives \draw[->, very thick] (2.3,0.75) -- (0.9,0.75); \node[anchor=west] at (2.4,0.75) {density: drives expansion}; % right: curvature opposes (for k>0) \draw[->, black, very thick] (-2.3,0.75) -- (-0.9,0.75); \node[black, anchor=east] at (-2.4,0.75) {curvature term}; % equation reminder below \node[black, anchor=north] at (0,-0.1) {H squared = (8 pi G / 3) rho minus k c squared / a squared}; \end{tikzpicture} $$ ## The fluid equation and the acceleration equation The Friedmann equation alone is not closed: we need to know how $\varepsilon$ changes as $a$ changes. That comes from the first law of thermodynamics applied to a comoving volume. Take a sphere of comoving radius, physical volume $V \propto a^{3}$, containing energy $E = \varepsilon V$. Cosmic expansion is adiabatic — there is no heat flow across a comoving boundary in a homogeneous universe, since every neighboring region is identical — so $\d E = -P\,\d V$. With $E = \varepsilon a^{3}$ (up to a constant volume factor) and $V \propto a^{3}$, $$ \d(\varepsilon a^{3}) = -P\,\d(a^{3}) . $$ Expanding the derivatives and dividing by $\d t$ gives the **fluid equation**: $$ \dot\varepsilon + 3\,\frac{\dot a}{a}\big(\varepsilon + P\big) = 0 . $$ The first term is the dilution of energy density by the growing volume; the second is the work done by the pressure as the universe expands. Differentiating the Friedmann equation with respect to time and eliminating $\dot\varepsilon$ with the fluid equation yields the second-order **acceleration equation**: $$ \frac{\ddot a}{a} = -\frac{4\pi G}{3 c^{2}}\big(\varepsilon + 3P\big) . $$ Three features stand out. There is no curvature term $k$: the acceleration depends only on the contents. Gravity decelerates the expansion, since $\ddot a < 0$ whenever $\varepsilon + 3P > 0$. Third and least intuitive, **pressure gravitates**: it is the combination $\varepsilon + 3P$, not the energy density alone, that sources the deceleration. A component with sufficiently negative pressure, $P < -\varepsilon/3$, makes $\ddot a > 0$ and drives the expansion to accelerate. This is the loophole that dark energy exploits. Of the three equations — Friedmann, fluid, acceleration — only two are independent; any one follows from the other two. In practice one solves the fluid equation for $\varepsilon(a)$ and substitutes into the Friedmann equation. > **Theorem (The acceleration equation).** In a Robertson-Walker universe the scale > factor obeys $\ddot a/a = -\tfrac{4\pi G}{3c^{2}}(\varepsilon + 3P)$. Deceleration is > sourced by $\varepsilon + 3P$; a component with $P < -\varepsilon/3$ accelerates the > expansion. ## The equation of state To close the system we relate pressure to energy density through an **equation of state**. For the components of cosmological interest the relation is linear, $$ P = w\,\varepsilon , $$ with a dimensionless constant $w$ characteristic of each component. Three values matter: - **Matter** (nonrelativistic "dust" — galaxies, cold dark matter, baryons): the random thermal pressure is utterly negligible compared to the rest-energy density, so $w = 0$. - **Radiation** (photons, and any relativistic species such as neutrinos in the early universe): an isotropic gas of massless particles has $P = \tfrac{1}{3}\varepsilon$, so $w = \tfrac{1}{3}$. - **Vacuum energy** (the cosmological constant $\Lambda$): a Lorentz-invariant vacuum has $P = -\varepsilon$, so $w = -1$. This is the negative-pressure component that accelerates expansion. The value $w = \tfrac{1}{3}$ for radiation follows from kinetic theory: for a relativistic gas the pressure is one-third of the energy density because momentum and energy are proportional for massless particles and the isotropic average of one Cartesian component is one-third. The value $w = -1$ for vacuum energy follows from requiring the stress-energy to look the same to every observer — only a term proportional to the metric itself is Lorentz invariant, and it carries $P = -\varepsilon$.[^ry-eos] $$ % caption: The equation-of-state parameter w for the three cosmic components on a % number line: vacuum at w = -1, matter at w = 0, radiation at w = +1/3. The dividing % value w = -1/3 separates the accelerating regime from the decelerating one. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % number line \draw[->, black] (-3.6,0) -- (3.6,0) node[right, black!70] {w}; \foreach \x in {-3,-1,0,3} \draw[black] (\x,0.1) -- (\x,-0.1); % vacuum at w = -1 \fill[black] (-3,0) circle (0.09); \node[black, anchor=south] at (-3,0.2) {vacuum}; % matter at w = 0 \fill[black] (0,0) circle (0.09); \node[black, anchor=south] at (0,0.2) {matter}; % radiation at w = 1/3 \fill[black] (3,0) circle (0.09); \node[black, anchor=south] at (3,0.2) {radiation}; % divider at w = -1/3 \draw[black, densely dashed] (-1,0.7) -- (-1,-0.75); \node[black, anchor=north, align=center] at (-1,-0.9) {accelerate : decelerate}; \end{tikzpicture} $$ ## How each component dilutes With $P = w\varepsilon$ the fluid equation becomes a simple ODE for $\varepsilon(a)$: $$ \dot\varepsilon + 3\,\frac{\dot a}{a}\,(1 + w)\,\varepsilon = 0 \quad\Longrightarrow\quad \frac{\d\varepsilon}{\varepsilon} = -3(1 + w)\,\frac{\d a}{a} , $$ which integrates to $$ \varepsilon(a) = \varepsilon_0\, a^{-3(1 + w)} . $$ Inserting the three values of $w$ gives the scaling laws that govern the whole thermal history: - **Matter** ($w = 0$): $\varepsilon_m \propto a^{-3}$. The energy density falls as the volume grows — pure dilution of a fixed number of particles. - **Radiation** ($w = \tfrac{1}{3}$): $\varepsilon_r \propto a^{-4}$. One factor of $a^{-3}$ from dilution and one more from the redshift of each photon's energy, $\varepsilon_{\text{photon}} \propto 1/\lambda \propto a^{-1}$. - **Vacuum** ($w = -1$): $\varepsilon_\Lambda \propto a^{0} = \text{const}$. The vacuum energy density does not dilute — as space grows, each new volume comes with the same energy density, since it is a property of space itself. These differing exponents mean the balance of the universe shifts with time. At early times, when $a$ is small, radiation with its steep $a^{-4}$ dominates. As the universe expands, radiation falls below matter at the epoch of **matter-radiation equality**, and the universe becomes matter-dominated. Much later, when matter has diluted enough, the constant vacuum energy takes over and the universe enters an accelerating, vacuum-dominated phase. The order — radiation, then matter, then vacuum — is dictated entirely by the exponents.[^co-friedmann] $$ % caption: Log-log energy density versus scale factor. Radiation falls steepest, matter % shallower, vacuum flat; the dominant component changes at each crossover. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {log scale factor}; \draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {log energy density}; % radiation: slope -4 (steepest), solid \draw[very thick] (0.4,4.1) -- (5.2,0.3); \node[anchor=south west] at (0.4,3.8) {radiation}; % matter: slope -3, dashed neutral \draw[black, very thick, densely dashed] (0.4,3.4) -- (6.6,0.3); \node[black, anchor=west] at (2.2,2.35) {matter}; % vacuum: flat, dotted neutral \draw[black, very thick, densely dotted] (0.4,1.2) -- (8.2,1.2); \node[black, anchor=south] at (6.9,1.25) {vacuum}; % crossovers \draw[black, densely dotted] (2.55,0) -- (2.55,3.6); \node[black, anchor=north, align=center] at (2.55,-0.1) {eq}; \draw[black, densely dotted] (6.05,0) -- (6.05,1.5); \node[black, anchor=north, align=center] at (6.05,-0.1) {Lambda}; \end{tikzpicture} $$ ## The critical density and the density parameters The Friedmann equation ties the total density to the geometry. Setting $k = 0$ defines the **critical density** — the density that makes the universe spatially flat at a given expansion rate: $$ \rho_{\text{crit}}(t) = \frac{3 H^{2}}{8\pi G} . $$ At the present epoch, with $H_0 = 70\ \text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$, $$ \rho_{\text{crit},0} = \frac{3 H_0^{2}}{8\pi G} \approx 9.2 \times 10^{-27}\ \text{kg}\,\text{m}^{-3} \approx 5.5\ \frac{\text{protons}}{\text{m}^{3}} , $$ an extraordinarily low density — a handful of hydrogen atoms per cubic meter averaged over the cosmos. Each component's density is then measured as a fraction of critical through the dimensionless **density parameter** $$ \Omega_i \equiv \frac{\rho_i}{\rho_{\text{crit}}} = \frac{8\pi G\,\rho_i}{3 H^{2}} . $$ Dividing the Friedmann equation by $H^{2}$ recasts it as a sum rule among the density parameters and a curvature term: $$ 1 = \Omega_m + \Omega_r + \Omega_\Lambda - \frac{k c^{2}}{a^{2} H^{2}} \equiv \Omega_{\text{tot}} + \Omega_k , $$ with $\Omega_k \equiv -k c^{2}/(a^{2} H^{2})$ the curvature contribution. The geometry follows directly from the total: $\Omega_{\text{tot}} > 1$ means $k > 0$ (closed), $\Omega_{\text{tot}} = 1$ means $k = 0$ (flat), and $\Omega_{\text{tot}} < 1$ means $k < 0$ (open). Measuring the total density is thus equivalent to measuring the curvature. The observed values from Planck are $\Omega_{m,0} \approx 0.31$ (of which only about $0.05$ is ordinary baryonic matter, the rest cold dark matter), $\Omega_{\Lambda,0} \approx 0.69$, and $\Omega_{r,0} \approx 9 \times 10^{-5}$, summing to $\Omega_{\text{tot},0} = 1.000 \pm 0.005$ — a flat universe.[^ry-omega] $$ % caption: The present-epoch energy budget as fractions of the critical density: dark % energy the largest share, then dark matter, then baryons, radiation negligible. \begin{tikzpicture}[scale=1.0, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % pie: Lambda 69, dark matter 26, baryons 5. Start at 90 deg, go clockwise. % Lambda: 0.69 -> 248.4 deg. from 90 to 90-248.4 = -158.4 \fill[acc!18] (0,0) -- (90:2.2) arc (90:-158.4:2.2) -- cycle; % dark matter: 0.26 -> 93.6 deg, from -158.4 to -252 % baryons: 0.05 -> 18 deg, from -252 to -270 (=90) \draw[black] (0,0) -- (90:2.2); \draw[black] (0,0) -- (-158.4:2.2); \draw[black] (0,0) -- (-252:2.2); \draw[black] (0,0) circle (2.2); % labels \node[black!70, anchor=west] at (2.5,1.4) {dark energy: 69 percent}; \node[black!70, anchor=west] at (2.5,0.4) {dark matter: 26 percent}; \node[black!70, anchor=west] at (2.5,-0.6) {baryons: 5 percent}; \node[black!70, anchor=west] at (2.5,-1.6) {radiation: tiny}; \end{tikzpicture} $$ ## The deceleration parameter A single dimensionless number summarizes whether the expansion is speeding up or slowing down. The **deceleration parameter** is defined by $$ q \equiv -\frac{\ddot a\, a}{\dot a^{2}} = -\frac{\ddot a/a}{H^{2}} , $$ with the minus sign chosen so that a decelerating universe (the historical expectation) has $q > 0$. Substituting the acceleration equation and expressing densities through the density parameters gives a compact formula. For a mix of components each with equation of state $w_i$, $$ q = \frac{1}{2}\sum_i \Omega_i\,(1 + 3 w_i) = \Omega_r + \frac{1}{2}\Omega_m - \Omega_\Lambda , $$ using $w_r = \tfrac{1}{3}$, $w_m = 0$, $w_\Lambda = -1$. Radiation and matter contribute positive deceleration; vacuum energy contributes acceleration with a coefficient twice as strong per unit density. With the observed present values $\Omega_{m,0} \approx 0.31$ and $\Omega_{\Lambda,0} \approx 0.69$, the radiation term negligible, $$ q_0 \approx \tfrac{1}{2}(0.31) - 0.69 \approx -0.53 , $$ a **negative** deceleration parameter: the expansion is accelerating today. This is the quantitative statement of the 1998 supernova discovery, treated in the dark-energy lesson. Earlier in cosmic history, before vacuum energy dominated, $q$ was positive and the expansion decelerated; the transition from $q > 0$ to $q < 0$ occurred at redshift $z \approx 0.6$, when the accelerating vacuum term overtook the decelerating matter term. ## Summary The scale factor is governed by three equations, two independent: the Friedmann equation $H^2 = 8\pi G\rho/3 - kc^2/a^2$ relating expansion rate to density and curvature, the fluid equation $\dot\varepsilon + 3H(\varepsilon + P) = 0$ enforcing adiabatic energy conservation, and the acceleration equation $\ddot a/a = -\tfrac{4\pi G}{3c^2}(\varepsilon + 3P)$ in which pressure gravitates. A linear equation of state $P = w\varepsilon$ fixes each component's dilution, $\varepsilon \propto a^{-3(1+w)}$, giving radiation $a^{-4}$, matter $a^{-3}$, and vacuum energy constant, so the universe passes from radiation to matter to vacuum domination. The critical density $\rho_{\text{crit}} = 3H^2/8\pi G$ defines the density parameters $\Omega_i$, whose sum fixes the geometry, and the deceleration parameter $q = \Omega_r + \tfrac{1}{2}\Omega_m - \Omega_\Lambda \approx -0.53$ today records that the expansion now accelerates. The next lesson integrates these equations for specific mixtures to produce the [benchmark cosmological models](/astrophysics-cosmology/cosmology-expansion-and-dynamics/cosmological-models-and-distances) and the distance-redshift relations they predict. [^ry-friedmann]: Ryden, *Introduction to Cosmology*, Ch. 4 — Cosmic Dynamics: the Friedmann equation and its energy interpretation. [^ry-eos]: Ryden, Ch. 4 — the equation of state $P = w\varepsilon$ for matter, radiation, and the cosmological constant. [^co-friedmann]: Carroll & Ostlie §29.3 — the scaling of each component's density with the scale factor and the sequence of dominant eras. [^ry-omega]: Ryden, Ch. 4–5 — the critical density, the density parameters, and the flatness constraint; Planck 2018 values, https://arxiv.org/abs/1807.06209.