Friedmann Dynamics and Cosmic Components

FLRW symmetry leaves the scale factor undetermined; Einstein’s equation and a matter model supply its dynamics. This chapter derives the Friedmann constraint, acceleration equation, and local conservation law, then uses equations of state to build radiation-, matter-, curvature-, and vacuum-dominated histories. The recurring rule is to declare one curvature normalization and one cosmological-constant convention, keep constraints distinct from evolution equations, and test every mixture against closure, transition scales, and analytic limits.

Helpful background. The Einstein Equation supplies the Einstein Equations and Bianchi consistency; Flat-Background Hilbert Stress-Energy from Metric Variation supplies Stress–Energy and Conservation; Relativistic Matter and Compact Objects supplies fluid equations of state and closure.

Six routes through background dynamics

The pages are ordered from field equations to complete expansion histories.

1. The Friedmann Equations computes the FLRW Einstein tensor, separates the time–time constraint from the spatial evolution equation, and checks how conservation propagates the constraint.
2. Continuity Equation and Adiabatic Expansion projects $\nabla_\mu T^{\mu\nu}=0$ along the fluid velocity, derives pressure work in a comoving volume, and states the additional assumptions behind entropy conservation.
3. Equations of State and Scaling Laws integrates general and constant-$w$ backgrounds while separating geometric scaling, background pressure, and perturbation sound speed.
4. Cosmological Constant and Accelerated Expansion relates acceleration to $\rho+3p$, translates between geometric $\Lambda$ and vacuum energy, and distinguishes equality from acceleration onset.
5. Spatial Curvature and Critical Density derives $\Omega-1=K/(a^2H^2)$, fixes its signs and domain, and explains why curvature is not a material fluid.
6. Multi-Component Expansion Histories combines the components into $E(a)$, solves equality and transition conditions, and validates the age quadrature against analytic limits and a convergent numerical example.

For a derivation-first route, read pages 1–3 before choosing components. For a model-building route, review the first page’s conventions and then use pages 4–6 to construct and test a background budget.

One convention package

Throughout this chapter, $a(t_0)=1$ is dimensionless and $K$ has dimensions $L^{-2}$. The spatial Ricci scalar is

$${}^{(3)}R=\frac{6K}{a^2}, \qquad H=\frac{\dot a}{a}.$$

The default field equation keeps the cosmological constant geometric,

$$G_{\mu\nu}+\Lambda g_{\mu\nu} =8\pi G\,T_{\mu\nu}.$$

Whenever $\Lambda$ is moved to the source side, the pages say so explicitly and use

$$\rho_\Lambda=\frac{\Lambda}{8\pi G}, \qquad p_\Lambda=-\rho_\Lambda.$$

These are equivalent descriptions. A component sum that includes $\rho_\Lambda$ must omit the separate $\Lambda/3$ term. Likewise, $\Omega_K=-K/(a^2H^2)$ is geometric bookkeeping, not a stress-energy density.

The coupled equation system

For a comoving perfect fluid, the chapter’s three central equations are

$$H^2+\frac{K}{a^2} =\frac{8\pi G}{3}\rho+\frac{\Lambda}{3},$$ $$\frac{\ddot a}{a} =-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}, \qquad \dot\rho+3H(\rho+p)=0.$$

They do not represent three freely independent evolution laws. The first is a constraint on initial data. The acceleration equation and conservation propagate it according to

$$\dot{\mathcal C}=-2H\mathcal C,$$

where $\mathcal C$ is the difference between the two sides of the Friedmann constraint. Differentiating the constraint and dividing by $H$ recovers the remaining evolution equation only on intervals with $H\ne0$. At a bounce, turnaround, or static event, the undivided spatial Einstein equation is essential.

The pressure term is physical rather than a convention: positive pressure adds to deceleration, while sufficiently negative total pressure permits $\ddot a\gt0$. Curvature cancels explicitly from the acceleration equation but still restricts $H$ through the constraint.

From components to a history

Separate conservation of a constant-$w_i$ component gives

$$\rho_i(a)=\rho_{i0}a^{-3(1+w_i)}.$$

The conserved fluids supply the radiation, matter, and constant vacuum terms. Geometric curvature separately supplies the $a^{-2}$ contribution, so the dimensionless expansion function is

$$E^2(a) =\Omega_{r0}a^{-4} +\Omega_{m0}a^{-3} +\Omega_{K0}a^{-2} +\Omega_{\Lambda0}.$$

Closure requires $E^2(1)=1$. Equality scales follow by equating the relevant terms; acceleration instead solves the pressure-weighted condition $2\rho_r+\rho_m-2\rho_\Lambda=0$. Time follows from

$$H_0t(a)=\int_0^a\frac{da'}{a'E(a')}$$

for a big-bang branch. Radiation and matter limits must recover $a\propto t^{1/2}$ and $a\propto t^{2/3}$, while vacuum domination approaches exponential expansion. The standard equations and component definitions are collected in Particle Data Group 2026, §§22.1.3–22.1.5, pp. 2–5 (PDF); Tong 2019, §§1.2–1.4, pp. 24–53 (PDF) develops the analytic histories and limiting solutions.

Boundaries of the model

The chapter treats general relativity, exact FLRW symmetry, and background perfect-fluid sources. It does not derive particle abundances, recombination, or nonequilibrium kinetics; those belong to the thermal-history chapter. It does not infer numerical parameters from data; likelihoods and model appraisal appear later. Interacting fluids require explicit transfer terms, and modified gravity requires a different background field equation rather than relabeling its terms as arbitrary fluids.

Density parameters are also poor coordinates at $H=0$, and a constant-$w$ power law cannot cross a physical threshold that changes the component’s microphysics. In those cases, return to dimensional equations and state the source model directly.

Review the chapter

Consider a flat, separately conserved mixture of pressureless matter and positive vacuum energy. A student claims that acceleration begins at matter–vacuum equality. They also claim that dividing a differentiated Friedmann constraint by $H$ proves the acceleration equation even through an arbitrary bounce or turnaround. Identify the physical error and the domain error, and give the correct transition scales.

Solution

With $a_0=1$,

$$\rho_m=\rho_{m0}a^{-3}, \qquad \rho_\Lambda=\rho_{\Lambda0}.$$

Matter–vacuum equality occurs at

$$a_{m\Lambda} =\left(\frac{\Omega_{m0}}{\Omega_{\Lambda0}}\right)^{1/3}.$$

Acceleration tests $\rho_m+3p_m+\rho_\Lambda+3p_\Lambda =\rho_m-2\rho_\Lambda$. Hence it begins at

$$a_{\rm acc} =\left(\frac{\Omega_{m0}} {2\Omega_{\Lambda0}}\right)^{1/3} =2^{-1/3}a_{m\Lambda},$$

which is earlier on the expanding branch.

In the declared flat model with positive matter and vacuum densities, the constraint gives $H^2\gt0$, so division by $H$ is valid on either regular expanding or contracting branch. It does not, however, prove the equation at an $H=0$ event in a more general model: differentiating the constraint produces an equation proportional to $H$, and division then loses the static or turning case. The spatial Einstein equation remains valid at $H=0$; together with continuity it propagates the constraint without that division.

References

• Particle Data Group. “Big-Bang Cosmology.” In Review of Particle Physics 2026, revised August 2025. Official PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas 2019. Official PDF.