The Friedmann Equations

The Friedmann equations are Einstein’s equation specialized to an FLRW spacetime and an isotropic stress tensor. The time–time component is a constraint on the expansion rate, while a spatial component governs its change. Combining them gives the acceleration equation. Stress-energy conservation ties the two components together but does not make the constraint dispensable, especially at a turnaround where $H=0$. This page derives every sign with a dimensionless scale factor and a dimensionful curvature constant.

Required background. The Einstein Equation supplies the Einstein Equations and the contracted Bianchi identity; Flat-Background Hilbert Stress-Energy from Metric Variation supplies the Stress–Energy and Conservation interpretation of $\rho$ and $p$.

Helpful background. Relativistic Matter and Compact Objects supplies the perfect-fluid closure and its limits.

FLRW metric and perfect-fluid source

Use cosmic time and write

$$ds^2=-dt^2+a^2(t)\gamma_{ij}dx^i dx^j, \qquad {}^{(3)}R[\gamma]=6K.$$

Here $a(t_0)=1$ is dimensionless, $K$ has dimensions of inverse length squared, and the physical spatial slices have $^{(3)}R=6K/a^2$. This is the dimensionless-scale-factor version of the normalized-curvature metric constructed on FLRW Geometry. The two parameterizations describe the same geometry but must not be mixed inside one equation.

The comoving four-velocity is $u^\mu=(1,0,0,0)$ and $u^\mu u_\mu=-1$. Exact spatial isotropy permits an energy density and one isotropic pressure but no comoving energy flux or anisotropic stress:

$$T_{\mu\nu} =(\rho+p)u_\mu u_\nu+p\,g_{\mu\nu}, \qquad T_{00}=\rho, \qquad T_{ij}=p\,g_{ij}.$$

With $c=1$ and explicit $G$, the field equation is

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

The cosmological constant is kept on the geometric side throughout this derivation. Moving it into an effective fluid is valid, but counting both representations at once is not.

Curvature and Einstein-tensor components

Let $H=\dot a/a$. In addition to the intrinsic connection of $\gamma_{ij}$, the time-dependent metric gives

$$\Gamma^0{}_{ij}=H g_{ij}, \qquad \Gamma^i{}_{0j}=H\delta^i_j.$$

Direct contraction of the Riemann tensor then yields

$$R_{00}=-3\left(\dot H+H^2\right),$$ $$R_{ij} =\left(\dot H+3H^2+\frac{2K}{a^2}\right)g_{ij}, \qquad R=6\left(\dot H+2H^2+\frac{K}{a^2}\right).$$

The $00$ component of $G_{\mu\nu}=R_{\mu\nu}-\tfrac12Rg_{\mu\nu}$ is therefore

$$\begin{aligned} G_{00} &=-3(\dot H+H^2) +3\left(\dot H+2H^2+\frac{K}{a^2}\right)\\ &=3\left(H^2+\frac{K}{a^2}\right), \end{aligned}$$

while any spatial component gives

$$G_{ij} =-\left(2\dot H+3H^2+\frac{K}{a^2}\right)g_{ij}.$$

Equivalently,

$$G^i{}_j =-\left(2\dot H+3H^2+\frac{K}{a^2}\right)\delta^i_j.$$

These expressions are the entire tensor reduction required by exact homogeneity and isotropy. The mixed time–space components vanish, and the three orthonormal-frame spatial equations reduce to one scalar equation. In a non-orthogonal coordinate chart, lower-index $G_{ij}$ can inherit off-diagonal components from $g_{ij}$ without introducing physical anisotropy.

Constraint, evolution, and acceleration

Since $g_{00}=-1$, the time–time equation gives the Friedmann constraint,

$$\boxed{ H^2+\frac{K}{a^2} =\frac{8\pi G}{3}\rho+\frac{\Lambda}{3} }.$$

The spatial equation, after dividing by $g_{ij}$, is

$$2\dot H+3H^2+\frac{K}{a^2} =\Lambda-8\pi Gp.$$

Eliminating $H^2$ between these equations produces a useful first-order evolution law,

$$\dot H =-4\pi G(\rho+p)+\frac{K}{a^2}.$$

Finally, $\ddot a/a=\dot H+H^2$ gives the acceleration equation,

$$\boxed{ \frac{\ddot a}{a} =-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3} }.$$

Curvature cancels from this last equation explicitly, but it still affects which $H$ and $a$ satisfy the constraint. Positive pressure contributes to deceleration because pressure gravitates in general relativity. The same equations, translated from an opposite metric signature and a dimensionful scale factor, appear in Particle Data Group 2026, §22.1.3, pp. 2–3, eqs. (22.6)–(22.10) (PDF). Ellis and van Elst give the covariant FLRW reduction in Ellis and van Elst 1999, §4.2, p. 22, eqs. (106)–(107) (PDF), using $8\pi G=1$ and displaying the $\Lambda=0$ form.

Conservation and mutual consistency

The contracted Bianchi identity and constant $\Lambda$ imply $\nabla_\mu T^{\mu\nu}=0$. For the comoving perfect fluid, its time component is

$$\dot\rho+3H(\rho+p)=0.$$

The next page derives this continuity equation covariantly and thermodynamically. Here it supplies a consistency check. Differentiate the Friedmann constraint without dividing by $H$:

$$2H\dot H-\frac{2KH}{a^2} =\frac{8\pi G}{3}\dot\rho.$$

Substitution of the continuity equation gives

$$2H\left[ \dot H-\frac{K}{a^2}+4\pi G(\rho+p) \right]=0.$$

On every interval where $H\ne0$, this recovers the first-order evolution law. At an isolated turnaround or bounce, however, division by $H$ is invalid. The spatial Einstein equation remains meaningful there, and a smooth solution obtained on neighboring intervals extends through the event.

Constraint propagation can be stated without this division. Define

$$\mathcal C =H^2+\frac{K}{a^2} -\frac{8\pi G}{3}\rho-\frac{\Lambda}{3}.$$

Using the acceleration and continuity equations gives

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

Thus initial data satisfying $\mathcal C=0$ continue to satisfy it, including through $H=0$. The acceleration equation plus conservation propagates the constraint; it does not choose the zero-constraint initial data. This is the precise sense in which the equations are mutually consistent rather than three unrelated laws.

Limiting and dimensional checks

Every term in the constraint has dimensions $L^{-2}$. Four limits expose the signs:

• Minkowski: $K=\rho=p=\Lambda=0$ and constant $a$ give $H=0$.
• Milne: with $\rho=p=\Lambda=0$, $K\lt0$, and $a(t)=\sqrt{-K}\,t$, the curvature term cancels $H^2$ and $\ddot a=0$. The spacetime is flat despite negatively curved slices.
• Flat de Sitter: $K=\rho=p=0$ and $\Lambda\gt0$ give $H^2=\Lambda/3$ and $a\propto e^{\sqrt{\Lambda/3}\,t}$ on the expanding branch.
• Einstein static: $H=\dot H=0$ requires both the constraint and spatial equation. It is the clearest warning that the differentiated constraint alone loses information at $H=0$.

Tong obtains the same constraint through a deliberately Newtonian heuristic in Tong 2019, §1.2.3, pp. 29–31, eqs. (1.42)–(1.44) (PDF). That argument is useful intuition for its dust, $\Lambda=0$ regime but does not replace the tensor calculation above. It neither derives pressure’s gravitational role nor justifies geometrically identifying its integration constant as spatial curvature; the relativistic acceleration equation and that identification require Einstein’s equation and the FLRW geometry.

Exercises

Starting from the $00$ and spatial Einstein equations above, eliminate $H^2$ and recover both $\dot H$ and $\ddot a/a$. Explain why $K$ disappears from only the second expression.

Solution

Subtracting three times the constraint from the spatial equation gives

$$2\dot H-2\frac{K}{a^2} =-8\pi G(\rho+p),$$

so $\dot H=-4\pi G(\rho+p)+K/a^2$. Adding the constraint expression for $H^2$ cancels the two curvature terms and yields

$$\frac{\ddot a}{a} =-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}.$$

The cancellation says that intrinsic curvature is not a local active-mass term in the acceleration equation. Curvature still restricts $H$ through the constraint and therefore affects the full expansion history.

At an event with $H=0$, show what the differentiated constraint and continuity equation imply. Why is the spatial Einstein equation still needed?

Solution

Both sides of

$$2H\dot H-\frac{2KH}{a^2} =\frac{8\pi G}{3}\dot\rho$$

vanish: continuity gives $\dot\rho=-3H(\rho+p)=0$. The differentiated constraint therefore reduces to $0=0$ and places no condition on $\dot H$. The spatial Einstein equation fixes $\dot H$ from $\rho$, $p$, $K/a^2$, and $\Lambda$. Dividing the differentiated equation by $H$ would silently discard precisely this static or turning-point case.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models (Cargèse Lectures 1998).” In Marc Lachièze-Rey, ed., Theoretical and Observational Cosmology, pp. 1–116. Dordrecht: Springer, 1999. DOI. Open PDF, arXiv:gr-qc/9812046v5.
• 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.