The FLRW Metric

The FLRW metric is the sentence in which modern cosmology writes down the universe. It answers a sharply posed geometrical question: of all conceivable spacetimes, which are compatible with the Cosmological Principle — the assumption that the universe looks the same from every vantage point (homogeneity) and in every direction (isotropy) at each instant of cosmic time? The answer is startlingly rigid. Homogeneity means the spatial slices admit a transitive group of isometries, isotropy means the rotation group acts on directions at each point, and a classical theorem then forces every spatial slice to be a maximally symmetric three-space of constant curvature. Up to scale there are exactly three: the 3-sphere $S^3$ (positive curvature), Euclidean space $\mathbb{E}^3$ (zero), and hyperbolic space $H^3$ (negative). The full spacetime is a stack of these identical-shaped slices, threaded by a single time-dependent dilation — the scale factor $a(t)$ — and labelled by one discrete number, the curvature constant $k\in\{+1,0,-1\}$ . Two continuous fields' worth of gravitational freedom collapse to one function of one variable. That is the miracle the symmetry buys: the ten coupled Einstein equations reduce to two ordinary differential equations for $a(t)$ , the Friedmann equations. The line element carries all of cosmological kinematics inside it — freely-falling comoving observers, the redshift of light as $a$ grows, proper distances and horizons, the topology set by $k$ . The FLRW metric is not itself a solution of Einstein's equations; it is the symmetry-fixed ansatz whose one free function the field equations, fed a cosmic fluid, are left to determine. Everything downstream in cosmology is a corollary of this one geometry.

Key takeaways

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is the unique (up to coordinates) spacetime geometry consistent with spatial homogeneity and isotropy at each cosmic time.
Its line element splits a global cosmic time $t$ from comoving spatial coordinates whose proper scale is set by one function, the scale factor $a(t)$ .
The spatial sections are maximally symmetric three-spaces of constant curvature, labelled by the discrete constant $k=+1,0,-1$ (closed, flat, open).
All the dynamics lives in $a(t)$ : feeding the metric into Einstein's equations with a perfect fluid yields the Friedmann equations.
Physical distances, redshift $1+z=a_0/a_e$ , and horizons all follow from $a(t)$ and $k$ , making FLRW the backbone of modern cosmology.
Planck data give $\Omega_k=0.001\pm0.002$ : the observable universe is spatially flat ( $k=0$ ) to a curvature radius exceeding 260 billion light-years.
Key idea: impose homogeneity and isotropy on spacetime and the geometry collapses to a single function of time, the scale factor $a(t)$ , multiplying a constant-curvature spatial metric labelled by $k\in\{+1,0,-1\}$ .

Where this lives today

The FLRW metric is the working coordinate system of observational cosmology, and every precision measurement is quoted in its language. Planck's map of the microwave background pins the geometry flat to a fraction of a percent through the angular scale of the first acoustic peak — a sound horizon divided by an angular-diameter distance that depends on $k$ through $S_k(\chi)$ . Supernova and baryon-acoustic-oscillation surveys (DESI and its predecessors) chart the luminosity distance $d_L=(1+z)\,a_0 S_k(\chi)$ and angular-diameter distance $d_A=a_0 S_k(\chi)/(1+z)$ to $z\gtrsim2$ ; their ratio $d_L=(1+z)^2 d_A$ , the Etherington distance-duality relation, is a direct null test of FLRW geometry. The metric is also where today's tensions are staged: the Hubble tension, a persistent $\sim5\sigma$ disagreement between the local $H_0\equiv\dot a/a$ and the value inferred from the CMB in flat FLRW, plays out entirely inside this line element. And the particle horizon computed from the metric exposes the horizon problem — patches of the CMB more than a degree apart were causally disconnected at recombination — the puzzle inflation was invented to solve. The one function $a(t)$ the metric leaves open is exactly what a decade of surveys is built to measure.

The physical picture

Picture a rising loaf of raisin bread. Each raisin — a galaxy — stays at a fixed coordinate in the dough, but the dough swells, so the proper distance between every pair of raisins grows in lockstep. The scale factor $a(t)$ is the size of the dough; the comoving coordinates are the fixed labels stamped on each raisin. No raisin is the centre; each sees every other receding, faster the farther apart they are. That single image — fixed labels, breathing ruler — is the whole of FLRW kinematics, and the first figure makes it quantitative: double the scale factor and every proper separation, and every photon wavelength, doubles with it.

Figure

A comoving lattice of galaxies at two epochs. The galaxies keep fixed comoving coordinates while the scale factor grows from a = 0.5 to a = 1, so every proper separation doubles: the A–B distance goes from 1.5 Gly to 3.0 Gly. Light emitted at a = 0.5 arrives with 1 + z = a₀/aₑ = 1/0.5 = 2, a redshift z = 1.

Isotropy at every point — not just ours — is the strong assumption, and it does all the geometric work: with homogeneity it is equivalent to maximal symmetry, which forces the spatial slices to have constant curvature, leaving no room for anything but the three model geometries. The curvature constant $k$ is a global label, not a local force: a small patch of any FLRW universe looks nearly flat, but $k$ decides whether parallel comoving geodesics converge, stay parallel, or diverge — the spherical, Euclidean, and hyperbolic trichotomy. The second figure shows all three and the relation $^{(3)}R = 6k/a^2$ that quantifies them.

Figure

The three constant-curvature spatial geometries allowed by homogeneity and isotropy. The intrinsic Ricci scalar of a slice is ³R = 6k/a², positive for the closed 3-sphere (k = +1, geodesics converge, triangle angles exceed 180°), zero for flat space (k = 0, geodesics stay parallel, angles sum to 180°), and negative for the open hyperbolic space (k = −1, geodesics diverge, angles fall short of 180°).

The core equations

The metric is written in two coordinate systems that see the same geometry from different angles. The first, the reduced-circumference form, uses a radial coordinate $r$ scaled so that a sphere at coordinate $r$ has proper area $4\pi a^2 r^2$ ; the curvature $k$ then hides in the radial denominator.

\mathrm{d}s^2 = -c^2\,\mathrm{d}t^2 + a^2(t)\left[\frac{\mathrm{d}r^2}{1-k r^2} + r^2\left(\mathrm{d}\theta^2 + \sin^2\theta\,\mathrm{d}\phi^2\right)\right] \qquad (1)

where: $t$ — cosmic (proper) time of comoving observers; $a(t)$ — scale factor, setting the proper size of comoving separations; $r,\theta,\phi$ — comoving spherical coordinates, fixed for free-falling galaxies; $k$ — curvature constant, normalized to $+1,0,-1$ ; $c$ — speed of light. Reading: time is orthogonal to space (no $g_{0i}$ ), and all the dynamics is packed into how the spatial metric breathes through $a(t)$ . The second, the geodesic-radial form, trades $r$ for the proper radial comoving distance $\chi$ , so $a\chi$ is the distance a ruler-carrying chain of observers reads, and the curvature is carried by a single function $S_k(\chi)$ on the angular part.

\mathrm{d}s^2 = -c^2\,\mathrm{d}t^2 + a^2(t)\left[\mathrm{d}\chi^2 + S_k^2(\chi)\,\mathrm{d}\Omega^2\right] \qquad (2)

where: $\chi$ — comoving radial geodesic distance (proper radial distance is $a\chi$ ); $S_k(\chi)$ equals $\sin\chi$ for $k=1$ , $\chi$ for $k=0$ , $\sinh\chi$ for $k=-1$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ is the unit 2-sphere metric. Reading: $S_k$ is the curvature-corrected radius converting a comoving distance into the proper size of the sphere at that distance — the geometry distance measures actually feel. The intrinsic curvature of a constant-time slice is captured by one number, its three-dimensional Ricci scalar.

{}^{(3)}R = \frac{6k}{a^2(t)} \qquad (3)

where: ${}^{(3)}R$ — intrinsic curvature scalar of the constant- $t$ hypersurface; $k/a^2$ — physical (proper) spatial curvature. Reading: because the denominator grows with $a$ , the proper curvature dilutes as the universe expands, making a curved universe ever harder to distinguish from a flat one. Galaxies at fixed comoving coordinates are in free fall: their four-velocity points purely along the time direction and satisfies the geodesic equation identically, because $\Gamma^\mu_{00}=0$ (the lapse is constant and time is orthogonal to space).

u^\mu = (c,0,0,0),\qquad \frac{\mathrm{D}u^\mu}{\mathrm{d}\tau}=0 \qquad (4)

where: $u^\mu$ — four-velocity of a comoving observer; $\mathrm{D}/\mathrm{d}\tau$ — covariant derivative along the worldline. Reading: comoving observers follow geodesics, and their proper time is exactly the cosmic time $t$ labelling the slices — the clock in the metric is a real, freely-falling clock. The single most consequential observable is the cosmological redshift: light emitted when the universe was smaller arrives stretched by the ratio of scale factors.

1+z = \frac{a(t_0)}{a(t_e)} \qquad (5)

where: $z$ — redshift of light emitted at $t_e$ and observed at $t_0$ ; $a(t_0)$ — scale factor today; $a(t_e)$ — scale factor at emission. Reading: redshift is a direct ruler of the size of the universe at emission — measure $z$ and you have measured $a(t_e)/a(t_0)$ , no dynamics required. The rate and acceleration of expansion are summarized by two quantities built from $a(t)$ : the Hubble and deceleration parameters.

H(t) \equiv \frac{\dot a(t)}{a(t)},\qquad q \equiv -\frac{\ddot a\, a}{\dot a^2} \qquad (6)

where: $H$ — Hubble parameter, present value $H_0$ ; $\dot a$ — time derivative of the scale factor; $q$ — dimensionless deceleration parameter. Reading: $H$ sets the recession law $v=H d_p$ ; the sign of $q$ says whether expansion slows ( $q>0$ ) or accelerates ( $q<0$ ). Because comoving galaxies are not at fixed proper distances, the metric supplies its own distance measure.

d_p(t) = a(t)\,\chi = a(t)\int_{t_e}^{t}\frac{c\,\mathrm{d}t'}{a(t')} \qquad (7)

where: $d_p$ — proper distance along a slice of constant $t$ ; $\chi$ — comoving distance to the source, fixed once chosen. Reading: differentiating $d_p$ and using $v=H d_p$ recovers Hubble's law and shows recession can exceed $c$ at large $d_p$ . The metric is only an ansatz until Einstein's equations, fed a source, turn its one free function into a history. A perfect fluid is the only stress-energy consistent with the symmetry.

G_{\mu\nu}+\Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu},\quad T^{\mu}{}_{\nu}=\mathrm{diag}(-\rho c^2,p,p,p) \qquad (8)

where: $G_{\mu\nu}$ — Einstein tensor of the FLRW metric; $\Lambda$ — cosmological constant; $\rho,p$ — energy density and isotropic pressure of the cosmic fluid; $G$ — Newton's constant. Reading: substituting (1) collapses ten equations to two — the Friedmann equation $H^2 = \tfrac{8\pi G}{3}\rho - \tfrac{kc^2}{a^2} + \tfrac{\Lambda c^2}{3}$ and its acceleration companion — which govern $a(t)$ .

Derivation AUniqueness of the spatial metric as constant-curvature slices

The goal of this derivation is to prove that the Cosmological Principle does not merely suggest but strictly forces the spatial metric into the single family $\mathrm{d}\sigma^2 = \mathrm{d}r^2/(1-kr^2)+r^2\mathrm{d}\Omega^2$ , fixed up to the sign of $k$ and an overall scale absorbed into $a(t)$ . We show homogeneity plus isotropy is equivalent to maximal symmetry, that this pins the Riemann tensor to a constant-curvature form, and that solving the resulting differential equation with regularity at the origin leaves exactly the three geometries and nothing else.

Step 1. Count the symmetries. Homogeneity supplies three independent translations and isotropy three rotations about a point, for six Killing vectors — precisely the maximum $n(n+1)/2 = 6$ a three-dimensional Riemannian space can possess. Why: a space with the maximal number of isometries is by definition maximally symmetric, so the cosmological assumptions translate directly into a strong geometric statement.

Step 2. Fix the Riemann tensor. A maximally symmetric space has a curvature tensor that looks the same in every direction and at every point, forcing $R_{ijkl}=K\,(g_{ik}g_{jl}-g_{il}g_{jk})$ with $K$ a single constant, the sectional curvature. Why: any freedom beyond this one number would single out a direction or location, contradicting isotropy or homogeneity.

Step 3. Adopt an isotropic ansatz. Isotropy about the origin lets us write the slice in spherical coordinates with one unknown radial function, $\mathrm{d}\sigma^2 = e^{2\beta(r)}\mathrm{d}r^2 + r^2\mathrm{d}\Omega^2$ , with $r$ the reduced-circumference radius. Why: isotropy forbids any angular dependence in the metric functions, collapsing the most general slice to a single function of $r$ .

Step 4. Impose constant curvature. Computing the three-dimensional Ricci tensor and demanding the maximal-symmetry trace condition $R_{ij}=2K g_{ij}$ yields the first-order equation $\tfrac{1}{r}\tfrac{\mathrm{d}}{\mathrm{d}r}\big(1-e^{-2\beta}\big)=2K$ . Why: this is "the curvature is the same everywhere" written for the one unknown function, and it is exactly integrable.

Step 5. Solve and normalize. Integrating with regularity $\beta(0)=0$ gives $g_{rr}=1/(1-Kr^2)$ ; rescaling $r\to r/\sqrt{|K|}$ absorbs the magnitude of $K$ into $a(t)$ and leaves only its sign $k=+1,0,-1$ . Why: the curvature scale is not an invariant separate from $a$ — only the sign survives, which is why $k$ is discrete.

Step 6. Assemble the spacetime. Append $-c^2\mathrm{d}t^2$ and the dilation $a^2(t)$ multiplying the spatial part; isotropy kills any $g_{0i}$ term and a position-independent lapse is reabsorbed into $t$ . The result is equation (1). Why: the same symmetry orthogonalizes time from space and freezes the lapse, leaving nothing undetermined but $a(t)$ .

Go deeper · Derivation B · 1 minCosmological redshift from a radial null geodesic

Derivation BCosmological redshift from a radial null geodesic

The goal here is to derive the exact relation $1+z=a(t_0)/a(t_e)$ directly from the metric, and to show it is not a Doppler or gravitational shift but a distinct third effect — a wavelength stretched by the growth of space between emission and reception. We trace a radial light ray through the geodesic-radial form, use the constancy of the comoving distance to relate two successive wave crests, and read off the wavelength ratio, valid for any expansion history $a(t)$ .

Step 1. Write the null condition. For light $\mathrm{d}s^2=0$ ; for a radial ray $\mathrm{d}\theta=\mathrm{d}\phi=0$ , so metric (2) gives $0=-c^2\mathrm{d}t^2+a^2(t)\,\mathrm{d}\chi^2$ , hence $c\,\mathrm{d}t = a(t)\,\mathrm{d}\chi$ . Why: reducing the metric to a photon's worldline turns the geometry into a single relation between elapsed coordinate time and comoving distance covered.

Step 2. Integrate to the comoving distance. Integrating from emission to observation gives $\chi = \int_{t_e}^{t_0} c\,\mathrm{d}t/a(t)$ , a fixed number set only by the source's location. Why: the comoving separation between a galaxy and us does not change as the universe expands, so this integral is the same for every crest that makes the trip.

Step 3. Send two successive crests. A crest emitted at $t_e$ arrives at $t_0$ ; the next, emitted one period later at $t_e+\delta t_e$ , arrives at $t_0+\delta t_0$ . Both traverse the identical $\chi$ , so $\int_{t_e}^{t_0} c\,\mathrm{d}t/a = \int_{t_e+\delta t_e}^{t_0+\delta t_0} c\,\mathrm{d}t/a$ . Why: equating the two fixed integrals encodes the expansion into the timing of arrivals.

Step 4. Subtract the integrals. Cancelling the common interior interval leaves the boundary terms $\delta t_e/a(t_e)=\delta t_0/a(t_0)$ . Why: over one wave period the scale factor is essentially constant, so the difference of the integrals reduces cleanly to their endpoints.

Step 5. Convert periods to wavelengths. Proper wavelengths are $\lambda=c\,\delta t$ , so $\lambda_{\rm obs}/\lambda_{\rm em}=\delta t_0/\delta t_e=a(t_0)/a(t_e)\equiv1+z$ . Why: the wavelength is dragged along in proportion to the scale factor, so the redshift measures $a$ at emission relative to today directly and exactly. *The result $1+z=a(t_0)/a(t_e)$ is exact for arbitrary $a(t)$ , not a low-velocity Doppler approximation.*

Limiting cases

The FLRW metric contains the familiar cosmological models as special cases:

Case	Result	Meaning
Flat spatial sections	$\mathrm{d}s^2 = -c^2\mathrm{d}t^2 + a^2(t)\left[\mathrm{d}r^2 + r^2\mathrm{d}\Omega^2\right]$	$k=0$ : comoving space is Euclidean $\mathbb{E}^3$ ; favored by current data, $\Omega_k\approx0$
Static empty space	$a=\text{const},\,k=0 \Rightarrow \mathrm{d}s^2=-c^2\mathrm{d}t^2+\delta_{ij}\mathrm{d}x^i\mathrm{d}x^j$	Minkowski spacetime: no expansion, no curvature
Einstein–de Sitter	$k=0,\,\Lambda=0$ , matter-dominated $\Rightarrow a(t)\propto t^{2/3}$	Flat, matter-only universe; the simple decelerating benchmark model
de Sitter	$a(t)=a_0 e^{Ht},\;H=\sqrt{\Lambda c^2/3}$	Flat FLRW driven by $\Lambda$ ; exponential expansion, event horizon at $c/H$
Closed universe	$\mathrm{d}\Sigma^2=\mathrm{d}\chi^2+\sin^2\chi\,\mathrm{d}\Omega^2,\;0\le\chi\le\pi$	$k=+1$ : spatial slices are finite 3-spheres of volume $2\pi^2 a^3$

Common trapGalaxies receding faster than light do not violate relativity

The recession speed of a comoving galaxy is $v=H d_p$ , and at large enough proper distance this exceeds $c$ — right now, everything beyond about 14 billion light-years recedes superluminally. This breaks no law. Special relativity's speed limit governs relative velocities measured within a single local inertial frame, and no such frame spans cosmological distances: comoving coordinates stitch together infinitely many local frames, and $H d_p$ is a rate of growth of proper distance, not a locally measured speed of anything passing anything. Photons from superluminally receding galaxies still reach us, because the Hubble sphere itself expands. A related trap: comoving coordinates are not an inertial frame, and the scale factor $a(t)$ means something only through its ratios — only $k/a^2$ , the proper curvature, is a genuine invariant, while $k$ alone is merely its sign.

Worked examples

1 · Redshift, scale factor, and the temperature of the cosmic microwave background

The cosmic microwave background was released at recombination, observed today at redshift $z\approx1100$ , present temperature $T_0=2.725$ K. Find the scale factor at emission (with $a_0=1$ ), the wavelength stretch, and the temperature of the plasma when the light was freed. As a warm-up, do the same for a galaxy at $z=1$ .

Step 1. Scale factor from redshift. Invert equation (5): $a_e = a_0/(1+z)$ . For the galaxy at $z=1$ , $a_e = 1/2$ — the universe was half its present size, all distances and wavelengths since doubled. For the CMB at $z=1100$ , $a_e = 1/1101 = \mathbf{9.08\times10^{-4}}$ , the universe about 1100 times smaller in every linear dimension.

Step 2. Wavelength stretch. Since $\lambda_{\rm obs}/\lambda_{\rm em}=1+z$ , every CMB photon has been stretched by a factor of 1101 — light emitted near $\lambda_{\rm em}\approx1\ \mu\mathrm{m}$ arrives at roughly $1.1$ mm, the microwave band where the CMB is observed.

Step 3. Temperature scaling. A blackbody stays a blackbody as space expands, with wavelengths stretching as $a$ and temperature falling as $T\propto1/a\propto(1+z)$ . Hence $T_e = T_0(1+z) = 2.725\times1101 = \mathbf{3000\ K}$ — precisely the temperature at which hydrogen recombines and the universe turns transparent. With no free parameters, the redshift is a thermometer reading the size of the universe at last scattering.

2 · How flat is space? The curvature radius from Planck

Planck measures the curvature density parameter $\Omega_k = 0.001\pm0.002$ . Using $H_0=67.4$ km s⁻¹ Mpc⁻¹, translate this into a lower bound on the present curvature radius $R_{\rm curv}=a_0/\sqrt{|k|}$ , and compare it to the size of the observable universe.

Step 1. Relate $\Omega_k$ to the curvature radius. The definition $\Omega_k \equiv -kc^2/(a_0^2 H_0^2)$ rearranges to $R_{\rm curv}=a_0/\sqrt{|k|}=c/(H_0\sqrt{|\Omega_k|})$ : a small $|\Omega_k|$ means a large curvature radius, space nearly flat within our horizon.

Step 2. Compute the Hubble length. $c/H_0 = (2.998\times10^5)/(67.4)\ \mathrm{Mpc} = 4448\ \mathrm{Mpc} = \mathbf{1.37\times10^{26}\ m}$ , about 14.5 billion light-years — the Hubble radius, the natural yardstick for cosmic curvature.

Step 3. Bound the curvature radius. Taking the $1\sigma$ upper edge $|\Omega_k|\lesssim0.003$ gives $\sqrt{|\Omega_k|}\lesssim0.0548$ , so $R_{\rm curv} \gtrsim (c/H_0)/0.0548 = 14.5/0.0548 \approx \mathbf{265\ billion\ light\text{-}years}$ ( $2.5\times10^{27}$ m). The observable universe is only about 93 billion light-years across, so any curvature bends space on a scale at least three times larger than everything we can see — which is why cosmologists set $k=0$ with a clear conscience, and why such fine-tuned flatness is itself a puzzle that inflation was built to explain.

Physics of the portrait

The FLRW metric wears four names because four people converged on the same geometry from different directions — two solving the dynamics, two proving the geometry unique. The portrait belongs to the man who, with Walker, turned a physicists' ansatz into a theorem.

Origins · 1 minHoward Percy Robertson and how this was discovered

Howard Percy Robertson

1903–1961 · relativistic cosmology and the geometry of the expanding universe

When Alexander Friedmann in 1922 and Georges Lemaître in 1927 wrote down expanding-universe solutions of Einstein's equations, they assumed the now-standard line element — a scale factor multiplying a constant-curvature space — as the natural way to build a homogeneous, isotropic model. What they did not prove was that it was the only way. That proof is the achievement of Howard Percy Robertson and Arthur Geoffrey Walker. Robertson, an American mathematical physicist at Princeton and Caltech, established in a series of papers ("Kinematics and World-Structure," 1935–36) that any spacetime obeying the Cosmological Principle must take exactly the form now bearing the four names; Walker gave an independent, fully rigorous derivation in 1937 from the theory of spaces with a maximal group of isometries. Their contribution was not a new solution but a uniqueness theorem — the guarantee that assuming homogeneity and isotropy leaves no geometric freedom beyond the single function $a(t)$ and the discrete label $k$ . Robertson's career ranged from the foundations of quantum mechanics (the Robertson uncertainty relation) to wartime scientific intelligence, but it is this theorem that made cosmology predictive: it lets the Friedmann equations claim to describe the universe, not merely a model of it. Friedmann found the dynamics, Lemaître the physics of expansion and redshift, and Robertson and Walker proved the stage they act on is unique.

History

1917

— Einstein and de Sitter publish the first relativistic cosmological models

1922

— Friedmann derives the dynamical expanding and contracting solutions

1927

— Lemaître independently finds cosmic expansion and predicts the redshift–distance law

1929

— Hubble establishes the distance–velocity relation, evidence for expansion

1935

— Robertson and Walker prove the metric is the unique homogeneous, isotropic form

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27.6 (cosmological models) · Wald, General Relativity, §5.1–5.3 · Carroll, Spacetime and Geometry, §8.1–8.2 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, Ch. 12 · Weinberg, Cosmology, Ch. 1. Key relations used throughout: the maximal-symmetry Riemann form $R_{ijkl}=K(g_{ik}g_{jl}-g_{il}g_{jk})$ , spatial curvature ${}^{(3)}R=6k/a^2$ , exact redshift $1+z=a_0/a_e$ , and geodesic property $\Gamma^\mu_{00}=0$ . Anchor numbers: CMB $z\approx1100$ gives $a_e=9.08\times10^{-4}$ and $T_e=T_0(1+z)=3000$ K ( $T_0=2.725$ K); the curvature bound uses $c/H_0=4448$ Mpc $=14.5$ Gly at $H_0=67.4$ km s⁻¹ Mpc⁻¹ and $\Omega_k=0.001\pm0.002$ (Planck 2018), giving $R_{\rm curv}\gtrsim265$ Gly. The de Sitter $H=\sqrt{\Lambda c^2/3}$ , horizon $c/H$ , and closed-universe volume $2\pi^2 a^3$ are standard results.

Prerequisites. The Cosmological Principle · Comoving and Proper Coordinates · The Scale Factor and Hubble Parameter. Leads to. The Friedmann Equations · Cosmological Redshift · Cosmological Horizons · Critical Density and the Density Parameters.

Ask the tutor: Ask me to feed this metric into Einstein's equations and watch the ten field equations collapse to the two Friedmann equations.

Is the FLRW metric a solution of Einstein's equations?

Not by itself. It is an ansatz — a form fixed entirely by symmetry, containing one undetermined function $a(t)$ and one discrete label $k$ . Homogeneity and isotropy alone, with no reference to gravity, force this shape (the Robertson–Walker uniqueness theorem). To get an actual solution you supply Einstein's equations and a source: feeding the metric into $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}/c^4$ with a perfect fluid yields the Friedmann equations, which determine $a(t)$ once density and pressure are specified. FLRW is the stage; the Friedmann equations are the script its one moving part must follow.

What is the difference between comoving distance and proper distance?

Comoving distance $\chi$ is a fixed label: the coordinate separation between two galaxies that stays constant as the universe expands, like two raisins measured in "dough units." Proper distance $d_p=a(t)\chi$ is what a chain of rulers laid along a constant-time slice reads, and it grows with the scale factor. With $a_0=1$ the two coincide today, which is why catalogues quote comoving distances. The distinction is essential for redshift and horizons: light travels a fixed comoving distance, but the proper distance to its source was smaller at emission and is larger now — and it is that change that stretches the wavelength.

Why is the curvature constant restricted to just three values?

Because only the sign of the spatial curvature is a coordinate-independent fact; its magnitude is not separate from the scale factor. The constant-curvature slice gives $g_{rr}=1/(1-Kr^2)$ with $K$ any real constant, but rescaling $r\to r/\sqrt{|K|}$ folds the size of $K$ into $a(t)$ , leaving only whether $K$ was positive, zero, or negative — the closed 3-sphere ( $k=+1$ ), flat space ( $k=0$ ), or open hyperbolic space ( $k=-1$ ). The physical, dimensionful curvature is $k/a^2$ , which varies continuously in time; $k$ itself is just the sign that survives the rescaling.

Is cosmological redshift a Doppler shift?

No, though the two are often conflated. A Doppler shift arises from relative motion in one inertial frame and a gravitational shift from a potential difference; cosmological redshift is a distinct third effect — a photon's wavelength is stretched in proportion to the scale factor while in flight, so $\lambda_{\rm obs}/\lambda_{\rm em}=a_0/a_e$ exactly, for any expansion history. There is no single relative velocity to plug into a Doppler formula, because the photon passes through a continuum of comoving frames. For nearby sources it reduces to $z\approx v/c$ , which is why the two are confused, but at large $z$ only the exact scale-factor relation is correct.

Where is the centre of an FLRW universe, and does it have an edge?

Neither. Homogeneity means every point is equivalent, so there is no centre — expansion looks the same from every galaxy. Whether there is an edge depends on $k$ . The closed universe ( $k=+1$ ) is a finite 3-sphere: finite volume $2\pi^2 a^3$ but no boundary, as the surface of a globe is finite yet edgeless. The flat ( $k=0$ ) and open ( $k=-1$ ) universes are spatially infinite, again with no edge. The expansion is not galaxies flying apart into a void; it is the growth of proper distance everywhere at once, the scale factor multiplying an edgeless space.

How does the metric alone tell you the universe is flat, without solving any dynamics?

It does so through geometry read off the sky. The metric fixes how a known physical length at redshift $z$ subtends an angle: that angle depends on the angular-diameter distance, which contains the curvature function $S_k(\chi)$ . The cosmic microwave background supplies exactly such a standard ruler — the sound horizon at recombination, whose size is known from atomic physics. Comparing its measured angular scale (about one degree) with each geometry's prediction, Planck finds the flat case $k=0$ fits to within a percent: $\Omega_k=0.001\pm0.002$ . No expansion history is solved; the curvature enters purely through the shape of the metric, which is why a geometry measurement constrains $k$ almost independently of the fluid content.