Cosmological Redshift and Hubble's Law

Derivation

Cosmological Redshift and Hubble's Law

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Statement

In a spatially homogeneous and isotropic (FLRW) spacetime with scale factor \(a(t)\), the wavelength of a freely propagating photon stretches in exact proportion to the scale factor, giving the exact relation \(1+z = a(t_{\text{obs}})/a(t_{\text{emit}})\); for sources near enough that light travel time is short compared with the Hubble time, this reduces to the linear Hubble law \(v = H_0 d\), where \(H_0 \equiv \dot a(t_0)/a(t_0)\) is the present-day Hubble rate.

Why it matters

The redshift-scale-factor relation is the single most-used equation in observational cosmology: every spectroscopic redshift is a direct measurement of how much the universe has expanded since the light was emitted, converting an otherwise inaccessible quantity, \(a(t_{\text{emit}})\), into a number read off a spectrograph. It underpins distance ladders, the interpretation of the cosmic microwave background at \(z \approx 1100\), and the entire framework of expansion history \(H(z)\).

The linear Hubble law is the low-redshift limit of that relation. Its discovery established that the universe is expanding, fixed the age scale \(H_0^{-1}\), and to this day the tension between local and early-universe determinations of \(H_0\) is one of the sharpest open problems in physics.

Assumptions

The metric is FLRW (homogeneous and isotropic).If dropped, there is no single global scale factor \(a(t)\); redshift becomes direction-dependent and the tidy \(1+z=a_0/a_e\) relation fails, replaced by an integral of the local expansion and shear along the ray.

Photons travel on radial null geodesics.If light did not follow \(ds^2=0\) (e.g. massive carriers, or a medium with refractive index \(\neq 1\)), the coordinate propagation relation \(c\,dt = a\,d\chi\) would acquire extra terms and the wavelength-stretch argument would break.

Source and observer are comoving.If either has peculiar velocity \(v_{\text{pec}}\), an extra special-relativistic Doppler factor multiplies the cosmological redshift; observed \(z\) is then \((1+z_{\text{cos}})(1+z_{\text{pec}})-1\), not the pure scale-factor ratio.

The scale factor is a smooth, monotonic function of cosmic time over the light path.If \(a(t)\) were non-monotonic or discontinuous (turnaround, or a singular epoch inside the interval), the ratio still defines \(1+z\) but the linear Taylor expansion leading to Hubble's law is invalid.

Derivation

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

Start from the FLRW line element; homogeneity and isotropy fix the metric up to \(a(t)\) and curvature \(k\). A

\[ d\Omega = 0, \qquad ds^2 = -c^2\,dt^2 + \frac{a(t)^2\,dr^2}{1-kr^2} = 0 \]

Place the observer at the origin; by isotropy a photon reaching us travels radially, so angular displacement vanishes and the geodesic is null. A

\[ \frac{c\,dt}{a(t)} = -\frac{dr}{\sqrt{1-kr^2}} \equiv -d\chi \]

Solve the null condition for the ratio; the incoming ray has \(dr<0\), and \(d\chi\) is the comoving radial distance element, independent of \(t\). A

\[ \int_{t_e}^{t_o} \frac{c\,dt}{a(t)} = \int_0^{\chi_e} d\chi = \chi_e \]

Integrate along the ray from emission \((t_e)\) to observation \((t_o)\); the right side is a fixed comoving distance because both source and observer are comoving. B

\[ \int_{t_e+\delta t_e}^{t_o+\delta t_o} \frac{c\,dt}{a(t)} = \chi_e \]

Consider a second wavecrest emitted one period \(\delta t_e\) later and received \(\delta t_o\) later; it covers the same comoving distance \(\chi_e\), since the geometry is static in comoving coordinates. B

\[ \int_{t_e}^{t_o}\frac{c\,dt}{a} = \int_{t_e+\delta t_e}^{t_o+\delta t_o}\frac{c\,dt}{a} \;\Rightarrow\; \int_{t_e}^{t_e+\delta t_e}\frac{c\,dt}{a} = \int_{t_o}^{t_o+\delta t_o}\frac{c\,dt}{a} \]

Equate the two path integrals and cancel the common interval \([t_e+\delta t_e,\,t_o]\); what remains equates the tiny end-intervals. B

\[ \frac{c\,\delta t_e}{a(t_e)} = \frac{c\,\delta t_o}{a(t_o)} \qquad\Longrightarrow\qquad \frac{\delta t_o}{\delta t_e} = \frac{a(t_o)}{a(t_e)} \]

Over one period \(a(t)\) is essentially constant (\(\delta t \ll H^{-1}\)), so each small integral is (interval)/(scale factor); rearrange. B

\[ \lambda = c\,\delta t \;\Rightarrow\; \frac{\lambda_o}{\lambda_e} = \frac{\delta t_o}{\delta t_e} = \frac{a(t_o)}{a(t_e)} \]

A wavelength is \(c\) times a period; substitute to convert the period ratio into a wavelength ratio. A

\[ 1+z \equiv \frac{\lambda_o}{\lambda_e} = \frac{a(t_o)}{a(t_e)} \]

Apply the definition of redshift \(z=(\lambda_o-\lambda_e)/\lambda_e\); this is the exact redshift-scale-factor relation, valid for any \(z\). A

\[ a(t_e) = a(t_o)\Big[1 + H_0\,(t_e-t_o) + \mathcal{O}\big((t_e-t_o)^2\big)\Big],\qquad H_0 \equiv \frac{\dot a(t_o)}{a(t_o)} \]

For a nearby source, Taylor-expand \(a\) about the present \(t_o\); the leading coefficient defines the Hubble rate. B

\[ 1+z = \frac{a(t_o)}{a(t_e)} \approx \frac{1}{1+H_0(t_e-t_o)} \approx 1 + H_0\,(t_o-t_e) \]

Insert the expansion and use \((1+x)^{-1}\approx 1-x\) for small \(x\); thus \(z \approx H_0(t_o-t_e)\) to first order. B

\[ d \approx c\,(t_o-t_e) \quad\Rightarrow\quad z \approx \frac{H_0}{c}\,d, \qquad cz \approx H_0\,d \]

For small look-back time the proper distance is \(c\) times the light-travel time; substitute. B

\[ v \equiv cz \quad\Longrightarrow\quad v = H_0\,d \]

Interpret \(cz\) as a recession velocity in the non-relativistic limit \(z\ll 1\); this is Hubble's law. A

Result

\[ \boxed{\,1+z = \dfrac{a(t_o)}{a(t_e)}\,}\qquad\qquad \boxed{\,v = H_0\,d \;\;(z\ll 1)\,} \]

Reading. The first relation is exact: light emitted when the universe was a factor \(a(t_e)/a(t_o)\) smaller arrives stretched by exactly the inverse of that factor. A galaxy at \(z=1\) emitted its light when the universe was half its present size. The second relation is the nearby limit: recession speed grows linearly with distance, with slope the present expansion rate \(H_0\). Redshift is not a Doppler shift through space but the accumulated stretching of the wave by the expansion of space itself.

Units check. \(1+z\) and \(a(t_o)/a(t_e)\) are both dimensionless (scale factor is a pure ratio). In Hubble's law, \([H_0]=\text{s}^{-1}\) (commonly \(\text{km s}^{-1}\text{Mpc}^{-1}\)), \([d]=\text{m}\), so \([H_0\,d]=\text{m s}^{-1}=[v]\). Also \([cz]=(\text{m s}^{-1})(\text{dimensionless})=\text{m s}^{-1}\), consistent.

Limiting cases

\(z\to 0\): \(a(t_e)\to a(t_o)\), no expansion between emission and reception; recovers \(v=H_0 d\) exactly and Euclidean, static intuition.
\(z\to\infty\): \(a(t_e)\to 0\), light from the initial singularity; \(z\approx 1100\) is the CMB last-scattering surface at \(a\approx 1/1101\).
Static universe (\(\dot a=0\)): \(H_0=0\), so \(z=0\) for all comoving sources; any observed shift must then be peculiar-velocity Doppler.
de Sitter (\(a\propto e^{Ht}\)): \(H\) constant, \(1+z=e^{H(t_o-t_e)}\); redshift grows exponentially with look-back time.
Matter-dominated (\(a\propto t^{2/3}\)): \(1+z=(t_o/t_e)^{2/3}\), tying redshift directly to the age ratio.

Breaks when

Large redshift with the linear law: \(v=H_0 d\) is only the first Taylor term. For \(z\gtrsim 0.1\) the deceleration/acceleration term \(\tfrac12(1-q_0)z\) matters, and treating \(cz\) as a true velocity gives \(v>c\) for \(z>1\), which is unphysical as a special-relativistic speed. One must use \(H(z)\) and a proper distance measure.
Peculiar velocities dominate: for nearby galaxies (e.g. in the Local Group or infalling into a cluster), peculiar motions of hundreds of km/s swamp the cosmological \(H_0 d\) term; some show blueshift (Andromeda). Hubble's law only emerges after averaging over many sources or at large \(d\).
Inhomogeneous geometry: along a ray passing through deep voids or strong potential wells, integrated Sachs-Wolfe and lensing shifts add to the pure \(a\)-stretch; the single global \(a(t)\) picture is an average, not exact per-ray.
Bound systems: gravitationally bound objects (atoms, the Solar System, our Galaxy) do not expand; their photons carry no cosmological redshift from internal scale, so applying \(1+z=a_o/a_e\) to intra-galactic distances is a category error.

Failure modes

Doppler conflation: claiming redshift is the ordinary velocity Doppler formula \(1+z=\sqrt{(1+\beta)/(1-\beta)}\). Cosmological redshift comes from metric expansion during transit, not a velocity at emission; the two only agree to first order in \(z\).
Inverting the scale-factor ratio: writing \(1+z=a(t_e)/a(t_o)\) instead of \(a(t_o)/a(t_e)\). Since \(a\) grows, the correct ratio exceeds 1 for distant past sources; the inverted form gives \(z<0\).
Superluminal panic: concluding a \(z=3\) quasar recedes faster than light "impossibly." Recession is proper-distance growth, not motion through space; no local frame sees a superluminal object.
Using \(v=H_0 d\) at cosmological \(z\): plugging \(z=1\) into \(cz=H_0 d\) and reporting a "distance." The linear law is a \(z\ll1\) expansion; at \(z=1\) it errs by tens of percent depending on \(q_0\).
Confusing \(H_0\) with a constant of nature: treating \(H\) as fixed for all time. \(H(t)=\dot a/a\) evolves; \(H_0\) is merely its present value, and Hubble's "constant" changes over cosmic history.
Distance-measure muddle: equating luminosity distance, angular-diameter distance, and proper distance at high \(z\). They differ by factors of \((1+z)\) and \((1+z)^2\); only proper distance appears in the naive \(cz=H_0 d\).

Discussion

The derivation makes precise what "expansion of space" means operationally. Nothing in the argument invokes a velocity of the source at emission; the photon simply propagates on a null geodesic while the comoving grid it lives on is stretched. The wavelength, being carried by the field, stretches with the grid. This is why the result \(1+z=a_o/a_e\) is exact and geometric, whereas the Doppler formula is only its low-order shadow. The equivalence to first order in \(z\) is genuine and useful, but it is a coincidence of the expansion, not a statement that redshift "is" a Doppler shift.

The transition from the exact relation to Hubble's law is a statement about locality in time. Over a short look-back interval the expansion history is well approximated by its instantaneous rate \(H_0\), and distances are Euclidean; both approximations degrade together as \(z\) grows. The next-order term brings in the deceleration parameter \(q_0=-a\ddot a/\dot a^2\), and measuring the departure from linearity in the Hubble diagram at \(z\sim 0.5\text{--}1\) is exactly how the accelerating expansion (dark energy) was discovered in 1998. Hubble's law is thus the flat baseline against which the dynamics of the universe are read.

Because \(1+z\) directly measures \(a\), redshift doubles as a clock and a ruler for the expansion. The relation \(H(z)^2 = H_0^2[\Omega_m(1+z)^3+\Omega_\Lambda+\dots]\) — the Friedmann equation rewritten in terms of \(z\) via \(a=1/(1+z)\) — turns a spectroscopic survey into a reconstruction of the entire expansion history. Every standard candle and standard ruler in cosmology is ultimately calibrated against this identity.

At the level of the photon's phase, the cleanest statement is that the comoving frequency \(a(t)\,\nu(t)\) is conserved along the null geodesic. This follows from the geodesic equation with the FLRW Killing structure: the photon's momentum satisfies \(p^\mu p_\mu=0\) and the projection onto the (conformal) time-translation of the comoving frame gives \(a p^0=\text{const}\), i.e. \(a\nu=\text{const}\), from which \(\nu_o/\nu_e=a_e/a_o\) and hence \(1+z=a_o/a_e\) follow immediately. This covariant derivation shows the result is not an artifact of the crest-counting argument but a conservation law rooted in the symmetry of the spacetime.

Common misconceptions. (i) "Galaxies fly apart through space" — no; comoving galaxies are at rest in the cosmic frame, and it is proper distances between them that grow. (ii) "Cosmological redshift is the Doppler effect" — only to first order; the exact effect is metric stretching. (iii) "\(z>1\) means faster than light and is forbidden" — recession velocity can exceed \(c\) with no violation of relativity, because it is not motion through a local frame. (iv) "\(H_0\) sets the size of the universe" — it sets a rate; \(cH_0^{-1}\) is a characteristic scale (the Hubble radius), not a physical edge.

Worked examples

Find the scale factor at emission for a quasar observed at \(z=3.20\), and state the recession velocity implied by the naive linear law versus the correct relativistic recession — commenting on validity.

Given \(z=3.20\). Use the exact relation, then compare velocity interpretations. A

\[ \frac{a(t_e)}{a(t_o)} = \frac{1}{1+z} = \frac{1}{4.20} = 0.238 \]

Invert \(1+z=a_o/a_e\); the universe was \(23.8\%\) of its present size when the light left. A

\[ v_{\text{naive}} = cz = (3.00\times10^5\ \text{km/s})(3.20) = 9.6\times10^5\ \text{km/s} = 3.2\,c \]

Apply \(cz\) literally; the absurd \(3.2c\) flags that \(cz\) is not a physical velocity here. A

\[ v_{\text{SR}} = c\,\frac{(1+z)^2-1}{(1+z)^2+1} = c\,\frac{17.64-1}{17.64+1} = 0.893\,c \]

The special-relativistic Doppler inversion gives a sub-luminal number, but even this is only the "apparent velocity" — the true cosmological recession from proper distance can exceed \(c\). B

\[ a_e/a_o = 0.238,\qquad v_{\text{naive}}=3.2c\ (\text{invalid}),\qquad v_{\text{SR,app}}=0.893c \]

Reading. The exact scale-factor statement is unambiguous and physical; the "velocity" is model-dependent and the linear \(cz\) badly overshoots at \(z=3.2\). At such redshift one must integrate \(H(z)\), never use \(v=H_0 d\).

A galaxy has heliocentric recession velocity \(v=1250\ \text{km/s}\). With \(H_0=70\ \text{km s}^{-1}\text{Mpc}^{-1}\), estimate its distance and its redshift, and assess whether the linear law is safe here.

Given \(v=1250\ \text{km/s}\), \(H_0=70\). Use \(v=H_0 d\) and \(z\approx v/c\). A

\[ d = \frac{v}{H_0} = \frac{1250\ \text{km/s}}{70\ \text{km s}^{-1}\text{Mpc}^{-1}} = 17.9\ \text{Mpc} \]

Solve Hubble's law for distance; the \(\text{km/s}\) cancels, leaving Mpc. A

\[ z \approx \frac{v}{c} = \frac{1250}{3.00\times10^5} = 4.17\times10^{-3} \]

In the small-\(z\) regime redshift is \(v/c\); here \(z\approx 0.0042\ll1\). A

\[ \frac{1}{2}(1-q_0)z \sim \frac{1}{2}(1.55)(0.0042) \approx 3\times10^{-3}\ \text{relative correction} \]

Estimate the next-order term with \(q_0\approx-0.55\); it is \(\sim0.3\%\), so the linear law is safe to well under a percent. B

\[ d \approx 17.9\ \text{Mpc},\qquad z \approx 4.2\times10^{-3} \]

Reading. At \(z\sim0.004\) the second-order correction is sub-percent, so \(v=H_0 d\) is fully adequate. The dominant real-world uncertainty here is the galaxy's unknown peculiar velocity (\(\sim\!300\ \text{km/s}\), i.e. \(\sim\!24\%\) of \(v\)), which limits the distance to roughly \(\pm4\ \text{Mpc}\) regardless of the law's accuracy.

Problems

The universe was one-quarter its present size when certain light was emitted. What redshift is observed today?
Solution
\(a_e/a_o=1/4\), so \(1+z=a_o/a_e=4\), giving \(z=3\). The light comes from an epoch of quarter-size expansion.
A galaxy is measured at \(z=0.024\). Using \(H_0=70\ \text{km s}^{-1}\text{Mpc}^{-1}\) and the linear law, estimate its distance.
Solution
\(v\approx cz=(3.00\times10^5)(0.024)=7200\ \text{km/s}\). Then \(d=v/H_0=7200/70=103\ \text{Mpc}\). (At \(z=0.024\) the linear law is good to \(\sim1\%\).)
The CMB is observed at \(z=1100\). By what factor has the universe expanded since last scattering, and what was the temperature then, given \(T_0=2.725\ \text{K}\) and \(T\propto (1+z)\)?
Solution
Expansion factor \(=a_o/a_e=1+z=1101\). Temperature at emission \(T_e=T_0(1+z)=2.725\times1101=3.00\times10^3\ \text{K}\approx3000\ \text{K}\), the hydrogen recombination scale, as expected.
In a matter-dominated universe \(a\propto t^{2/3}\). If a source is seen at \(z=1\), what was the cosmic time \(t_e\) as a fraction of the present time \(t_0\)?
Solution
\(1+z=(t_0/t_e)^{2/3}\Rightarrow t_e/t_0=(1+z)^{-3/2}=2^{-3/2}=0.354\). The light was emitted when the universe was about \(35\%\) of its present age.
A galaxy in the Virgo cluster is at true distance \(d=16.5\ \text{Mpc}\) but has a peculiar velocity of \(-400\ \text{km/s}\) (toward us). With \(H_0=73\ \text{km s}^{-1}\text{Mpc}^{-1}\), what recession velocity is observed, and what fractional error would result if you inverted \(v_{\text{obs}}=H_0 d\) to get distance?
Solution
Hubble flow: \(v_{\text{Hubble}}=H_0 d=73\times16.5=1205\ \text{km/s}\). Observed: \(v_{\text{obs}}=1205-400=805\ \text{km/s}\). Naive distance: \(d_{\text{naive}}=v_{\text{obs}}/H_0=805/73=11.0\ \text{Mpc}\). Fractional error \(=(11.0-16.5)/16.5=-33\%\). Peculiar velocity causes a large distance error for nearby objects — the reason Hubble's law needs averaging or large \(d\).