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At low redshift, the recession velocity of a sufficiently distant galaxy is approximately proportional to its distance.
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Formula
vᵣ in km/s; H0 in km/(s·Mpc); d in Mpc; z dimensionless
| Symbol | Meaning | Unit |
|---|---|---|
| Low-redshift recession velocity | km/s | |
| Present-day Hubble parameter | km/(s·Mpc) | |
| Distance in the adopted cosmological convention | Mpc | |
| Cosmological redshift | dimensionless |
v_r \approx H_0 d \quad (z\ll 1)The linear relation vᵣ≈H0d is a local, low-redshift approximation. For small cosmological redshift, v≈cz can also be used approximately, but nearby galaxies have peculiar velocities that may be comparable to their expansion velocity. At higher redshift, the mapping between redshift, distance and recession rate depends on the cosmological model; do not extrapolate this simple equation as an exact law.
Use as a low-redshift linear approximation; account for peculiar velocities, higher redshifts and cosmological model dependence.
At small z, v≈cz and locally v is proportional to d.
Distance from velocity
Only at low redshift and with local peculiar motions considered.
Hubble parameter from a pair
One galaxy pair is insufficient for a precise measurement.
A nearby galaxy has v≈700 km/s. Estimate d for H0=70 km/(s·Mpc).
d≈v/H0=700/70=10 Mpc, assuming the low-z approximation and negligible peculiar velocity.
Why is H0d not automatically the exact recession velocity for a strongly redshifted galaxy?
At large z, mapping redshift to distance and velocity depends on the cosmological expansion model; the local linear approximation is insufficient.
Extrapolating the linear relation to arbitrarily high redshift.
For higher z, specify a cosmological distance definition and expansion model.
Ignoring peculiar velocity for nearby galaxies.
Peculiar velocity can be comparable to the local expansion component.
Formula cluster
The Hubble approximation links spectral shift to distance, but higher z requires an expansion model.
With vᵣ≈700 km/s and H0≈70 km/(s·Mpc), d≈10 Mpc, provided the low-redshift approximation is suitable and peculiar velocity is small.
Make first-order distance estimates for nearby galaxies and explain why cosmological distance inference needs a fuller model at larger redshift.
Here is how to work through a typical Hubble–Lemaître law (low-redshift approximation) (vᵣ ≈ H₀d (z ≪ 1)) task step by step:
Task
A nearby galaxy has v≈700 km/s. Estimate d for H0=70 km/(s·Mpc).
Solution path
d≈v/H0=700/70=10 Mpc, assuming the low-z approximation and negligible peculiar velocity.
Task
Why is H0d not automatically the exact recession velocity for a strongly redshifted galaxy?
Solution path
At large z, mapping redshift to distance and velocity depends on the cosmological expansion model; the local linear approximation is insufficient.
H0 is the present-day slope between cosmological recession velocity and distance in the local linear regime. It is not the velocity of one galaxy and is not dimensionless. Its unit km/(s·Mpc) indicates the extra recession velocity per megaparsec of distance. A reliable estimate needs distance and velocity from compatible measurements. H0 is inferred from many objects and depends on calibration. For nearby galaxies, peculiar motion can overwhelm cosmic expansion; at high redshift, cosmological distance calculations are needed. State the low-redshift regime and the adopted H0 units when reporting a result.
The linear relation applies locally when cosmological redshift is small. At higher redshift, distance and recession velocity are not linked by the same simple linear mapping. One must then specify a cosmological distance definition and expansion model instead of treating H0d as exact. A reliable estimate needs distance and velocity from compatible measurements. H0 is inferred from many objects and depends on calibration. For nearby galaxies, peculiar motion can overwhelm cosmic expansion; at high redshift, cosmological distance calculations are needed. State the low-redshift regime and the adopted H0 units when reporting a result.
Rearrange vᵣ≈H0d to d≈vᵣ/H0. A velocity of 700 km/s and H0=70 km/(s·Mpc) give about 10 Mpc. First check that units are compatible and that the measured velocity is not mainly local peculiar motion. A reliable estimate needs distance and velocity from compatible measurements. H0 is inferred from many objects and depends on calibration. For nearby galaxies, peculiar motion can overwhelm cosmic expansion; at high redshift, cosmological distance calculations are needed. State the low-redshift regime and the adopted H0 units when reporting a result.
It is a galaxy’s motion relative to the background following cosmic expansion. The observed Doppler shift can therefore combine expansion and local motion. For nearby galaxies, the local component may be comparable to H0d, so one measurement does not yield a precise Hubble distance. A reliable estimate needs distance and velocity from compatible measurements. H0 is inferred from many objects and depends on calibration. For nearby galaxies, peculiar motion can overwhelm cosmic expansion; at high redshift, cosmological distance calculations are needed. State the low-redshift regime and the adopted H0 units when reporting a result.
One pair gives at most a rough ratio. Errors in distance, redshift and peculiar velocity can shift the result substantially. H0 is therefore inferred from many objects using an explicit distance ladder or other cosmological methods, with local and systematic uncertainties assessed separately. A reliable estimate needs distance and velocity from compatible measurements. H0 is inferred from many objects and depends on calibration. For nearby galaxies, peculiar motion can overwhelm cosmic expansion; at high redshift, cosmological distance calculations are needed. State the low-redshift regime and the adopted H0 units when reporting a result.
Quanta exam set
Curated exam set for "Hubble–Lemaître law (low-redshift approximation)":
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Which formula describes Hubble–Lemaître law (low-redshift approximation)?
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How do you rearrange Hubble–Lemaître law (low-redshift approximation) for Distance from velocity?
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Which common mistake happens with Hubble–Lemaître law (low-redshift approximation)?
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