Hubble-Lemaître Law

Equivalent forms

v = H_0\, d

H_0 \approx 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}

A one-line law whose slope sets the age and size of the observable universe (t ~ 1/H_0).

Symbol	Name	SI unit	Dimension	Range
$v$	Recession Velocityoutput Speed at which a galaxy recedes	m/s	LT^-1	0 – 300000000
$H0$	Hubble Constant Present-day expansion rate	s^-1	T^-1	1.5e-18 – 3e-18
$d$	Proper Distance Present distance to the galaxy	m	L	0 – 1.3e+26

Dimensional analysis

v -> [L T^-1]

H_0 d ->

[T^-1][L] = [L T^-1]

Velocity.

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Discovery

Georges Lemaître / Edwin Hubble · 1929

Lemaître derived the velocity-distance relation in 1927 from Einstein's equations; Hubble's 1929 galaxy data confirmed it. The IAU renamed it the Hubble-Lemaître law in 2018.

Try this

A galaxy sits 100 Mpc away. How fast is it rushing from us just because the universe is expanding?

v = H_0 d with H_0 = 70 km/s/Mpc gives v = 7,000 km/s — and it grows with distance without limit.

Research status: stable

Real-world applications

Measuring the age of the universe $(\sim 1/H_0)$
Cosmic distance ladder calibration
Redshift-to-distance conversion for surveys

Common misconceptions

Galaxies are not moving THROUGH space away from us — space between them expands
There is no centre of expansion; every observer sees the same law
Recession faster than c is allowed and does not violate relativity

Experimental verification

Hubble's original data, then Type Ia supernovae (which also revealed acceleration in 1998), Cepheid distance ladders, and the CMB all measure H_0. A persistent

\sim 9

% 'Hubble tension' between local (73) and CMB (67) values is a leading open problem.

Derivation

In an expanding FLRW universe a galaxy's proper distance is

d(t) = a(t) x

, with x the fixed comoving coordinate.

Differentiating,

v = d-dot = a-dot x

= (a-dot/a

) d = H d

, where

H = a-dot

/a.

Evaluated today,

H = H_0

, giving

v = H_0\,\mathrm{d}

exactly for small distances.

Limiting cases

d small⟶ v small, linearNearby galaxies obey a clean straight line; peculiar velocities add scatter.

v -> c⟶

d =

Hubble distance

\sim 4400\,\mathrm{Mpc}

Beyond this, recession is 'faster than light' — allowed because it is expansion of space, not motion through it.

H_0 -> 0⟶ v -> 0A static (non-expanding) universe.

What if…

What if we measure H_0 more precisely?

It pins down the universe's age and fate; the current tension between methods may point to new physics.

What if the expansion accelerates?

H is not truly constant in time; dark energy makes distant galaxies eventually cross beyond our horizon.

1

Recession at 100 Mpc

Given ·

H0:: 70 km/s/Mpc
d:: 100 Mpc

Find · v

Steps

$v = H_{0} * d$
$= 70 * 100\,\mathrm{km/s}$
$= 7000\,\mathrm{km/s}$

Result ·

v = 70\,\mathrm{km/s/Mpc} * 100\,\mathrm{Mpc} = 7000\,\mathrm{km/s} = 7e_{6} m/s

.

Cosmological Redshift→friedmann equationsscale factor evolutioncritical density universe