Hubble’s law is one of the simplest and most important relations in cosmology. It connects the recession velocity of a
distant galaxy to its distance from the observer. In its linear, low-redshift form, it says that farther galaxies recede
faster:
Hubble’s law.
\[
\begin{aligned}
v &= H_0 d.
\end{aligned}
\]
Here \(v\) is the recession velocity, \(d\) is the distance, and \(H_0\) is the Hubble constant. A common reference value
is about \(70\ \mathrm{km/s/Mpc}\), meaning that for every megaparsec of distance, the recession velocity increases by
about \(70\ \mathrm{km/s}\).
Velocity from distance
The most direct use of the law is to compute the recession velocity of a galaxy when its distance is known.
For example, if
\[
\begin{aligned}
H_0 &= 70\ \mathrm{km/s/Mpc},\\
d &= 100\ \mathrm{Mpc},
\end{aligned}
\]
then
Sample step.
\[
\begin{aligned}
v &= H_0 d \\
&= 70 \cdot 100 \\
&= 7000\ \mathrm{km/s}.
\end{aligned}
\]
This is the standard sample result quoted for a galaxy at \(100\ \mathrm{Mpc}\).
Redshift and the small-\(z\) approximation
At low redshift, the observed redshift \(z\) is approximately related to recession velocity by
\[
\begin{aligned}
v &\approx zc,
\end{aligned}
\]
where \(c\) is the speed of light. Combining this with Hubble’s law gives a simple distance estimate:
Distance from small redshift.
\[
\begin{aligned}
d &\approx \frac{zc}{H_0}.
\end{aligned}
\]
This approximation is useful for nearby and moderately distant galaxies where \(z\) is small. It becomes less accurate
at higher redshift, where the expansion history of the Universe and relativistic cosmology must be treated more carefully.
Look-back time estimate
In the same low-redshift regime, one can estimate the look-back time with
\[
\begin{aligned}
t_{\text{look-back}} &\approx \frac{z}{H_0}.
\end{aligned}
\]
Since \(v \approx zc\) and \(v = H_0 d\), this is also consistent with the rough light-travel estimate
\[
\begin{aligned}
t_{\text{look-back}} &\approx \frac{d}{c}.
\end{aligned}
\]
This gives a simple physical interpretation: in the linear regime, the look-back time is approximately the travel time
of light from the source to the observer. The calculator reports this only as an estimate, not as a full cosmological
integral.
The Hubble time
Another useful scale is the inverse of the Hubble constant:
Hubble time.
\[
\begin{aligned}
t_H &= \frac{1}{H_0}.
\end{aligned}
\]
When converted into years, this gives a characteristic cosmological timescale. For \(H_0 \approx 70\ \mathrm{km/s/Mpc}\),
the Hubble time is on the order of \(14\) billion years, which is close to the age scale of the Universe.
Worked comparison
Suppose you observe a small redshift \(z = 0.02335\) and take \(H_0 = 70\ \mathrm{km/s/Mpc}\).
Then the low-redshift approximation gives
Step 1. Convert redshift to velocity.
\[
\begin{aligned}
v &\approx zc \\
&\approx 0.02335 \cdot 299792.458 \\
&\approx 7000\ \mathrm{km/s}.
\end{aligned}
\]
Step 2. Convert velocity to distance.
\[
\begin{aligned}
d &\approx \frac{v}{H_0} \\
&\approx \frac{7000}{70} \\
&\approx 100\ \mathrm{Mpc}.
\end{aligned}
\]
This matches the \(100\ \mathrm{Mpc}\) sample from the direct Hubble-law calculation, showing how the distance and
redshift forms fit together in the low-\(z\) limit.
Why the linear law matters
Hubble’s law was historically crucial because it revealed that the Universe is expanding. A linear velocity–distance plot,
often called a Hubble diagram, shows that galaxies are not moving randomly overall. Instead, the farther
they are, the faster they recede on average.
This does not mean every galaxy is simply “flying through space” in the ordinary sense. In modern cosmology, the more
accurate picture is that large-scale space itself expands, increasing the separation between distant galaxies.
Important limitation
The formulas in this calculator are intentionally the simple, educational versions. For large redshift, the relation
\(v \approx zc\) breaks down, and the connection between redshift, distance, and look-back time depends on cosmological
parameters such as matter density, dark energy, and the expansion history of the Universe.
Advanced note
At university level, one replaces the linear approximation with the Friedmann–Lemaître cosmological framework and uses
proper cosmological distance measures such as comoving distance, luminosity distance, and angular-diameter distance.
One also studies the accelerating Universe and the role of dark energy. This calculator deliberately stays in the linear
Hubble-law regime so that the core ideas remain transparent:
\[
\begin{aligned}
v &= H_0 d,\\
v &\approx zc,\\
d &\approx \frac{zc}{H_0}.
\end{aligned}
\]
| Idea |
Main relation |
Meaning |
| Hubble’s law |
\(v = H_0 d\) |
Linear relation between recession velocity and distance |
| Small-redshift velocity |
\(v \approx zc\) |
Low-\(z\) approximation for recession velocity |
| Distance from redshift |
\(d \approx zc/H_0\) |
Linear distance estimate in the low-\(z\) regime |
| Look-back estimate |
\(t \approx z/H_0 \approx d/c\) |
Approximate low-redshift look-back time |
| Hubble time |
\(t_H = 1/H_0\) |
Characteristic cosmological timescale |
| Sample benchmark |
\(d=100\ \mathrm{Mpc} \Rightarrow v \approx 7000\ \mathrm{km/s}\) |
Standard Hubble-law example |