What causes cosmological redshift?

At A-Level it is treated as a Doppler shift from galaxies moving away. Strictly, it arises because space itself stretches while the light travels.

Does Hubble’s law mean we are at the centre of the universe?

No. In uniform expansion, every observer sees all other galaxies receding with speed proportional to distance.

Why is 1/H_0 only an estimate of the age?

It assumes the expansion rate has always been constant, but gravity and dark energy have changed it over time.

What is the evidence for dark matter?

Stars in the outer parts of galaxies orbit too fast for the visible mass to hold them, so extra unseen mass must be present.

Back to course

161–163SpaceStretch

The Expanding Universe

In the 1920s Edwin Hubble found that almost every galaxy is moving away from us, and the further away it is, the faster it recedes. That single observation means the universe is $\textbf{expanding}$ — and running the expansion backwards leads to a beginning.

35 min Topic 10 · Space

What you'll be able to do

Use the Doppler equation $z = \Delta\lambda/\lambda \approx v/c$ for galaxies
State and use Hubble’s law, $v = H_{0}d$
Estimate the age of the universe from $1/H_{0}$
Discuss the possible fates of the universe and the evidence for dark matter

Redshift

Elements absorb and emit light at precise wavelengths, creating line spectra. In light from distant galaxies, the familiar pattern of lines is still there but shifted towards longer wavelengths — $\textbf{redshift}$ .

The shift is caused by the galaxy moving away from us. For speeds much less than light, the fractional change in wavelength equals the ratio of the recession speed to the speed of light.

A light source moving towards us is blueshifted instead. Nearby galaxies such as Andromeda show a small blueshift, but beyond our local group virtually all galaxies are redshifted.

z = \dfrac{\Delta\lambda}{\lambda} \approx \dfrac{\Delta f}{f} \approx \dfrac{v}{c}

Redshift, valid for

v \ll c

Example 1A hydrogen line with laboratory wavelength 656.3 nm is observed at 669.4 nm in a galaxy’s spectrum. Find the galaxy’s recession speed.

\Delta\lambda = 669.4 - 656.3 = 13.1

nm.

z = 13.1/656.3 = 0.01996

v = zc = 0.01996 \times 3.00\times10^{8}

Answer

6.0\times10^{6}\ \mathrm{m\,s^{-1}}

Tip — Divide by the $\textit{emitted}$ (laboratory) wavelength, not the observed one.

Hubble’s law

$\textbf{Hubble’s law}$ states that the recession velocity of a galaxy is proportional to its distance from us. The constant of proportionality is the $\textbf{Hubble constant}$ $H_{0}$ .

Current measurements give $H_{0}$ around $70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , or about $2.3\times10^{-18}\ \mathrm{s^{-1}}$ in SI units. Different methods disagree slightly, and the precise value is an active research question.

The law does not mean we are at the centre. In a uniformly expanding universe, observers in every galaxy see all others receding with speed proportional to distance — like dots on an inflating balloon.

v = H_{0}d

Hubble’s law.

Example 1Using

H_{0} = 2.3\times10^{-18}\ \mathrm{s^{-1}}

, find the distance to the galaxy in the previous example.

d = \dfrac{v}{H_{0}} = \dfrac{6.0\times10^{6}}{2.3\times10^{-18}}

Answer

2.6\times10^{24}\ \mathrm{m}

, about 85 Mpc.

The age of the universe

If galaxies have always moved apart at constant speed, a galaxy now at distance $d$ moving at $v$ set off from our location a time $d/v$ ago. Hubble’s law makes $d/v = 1/H_{0}$ the same for every galaxy.

So $1/H_{0}$ estimates the time since everything was together — the age of the universe. That is the basis of the $\textbf{Big Bang}$ model.

The estimate assumes constant expansion. In reality gravity slowed the expansion early on, and it now appears to be accelerating, but $1/H_{0}$ still lands close to the accepted age of about 13.8 billion years.

t \approx \dfrac{1}{H_{0}}

Age of the universe estimate.

Example 1Estimate the age of the universe in years with

H_{0} = 2.3\times10^{-18}\ \mathrm{s^{-1}}

t = 1/2.3\times10^{-18} = 4.35\times10^{17}

2Divide by

3.16\times10^{7}

s per year.

Answer

1.4\times10^{10}

years

Tip — To use $1/H_{0}$ , first convert $H_{0}$ into $\mathrm{s^{-1}}$ : divide km s⁻¹ Mpc⁻¹ by the number of km in a megaparsec.

Dark matter and the fate of the universe

Whether expansion continues forever depends on the average density of the universe. Above a $\textbf{critical density}$ , gravity would eventually halt the expansion and reverse it; below it, expansion continues.

Estimating density is hard because much of the mass cannot be seen. Stars at the edges of spiral galaxies orbit much faster than the visible mass could hold them. There must be far more mass present than we observe — $\textbf{dark matter}$ , which interacts gravitationally but emits no light.

Observations of distant Type Ia supernovae show that expansion is currently $\textbf{accelerating}$ , attributed to an unexplained dark energy. Together, dark matter and dark energy make up about 95% of the universe, and neither is understood.

Equation recap

z = \Delta\lambda/\lambda \approx v/c

Redshift.

v = H_{0}d

Hubble’s law.

t \approx 1/H_{0}

Age of the universe.

Common mistakes to avoid

Dividing

\Delta\lambda

by the observed wavelength.

Use the emitted, laboratory wavelength.

Leaving

H_{0}

in km s⁻¹ Mpc⁻¹ when finding age.

Convert to s⁻¹ first.

Saying Hubble’s law puts Earth at the centre.

Every observer sees the same pattern of recession.

Using

z \approx v/c

at speeds close to light.

The approximation is only valid for

v \ll c

Key takeaways

Redshift $z = \Delta\lambda/\lambda \approx v/c$ gives recession speed.
Hubble’s law: $v = H_{0}d$ .
The age of the universe is roughly $1/H_{0}$ , about 14 billion years.
The fate of the universe depends on its density relative to a critical value.
Galaxy rotation curves are evidence for dark matter; supernovae show accelerating expansion.

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