The age and history of the Universe

Builds on: Sections 13.3 and 13.4
Reading time: 40 min
Exercises: 3
Olympiad problems: 3

The oldest stars in the Milky Way, in its globular clusters, are about 12 to 13 billion years old. In the early 1990s this was a puzzle. A flat universe of matter alone, which most cosmologists then expected, would be only two thirds of the Hubble time old, about 9.3 billion years for a Hubble constant of 70 km/s/Mpc, younger than its own stars. Dark energy removed the conflict. Because the expansion has been speeding up, it used to be slower, so it took longer to reach today's size. In this section we work out the age of a model universe from the Friedmann equation.

13.5.1The three simple universes

Before you readMake a guess first

Two flat universes have the same Hubble constant today. One holds only matter, and the other also holds dark energy. Which is older?

When one part rules the density, the Friedmann equation ((13.8)) can be solved directly. In a flat universe of matter alone, ${\dot{a}}^{2} = H_{0}^{2} / a$ , so $\sqrt{a} d a = H_{0} d t$ and $a^{3 / 2}$ grows in proportion to time. Radiation, falling as $a^{- 4}$ , gives $a \propto t^{1 / 2}$ . Dark energy alone keeps $H$ constant, and the scale factor grows exponentially.

ResultExpansion when one part rules(13.9)

\begin{aligned} radiation: a & \propto t^{1 / 2}, H = \frac{1}{2 t} \\ matter: a & \propto t^{2 / 3}, H = \frac{2}{3 t} \\ dark energy: a & \propto e^{H t}, H = constant \end{aligned}

$a$	the scale factor
$t$	the time since the Big Bang
$H$	the Hubble parameter at that time

These hold for a flat universe in which one part dominates. A flat universe of matter alone has the age

t_{0} = 2 / 3 H_{0}

, two thirds of the Hubble time. In our Universe radiation ruled until about

z = 3570

, matter until a few billion years ago, and dark energy since.

Show the derivation

In a flat universe whose density falls as $a^{- n}$ , the Friedmann equation reads ${\dot{a}}^{2} = H_{0}^{2} a^{2 - n}$ , so $a^{n / 2 - 1} d a = H_{0} d t$ . Integrating from the Big Bang, where $a = 0$ , gives

\frac{2}{n} a^{n / 2} = H_{0} t,

so $a \propto t^{2 / n}$ and $H = \dot{a} / a = 2 / (n t)$ . Radiation has $n = 4$ and matter $n = 3$ . Dark energy has $n = 0$ , for which $\dot{a} = H_{0} a$ and the growth is exponential.

13.5.2The age of our Universe

For a flat universe of matter and dark energy, the Friedmann equation can still be solved exactly. The age comes out as

ResultThe age of a flat universe of matter and dark energy(13.10)

t_{0} = \frac{2}{3 H_{0} \sqrt{Ω_{Λ}}} arsinh \sqrt{\frac{Ω_{Λ}}{Ω_{m}}}

$t_{0}$	the age of the Universe
$Ω_{m}, Ω_{Λ}$	the present density parameters of matter and dark energy, with $Ω_{m} + Ω_{Λ} = 1$

The function arsinh is the inverse of sinh, and

arsinh x = \ln (x + \sqrt{x^{2} + 1})

. Radiation is left out, which changes the age by less than a thousandth. With

Ω_{m} = 0.3

and

Ω_{Λ} = 0.7

the age comes out as 0.964 Hubble times, 13.47 billion years for

H_{0} = 70

km/s/Mpc. Measurements of the cosmic microwave background give 13.8 billion years, with a slightly smaller Hubble constant.

Figure 13.4The expansion history of a model universe

Matter

Ω_{m}

Dark energy

Ω_{Λ}

Space is
Age
Expansion
Future

The figure shows the scale factor against time since the Big Bang for a model universe with the chosen matter and dark energy, and

H_{0} = 70

km/s/Mpc. The point marked now is where the scale factor is 1. The dashed line is expansion at today's rate kept steady, which meets

a = 0

one Hubble time ago, and the dotted line marks the Big Bang of the model, so the gap between where it and the curve start shows how the age differs from the Hubble time. A curve that bends down has a slowing expansion, and one that bends up a speeding one. The figure can be set to , , , or , which is close to the dashed line.

13.5.3The age at a redshift

The time since the Big Bang at which light now seen at redshift $z$ set out is the age of the Universe at that redshift. The difference between it and the present age is the lookback time, how long the light has been travelling. For our model, light from $z = 1$ set out when the Universe was 5.75 billion years old, 7.72 billion years ago, and light from $z = 3$ when it was 2.11 billion years old. At redshifts well above 1 matter rules, and the matter solution gives a simple rule.

ResultThe age at a high redshift(13.11)

t (z) \approx \frac{2}{3 H_{0} \sqrt{Ω_{m}}} (1 + z)^{- 3 / 2}

t (z)

the time since the Big Bang at which light now seen at redshift

z

set out

This holds for

z

well above 1, where dark energy no longer matters, and well below 3570, where radiation begins to. For a flat universe of matter alone,

Ω_{m} = 1

, it holds at every redshift where radiation can be ignored, as

t (z) = t_{0} (1 + z)^{- 3 / 2}

Example 13.7The age at redshift 3

Use (13.11) to find the age of the Universe at $z = 3$ , first for a flat universe of matter alone and then for our model, with $Ω_{m} = 0.3$ . Take $H_{0} = 70$ km/s/Mpc.

Solution

For matter alone $t_{0} = 2 / 3 H_{0} = 9.31$ billion years, so $t = 9.31 \times 4^{- 3 / 2} = 1.16$ billion years. With $Ω_{m} = 0.3$ the front factor is larger by $1 / \sqrt{0.3}$ , giving $t = 2.13$ billion years.

AnswerA universe of matter alone was 1.16 billion years old at

z = 3

, while our Universe was about 2.13 billion years old. The full calculation for our model gives 2.11 billion years, so the rule is already good at

z = 3

Example 13.8Half the present agefinish it

In a flat universe of matter alone, at what redshift was the Universe half its present age?

The start

By (13.9), $a \propto t^{2 / 3}$ , so at half the present age $a = (1 / 2)^{2 / 3} = 0.63$ .

Your stepNow find the redshift.

Common errorsSection 13.5

The age of the Universe is not the Hubble time. It equals $1 / H_{0}$ only for steady expansion. A flat universe of matter alone is $2 / 3 H_{0}$ old, and our Universe is about $0.96 / H_{0}$ old.
The rule $t \propto (1 + z)^{- 3 / 2}$ holds only while matter rules. Using it at $z = 0.5$ for our Universe, where dark energy matters, gives a wrong age.
Time and redshift run in opposite directions. A larger redshift means an earlier time and a younger Universe.
In a universe of dark energy alone there is no Big Bang at a finite time, since $a$ never reaches zero. Real universes have one because matter and radiation dominate early on.

ExercisesSection 13.53 questions

13.5.1
How old would a flat universe of matter alone be if $H_{0} = 50$ km/s/Mpc?
billion years
13.5.2
In a flat universe of matter alone with $H_{0} = 70$ km/s/Mpc, how old was the Universe at $z = 8$ ?
billion years
13.5.3
In the far future our Universe will be ruled by dark energy, with $H = H_{0} \sqrt{Ω_{Λ}}$ . How long will it then take for every distance to double? Take $H_{0} = 70$ km/s/Mpc and $Ω_{Λ} = 0.7$ .
billion years

Olympiad problemsPast IOAA problems, restated3 problems

2011When a distant galaxy formed its stars10 points

Adapted from IOAA 2011 (Katowice), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The spectrum of a galaxy at redshift $z = 6.03$ shows that its stars are 560 to 600 million years old. At what redshift did the galaxy form its stars? Take the present age of the Universe as $t_{0} = 13.7 \times 10^{9}$ years, and assume a flat universe with no cosmological constant, in which the scale factor grows as $t^{2 / 3}$ .

I solved it without the solution

AnswerThe stars formed between $z = 17.3$ and $z = 20.8$ .

2012How long the Universe takes to cool by a tenth of a kelvin10 points

Adapted from IOAA 2012 (Rio de Janeiro), theory problem 10. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

Assume that the Universe is flat with $Ω_{0} = 1$ and no dark energy, that the present temperature of the background radiation is 2.73 K, and that the temperature is inversely proportional to the scale factor. Take $H_{0} = 72$ km/s/Mpc. How long, from now, will the Universe take to cool by 0.1 K?

I solved it without the solution

AnswerThe Universe takes about 520 million years to cool by 0.1 K.

2024When the background radiation was infrared10 points

Adapted from IOAA 2024 (Rio de Janeiro), theory problem 5. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The cosmic microwave background is close to a blackbody spectrum, now at about 3 K.

(a) At what redshift did its spectrum peak at an infrared wavelength of about 0.1 mm? [3 points]

(b) Assuming a flat universe of matter alone with $H_{0} = 70$ km/s/Mpc, how old was the Universe at that redshift? [7 points]

I solved it without the solution

Answer(a) $z \approx 9$ . (b) The Universe was about 0.31 billion years old, 0.29 billion years if $z = 9$ is used. Our actual Universe, with dark energy, was about 0.56 billion years old at that redshift.