Matter, radiation and dark energy

Builds on: Sections 13.2 and 13.3
Reading time: 40 min
Exercises: 3
Olympiad problems: 3

In 1998 two teams measuring distant supernovae found them fainter than expected, and therefore further away than a slowing expansion would have put them. The expansion of the Universe has been speeding up for about the last 6.1 billion years. Something that does not thin out as space grows, now called dark energy, makes up about 70 per cent of the density of the Universe today. In this section we see how matter, radiation and dark energy each thin out as the Universe expands, and why each of them in turn has ruled its history.

13.4.1How each part thins out

Before you readMake a guess first

The Universe doubles in size. By what factor does the energy density of the radiation in it fall?

Matter, both ordinary and dark, is made of particles that are neither created nor destroyed. When every length grows by the factor $a$ , a volume grows by $a^{3}$ , so the density of matter falls as $1 / a^{3}$ . Radiation thins out in the same way, and on top of that each photon is stretched and loses energy as $1 / a$ ((13.4)), so its energy density falls as $1 / a^{4}$ . Dark energy behaves differently. In the simplest picture it is a property of empty space itself, the energy of the vacuum, and every cubic metre of new space comes with the same amount, so its density stays the same. Einstein called a term of this kind in his equations the cosmological constant, written $Λ$ .

ResultHow densities change with the scale factor(13.7)

ρ_{m} \propto a^{- 3}, ρ_{r} \propto a^{- 4}, ρ_{Λ} = constant

$ρ_{m}$	the density of matter, ordinary and dark
$ρ_{r}$	the mass equivalent of the energy density of radiation, photons and neutrinos
$ρ_{Λ}$	the mass equivalent of the energy density of dark energy, taken as a cosmological constant

With

a = 1 / (1 + z)

, matter was

(1 + z)^{3}

times as dense at redshift

z

as today and radiation

(1 + z)^{4}

times. The radiation today is mostly the cosmic microwave background, whose energy density at 2.725 K is

4.17 \times 10^{- 14}

J/m³, which gives

Ω_{γ} = 5.04 \times 10^{- 5}

. With the neutrinos left from the Big Bang, the total is about

8.5 \times 10^{- 5}

. This course uses

8.4 \times 10^{- 5}

, and the difference never matters in a problem.

Putting these into the Friedmann equation ((13.5)), and writing each density as its present value times its scaling, gives the form used in nearly every problem.

ResultThe Friedmann equation with density parameters(13.8)

\frac{H^{2}}{H_{0}^{2}} = \frac{Ω_{r}}{a^{4}} + \frac{Ω_{m}}{a^{3}} + \frac{Ω_{k}}{a^{2}} + Ω_{Λ}

$H, H_{0}$	the Hubble parameter at scale factor $a$ and today
$Ω_{r}, Ω_{m}, Ω_{Λ}$	the present density parameters of radiation, matter and dark energy
$Ω_{k}$	the curvature term, $1 - Ω_{r} - Ω_{m} - Ω_{Λ}$ , zero for flat space

The present values are about

Ω_{m} = 0.3

Ω_{Λ} = 0.7

Ω_{r} = 8.4 \times 10^{- 5}

and

Ω_{k} = 0

, which this course uses as its standard model. Setting

a = 1

gives

H = H_{0}

, as it should, since the four terms then add up to 1.

Quick checkMatter in the past

How many times as dense was matter at $z = 1$ as today?

times

13.4.2Three eras

Because the three densities fall at different rates, each term of (13.8) has its turn to be the largest. Early on, when $a$ was tiny, radiation ruled, since $1 / a^{4}$ grows fastest into the past. Matter took over when $Ω_{r} / a^{4} = Ω_{m} / a^{3}$ , at $a = Ω_{r} / Ω_{m} = 2.80 \times 10^{- 4}$ , or $z = 3570$ . Dark energy overtook matter at $a = (Ω_{m} / Ω_{Λ})^{1 / 3} = 0.754$ , or $z = 0.33$ , about 3.7 billion years ago.

Figure 13.3Radiation, matter and dark energy

Matter

Ω_{m}

Dark energy

Ω_{Λ}

Space is
Radiation and matter equal at
Matter and dark energy equal at

The figure shows the densities of radiation (dashed), matter (heavy) and dark energy (dotted) against the scale factor, both on logarithmic axes, as fractions of today's critical density. Each is a straight line, with slopes −4, −3 and 0. The point R marks the moment when radiation and matter were equally dense, and L the moment when matter and dark energy were. The sliders change today's matter and dark energy, and the table gives the redshifts of R and L. The figure can be set to , , or .

The pull of matter and radiation slows the expansion, while dark energy speeds it up. The expansion began to speed up a little before dark energy overtook matter, at $z = 0.67$ , when the density of dark energy reached half that of matter. Section 13.5 shows how each era shapes the growth of the scale factor.

Example 13.6The Universe at redshift 9

How fast was the Universe expanding at $z = 9$ , and what fraction of its density was dark energy? Use the standard model and $H_{0} = 70$ km/s/Mpc.

Solution

At $z = 9$ , $1 / a = 10$ . The terms of (13.8) are $Ω_{m} \times 10^{3} = 300$ for matter, $Ω_{r} \times 10^{4} = 0.84$ for radiation and 0.7 for dark energy, so

\frac{H^{2}}{H_{0}^{2}} = 300 + 0.84 + 0.7 = 301.5, \frac{H}{H_{0}} = 17.4 .

AnswerThe Universe was expanding at about 1220 km/s/Mpc, and dark energy was only 0.23 per cent of the density. At that time the Universe was almost all matter.

Common errorsSection 13.4

Radiation thins out as $1 / a^{4}$ , not $1 / a^{3}$ . The extra factor comes from the stretching of each photon, which matter does not suffer.
Dark energy, taken as a cosmological constant, does not thin out at all. Its density is the same at every redshift, and only its share of the total grows.
The density parameters in (13.8) are today's values. The share of each part at another time comes from its term divided by the sum of all the terms.
The moment when two parts were equal is not the moment when the expansion began to speed up. Speeding up needs dark energy to reach half the density of matter, which happened earlier, at $z = 0.67$ against $z = 0.33$ .

ExercisesSection 13.43 questions

13.4.1
At $z = 999$ , what was the ratio of the density of radiation to that of matter? Take $Ω_{r} = 8.4 \times 10^{- 5}$ and $Ω_{m} = 0.3$ .
13.4.2
What was the Hubble parameter at $z = 1$ in the standard model, with $H_{0} = 70$ km/s/Mpc?
km/s/Mpc
13.4.3
In the far future, as $a$ grows without limit in our Universe, which term of the Friedmann equation is largest?

Olympiad problemsPast IOAA problems, restated3 problems

2016When radiation and matter were equal10 points

Adapted from IOAA 2016 (Bhubaneswar), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.

The energy density of radiation in the Universe grows into the past as $(1 + z)^{4}$ , and that of matter as $(1 + z)^{3}$ . Today the density parameters of radiation and matter are $Ω_{r, 0} = 10^{- 4}$ and $Ω_{m, 0} = 0.3$ .

(a) Find the redshift $z_{e}$ at which radiation and matter were equally dense. [3 points]

(b) The radiation has a blackbody spectrum, now at 2.732 K. Estimate its temperature $T_{e}$ at $z_{e}$ . [4 points]

I solved it without the solution

Answer(a) $z_{e} \approx 3000$ . (b) $T_{e} \approx 8200$ K. (c) A typical photon had about 3.5 eV, the energy at the peak of the spectrum, about $5 k T$ . The official solution also accepts $k T = 0.71$ eV or $2.7 k T = 1.9$ eV, and $2.7 k T$ is the mean energy of a photon.

2008Dark matter against dark energy at the time of the background20 points

Adapted from IOAA 2008 (Bandung), 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 now has a temperature of 2.73 K, and it set out at $z = 1100$ . Today the densities of dark energy, dark matter and ordinary matter are $7.1 \times 10^{- 30}$ , $2.4 \times 10^{- 30}$ and $0.5 \times 10^{- 30}$ g/cm³. Assuming that dark energy is the energy of the vacuum, what was the ratio of the density of dark matter to that of dark energy when the background radiation set out?

I solved it without the solution

AnswerDark matter was about $4.5 \times 10^{8}$ times as dense as dark energy. The official solution gives $4.8 \times 10^{8}$ because it divides by $6.7 \times 10^{- 30}$ g/cm³ in place of the given $7.1 \times 10^{- 30}$ .

2023How much dark energy12 points

Adapted from IOAA 2023 (Katowice), 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 fluctuations of the cosmic microwave background point to flat space, in which the total density, counting the mass equivalent of every form of energy, equals the critical density $ρ_{cr} = 3 H_{0}^{2} / 8 π G$ , which is $9.202 \times 10^{- 27}$ kg/m³ for $H_{0} = 70$ km/s/Mpc. The total density of matter, luminous and dark, is estimated at $ρ_{m, 0} = 2.8 \times 10^{- 27}$ kg/m³. The standard model makes up the difference with dark energy of constant energy density $ε_{Λ}$ . Find $ε_{Λ}$ , and the redshift at which the density of matter, as energy, equalled that of dark energy. Ignore radiation.

I solved it without the solution

Answer $ε_{Λ} = 5.75 \times 10^{- 10}$ J/m³, and the two were equal at $z = 0.32$ .