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 Incoming links: Curvature-dominated universe

Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 310 / 1 / b / Solution Created 2026-10-03 Updated 2026-10-06

Insert the component-density laws into the Friedmann equation, using

ρ_{i, 0} = 3 H_{0}^{2} Ω_{i, 0} / (8 π G)

and

a_{0} = 1

. Multiplying by

a^{2}

gives

\overset{a}{˙}^{2} = H_{0}^{2} (\frac{Ω _{r, 0}}{a ^{2}} + \frac{Ω _{m, 0}}{a} + Ω_{Λ, 0} a^{2}) - K .

The radiation in cosmology and pressureless matter contributions add; the printed first formula has lost the plus sign between them. In the Friedmann acceleration equation,

ρ + 3 P = 2 ρ_{r} + ρ_{m} - 2 ρ_{Λ}

, so

\overset{a}{¨} = - H_{0}^{2} (\frac{Ω _{r, 0}}{a ^{3}} + \frac{Ω _{m, 0}}{2 a ^{2}} - Ω_{Λ, 0} a) .

Evaluating the first equation today gives

K = H_{0}^{2} (Ω_{r, 0} + Ω_{m, 0} + Ω_{Λ, 0} - 1)

.

Now remove dark energy and assume a positive pressureless matter or radiation in cosmology density, starting on an expanding branch. Let

f (a) = H_{0}^{2} (Ω_{r, 0} / a^{2} + Ω_{m, 0} / a)

. It decreases from infinity to zero, and

\overset{a}{˙}^{2} = f (a) - K

, while

\overset{a}{¨} < 0

. This immediately distinguishes all three cases of spatial curvature of an FLRW universe.

Closed, $K > 0$ . There is a unique maximum size $a_{m a x}$ with $f (a_{m a x}) = K$ . Expansion from a Big Bang slows to $\overset{a}{˙} = 0$ and then becomes contraction, ending in a Big Crunch. The turning point is reached in finite time because $f (a) - K$ has a simple zero there. The acceleration remains negative at the maximum, so it is a recollapse rather than a static state.
Open, $K < 0$ . The expanding solution never stops because $\overset{a}{˙}^{2} = f (a) + ∣ K ∣ > 0$ . It expands forever, eventually approaching a curvature-dominated universe with $\overset{a}{˙} \to ∣ K ∣$ and $a (t) \sim ∣ K ∣ t$ . Locally its late-time behavior approaches the empty Milne universe.
Spatially flat, $K = 0$ . Expansion also continues forever, but $\overset{a}{˙} \to 0$ instead of approaching a nonzero constant. With pressureless matter present, the late-time solution has $a \propto t^{2/3}$ , the Einstein-de Sitter universe behavior; a flat solution containing only radiation in cosmology has $a \propto t^{1/2}$ . With both components, an early radiation domination era gives way to matter domination.

The early small-

a

solution has

a \propto t^{1/2}

if radiation in cosmology is present, and

a \propto t^{2/3}

for pressureless matter alone. The figure isolates the effect of curvature in three matter-only examples, rather than comparing models with the same present density parameters.

Figure 1.
Matter-only scale-factor evolution: closed recollapse, flat power-law expansion, and open expansion tending to a constant speed.

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 310 / 1 / c / Solution Created 2026-10-03 Updated 2026-10-06

For the open matter-dominated Friedmann solution, write

Ω = Ω_{m, 0}

, so

0 < Ω < 1

. The present Friedmann equation gives

- K = H_{0}^{2} (1 - Ω)

. With

A = H_{0}^{2} Ω

and

B = H_{0}^{2} (1 - Ω) > 0

, the expanding equation becomes

\overset{a}{˙}^{2} = \frac{A}{a} + B, d t = \frac{1}{B} \frac{a}{a + A / B} d a .

Choose the hyperbolic substitution

a = \frac{A}{B} sinh^{2} \frac{η}{2} = \frac{A}{2 B} (cosh η - 1), η \geq 0.

Then

d a = (A / (2 B)) sinh η d η

and

a / (a + A / B) = tanh (η /2)

. Their product gives

\frac{d t}{d η} = \frac{A}{2 B ^{3/2}} (cosh η - 1) = \frac{a}{B} .

Take the Big Bang to be

t = 0

,

η = 0

, and integrate. The parametric solution is

a (η) = \frac{Ω}{2 ( 1 - Ω )} (cosh η - 1), t (η) = \frac{Ω}{2 H _{0} ( 1 - Ω ) ^{3/2}} (sinh η - η) .

The parameter is determined explicitly by

η = 2 arsinh \frac{( 1 - Ω ) a}{Ω}, η = - K \int_{0}^{t} \frac{d t ^{'}}{a ( t ^{'} )} .

Thus

η

is conformal time multiplied by

- K

, with its origin at the Big Bang; it is not equal to unscaled cosmic time. In particular today

η_{0} = arcosh (2/Ω - 1)

. Small

η

gives

a \propto η^{2}

and

t \propto η^{3}

, hence

a \propto t^{2/3}

. Large

η

gives

a / t \to H_{0} 1 - Ω = - K

, in agreement with the curvature-dominated universe limit.

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Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 14D / iii / Solution Created 2026-09-24 Updated 2026-09-29

Under the stated approximation the perturbation equation is

\ddot{δ} + 2 H \dot{δ} = 0, \dot{δ} \propto a^{- 2} .

During radiation domination,

a \propto t^{1/2}

, so

\dot{δ} \propto t^{- 1}

and

δ = C_{1} + C_{2} lo g t = C_{1} + 2 C_{2} lo g a .

Matter perturbations therefore have only logarithmic growth of matter perturbations during radiation domination.

In a curvature-dominated universe,

a \propto t

, so

\dot{δ} \propto t^{- 2}

and

δ = C_{1} + C_{2} t^{- 1} = C_{1} + C_{2} a^{- 1} .

The nondecaying mode is constant: curvature domination gives freezing of matter perturbations during curvature domination, rather than continued growth.

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