The graph shows the variation with time t of the cosmic scale factor R in the flat model of the universe in which dark energy is ignored.

On the axes above draw a graph to show the variation of R with time, when dark energy is present.

Suggest how fluctuations in the cosmic microwave background (CMB) radiation are linked to the observation that galaxies collide.

At critical density there is zero total energy. Show that the critical density of the universe is: ${\text{r}}_c=\frac{3H_0^2}{8\pi G}$.

Show that the temperature of the universe is inversely proportional to the cosmic scale factor.

Curve A shows the actual rotation curve of a nearby galaxy. Curve B shows the predicted rotation curve based on the visible stars in the galaxy.

Explain how curve A provides evidence for dark matter.

The mass of visible matter in the galaxy is M.

Show that for stars where r > R₀ the velocity of orbit is v = $\sqrt{\frac{GM}{r}}$.

Describe what is meant by dark matter.

The density of the observable matter in the universe is only 0.05 ρ_c. Suggest how the remaining 0.27 ρ_c is accounted for.

Compare and contrast, the variation with time of the temperature of the cosmic background (CMB) radiation, for the two models from the present time onward.

Explain the evidence that indicates the location of dark matter in galaxies.

The density of dark energy is ρ_Λc² where ρ_Λ = ρ_c – ρ_m. Calculate the amount of dark energy in 1 m³ of space.

Calculate the rotation velocity of stars 4.0 kpc from the centre of the galaxy. The average density of the galaxy is 5.0 × 10^–21 kg m^–3.

Calculate the density of matter in the universe, using the Hubble constant 70 km s^–1Mpc^–1.

State how the anisotropies in the CMB distribution are interpreted.

Show that the critical density of the universe is

$$\frac{3H^2}{8\pi G}$$

where H is the Hubble parameter and G is the gravitational constant.

Identify two possible causes of the anisotropies in (a).

Draw on the axes the observed variation with r of the orbital speed v of stars in a galaxy.

Derive, using the concept of the cosmological origin of redshift, the relation

T $\propto \frac{1}{R}$

between the temperature T of the cosmic microwave background (CMB) radiation and the cosmic scale factor R.

The present temperature of the CMB is 2.8 K. This radiation was emitted when the universe was smaller by a factor of 1100. Estimate the temperature of the CMB at the time of its emission.

Justify that the total energy of this particle is $E=\frac{1}{2}m{v}^2-\frac{4}{3}\pi G\text{r}{r}^{2}m$.

Outline why a hypothesis of dark energy has been developed.

The distribution of mass in a spherical system is such that the density ρ varies with distance r from the centre as

ρ = $\frac{k}{r^2}$

where k is a constant.

Show that the rotation curve of this system is described by

v = constant.

Explain, using the equation in (a) and the graphs, why the presence of visible matter alone cannot account for the velocity of stars when r > R₀.