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Date	May 2017	Marks available	2	Reference code	17M.3.HL.TZ2.19
Level	Higher level	Paper	Paper 3	Time zone	2
Command term	Derive	Question number	19	Adapted from	N/A

Question

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.

[2]

a.i.

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.

[2]

a.ii.

State how the anisotropies in the CMB distribution are interpreted.

[1]

Markscheme

the cosmological origin of redshift implies that the wavelength is proportional to the scale factor: $\lambda$ $\propto$ R

combining this with Wien’s law $\lambda$ $\propto$ $\frac{1}{T}$

use of kT $\propto \frac{{hc}}{\lambda }$

«gives the result»

Evidence of correct algebra is needed as relationship T = $\frac{k}{R}$ is given.

[2 marks]

a.i.

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

= 2.8 x 1100 x 3080 ≈ 3100 «K»

[2 marks]

a.ii.

CMB anisotropies are related to fluctuations in density which are the cause for the formation of structures/nebulae/stars/galaxies

OWTTE

[1 mark]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

Syllabus sections

Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics) » D.5 – Further cosmology (HL only)

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17M.3.HL.TZ1.17a:
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.
16N.3.HL.TZ0.25b: Explain why the rotation curves are evidence for the existence of dark matter.
19M.3.HL.TZ2.22c: The accepted value for the Hubble constant is 2.3 × 10−18 s−1. Estimate the critical density...
17N.3.HL.TZ0.20b:
Suggest how fluctuations in the cosmic microwave background (CMB) radiation are linked to the observation that galaxies collide.
19M.3.HL.TZ2.22b:
At critical density there is zero total energy. Show that the critical density of the universe is: ${{\text{r}}_c} = \frac{{3H_0^2}}{{8\pi G}}$ .
18N.3.HL.TZ0.20a: Outline what is meant by dark energy.
18N.3.HL.TZ0.21a:
Show that the temperature of the universe is inversely proportional to the cosmic scale factor.
18N.3.HL.TZ0.20b: State two candidates for dark matter.
19N.3.HL.TZ0.16b(iii): It is estimated that less than 20 % of the matter in the universe is observable. Discuss how...
17M.3.HL.TZ2.20c:
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.

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18M.3.HL.TZ1.19a:
The mass of visible matter in the galaxy is M.

Show that for stars where r > R₀ the velocity of orbit is v = $\sqrt {\frac{{GM}}{r}}$ .
17M.3.HL.TZ2.20a:
Describe what is meant by dark matter.
17M.3.HL.TZ1.17b.i:
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.
19M.3.HL.TZ1.20b.ii:
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.
18M.3.HL.TZ2.19a:
Explain the evidence that indicates the location of dark matter in galaxies.
17M.3.HL.TZ1.17b.ii:
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.
19N.3.HL.TZ0.16b(i): State what is meant by the critical density.
16N.3.HL.TZ0.25a:
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.
18N.3.HL.TZ0.21b:
The present temperature of the cosmic microwave background (CMB) radiation is 3 K. Estimate the size of the universe relative to the present size of the universe when the temperature of the CMB was 300 K.
20N.3.HL.TZ0.21c: Hubble originally linked galactic redshift to a Doppler effect arising from galactic...
20N.3.HL.TZ0.24a: State the nature of the anisotropies observed in the CMB radiation.
19N.3.HL.TZ0.16b(ii):
Calculate the density of matter in the universe, using the Hubble constant 70 km s^–1Mpc^–1.
17M.3.HL.TZ2.19b:
State how the anisotropies in the CMB distribution are interpreted.
17N.3.HL.TZ0.20c:
Show that the critical density of the universe is

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

where H is the Hubble parameter and G is the gravitational constant.
20N.3.HL.TZ0.24b:
Identify two possible causes of the anisotropies in (a).
18M.3.HL.TZ1.19b:
Draw on the axes the observed variation with r of the orbital speed v of stars in a galaxy.
17M.3.HL.TZ2.19a.ii:
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.
19M.3.HL.TZ2.22a:
Justify that the total energy of this particle is $E = \frac{1}{2}m{v^2} - \frac{4}{3}\pi G{\text{r}}{r^2}m$ .
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18M.3.HL.TZ2.19b:
Outline why a hypothesis of dark energy has been developed.
17M.3.HL.TZ2.20b:
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.
18M.3.HL.TZ1.19c:
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.
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Option D: Astrophysics » Option D: Astrophysics (Additional higher level option topics)

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