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Exercises for Cosmology

Prof. Dr. P. Schneider & Dr. Hendrik Hildebrandt

Homework 2 (27th Oct - 31st Oct 2014)

1. Quickies

(a) Write down the relation between redshiftzand the scale factor a!

(b) Explain why there is no unique distance measure in Cosmology!

(c) What is the difference between proper and comoving volumes? What is thecomoving volume of a redshift shell at redshift zwith (redshift) thickness dzaround us?

2. Elliptical coordinates

Let x1=r cos , x2 = f r sin .

(a) What are the curves of constant r? What are curves of constant ?

(b) If ds2 = dx21+ dx2

2, what is ds2 in terms ofr and ?

3. Angular diameter distance on the sphere

(a) Write down the metric on a three-dimensional sphere in spherical polar coordi-nates.

(b) What is the distanceL between two great circles that intersect at the poles atan angle of (i.e. Lis measured along lines of constant latitude), as functionof the distance to the poles (measured along lines of constant longitude)?

4. Age of the Universe at its milestones

Use equation 4.26 to find the age of the Universe...

(a) when matter and radiation had the same energy density, and

(b) when recombination occurred (i.e. at redshift z 1100).

For this, take the Hubble parameter to beh = 0.72 and the cosmic density parameterfor matter in the present to be m= 0.3.

5. Derivation of the Mattig relation

The Mattig relation (equation 4.34) gives the angular-diameter distance as a functionof redshift for a Universe with = 0. We talk through the derivation here, with a

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couple of easy questions at the end. Feel free to do the algebra in between the stepswe outline here!

Although not the most direct derivation, this method provides, as a side product,a differential equation for the angular-diameter distance D(z) for all values of thedensity parameters. We start by noting thatD = fk/(1 +z) (eqn 4.16), where fkis the comoving angular-diameter distance, which satisfies the differential equationd2fk/d

2 = K fk (see eqn 4.2). Well simplify the notation by writing

H(z) =H0H(z), with H(z) =

m(1 +z)3 + (1 m )(1 +z)2 + .

One can show thatdD

dz

= 1

(1 +z)

c

H0H

dfk

d

D .By taking a further derivative, we arrive at

d2D

dz2 =

2

1 +z+H

H

c

H0H (1 +z)

dfk

d +

2

(1 +z)2

m+ 1

H2

D ,

where H = dH/dz.

Finally by combining the two preceding equations, it is possible derive the second-order differential equation for D,

d2D

dz2 + 2

1 +z+1 +z

2H2 [3m(1 +z) + 2(1 m )]dD

dz +3(1 +z)

2H2 mD= 0.

(a) The angular-diameter distance that we use in practice, D(z), is a particularsolution of this differential equation. State the corresponding boundary condi-tions which fix this solution. What would the boundary conditions be for theangular-diameter distance D(z1, z2) for a source at redshift z2 as seen by anobserver at redshift 0 z1 < z2?

(You may then want to try solving the equation in the case of = 0, yieldingthe Mattig relation.)

(b) Given the Mattig relation itself, show that it reduces to the expression in equa-tion 4.25 in the case of an Einsteinde Sitter universe.

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