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Cosmology
The word Universe refers to all existing regions of space.
Cosmology is the study of the properties of the Universe.
Cosmology includes the study of how the Universe's properties change with time.
Observed Properties of the Universe
1. The Sky i

ASSIGNMENT 1 solutions
Note: this assignment will have an extra low weight in the nal mark, unless you do it perfectly !
Problem I: In the homogenenous and isotropic Universe which evolution is described by the scale factor
a() and the conformal time is u

ASSIGNMENT 6, due date April 17th, 2013
1. Let us return to the spherical collapse problem of the Assignment 5. To remind you, we have
considered matter-dominated at cosmology with H0 = 67.3km/s/M pc. and the level of linear
inhomogeneities at the present

ASSIGNMENT 4, solutions
2 2
1
Let us consider inationary model with scalar eld described by the largrangian 2 , , 1 m 2c 2 . Here
2
h
I have restored the dimensional units for the potential in order the mass parameter to have units of
kilograms. (One may

Solutions to ASSIGNMENT 5
1. Matter perturbations in the Universe dominated by another component that cannot cluster.
One of the forms of the equation for perturbations for matter that we have derived is
a
m + 2 m 4Gm m = 0
a
(1)
One can show that when m

ASSIGNMENT 3, solutions
Problem I Consider positively curved Universe with either matter or radiation (look at both cases). When scale
factor is zero at t = 0, the ray of light is emitted. How many times will it circumnavigate the space
by the moment of r

Midterm Exam
The midterm is worth 20% of the total mark. All Problems are worth (20/3)%.
Problem I Enstein static Universe is the model of the homogenous and isotropic Universe with matter
and Lambda term values arranged in such way that the Universe does

Solutions to Midterm Exam problems
Problem I If matter density = 41027 kg/m3 , what is the curvature radius R0 of the space in Einstein
static Universe ? How long it will take a photon to circumnavigate such a universe ?
Solution: Einstein static universe

ASSIGNMENT 5, due date March 30st, 2012
1. We have mentioned that the inationary model with V () = 1 m2 2 is a model of chaotic ination.
2
Let us investigate what is the origin of this name
Solution: Let us rst write out some information about the slow ro

ASSIGNMENT 3, solutions
Problem I In a single component at universe with the equation of state given by parameter w nd several
distance measures.
Solution: The expansion law of such Universe is given by (here I use dimensional time units)
1
2
H0
a
a
2
= a

ASSIGNMENT 1 solutions
Problem I: In the homogenenous and isotropic Universe which evolution is described by the scale factor
a() and the conformal time is used
2
ds2 = a2 () d 2 dr2 Sk (r)d2
(1)
rederive the relation between the scale factors of emission

Assignment 4, due March 14th
Problem I We discussed the nucleosynthesis from the modern perspective where we now the properties
of CMB. However, the theory of hot Universe as the mechanism for creating of heavy elements was
proposed by George Gamow when C

ASSIGNMENT 2 solutions
The problem
In this problem you solve for evolution of the scale factor in the spatially at universe lled by
radiation and matter. So we consider Friedman equation
a
a
2
8G
3c2
=
m0 r0
+ 4
a3
a
(1)
Analytical solutions exists if one

ASSIGNMENT 6, due date April 11th, 2014
1. We are considering spherical collapse in the matter-dominated at cosmology with H0 = 67.3km/s/M pc.
and the level of linear inhomogeneities at the present time was set by (8h1 M pc) = 0.83. We also
suppose that 2

ASSIGNMENT 3, solutions
Problem I In a single component at universe with the equation of state given by parameter w nd several
distance measures.
Solution: The expansion law of such Universe is given by (here I use dimensional time units)
1
2
H0
a
a
2
= a