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13 Cosmological Preliminaries
13.2 Look up the nature and size of the great walls and voids in the matter distribution of the universe. Compare to the cosmological scale of about 10 billion
light years.
13.3 In Euclidean plane geometry parallel lines never meet, and only parallel lines
never meet. What is the analog of this statement for the surface of a sphere
and a pseudosphere?
13.4 Calculate the Riemann scalar for the surface of a sphere and a plane and a
pseudosphere. How is it related to the parameters R and k.
13.5 Consider the metric on a sphere in (13.4) using cylindrical coordinates. Transform to the usual spherical coordinates using ρ = R sin θ and get the more
standard form
d
2
= R
2
(dθ
2
+ sin
2
θ dϕ
2
).
13.6 Let us do the analog of Exercise 13.5 in 3 dimensions. Consider the 3-sphere
metric in (13.8). Introduce a hyperspherical angle ψ defined by r = R sin ψ,
and show that the metric becomes
d
2
= R
2
dψ
2
+ sin
2
ψ
dθ
2
+ sin
2
θ dϕ
2
.
This is useful for many geometric calculations. Can you construct the metric
for a 4-sphere in a similar way? Do you see the pattern?
13.7 Calculate the circumference of the hyperspherical universe.
13.8 Calculate the total volume of the hyperspherical universe.
13.9 For the pseudo-hypersphere, k = −1, there is a singularity in the metric
(13.11) at u = 2. Discuss briefly the behavior of the metric and the space
there.
13.10 Consider a de Sitter universe with curvature parameter k = 0 and Hubble
constant H 0 . If a galaxy has a redshift of z how far away is it? What numbers
do you get for a Hubble time of 14 billion years and a redshift of z = 2? Is
there any upper limit to the redshift and the distance of the galaxy?
13.11 In the text we discussed the cosmological metric as the 3-dimensional generalization of the plane and sphere and pseudosphere. It is possible to also
construct other 2-spaces that are homogeneous and isotropic but topologically more complex. For example, consider a flat square torus; it is constructed
by identifying the opposite sides of an ordinary square. Simply glue them
together! This space is clearly locally homogeneous and finite. Can you do
an analogous construction for the surface of a sphere and pseudosphere?
13.12 Suppose the space part of the metric of the universe is in fact a cubic torus,
the three dimensional analog of the square torus in Exercise 13.11. What
observations could you make to test this idea? Can you think of any problems
with such a theoretical speculation?
13.13 In the text we related the radial marker r used in (13.12b) to the physical
distance . Do the same for the radial marker w used in (13.12a).
13 Cosmological Preliminaries
13.2 Look up the nature and size of the great walls and voids in the matter distribution of the universe. Compare to the cosmological scale of about 10 billion
light years.
13.3 In Euclidean plane geometry parallel lines never meet, and only parallel lines
never meet. What is the analog of this statement for the surface of a sphere
and a pseudosphere?
13.4 Calculate the Riemann scalar for the surface of a sphere and a plane and a
pseudosphere. How is it related to the parameters R and k.
13.5 Consider the metric on a sphere in (13.4) using cylindrical coordinates. Transform to the usual spherical coordinates using ρ = R sin θ and get the more
standard form
d
2
= R
2
(dθ
2
+ sin
2
θ dϕ
2
).
13.6 Let us do the analog of Exercise 13.5 in 3 dimensions. Consider the 3-sphere
metric in (13.8). Introduce a hyperspherical angle ψ defined by r = R sin ψ,
and show that the metric becomes
d
2
= R
2
dψ
2
+ sin
2
ψ
dθ
2
+ sin
2
θ dϕ
2
.
This is useful for many geometric calculations. Can you construct the metric
for a 4-sphere in a similar way? Do you see the pattern?
13.7 Calculate the circumference of the hyperspherical universe.
13.8 Calculate the total volume of the hyperspherical universe.
13.9 For the pseudo-hypersphere, k = −1, there is a singularity in the metric
(13.11) at u = 2. Discuss briefly the behavior of the metric and the space
there.
13.10 Consider a de Sitter universe with curvature parameter k = 0 and Hubble
constant H 0 . If a galaxy has a redshift of z how far away is it? What numbers
do you get for a Hubble time of 14 billion years and a redshift of z = 2? Is
there any upper limit to the redshift and the distance of the galaxy?
13.11 In the text we discussed the cosmological metric as the 3-dimensional generalization of the plane and sphere and pseudosphere. It is possible to also
construct other 2-spaces that are homogeneous and isotropic but topologically more complex. For example, consider a flat square torus; it is constructed
by identifying the opposite sides of an ordinary square. Simply glue them
together! This space is clearly locally homogeneous and finite. Can you do
an analogous construction for the surface of a sphere and pseudosphere?
13.12 Suppose the space part of the metric of the universe is in fact a cubic torus,
the three dimensional analog of the square torus in Exercise 13.11. What
observations could you make to test this idea? Can you think of any problems
with such a theoretical speculation?
13.13 In the text we related the radial marker r used in (13.12b) to the physical
distance . Do the same for the radial marker w used in (13.12a).