# Problem Set #2 Popular cosmological models have Ωm

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Ast 8110 (LSS & GL)
Problem Set #2
1. Popular cosmological models have Ωm + ΩΛ = 1, i.e., curvature is zero, and the contribution
of radiation is negligible in today’s Universe. For these models the angular diameter distance
between the observer and source at z reduces to a simpler expression,
DA(z) = c
H0
a(z) χ =
c
H0
1
(1 + z)
Z z
0
dz
[Ωm(1 + z)
3 + ΩΛ]
1/2
(1)
Angular diameter distance between lens at zl and source at zs is
DA(zl
, zs) = c
H0
a(zs) χls =
c
H0
1
(1 + zs)
Z zs
zl
dz
[Ωm(1 + z)
3 + ΩΛ]
1/2
(2)
(a) Write a code to calculate DA, in units of c/H0, as a function of Ωm, ΩΛ, and z. (You don’t
need to turn in the source code.) Generate a curve of DA(z)/(c/H0) using Ωm = 0.3, ΩΛ = 0.7, for
observer at z = 0, and redshift ranging from just above 0, and up to z = 5.
(b) For a fixed lens redshift, zl = 0.4, calculate and plot how Σcrit varies with zs, for zs values
between 0.41 and 7. Use units of g/cm2
, and this value, c
2/(4πG)/(c/H0) = 0.115 g/cm2
.
(c) For a fixed source redshift, zs = 2, calculate and plot how Σcrit varies with zl
. From your
plot, or the data generated by your code, determine zl at whch Σcrit attains a minimum. What
is that value of Σcrit? This zl
is where a lens of a fixed g/cm2
surface mass density will be most
effective as a lens for source at zs = 2, typical of lensed sources.
(d) For a fixed source redshift, zs = 1000, calculate and plot how Σcrit varies with zl
. From your
plot, or the data generated by your code, determine zl at which Σcrit attains a minimum. What
is that value of Σcrit? This zl
is where a lens of a fixed g/cm2
surface mass density, will be most
effective as a lens for the Cosmic Microwave Background as the source.

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