17.4 The Anisotropies of the CMB
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17.2 Show that the limit of (17.13) for small values of the scale factor is the same as
(17.12) for a pure radiation filled universe. Show that the limit for large values
of the scale factor is the same as (17.10) the pure matter filled universe.
17.3 What is the kinetic energy of an electron in thermal equilibrium with radiation
at the time of decoupling? What is its velocity as a fraction of the speed of
light? Is the gas of electrons hot or cold?
17.4 Estimate the time of decoupling as we did in Sect. 17.2 but use the assumption
that the scale factor is that for matter given in (17.10). Repeat for the pure
radiation scale factor in (17.12). Compare the results.
17.5 Repeat the analysis of Sect. 17.3 if the scale factor of the very early universe
is that of flat de Sitter space with zero curvature. Note that the universe for this
case could begin at any negative time. Obtain the analog of (17.19). Is there a
horizon puzzle for this choice of scale factor? We will return to this problem
in Chap. 19.
17.6 Repeat Exercise 17.5 for the cases of positive and negative curvature parameter
k. Is there a horizon puzzle for these cases?
17.7 Show that features in the CMB with an angular size of about θ will show up
in the power spectrum at values of about = π//θ.
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17.2 Show that the limit of (17.13) for small values of the scale factor is the same as
(17.12) for a pure radiation filled universe. Show that the limit for large values
of the scale factor is the same as (17.10) the pure matter filled universe.
17.3 What is the kinetic energy of an electron in thermal equilibrium with radiation
at the time of decoupling? What is its velocity as a fraction of the speed of
light? Is the gas of electrons hot or cold?
17.4 Estimate the time of decoupling as we did in Sect. 17.2 but use the assumption
that the scale factor is that for matter given in (17.10). Repeat for the pure
radiation scale factor in (17.12). Compare the results.
17.5 Repeat the analysis of Sect. 17.3 if the scale factor of the very early universe
is that of flat de Sitter space with zero curvature. Note that the universe for this
case could begin at any negative time. Obtain the analog of (17.19). Is there a
horizon puzzle for this choice of scale factor? We will return to this problem
in Chap. 19.
17.6 Repeat Exercise 17.5 for the cases of positive and negative curvature parameter
k. Is there a horizon puzzle for these cases?
17.7 Show that features in the CMB with an angular size of about θ will show up
in the power spectrum at values of about = π//θ.