Exercises
27
determines 71 once the helium abundance has been measured. However. since
Yp (4He) depends only logarithmically on 71. as is apparent from (1.158). this does
not yield a very precise value for 7' /. In contrast. when similar techniques are
applied to the (much smaller) primordial abundances of the other light nuclei.
specifically D. 3He. and 7Li. the dependence upon 7' / becomes a power. It is a
testament to the success of the standard cosmological model that a single value
of 71 simultaneously fits the data on all primordial abundances and allows a much
more precise determination of the parameter. The value (1.129) of the parameter
that we use in chapter 4 is obtained from a simultaneous fit of the cosmological
parameters to all of these and other data. including the recent WMAP microwave
anisotropy data ll].
1.12 Exercises
I. Show that the radial path from coordinate (r. (J. r/J) at t = 0 to coordinate
(0. (J. r/J) at time t is a geodesic.
2. Verify that the Ricci tensor for the Robertson-Walker metric (1.1) has the
non-zero components given in (1.30).
3. Verify that the IL = 0 component of (1.43) gives (1.45). What information is
provided by the IL = i components?
4. For a flat FRW universe with pressureless matter and a negative cosmological
constant A. show that the universe collapses to a point singularity in a time
21f
T = J3IAI.
5. Show that the solution of the balance equation (1.141) for the relative neutron
abundance has the form (1.148) with the integrating factor as given in
(1.150). Using the approximation (1.147). verify that the function K(y) is
given by (1.151).
1.13 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Kolb E Wand Turner M S 1990 The Early Universe (Reading. MA:
Addison-Wesley)
• Weinberg S 1972 Gravitation and Cosmology: Principles and Applications
of the General Theory of Relativity (New York: WHey)
• Weinberg S 1989 Rev. Mod. Phys. 61 I
• Sarkar S 1996 Rep. Prog. Phys. 59 1493. arXiv:hep-phl9602260
27
determines 71 once the helium abundance has been measured. However. since
Yp (4He) depends only logarithmically on 71. as is apparent from (1.158). this does
not yield a very precise value for 7' /. In contrast. when similar techniques are
applied to the (much smaller) primordial abundances of the other light nuclei.
specifically D. 3He. and 7Li. the dependence upon 7' / becomes a power. It is a
testament to the success of the standard cosmological model that a single value
of 71 simultaneously fits the data on all primordial abundances and allows a much
more precise determination of the parameter. The value (1.129) of the parameter
that we use in chapter 4 is obtained from a simultaneous fit of the cosmological
parameters to all of these and other data. including the recent WMAP microwave
anisotropy data ll].
1.12 Exercises
I. Show that the radial path from coordinate (r. (J. r/J) at t = 0 to coordinate
(0. (J. r/J) at time t is a geodesic.
2. Verify that the Ricci tensor for the Robertson-Walker metric (1.1) has the
non-zero components given in (1.30).
3. Verify that the IL = 0 component of (1.43) gives (1.45). What information is
provided by the IL = i components?
4. For a flat FRW universe with pressureless matter and a negative cosmological
constant A. show that the universe collapses to a point singularity in a time
21f
T = J3IAI.
5. Show that the solution of the balance equation (1.141) for the relative neutron
abundance has the form (1.148) with the integrating factor as given in
(1.150). Using the approximation (1.147). verify that the function K(y) is
given by (1.151).
1.13 General references
The books and review articles that we have found most useful in preparing this
chapter are:
• Kolb E Wand Turner M S 1990 The Early Universe (Reading. MA:
Addison-Wesley)
• Weinberg S 1972 Gravitation and Cosmology: Principles and Applications
of the General Theory of Relativity (New York: WHey)
• Weinberg S 1989 Rev. Mod. Phys. 61 I
• Sarkar S 1996 Rep. Prog. Phys. 59 1493. arXiv:hep-phl9602260