# An Introduction to Modern Cosmology, 2nd edition: errata

If you have the revised second edition (from the December 2008 reprinting onwards, identified by having 6 Advanced Topic chapters rather than 5), the only known errors are:

Page 59, 3 lines after Eq8.6: For the quoted parameter values the age should be given as approximately thirteen billion years (not fourteen). To get the commonly-quoted fourteen billion years, the matter density has to be taken a little smaller.

Page 80, line 13: The bracket should refer to Problem 10.6, not 10.5.

Page 109: The formula given in Problem 13.3 differs slightly from the version given as Equation 11.12; ideally they would have matched. The latter is more accurate, but for the purposes of the Problem the numerical difference is irrelevant.

Page 171: The answer to Problem 12.2 should be given as Y4 = 0.04 (I changed tnuc from the earlier editions and forgot to correct this).

Errata for the original second edition (pre December 2008 reprinting):

In addition to those listed above, the known errors in the original printing are as follows

Page 16: The answer to Problem 2.5 seems to be wrong -- see item for Page 163.

Page 17: The last sentence of the second paragraph would be clearer if it said something like `Because a force on an object induces an acceleration which is also proportional to its mass ...'

Page 22: After Eq3.11, the gas should be in a `cylinder', not a `piston'. Before Eq3.13, it is stated that the chain rule is used for its derivation; while the chain rule is used for the first term, the main rule being used is the product rule for derivatives. After Eq3.13 it should be *rate of* change of volume, and after Eq3.14 the expansion could be additionally described as adiabatic.

Page 38: I'm not at all sure `diminishment' is a word in modern English (though the OED does recognize it). `Diminution' would be better.

Page 74: Note the answer to the first part of Problem 9.1 given on page 164 is wrong (see below).

Page 76: The third last sentence on this page should be replaced by `The second is because the f3 on the numerator scales as inverse volume, corresponding to the evolution of the photon number density as the Universe expands.' since it doesn't make much sense as it stands.

Page 81:  In the paragraph following Eq10.17, the second sentence is wrong. The exponential clearly doesn't dominate, since in that limit is simply equals one. However the Saha equation only makes sense if the temperature is below the electron mass energy (to avoid electron-positron pair creation), and that coupled with the small baryon-to-photon ratio is enough to ensure that the right-hand side of the Saha equation is small at relevant temperatures well above the binding energy.

Page 87: The numerical values given in Eq11.11 are both incorrect and specified to too low an accuracy. Rather than substituting the values of the temperature and time at equality (which is approximate and creates an illusory dependence on Omega0 and h), it is better to immediately use Eq11.12. It gives the constants of proportionality in Eq11.11 as 1.3x1010 K and 1.1 MeV. These constants are valid only for energies less than about 1MeV; at higher energies electrons and positrons would be relativistic and contribute to the energy density of the radiation bath. See Kolb and Turner section 3.3 for a much more detailed analysis. This correction has implications for the nucleosynthesis calculation detailed in the item for page 93.

Page 91: It would be better if the first sentence of Section 12.1 said ` ... helium-4, the most stable of the light nuclei, was formed ...'

Page 93: In Eq12.6, the `D' should not be in italics.

Page 93: Eq12.9 uses the temperature-time relation of Eq11.11 to compute the effect of neutron decays on the neutron-to-proton ratio. The erratum for page 87 corrects that relation, which changes the time at which this ratio is to be evaluated. Tracked through the calculation, this significantly increases the final value of Y4 to well above the observed value. A suitable compensation would take the build-up of the nuclei to happen at 0.06 MeV rather than 0.1 MeV, leading eventually to Y4 = 0.24. Tuning the build-up energy to get the right answer is really rather a fiddle though; a full calculation of the deuterium abundance should include far more physics than this simple estimate (eg see Kolb and Turner Chapter 4).

Page 111: In Eq14.1, the strong energy condition would be better written with a `greater than or equals' sign, rather than just `greater than'.

Page 116: Missing word: `fourteen billion *years* old'.

Page 122: The claim that a torus has a flat geometry is oversimplistic. It is possible to have a torus with a flat geometry, most easily realised by identifying the edges of a cube, and this `flat torus' is a valid model for the Universe. However bending a two-dimensional torus to embed it in 3D as a bagel will distort it meaning it is no longer flat. It also no longer has constant curvature and such an embedded torus is not a good model for the Universe. For more details on topology see Janna Levin's review article.

Page 139: A more accurate version of EqsA3.3 and A3.4 would have 94 on the denominator rather than 90. The difference arises because the mean thermal energy of a fermion in a thermal distribution is actually 3.15 kBT, rather than the 3kBT I had taken.

Page 140: The Boltzmann suppression factor in A3.2.2 has been inadvertently written the wrong way up. It should be
exp(-mnuc2/kBT). This discussion is also oversimplistic in implying that the exponential is all that is needed; a full calculation needs to account for the breakdown of thermal equilibrium once the particle density becomes too low, which cuts off the exponential decrease in a process known as `freeze-out'. This is discussed for instance in Kolb and Turner's book "The Early Universe".

Page 141: There is a typo on the 6th line of Problem A3.1; the word `put' should be replaced with `by'.

Page 158: There is a typo in the first line of Problem A5.2, with the phrase `the Universe' duplicated.

Page 163: The answer to Problem 2.5 blithely states that the peak of the frequency spectrum of the Sun will `unsurprisingly' lie in the visible part of the electromagnetic spectrum. After all, everyone knows Wien's displacement law placing the wavelength  at 500nm. However, because of the non-linear transformation between wavelength and frequency, the peaks don't actually satisfy the relation f = c/lambda, and when expressed in frequency the maximum in fact lies slightly into the infrared.

Page 164: The first answer to 9.1 should be 28h2 (or maybe 29h2 depending on the rounding used for the Boltzmann constant). Anyway, not 31h2 as given. This changes the upper limit on the Hubble constant to h<0.60. Note also that the upper limit on the neutrino mass has strengthened considerably as compared to the value used in this question, and is now below 1eV.

Page 165: In the answer to A1.1, the sinh should not be squared. The square root inside the sinh covers the |k| but not the xi. The numerical answer is correct.

If you've spotted something else, why not tell me at a.liddle@sussex.ac.uk?