Cosmological inflation and large-scale structure: errata

Here are the errors we know about; if you spot anything that's not here, we'd appreciate an e-mail to

Page 21: Although not a typo, you might stress that the rhoeq in Eqs 2.42 and 2.44 is the density of each component, rather than the total, by underlining the word `each' two lines after Eq2.42.

Page 22:  In Eq2.45, the central and right-hand terms should both be multiplied by a03. This is required to correctly allow for the inability to set a = 1 after instead deciding to set K = -1. The parameter psi in Eqs2.46 and 2.47 actually coincides with conformal time tau.

Page 24: In Eq2.54 the final quantity, dphi, should be squared.

Page 46: In Eq3.19 the sign of the last term on the right-hand side should be positive. See this paper for a much more detailed treatment of the e-foldings relation.

Page 47: In mentioning eternal inflation, the citations should reflect that the idea originated in a 1983 paper by Alex Vilenkin, "Birth of inflationary universes", Phys. Rev. D27, 2848 (1983), who discussed the scenario in the context of `new inflation'. The Linde papers cited extended the picture to chaotic inflation.

Page 49: Last paragraph of 3.5.1, the formula should be w=2/3p - 1, and refer to Eq2.38 rather than Eq2.7.

Page 57: for clarity, Example 3.6 should say `for the slow-roll approximations Eqs3.7 and 3.8 to be valid

Page 66: The second line of Eq4.21 has got jumbled up. The term inside the square brackets should read
   Sum over n of [ ln Nn - (Rn2 + In2)/2sigman2 ]

Page 69: In Eqs4.37 and 4.38, the Ylm(k) should be complex conjugates (note, not the Ylm(x) in the former equation).

Page 79: Three lines after Eq4.77, the equation for Phigr should contain an additional factor G. Likewise, the first term on the right-hand side of Eq4.83 should also have an additional G, to that the numerator reads `2 pi G'.

Page 80: In Eq4.92, the factor Omega should be raised to the power -1. That is, for a fixed value of the potential perturbation, the density contrast should be greater if the density parameter is reduced. The correct version was used where this equation was needed later, notably in Section 6.2.1.

Page 82: In Eqs4.102 and 4.103, the prefactors a should be raised to the power -1  (in both the central and right-hand parts of the equalities). In Eqs 4.104 and 4.105, the powers of H should not be raised to the power -1.

Page 82: In Eq4.109, delta H should be just H.

Page 83: In the sentence between Eqs4.117 and 4.118, vk should be vksc.

Page 85: In Eq4.126, the right-hand side should be (1+z), not its reciprocal.

Page 92: Eqns 4.138 and 4.139 would be better written using a single function \rho(x,\tilde t), whose average (ie. unperturbed) value is \rho(\tilde t). Then they giving the splitting into perturbed and unperturbed values on the two slices as
     \rho(x, \tilde t(x,t) ) = \rho(t) + \delta \rho(x,t)
     \rho(x,\tilde t) = \rho(\tilde t) + \tilde{\delta \rho(x,\tilde t) }
With this notation, Eq4.141 can be derived by substituting Eq4.137 into Eq4.138, and then setting t=\tilde t so that we are dealing with slices corresponding to the same unperturbed value of \rho.

This direct argument is an alternative to the one in the text, invoking the interpretation of $\delta t$ as a time displacement between slices corresponding to the same unperturbed value of \rho (Figure 4.5). That interpretation indeed follows from Eq4.140, in which a final equality `=t(x,t)' would have been helpful.

Page 93: The Raychaudhuri equation, Eq4.146, is not correct. The first term on the right-hand side should be (rho + 3P) rather than just rho. Accordingly, the second sentence on page 94 should be modified to read "The Euler equation is the same as in the Newtonian case, except that the inertial mass ...". The correct version of the Raychaudhuri equation is used later on, e.g. Eq4.151.

Page 99: In Eq4.169, the first phi should be upper case, not lower case. Two lines above that, `Eq. (4.90)' should be replaced by `Eq. (4.91)'.

Page 99: Footnote 11: in the equation the delta P should have a plus sign, not a minus. The last sentence of the footnote would be much less obscure if it read `Eq (4.171) is obtained by inserting this result into the right-hand side of Eq (4.166), and for the left-hand side using Eq (7.81) which gives the time-dependence of  R_k.'.

Page 101: In Eq4.180, the second term on the left-hand side should be multiplied by 2.

Page 103: In the line after Eq4.187, `consider' to `considered'.

Page 107: In the first line of Eq5.9, where keq is defined, there should be additional brackets enclosing the phrase `14 ... Mpc', which is all to be raised to the power minus one (see Eq2.37).

Page 107: Two lines before Eq5.10, `matter-density contrast' should be just `matter density'.

Page 107-108: Contrary to the implication of the text, the logarithmic growth of the power spectrum within the horizon during radiation domination does not require the Meszaros equation to hold. The Meszaros equation however remains useful because (again contrary to the text) it remains valid in matter domination and should be derived using Eq2.42, not the approximation H=1/2t. For a fuller discussion click here.

Page 108: Not strictly an error, but the reader might wish to know that there is no physical motivation for the functional form of Eq5.14, which has arisen as a historical accident. The baryonic damping could only depend on the physical parameters at that epoch, and so in particular OmegaB couldn't really appear on its own.

Page 120: The equation two lines before Eq5.38 should read  theta simeq 2pi/l,  and Eq5.38 should read
      k-1  simeq  x/2pi   simeq  2/H0l   equals  6,000 h-1 l-1 Mpc
The 2pi factors cancel, leaving the correspondence between k and physical distance unchanged.
This change additionally means that the l corresponding to the thickness of the last-scattering surface should be approx 5000, and that for decoupling as approx 400. In each case, the identification of an l with a scale is imprecise and only for guidance.

Page 121: The last sentence in the figure caption should read `... is just proportional to the area under the curves.'

Page 127: In the first line of the paragraph containing Eq5.62, it should be `x-y plane', not `x-z plane'.

Page 132/133: The algebra in Eq5.88 is not quite right. First, it would be better if a factor of the speed of light c were written explicitly in the numerator of the central expression. Secondly, the 0.050 should be 0.061. This then leads to corrections in Eq5.89, where 0.033 should be 0.041, and in Eq5.90 where the 5 should be a 7. You might also note that the current fashion is for a higher nucleosynthesis value of OmegaBh2=0.02 which would lead to a yet higher number in Eq5.90 by a factor of 5/4. For both these reasons, the curves in Figure 5.12 could be shifted upwards.

Page 138: In Eq6.2, the numerical prefactor should be 2x105, not 6x104.

Page 144: In Eq6.10, there should be no Omega on the left-hand side, which should just read ag = ...
To make this clearer, you might consider modifying the two sentences before this equation to read
    "From Eq4.92, the time dependence of deltak is  g(aH)-2/Omega, which for a
     matter-dominated Universe is simply proportional to ag. Using the appropriately
     normalized Eq 4.104 determines g as"
Notice that there are corrections to both these quoted equations given earlier in this errata.

Page 146: We didn't mention explicitly that the horizon distance xls changes in the low-density case from the value 2H0 which applies in the critical density case (page 119). The horizon distance for LambdaCDM is given by Eq2.60; in particular this is what now appears in the generalization of Eq5.38. From this one can conclude that in the spatially-flat case the  values of the oscillatory features in the potential hardly change as Omega0 is varied, according to the argument at the end of Section 2.4, as their location is governed by the horizon size at decoupling.

Page 148: In Eq6.22, both instances of the scale factor a should be replaced by a/a0, as we have chosen to set the curvature K equal to minus one rather than the present scale factor to unity. Also, the minus sign between the two terms in the brackets should be a plus.

Page 148: In Eq6.27, the second term in the square bracket [the one with l(l+1)] should have a minus sign in front, not plus. This error seems to be quite widespread in the literature.

Page 149: In Eq6.33, the subscript f on the right-hand side should be g.

Page 150: To remove ambiguity, replace `growth suppression factor is larger' with `growth suppression is more pronounced'.

Page 151: In the caption, change Omega0h to Omegabh2.

Page 158: In the expression for the curvature scalar, the overall sign should be positive, not negative, with our conventions.

Page 159: In Eq6.49, to allow for the baryons the third equality sign should be a `simeq', and the equation in text after it ought to read
        delta rhor = - delta rhoc - delta rhob \simeq - delta rhoc
Also immediately after that, change `and the last equality holds' to `where the approximate equalities hold'.

Page 172: In Eq7.48, the argument of a-dagger should be -p rather than p. In this equation and in Eq7.49, w(p) should be replaced by w(p.t).

Page 174: In sec 7.3.2, fourth line, Z1should be Z2.

Page 177: In Eq7.64, the third term (Nabla2phi) should have a minus sign.

Page 177: After Eq7.65, in both the equations for the Ricci scalar R the overall sign should be positive with our conventions.

Page 178:  In Eq7.68, there should be no factor 1/2 in the last term, as is obvious from comparison with Eq7.67.

Page 180: After Eq7.78, add the phrase "where k1=a0H0 is the wavenumber corresponding to the present horizon."

Page 182: Eq7.92, the argument on the left-hand side should be tau, not t.

Page 187: Although not strictly a typo, it would have been useful to remind the reader that the COBE normalization of Eq7.106 is for critical density only, and would be modified for a low-density cosmology as described in Chapter 9.

Page 188: The mathematical expression ending the paragraph after Eq7.116, giving the `new quantity' should have an additional multiplication by V-n (for example to reproduce Eq7.116 when n=3).

Pages 188/189: There is an overall sign error in all of Eqs7.112, 7.113, 7.114, 7.118 and 7.123. This is most easily corrected by adding a minus sign to the left-hand sides. Additionally, in Eq7.114 the first term on the right-hand side should have a 4 rather than a 2.

Page 190: The left-hand side Eq7.128 should be multiplied by (1/z).

Page 191: The left-hand side Eq7.133 should be multiplied by (1/z).

Page 191: Eq7.138 should have eta-epsilon on the right-hand side rather than epsilon-eta.

Page 191: Eq7.139, replace `2C' with `3C'.

Page 192: In Eq7.146, the final denominator should contain phi-dot within the modulus signs, not H-dot.

Page 211: On the sixth line, there should be no `2' in the expression giving the contribution of the mass to eta.

Page 214: In Eq8.30, the phi(N) should be divided by MPl.

Page 217: Two lines after Eq8.44, the term lambda phi4/4 should be replaced by lambda-prime psi2phi2.

Page 219: In Eq8.53, the lambda should not be squared, while in Eq8.55 the square root of lambda should instead be the fourth root of lambda multiplied by the square root of 2.

Page 222: In Eq8.67, the factor g should be raised to the power one half.

Page 223: The right-hand side of Eq8.71 should be divided by MPl (so as to make both sides dimensionless).

Page 232: The second part of Eq8.101 should be xi=1/4omega, not 1/8omega.

Page 325: Eq14.48 is wrong. On the right-hand side the index mu should be replaced by beta, and then the rhs multiplied by a term (partial x^mu/partial x'^beta). Incidentally Eq14.46 could be simplified by cancelling the partial x'^alpha.

Page 326: Immediately above Eq14.56, there shoud be a minus sign in the equation for f, i.e. f = - dt/dtpr.

Page 328: Two lines after Eq14.67, the reference should be to Eq14.35, not Eq14.34.

Page 330: Concerning Eq14.77, although it is valid as it stands, it would make more logical sense if the subscript on the partial derivative were a mu rather than a nu, to mimic Eq14.76.

Page 334: In the line preceding Eq14.90, the curvature K should be raised to the power of minus a half.

Page 334: 2nd last line, the section referred to should be 2.3.1, not 2.2.3. On the last line, the equation should read R(3) = 6K/a2.

Page 335: In Eq14.92, the entire denominator of the last term, including the `1+', should be squared.

Page 335: In Eq14.95, there should be a factor 2 before the Bi.

Page 336: The left-hand side of Eq14.96 should be multiplied by 1/a(tau).

Page 336: The very last dt on the page should be dtpr.

Page 337: The right hand sides of Eqs 14.99 and 14.101 should be multiplied by -1. The left-hand sides of Eqs14.99 and 14.100 should be multiplied by a, and the left-hand sides of Eqs14.101 and 14.102 by 1/a.

Page 340: In the second line of Eq14.120, the second and third terms (the two terms of the form B partial delta x) should both have minus signs. In fact this error is also in Kodama and Sasaki (1984) and probably many other places too.

Page 341: In Eq14.135 the right-hand side should be multiplied by a factor 2.

Page 342: The third line of Section 14.6.2 should say Chapter 15, not Chapter 10.

Page 343: In Eq14.145, and three lines after it, it makes better logical sense to replace V with VN (though the two quantities are the same in this case).

Page 344: Second line of the paragraph after Eq14.152 should begin "Eqs. (14.149) and (14.150) are identical ...".

Page 355:  Contrary to what is stated in the text, observation still allows significant neutrino asymmetry corresponding to a non-negligible chemical potential which would modify Eq 15.24.

Page 361: In Eq15.62, on the left hand side it should be delta Fl, not delta fl.

Pages 362, 364 and 368: In Eqs 15.70, 15.79 and 15.107, the right-hand side should have an additional term   +4 Phi-dot. In Eq15.71, the factor -aHVnu should not be there (ie the bracketed term should be multiplied by k alone). In Eqs 15.80 and 15.108 the first term on the right-hand side (i.e. aHVgamma) should not be there. In all of Eqs15.70, 15.79, 15.80, 15.107 and 15.108, the quantity deltagamma should also have a superscript N to indicate conformal Newtonian gauge.

Page 369: At the end of the line following Eq15.114, insert the phrase `and evaluating Pik from Eqs15.72-15.74'.

Page 369: Before the last sentence, insert `The Newtonian-gauge density contrasts $\delta^N_i$ can then be calculated from the comoving gauge density contrasts $\delta_i$ using Eq 14.162.'

Page 370: Eq.~(15.120) should be replaced by the pair $\Theta(\bfk,\ell)= \int^{\tau_0}_0 S_\Theta(\bfk,\tau) j_\ell(x) d\tau$ and
$E(\bfk,\ell) = \int^{\tau_0}_0 S_E(\bfk,\tau) j_\ell(x)}/x^2 d\tau$, with $x\equiv k(\tau_0-\tau)$, and $k$ should be replaced
by $\bfk$ in Eqs~(15.121) to (15.125).

Page 371: After Eq15.122, replace the first part of the sentence with `Here A is defined by Eq. (15.104) and is gauge-invariant, ...'

Page 371: The variable A2 in Eqs15.122 to 15.125 should just be A. In Eq15.123 the last two terms should have a plus sign rather than a minus sign, the last term in Eq15.124 should have its sign changed to minus and its denominator should be `2' and not `4', and Eq15.125 should also have a plus sign on its right-hand side. [Thanks to Uros Seljak for pointing out the errors in this and the previous two items, and even admitting that Seljak & Zaldarriaga (1996) might partly be to blame (though CMBFAST works fine)!]

Page 373: The exponent in the conversion of grams to GeV should be 23, not 25 as given.

Page 374: The answer to Example 3.6 should specify that Eqs 3.7 and 3.8 are the slow-roll equations to be used.

We are indebted to Toshiyuki Kanazawa for a comprehensive reading of the book uncovering many of these errors.

If you are the sort of person who likes to correct minor typographical errors in a book, click here.

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Last updated:  June 2009
Andrew Liddle