First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters

D. N. Spergel, L. Verde, H. V. Peiris, E. Komatsu, M. R. Nolta, C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer et al.

The Astrophysical Journal Supplement Series, 2003 · paper · Astronomy

Fits cosmological models to first-year WMAP microwave-background data, alone and combined with other measurements. Reports that a flat, dark-energy-dominated universe fits well, with Hubble constant about 71, age 13.7 billion years, and total density near critical.

Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.

14 findings

Substance The abstract says "The combination of WMAP and 2dFGRS data constrains the energy density in stable neutrinos: Ωνh2 < 0.0076". §6.3 gives "Ωνh2 < 0.0067" for WMAPext+2dFGRS, and 0.0076 only "If we add the Lyman α data". Also, the solid 95% line in Fig. 14 (the Lyα case) sits near 0.0072, not 0.0076.

p. 2 Abstract
The combination of WMAP and 2dFGRS data constrains the energy density in stable neutrinos: Ωνh2 < 0.0076
p. 23 §6.3
that Ωνh2 < 0.0067 (95% conﬁdence limit). If we add the Lyman α data, then the limit slightly weakens to Ωνh2 < 0.0076.
p. 23 §6.3
Ωνh2 < 0.0067 (95% conﬁdence limit). If we add the Lyman α data, then the limit slightly weakens to Ωνh2 < 0.0076
p. 2 Abstract
Ωνh2 < 0.0076 (95%

Only the authors can say what is right here.

Judge's note: Page 2 (abstract) attributes 0.0076 to WMAP+2dFGRS only. Page 23 gives 0.0067 for WMAP/CBI/ACBAR/2dFGRS and says the limit 'weakens to Ωνh2 < 0.0076' when Lyα is added. The Fig. 14 caption (page 48) gives 0.21 and 0.23 eV for the two cases, so the abstract's 0.23 eV is the Lyα case. That means the abstract names the wrong data combination for its number. On the page 48 image, the dashed line sits at about 0.0067. By pixel interpolation between the 0.006 and 0.008 ticks, the solid line sits at about 0.0072, which is less than the 0.0076 printed in the text. Rests on outside knowledge: the mass conversion Ωνh2 = Σmν/94 eV, which the paper does not state. Under it, 0.0072 gives about 0.23 eV and 0.0076 gives about 0.24 eV. Only the authors can say whether the abstract should name Lyα or quote 0.0067, and whether 0.0076 or about 0.0072 is the intended Lyα limit.

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Substance The text and abstract give 'Ωνh2 < 0.0076' and 'mν < 0.23 eV'. With Eq. (6), 0.0076 gives ≈0.24 eV per species, and Fig. 14's solid 95% line sits at ≈0.0072, which does give 0.23 eV.

p. 23 §6.3
weakens to Ωνh2 < 0.0076. For three degenerate neutrino species, this implies that mν < 0.23 eV
p. 48 Fig. 14 plot, solid vertical line
limits for each case (0.21 and 0.23 eV)
p. 2 Abstract
Ωνh2 < 0.0076
p. 23 §6.3, first paragraph
the limit slightly weakens to Ωνh2 < 0.0076. For three degenerate neutrino species, this implies that mν < 0.23 eV

Only the authors can say what is right here.

Judge's note: Page 23 image confirms 0.0067 and 0.0076 and mν < 0.23 eV. Page 22 gives Eq. 6 as Ωνh² = Σm/94.0 eV. On page 48 I measured the axis (0.000 at x≈217, 0.010 at x≈791). The dashed line falls at ≈0.0067, matching the text. The solid line falls at ≈0.0072, not 0.0076. The caption gives 0.21 and 0.23 eV, consistent with 0.0067 and 0.0072. The printed 0.0076 is inconsistent with both the figure and the derived mass.

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Substance For the WMAP-only power-law ΛCDM fit, χ²eff/ν is "1431/1342" in Table 1 (the text says "1.066 for 1342 degrees of freedom") but "1429/1341" in the Table 7 WMAP column. The Table 8 running-index WMAP fit, which has one more parameter, is listed as "1431/1342".

p. 6 Table 1, χ²_eff/ν row
1431/1342
p. 19 Table 7, WMAP column, χ²_eff/ν row
1429/1341
p. 4 §3 text
is 1.066 for 1342 degrees of freedom
p. 4 Sec 3
1.066 for 1342 degrees of freedom

Only the authors can say what is right here.

Judge's note: Page 6 (Table 1) prints 1431/1342. The Table 7 WMAP column on page 19 prints 1429/1341, with the same parameter means as Table 1 (A 0.9±0.1, ns 0.99±0.04, τ 0.166, h 0.72, Ωmh2 0.14, Ωbh2 0.024). Table 8's WMAP running model prints 1431/1342. I found nothing in the paper describing a different analysis set for Table 7, and the dof count is reversed relative to the extra running parameter. The values may be swapped between Tables 7 and 8; only the authors can say.

Second check: another model agreed

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Substance In Fig. 3 the Ω_mh² panel has its dashed line ("The dashed lines show the maximum likelihood values of the global six dimensional fit") at about 0.15, right of the peak, while Table 1 lists the Ω_mh² maximum-likelihood value as 0.13.

p. 38 Fig. 3, Ω_mh² panel
The dashed lines show the maximum likelihood values of the global six dimensional ﬁt.
p. 6 Table 1, Matter Density row, Maximum Likelihood column
0.13
p. 6 Table 1, Maximum Likelihood column
Matter Density Ωm h2 0.14 ± 0.02 0.13

Only the authors can say what is right here.

Judge's note: On p. 38 the Ω_mh² panel's dashed line falls between the 0.14 and 0.16 ticks, about 60% of the way to 0.16, which is roughly 0.15. A value of 0.13 would sit left of the 0.14 tick. In the other five panels the dashed lines agree with Table 1's ML column (0.023, 0.68, 0.78, 0.10, 0.97). The position is read from a raster image, but the gap is far larger than the reading error.

Second check: another model agreed

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Substance §4.7: 'fburn ≲ 25%, fesc ≲ 50%, fUV ≲ 50%, fion ≲ 90%, and 1 ≲ Cclump ≲ 100, thus fnet ≲ 5.6 × 10−3'. With fnet = fburn fUV fesc fion/Cclump, these bounds give fnet ≲ 5.6 × 10−2.

p. 16 §4.7, paragraph on reionization efficiency
thus fnet ≲5.6 × 10−3
p. 16 §4.7, second paragraph
fburn ≲25%, fesc ≲50%, fUV ≲50%, fion ≲90%, and 1 ≲Cclump ≲100, thus fnet ≲5.6 × 10−3

Only the authors can say what is right here.

Judge's note: Page 16 image confirms that the printed exponent is −3 and that fnet = fburn fUV fesc fion/Cclump. With the stated upper bounds and the lower bound Cclump = 1, the product is 0.05625. Cclump = 100 would give 5.6e-4, so no combination of the stated limits yields 5.6e-3. The exponent is off by a factor of 10 unless some unstated Cclump was assumed.

Second check: another model agreed

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Substance For the running-index fit to all data, the lower error on dn_s/d ln k is printed as "−0.031+0.016 −0.018" in Table 10 but "−0.031+0.016 −0.017" in Table 8 (last column) and in the §5.2 text.

p. 26 Table 10, Derivative of Spectral Index
dns/d ln k = −0.031+0.016 −0.018
p. 19 Table 8, WMAPext+2dFGRS+Lyman α column
−0.031+0.016 −0.017
p. 18 §5.2 text
−0.031+0.016 −0.017
p. 18 §5.2
−0.031+0.016 −0.017: fewer than 5% of the models

Only the authors can say what is right here.

Judge's note: Table 10's footnote reads 'Fit to the WMAP, CBI, ACBAR, 2dFGRS and Lyman α forest data', the same data and model as the Table 8 WMAPext+2dFGRS+Lyman α column. Every other shared entry agrees exactly in the images (A 0.83+0.09−0.08, ns 0.93±0.03, h 0.71+0.04−0.03, Ωbh² 0.0224±0.0009, Ωmh² 0.135+0.008−0.009, τ 0.17±0.06). Only the lower error on the running differs, 0.018 against 0.017. I found nothing in the paper that explains the difference.

Second check: another model agreed

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Substance Table 8 gives χ²eff/ν = '1431/1342' for the 7-parameter running model fitted to WMAP. That equals Table 1's 6-parameter power-law value (1431/1342; text: 1.066 for 1342 dof), while Table 7's power-law WMAP column gives '1429/1341'. The WMAP χ² entries of Tables 7 and 8 appear interchanged.

p. 19 Table 8, WMAP column, χ²_eff/ν
1431/1342
p. 19 Table 8 vs Table 7, WMAPext column
1437/1350

Only the authors can say what is right here.

Judge's note: Page 19 image: Table 7 WMAP 1429/1341; Table 8 WMAP 1431/1342. Table 1 (text layer, p6) gives 1431/1342 for the power-law WMAP-only fit, and §3 says 1.066 for 1342 dof. Footnote 13 defines ν as the number of data minus the number of parameters. On the same WMAP data the running model should therefore have one fewer dof than the power law. As printed, it has one more, plus a higher χ². Tables 1 and 7 also conflict for what appears to be the same model and data. The paper gives no explanation; only the authors can say which entries are right.

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Reference or label The Kirkman et al. (2003) D/H = 2.42 (+0.35/−0.25)×10−5 system is named "QSO HS 243+3057" in §4.3 but "Q1234+3047" in Table 5.

p. 12 §4.3
QSO HS 243+3057
p. 13 Table 5, last row
Q1234+3047
p. 12 §4.3, second paragraph
The Kirkman et al. (2003) analysis of QSO HS 243+3057 yields a D/H ratio of 2.42

Only the authors can say what is right here.

Judge's note: Page 12 text and page 13 Table 5 give the same value and reference under two designations whose RA and Dec digits both differ, so they cannot be the same name in different conventions. Rests on outside knowledge: the object's true designation, said by reviewers to be Q1243+3047 / HS 1243+3047, which cannot be verified from the files.

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Reference or label §5.1: 'While it overpredicts the amplitude of fluctuations on large angular scales (see §6)'. The large-angle discrepancy is treated in §7 ('Intriguing Discrepancies'), not §6.

p. 17 §5.1, first paragraph
the amplitude of ﬂuctuations on large angular scales (see §6)
p. 24 §7 Intriguing Discrepancies
a possible discrepancy between predictions and observations on the largest angular scales
p. 17 §5.1, first sentence
While it overpredicts the amplitude of ﬂuctuations on large angular scales (see §6)
p. 17 §5.1
large angular scales (see §6)

Judge's note: Crop C020-c confirms the '§6' pointer. §6's introduction lists its contents as dark energy (§6.1), non-flat models (§6.2), neutrinos (§6.3) and tensors (§6.4). The Introduction ('In §7, we note an intriguing discrepancy...') and §3 ('the low value of the quadrupole (see Figure 1 and §7)') both place this topic in §7.

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Reference or label §6.3: 'the square mass difference between the electron and muon neutrino is ∼10−9 eV' gives a squared mass difference in eV rather than eV². The next sentence uses eV² ('10−5eV2').

p. 22 §6.3, second paragraph
the square mass diﬀerence between the electron and muon neutrino is ∼10−9 eV
p. 22 §6.3
is ∼10−9 eV
p. 22 §6.3
muon and tau neutrinos is 10−5eV2
p. 22 §6.3
is ∼10−9 eV. The deﬁcit of muon neutrinos in atmospheric showers

Judge's note: Page 22 image confirms the unit 'eV' for a quantity the sentence calls a 'square mass difference'. This verdict confirms only the internal unit inconsistency. The reviewers' claim that the magnitudes (10^-9, 10^-5) are misquoted rests on outside knowledge of the oscillation parameters and is not judged here.

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Reference or label §7 says "Figure 16 shows the measured angular power spectrum and the predictions of our best fit Λ−CDM model ... The figure also shows the measured angular correlation function", but Fig. 16 (p. 50) plots only C(cos θ) against cos θ, with no angular power spectrum panel.

p. 24 §7, second paragraph
Figure 16 shows the measured angular power spectrum and the predictions of our best ﬁt
p. 50 Fig. 16 caption
Angular correlation function of the best ﬁt ΛCDM model, toy ﬁnite universe model, and WMAP data

Only the authors can say what is right here.

Judge's note: Viewed the §7 crop and all of p. 50. Fig. 16 has a single panel with C(cos θ) curves for ΛCDM, the toy model and the WMAP data. Its caption describes only the angular correlation function. No power-spectrum panel appears, so the text's description of the figure does not match what is printed. The authors need to say whether a panel is missing or the text should be changed.

Second check: another model agreed

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Reference or label §7 says "Figure 16 shows the measured angular power spectrum and the predictions of our best fit Λ−CDM model", but Fig. 16 (p. 50) is one panel of the angular correlation function C(cos θ) against cos θ only. No angular power spectrum is plotted.

p. 50 Fig. 16, y-axis C(cos θ)
Angular correlation function of the best ﬁt ΛCDM model
p. 24 §7, second paragraph
Figure 16 shows the measured angular power spectrum and the predictions of our best ﬁt Λ−CDM model
p. 50 Fig. 16
Angular correlation function of the best ﬁt ΛCDM model, toy ﬁnite universe model, and WMAP data on large angular scales.
p. 24 Sec 7, para 2
Figure 16 shows the measured angular power spectrum and the predictions of our best ﬁt

Judge's note: I opened the p. 24 text crop and the full p. 50 page image. Fig. 16 has one panel: C(cos θ) against cos θ with ΛCDM, Toy Model and WMAP data curves. Its caption mentions only the angular correlation function. Line 1140 of the text also describes Fig. 16 as the angular correlation function. The sentence on p. 24 is therefore a wrong pointer or a wrong description of the figure. The power spectrum is shown in Fig. 1, which the text cites elsewhere. The missing minus signs on the y-axis ticks (reviewer note 1) are a separate defect and are judged under C047.

Second check: another model agreed

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Reference or label §7 cites "Cornish et al. (1998)" and "topology of the universeCornish et al. (1998b); Weeks (1998)", but the reference list has two unlabelled 'Cornish, N. J., Spergel, D., & Starkman, G. 1998' entries (Phys. Rev. D, 57, 5982 and Proc. Nat. Acad. Sci., 95, 82), so neither '1998' nor '1998b' can be resolved; the citation is also run onto 'universe'.

p. 25 §7
topology of the universeCornish et al. (1998b); Weeks (1998)
p. 29 Reference list
Cornish, N. J., Spergel, D., & Starkman, G. 1998, Proc. Nat. Acad. Sci., 95, 82
p. 25 Sec 7
Cornish et al. (1998b); Weeks (1998)
p. 29 References
Cornish, N. J., Spergel, D., & Starkman, G. 1998, Phys. Rev. D, 57, 5982

Judge's note: Viewed the p. 25 text and the p. 29 reference list. Neither 1998 entry carries an a/b suffix. The text uses both '(1998)' and '(1998b)', and there is no '1998a' anywhere. The missing space and parentheses in 'universeCornish' are visible on the page.

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Reference or label Reference entries are malformed. "Levshakov, S.A., ... & Dessauges-Zavadsky, M., apj, 582, 596." has no year and a lowercase 'apj'. Both Schuecker entries ("Schuecker,P., Bohringer,H., Collins, C.A., & Guzzo, L., å, 398, 867" and "... Weinberg, N.N., å, 402, 53.") have no year and show 'å' where the journal name should be, although the text cites them as 2003, 2003a and 2003b.

p. 32 References, Levshakov
Levshakov, S.A., Agafonova, I.I, D’Odorico, S., Wolfe, A.M., & Dessauges-Zavadsky, M., apj, 582,
p. 34 References, Schuecker
Schuecker,P., Bohringer,H., Collins, C.A., & Guzzo, L., ˚a, 398, 867

Judge's note: Seen in the p. 32 and p. 34 images. The 'å' glyph is in the rendered PDF, so it is not an extraction artefact. In the text the citations read 'Levshakov et al. (2003)' and 'Schuecker et al. (2003a)'.

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27 minor (typos and the like; most checked by the judge only)

Wording §4.5 says 'Figure 6 compares the predicted power spectra ... to the linear power spectra inferred by Gnedin & Hamilton (2002) and by Croft et al. (2002)', but Fig. 6 (right) shows a single data set, and its caption cites only Croft et al. (2002).

p. 15 §4.5
to the linear power spectra inferred by Gnedin & Hamilton (2002) and by Croft et al. (2002)
p. 40 Fig. 6 caption, right panel
1 σ calibration uncertainty (Croft et al. 2002)
p. 40 Fig. 6 right panel and caption
The data points have been scaled downwards by 20%, which is consistent with the 1 σ calibration uncertainty (Croft et al. 2002).

Only the authors can say what is right here.

Judge's note: Crop C016-a shows the §4.5 sentence naming both groups. On full page 40 the right panel of Fig. 6 has one set of points ('Amplitude correction of 20%'). The caption says only 'the power spectrum at z = 3 inferred from the Lyman α forest data' and cites Croft et al. 2002 for the calibration uncertainty. No Gnedin & Hamilton data are identified. Only the authors can say whether the points are Croft's alone or a combination.

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Wording The WMAPext dof fall from '1440/1352' (Table 7, power law) to '1437/1350' (Table 8, running), a drop of 2 for a model with one extra parameter. For WMAPext+2dFGRS the drop is 1 (1381 → 1380).

p. 19 Table 7, WMAPext column, χ²_eff/ν
1440/1352
p. 19 Table 8, WMAPext column, χ²_eff/ν
1437/1350

Only the authors can say what is right here.

Judge's note: Page 19 image confirms the entries. With footnote 13's definition, these imply 1358 vs 1357 data points for the same WMAPext data set. The paper states no different data selection, so the authors need to clarify.

Second check: another model agreed

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Wording The prior on the running behind the r < 0.43 limit is given as 'dns/d ln k = 0' (Table 9) and '|dn/d ln k| = 0' (§6.4), but as '|dn/d ln k| < 0.005' in the Fig. 15 caption and legend.

p. 23 Table 9 row label and §6.4 text
|dn/d ln k| = 0
p. 49 Fig. 15 caption and legend
for models with |dn/d ln k| < 0.005
p. 23 Table 9 row label and §6.4 text
dns/d ln k = 0
p. 49 Fig. 15 caption
|dn/d ln k| < 0.005

Only the authors can say what is right here.

Judge's note: Page 23 image: Table 9 row 'dns/d ln k = 0', value 0.43; the text says '|dn/d ln k| = 0'. Page 49: the legend reads 'r<0.43 (|dns/dlnk|<0.005)' and the caption says '|dn/d ln k| < 0.005'. The same limit is attached to two different stated priors. The window may be how zero running was implemented in the chain, but the paper does not say so.

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Wording The Fig. 4 caption reads "This plot shows the contribution to 2 ln L per multipole", whereas the paper defines χ²_eff ≡ −2 ln L and the y-axis is the positive Δχ²_eff.

p. 39 Fig. 4 caption
This plot shows the contribution to 2 ln L per multipole

Judge's note: The p. 39 image shows '2 ln ℒ' with no minus sign. The definition χ²_eff ≡ −2 ln L is in the text (main.txt line 162), and the axis label is Δχ²_eff.

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Typo Affiliation footnote 3 reads "Chandra Postdoctral Fellow"; it should read "Postdoctoral".

p. 1 Affiliation footnote 3
Chandra Postdoctral Fellow

Judge's note: The crop of page 1 shows the misspelling "Postdoctral" in footnote 3.

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Typo The abstract reads "a flat Λ−dominated universe seeded by a nearly scale-invariant adiabatic Gaussian fluctuations": the article "a" does not agree with the plural noun.

p. 1 Abstract, first sentence
seeded by a nearly scale-invariant adiabatic Gaussian ﬂuctuations

Judge's note: Checked in the abstract on the page 1 crop. Either 'a' should be dropped or the noun made singular.

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Typo §6.2: 'Ωtot < 1.02 ± 0.02 Figure 13 shows' pairs an inequality with a value ± error and lacks a full stop. The abstract gives 'Ωtot = 1.02 ± 0.02'.

p. 22 §6.2
Ωtot < 1.02 ± 0.02
p. 2 Abstract
Ωtot = 1.02 ± 0.02
p. 22 §6.2, third paragraph
Ωtot < 1.02 ± 0.02 Figure 13
p. 22 Sec. 6.2, last paragraph
Ωtot < 1.02 ± 0.02 Figure 13 shows

Judge's note: Page 22 image confirms '<' and the missing full stop. The abstract (text layer, p2) reads 'Ωtot = 1.02 ± 0.02'. The value matches, so this is typographic.

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Typo Unclosed parentheses: §2 "(these limits lie at such low likelihood that they are unimportant for the models." and the Fig. 13 caption "(supernova data (Riess et al. 1998, 2001) is shown but not used in the likelihood in this part of the panel;".

p. 3 §2
(these limits lie at such low likelihood that they are unimportant for the models.
p. 47 Fig. 13 caption
(supernova data (Riess et al. 1998, 2001) is shown but not used in the likelihood in this part of the panel;
p. 47
WMAPext+ HST Key Project (supernova data (Riess et al. 1998, 2001) is shown but not used in the likelihood in this part of the panel;

Judge's note: On the page 3 crop, the §2 parenthesis is never closed before the sentence ends. On the page 47 crop, the outer parenthesis in the Fig. 13 caption is never closed; only the inner citation parenthesis closes.

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Typo Table 1 has the footnote "aFit to WMAP data only" but no 'a' marker anywhere in the table.

p. 6 Table 1 footnote
aFit to WMAP data only

Judge's note: Viewed the full page 6: there is no superscript a in the title, headers or cells. The footnote is orphaned; it is also redundant with the title 'WMAP Data Only'.

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Typo Table 2 has the footnote "aFit to the WMAP data only" but no 'a' marker in the table; only the b and c markers appear.

p. 7 Table 2 footnote
aFit to the WMAP data only

Judge's note: Viewed page 7: the superscripts b (Redshift of Reionization) and c (Acoustic Angular Scale) are present, and there is no 'a' marker.

Second check: none (judge only)

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Typo The §4 heading reads "COMPARSION WITH ASTRONOMICAL PREDICTIONS"; it should read "COMPARISON".

p. 8 §4 heading
COMPARSION WITH ASTRONOMICAL PREDICTIONS

Judge's note: The misspelling is visible in the heading on the page 8 crop.

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Typo The citation in §4.4 is garbled: "(See Hu et al. 2001 Hu (Fukugita) and Knox, Christensen & Skordis (2001)".

p. 12 §4.4, first paragraph
(See Hu et al. 2001 Hu (Fukugita) and Knox

Judge's note: Visible on the page 12 image. The stray 'Hu (Fukugita)' looks like leftover citation markup.

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Typo §4.3 reads "more distant measurements by IMAP and FUSE"; the instrument is presumably IMAPS.

p. 12 §4.3
more distant measurements by IMAP and FUSE

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: the instrument behind Jenkins et al. (1999) and Sonneborn et al. (2000) is IMAPS. The page 12 image does print 'IMAP'.

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Typo §4.4 has a stray full stop mid-sentence: "Jimenez et al. (1996). using a distance-independent method".

p. 12 Sec. 4.4
Jimenez et al. (1996). using a distance-independent

Judge's note: Visible on page 12. The period comes before a lowercase 'using' that continues the sentence.

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Typo In §4.3 the parenthesis opened at "(although a reanalysis of the system by Levshakov et al. (2003) finds a higher D/H value: 3.75 ± 0.25." is never closed, and the value lacks the ×10^−5 factor that Table 5 gives (3.75 ± 0.25 × 10−5).

p. 32 Reference list
Dessauges-Zavadsky, M., apj, 582,
p. 12 §4.3
ﬁnds a higher D/H value: 3.75 ± 0.25. Pettini
p. 12 §4.3
ﬁnds a higher D/H value: 3.75 ± 0.25. Pettini & Bowen
p. 12 Sec. 4.3
(although a reanalysis of the system by Levshakov et al. (2003) ﬁnds a higher D/H value: 3.75 ± 0.25.

Judge's note: In the p. 12 image no closing parenthesis follows '0.25.' before 'Pettini & Bowen'. Every other D/H value in the paragraph carries ×10^−5, and Table 5 lists 3.75 ± 0.25 × 10−5 for Levshakov et al. (2003). The missing year in the reference entry is handled in C039.

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Typo §4.7: 'M_min^HRL(z) = 10^8(1 + z)/10)^−3/2' has one opening parenthesis and two closing ones; it should read 10^8((1 + z)/10)^−3/2.

p. 16 §4.7
108(1 + z)/10)−3/2
p. 16 §4.7
then the Jean Mass, Mj = 106M⊙

Judge's note: Page 16 image shows the unbalanced parentheses. This verdict covers only the parenthesis defect; the 'Jean Mass' misspelling also quoted here is judged in C019.

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Typo §4.7: 'then the Jean Mass, Mj = 10^6 M⊙' should be 'Jeans mass'; the next sentence spells it 'Jeans mass'.

p. 16 §4.7
then the Jean Mass
p. 16 Sec. 4.7
then the Jean Mass, Mj = 106M⊙

Judge's note: Page 16 image shows 'Jean Mass', and two lines later 'the Jeans mass before reionization'.

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Typo §5.2: 'Figure 9 shows the the power spectrum' has a duplicated 'the'.

p. 18 §5.2
Figure 9 shows the the power spectrum
p. 18
ns/d ln k > 0. Figure 9 shows the the power spectrum as a

Judge's note: Crop C021-a shows the duplicated word.

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Typo The Table 7 column header reads 'WMAPext 16a'. Superscript 16 points to no footnote; the paper's footnotes end at 15, which defines WMAPext.

p. 19 Table 7 column header
WMAPext 16a

Judge's note: Page 19 image shows '16a' in the header. A search of the text finds no footnote 16; footnote 15 (p17) defines WMAPext.

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Typo §5.2 (p20): 'unless H2 cooling enables mass dark halos to collapse' appears to be missing a qualifier (e.g. 'low mass').

p. 20 §5.2 continuation, item (a)
unless H2 cooling enables mass dark halos to collapse

Judge's note: Crop C025-a confirms the wording. In context (§4.7), H2 cooling lowers the minimum collapse mass, so 'low mass' is the evident intent.

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Typo §6.3: 'neutrino mass eiginstates' should be 'eigenstates'.

p. 22 §6.3
neutrino mass eiginstates
p. 22
mass eiginstates

Judge's note: Page 22 image shows 'eiginstates'.

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Typo §7 misspells 'a posteriori' as "determined a posteori in response to the shape of the correlation function".

p. 24 §7
a posteori
p. 24 Sec. 7
determined a posteori in response
p. 24
determined a posteori in response to

Judge's note: The page image on p. 24 shows the italic 'a posteori'. The correct form is 'a posteriori'.

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Typo The Acknowledgments have 'form' for 'from' in "providing us with the likelihood surface form the SNIA data".

p. 28 Acknowledgments
likelihood surface form the SNIA data
p. 28 Acknowledgments
the support of the Oﬃce Space at NASA Headquarters
p. 28 Acknowledgments
the Oﬃce Space at NASA Headquarters

Judge's note: Seen in the p. 28 image. The point about the 'Office Space' name is handled separately in C037.

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Typo The Acknowledgments read "the support of the Office Space at NASA Headquarters", which looks like a garbled office name.

p. 28 Acknowledgments
support of the Oﬃce Space at NASA Headquarters

Only the authors can say what is right here.

Judge's note: Rests on outside knowledge: that the NASA unit was called the Office of Space Science. The p. 28 image does read 'Office Space at NASA Headquarters'.

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Typo The reference "Turner, M. S. & Riess, A. G. 2002, ApJ, 569, 18" is listed but never cited in the text or in any caption.

p. 34 Reference list
Turner, M. S. & Riess, A. G. 2002, ApJ, 569, 18

Judge's note: I searched main.txt and main.layout.txt. 'Turner' appears only in Kosowsky & Turner (1995) and Burles, Nollett & Turner, and 'Riess' only in Riess et al. 1995/1998/2001. Turner & Riess 2002 is never cited.

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Typo The Fig. 9 caption repeats a phrase in "for the amplitude of the amplitude of mass fluctuations" and has an unbalanced parenthesis in "∆²(k) = (k³/(2π²)P(k)".

p. 43 Fig. 9 caption, right panel
for the amplitude of the amplitude of mass ﬂuctuations
p. 43 Fig. 9 caption
the amplitude of the amplitude of mass ﬂuctuations, ∆2(k) = (k3/(2π2)P(k)

Judge's note: Both defects are visible in the p. 43 caption image: the repeated words, and two opening parentheses against one closing parenthesis before P(k).

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Typo Fig. 16: the y-axis ticks read "400, 200, 0, 200, 400" from top to bottom. The two ticks below the dashed zero line have no minus signs, so they should read −200 and −400.

p. 50 Fig. 16, y-axis
C(cos θ)

Judge's note: I checked the p. 50 page image. The y-axis labels are 400, 200, 0, 200, 400, with the dashed line at 0. The ΛCDM curve (minimum near the lower 200 tick) and the toy-model curve go below the zero line, so the lower ticks must be negative. The x-axis on the same figure prints minus signs correctly (-0.5, -1.0), which makes a rendering loss unlikely. The dashed zero line makes the intended values easy to infer, so I rate this cosmetic.

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Something wrong with this page, or a finding we missed?