Measuring the cosmological constant with redshift surveys

A. F. Heavens (Institute for Astronomy; Edinburgh); J. A. Peacock; Royal Observatory; W. E. Ballinger

REVIEW

Measuring the cosmological constant with redshift surveys

W. E. Ballinger, J. A. Peacock, A. F. Heavens (Institute for Astronomy, Royal Observatory, Edinburgh)

Astrophysics · Submitted 3 May 1996 · 2 references

Pith's verdict: A careful, honest feasibility study of the Alcock–Paczyński test that mostly holds up; the one real soft spot is that the forecast errors assume a finger-of-God damping form the paper never tests.

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arXiv IDastro-ph/9605017 (version 1)

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CitationA. F. Heavens (Institute for Astronomy, Edinburgh), J. A. Peacock, Royal Observatory, W. E. Ballinger. "Measuring the cosmological constant with redshift surveys." arXiv:astro-ph/9605017 (1996). https://pith.science/paper/SDYOEV4F

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Editor's note

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Bottom line: this is the paper that made people take the Alcock–Paczyński test seriously as a practical measurement rather than a slogan, and it did so by arguing the effect is smaller than advertised. The two results worth carrying around are the saturation of the flattening factor — F stays below about 1.3 for Ω_m ≥ 0.2, so there is no point going deeper than z ≈ 1 — and the explicit demonstration that geometric flattening masquerades as a velocity distortion with β_eff ≈ 0.5(F−1). That second point is the one that reframed the field: it told everyone the AP test is a joint F–β estimation problem, not a clean geometric measurement. The combined geometric + Kaiser + finger-of-God power spectrum in Eq. (20) and the likelihood machinery in Section 4, with the covariance tables in Appendix C, are what later forecast papers actually reuse. Appendix A is clean and I checked the coordinate transform and the μ-dependence; it is right. The paper is also candid about its own limits, including the admission in Section 3.3 that the damping model is a placeholder for empirical simulation fits. Credit for that; most forecast papers of the era were not so forthright. On the stress-test note: it lands, but smaller than it sounds. At k_max σ_p ≈ 1.2 the Lorentzian, Gaussian and Lorentzian-squared forms differ by roughly 20–40% in damping, which is right at the paper's own stated threshold of a factor ~2, so the 'very little difference' claim in Section 3.2 is marginal at the adopted k_max rather than wrong. Appendix C does marginalise over σ_p and shows σ(F) barely moves, so the amplitude uncertainty is handled; it is the functional form that is untested, and the Section 5 numbers do not propagate that. That is a real but bounded caveat, not a load-bearing flaw — the qualitative conclusion (2df marginal, Sloan/quasar surveys plausible) survives a 20% shift in σ(F). Minor: the low-redshift measurement really constrains Ω_Λ − Ω_m/2, which the paper says plainly, so the Ω_Λ error bars in Fig. 7 should be read with Ω_m fixed by flatness. Who should read it: anyone forecasting or interpreting AP/RSD measurements, and anyone wanting the canonical statement of the F–β degeneracy. Worth a serious referee, and worth citing.

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Major objections (1)

[Section 3.2 and Section 5, Eqs. (17)-(19), Fig. 3] The forecast errors on Ω_Λ in Figs. 7-8 are computed with k_max = 0.35 h Mpc⁻¹ and σ_p = 350 km s⁻¹, so k_max σ_p ≈ 1.2. At this value the Lorentzian damping of Eq. (19) gives D ≈ 0.57, the Gaussian of Eq. (18) gives ≈ 0.47, and a Lorentzian-squared form gives ≈ 0.33, differences of 20-40% in the damping alone. Because the β-F separation relies on the μ⁴ term measured at these wavenumbers (Section 4.1: 'To separate β and ε requires at least the μ⁴ term'), a change of this size in the damping could shift the marginalised σ(F), and hence σ(Ω_Λ), by a comparable amount. Fig. 3 shows only one parameter set (β = 0.5, σ_p = 350 km s⁻¹, n = -1.5), and Appendix C floats σ_p but never the damping form. The paper is candid that the model is approximate (Section 3.3), but the quantitative robustness of the Section 5 forecasts to the form choice is not demonstrated. I recommend repeating the Fig. 7/8 forecast with at least one alternative damping form, or stating explicitly that the quoted σ(Ω_Λ) ≈ 0.35 carries an additional systematic of order 20-30% from the damping form. This does not undermine the qualitative conclusion that the test is feasible with next-generation surveys.

Minor objections (8)

[Abstract vs Eq. (21)] The abstract states β_eff ≈ 0.5(F-1) without the spectral-index dependence; Eq. (21) gives β_a ≈ -(n/2)(F-1), which equals 0.5(F-1) only for n = -1. Please state the n dependence in the abstract or note the n = -1 assumption explicitly.
[Section 3.4, Eqs. (24) and (27)] The sign of the geometric term in Eq. (27) appears opposite to what Eq. (24) implies under the stated definition of D_a. With D_a = (b(z)/b(0)) D(z) [f⊥^{3+n} F]^{1/2}, d ln D_a/dz at z = 0 includes +(n+4)/8 (1-2q0), whereas Eq. (27) has -(n+4)/8 [1-Ω_m+2Ω_Λ]. The paper's Eq. (29) is internally consistent with Eq. (27), so the issue is the sign convention linking (24) and (27); please reconcile. This does not affect the main conclusions.
[Section 4.1, Eq. (35)] The linear interpolation between the shot-free and shot-dominated covariance matrices omits the cross term; the correct scaling in the shot-dominated regime is quadratic in P_S/P_T, not linear. The paper presents both limits in Figs. 5-6, so the practical impact may be small, but the interpolation should be justified or replaced by the exact (1 + P_S/P_T)² scaling.
[Section 5.1] The text says 'assuming the same parameter values as in section 4.2', but Section 4.2 is 'Evolutionary effects' and does not contain the parameter values; this appears to be a cross-reference to Section 4.1.
[Section 5.1] With σ(Ω_Λ) ≈ 0.35, a true Ω_Λ = 0 would rule out Ω_Λ = 0.8 at only about 2.3σ; 'ruling out' is stronger than the forecast supports. Please soften the wording or quantify the significance.
[Section 4.1, Eq. (36)] The small-β limit C_ββ ∝ β^{-2} diverges as β → 0, reflecting the perfect degeneracy between β and F at leading order. This is correct, but it should be stated explicitly as the reason the small-β limit is not useful for forecasting.
[Fig. 1 caption] Panel (b) says 'Ω_Λ varies as for (a)' with Ω_m = 0.2 fixed; for Ω_Λ = -1 this gives Ω = -0.8, which is not a physically relevant model. Please clarify that panel (b) explores open models with fixed Ω_m.
[Section 6] The statement that 'the full Sloan survey would only just suffice if β ≃ 0.2' is consistent with Fig. 7, but the figure shows β = 0.25 and 0.75; please clarify the interpolation to β = 0.2.

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The paper

The paper claims

The Alcock–Paczyński test for a cosmological constant is weaker than advertised, and its signal overlaps with velocity distortions, but the two can be separated with a large enough survey.

The reading

A positive vacuum density makes the universe's comoving geometry differ from the matter-dominated one we usually assume, so a sphere of clustering in real space appears squashed along the line of sight when its redshifts are converted to distances with the wrong cosmology. This paper asks how big that squashing really is and whether it can be told apart from the squashing that galaxy peculiar velocities produce on their own. The answer to the first question is discouraging: for any plausible matter density the flattening factor $F$ stays below about 1.3 and stops growing beyond $z$ of roughly 1, because matter outpaces vacuum energy at early times. The answer to the second is more hopeful: a geometric flattening acts at leading order like a spurious growth-of-structure signal with $F-1$ playing the role of $2.1$ times an effective $F-1$ in the velocity parameter, but the two distortions have different angular dependence, so a maximum-likelihood fit to an anisotropic power spectrum can in principle recover both. The authors conclude that surveys of the size then being planned — hundreds of thousands of galaxies, or tens of thousands of quasars at higher redshift — could detect a vacuum density at a cosmologically interesting level if the velocity distortions are strong enough.

What carries the argument

The load-bearing object is equation (20), the apparent anisotropic power spectrum in the assumed geometry: $P_a(k,mu) = f_perp^{-(3+n)} F^{-1} P_t(k) [1 + mu^2(F^{-2}-1)]^{(n-4)/2} [1 + mu^2(beta + F^{-2} - 1)]^2 D(k mu sigma'_p)$, with $sigma'_p = sigma_p/f_parallel$. It combines three multiplicative effects: the coordinate rescaling that turns true wavenumbers into apparent ones, the Kaiser factor $(1+beta mu^2)^2$ from linear infall, and a damping factor $D$ for virialised velocities. Expanding this in powers of $mu$ shows the $mu^2$ coefficient mixes $beta$ and $F-1$ linearly, while the $mu^4$ coefficient is quadratic in both — that is exactly why the two cannot be separated from a single quadrupole measurement, and why the fit must reach into the mildly nonlinear regime. The fitting procedure is a likelihood on the logarithm of the power, $ln L = const. - rho ∫ pi k^2 dk ∫ (ln P - ln P')^2 dmu$, with the effective density of states set by the survey's selection function and optimal weighting, and the amplitude of $P_t(k)$ floated and integrated over so that the anisotropy, not the normalisation, carries the signal.

Extended reading notes

Core claim

The paper's central claim is a qualified yes for the Alcock–Paczyński geometric test, with two corrections to earlier optimism. First, the flattening factor $F(z) = f_parallel/f_perp$, defined relative to an Einstein–de Sitter assumed geometry, is bounded: $F$ saturates near $z = 1$ and stays at or below roughly 1.3 for $Omega_m >= 0.2$, so the de Sitter case that earlier work used to argue for easy detectability is unrepresentative. Second, and more seriously, the geometric squashing is degenerate with the Kaiser redshift-space distortion at leading order: for a power spectrum with local slope $n$, a pure geometric flattening mimics gravitational instability with $beta_eff = -(n/2)(F-1)$, which for $n = -1$ is about $0.5(F-1)$. The degeneracy is broken only by the $mu^4$ term in the anisotropy, which is second order in both $beta$ and $F-1$. The authors derive the full combined model — geometry, linear Kaiser boost, and a finger-of-God damping term — and show that the $mu$-dependence of the two effects is formally distinct, so a likelihood fit to $P(k, mu)$ over a wide range of wavenumbers can recover $F$ and $beta$ separately, provided the survey has enough modes and the small-scale damping is modelled well enough.

Load-bearing premise

The claim that a survey can separate the geometric flattening from the velocity distortion rests on the phenomenological finger-of-God damping model remaining accurate out to $k sigma_p$ of about 1.6, the wavenumber range where the $mu^4$ term that breaks the degeneracy becomes measurable; the authors themselves note that one should not assume the precise correctness of that model and that empirical relations from numerical simulations would be more appropriate.

Editorial extensions

If this is right

A measurement of $F$ at a single redshift determines the combination $Omega_Lambda - Omega_m/2$ rather than $Omega_Lambda$ alone, since the lowest-order distance-redshift correction depends only on the deceleration parameter.
Because $F$ evolves differently from $beta(z)$, splitting a survey into small solid-angle cubes and combining the per-shell likelihoods on $Omega_Lambda$ gives an estimate that is by construction independent of assumptions about how bias evolves.
For a 2dF-like survey of 250,000 galaxies, the expected rms error on $Omega_Lambda$ is about 0.35 if $beta = 1$, roughly doubling if $beta = 0.3$; the full Sloan survey would barely detect the effect in the low-$beta$ case.
Quasar samples out to $z = 2.2$ constrain a positive $Omega_Lambda$ more tightly than low-redshift galaxies at fixed $beta$, despite being shot-noise limited, because the geometric signal and the surveyed volume are both larger.
The same fitting machinery yields a precise measurement of $beta(z)$ as a by-product, which constrains bias histories even if no flattening is detected.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The forecast errors on $Omega_Lambda$ are quoted for a fixed, assumed damping model; if the true finger-of-God damping differs from the Gaussian or Lorentzian forms used here, the usable $k_max$ shrinks and the quoted uncertainties should be read as lower bounds rather than expectations.
The paper's remark that the squashing saturates because $Omega_m(z)/Omega_Lambda(z)$ grows as $(1+z)^3$ suggests a general rule for any geometric dark-energy test: the signal per unit redshift peaks where the two densities are comparable, so survey design should target that epoch rather than the largest available depth.
Because the geometric and velocity distortions enter with different powers of $mu$, a measurement of the full anisotropic power spectrum at several well-separated narrow redshift shells is a stronger test than a single wide-bin measurement of the same total volume, since the redshift evolution of the two effects acts as an extra discriminant.

Original abstract

It has been proposed that the cosmological constant $\Lambda$ might be measured from geometric effects on large-scale structure. A positive vacuum density leads to correlation-function contours which are squashed in the radial direction when calculated assuming a matter-dominated model. We show that this effect will be somewhat harder to detect than previous calculations have suggested: the squashing factor is likely to be $<1.3$, given realistic constraints on the matter contribution to $\Omega$. Moreover, the geometrical distortion risks being confused with the redshift-space distortions caused by the peculiar velocities associated with the growth of galaxy clustering. These depend on the density and bias parameters via the combination $\beta\equiv \Omega^{0.6}/b$, and we show that the main practical effect of a geometrical flattening factor $F$ is to simulate gravitational instability with $\beta_{\rm eff}\simeq 0.5(F-1)$. Nevertheless, with datasets of sufficient size it is possible to distinguish the two effects; we discuss in detail how this should be done. New-generation redshift surveys of galaxies and quasars are potentially capable of detecting a non-zero vacuum density, if it exists at a cosmologically interesting level.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

It has been proposed that the cosmological constant $\Lambda$ might be measured from geometric effects on large-scale structure. A positive vacuum density leads to correlation-function contours which are squashed in the radial direction when calculated assuming a matter-dominated model. We show that this effect will be somewhat harder to detect than previous calculations have suggested: the squashing factor is likely to be $<1.3$, given realistic constraints on the matter contribution to $\Omega$. Moreover, the geometrical distortion risks being confused with the redshift-space distortions caused by the peculiar velocities associated with the growth of galaxy clustering. These depend on the density and bias parameters via the combination $\beta\equiv \Omega^{0.6}/b$, and we show that the main practical effect of a geometrical flattening factor $F$ is to simulate gravitational instability with $\beta_{\rm eff}\simeq 0.5(F-1)$. Nevertheless, with datasets of sufficient size it is possible to distinguish the two effects; we discuss in detail how this should be done. New-generation redshift surveys of galaxies and quasars are potentially capable of detecting a non-zero vacuum density, if it exists at a cosmologically interesting level.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It has been proposed that the cosmological constant $\Lambda$ might be measured from geometric effects on large-scale structure.	background 1.00	0.04	0.04	speculative 0.81
s1	A positive vacuum density leads to correlation-function contours which are squashed in the radial direction when calculated assuming a matter-dominated model.	background 0.88	0.11	0.06	asserted 0.99
s2	We show that this effect will be somewhat harder to detect than previous calculations have suggested: the squashing factor is likely to be $<1.3$, given realistic constraints on the matter contribution to $\Omega$.	result 0.94	0.84	0.24	hedged possibility 0.79
s3	Moreover, the geometrical distortion risks being confused with the redshift-space distortions caused by the peculiar velocities associated with the growth of galaxy clustering.	limitation 0.90	0.17	0.05	hedged possibility 0.87
s4	These depend on the density and bias parameters via the combination $\beta\equiv \Omega^{0.6}/b$, and we show that the main practical effect of a geometrical flattening factor $F$ is to simulate gravitational instability with $\beta_{\rm eff}\simeq 0.5(F-1)$.	result 0.75	0.90	0.10	asserted 1.00
s5	Nevertheless, with datasets of sufficient size it is possible to distinguish the two effects; we discuss in detail how this should be done.	contribution 0.49	0.53	0.27	hedged possibility 0.78
s6	New-generation redshift surveys of galaxies and quasars are potentially capable of detecting a non-zero vacuum density, if it exists at a cosmologically interesting level.	interpretation 0.60	0.19	0.70	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.87 · evidence from proof or derivation 0.78.

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sources read

arXiv listing abstract v1

not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** (a) Flattening factor F as a function of redshift for flat models (Ωm + ΩΛ = 1). The curves range from ΩΛ = 1 (top curve) to ΩΛ = −1 in steps of 0.1. (b) F against redshift for models with a fixed mass density Ωm = 0.2 and a cosmological constant. ΩΛ varies as for (a). (c) F against redshift for models with no cosmological constant. Ωm varies from 0 (top) to 2, again in steps of 0.1. The calculations can be repeated for open models with no cosmological constant, but the effects are much smaller (see Fig. 1c). Hence a large value of F indicates a cosmological constant, and would be a…

**Figure 2.** Contours of the power spectrum in the kk and k⊥ plane, assuming a power-law index n = −1.5 for the true power spectrum. The contour interval is ∆ln P = 1/2. The models are linear (Kaiser) redshift distortion only with β = 0.5 (full contours) and geometric squashing effect only, with F = 1.5 (dotted contours). 3.2 Fingers of God Additionally, we consider the nonlinear finger-of-God effect caused by velocity dispersions in virialized clusters. A simple model for the effects of such velocities on the power spectrum was introduced by Peacock (1992; see also Peacock & Dodds 1994). This…

**Figure 4.** Solid contours for a model with F = 1.1, β = 0.5 and σp = 350 kms−1 . Dotted contours are for the best fit to this model with redshift distortions only (F = 1). βa ≃ − n 2 (F − 1). (21) If the true β is small but non-zero, then the true distortion adds linearly to the geometrical signature; for large β, things are more complex, since F also modifies the effect of β. Since the effective n ≃ −1 on the scales of interest where β can be measured, we see that any experiment which hopes to detect Λ through F ≃ 1.3 will need to be able to measure βa to a precision of rather better than ±0.1

**Figure 5.** Expected contours of likelihood in the F −β plane, for the case of a survey with negligible shot noise, for three values of β. The scaling is set so that σ(β) = 0.1 for β = 1. At each (β, F) point, the maximum-likelihood value of the power-spectrum amplitude has been chosen. The contour interval is ∆ln L = 1/2. The position of the true values of the parameters is marked by the cross; contours are shown for β = 0.3, 0.6 and 1.0

**Figure 6.** Expected contours of likelihood in the F − β plane, as for

**Figure 7.** Expected contours of likelihood in the ΩΛ − β plane for a 2df-type galaxy redshift survey of 250,000 galaxies to BJ < 19.5, obtained by combining cubic subsamples over a range of redshifts. Spatial flatness (Ωm + ΩΛ = 1) is assumed. The models shown assume true values for β of 0.25 and 0.75, and ΩΛ = 0, as indicated by the crosses. a constant dN/dz over the redshift range 0.3 ≤ z ≤ 2.2. For a given value of β, quasars put tighter constraints on a positive ΩΛ than galaxies. The larger volume, and in particular the higher redshift, more than offset the increased shot noise,…

Showing 6 of the paper's 7 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/9605017 by the authors.

References

Works cited (2)

[1]

Alcock C., Paczy´ nski B., 1979 Nature, 281, 358 Baugh C.M., Efstathiou G., 1993, MNRAS, 265, 145 Bucher M., Goldhaber A.S., Turok N., 1995, Phys. Rev. D, 52, 3314 Carroll S.M., Press W.H., Turner L.E., 1992, ARA&A, 30, 499 Cole S., Fisher K.B., Weinberg D.H., 1994, MNRAS, 267, 785 Cole S., Fisher K.B., Weinberg D.H., 1995, MNRAS, 275, 515 Davis M., Peebl...

View on Pith 1979
[2]

k′2‖ f 2‖ ] D ( k‖σp f‖ ) . (A7) Now we set F = f‖/f⊥, k′‖ = µ′k′ and σ′p = σp/f‖, which yields P ′(k′) = 1f 2⊥f‖ P0 [ k′ f⊥ √ 1 + µ′2 ( 1 F 2 − 1 )] × [ 1 + µ′2 ( 1 F 2 − 1 )]−2 × [ 1 + µ′2 ( β + 1 F 2 − 1 )]2 D ( k′µ′σ′p) . (A8) For a power spectrum which is locally close to a power law, with index n = d ln P/d ln k, we have P ′(k′) = 1f 3+n⊥ F P (k′) [...

View on Pith 1980

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Neighbor-density-weighted marked correlation functions improve FoM for Ωm–σ8 by 1.7–2.5× over standard 2PCF using Gaussian-process emulators on 129 w0waCDM+∑mν simulations.
The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Anisotropic galaxy clustering in Fourier-space
astro-ph.CO 2016-07 accept novelty 5.0 of 10

BOSS DR12 power-spectrum multipoles, including the hexadecapole, yield fσ8 = 0.482±0.053, 0.455±0.050, and 0.410±0.042 at z = 0.38, 0.51, and 0.61, with the highest-redshift point about 1.4 standard deviations below Planck.
The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: measurements of the growth of structure and expansion rate at z=0.57 from anisotropic clustering
astro-ph.CO 2012-03 accept novelty 5.0 of 10

Anisotropic clustering of BOSS CMASS galaxies at z=0.57 yields a 2.8 percent measurement of the Hubble rate, a 2.8 percent measurement of the comoving angular diameter distance, and a 16 percent measurement of the gro...
Constraining Scale-Dependent Growth in $f(R)$ Gravity with Future 21 cm Surveys
astro-ph.CO 2026-06 unverdicted novelty 3.0 of 10

Forecasts show that future 21 cm surveys can deliver moderate constraints on the scale-dependent growth index and HI bias in viable f(R) models.
Unveiling the Universe with Emerging Cosmological Probes
astro-ph.CO 2022-01 conditional novelty 3.0 of 10

A structured review of twelve emerging cosmological probes, each described through the same template of method, sample selection, systematics, current results, and forecasts, with an emphasis on combining probes to cr...
Cosmology and Fundamental Physics with the Euclid Satellite
astro-ph.CO 2016-06 accept novelty 2.0 of 10

Euclid's weak lensing and galaxy clustering surveys are forecast to pin the dark energy equation of state to about one percent and to test modified gravity, neutrino mass, and primordial non-Gaussianity.
The Cosmological Constant
astro-ph 2000-04 accept novelty 2.0 of 10

Independent supernova, microwave-background, and cluster-mass measurements converge on a flat universe with about 30% matter and 70% vacuum energy, leaving the origin of that vacuum energy unexplained.
Growth of Cosmic Structure: Probing Dark Energy Beyond Expansion
astro-ph.CO 2013-09 conditional novelty 1.0 of 10

Combining growth-of-structure measurements with distance measurements tests whether cosmic acceleration comes from dark energy or from modified gravity, and the planned surveys can reach roughly percent-level growth c...

How later papers used it

116 later papers cite it

Evidence: substantial. 110 of the citing papers share no author with it, and 21 of those carry a confident label from Pith's citation classifier.

Pith read 16 of the 132 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

7 later papers by other authors are listed below: 7 used its method or tool. 2 more papers by the same authors are listed: 2 used its method or tool.

Uses its method or tool 9 of 21 papers with this label · 2 by the same authors

Gave the value of q using this paper, with the same equations but inverse distortion parameters for LAEs misclassified as galaxies.

Correcting correlation functions for redshift-dependent interloper contamination (2021) · 0.83 · paraphrased: the licence does not allow quoting
Their relationship is given by [ref] [equation] Here, $α_iso$ and $α_AP$ are the isotropic dilation and the anisotropic distortion in the distances, respectively, and are related to rescaling of scales along and transverse to the line of sight as [equation] respectively, where $H$ is the Hubble parameter, $D_M$ is the comoving angular diameter distance, and $r_d$ is the sound horizon at radiation drag, and we distinguish between the fiducial choices and the actual values.[We need to include the difference between the fiducial and actual $r_d$, because it determines the position of the BAO in our template, and changes in its value are completely degenerate with $α_iso$.] We do the forecast in terms of $α_iso$ and $αAP$, which present lower degeneracies, and then transform to $α_∥$ and $α_perp$.

Cosmology with HI Intensity Mapping (2026) · Baryon Acoustic Scale · 0.92
Related quantities in the fiducial cosmology to those in the true cosmology as given by this paper.

COMET: Clustering Observables Modelled by Emulated perturbation Theory (2022) · 0.90 · paraphrased: the licence does not allow quoting
Gave the mapping from (k, mu) to (k', mu') using this paper.

The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: measurement of the BAO and growth rate of structure of the emission line galaxy sample from the anisotropic power spectrum between redshift 0.6 and 1.1 (2020) · 0.89 · paraphrased: the licence does not allow quoting
Used equation (A8) of this paper.

Geometric and growth rate tests of General Relativity with recovered linear cosmological perturbations (2016) · 0.89 · paraphrased: the licence does not allow quoting
Computed the distorted power spectrum wedges using relations corresponding to those of this paper expressed in terms of k and mu.

The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: Cosmological implications of the Fourier space wedges of the final sample (2016) · 0.88 · paraphrased: the licence does not allow quoting
We now define the different elements of [ref]: 4pt * $O(k,μ)$: This function encodes the BAO signal and can be written as O(k,μ) ≡B(k) O_lin(k'(k,μ)) + A(k) , where $O_lin(k') ≡_lin(k')/_lin(k')$ is the normalized wiggle spectrum evaluated at the rescaled wavenumbers [ref] k' = k √((1-μ^2)/q_⊥^2 + μ^2/q_∥^2) , with q_⊥≡D_A(z)/D_A^fid(z) , q_∥≡H^fid(z)/H(z) .

Searching for Light Relics with Large-Scale Structure (2017) · Modeling the Power Spectrum · 0.80

By the same authors

Recast the apparent power spectrum in a form outlined in equation (A8) from this paper.

Difficulties Distinguishing Dark Energy from Modified Gravity via Redshift Distortions (2009) (same authors) · 0.91 · paraphrased: the licence does not allow quoting
Then, we transform our measurements to how they would have appeared if we had assumed the same cosmology as the BOSS collaboration [ref] :

Cosmological constraints from the full-shape galaxy power spectrum in SDSS-III BOSS using the BACCO hybrid Lagrangian bias emulator (2024) (same authors) · The Alcock-Paczynski effect · 0.91

How the literature uses it

70 arXiv papers cite this in 83 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 46 sentences without a confident label

Uses its method or tool 12 papers

Modeling the signal.— Following the standard treatment of interloper projection in LIM [ref] , with the Alcock-Paczynski Jacobian [ref] and linear Kaiser redshift-space distortions [ref] , we model the auto power spectrum as [equation] where $P_N$ is the instrumental noise power spectrum and the index $L$ labels the spectral line contributing to the observed band, including the target and all interlopers.

Interlopers as Signal in Line Intensity Mapping · Introduction · 0.93
As a first step, we compute the Fisher Matrix for the set of parameters $p_i = { β_s, β_L, lnH, lnA, n_s, b, σ_v, Ω_m,0 H^2_0 }$; $A$ and $n_s$ represent the amplitude and scalar spectral index of the primordial perturbations, and $σ_v$ is the peculiar velocity dispersion we use to model the Finger of God effect [ref] , following [ref] .

How to Falsify the GR+LambdaCDM Model with Galaxy Redshift Surveys · Figure of Merit · 0.89
We fit the BAO power spectrum with the widely used model [ref] , i.e. [equation] where $δP_lin$ is the linear BAO power spectrum calculated from the difference between the input linear power spectra $P_wig$ and $P_now$. $G_model$ is given by equation ([ref]). $D_FoG$ denotes the Fingers of God (FoG) damping term, which is usually modelled as a Lorentzian function [ref] , i.e. [equation] The FoG effect originates from the virial motions of galaxies in dark matter haloes [ref] .

The Effect of Slitless Spectroscopic Redshift Uncertainty and Interlopers on BAO Measurements from CSST-like Samples · BAO fitting model · 0.97

Uses its data or values 1 paper

In this paper, we use $σ_p=350/H(z)$, as per [ref] .

Constraining Scale-Dependent Growth in $f(R)$ Gravity with Future 21 cm Surveys · Power Spectrum of 21cm Intensity · 0.94

Compares with it 1 paper

The bias is in agreement with values obtained by several authors [ref] and the velocity dispersion is consistent with the typical CMASS-galaxy-sample velocity-dispersion, $σ_p ≈240±50$ km/s [ref] .

Exploring cosmic homogeneity with the BOSS DR12 galaxy sample · Determining the bias and the velocity dispersion · 0.94

Cites it as background 22 papers

Because radial information comes from redshifts, AP measurements are traditionally limited by peculiar velocities, also known as comoving velocities [ref] .

Using galaxy pairs as cosmological tracers · Introduction · 0.83
In the autocorrelation, these two effects are degenerate and hard to separate [ref] , a difficulty compounded by the fact that non-linearities in the coupling of density and velocity fields limit the range of scales over which RSD models can be applied, e.g. [ref] .

Testing Low-Redshift Cosmic Acceleration with Large-Scale Structure · Introduction · 0.81
Such geometrical distortions, called the Alcock-Paczynski (AP) effect [ref] , must be accurately evaluated and calibrated for precise data analysis, and can be served as a cosmological probe on their own right (e.g., [ref] ).

Alcock-Paczynski effects on wide-angle galaxy statistics · Introduction · 0.91

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Alcock C., Paczy´ nski B., 1979 Nature, 281, 358 Baugh C.M., Efstathiou G., 1993, MNRAS, 265, 145 Bucher M., Goldhaber A.S., Turok N., 1995, Phys. Rev. D, 52, 3314 Carroll S.M., Press W.H., Turner L.E., 1992, ARA&A, 30, 499 Cole S., Fisher K.B., Weinberg D.H., 1994, MNRAS, 267, 785 Cole S., Fisher K.B., Weinberg D.H., 1995, MNRAS, 275, 515 Davis M., Peebl...

View on Pith 1979

[2] [2]

k′2‖ f 2‖ ] D ( k‖σp f‖ ) . (A7) Now we set F = f‖/f⊥, k′‖ = µ′k′ and σ′p = σp/f‖, which yields P ′(k′) = 1f 2⊥f‖ P0 [ k′ f⊥ √ 1 + µ′2 ( 1 F 2 − 1 )] × [ 1 + µ′2 ( 1 F 2 − 1 )]−2 × [ 1 + µ′2 ( β + 1 F 2 − 1 )]2 D ( k′µ′σ′p) . (A8) For a power spectrum which is locally close to a power law, with index n = d ln P/d ln k, we have P ′(k′) = 1f 3+n⊥ F P (k′) [...

View on Pith 1980