Cluster Abundance Constraints on Quintessence Models

Limin Wang; Paul J. Steinhardt (University of Pennsylvania)

REVIEW

Cluster Abundance Constraints on Quintessence Models

Limin Wang, Paul J. Steinhardt (University of Pennsylvania)

Astrophysics · Submitted 2 Apr 1998 · 22 references

Pith's verdict: The w-dependent γ fit and the quintessence growth index are worth having; the fit ships without residuals or a stated domain, and Eq. 13 contradicts its own ΛCDM limit.

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CitationLimin Wang, Paul J. Steinhardt (University of Pennsylvania). "Cluster Abundance Constraints on Quintessence Models." arXiv:astro-ph/9804015 (1998). https://pith.science/paper/3R4PQ33T

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The part worth keeping is Eq. 19 and Eq. 13. Prior cluster-normalization work (Eke-Cole-Frenk, Viana-Liddle, Pen) was ΛCDM-only; extending γ to w, and adding explicit n and h dependence through Θ, is what turns cluster counts into a probe of the dark-energy equation of state rather than just Ωm. The growth-index derivation in Appendix B and the constant-w spherical-collapse fit in Eq. A11 are the pieces I would actually pull into my own work. Section 3.2 is also unusually candid for this literature: it itemizes why their σ8 differs from five prior groups, including the observation that other groups folded the redshift integration into fβ. That bookkeeping raised my confidence in their care, and it is the reason I would take the numbers seriously at all. Now the soft spots, roughly in order of how much they matter. First, Eq. 13 as printed fails its own stated limit. Put w = -1 into the leading terms: 3/5 + 1/2 = 1.1, but Appendix B says α ≈ 6/11 for ΛCDM. Either the coefficient or the limit is misprinted. Since Eq. 14 feeds every growth factor and hence every σ8 here, this needs checking against the numerical integration they claim agrees to 1%. It is probably a typo, but it is the one place a reader with a pencil can catch them out. Second, the fit arrives with no residuals, no grid, and no stated range, while the abstract says 'any models.' That is an overreach for a four-coefficient linear form; give the residuals and the Ωm, w, n, h box it was fit over. Third, the non-clustering assumption in §2.1 is what makes the w-dependence meaningful, and it rests on Caldwell, Dave & Steinhardt 1998, listed as in preparation. That is the load-bearing citation and it is not yet in the literature. Related: §2.2 concedes BBKS fails for w > -1/3, yet Fig. 2 plots w = 0 and w = -1/6, and model (6) in Fig. 1 is w = -1/6. The shallow-w end mixes methods exactly where the smoothness assumption is weakest. Fourth, minor: 20%, 10% and 15% in quadrature is 27%, not the quoted 20%, and the 95% label is asserted rather than derived. This is for anyone normalizing cluster surveys or fitting w. Send it to a referee; the fixes are all addressable and the formula deserves to exist in citable, corrected form.

Referee report

Significance

If the fit holds over its claimed domain, this is a genuinely useful result: it converts cluster counts from a constraint on Ω_m alone into a constraint that carries the dark-energy equation of state, and it does so with an explicit, short formula that other groups can apply without re-running the pipeline. The paper is also unusually explicit about its own methodology, and several parts are strong on their own terms. The growth-index derivation in Appendix B, culminating in Eq. (13), is a self-contained analytic result accompanied by a stated accuracy claim (better than 1% for 1-Ω_m between 0 and 0.8), which is exactly the kind of checkable statement a fitting formula should carry. The comparison in §3.2 is candid and specific: the authors identify five distinct sources of disagreement with White et al., Eke et al., Viana & Liddle, Pen, and Kitayama & Suto, including the redshift-integration correction and the distinction between f_β and the simulation-calibrated f̃_β. That discussion is a real service to the literature. Finally, §4 and §5 make a falsifiable prediction: models degenerate in the CMB power spectrum are spread out in A(M_1.5) by nearly 2 dex, so a measurement of theA

Major objections (5)

[Abstract, sentence s2; §3.1, Eqs. (18)-(19)] The abstract claims a general expression for γ that applies to 'any models' with CDM plus Λ or quintessence, but Eq. (19) is presented with four fitted coefficients and no residuals, no fitting grid, and no stated domain in Ω_m, w, n_s, or h. This is the paper's central deliverable, so its accuracy over the claimed domain is load-bearing rather than a presentation detail. Please add a table of residuals (or fractional residuals in σ8) across the fitted parameter grid, state the grid spacing and the ranges actually sampled, and replace 'any models' in the abstract with the explicit domain. As written, a user cannot tell whether Eq. (19) is good to 1% or 10% at, say, w = -0.5 and Ω_m = 0.4.
[§2.2 and §3.1; Fig. 2; Fig. 1, model (6)] Section 2.2 states that the BBKS approximation is reliable only for constant or slowly varying w < -1/3, and that for w > -1/3 there is no general fitting formula and P(k) must be obtained numerically. Yet Eq. (19) is fitted across w from -1 to 0, Fig. 2 plots curves at w = -1/6 and w = 0, and model (6) of Fig. 1 is w = -1/6. The paper therefore applies a single fitting formula across a boundary where its own stated method changes. Either restrict Eq. (19) to w ≤ -1/3 and say so, or state explicitly how P(k) was computed for the w > -1/3 points and demonstrate that the two treatments match at the boundary. Without that, the shallow-w end of the fit and the corresponding best-fit selections in Fig. 2 are not established.
[§2.1, first paragraph after Eq. (4)] The w-dependence of γ rests on the assumption that the quintessence component does not cluster on scales below 100 Mpc, so that its only effect on cluster abundance is through the background expansion. This assumption is stated and supported by a citation to Caldwell, Dave, & Steinhardt 1998, which the reference list gives as 'in preparation.' The assumption is also least secure precisely where the fit is pushed hardest: as w → 0 the field behaves increasingly like matter, and clustering on cluster scales is no longer obviously negligible. Please either provide a quantitative estimate of the residual clustering of Q on the relevant scales as a function of w (even an order-of-magnitude argument from the sound speed or the Jeans length would help), or restrict the fit to the range of w where the assumption is demonstrably safe. As it stands, the physical input that generates the entire w-dependence of Eq. (19) is asserted rather than shown.
[§3.2, item (5); §3.3] The normalization f_β = 1.1 is obtained by calibrating the authors' calculation to the Eke et al. (1996) and Pen (1997) simulations, and §3.3 assigns it a 20% uncertainty. Those simulations were run for a limited set of cosmologies, and the paper does not state which. Applying the same f_β unchanged to QCDM models with w near 0 is an extrapolation that is not tested here, and it propagates directly into the σ8 normalization of Eq. (18). Please state the cosmologies against which f_β was calibrated and, if they do not span the QCDM range, either qualify the claim or assign an additional model-dependent systematic. Relatedly, Eq. (18) folds all sources of error into a single ±0.1 that is independent of Ω_m and w; §3.3's error propagation in Eq. (16) suggests the fractional error should vary with the model, so a constant absolute error deserves justification.
[§3.1, Eq. (18) and the definition of Θ] The dependence on spectral index and Hubble constant is collapsed into a single linear combination Θ = (n_s - 1) + (h - 0.65), with equal unit coefficients and no shown derivation. This asserts that n_s and h enter identically and linearly over the plausible ranges, which is a nontrivial claim about the shape of the temperature function. Please show the fit residuals when n_s and h are varied separately, or state the ranges over which the linearization was verified. If the degeneracy between n_s and h is only approximate, the single parameter Θ will hide a systematic that matters at the ±0.1 level quoted in Eq. (18).

Minor objections (9)

[Eq. (9)] The top-hat window function contains a stray 'r, .' after the closing parenthesis; the expression should be cleaned up so the Fourier transform of the real-space top hat is unambiguous.
[Eq. (17)] The left-hand side is written as dn(T,z)/dz while the right-hand side ends in a factor dT, and the temperature argument appears in the mass function as M(T,z)T. The differential bookkeeping here should be rewritten so that the quantity being differentiated and the integration variable are consistent.
[Eq. (16)] The quantity u3 is defined as -d ln σ(R)'/d ln R, but the prime on σ(R) is never defined. Presumably it denotes a derivative with respect to mass or a second logarithmic slope; please state it explicitly.
[§3.2] The enumerated list contains two items both labelled '(5)'. The second should be renumbered, and the surrounding text adjusted if any later sentence refers back to item numbers.
[Fig. 1 and its caption] Model (6) has n_s = 1.1 while models (1)-(5) all have n_s = 1. Since the figure is used to illustrate a trend with w, the reader cannot tell how much of the offset of the top curve is due to w and how much to the tilt. Either add a same-tilt comparison or note in the caption how much of the displacement is attributable to n_s.
[§4, final paragraph] The text states that current data indicate A(M_1.5 = 8×10^14 h^-1 Mpc) ≳ -5 at the 3σ level 'with mean equal to -1.7.' A mean of -1.7 is far above the quoted bound of -5, so the reader cannot tell which quantity is being compared to the model curves in Fig. 1. Please clarify whether -1.7 is the central value and -5 the 3σ lower limit, and state the corresponding upper limit.
[Eq. (21)] The relation between M and M_1.5 involves the ratio 178/Δ_c and the exponent p/(3-p) with p = 0.64 adopted from Carlberg, Yee, & Ellingson (1997). Please confirm that the same p is appropriate for the QCDM profiles considered here, and check that the normalization reproduces M = M_1.5 in the appropriate limit.
[References] Several results that the argument leans on are cited to works listed as 'in preparation' (Caldwell, Dave, & Steinhardt 1998; Wang, Caldwell, Ostriker, & Steinhardt 1998; Huey et al. 1998). Where these support specific quantitative claims rather than general context, please indicate what is already available in the literature so the reader can assess the support independently.
[Eq. (13) and Appendix B] The stated 1% accuracy of the growth index is quoted for 1-Ω_m between 0 and 0.8, which is presumably where the truncation of the expansion in (1-Ω_m) is adequate. It would help to say explicitly that the 1% figure is a property of that truncated expansion and to note how the accuracy degrades beyond 1-Ω_m = 0.8, since some of the models in Fig. 1 sit at Ω_m ≈ 0.5.

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The paper

The paper claims

Cluster counts can be compressed into one fitted exponent that tells dark energy apart from a cosmological constant.

The reading

Rich galaxy clusters are the largest gravitationally bound objects in the universe, so their number as a function of mass and redshift tracks the amplitude of the mass power spectrum. The paper's aim is to turn that fact into a single usable formula: the observed cluster abundance fixes the combination $0.5$ at the level of $0.5$ in the combination $0.5$, more precisely $0.5$, written as $0.5$, where the exponent $0.5$ is model-dependent. Wang and Steinhardt work out what $0.5$ actually is for cold dark matter mixed with either a cosmological constant or quintessence — a dynamical, slowly evolving negative-pressure component — and fit it as a function of the dark-energy equation of state $0.5$, the matter density $0.5$, the spectral index $0.5$ and the Hubble constant $0.5$. The payoff is that cluster counts stop being only a measurement of $0.5$ and become a constraint on whether the dark energy evolves. They then show that the change in cluster abundance between redshift zero and one differs by up to two orders of magnitude across models that a cosmic microwave background experiment alone cannot separate.

What carries the argument

The load-bearing object is the exponent $0.5$ in $0.5$. It is not derived in closed form but assembled from three pieces: the virial overdensity $0.5$ at collapse, computed by evolving a spherical overdensity in a background with constant $0.5$ and fitted as $0.5$; the linear growth index $0.5$, obtained from a series expansion of the growth equation in $0.5$ and giving $0.5$ for a cosmological constant and $0.5$ for an open universe; and the Press-Schechter temperature function integrated over the redshift at which each cluster virializes, with a merger correction averaged over two published estimates. The second key object is the evolution index $0.5$, the slope of $0.5$ versus redshift over $0.5$, which the paper proposes as the discriminator that survives when the microwave background cannot distinguish the models.

Extended reading notes

Core claim

The central result is a two-line fitting formula. For flat cold dark matter models with either a cosmological constant or quintessence, the cluster abundance requires $0.5$ with $0.5$ where $0.5$ collects the tilt and Hubble-constant dependence, $0.5$; for open models the corresponding exponent is $0.5$. Because $0.5$ enters with a coefficient of $-0.22$, a dark-energy component that is not exactly a cosmological constant shifts the required amplitude of density fluctuations in a way that is separable from a change in $0.5$ or $0.5$ over the range the authors consider. The paper also supplies the machinery behind the fit: a mass-temperature relation for a virialized cluster that accounts for the (non-clustering) dark-energy density, a growth index $0.5$ that reproduces the numerical growth factor to better than one percent for $0.5$ between zero and $0.8$, and a redshift-integrated Press-Schechter count rather than a single-epoch evaluation. Combining the resulting band with COBE normalization leaves a family of acceptable models at each $0.5$, and the paper identifies which ones sit in the overlap.

Load-bearing premise

The whole derivation assumes the quintessence field does not clump on scales smaller than about 100 Mpc, so that its only influence on a cluster is through the background expansion rate; the paper states this and cites a companion study still in preparation, and the entire dependence of $0.5$ on $0.5$ rests on it.

Editorial extensions

If this is right

Any flat dark-energy model with constant or slowly varying equation of state has its required $0.5$ fixed by $0.5$, $0.5$, $0.5$ and $0.5$ through Eq. (19), so cluster counts become a constraint on the dark energy rather than only on the matter density.
Models that produce identical microwave background power spectra still differ in the predicted slope of cluster abundance versus redshift, with the paper's worked examples spanning roughly two orders of magnitude in abundance at $0.5$.
If the evolution index $0.5$ can be measured to better than about $0.5$, cluster evolution becomes a practical way to separate quintessence from a cosmological constant and to pin down $0.5$ and $0.5$.
The reanalysis argues that earlier normalizations were biased high by up to about ten percent, because most groups evaluated the mass-temperature relation only at redshift zero instead of integrating over the virialization epoch.

Reading between the lines

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

The same fitting-formula approach could be inverted: given a future cluster survey that measures $0.5$ and its evolution precisely, Eq. (19) becomes a direct estimator of $0.5$ at low redshift, competitive with supernova distance measurements and statistically independent of them.
Because the exponent depends on $0.5$ only through the combination $0.5$, the cluster constraint alone cannot break the tilt-Hubble degeneracy; it needs an independent $0.5$ or $0.5$ measurement, which is worth stating explicitly when comparing cluster bounds to other probes.
If the dark energy does turn out to cluster at cluster scales, the failure mode is not random: it would show up as a systematic mismatch between the low-redshift normalization and the high-redshift evolution slope, which is a checkable signature rather than a silent bias.

Original abstract

The abundance of rich clusters is a strong constraint on the mass power spectrum. The current constraint can be expressed in the form $\sigma_8 \Omega_m^{\gamma} = 0.5 \pm 0.1$ where $\sigma_8$ is the $rms$ mass fluctuation on 8 $h^{-1}$ Mpc scales, $\Omega_m$ is the ratio of matter density to the critical density, and $\gamma$ is model-dependent. In this paper, we determine a general expression for $\gamma$ that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence (a time-evolving, spatially-inhomogeneous component with negative pressure) including dependence on the spectral index $n$, the Hubble constant $h$, and the equation-of-state of the quintessence component $w$. The cluster constraint is combined with COBE measurements to identify a spectrum of best-fitting models. The constraint from the evolution of rich clusters is also discussed.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.60 · compares with prior results 0.27 · mentions code or data availability 0.02.

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The abundance of rich clusters is a strong constraint on the mass power spectrum. The current constraint can be expressed in the form $\sigma_8 \Omega_m^{\gamma} = 0.5 \pm 0.1$ where $\sigma_8$ is the $rms$ mass fluctuation on 8 $h^{-1}$ Mpc scales, $\Omega_m$ is the ratio of matter density to the critical density, and $\gamma$ is model-dependent. In this paper, we determine a general expression for $\gamma$ that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence (a time-evolving, spatially-inhomogeneous component with negative pressure) including dependence on the spectral index $n$, the Hubble constant $h$, and the equation-of-state of the quintessence component $w$. The cluster constraint is combined with COBE measurements to identify a spectrum of best-fitting models. The constraint from the evolution of rich clusters is also discussed.

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	The abundance of rich clusters is a strong constraint on the mass power spectrum.	background 1.00	0.06	0.05	asserted 1.00
s1	The current constraint can be expressed in the form $\sigma_8 \Omega_m^{\gamma} = 0.5 \pm 0.1$ where $\sigma_8$ is the $rms$ mass fluctuation on 8 $h^{-1}$ Mpc scales, $\Omega_m$ is the ratio of matter density to the critical density, and $\gamma$ is model-dependent.	background 0.90	0.12	0.11	asserted 1.00
s2	In this paper, we determine a general expression for $\gamma$ that applies to any models with a mixture of cold dark matter plus cosmological constant or quintessence (a time-evolving, spatially-inhomogeneous component with negative pressure) including dependence on the spectral index $n$, the Hubble constant $h$, and the equation-of-state of the quintessence component $w$.	contribution 0.97	0.95	0.21	asserted 1.00
s3	The cluster constraint is combined with COBE measurements to identify a spectrum of best-fitting models.	method 0.65	0.88	0.19	asserted 1.00
s4	The constraint from the evolution of rich clusters is also discussed.	contribution 0.90	0.53	0.04	unclear 0.52

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from analysis of existing data 0.72.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** — The evolution index of cluster abundance for six mod

**Figure 2.** — COBE-normalized σ8 as a function of ΩQ for six constant w models with ns = 1 and h = 0.65. They are (from top to bottom): (1) w = −1; (2) w = −2/3; (3) w = −1/2; (4) w = −1/3; (5) w = −1/6; (6) w = 0. The highlighted regions indicate where the x-ray cluster abundance constraints imposed by Eq. (18) overlap the COBE constraint. Best-fit models correspond to the overlap region

**Figure 3.** — Cosmic microwave background (CMB) anisotropy meas

**Figure 4.** — The evolution of cluster abundance may break down th

Figures reproduced from arXiv:astro-ph/9804015 by the authors.

References

Works cited (22)

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Huey, G., Wang, L., Dave, R., Caldwell, R.R., & Steinhardt, P .J. 1998,in preparation

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[22]

White, S.D.M., Efstathiou, G., & Frenk, C.S. 1993,MNRAS,26 2,1023 This preprint was prepared with the AAS L ATEX macros v4.0. – 14 – A. Spherical Model We study a spherical overdensity with uniform matter densit y ρcluster and radius R in a background that satisﬁes the Friedmann equation: ( ˙a a )2 = 8πG3 (ρb + ρQ) (A1) where a is the scale factor, ρb is ...

View on Pith 1993

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astro-ph.CO 2014-09 conditional novelty 4.0 of 10

Vacuum energy models built from H-squared and H-dot plus a constant term fit current cosmological data about as well as Lambda-CDM, while models lacking the constant term are excluded by growth and cluster-count data.
How to optimally parametrize deviations from General Relativity in the evolution of cosmological perturbations
astro-ph.CO 2010-02 conditional novelty 4.0 of 10

The MGCAMB (mu, eta) parametrization of modified gravity is shown to be internally consistent across all linear scales, and a PCA-based argument says future surveys need scale-dependent parametrizations.
Measuring neutrino masses and dark energy with weak lensing tomography
astro-ph 2006-03 conditional novelty 4.0 of 10

Fisher forecasts show that LSST-like weak lensing tomography plus Planck can reach sigma(sum m_nu) around 0.04 eV, robust to running spectral index, time-varying dark energy, and Neff.
The effective field theory of inflation/dark energy and the Horndeski theory
gr-qc 2014-04 accept novelty 3.0 of 10

Horndeski scalar-tensor gravity is a sub-class of the effective field theory of cosmological perturbations, with its background dynamics fixed by three time-dependent functions, and this review derives the resulting i...
Spherical collapse with dark energy
astro-ph 2006-02 conditional novelty 3.0 of 10

Virialized structures larger than half their turnaround radius would indicate that dark energy is dynamical rather than a true cosmological constant, independent of measuring its equation of state.
Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution
gr-qc 2017-05 accept novelty 2.0 of 10

Modified gravity theories supply viable mathematical frameworks for inflation, bounces, and dark energy eras that match observational data.
The Physics of Cosmic Acceleration
astro-ph.CO 2009-03 accept novelty 2.0 of 10

Cosmic acceleration is established at high significance, and the leading explanations are a cosmological constant, a slowly rolling scalar field, or a modification of general relativity, none of which current data dis...
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

160 later papers cite it

Evidence: substantial. 160 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 242 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: 6 used its method or tool and 1 compared its own results with it.

Uses its method or tool 6 of 12 papers with this label

Constructed its new growth index following the same ansatz as prescribed by this paper.

GCG Parametrization for Growth Function and Current Constraints (2011) · 0.94 · paraphrased: the licence does not allow quoting
Followed this paper, supposing the solution only slightly differs from EdS, to solve the virial condition perturbatively.

Halo collapse: virialization by shear and rotation in dynamical dark-energy models. Effects on weak-lensing peaks (2018) · 0.82 · paraphrased: the licence does not allow quoting
Used the recipe by this paper and another work to define it.

On the implementation of the spherical collapse model for dark energy models (2017) · 1.00 · paraphrased: the licence does not allow quoting
Fitted f^(1,2) for all models using the fitting formulae proposed in this paper.

Non-linear Structure Formation for Dark Energy Models with a Steep Equation of State (2019) · 0.97 · paraphrased: the licence does not allow quoting
Adopted the notations in this paper to investigate the evolutions of a and R.

Spherical collapse model with and without curvature (2009) · 0.86 · paraphrased: the licence does not allow quoting
Followed the same procedure of this paper to define x and y and redefine the dark matter and radiation densities.

Unified Dark Matter Scalar Field Models (2010) · 0.86 · paraphrased: the licence does not allow quoting

Compares with it 1 of 6 papers with this label

Compared its COBE normalized sigma_8 with the one from this paper.

Cluster number counts in quintessence models (2010) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

103 arXiv papers cite this in 168 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 · 109 sentences without a confident label

Contests it 1 paper

(B11) of [ref] .] γ_smooth = 3(1-w)/5-6w + 3 (1-w) (2-3w) /2 (5 - 6 w)^2 (5 - 12 w) Ω_Q + O(Ω_Q^2) ≃0.545 + 0.007 Ω_Q + (1+w) ( 0.025 + 0.005 Ω_Q ) + … , where in the second line we have expanded as well with respect to $1+w$.

Cosmological structure formation with clustering quintessence · Linear growth rate · contest 0.80

Builds on it 1 paper

This construction generalizes to smooth and relativistic conditions the so called Newtonian “top hat” or “spherical collapse models” that are widely used in astrophysical literature [ref] , and that tend to overlook the details of the embedding of the overdensity or void into the FLRW background.

Quasi-local variables in spherical symmetry: numerical applications to dark matter and dark energy sources · Smooth and relativistic “top hat” models. · 0.86

Uses its method or tool 9 papers

Following [ref] ,[As also noted by [ref] , different strategies have also been adopted leading to the same results, at least for a $Λ$CDM model; see however [ref] .] supposing that the solution is only slightly different from the EdS, the virial condition $[U + R/2∂U/∂R]_vir = U_ta$, can be solved perturbatively, yielding at first order [equation] where [equation] Finally, the virial overdensity with respect to the background density is defined as in [ref] , [equation] where $ζ_ta$ has been defined above, and $x_c$ and $x_vir$ are respectively the scale factors at collapse and virialization normalised at the turn-around epoch.

Halo collapse: virialization by shear and rotation in dynamical dark-energy models. Effects on weak-lensing peaks · Ordinary SCM and virialization in Einstein-de Sitter and LCDM models · 0.82
To do so, we use the recipe by [ref] and [ref] .

On the implementation of the spherical collapse model for dark energy models · Spherical collapse model · 1.00
A simple phenomenological model to quantify such a deviation has been suggested by Linder [ref] (see also Wang & Steinhardt [ref] ), parameterising the growth of linear density perturbations as [equation] where $δ$ is the fractional density fluctuation and $a$ is the scale factor; this approximation typically holds to a precision of $∼0.1%$.

Difficulties Distinguishing Dark Energy from Modified Gravity via Redshift Distortions · Signatures of modified gravity · 0.85

Uses its data or values 5 papers

For the $Λ$CDM cosmology, Refs. [ref] found $γ=0.55$.

Can dark energy explain a high growth index? · Introduction · 0.98
This empirical formula gives a value of $γ_L≃0.55$ for $Λ$CDM [ref] .

Exploring the Growth Index $\gamma_L$: Insights from Different CMB Dataset Combinations and Approaches · Introduction · 0.86
It was found that $γ=6/11$ for the $ΛCDM$ [ref] .

Dynamical dark energy models with singularities in the view of the forthcoming results of the growth observations · Scale dependent growth function · 0.86

Compares with it 6 papers

In [ref] we have generalized the two original works developed for the EdS model to $Λ$CDM and dark-energy models with an arbitrary equation-of-state $w_de$, proving that the results obtained with the new shear-and-rotation-based virialization model are qualitatively similar to those obtained with the standard recipe of [ref] : $Δ_vir$ is time-independent for the EdS model and becomes time-dependent for the other models, actually returning smaller values with [ref] and [ref] virialization recipes instead of the virial theorem.

Tidal virialization of dark matter haloes with clustering dark energy · Introduction · 0.82
Since the qualitative evolution of the radius and peculiar velocity does not change with respect to the $Λ$CDM model, we limit the discussion to the redshift evolution of $Δ_vir$ and compare the standard results obtained by using the prescription by [ref] at the virialization time $z_vir$ with those of [ref] and [ref] .

Halo collapse: virialization by shear and rotation in dynamical dark-energy models. Effects on weak-lensing peaks · Evolution of the virial overdensity in smooth dark-energy models · 0.95
This contrasts with the conventional approximation for $f(a)$ [ref] given by

Unmodified Gravity · Background Dynamics · 0.96

Cites it as background 31 papers

Since then, the spherical collapse model has been applied to quintessence scenarios [ref] and more recently to interacting DE–CDM models [ref] .

Coupled Quintessence and the Halo Mass Function · Critical density for collapse, $δ_⋆(z)$ · 0.80
This parametrization [ref] provides an excellent fit to the evolution equation for $δ(a)$ in the small scale (sub-Hubble) approximation δ̈ + 2 H δ̇ - 4πG ρ_m δ=0 where an overdot denotes the derivative with respect to time and $ρ_m$ is the matter density.

New Parametrization for the Scale Dependent Growth Function in General Relativity · Introduction · 0.80
The growth of cosmic structure is commonly characterized by the growth index $γ$, defined through the relation $f ≡Ω_m^γ$, where $f$ is the growth rate and $Ω_m$ is the matter density parameter [ref] .

Linear matter density perturbations in the $\Lambda_{\rm s}$CDM model: Examining growth dynamics and addressing the $S_8$ tension · INTRODUCTION · 0.99

Discussion

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