Spherical collapse with dark energy

Irit Maor

REVIEW

Spherical collapse with dark energy

Irit Maor

Astrophysics · Submitted 20 Feb 2006 · 17 references

Pith's verdict: A clear and honest review of the authors' own spherical-collapse model, but the claimed Rvir > Rta/2 signature is conditional on unresolved dark-energy clustering and virialization assumptions and is overstated as 'strong evidence'.

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arXiv IDastro-ph/0602441 (version 1)

Pith IDCKIRVJK3, Pith's permanent record of this paper and its review

CitationIrit Maor. "Spherical collapse with dark energy." arXiv:astro-ph/0602441 (2006). https://pith.science/paper/CKIRVJK3

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Claim paper

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Editor's note

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Colleague,

The short version: this paper is a well-written, honest review of Maor & Lahav (2005), and it does a good job of laying out the parameter space for dark energy in spherical collapse. But the headline signature — that observed Rvir/Rta > 1/2 would be strong evidence against a true cosmological constant — only holds if you accept two microphysical assumptions the paper itself says are unresolved: that dark energy clusters on the object scale (γ=0) and that it participates in virialization (Eq. 13). Without those, the effect disappears.

What is genuinely useful. The paper clearly defines the γ interpolation between homogeneous and fully clustered dark energy, and it carefully tracks the energy non-conservation that arises when a homogeneous, non-Λ fluid is included in the virial budget. The ENC correction (Eq. 17) is a nice, quantitative point that is easy to overlook. The figures are honest about the sensitivity: Figure 1 shows the full range of choices, and the text states plainly that a top-hat dark-energy profile is not stable and that dark-energy virialization is an open question.

Where it wobbles. The central claim in Figure 4 compares two different physical models, not two values of w. The dashed curve is the Lahav et al. Λ-CDM solution, where only matter virializes; the solid curve is a dynamical quintessence field with γ=0 that virializes with the matter. The 'discontinuity' at w=-1 is therefore not a property of the field equations — it is the result of the assumption that a dynamical mimic of Λ virializes while a true constant does not. The paper acknowledges this in the text, but the conclusion ('strong evidence') overstates the certainty. The signature is conditional, not decisive. Also, the novelty is limited: the main result comes from the authors' earlier JCAP paper, and the new material is an interpretive extension. That is fine for a review, but it should not be advertised as a new independent probe.

The citation pattern looks okay: the self-citation is appropriate, and the in-progress reference [15] is flagged as such.

Who should read it. Cosmologists who work on structure formation and dark energy, and who want a concise map of what changes when you let DE cluster or virialize. It is a useful pedagogical and reference document, not a settled observational test.

Recommendation. I would send it to a serious referee — the claim is important if true, and the referee should push for a rewriting that makes the conditional nature of the signature front and center. If the authors agree to state plainly that the signature is a prediction of a particular (currently unverified) clustering/virialization scenario, the paper would be a respectable contribution. As it stands, I would not cite it as evidence for anything except the existence of this modeling choice.

Referee report

Significance

If the central claim were secure, it would provide a novel observational discriminator between a genuine cosmological constant and dynamical dark energy with w≈-1 that does not require measuring w. The paper's phenomenological framework usefully organizes the various choices that enter spherical collapse calculations with dark energy, and it is explicit about several of the limitations. The strength is the clear presentation of a potential signature and the honest cataloguing of the underlying modeling choices. However, the signature is not a prediction from a single physical model but the result of comparing different model assumptions; the paper repeatedly acknowledges that the key assumptions (DE clustering, DE virialization) are unresolved. Thus the headline conclusion considerably overstates the certainty of the result, and the paper's main value is conditional and exploratory rather than as a robust observational prediction.

Major objections (4)

[Section V, Figure 4; Section VI] The claimed observational signature is not derived within a single, well-justified physical model; it results from comparing two different modeling assumptions. At w=-1, the dashed curve with the Lahav et al. circle assumes only matter virializes and the cosmological constant does not enter the energy budget, while the solid curve uses the full-system virialization of Eq. (13) with ρtot = ρmc + ρQc. The gap in x at w→-1 is therefore a consequence of the chosen virialization prescription, not of the equation of state. Since Section III explicitly states that whether dark energy participates in virialization is an open question, the conclusion in Section VI that 'an observational detection of virialized objects with Rvir > Rta/2 would be a strong evidence against a cosmological constant which is a true constant' is too strong. At most, the paper can claim that, under the assumptions γ=0 and full-system virialization, the result would conflict with the matter-only virialization treatment of a true constant. This conditional framing must be made explicit in the abstract and conclusion.
[Section II, Eq. (7); Section III, Eq. (13)] The main signature depends on two ad hoc assumptions: that the dark energy clusters with γ=0 and that it participates in virialization as part of ρtot. Neither assumption is derived from a microphysical model. The paper itself notes in Section II that a top-hat profile for dark energy is not a stable configuration, and in Section III that the virialization properties of dark energy are unresolved. Because Figure 3 shows that x varies by roughly 0.1 as γ goes from 0 to 1, the predicted excess Rvir > Rta/2 is not robust to plausible alternative choices. The paper should either provide a physical justification for γ=0 and for DE virialization, or explicitly present the result as a conditional statement that applies only within a particular phenomenological scenario.
[Section IV, Eq. (17)] The energy non-conservation correction relies on an auxiliary energy-conserving field that is matched to the real field at turnaround (ρ̃Qc(tta)=ρQc(tta)). This matching point is chosen without physical justification; matching at any other time would produce a different ΔU and hence a different corrected Rvir. Since the ENC-corrected curves in Figures 1 and 3 differ appreciably from the uncorrected ones, the quantitative predictions are sensitive to this arbitrary choice. The paper should explain why turnaround is the preferred matching epoch, or present the correction as a schematic estimate rather than a quantitative prediction.
[Section III, Eqs. (11)–(13); Section V] The potential energy U used in the virialization equations is never explicitly defined. The reader cannot reproduce Figures 1, 3, or 4 without knowing the functional form of U(ρmc) and U(ρQc) (e.g., whether pressure contributions of the dark energy are included in the potential, and how the top-hat radius enters). Providing the explicit expressions for U(ρ) would make the calculation transparent and checkable, and is necessary for the paper to stand alone from the earlier work of Maor and Lahav [1].

Minor objections (5)

[Section VI (final paragraph)] There is a typo in the last line: 'exiting challenge' should be 'exciting challenge'.
[References] Reference [15] is to an unpublished work 'in progress' by the author; such references are generally not verifiable and should either be removed or marked as a private communication.
[Figure 1 caption] The caption describes the dotted line as showing the effect of the ENC correction, but the legend in the figure does not label this line; adding a legend entry or a note in the caption would improve clarity.
[Section III (paragraph after Eq. (10))] The phrase 'the above energy budget should be applied to those components that virialize' is central to the paper, but it is presented as an assumption; the paper would benefit from a fuller discussion of what determines which components virialize, rather than leaving this as a binary choice.
[Section I] The abstract and the introduction describe the paper as a 'review,' but the paper also presents new results; this wording is somewhat misleading and could be adjusted to reflect that it is a research article with review elements.

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The paper

The paper claims

Bigger collapsed objects would rule out a true cosmological constant

The reading

This paper works out what happens to the final size of a collapsed, virialized object when dark energy is allowed to cluster and to take part in virialization. Using the spherical top-hat collapse model, it compares the ratio of the virial radius to the turnaround radius, $R_{\rm vir}/R_{\rm ta}$, for two kinds of dark energy: a true cosmological constant, which stays smooth and does not virialize, and a dynamical field with equation of state $w$ that clusters and enters the energy budget. The central result is that the dynamical field leaves objects systematically larger, with $R_{\rm vir} > R_{\rm ta}/2$ for the parameter range of interest, while the true constant gives $R_{\rm vir} = R_{\rm ta}/2$ independent of $w$. So an observed population of relaxed structures with $R_{\rm vir}$ exceeding half their turnaround radius would count as strong evidence against a constant vacuum energy, without needing to measure $w$.

What carries the argument

The central tool is the spherical top-hat collapse model, extended by a clustering parameter $\gamma$ in the continuity equation for the dark energy inside the perturbation, interpolating between homogeneous ($\gamma = 1$) and fully clustered ($\gamma = 0$) dark energy, and by a choice of virial-energy budget: matter only or the total density including dark energy. The ratio $x = R_{\rm vir}/R_{\rm ta}$, computed by energy conservation between turnaround and virialization, is the diagnostic. The parameter $q = (\rho_{Qc}/\rho_{mc})_{\rm ta}$ measures the dark-energy share at turnaround, and the plots of $x(w,q,\gamma)$ carry the argument, including a correction for energy non-conservation when homogeneous dark energy participates in virialization.

Extended reading notes

Core claim

The paper's claim, stated in its conclusions, is that an observational detection of virialized objects with $R_{\rm vir} > R_{\rm ta}/2$ would be strong evidence against a cosmological constant that is a true constant, regardless of the measured value of the equation of state of dark energy. The mechanism is the choice of which components enter the virial theorem: if only matter virializes, the presence of dark energy makes objects smaller and denser, whereas if the dark energy clusters (clustering parameter $\gamma = 0$) and the whole system virializes together, objects are larger and less dense. As $w$ approaches $-1$, a dynamical field that mimics the constant can still participate in virialization, producing $R_{\rm vir}/R_{\rm ta}$ above $1/2$, while a true constant cannot. Hence the ratio of final to turnaround radius becomes a discriminating observable that does not rely on measuring the value of the equation of state.

Load-bearing premise

The prediction that virialized objects can be larger than half their turnaround radius depends on dark energy actually clustering inside the perturbation ($\gamma = 0$) and on including it in the virialization energy budget; if dark energy stays smooth or does not virialize, the distinguishing excess disappears.

Editorial extensions

If this is right

If the claim is correct, measuring $R_{\rm vir}/R_{\rm ta}$ on relaxed galaxy clusters offers a dark-energy probe that does not depend on the value of $w$.
A confirmed population of virialized objects with $R_{\rm vir} > R_{\rm ta}/2$ would rule out a fundamental cosmological constant, even if the measured equation of state is very close to $-1$.
For a true cosmological constant, the matter-only virialization treatment is the right one, and no excess over $1/2$ is expected in $R_{\rm vir}/R_{\rm ta}$.
The predicted discontinuity between the clustered and homogeneous dark-energy treatments means that precise forecasts of the signature require knowing how dark energy behaves during nonlinear collapse and virialization.

Reading between the lines

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

If dark energy clusters only partially ($0 < \gamma < 1$), figure 3 of the paper suggests $R_{\rm vir}/R_{\rm ta}$ may jump between the two limiting behaviors; real structures could then sit near one endpoint depending on the efficiency of dark-energy clustering.
The same ratio could be tested in simulations that let dark energy respond to the local metric rather than only to the global expansion, offering a numerical check of the predicted discontinuity.
Because the signature strengthens with larger $q$, old, massive clusters that turned around late are the most promising observational targets, and the predicted excess should correlate with turnaround redshift.
An implicit extension is to the mass function built from the collapse threshold: counts of massive halos could carry the same distinguishing signature even where individual virial radii are difficult to measure.

Original abstract

I discuss the work of Maor and Lahav [1], in which the inclusion of dark energy into the spherical collapse formalism is reviewed. Adopting a phenomenological approach, I consider the consequences of - a) allowing the dark energy to cluster, and, b) including the dark energy in the virialization process. Both of these issues affect the final state of the system in a fundamental way. The results suggest a potentially differentiating signature between a true cosmological constant and a dynamic form of dark energy. This signature is unique in the sense that it does not depend on a measurement of the value of the equation of state of dark energy.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,213 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.44 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

I discuss the work of Maor and Lahav [1], in which the inclusion of dark energy into the spherical collapse formalism is reviewed. Adopting a phenomenological approach, I consider the consequences of - a) allowing the dark energy to cluster, and, b) including the dark energy in the virialization process. Both of these issues affect the final state of the system in a fundamental way. The results suggest a potentially differentiating signature between a true cosmological constant and a dynamic form of dark energy. This signature is unique in the sense that it does not depend on a measurement of the value of the equation of state of dark energy.

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	I discuss the work of Maor and Lahav [1], in which the inclusion of dark energy into the spherical collapse formalism is reviewed.	background 0.99	0.08	0.03	asserted 0.86
s1	Adopting a phenomenological approach, I consider the consequences of - a) allowing the dark energy to cluster, and, b) including the dark energy in the virialization process.	contribution 0.54	0.83	0.05	asserted 0.62
s2	Both of these issues affect the final state of the system in a fundamental way.	result 0.61	0.41	0.14	asserted 1.00
s3	The results suggest a potentially differentiating signature between a true cosmological constant and a dynamic form of dark energy.	interpretation 0.77	0.71	0.70	hedged possibility 1.00
s4	This signature is unique in the sense that it does not depend on a measurement of the value of the equation of state of dark energy.	interpretation 0.88	0.65	0.23	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 0.53 · evidence from argument 0.45.

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.** FIG. 1: The ratio of final to turnaround radii,

Showing 1 of the paper's 4 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:astro-ph/0602441 by the authors.

References

Works cited (17)

[15]

C. Wetterich, Astron. Astrophys. 301, 321 (1995) [arXiv:hep-th/9408025]. L. Amendola, Phys. Rev. D 62, 043511 (2000) [arXiv:astro-ph/9908023]

View on Pith Pith review arXiv 1995
[1]

I. Maor and O. Lahav, JCAP 0507, 003 (2005) [arXiv:astro-ph/0505308]

View on Pith Pith review arXiv 2005
[2]

J. E. Gunn and J. R. I. Gott, Astrophys. J. 176, 1 (1972)

View on Pith 1972
[3]

formalism. It is, therefore, widely used in present day interpret ation of data sets. The generalization of the spherical collapse formalism to include addit ional forms of en- ergy has been subject to numerous studies [1, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Lahav et al [4] generalized the formalism to a universe composed of ordinary matte r and a cosmological...

View on Pith
[4]

W. H. Press and P. Schechter, Astrophys. J. 187, 425 (1974)

View on Pith 1974
[5]

O. Lahav, P. B. Lilje, J. R. Primack and M. J. Rees, Mon. Not . Roy. Astron. Soc. 251, 128 (1991)

View on Pith 1991
[6]

L. M. Wang and P. J. Steinhardt, Astrophys. J. 508, 483 (1998) [arXiv:astro-ph/9804015]

View on Pith Pith review arXiv 1998
[7]

I. T. Iliev and P. R. Shapiro, Mon. Not. Roy. Astron. Soc. 325, 468 (2001). [arXiv:astro-ph/0101067]

View on Pith Pith review arXiv 2001
[8]

N. N. Weinberg and M. Kamionkowski, Mon. Not. Roy. Astron . Soc. 341, 251 (2003) [arXiv:astro-ph/0210134]. 13

View on Pith Pith review arXiv 2003
[9]

R. A. Battye and J. Weller, Phys. Rev. D 68, 083506 (2003) [arXiv:astro-ph/0305568]

View on Pith Pith review arXiv 2003
[10]

D. F. Zeng and Y. H. Gao, arXiv:astro-ph/0412628

View on Pith Pith review arXiv
[11]

D. F. Mota and C. van de Bruck, Astron. Astrophys. 421, 71 (2004) [arXiv:astro-ph/0401504]

View on Pith Pith review arXiv 2004
[12]

C. Horellou and J. Berge, Mon. Not. Roy. Astron. Soc. 360, 1393 (2005) [arXiv:astro-ph/0504465]

View on Pith Pith review arXiv 2005
[13]

P. Wang, arXiv:astro-ph/0507195

View on Pith Pith review arXiv
[14]

R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. L ett. 80, 1582 (1998) [arXiv:astro-ph/9708069]

View on Pith Pith review arXiv 1998
[16]

S. Dutta and I. Maor, in progress

View on Pith
[17]

S. Engineer, N. Kanekar and T. Padmanabhan, Mon. Not. Ro y. Astron. Soc. 314, 279 (2000) [arXiv:astro-ph/9812452]

View on Pith Pith review arXiv 2000

How the literature uses it

4 arXiv papers cite this in 4 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 · 1 sentence without a confident label

Cites it as support 2 papers

This qualitative view is supported by the work of [ref] and [ref] , where they found that for a two-component system made up of matter and dark energy, the virialization radius will increase when the Dark Energy is allowed to dynamically participate in the virialization, a sign of an additional attractive force (due to the increased virial kinetic energies occurring in a deeper potential well).

Cosmic Acceleration from Causal Backreaction in a Smoothly Inhomogeneous Universe · INTRODUCTION AND MOTIVATIONS · 1.00
For example, energy conservation is violated between turn-around and collapse when $w≠-1, -1/3$ [ref] .

On the implementation of the spherical collapse model for dark energy models · Spherical collapse model · 0.97

Cites it as background 1 paper

Studies of non-linear structure formation in cosmology, namely spherical top hat collapse models, often assume that there is no influence of the cosmological background on a finite domain which has disconnected from the background dynamics to justify using the dynamics of a collapsing Friedmann-like patch embedded in a cosmological model (see e.g. [ref] ).

The role of shell crossing on the existence and stability of trapped matter shells in spherical inhomogeneous \Lambda-CDM models · Introduction · 0.96

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] [15]

C. Wetterich, Astron. Astrophys. 301, 321 (1995) [arXiv:hep-th/9408025]. L. Amendola, Phys. Rev. D 62, 043511 (2000) [arXiv:astro-ph/9908023]

View on Pith Pith review arXiv 1995

[2] [1]

I. Maor and O. Lahav, JCAP 0507, 003 (2005) [arXiv:astro-ph/0505308]

View on Pith Pith review arXiv 2005

[3] [2]

J. E. Gunn and J. R. I. Gott, Astrophys. J. 176, 1 (1972)

View on Pith 1972

[4] [3]

formalism. It is, therefore, widely used in present day interpret ation of data sets. The generalization of the spherical collapse formalism to include addit ional forms of en- ergy has been subject to numerous studies [1, 4, 5, 6, 7, 8, 9, 10, 11, 12]. Lahav et al [4] generalized the formalism to a universe composed of ordinary matte r and a cosmological...

View on Pith

[5] [4]

W. H. Press and P. Schechter, Astrophys. J. 187, 425 (1974)

View on Pith 1974

[6] [5]

O. Lahav, P. B. Lilje, J. R. Primack and M. J. Rees, Mon. Not . Roy. Astron. Soc. 251, 128 (1991)

View on Pith 1991

[7] [6]

L. M. Wang and P. J. Steinhardt, Astrophys. J. 508, 483 (1998) [arXiv:astro-ph/9804015]

View on Pith Pith review arXiv 1998

[8] [7]

I. T. Iliev and P. R. Shapiro, Mon. Not. Roy. Astron. Soc. 325, 468 (2001). [arXiv:astro-ph/0101067]

View on Pith Pith review arXiv 2001

[9] [8]

N. N. Weinberg and M. Kamionkowski, Mon. Not. Roy. Astron . Soc. 341, 251 (2003) [arXiv:astro-ph/0210134]. 13

View on Pith Pith review arXiv 2003

[10] [9]

R. A. Battye and J. Weller, Phys. Rev. D 68, 083506 (2003) [arXiv:astro-ph/0305568]

View on Pith Pith review arXiv 2003

[11] [10]

D. F. Zeng and Y. H. Gao, arXiv:astro-ph/0412628

View on Pith Pith review arXiv

[12] [11]

D. F. Mota and C. van de Bruck, Astron. Astrophys. 421, 71 (2004) [arXiv:astro-ph/0401504]

View on Pith Pith review arXiv 2004

[13] [12]

C. Horellou and J. Berge, Mon. Not. Roy. Astron. Soc. 360, 1393 (2005) [arXiv:astro-ph/0504465]

View on Pith Pith review arXiv 2005

[14] [13]

P. Wang, arXiv:astro-ph/0507195

View on Pith Pith review arXiv

[15] [14]

R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. L ett. 80, 1582 (1998) [arXiv:astro-ph/9708069]

View on Pith Pith review arXiv 1998

[16] [16]

S. Dutta and I. Maor, in progress

View on Pith

[17] [17]

S. Engineer, N. Kanekar and T. Padmanabhan, Mon. Not. Ro y. Astron. Soc. 314, 279 (2000) [arXiv:astro-ph/9812452]

View on Pith Pith review arXiv 2000