A semi-analytical approach to cosmic void evolution

A.G. Nikiforov; A.N. Baushev; M.V. Barkov

REVIEW 2 major objections 6 minor 22 references

Cosmic voids stay only moderately empty and keep their asphericity while growing, with underdensity almost independent of shape.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · grok-4.5

2026-07-10 07:48 UTC pith:QXY524O7

load-bearing objection Clean White–Silk integration for spheroidal voids; μ>0.5 and early non-linearity are solid inside the model, but the isolated-homogeneous exterior is load-bearing and untested against tides. the 2 major comments →

arxiv 2607.08414 v1 pith:QXY524O7 submitted 2026-07-09 astro-ph.CO

A semi-analytical approach to cosmic void evolution

A.N. Baushev , A.G. Nikiforov , M.V. Barkov This is my paper

classification astro-ph.CO

keywords cosmic voidsellipsoidal collapsenon-linear structure formationLambda-CDMdensity contrastvoid abundancesemi-analytic modeling

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper models a cosmic void as an isolated homogeneous spheroidal ellipsoid sitting in an otherwise smooth expanding Lambda-CDM universe, then integrates the equations of motion for its two semi-axes from redshift 500 to the present. The calculation shows that the void expands anisotropically and slowly becomes rounder, yet remains clearly aspherical; its final underdensity is essentially the same whether the void is prolate or oblate. Non-linear effects already appear by redshift about 8, when the density contrast is only about 10 percent. The central result is that most voids retain a substantial amount of matter, with residual density more than half the cosmic mean, rather than being nearly empty. That claim matters because voids fill much of the cosmic volume and are used as cosmological probes; if they are systematically denser than many simulations report, abundance statistics and dark-energy constraints drawn from them must be revised.

Core claim

When a void is evolved as a homogeneous spheroidal underdensity from a small initial contrast at z=500, its eccentricity falls only modestly (from roughly 0.87 to 0.81), its final underdensity is virtually independent of that eccentricity, and the residual matter fraction satisfies mu greater than 0.5 for typical initial conditions. Non-linearity already produces a 10 percent departure from linear theory by z approximately 8.

What carries the argument

The closed system of ordinary differential equations for the two semi-axes of a homogeneous spheroidal ellipsoid, driven by the exact gravitational potential coefficients A(e) and B(e) together with the Lambda-CDM background expansion.

Load-bearing premise

The exterior of the void is assumed to remain perfectly homogeneous and unperturbed for all time, so the boundary stays ellipsoidal and the interior density stays uniform.

What would settle it

Direct comparison of residual densities and shapes of moderate voids (smoothing scale roughly 8 h^{-1} Mpc) in high-resolution N-body simulations or in galaxy surveys against the predicted mu greater than 0.5 and the slow eccentricity decline; a population of voids that are systematically emptier than mu approximately 0.5 would falsify the central claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Volume-fraction statistics of voids can be computed with the fast spherical mapping without appreciable error from shape.
Linear extrapolation of void depth systematically overestimates how empty voids become by the present day.
Observed voids that appear extremely empty may still contain substantial unseen matter (dwarfs, gas, dark matter).
Constraints on dark energy or modified gravity that assume nearly empty voids need to be re-calibrated for mu greater than 0.5.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the homogeneous-exterior assumption is the main reason the model disagrees with simulations, controlled experiments that gradually add exterior tidal fields should recover the emptier voids seen in N-body work.
The early onset of non-linearity suggests that void-based cosmological forecasts should include non-linear corrections already at intermediate redshifts rather than only at z=0.
The same ODE machinery can be reused to map how residual density scales with cosmological parameters, offering a cheap prior for void abundance in non-standard cosmologies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Clean White–Silk integration for spheroidal voids; μ>0.5 and early non-linearity are solid inside the model, but the isolated-homogeneous exterior is load-bearing and untested against tides.

read the letter

The paper’s real addition is a careful numerical integration of the White–Silk ODEs for axisymmetric underdense spheroids from z=500 to today. You get three concrete, new numbers: eccentricity falls only slowly (≈0.87→0.81), final underdensity μ is essentially independent of shape (Fig. 7), and a 10 % departure from linear growth already appears by z≃8. Those trajectories are not in the earlier analytic papers and the spherical reduction (Appendix A) is cleanly derived.

What they do well is keep the math transparent. Equations (2.2)–(2.3), the A(e)/B(e) coefficients, mass conservation, and growing-mode initial velocities are standard and consistently integrated; prolate and oblate cases sit on top of the spherical result for μ. The abundance mapping with the spherical model is therefore justified inside their framework, and the claim that typical voids keep μ>0.5 follows directly from that mapping for R=8 h⁻¹ Mpc.

The soft spot is exactly the one the stress-test flags, and it is not minor: the entire μ>0.5 conclusion rests on the idealization that the exterior stays perfectly homogeneous and unperturbed for all time. That is stated up front (following White & Silk 1979) but never stress-tested against exterior tides or shell-crossing—the very effects that empty voids in N-body runs. Because the paper never quantifies how large a tidal field would have to be before the ODE trajectory for μ shifts by tens of percent, the discrepancy with simulations is left unresolved. Self-citation to the authors’ prior “voids are not empty” papers is present but the ODE work itself is independent, so circularity is mild.

This is for people who already work with analytic void models or who need a transparent benchmark against which to judge simulation emptiness. It is not yet a result that can be used to recalibrate void cosmology without further checks. I would send it to peer review; the calculation is reproducible enough and the limitation is clear enough that a referee can force the necessary discussion of exterior tides. Worth reading if you care about the analytic–simulation tension; not something I would cite as settled cosmology until that tension is addressed.

Referee Report

2 major / 6 minor

Summary. The paper models a cosmic void as an isolated homogeneous spheroidal (axisymmetric) ellipsoid in a flat ΛCDM background and numerically integrates the White–Silk equations of motion for the semi-axes from z=500 to z=0. It reports that eccentricity decreases only modestly (from ≈0.87 to ≈0.81 for typical initial conditions), that the final underdensity μ is essentially independent of initial eccentricity, that a ~10% departure from linear growth already appears by z≃8, and that the non-linear mapping from primordial ε₀ implies most voids retain μ>0.5 today. A spherical reduction is derived in Appendix A and used for a simple abundance estimate on the R=8 h⁻¹ Mpc scale.

Significance. If the homogeneous-isolated idealization is adequate, the work supplies a clean, computationally cheap non-linear mapping μ(ε₀) that can be inserted into excursion-set or Gaussian-peak statistics, and it quantifies early non-linearity and the slow isotropization of voids. The explicit demonstration that final density contrast is shape-independent (Fig. 7) and the transparent spherical reduction (Eqs. 3.2–3.3) are useful technical results. The tension with N-body residual densities is already well known; the paper’s main contribution is therefore the controlled semi-analytic calculation itself rather than a resolution of that tension.

major comments (2)

The central cosmological claim (Abstract and §4 point 4) that “the majority of voids … contain a significant amount of matter, μ>0.5” rests entirely on the closed ODE system (2.2)–(2.3) for a homogeneous spheroid whose exterior remains perfectly homogeneous and unperturbed (Introduction and §2, following White & Silk 1979). The manuscript never quantifies how large an exterior tidal field or density gradient must be before the trajectory for μ changes by tens of percent. Because N-body simulations, which include those effects, systematically produce μ~0.1, the discrepancy is left unresolved and the abundance statement remains conditional on an untested idealization. A short sensitivity test (e.g., an external tidal term added to (2.2)–(2.3), or a comparison with a known spherical top-hat plus shell-crossing calculation) is needed before the claim can be regarded as robust.
Section 3.1.1 converts the non-linear mapping into a volume fraction 2P(μ) via the Gaussian erfc formula (3.6)–(3.7) evaluated at a single smoothing scale R=8 h⁻¹ Mpc. No justification is given for the choice of R, no σ_R value is stated, and no comparison is made with the void size function measured in simulations or surveys. Without that calibration the statement that voids with μ≲0.6 occupy only a quarter of the volume cannot be assessed, and the link between the ODE results and the observational claim is incomplete.

minor comments (6)

Notation for eccentricity is inconsistent: the abstract and Fig. 2 use e, while the text of §2 switches between e and ε for the same quantity; ε is also used for density contrast. A single symbol for eccentricity should be fixed throughout.
Figure 1 caption states that b1 expands faster than b3 “because of the stronger outward gravitational pull in the equatorial plane of a prolate void,” yet the same figure also shows the oblate case; the sentence should be generalized or split.
In §3 the ratio ξ is defined once as D/(1-μ) and once as D/ε; both appear in the text and in the Fig. 6 caption. Clarify which definition is plotted.
The phrase “triaxial throughout the whole evolution” (Fig. 2 caption and §3) is imprecise for an axisymmetric (spheroidal) model; replace with “aspherical” or “spheroidal.”
Appendix A, Eq. (A.5): the factor 1/3 that converts ε into the peculiar expansion rate h is standard for the growing mode, but a one-sentence reminder that it follows from δ∝ a would help readers who do not immediately recall the derivation.
References [5] and [6] are the authors’ own prior analytic estimates; a brief comparison table or sentence quantifying how the present numerical μ differs from those earlier closed-form results would strengthen the narrative.

Circularity Check

1 steps flagged

Mild self-citation framing of the µ>0.5 claim as confirmation of the authors’ own prior analytic estimates; the ODE integration itself is independent and not circular by construction.

specific steps

self citation load bearing [Abstract; §3.1.1; Conclusions point 4]
"Notably, our calculations show that the majority of voids are not strongly underdense and contain a significant amount of matter, µ>0.5. … Our results suggest a less pronounced underdensity, µ∼0.5, and this result is supported by analytical models of the central void density [5, 6]. … Our semi-analytical calculation confirms that voids are not so empty, Δm ≃ -50% (µ > 0.5). This result is in agreement with theoretical estimates [5, 6]"

The central cosmological claim µ>0.5 is repeatedly framed as confirmation of the same authors’ prior analytic papers [5,6] that reached the identical conclusion under closely related homogeneous-void assumptions. While the present ODE trajectory is independently computed, the interpretive weight given to the result against N-body simulations rests partly on this self-consistent chain rather than on external, model-independent evidence.

full rationale

The core derivation is a numerical integration of the White–Silk ODEs (2.2)–(2.3) for a homogeneous spheroid (with potential coefficients A(e), B(e) taken from the literature, only lightly corrected in the authors’ concurrent [16]), started from linear growing-mode initial conditions at z=500 and evolved under mass conservation (2.9). Final underdensity µ(ε0), its near-independence of eccentricity (Fig. 7), and the early departure from linear growth (Figs. 5–6) are direct outputs of that integration; they are not fitted parameters, not redefined from the target, and not forced by any uniqueness theorem. The only mild circularity is interpretive: the abstract and Conclusions present the µ>0.5 result as “confirming” the authors’ earlier analytic estimates [5,6] that used related idealizations of void emptiness. That framing is not load-bearing for the calculation, so the score remains low (2). No self-definitional loops, no data fits renamed as predictions, and no ansatz smuggled solely via self-citation appear.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The central claims rest on a classical ellipsoidal collapse model plus standard ΛCDM background parameters. No new particles or forces are introduced. Free parameters are the usual cosmological numbers plus hand-chosen initial eccentricity, initial contrast, start redshift, and smoothing scale for abundance. Load-bearing domain assumptions are isolation, interior/exterior homogeneity, and growing-mode initial velocities.

free parameters (5)

Ωm,p = 0.306
Present-day matter density fixed at 0.306; enters every acceleration equation and the growth factor D(a).
initial eccentricity e0 = ≈0.866 (fiducial)
Set to ≈0.866 (and scanned 0.001–0.9) by hand; controls A(e),B(e) and the shape evolution that is a main result.
initial density contrast ε0 = 0.002 (fiducial)
Scanned values 0.002, 0.02, 0.2 at z=500; maps non-linearly to present μ and drives the abundance calculation.
start redshift z0 = 500
Evolution begun at z=500; chosen deep in matter era so linear growing-mode ICs apply.
smoothing scale R for abundance = 8 h^{-1} Mpc
R=8 h^{-1} Mpc chosen as ‘typical for moderate voids’; sets σR and therefore 2P(μ).

axioms (6)

domain assumption Exterior of the void remains homogeneous and unperturbed for all time; boundary stays ellipsoidal and interior density stays uniform.
Stated in Introduction and §2 as the White–Silk idealization; closes the system to two ODEs.
domain assumption Initial velocities equal the linear growing-mode solution in a matter-dominated universe (Eqs. 2.10–2.11).
Sets ḃi(t0); required for the early-time match to D(a).
domain assumption Mass inside the ellipsoid is conserved, yielding μ(t) from the product of semi-axes (Eq. 2.9).
Standard for homogeneous ellipsoids; converts axis evolution into density contrast.
domain assumption Background expansion is flat ΛCDM with negligible radiation (scale factor Eq. 2.1).
Standard cosmology; used throughout.
standard math Gravitational potential coefficients A(e), B(e) of a homogeneous spheroid (Lin–Mestel–Shu / Nikiforov et al.).
Taken from classical potential theory; enter the accelerations (2.2)–(2.3).
domain assumption Primordial density field is Gaussian; volume fraction given by erfc with a factor-of-two cloud-in-cloud fudge (Eqs. 3.6–3.7).
Standard Press–Schechter-style argument used for void abundance in §3.1.1.

pith-pipeline@v1.1.0-grok45 · 15735 in / 3623 out tokens · 36192 ms · 2026-07-10T07:48:33.194328+00:00 · methodology

0 comments

read the original abstract

We present a theoretical study of the non-linear evolution of cosmic voids -- underdense regions that occupy a large fraction of the observable Universe. We model a void as an isolated homogeneous spheroidal (axisymmetric) ellipsoid embedded in a homogeneous $\Lambda$CDM universe. Starting from a small initial density contrast at redshift $z=500$, we numerically integrate the equations of motion for the ellipsoid semi-axes and follow their evolution to the present epoch. We examine the anisotropic expansion of the void and the corresponding change in its shape, characterised by the eccentricity $e$. We find that the void non-sphericity always decreases, but rather slowly: the eccentricity drops from $e\approx0.87$ at $z=500$ to $e\approx0.81$ at $z=0$. Thus the void becomes rounder but remains aspherical throughout the evolution. The evolution and final value of the void underdensity are virtually independent of the void's eccentricity. The nonlinearity of void evolution becomes apparent very early: a ten percent deviation from the linear regime occurs already at $z\simeq8$, when $\varepsilon = \Delta\rho/\rho\sim10\%$. Notably, our calculations show that the majority of voids are not strongly underdense and contain a significant amount of matter, $\mu>0.5$.

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 11 internal anchors

[1]

Kirshner, J

R.P. Kirshner, J. Oemler, Augustus, P.L. Schechter and S.A. Shectman,A Survey of the Bootes Void,ApJ314(1987) 493

work page 1987
[2]

Clampitt and B

J. Clampitt and B. Jain,Lensing measurements of the mass distribution in sdss voids,Monthly Notices of the Royal Astronomical Society454(2015) 3357–3365

work page 2015
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Y. Fang, N. Hamaus and DES Collaboration,Dark Energy Survey year 1 results: the relationship between mass and light around cosmic voids,MNRAS490(2019) 3573 [1909.01386]

work page internal anchor Pith review Pith/arXiv arXiv 2019
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Contarini, G

S. Contarini, G. Verza and A. Pisani,The era of precision cosmology with voids,A&AR34 (2026) 1 [2601.14362]

work page arXiv 2026
[5]

The central region of a void: an analytical solution

A.N. Baushev,The central region of a void: an analytical solution,MNRAS504(2021) L56 [2104.01359]

work page internal anchor Pith review Pith/arXiv arXiv 2021
[6]

How empty are the voids?

A.N. Baushev,How empty are the voids?,arXiv e-prints(2024) arXiv:2409.02780 [2409.02780]

work page internal anchor Pith review Pith/arXiv arXiv 2024
[7]

Bertschinger,The self-similar evolution of holes in an Einstein-de Sitter universe,ApJS58 (1985) 1

E. Bertschinger,The self-similar evolution of holes in an Einstein-de Sitter universe,ApJS58 (1985) 1

work page 1985
[8]

Blumenthal, L.N

G.R. Blumenthal, L.N. da Costa, D.S. Goldwirth, M. Lecar and T. Piran,The Largest Possible Voids,ApJ388(1992) 234

work page 1992
[9]

van de Weygaert and E

R. van de Weygaert and E. van Kampen,Voids in Gravitational Instability Scenarios - Part One - Global Density and Velocity Fields in an Einstein - De-Sitter Universe,MNRAS263 (1993) 481

work page 1993
[10]

Disruption of Dark Matter Substructure: Fact or Fiction?

F.C. van den Bosch, G. Ogiya, O. Hahn and A. Burkert,Disruption of Dark Matter Substructure: Fact or Fiction?,MNRAS474(2018) 3043 [1711.05276]

work page internal anchor Pith review Pith/arXiv arXiv 2018
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The central cusps in dark matter halos: fact or fiction?

A.N. Baushev and S.V. Pilipenko,The central cusps in dark matter halos: Fact or fiction?, Physics of the Dark Universe30(2020) 100679 [1808.03088]

work page internal anchor Pith review Pith/arXiv arXiv 2020
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Precision cosmography with stacked voids

G. Lavaux and B.D. Wandelt,Precision Cosmography with Stacked Voids,ApJ754(2012) 109 [1110.0345]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[13]

Baryons in the CosmicWeb of IllustrisTNG - I: Gas in Knots, Filaments, Sheets and Voids

D. Martizzi, M. Vogelsberger, M.C. Artale, M. Haider, P. Torrey, F. Marinacci et al.,Baryons in the Cosmic Web of IllustrisTNG - I: gas in knots, filaments, sheets, and voids,MNRAS486 (2019) 3766 [1810.01883]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[14]

White and J

S.D.M. White and J. Silk,The growth of aspherical structure in the universe: Is the Local Supercluster an unusual system?,ApJ231(1979) 1

work page 1979
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C.C. Lin, L. Mestel and F.H. Shu,The Gravitational Collapse of a Uniform Spheroid.,ApJ 142(1965) 1431

work page 1965
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Nikiforov, A.N

A.G. Nikiforov, A.N. Baushev and M.V. Barkov,The Impact of the Eccentricity on the Collapse of an Ellipsoid into a Black Hole,Universe11(2025) 95 [2412.14358]

work page arXiv 2025
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Gorbunov and V.A

D.S. Gorbunov and V.A. Rubakov,Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory, World Scientific Publishing Co (2011)

work page 2011
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In the Beginning: The First Sources of Light and the Reionization of the Universe

R. Barkana and A. Loeb,In the beginning: the first sources of light and the reionization of the universe,Phys. Rep.349(2001) 125 [astro-ph/0010468]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[19]

Press and P

W.H. Press and P. Schechter,Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation,ApJ187(1974) 425

work page 1974
[20]

Voids in the 2dF Galaxy Redshift Survey

F. Hoyle and M.S. Vogeley,Voids in the Two-Degree Field Galaxy Redshift Survey,ApJ607 (2004) 751 [astro-ph/0312533]. – 12 –

work page internal anchor Pith review Pith/arXiv arXiv 2004
[21]

The abundance of voids and the excursion set formalism

E. Jennings, Y. Li and W. Hu,The abundance of voids and the excursion set formalism, MNRAS434(2013) 2167 [1304.6087]

work page internal anchor Pith review Pith/arXiv arXiv 2013
[22]

The radius, at which a galaxy group stops the Hubble stream, and the group mass: an exact analytical solution

A.N. Baushev,The radius, at which a galaxy group stops the Hubble stream, and the group mass: an exact analytical solution,MNRAS490(2019) L38 [1907.08716]. A Derivation of equation (3.2) We use the method proposed in [22] to derive equation (3.2). Let us consider a homogeneous flat Friedmann universe withΩm,p = 0.306,Ω Λ + Ωm,p = 1. There is a single smal...

work page internal anchor Pith review Pith/arXiv arXiv 2019

[1] [1]

Kirshner, J

R.P. Kirshner, J. Oemler, Augustus, P.L. Schechter and S.A. Shectman,A Survey of the Bootes Void,ApJ314(1987) 493

work page 1987

[2] [2]

Clampitt and B

J. Clampitt and B. Jain,Lensing measurements of the mass distribution in sdss voids,Monthly Notices of the Royal Astronomical Society454(2015) 3357–3365

work page 2015

[3] [3]

Y. Fang, N. Hamaus and DES Collaboration,Dark Energy Survey year 1 results: the relationship between mass and light around cosmic voids,MNRAS490(2019) 3573 [1909.01386]

work page internal anchor Pith review Pith/arXiv arXiv 2019

[4] [4]

Contarini, G

S. Contarini, G. Verza and A. Pisani,The era of precision cosmology with voids,A&AR34 (2026) 1 [2601.14362]

work page arXiv 2026

[5] [5]

The central region of a void: an analytical solution

A.N. Baushev,The central region of a void: an analytical solution,MNRAS504(2021) L56 [2104.01359]

work page internal anchor Pith review Pith/arXiv arXiv 2021

[6] [6]

How empty are the voids?

A.N. Baushev,How empty are the voids?,arXiv e-prints(2024) arXiv:2409.02780 [2409.02780]

work page internal anchor Pith review Pith/arXiv arXiv 2024

[7] [7]

Bertschinger,The self-similar evolution of holes in an Einstein-de Sitter universe,ApJS58 (1985) 1

E. Bertschinger,The self-similar evolution of holes in an Einstein-de Sitter universe,ApJS58 (1985) 1

work page 1985

[8] [8]

Blumenthal, L.N

G.R. Blumenthal, L.N. da Costa, D.S. Goldwirth, M. Lecar and T. Piran,The Largest Possible Voids,ApJ388(1992) 234

work page 1992

[9] [9]

van de Weygaert and E

R. van de Weygaert and E. van Kampen,Voids in Gravitational Instability Scenarios - Part One - Global Density and Velocity Fields in an Einstein - De-Sitter Universe,MNRAS263 (1993) 481

work page 1993

[10] [10]

Disruption of Dark Matter Substructure: Fact or Fiction?

F.C. van den Bosch, G. Ogiya, O. Hahn and A. Burkert,Disruption of Dark Matter Substructure: Fact or Fiction?,MNRAS474(2018) 3043 [1711.05276]

work page internal anchor Pith review Pith/arXiv arXiv 2018

[11] [11]

The central cusps in dark matter halos: fact or fiction?

A.N. Baushev and S.V. Pilipenko,The central cusps in dark matter halos: Fact or fiction?, Physics of the Dark Universe30(2020) 100679 [1808.03088]

work page internal anchor Pith review Pith/arXiv arXiv 2020

[12] [12]

Precision cosmography with stacked voids

G. Lavaux and B.D. Wandelt,Precision Cosmography with Stacked Voids,ApJ754(2012) 109 [1110.0345]

work page internal anchor Pith review Pith/arXiv arXiv 2012

[13] [13]

Baryons in the CosmicWeb of IllustrisTNG - I: Gas in Knots, Filaments, Sheets and Voids

D. Martizzi, M. Vogelsberger, M.C. Artale, M. Haider, P. Torrey, F. Marinacci et al.,Baryons in the Cosmic Web of IllustrisTNG - I: gas in knots, filaments, sheets, and voids,MNRAS486 (2019) 3766 [1810.01883]

work page internal anchor Pith review Pith/arXiv arXiv 2019

[14] [14]

White and J

S.D.M. White and J. Silk,The growth of aspherical structure in the universe: Is the Local Supercluster an unusual system?,ApJ231(1979) 1

work page 1979

[15] [15]

C.C. Lin, L. Mestel and F.H. Shu,The Gravitational Collapse of a Uniform Spheroid.,ApJ 142(1965) 1431

work page 1965

[16] [16]

Nikiforov, A.N

A.G. Nikiforov, A.N. Baushev and M.V. Barkov,The Impact of the Eccentricity on the Collapse of an Ellipsoid into a Black Hole,Universe11(2025) 95 [2412.14358]

work page arXiv 2025

[17] [17]

Gorbunov and V.A

D.S. Gorbunov and V.A. Rubakov,Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory, World Scientific Publishing Co (2011)

work page 2011

[18] [18]

In the Beginning: The First Sources of Light and the Reionization of the Universe

R. Barkana and A. Loeb,In the beginning: the first sources of light and the reionization of the universe,Phys. Rep.349(2001) 125 [astro-ph/0010468]

work page internal anchor Pith review Pith/arXiv arXiv 2001

[19] [19]

Press and P

W.H. Press and P. Schechter,Formation of Galaxies and Clusters of Galaxies by Self-Similar Gravitational Condensation,ApJ187(1974) 425

work page 1974

[20] [20]

Voids in the 2dF Galaxy Redshift Survey

F. Hoyle and M.S. Vogeley,Voids in the Two-Degree Field Galaxy Redshift Survey,ApJ607 (2004) 751 [astro-ph/0312533]. – 12 –

work page internal anchor Pith review Pith/arXiv arXiv 2004

[21] [21]

The abundance of voids and the excursion set formalism

E. Jennings, Y. Li and W. Hu,The abundance of voids and the excursion set formalism, MNRAS434(2013) 2167 [1304.6087]

work page internal anchor Pith review Pith/arXiv arXiv 2013

[22] [22]

The radius, at which a galaxy group stops the Hubble stream, and the group mass: an exact analytical solution

A.N. Baushev,The radius, at which a galaxy group stops the Hubble stream, and the group mass: an exact analytical solution,MNRAS490(2019) L38 [1907.08716]. A Derivation of equation (3.2) We use the method proposed in [22] to derive equation (3.2). Let us consider a homogeneous flat Friedmann universe withΩm,p = 0.306,Ω Λ + Ωm,p = 1. There is a single smal...

work page internal anchor Pith review Pith/arXiv arXiv 2019