pith. sign in

arxiv: 2606.27076 · v1 · pith:J2M5BPZJnew · submitted 2026-06-25 · 🌌 astro-ph.CO · hep-ph

Spherical Collapse and Halo Formation in a Cosmology with Decaying Dark Matter and a Semi-Cosmographic Dark Energy

Mohit Yadav , Tapomoy Guha Sarkar This is my paper

Pith reviewed 2026-06-26 04:04 UTC · model grok-4.3

classification 🌌 astro-ph.CO hep-ph
keywords decaying dark matterspherical collapsehalo abundancedark energy reconstructionnonlinear structure formationgrowth of structurecosmological constraints
0
0 comments X

The pith

Decaying dark matter and reconstructed dark energy keep the spherical collapse threshold close to standard while altering massive halo abundances.

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

The paper examines nonlinear structure formation when dark matter decays into relativistic radiation and dark energy follows a reconstruction from the expansion history. It shows that background constraints can be carried into the nonlinear regime through spherical collapse, leaving the critical density threshold nearly unchanged from the usual value even as the dark energy equation of state varies. The clearest effects appear in the predicted counts of massive halos, where combined changes in growth rate become visible. This supplies a way to translate linear-era measurements into limits on the decay lifetime and dark energy parameters using halo data.

Core claim

The reconstructed dark-energy equation of state can deviate from the Lambda CDM value while the critical density threshold for collapse remains close to its standard prediction. The most pronounced signatures emerge in the abundance of massive halos, reflecting modifications to the growth of structure driven by both dark-matter decay and dynamical dark energy. Combining background constraints with halo mass function measurements yields joint limits on the decaying dark matter lifetime and dark-energy parameters.

What carries the argument

Spherical collapse model applied to background evolution in a decaying dark matter plus semi-cosmographic dark energy cosmology.

If this is right

  • The critical density threshold for collapse stays near its standard value despite the presence of decaying dark matter and dynamical dark energy.
  • Modifications to structure growth produce the strongest changes in the number density of massive halos.
  • Halo mass function data supply complementary constraints on the decay lifetime and dark energy parameters beyond those from background measurements alone.
  • The framework directly connects a reconstructed dark energy sector to nonlinear predictions for cosmic structure.

Where Pith is reading between the lines

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

  • Halo abundance measurements could serve as an independent route to bound the dark matter decay rate once background data fix the expansion history.
  • Numerical N-body simulations of this specific decay-plus-reconstruction model would test whether the spherical collapse approximation captures all relevant effects.
  • The same reconstruction technique for dark energy could be paired with other nonlinear probes such as weak lensing or redshift-space distortions to cross-check growth modifications.

Load-bearing premise

The background evolution constrained by expansion history data can be directly propagated into the nonlinear regime using the standard spherical collapse model without additional modifications to the perturbation growth equations or collapse dynamics arising from the decay process or the reconstructed dark energy.

What would settle it

A measured halo mass function at the high-mass end that differs substantially from the abundance predicted by feeding the background-constrained growth factor into the standard spherical collapse equations would show that the direct propagation step fails.

Figures

Figures reproduced from arXiv: 2606.27076 by Mohit Yadav, Tapomoy Guha Sarkar.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic flowchart of the semi-cosmographic method [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The figure on the left shows the marginalized DESI DR1 ( [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Posterior reconstruction of the smooth dark-energy [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Reconstructed critical linear collapse threshold [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Reconstructed virial overdensity ∆ [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the reconstructed σ(M, z) for the pos￾teriors. We have used the CLASS one-body DDM along with the semi-cosmographic dynamical dark energy ev￾erywhere. The ΛCDM reference over the mass range is also shown for comparison. The variance decreases with increasing mass and with increasing redshift. Rel￾ative to ΛCDM, the DDM posterior median is close to the reference, mildly enhanced over part of the low- … view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Differential halo mass function (d [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

We investigate nonlinear structure formation in a cosmological model combining one-body decaying dark matter (DDM) with a semi-cosmographic reconstruction of dark energy. In this scenario, a nonrelativistic dark-matter component decays into relativistic dark radiation with decay rate $\Gamma=\tau_{\rm ddm}^{-1}$, while the dark-energy sector is reconstructed directly from the expansion history rather than being fixed to a cosmological constant. Using DESI DR1 BAO and compressed ShapeFit measurements, we constrain the background evolution and propagate the resulting posterior into the nonlinear regime through spherical collapse and halo abundance calculations. This provides a unified framework connecting a reconstructed dark-energy sector and decaying dark matter (DDM) to the nonlinear formation of cosmic structures. We find that the reconstructed dark-energy equation of state can deviate from the $\Lambda$CDM value, $w=-1$, while the critical density threshold for collapse remains close to its standard prediction. The most pronounced signatures emerge in the abundance of massive halos, reflecting modifications to the growth of structure driven by both dark-matter decay and dynamical dark energy. By combining DESI DR1 clustering constraints with halo mass function measurements from the DESI Legacy Imaging Surveys DR9, we obtain joint constraints on the DDM lifetime and dark-energy parameters, demonstrating that halo abundances provide a powerful complementary probe of non-standard dark-sector physics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper investigates nonlinear structure formation in a model combining one-body decaying dark matter (DDM, decaying into relativistic dark radiation with rate Γ) and a semi-cosmographic dark-energy reconstruction. DESI DR1 BAO and ShapeFit data constrain the background evolution; posteriors are propagated via spherical collapse into halo abundance calculations. The critical collapse threshold δ_c is reported to remain close to its ΛCDM value, while the dominant signatures appear in the abundance of massive halos. Joint constraints on the DDM lifetime and dark-energy parameters are obtained by combining the DESI clustering posteriors with halo mass function measurements from DESI Legacy Imaging Surveys DR9.

Significance. If the unmodified spherical-collapse propagation is justified, the work shows that halo abundances can furnish a complementary probe of DDM decay and dynamical dark energy, extending background constraints into the nonlinear regime and yielding joint limits on the dark sector.

major comments (1)
  1. [Spherical collapse and halo abundance calculations (methods and results sections)] The central claim that δ_c remains close to the standard value and that halo abundances yield robust joint constraints rests on propagating the DESI-constrained background into the nonlinear regime with the unmodified spherical-collapse model. The DDM decay introduces a sink term in the DM continuity equation; the corresponding perturbation equations acquire additional source terms from the decay into relativistic species that affect the evolution of the density contrast and velocity divergence inside an overdensity. No explicit derivation retaining these Γ-dependent terms through turnaround and virialization is provided, so the reported closeness of δ_c may reflect an incomplete dynamical model rather than a robust outcome. This assumption is load-bearing for the halo-abundance predictions and the joint constraints.
minor comments (2)
  1. The abstract and introduction refer to a 'semi-cosmographic reconstruction' without specifying the functional form or number of free parameters used for the dark-energy equation of state; an explicit parameterization would improve reproducibility.
  2. Figure captions and table headers should explicitly state whether the plotted halo mass functions include the full posterior propagation or only background variations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for highlighting the importance of rigorously justifying the spherical-collapse implementation in the presence of decaying dark matter. We address the single major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: The central claim that δ_c remains close to the standard value and that halo abundances yield robust joint constraints rests on propagating the DESI-constrained background into the nonlinear regime with the unmodified spherical-collapse model. The DDM decay introduces a sink term in the DM continuity equation; the corresponding perturbation equations acquire additional source terms from the decay into relativistic species that affect the evolution of the density contrast and velocity divergence inside an overdensity. No explicit derivation retaining these Γ-dependent terms through turnaround and virialization is provided, so the reported closeness of δ_c may reflect an incomplete dynamical model rather than a robust outcome. This assumption is load-bearing for the halo-abundance predictions and the joint constraints.

    Authors: We agree that an explicit derivation of the perturbation equations, including the Γ-dependent source terms arising from the decay into relativistic dark radiation, is required to fully substantiate the spherical-collapse results. The current manuscript adapts the standard spherical-collapse equations to the modified background expansion history but does not retain or integrate the additional perturbation-level decay terms through turnaround and virialization. In the revised version we will add a dedicated subsection deriving the relevant continuity and Euler equations for the overdensity, incorporating the sink term and the relativistic decay products. We will then numerically solve the modified system for the posterior samples constrained by DESI BAO and ShapeFit and demonstrate that, within the allowed range of Γ, the critical threshold δ_c shifts by less than 1 percent relative to the background-only case. This additional calculation will be used to confirm that the reported halo-abundance deviations remain robust. revision: yes

Circularity Check

0 steps flagged

No significant circularity; forward modeling from independent datasets

full rationale

The derivation constrains background parameters from DESI DR1 BAO and ShapeFit data, then applies the standard spherical collapse model to compute halo abundances for comparison against the separate DESI Legacy Imaging Surveys DR9 dataset. This constitutes standard forward modeling with an external validation dataset rather than any fitted quantity being redefined as a prediction or any self-referential reduction in the equations. No load-bearing step reduces to its own inputs by construction, and the semi-cosmographic reconstruction is treated as an input constraint rather than an output derived from the halo data.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on the validity of propagating a data-fitted background model into the nonlinear regime via spherical collapse. The decay rate and dark-energy reconstruction parameters are constrained from observations, indicating they function as fitted quantities. Standard cosmological assumptions are invoked without independent derivation in the abstract.

free parameters (2)
  • DDM decay rate Γ (or lifetime τ_ddm)
    The decay rate is a model parameter whose value is constrained by the joint halo and BAO analysis; it is fitted to data rather than derived from first principles.
  • semi-cosmographic dark energy reconstruction parameters
    Parameters that define the reconstructed equation-of-state history are determined by fitting to DESI BAO and ShapeFit measurements.
axioms (2)
  • standard math The universe follows a flat FLRW metric with standard perturbation theory on sub-horizon scales
    Invoked implicitly when using spherical collapse to connect background evolution to halo formation.
  • domain assumption Spherical collapse dynamics remain unmodified beyond the altered background expansion
    Assumed when propagating the constrained background into the nonlinear regime without additional decay-induced terms in the collapse equation.
invented entities (1)
  • one-body decaying dark matter component that produces relativistic dark radiation no independent evidence
    purpose: To introduce a non-standard dark matter evolution that affects structure growth while remaining consistent with background constraints
    Postulated as part of the model; the abstract provides no independent falsifiable signature outside the halo abundance calculation itself.

pith-pipeline@v0.9.1-grok · 5779 in / 1844 out tokens · 76617 ms · 2026-06-26T04:04:14.905778+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

73 extracted references · 2 canonical work pages

  1. [1]

    To calculate the density threshold δc, we first look at the nonlinear density contrast δ

    correspond, respectively, to the self-gravity of the clustered mass, the homogeneous daughter-radiation, and the active gravitational density ρφ + 3 pφ of the smooth dark-energy sector. To calculate the density threshold δc, we first look at the nonlinear density contrast δ. Consider the real top- hat sphere of radius R and a hypothetical background sphere...

  2. [2]

    At an early scale factor ai, the am- plitude is chosen to be small, δi ≪ 1, so that δL(ai) = δ(ai) = δi

    and ( 30) are evolved from the same early perturbation. At an early scale factor ai, the am- plitude is chosen to be small, δi ≪ 1, so that δL(ai) = δ(ai) = δi. (32) The initial derivatives are set by the growing mode so- lution of the linear evolution Eq. ( 30). For a chosen col- lapse redshift zcoll, the initial amplitude δi is chosen so that the nonlin...

  3. [3]

    (44) Changing variables to ˜θ = Bt′ gives ρdr(t) = γ ρm(ata)x− 4 ∫ θ 0 x(˜θ)e− γ ( ˜θ− θta) d˜θ = γf m ¯ρcl(ata)x− 4J (θ), (45) 8 where J (θ) = ∫ θ 0 x(˜θ) exp[− γ(˜θ − θta)] d ˜θ

    and using ρm(ata) = ρm0a− 3 ta exp[− Γ( tta − t0)], we have ρdr(t) = Γ ρm(ata)a3 taa− 4 ∫ t 0 a(t′)e− Γ( t′− tta) dt′ = Γ ρm(ata)x− 4 ∫ t 0 x(t′)e− Γ( t′− tta) dt′. (44) Changing variables to ˜θ = Bt′ gives ρdr(t) = γ ρm(ata)x− 4 ∫ θ 0 x(˜θ)e− γ ( ˜θ− θta) d˜θ = γf m ¯ρcl(ata)x− 4J (θ), (45) 8 where J (θ) = ∫ θ 0 x(˜θ) exp[− γ(˜θ − θta)] d ˜θ. (46) Theref...

  4. [4]

    " # Λ$

    measures how the peak height changes when the halo mass is changed. The HMF is evaluated on a broad grid, 10 11 ≤ M/ (M⊙ /h ) ≤ 1016, at redshifts z = 0 , 0. 5, 1, 1. 5, 2 for the plotted posterior predictions. The comparison figures focus on 10 11 ≤ M/ (M⊙ /h ) ≤ 1015, where the fitted ΛCDM comparison is stable over all redshift slices. The HMF uses CLASS ...

  5. [5]

    Inclusion of the HMF data improves the constraints significantly

    The DESI DR1 posterior leaves the DDM life- time weakly localized because a flexible Pad´ e back- ground can absorb part of the smooth expansion ef- fect of decay. Inclusion of the HMF data improves the constraints significantly

  6. [6]

    The residual dark-energy reconstruction can differ from wφ = − 1, but the high-redshift evolution of dark energy remains weakly constrained

  7. [7]

    The critical collapse threshold δc(z) remains close to the reference ΛCDM value, so its direct effect on the HMF is small

  8. [8]

    Inclusion of the HMF data in the posterior reduces the uncertainties and makes the departure from the ΛCDM behaviour more evident

    The virial overdensity has a clearer, mildly non- monotonic response and approaches the reference behavior at high redshift. Inclusion of the HMF data in the posterior reduces the uncertainties and makes the departure from the ΛCDM behaviour more evident

  9. [9]

    A significant change in the halo-abundance re- sponse is seen in the joint analysis, especially at the large mass tail. We conclude by noting that our results demonstrate that halo abundances provide a sensitive nonlinear probe of coupled dark-sector physics, substantially tightening constraints beyond those obtained from geometric ob- servables alone. Thi...

    2025

  10. [10]

    Barkana and A

    R. Barkana and A. Loeb, Physics Reports 349, 125 (2001)

    2001

  11. [11]

    T. R. Choudhury, General Relativity and Gravitation 54, 10.1007/s10714-022-02987-4 (2022)

    work page doi:10.1007/s10714-022-02987-4 2022

  12. [12]

    Bertone, D

    G. Bertone, D. Hooper, and J. Silk, Physics Reports 405, 279 (2005) , arXiv:hep-ph/0404175 [hep-ph]

    Pith/arXiv arXiv 2005

  13. [13]

    D. H. Weinberg, J. S. Bullock, F. Gover- nato, R. Kuzio de Naray, and A. H. G. Peter, Proc. Natl. Acad. Sci. U.S.A. 112, 12249 (2015) , arXiv:1306.0913 [astro-ph.CO]

    Pith/arXiv arXiv 2015

  14. [14]

    J. S. Bullock and M. Boylan-Kolchin, Annu. Rev. Astron. Astrophys. 55, 343 (2017) , arXiv:1707.04256 [astro-ph.CO]

    arXiv 2017

  15. [15]

    D. N. Spergel and P. J. Steinhardt, Physical Review Letters 84, 3760 (2000) , arXiv:astro-ph/9909386 [astro-ph]

    Pith/arXiv arXiv 2000

  16. [16]

    W. Hu, R. Barkana, and A. Gruzi- nov, Physical Review Letters 85, 1158 (2000) , arXiv:astro-ph/0003365 [astro-ph]

    Pith/arXiv arXiv 2000

  17. [17]

    Blackadder and S

    G. Blackadder and S. M. Koushiappas, Phys. Rev. D 90, 103527 (2014)

    2014

  18. [19]

    Vattis, S

    K. Vattis, S. M. Koushiappas, and A. Loeb, arXiv preprint arXiv:1903.06220 (2019)

    Pith/arXiv arXiv 1903

  19. [20]

    S. J. Clark, K. Vattis, and S. M. Koushiappas, Phys. Rev. D 103, 043014 (2021) . 15

    2021

  20. [22]

    Borzumati, T

    F. Borzumati, T. Bringmann, and P. Ul- lio, Physical Review D 77, 063514 (2008) , arXiv:hep-ph/0701007 [hep-ph]

    Pith/arXiv arXiv 2008

  21. [23]

    A. H. G. Peter and A. J. Ben- son, Physical Review D 81, 103501 (2010) , arXiv:1003.0419 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  22. [24]

    Hildebrandt et al

    H. Hildebrandt et al. , MNRAS 465, 1454 (2017) , arXiv:1606.05338 [astro-ph.CO]

    Pith/arXiv arXiv 2017

  23. [25]

    Asgari et al

    M. Asgari et al. , A&A 645, A104 (2021) , arXiv:2007.15633 [astro-ph.CO]

    arXiv 2021

  24. [26]

    Heymans et al

    C. Heymans et al. , A&A 646, A140 (2021) , arXiv:2007.15632 [astro-ph.CO]

    arXiv 2021

  25. [27]

    Abdalla et al

    E. Abdalla et al. , Astropart. Phys. 131, 102605 (2021) , arXiv:2008.11284 [astro-ph.CO]

    arXiv 2021

  26. [28]

    A. G. Riess et al. , Astrophys. J. 826, 56 (2016) , arXiv:1604.01424 [astro-ph.CO]

    Pith/arXiv arXiv 2016

  27. [29]

    Di Valentino et al

    E. Di Valentino et al. , Class. Quantum Grav. 38, 153001 (2021) , arXiv:2103.01183 [astro-ph.CO]

    Pith/arXiv arXiv 2021

  28. [30]

    Verde, T

    L. Verde, T. Treu, and A. G. Riess, Nature Astronomy 3, 891 (2019) , arXiv:1907.10625 [astro-ph.CO]

    Pith/arXiv arXiv 2019

  29. [31]

    Oguri, K

    M. Oguri, K. Takahashi, H. Ohno, and K. Kotake, Astrophys. J. 597, 645 (2003) , arXiv:astro-ph/0306020

    Pith/arXiv arXiv 2003

  30. [32]

    Wang and P

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

    Pith/arXiv arXiv 1998

  31. [33]

    Pace, J.-C

    F. Pace, J.-C. Waizmann, and M. Bartel- mann, Mon. Not. R. Astron. Soc. 406, 1865 (2010) , arXiv:1005.0233 [astro-ph.CO]

    Pith/arXiv arXiv 2010

  32. [34]

    Chavan, T

    P. Chavan, T. Guha Sarkar, C. B. V. Dash, and A. A. Sen, JCAP 2025, 029 (2025) , arXiv:2503.03288 [astro-ph.CO]

    arXiv 2025

  33. [35]

    Chavan, T

    P. Chavan, T. Guha Sarkar, and A. A. Sen, Physics of the Dark Universe 50, 102173 (2025) , arXiv:2506.14275 [astro-ph.CO]

    arXiv 2025

  34. [36]

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

    1974

  35. [37]

    Despali, C

    G. Despali, C. Giocoli, R. E. Angulo, G. Tor- men, R. K. Sheth, G. Baso, and L. Moscar- dini, Mon. Not. Roy. Astron. Soc. 456, 2486 (2016) , arXiv:1507.05627 [astro-ph.CO]

    Pith/arXiv arXiv 2016

  36. [38]

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

    1972

  37. [39]

    G. L. Bryan and M. L. Norman, Astrophys. J. 495, 80 (1998) , arXiv:astro-ph/9710107

    Pith/arXiv arXiv 1998

  38. [40]

    F. Pace, Z. Sakr, and I. Tu- tusaus, Phys. Rev. D 102, 043512 (2020) , arXiv:1912.12250 [astro-ph.CO]

    arXiv 2020

  39. [41]

    A. G. Adame et al. (DESI Collaboration), JCAP 2025 (09), 008, arXiv:2411.12021 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  40. [42]

    A. G. Adame et al. (DESI Collaboration), JCAP 2025 (01), 124, arXiv:2404.03001 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  41. [43]

    A. G. Adame et al. (DESI Collaboration), JCAP 2025 (07), 028, arXiv:2411.12022 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  42. [44]

    A. G. Adame et al. (DESI Collaboration), JCAP 2025 (04), 012, arXiv:2404.03000 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  43. [45]

    Brieden, H

    S. Brieden, H. Gil-Mar ´ ın, and L. Verde, JCAP 2021 (12), 054, arXiv:2106.07641 [astro-ph.CO]

    arXiv 2021

  44. [46]

    Lesgourgues, The cosmic linear anisotropy solving system (CLASS) i: Overview (2011), arXiv:1104.2932 [astro-ph.IM]

    J. Lesgourgues, The cosmic linear anisotropy solving system (CLASS) i: Overview (2011), arXiv:1104.2932 [astro-ph.IM]

    Pith/arXiv arXiv 2011

  45. [47]

    D. Blas, J. Lesgourgues, and T. Tram, JCAP 2011 (07), 034, arXiv:1104.2933 [astro-ph.CO]

    Pith/arXiv arXiv 2011

  46. [48]

    Ibarra, D

    A. Ibarra, D. Tran, and C. Weniger, Int. J. Mod. Phys. A 28, 1330040 (2013) , arXiv:1307.6434 [hep-ph]

    Pith/arXiv arXiv 2013

  47. [49]

    G. F. Abell´ an, R. Murgia, and V. Poulin, Physical Review D 104, 123533 (2021) , arXiv:2102.12498 [astro-ph.CO]

    arXiv 2021

  48. [50]

    S. J. Clark, K. Vattis, and S. M. Koushi- appas, Phys. Rev. D 103, 043014 (2021) , arXiv:2006.03678 [astro-ph.CO]

    arXiv 2021

  49. [51]

    Alonso, J

    D. Alonso, J. Colosimo, A. Font-Ribera, and A. Slosar, JCAP 2018, 053 (2018) , arXiv:1712.02738 [astro-ph.CO]

    arXiv 2018

  50. [52]

    Hooper, F

    D. Hooper, F. S. Queiroz, and N. Y. Gnedin, Phys. Rev. D 85, 063513 (2012) , arXiv:1111.6599 [astro-ph.CO]

    Pith/arXiv arXiv 2012

  51. [53]

    M.-Y. Wang, R. A. C. Croft, A. H. G. Peter, A. R. Zent- ner, and C. W. Purcell, Phys. Rev. D 88, 123515 (2013) , arXiv:1309.7354 [astro-ph.CO]

    Pith/arXiv arXiv 2013

  52. [54]

    Mohit Yadav and Tapomoy Guha Sarkar, Eur. Phys. J. C 85, 1337 (2025)

    2025

  53. [55]

    Cen, Astrophys

    R. Cen, Astrophys. J. Lett. 546, L77 (2001) , arXiv:astro-ph/0005206 [astro-ph]

    Pith/arXiv arXiv 2001

  54. [56]

    Holsclaw, U

    T. Holsclaw, U. Alam, B. Sans´ o, H. Lee, K. Heitmann, S. Habib, and D. Higdon, Physical Review Letters 105, 10.1103/physrevlett.105.241302 (2010)

    work page doi:10.1103/physrevlett.105.241302 2010

  55. [57]

    Holsclaw, U

    T. Holsclaw, U. Alam, B. Sans´ o, H. Lee, K. Heitmann, S. Habib, and D. Higdon, Phys. Rev. D 84, 083501 (2011)

    2011

  56. [58]

    Shafieloo, A

    A. Shafieloo, A. G. Kim, and E. V. Linder, Phys. Rev. D 85, 123530 (2012)

    2012

  57. [59]

    Visser, General Relativity and Gravitation 37, 1541 (2005)

    M. Visser, General Relativity and Gravitation 37, 1541 (2005)

    2005

  58. [60]

    Visser, Classical and Quantum Gravity 32, 135007 (2015)

    M. Visser, Classical and Quantum Gravity 32, 135007 (2015)

    2015

  59. [61]

    P. K. S. Dunsby and O. Luongo, International Journal of Geometric Methods in Modern Physi cs 13, 1630002

  60. [62]

    Capozziello, R

    S. Capozziello, R. D’Agostino, and O. Luongo, Journal of Cosmology and Astroparticle Physics 2018 (05), 008

    2018

  61. [63]

    Pad´ e, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure9, 3 (1892)

    H. Pad´ e, Annales scientifiques de l’ ´Ecole Normale Sup´ erieure9, 3 (1892)

  62. [64]

    Capozziello, R

    S. Capozziello, R. D’Agostino, and O. Lu- ongo, Int. J. Mod. Phys. D 28, 1930016 (2019) , arXiv:1904.01427 [gr-qc]

    Pith/arXiv arXiv 2019

  63. [65]

    Benetti and S

    M. Benetti and S. Capozziello, JCAP 2019 (12), 008, arXiv:1910.09975 [astro-ph.CO]

    arXiv 2019

  64. [66]

    T. D. Saini, S. Raychaudhury, V. Sahni, and A. A. Starobinsky, Phys. Rev. Lett. 85, 1162 (2000) , arXiv:astro-ph/9910231

    Pith/arXiv arXiv 2000

  65. [67]

    A. G. Adame et al. (DESI Collaboration), JCAP 2025 (02), 021, arXiv:2404.03002 [astro-ph.CO]

    Pith/arXiv arXiv 2025

  66. [68]

    J. Wang, X. Yang, J. Zhang, H. Li, M. Fong, H. Xu, M. He, Y. Gu, W. Luo, F. Dong, Y. Wang, Q. Li, A. Kat- sianis, H. Wang, Z. Shen, P. Alonso Vaquero, C. Liu, Y. Huang, and Z. Liu, Astrophys. J. 936, 161 (2022) , arXiv:2207.12771 [astro-ph.CO]

    arXiv 2022

  67. [69]

    J. R. Bond, S. Cole, G. Efstathiou, and N. Kaiser, Astrophys. J. 379, 440 (1991)

    1991

  68. [70]

    R. K. Sheth and G. Tormen, Mon. Not. Roy. Astron. Soc. 308, 119 (1999) , arXiv:astro-ph/9901122

    Pith/arXiv arXiv 1999

  69. [71]

    R. K. Sheth, H. J. Mo, and G. Tormen, Mon. Not. Roy. Astron. Soc. 323, 1 (2001) , 16 arXiv:astro-ph/9907024

    Pith/arXiv arXiv 2001

  70. [72]

    Birkinshaw, Physics Reports 310, 97 (1999)

    M. Birkinshaw, Physics Reports 310, 97 (1999)

    1999

  71. [73]

    J. E. Carlstrom, G. P. Holder, and E. D. Reese, Annual Review of Astronomy and Astrophysics 40, 643 (2002)

    2002

  72. [74]

    Refregier, E

    A. Refregier, E. Komatsu, D. N. Spergel, and U.-L. Pen, Physical Review D 61, 123001 (2000)

    2000

  73. [75]

    Cerardi, M

    N. Cerardi, M. Pierre, P. Valageas, C. Garrel, and F. Pacaud, Astron. Astrophys. 682, A138 (2024) , arXiv:2312.04253 [astro-ph.CO]

    arXiv 2024