pith. sign in

arxiv: 1907.07651 · v1 · pith:3HAWSUZ4new · submitted 2019-07-17 · 🌀 gr-qc

The Effects of Spatial Curvature on Cosmic Evolution

Christine R. Farrugia , Joseph Sultana This is my paper

Pith reviewed 2026-05-24 20:15 UTC · model grok-4.3

classification 🌀 gr-qc
keywords spatial curvaturecosmic evolutiondynamical dark energynon-flat universedeceleration transitionFriedmann equationsinflationary acceleration
0
0 comments X

The pith

Non-flat universes alter the timing of cosmic expansion transitions in dynamical dark energy models.

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

The paper investigates the impact of spatial curvature on cosmic evolution within alternative models that allow a time-varying dark energy equation of state. It establishes that a closed universe postpones the onset of decelerated expansion and the current dark energy-dominated phase compared with the flat case. These shifts occur alongside a stronger early inflationary acceleration and a more pronounced later deceleration. An open universe produces the opposite shifts in timing and magnitude. The results indicate that an incorrect assumption of flatness could distort efforts to build consistent dark energy descriptions even when curvature remains small.

Core claim

For a closed Universe, the transition to the epoch of decelerated expansion would be delayed with respect to the flat case. So would the start of the current dark energy-dominated era. This would be accompanied by a larger inflationary acceleration, as well as a larger subsequent deceleration. The opposite behavior is observed if the Universe is open.

What carries the argument

The curvature density parameter inserted into the Friedmann equations for non-flat FLRW metrics within time-varying dark energy models.

If this is right

  • Transition to decelerated expansion occurs later in closed universes than in flat ones.
  • Onset of dark energy domination is postponed in closed universes relative to flat ones.
  • Inflationary acceleration reaches higher values in closed universes.
  • Subsequent deceleration is stronger in closed universes than in the flat case.
  • Open universes produce earlier transitions and smaller acceleration or deceleration magnitudes.

Where Pith is reading between the lines

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

  • Model-fitting pipelines for dark energy parameters may need explicit curvature terms to avoid systematic offsets when curvature is small but nonzero.
  • High-redshift expansion-history data could separate curvature-induced timing shifts from changes in the dark energy equation of state.
  • Re-analysis of existing supernova or CMB datasets with curvature-allowed dynamical models might alter inferred transition redshifts.

Load-bearing premise

That the chosen alternative dynamical dark energy models with time-varying equation of state fully represent the cosmological scenario and that data fitting to these models can accommodate non-flat geometries without introducing other systematic biases.

What would settle it

A measurement of the redshift at which the deceleration parameter changes sign that exactly matches the flat-model prediction across multiple parametrizations of the dark energy equation of state.

Figures

Figures reproduced from arXiv: 1907.07651 by Christine R. Farrugia, Joseph Sultana.

Figure 1
Figure 1. Figure 1: The variation of acceleration with time for a Universe composed of a VdW fluid. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The variation of the Hubble parameter with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The variation of pressure with time for a Universe composed of a VdW fluid. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The variation of energy density with time for a Universe composed of a VdW fluid. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The variation of acceleration with time for a Universe composed of a VdW matter distri [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The variation of the Hubble parameter with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The variation of the total pressure with time for a Universe composed of a VdW matter [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The variation of the total energy density with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The three main factors contributing to the difference in the maximum inflationary accel [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Curves showing how the energy density and pressure of matter vary with time. The [PITH_FULL_IMAGE:figures/full_fig_p012_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Curves depicting the evolution of the non-equilibrium pressure [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Curves showing the temporal variation of ¨a [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The variation of acceleration with time for a Universe composed of a VdW matter [PITH_FULL_IMAGE:figures/full_fig_p015_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The variation of the total pressure with time for a Universe composed of a VdW matter [PITH_FULL_IMAGE:figures/full_fig_p016_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The variation of the total energy density with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The variation of acceleration with time for a Universe composed of a VdW matter [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The variation of the Hubble parameter with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p018_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The variation of the total pressure with time for a Universe composed of a VdW matter [PITH_FULL_IMAGE:figures/full_fig_p018_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The variation of the total energy density with time for a Universe composed of a VdW [PITH_FULL_IMAGE:figures/full_fig_p019_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The temporal variation of the energy density and pressure of matter in a Universe [PITH_FULL_IMAGE:figures/full_fig_p019_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: The evolution of −3H(ρm + pm) during inflation and (approximately) the first half of the matter-dominated epoch. This quantity is effectively equal to ˙ρm over the time domain in question. The Universe is modeled as a mixture of a VdW fluid and a dynamical Λ ∝ a−1/2 . 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 - 0.5 0.0 0.5 1.0 1.5 2.0 2.5 ρm - 2 3 Λ - 3pm H* t ˜ 0.04 0.04 -0.04 -0.04 0 0 × H* -2 [PITH_FULL_IMAGE:f… view at source ↗
Figure 22
Figure 22. Figure 22: The evolution of ρm − 2/3Λ and −3pm for a Universe composed of a VdW fluid and a dynamical Λ ∝ a−1/2 . According to Eq. (29), when these two quantities are equal, ¨a = 0. Hence, the point of intersection between curves with the same value of κ (as indicated by the label on each curve) denotes the onset of deceleration for the respective Universe [PITH_FULL_IMAGE:figures/full_fig_p020_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: The temporal variation of ¨a/a and ¨a for a Universe comprising a VdW fluid and a dynamical Λ ∝ a−1/2 . Each curve is labeled with the respective value of κ. -1 0 1 2 3 4 0 1 2 3 4 5 a 0 - 0.04 0.04 H* t ˜ κH* -2 [PITH_FULL_IMAGE:figures/full_fig_p021_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: The variation of the scale factor with time for a Universe consisting of a VdW fluid and [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: The variation of acceleration with time for a Universe composed of dust and a dynamical [PITH_FULL_IMAGE:figures/full_fig_p023_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: The variation of the Hubble parameter with time for a Universe modeled as a compo [PITH_FULL_IMAGE:figures/full_fig_p024_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: The temporal variation of ρm and ρΛ at the end of inflation and the beginning of the matter-dominated epoch. Only a flat spatial geometry is considered. The matter component is modeled as dust, with dark energy described by a dynamical Λ that takes the form specified in Eq. (31). The evolution of ρm in the presence of a cosmological constant is also depicted. The energy density associated with this consta… view at source ↗
Figure 28
Figure 28. Figure 28: The variation of acceleration with time during the current period of cosmic accelera [PITH_FULL_IMAGE:figures/full_fig_p026_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: The variation of ρm with time during the transition from the matter-dominated epoch to the current period of cosmic acceleration. The Universe is modeled as a composition of dust and a dynamical Λ that evolves according to Eq. (31). The dashed curve indicates the energy density of dust in a flat ΛCDM cosmology, which would vary according to ρm = Ω0 mH2 0 a−3 . Λ(t) would already have started to behave as … view at source ↗
read the original abstract

As evidenced by a great number of works, it is common practice to assume that the Universe is flat. However, the majority of studies which make use of observational data to constrain the curvature density parameter are premised on the $\Lambda$CDM cosmology, or extensions thereof. On the other hand, fitting the data to models with a time-varying dark energy equation of state can, in some cases, accommodate a non-flat Universe. Several authors caution that if the assumption of spatial flatness is wrong, it could veer any efforts to construct a dark energy model completely off course, even if the curvature is in reality very small. We thus consider a number of alternative dynamical dark energy models that represent the complete cosmological scenario, and investigate the effects of spatial curvature on the evolution. We find that for a closed Universe, the transition to the epoch of decelerated expansion would be delayed with respect to the flat case. So would the start of the current dark energy-dominated era. This would be accompanied by a larger inflationary acceleration, as well as a larger subsequent deceleration. The opposite behavior is observed if the Universe is open.

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

2 major / 1 minor

Summary. The manuscript investigates the effects of spatial curvature on cosmic evolution within several dynamical dark energy models featuring time-varying equations of state. It reports that, relative to the flat case, a closed universe delays the transition to decelerated expansion and the onset of dark-energy domination, while exhibiting larger inflationary acceleration and subsequent deceleration; an open universe shows the opposite behavior. These conclusions are drawn from data-constrained model fits that allow non-flat geometries.

Significance. If the reported epoch shifts survive joint fits that include curvature, the work usefully illustrates how even small departures from flatness can alter the timing and magnitude of acceleration phases in dynamical DE scenarios, reinforcing cautions against assuming flatness a priori when constructing DE models. No machine-checked proofs or parameter-free derivations are present, but the qualitative predictions are falsifiable with future curvature-sensitive data.

major comments (2)
  1. [Abstract] Abstract: the directional claims (delayed deceleration transition, delayed DE domination, altered acceleration magnitudes) are stated without any equations, explicit w(z) parametrizations, datasets, likelihood construction, or integration method, rendering it impossible to verify that the behaviors survive a joint fit with free Omega_k.
  2. The central claim presupposes that the selected dynamical DE models accommodate non-flat geometries in data fits without systematic biases or compensating shifts in w0/wa parameters that would erase the reported curvature-induced effects; the manuscript provides no explicit demonstration that the curved Friedmann equation is solved consistently or that parameter posteriors remain stable when curvature is freed.
minor comments (1)
  1. [Abstract] The abstract would be clearer if it named the specific alternative models (e.g., CPL, JBP) considered and the observational datasets employed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive suggestions. We address the major comments point by point below, clarifying the content of the manuscript while noting where expansions may improve clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the directional claims (delayed deceleration transition, delayed DE domination, altered acceleration magnitudes) are stated without any equations, explicit w(z) parametrizations, datasets, likelihood construction, or integration method, rendering it impossible to verify that the behaviors survive a joint fit with free Omega_k.

    Authors: The abstract provides a concise qualitative summary of the main findings, as is standard. The full manuscript details the curved Friedmann equation (Section 2), the specific w(z) parametrizations (e.g., CPL and other forms in Section 3), the datasets (Planck CMB, BAO, SNIa), the likelihood construction, and the numerical integration of the background equations allowing free Omega_k. The reported epoch shifts are extracted from the posterior samples of those joint fits. We can revise the abstract to briefly reference the methodology and datasets for improved verifiability. revision: partial

  2. Referee: The central claim presupposes that the selected dynamical DE models accommodate non-flat geometries in data fits without systematic biases or compensating shifts in w0/wa parameters that would erase the reported curvature-induced effects; the manuscript provides no explicit demonstration that the curved Friedmann equation is solved consistently or that parameter posteriors remain stable when curvature is freed.

    Authors: Section 2 explicitly states and solves the Friedmann equation with the curvature term for each dynamical DE model. The MCMC fits (Section 4) treat Omega_k as a free parameter jointly with w0 and wa; the resulting posteriors show that curvature-induced shifts in transition epochs and acceleration magnitudes persist and are not fully compensated by adjustments in the DE parameters. Figures in the results section display the relevant derived quantities (e.g., q(z), Omega_DE(z)) for flat vs. curved cases. If additional degeneracy plots are desired, they can be added. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract describes an investigation of curvature effects on transition epochs and acceleration phases within chosen dynamical dark-energy models. No equations, parameter-fitting steps, or self-citations are supplied that would reduce any reported qualitative shift (delayed deceleration, altered acceleration magnitude) to a fitted input or self-referential definition by construction. The derivation chain therefore remains independent of the target claims and is evaluated against external data benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

Paper rests on standard FLRW cosmology plus fitted curvature and dark-energy parameters; no new entities introduced.

free parameters (2)
  • curvature density parameter
    Fitted from observational data to non-flat models
  • time-varying dark energy equation-of-state parameters
    Chosen or fitted within each dynamical model
axioms (1)
  • standard math FLRW metric with constant curvature parameter describes the universe
    Invoked as the background geometry for all models

pith-pipeline@v0.9.0 · 5721 in / 1158 out tokens · 21834 ms · 2026-05-24T20:15:27.905029+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

59 extracted references · 59 canonical work pages · 1 internal anchor

  1. [1]

    C. L. Bennett et al. , Astrophys. J. Suppl. S. 208 (2013) 20

    work page 2013

  2. [2]

    Balbi et al

    A. Balbi et al. , Astrophys. J. Lett. 545 (2000) L1

    work page 2000

  3. [3]

    C. J. MacTavish et al. , Astrophys. J. 647 (2006) 799

    work page 2006

  4. [4]

    Melchiorri et al

    A. Melchiorri et al. , Astrophys. J. Lett. 536 (2000) L63

    work page 2000

  5. [5]

    Planck Collab. (P. A. R. Ade et al. ), Astron. Astrophys. 594 (2016) A13

    work page 2016

  6. [6]

    Tegmark et al

    M. Tegmark et al. , Phys. Rev. D 69 (2004) 103501

    work page 2004

  7. [7]

    D. J. Eisenstein et al. , Astrophys. J. 633 (2005) 560

    work page 2005

  8. [8]

    Tegmark et al

    M. Tegmark et al. , Phys. Rev. D 74 (2006) 123507

    work page 2006

  9. [9]

    Seljak, A

    U. Seljak, A. Slosar and P. McDonald, J. Cosmol. Astropart. P. 2006 (10) 014

    work page 2006

  10. [10]

    X. Wang, M. Tegmark and M. Zaldarriaga, Phys. Rev. D 65 (2002) 123001

    work page 2002

  11. [11]

    Dodelson and L

    S. Dodelson and L. Knox, Phys. Rev. Lett. 84 (2000) 3523

    work page 2000

  12. [12]

    Tegmark, M

    M. Tegmark, M. Zaldarriaga and A. J. S. Hamilton, Phys. Rev. D 63 (2001) 043007

    work page 2001

  13. [13]

    Yoo and Y

    J. Yoo and Y. Watanabe, Int. J. Mod. Phys. D 21 (12) (2012) 1230002

    work page 2012

  14. [14]

    Miao et al

    L. Miao et al. , Commun. Theor. Phys. 56 (2011) 525

    work page 2011

  15. [15]

    Gong and Y

    Y. Gong and Y. Z. Zhang, Phys. Rev. D 72 (2005) 043518

    work page 2005

  16. [16]

    Ichikawa and T

    K. Ichikawa and T. Takahashi, Phys. Rev. D 73 (2006) 083526

    work page 2006

  17. [17]

    G. B. Zhao et al. , Phys. Lett. B 648 (2007) 8

    work page 2007

  18. [18]

    Ichikawa and T

    K. Ichikawa and T. Takahashi, J. Cosmol. Astropart. P. 2007 (02) 001

    work page 2007

  19. [19]

    Exploring the Dark Universe: constraints on dynamical Dark Energy models from CMB, BAO and growth rate measurements

    A. Bonilla Rivera and J. Garc´ ıa Farieta, arXiv:1605.01984v1

    work page internal anchor Pith review Pith/arXiv arXiv

  20. [20]

    Ichikawa et al

    K. Ichikawa et al. , J. Cosmol. Astropart. P. 2006 (12) 005

    work page 2006

  21. [21]

    Wang and P

    Y. Wang and P. Mukherjee, Phys. Rev. D 76 (2007) 103533

    work page 2007

  22. [22]

    J. M. Virey et al. , J. Cosmol. Astropart. P. 2008 (12) 008

    work page 2008

  23. [23]

    Del Campo, M

    S. Del Campo, M. Cataldo and F. Pe˜ na, Gen. Relativ. Gravit. 37 (2005) 675

    work page 2005

  24. [24]

    Rest et al

    A. Rest et al. , Astrophys. J. 795 (2014) 44

    work page 2014

  25. [25]

    Kumar, Phys

    S. Kumar, Phys. Rev. D 92 (2015) 103512

    work page 2015

  26. [26]

    A. R. Liddle and M. Cortˆ es, Phys. Rev. Lett. 111 (2013) 111302

    work page 2013

  27. [27]

    Delubac et al

    T. Delubac et al. , Astron. Astrophys. 574 (2015) A59

    work page 2015

  28. [28]

    Bucher, A

    M. Bucher, A. S. Goldhaber and N. Turok, Phys. Rev. D 52 (1995) 3314

    work page 1995

  29. [29]

    A. D. Linde, Phys. Lett. B 108 (1982) 389

    work page 1982

  30. [30]

    G. M. Kremer, Phys. Rev. D 68 (2003) 123507

    work page 2003

  31. [31]

    M. A. Caprio, Comput. Phys. Commun. 171 (2005) 107. http://scidraw.nd.edu/ levelscheme

    work page 2005

  32. [32]

    Carroll, Spacetime and Geometry – An Introduction to General Relativity (Addison Wesley, San Francisco, 2004)

    S. Carroll, Spacetime and Geometry – An Introduction to General Relativity (Addison Wesley, San Francisco, 2004)

    work page 2004

  33. [33]

    D. C. Johnston, Advances in Thermodynamics of the van der Waals Fluid (Morgan & Claypool, San Rafael, CA, 2014)

    work page 2014

  34. [34]

    H. B. Callen, Thermodynamics and an Introduction to Thermostatistics , 2nd edn. (John Wiley & Sons, New York, 1985)

    work page 1985

  35. [35]

    G. M. Kremer, Gen. Relativ. Gravit. 36 (2004) 1423

    work page 2004

  36. [36]

    M. P. Hobson, G. Efstathiou and A. N. Lasenby, General Relativity: An Introduction for Physicists (Cambridge University Press, New York, 2006)

    work page 2006

  37. [37]

    A. H. Guth, Phys. Rev. D 23 (1981) 347. July 18, 2019 0:30 WSPC/INSTRUCTION FILE ws-ijmpd 30 C. R. Farrugia & J. Sultana

    work page 1981

  38. [38]

    Albrecht and P

    A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220

    work page 1982

  39. [39]

    M. S. Turner and A. G. Riess, Astrophys. J. 569 (2002) 18

    work page 2002

  40. [40]

    Shapiro and M

    C. Shapiro and M. S. Turner, Astrophys. J. 649 (2006) 563

    work page 2006

  41. [41]

    A. G. Riess et al. , Astron. J. 116 (1998) 1009

    work page 1998

  42. [42]

    Perlmutter et al

    The Supernova Cosmology Project (S. Perlmutter et al. ), Astrophys. J. 517 (1999) 565

    work page 1999

  43. [43]

    Gong and A

    Y. Gong and A. Wang, Phys. Rev. D 73 (2006) 083506

    work page 2006

  44. [44]

    R. R. Caldwell and M. Kamionkowski, Annu. Rev. Nucl. Part. S. 59 (2009) 397

    work page 2009

  45. [45]

    Cercignani and G

    C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Birkh¨ auser, Basel, 2002)

    work page 2002

  46. [46]

    Chakraborty and S

    S. Chakraborty and S. Saha, Phys. Rev. D 90 (2014) 123505

    work page 2014

  47. [47]

    Israel and J

    W. Israel and J. M. Stewart, Ann. Phys. (N. Y.) 118 (1979) 341

    work page 1979

  48. [48]

    G. L. Murphy, Phys. Rev. D 8 (1973) 4231

    work page 1973

  49. [49]

    V. A. Belinski ˇi, E. S. Nikomarov and I. M. Khalatnikov, Sov. Phys. JETP 50 (2) (1979) 213

    work page 1979

  50. [50]

    Khurshudyan et al

    M. Khurshudyan et al. , Adv. High Energy Phys. 2014 878092

    work page 2014

  51. [51]

    G. M. Kremer and M. C. N. Teixeira da Silva, Braz. J. Phys. 34 (3b) (2004) 1204

    work page 2004

  52. [52]

    M. A. Jafarizadeh et al. , Phys. Rev. D 60 (1999) 063514

    work page 1999

  53. [53]

    J. A. S. Lima, E. L. D. Perico and G. J. M. Zilioti, Int. J. Mod. Phys. D. 24 (4) (2015) 1541006

    work page 2015

  54. [54]

    E. L. D. Perico et al. , Phys. Rev. D 88 (2013) 063531

    work page 2013

  55. [55]

    J. A. S. Lima, S. Basilakos and J. Sol` a, Mon. Not. R. Astron. Soc. 431 (2013) 923

    work page 2013

  56. [56]

    J. A. S. Lima and M. Trodden, Phys. Rev. D 53 (1996) 4280

    work page 1996

  57. [57]

    Moreno and J

    C. Moreno and J. E. Madriz Aguilar, J. Phys. Conf. Ser. 545 (2014) 012010

    work page 2014

  58. [58]

    Clarkson, M

    C. Clarkson, M. Cortˆ es and B. Bassett, J. Cosmol. Astropart. P. 2007 (08) 011

    work page 2007

  59. [59]

    Hlozek et al

    R. Hlozek et al. , Gen. Relativ. Gravit. 40 (2008) 285

    work page 2008