Cosmographic analysis of sign-switching dark energy

Be\~nat Ibarra-Uriondo; Mariam Bouhmadi-L\'opez

REVIEW 3 major objections 4 minor 6 cited by

Cosmographic analysis of sign-switching dark energy

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A continuous sign change in dark energy removes the sudden singularity of $\Lambda_{\rm s}$CDM, replacing it with a milder w-singularity.

desk verdict A useful set of sign-switching dark-energy templates whose main singularity claim is qualitatively right but whose written derivation has concrete algebra errors that need fixing. read the letter →

arxiv 2506.12139 v2 pith:K6OVV3UM submitted 2025-06-13 gr-qc astro-ph.COhep-phhep-th

Mariam Bouhmadi-López , Beñat Ibarra-Uriondo This is my paper

classification gr-qcastro-ph.COhep-phhep-th PACS 98.80.-k95.36.+x

keywords sign-switchingcosmologicalconstantLambda_sCDMmodeldarkenergyequationofstatecosmographystatefinderhierarchysuddensingularityw-singularitytensions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a family of dark-energy models in which the cosmological constant switches from a negative value to a positive value around redshift $z \approx 1.7$, and compares abrupt, ladder-like, smooth-step, and error-function transitions. Its central result is that a continuous sign change removes the sudden (type II) singularity that the abrupt $\Lambda_{\rm s}$CDM transition produces, leaving only a milder w-singularity in which the dark-energy equation of state diverges while density and pressure stay finite. The authors also build analytic background solutions for the ladder models and compute cosmographic and statefinder diagnostics, showing that the smooth models produce characteristic oscillations in deceleration, jerk, snap, and lerk, while the constant-density models mimic $\Lambda$CDM away from the jumps. If the universe really switched the sign of its effective cosmological constant, the singularity type determines whether bound structures are disrupted at the transition.

What carries the argument

The key objects are the prescribed dark-energy density profiles as functions of $x=\ln(a/a_0)$: a Heaviside-ladder profile $\Lambda_{\rm le}(z)$, a smooth polynomial-step profile $\Lambda_{\rm ss}(x)$ (Eq. 19), and an error-function profile $\Lambda_{\rm e}(x)=\Lambda\,\mathrm{Erf}[\eta(x-x_{\dagger,\rm e})]$ (Eq. 21). The argument runs through the conservation equation $\dot{\rho}_d+3H(\rho_d+p_d)=0$, which converts the chosen density profile into an effective dark-energy equation of state $w_d$; the profile's continuity determines the singularity class. The scale-factor solutions for the ladder models and the explicit pressure and density evaluation in Appendix A carry the classification: finite density with divergent pressure gives a type II singularity, while finite density and pressure with divergent $w_d$ gives a type V w-singularity.

What would settle it

A precise measurement of $H(z)$ in the redshift window around $z \approx 1.7$ would settle the central claim: the smooth models predict a continuous $H(z)$ with finite total density and pressure, whereas $\Lambda_{\rm s}$CDM and the ladder models predict a discontinuous $H(z)$ and divergent pressure. Observing a sharp jump in $H(z)$ would falsify the smoothing claim, while observing continuity with a divergent dark-energy equation of state is the signature of the w-singularity.

Watch

Extended reading notes

Core claim

The paper's central claim is that the abrupt sign switch of the $\Lambda_{\rm s}$CDM model, where the dark-energy density jumps from a negative constant to a positive constant at a transition redshift, produces a sudden (type II) singularity with finite density but divergent pressure, and that this singularity is an artifact of the discontinuity. For the two smooth models, SSCDM and ECDM, the dark-energy density is continuous through zero, so at the crossing the total density and total pressure remain finite; the only divergence is in the dark-energy equation of state, which goes to $\pm\infty$ as the density passes through zero. This is a w-singularity (type V), which is milder because no physical quantity that enters the Friedmann equations diverges. The abrupt and ladder models, by contrast, have discontinuous Hubble parameters and hence sudden singularities at each jump.

Load-bearing premise

The load-bearing premise is that the smooth density profiles in Eqs. (19) and (21) are legitimate dark-energy fluids whose pressure follows from energy conservation even though their equation of state diverges at the crossing; the paper gives no microphysical mechanism that realizes them, so without such a realization the singularity classification describes the chosen interpolation rather than a physical cosmology.

Editorial extensions

If this is right

If the smooth models are correct, the universe crosses from negative to positive dark-energy density without any instant of infinite pressure; the only singular behaviour is the divergence of the dark-energy equation of state at the density-zero crossing.
The ladder and abrupt models behave like $\Lambda$CDM at all times except at jumps, where $H$ and higher derivatives are discontinuous; with radiation included, the statefinder hierarchy can distinguish the even-step from the odd-step ladder because the even ladder crosses $\rho_d=0$ and makes the $\tilde{s}$ parameter discontinuous.
For the smooth models, the deceleration parameter and higher cosmographic parameters oscillate around the transition redshift and then return to $\Lambda$CDM values, so detecting or ruling out these oscillations is a direct observational test.
In the limit where the smooth transition is made instantaneous, the pressure diverges and the sudden singularity of $\Lambda_{\rm s}$CDM is recovered, showing that the singularity classification is continuous in the transition width.
Asymptotically in the future, all the models converge to the $\Lambda$CDM de Sitter state, with statefinder parameters $S^{(1)}_3=S^{(1)}_4=S^{(1)}_5=1$ and $\tilde{s}=0$.

Reading between the lines

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

Beyond the paper: a natural next step is to promote the smooth density profiles to a microphysical field theory; if such a realization exists, the w-singularity would survive as an effective description, but if only abrupt transitions are realizable, the sudden singularity remains the physically relevant one.
Beyond the paper: the same classification scheme should apply to other sign-switching cosmological quantities, such as an effective curvature term or a scalar-field potential crossing zero, where continuity of the energy density generically converts a type II singularity into a type V singularity.
Beyond the paper: the predicted oscillations in $q(z)$, $j(z)$, $s(z)$, and $l(z)$ could be searched for with model-independent reconstructions of $H(z)$ from cosmic chronometers or BAO data; a null detection would disfavour smooth transitions without a dedicated fit.
Beyond the paper: the even-versus-odd ladder comparison suggests that whether the dark-energy density ever crosses exactly zero is itself observable in statefinder diagnostics, so future data may constrain not just the transition redshift but the continuity class of the transition.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take: the paper's headline claim — that smooth sign-switching dark energy replaces the sudden singularity of ΛsCDM with a milder w-singularity — is almost certainly correct, but the written derivation is not reliable as it stands. There are concrete algebraic errors in two key equations (Eq. (10) and Eq. (20)) and a misstated limit in Appendix A.1. The qualitative conclusion survives the corrections, but the paper needs a careful revision before I'd trust its details.

What's genuinely new: the ladder, smooth-step, and error-function profiles are explicit constructions not present in the cited ΛsCDM/ΛsVCDM literature. The cosmographic analysis is thorough — deceleration, jerk, snap, lerk, and statefinder parameters are computed and compared across the models. The singularity classification appendix is a useful addition; the distinction between type II and type V singularities for these models is worth having on record. The paper is also honest about its own limits: the "fit" in Sec. II E is flagged in a footnote as approximate, and the models are explicitly phenomenological with no microphysical mechanism. That's fine for a template paper.

Where it's soft. Eq. (10) for the ΛsCDM EoS is wrong: conservation gives the delta term as (2/3)(1+z) δ(z†−z)/sgn, not (1/3) δ/(1+z). Eq. (20) for the smooth-step EoS has the wrong sign and coefficient: the correct result is −1 + 420 t^4(1−t)^4/(Δx f), not −1 − 1260 t^4(1−t)^4/(Δx f). In both cases the qualitative behavior (divergence, phantom crossing) is unchanged, but the displayed formulas are incorrect. Appendix A.1 takes the sudden-singularity limit as Δx → ∞; the abrupt limit is actually Δx → 0, since that's when the transition becomes instantaneous. The claim that Δx → ∞ reduces to ΛsCDM is backwards. These are fixable, but a referee should demand the corrections before publication.

The deeper caveat is the one the authors themselves acknowledge: the smooth profiles are prescribed at the level of the energy density, with no action or field theory realization. The w-singularity classification is therefore a statement about the chosen interpolations, not necessarily about a physical dark-energy component. That limits the impact to model-building, which is fine, but it should be said clearly.

Bottom line: this is a solid, honest model-comparison paper in a narrow subfield, with a load-bearing but correctable derivation flaw. It deserves a serious referee — not a desk reject — and the referee should check the algebra carefully. I would not cite it in its current form, but I'd be happy to see a corrected version.

Referee Report

3 major / 4 minor

Summary. The paper introduces three phenomenological extensions of the Lambda_sCDM model with a late-time sign switch of the effective cosmological constant: a ladder model with an even or odd number of steps (LLambdaCDM), a smooth polynomial-step model (SSCDM), and an error-function model (ECDM). For each model the authors impose a dark-energy density as a function of x = ln a, obtain the dark-energy equation of state from covariant conservation, give analytical scale-factor solutions for the discontinuous models, and compare the background Hubble rate with cosmic chronometer data. The cosmographic parameters q, j, s, l and the statefinder hierarchy parameters S_3^(1), S_4^(1), S_5^(1), and s-tilde are computed with and without radiation. The paper's central claim is that a continuous sign change in the dark-energy density removes the Type II sudden singularity of Lambda_sCDM and replaces it with a milder Type V w-singularity at the density-zero crossing, with the sudden singularity recovered only in the instantaneous transition limit.

Significance. The singularity classification for smooth sign-switching dark energy is the paper's most valuable contribution: if the derivation is corrected, the claim that continuous profiles replace a sudden singularity by a finite-pressure w-singularity is a concrete and, in principle, falsifiable statement. The paper also provides useful analytic reference formulas for cosmographic and statefinder diagnostics in the presence of radiation, and the explicit scale-factor solutions for the ladder models are a genuine addition. The authors are transparent that the density profiles are prescribed phenomenologically rather than derived from an action, and that the comparison with data is illustrative rather than a proper fit (Sec. II E, footnote 3). These limitations are acknowledged and do not invalidate the background analysis, but they should be kept in mind when interpreting the singularity classification.

major comments (3)

[II C, Eq. (20)] Equation (20) is not the conservation result for the profile (19). With S(t)=126t^5-420t^6+540t^7-315t^8+70t^9 and t=(x-x_f)/(x_i-x_f), Eq. (3) gives 1+w_d = +420 t^4(1-t)^4 / (Delta x * Lambda_ss/Lambda), so w_d = -1 + 420 t^4(1-t)^4 / (Delta x * Lambda_ss/Lambda), not the -1260 expression with the opposite sign displayed in Eq. (20). This error propagates into the pressure evaluation in Appendix A and must be corrected.
[Appendix A.1, Eq. (A3)] The total pressure at the crossing Lambda_ss=0 should read P_tot(x_dagger) = (1/3) rho_r,0 e^{-4x_dagger} + rho_d,0 [420 t_dagger^4 (1-t_dagger)^4] / Delta x, which is finite and negative for the chosen Delta x<0; Eq. (A3) has the wrong sign and an extra factor of 3. The finiteness conclusion, and hence the w-singularity classification, is correct only after this replacement.
[Appendix A.1, Eqs. (A4) and (A7)] The sudden-singularity limit is taken in the wrong direction. The transition becomes instantaneous when Delta x = x_i - x_f tends to 0, not to infinity; with the corrected pressure, Delta x -> 0^- makes P_tot diverge to -infinity, which is the Type II limit. The sentence 'when taking Delta x -> infinity ...' and the limit lim_{Delta x -> infinity} P_tot = -infinity in Eq. (A4) should be replaced by Delta x -> 0. Similarly, Eq. (A7) should be lim_{eta -> infinity}, not lim_{Delta x -> infinity}. This is a load-bearing step for the claimed reduction of the smooth models to model (A).

minor comments (4)

[II A, Eq. (10)] Equation (10) does not follow from Eq. (5); the correct expression is w_d,s = -1 - (2/3)(1+z) delta(z_dagger,s - z) / sgn(z_dagger,s - z), with the factor (1+z) in the numerator rather than the denominator.
[II E, Fig. 1 and footnote 3] The sentence 'Our models fit most of the cosmic chronometers' is stronger than what footnote 3 supports, since the model parameters are taken from prior Lambda_sCDM fits and the curves are only visually compared with the data; 'are compatible with' would be more accurate.
[II C, notation after Eq. (20)] The notation Delta x = x_i - x_f is negative for the chosen redshifts (since x_i < x_f), and this convention is a recurring source of sign confusion in Eqs. (20), (A3), and (A4); consider defining Delta x as a positive transition width and adjusting signs consistently.
[Eq. (4)] The dark-energy equation of state is written as w_d = rho_d / p_d; it should be w_d = p_d / rho_d.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the singularity classification is derived from the prescribed profiles and conservation, and externally fitted quantities are used only as benchmark inputs.

full rationale

I traced the main derivations: model definitions (Eqs. 13, 19, 21), the conservation-derived DE equations of state (Eqs. 20 and 22), the singularity analysis in Appendix A, and the cosmographic/statefinder computations in Section III and Appendices C-D. The central claim that a continuous sign change replaces the sudden singularity with a w-singularity follows from the assumed smooth density profile and the conservation equation; the pressure remains finite at the density-zero crossing while w_d diverges. This is a mathematical consequence of the ansatz, not a fitted parameter renamed as a prediction, and no equation in the chain is defined in terms of the result it is used to establish. The values H0 = 69.68 km/s/Mpc and z_dagger ~ 1.7 are imported from the prior Lambda_sCDM fit in Ref. [40] for comparison plots, and the paper explicitly labels its cosmic-chronometer comparison as an approximate fit with proper fitting deferred (Sec. II E footnote); those inputs are not presented as predictions of the proposed models. The only citations that overlap with the present authors (Refs. [70] and [86]) are announced future companion papers and are not load-bearing for the singularity or cosmographic results; the w-singularity taxonomy is cited to the independent literature [79,80]. The paper also openly notes where analytical expressions are omitted (scale factor for models C and D, SSCDM cosmographic parameters), which are acknowledged limitations rather than circular substitutions. I therefore find no step that reduces, by construction or by self-citation, to its own input; a sign/factor issue in Eq. (20) noted by an external reader would be a correctness matter, not a circularity matter.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The singularity claim uses only the smoothness of the prescribed profiles and the conservation equation; the specific parameter values enter only the illustrative plots.

free parameters (6)

sign-switch redshift z_dagger = 1.7
Chosen from Ref. [42] (models A and D) and about 1.72 for model C; sets the crossing position in all figures.
ladder step parameters (N, M, Delta z_step) = N=20, M=19, Delta z_step=0.1
Hand-picked to make a smooth-looking ladder while keeping an odd/even comparison.
ladder transition redshifts z_i, z_f = z_i,le=2.7, z_f,le=0.7; z_i,lo=2.6, z_f,lo=0.7
Derived from z_dagger and the step count; not fitted.
smooth-step transition redshifts z_i,ss, z_f,ss = 2.7, 1.0
Chosen to mimic the ladder transition window.
error-function smoothness eta = 10
Hand-chosen to approximate the transition speed of models B and C.
background density and Hubble inputs = Omega_m0=0.3, Omega_r0=0 or 8e-5, Omega_d0=0.7, H0=69.68
Taken from the Lambda_sCDM Planck+Pantheon fit in Ref. [40]; used to set all plots.

assumptions (4)

standard math FLRW background with flat k=0 and standard Friedmann/Raychaudhuri equations (Eq. 1).
Assumed for all models.
standard math Each fluid obeys the separate conservation equation (Eq. 3).
Used to convert energy-density profiles into EoS parameters.
ad hoc to paper The sign-switching energy density is a prescribed function of x=ln a (Eqs. 19 and 21), not derived from a field theory.
The paper states these are phenomenological alternatives and defers microphysical embedding to future work.
domain assumption The abrupt Lambda_sCDM model from the literature is taken as the benchmark.
All new models are designed to mimic it; the singularity comparison relies on its known Type II classification.

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Cite this review

Pith. "Pith review of Cosmographic analysis of sign-switching dark energy." pith.science (2026). https://pith.science/paper/K6OVV3UM

@misc{pith2026250612139,
  author       = {Pith},
  title        = {Pith review of: Cosmographic analysis of sign-switching dark energy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K6OVV3UM}},
  note         = {Machine review of arXiv:2506.12139}
}

abstract

Inspired by the well-studied $\Lambda_{\rm s}$CDM model, we propose and investigate a class of dynamical dark energy models that evolve from a negative cosmological constant, transitioning to a positive value at low redshifts. Specifically, we introduce a dark energy framework based on a generalised ladder-step function for the cosmological constant. Furthermore, we present two additional models in which the cosmological constant undergoes a smooth sign change at a specified redshift. We provide a detailed discussion of the construction and theoretical properties of these models, analysing their background cosmological evolution using a cosmographic approach and the statefinder hierarchy parameters. Our results are compared with those obtained for the standard $\Lambda$CDM and $\Lambda_{\rm s}$CDM models. We also perform a careful analysis of the types of singularities that may arise in these models due to a sign change in the cosmological constant. Notably, we show that a continuous sign change removes the sudden singularity present in the $\Lambda_{\rm s}$CDM model, replacing it with a milder $w$-singularity.

Figures

Figures reproduced from arXiv: 2506.12139 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

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Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Fitting a Gaussian-potential 3-form dark-energy model to Planck, DESI, Pantheon+, Cepheid and DES data shifts H0 up by only ~0.4 km/s/Mpc and yields a weak statistical preference over ΛCDM.
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astro-ph.CO 2026-03 conditional novelty 5.5 of 10

A background reconstruction maps prescribed ρ_de(z) histories to V(φ) and ranks analytic potentials by Bayesian evidence, with exponential preferred for CPL and shifted-tanh for sign-switching targets.
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Reference graph

Works this paper leans on

94 extracted references · 5 canonical work pages · cited by 6 Pith papers

[1]

3 2 r Λ 3 t0 +B 1 # , a10 =D −1/2

Even ladder-like DE model (LeΛCDM) This is a two-parameters extension of the ΛCDM model, in which we consider an initial and final redshifts, denoted respectively byz i,le andz f,le. Those parameters determine when the cosmological constant starts growing from a minimum negative value atz i,le to a maximum positive value atz f,le. The growth of the cosmol...
[2]

The difference lies in the require- ment thatMmust be odd, ensuring that there is no moment in time when the DE density becomes zero

Odd ladder-like DE (LoΛCDM) Similar to the previous case, we introduce a two- parameters extension of the ΛCDM model, with anM- step ladder-function. The difference lies in the require- ment thatMmust be odd, ensuring that there is no moment in time when the DE density becomes zero. We denote the initial and final redshifts of the transition as zi,lo andz...
[3]

The stars indicate the present day values of the statefinder parameters for each of the models

The assumed values of the actual energy densities areΩ m,0 = 0.3,Ω r,0 = 0andΩ d,0 = 0.7. The stars indicate the present day values of the statefinder parameters for each of the models. As can be seen, they all overlap in the same region. j= 1 holds consistently, provided that radiation is ne- glected. In contrast, models with a time varying DE den- sity ...
[4]

(20), we can compute the total pressure and the total density ρtot =ρ r,0e−4x +ρ m,0e−3x +ρ d,0Λss(x), Ptot = 1 3 ρr,0e−4x +w d,ss(x)ρd,0Λss(x)

SSCDM singularities Using Eq. (20), we can compute the total pressure and the total density ρtot =ρ r,0e−4x +ρ m,0e−3x +ρ d,0Λss(x), Ptot = 1 3 ρr,0e−4x +w d,ss(x)ρd,0Λss(x). (A2) Upon evaluatingρ tot(x†,ss) andP tot(x†,ss), we find ρtot(x†,ss) =ρ r,0e−4x†,ss +ρ m,0e−3x†,ss , Ptot(x†,ss) = 1 3 ρr,0e−4x −ρ d,0 1260 ∆x (t4 † −4t 5 † + 6t6 † −4t 7 † +t 8 †),...
[5]

(22) we can compute the total pressure and the total density ρtot =ρr,0e−4x +ρ m,0e−3x +ρ d,0 Erf[η(x−x †)], Ptot = 1 3 ρr,0e−4x −ρ d,0Erf[η(x−x †)] −ρ d,0 2η 3√π e−η2(x−x†)2

ECDM singularities Using Eq. (22) we can compute the total pressure and the total density ρtot =ρr,0e−4x +ρ m,0e−3x +ρ d,0 Erf[η(x−x †)], Ptot = 1 3 ρr,0e−4x −ρ d,0Erf[η(x−x †)] −ρ d,0 2η 3√π e−η2(x−x†)2 . (A5) FIG. 6:Total density and total pressure of the SSCDM and ECDM universes (see Eqs. (A2) and (A5)) with respect to scale factor. For∆x→ ∞andη→ ∞, we...
[6]

(C12) 15

Ladder models (LeΛCDM and LoΛCDM) We will refer to both the even, Λ le(z), and odd, Λ lo(z), cases as Λ L(e−x −1) for simplicity q(x) =−1 + 3Ωm,0ex + 4Ωr,0 2 (Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0) ,(C9) j(x) = 1 + 2Ωr,0 Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0 ,(C10) s(x) = 1− 2Ωr,0(3Ωm,0ex + 4Ωr,0) + 3(Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0)(3Ωm,0ex + 8Ωr,0) 2(Ωd,0Λ...
[7]

All the results in this section are presented for a spatially flat FLR W universe filled with DE, matter, and radiation

ECDM q(x) =−1 + −Ωd,0 2√π e4x−(x−x†,e)2η2 η+ 3Ω m,0ex + 4Ωr,0 2 (Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)]) ,(C13) j(x) = 1 + e−(x−x†,e)2η2 −Ωd,0e4xη(−3 + 2(x−x †,e)η2) + 2√πΩr,0 √π(Ω m,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)]) ,(C14) s(x) = h 2π Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)] 2i−1 ( e−2(x−x†,e)2η2 −2e8xη2(−3 + 2(x−x †,e)η2)Ω2 d,0 +e4x+(x−x†,e)2η2 √πηΩ...
[8]

Ladder models (LeΛCDM and LoΛCDM) We will refer to both the even, Λ le(z), and odd, Λ lo(z), cases as Λ L(e−x −1) for simplicity S(1) 3 = 1 + 2Ωr,0e−x Ωr,0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x ,(D9) S(1) 4 = 1− Ωr0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x −2 10Ωr,0e−x + 9Ωm,0 + 6Ωd,0ΛL(e−x −1)e −3x Ωr,0e−x , (D10) S(1) 5 = 1 + Ωr,0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x −2 76...

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ECDM S(1) 3 = 1 + e−(x−x†,e)2η2 (−e4xη(−3 + 2(x−x †,e)η2)Ωd,0) + 2√πΩr,0√π(Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[(x−x †,e)η]) ,(D13) S(1) 4 = 1 2π(Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[(x−x †,e)η])2 e−2(x−x†,e)2η2 −2e 8xη2(−3 + 2(x−x †,e)η2)Ω2 d,0 +e 4x+(x−x†,e)2η2 √πηΩ d,0(ex(−3−2(2 + 5x−5x †,e)...
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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

3 2 r Λ 3 t0 +B 1 # , a10 =D −1/2

Even ladder-like DE model (LeΛCDM) This is a two-parameters extension of the ΛCDM model, in which we consider an initial and final redshifts, denoted respectively byz i,le andz f,le. Those parameters determine when the cosmological constant starts growing from a minimum negative value atz i,le to a maximum positive value atz f,le. The growth of the cosmol...

[2] [2]

The difference lies in the require- ment thatMmust be odd, ensuring that there is no moment in time when the DE density becomes zero

Odd ladder-like DE (LoΛCDM) Similar to the previous case, we introduce a two- parameters extension of the ΛCDM model, with anM- step ladder-function. The difference lies in the require- ment thatMmust be odd, ensuring that there is no moment in time when the DE density becomes zero. We denote the initial and final redshifts of the transition as zi,lo andz...

[3] [3]

The stars indicate the present day values of the statefinder parameters for each of the models

The assumed values of the actual energy densities areΩ m,0 = 0.3,Ω r,0 = 0andΩ d,0 = 0.7. The stars indicate the present day values of the statefinder parameters for each of the models. As can be seen, they all overlap in the same region. j= 1 holds consistently, provided that radiation is ne- glected. In contrast, models with a time varying DE den- sity ...

[4] [4]

(20), we can compute the total pressure and the total density ρtot =ρ r,0e−4x +ρ m,0e−3x +ρ d,0Λss(x), Ptot = 1 3 ρr,0e−4x +w d,ss(x)ρd,0Λss(x)

SSCDM singularities Using Eq. (20), we can compute the total pressure and the total density ρtot =ρ r,0e−4x +ρ m,0e−3x +ρ d,0Λss(x), Ptot = 1 3 ρr,0e−4x +w d,ss(x)ρd,0Λss(x). (A2) Upon evaluatingρ tot(x†,ss) andP tot(x†,ss), we find ρtot(x†,ss) =ρ r,0e−4x†,ss +ρ m,0e−3x†,ss , Ptot(x†,ss) = 1 3 ρr,0e−4x −ρ d,0 1260 ∆x (t4 † −4t 5 † + 6t6 † −4t 7 † +t 8 †),...

[5] [5]

(22) we can compute the total pressure and the total density ρtot =ρr,0e−4x +ρ m,0e−3x +ρ d,0 Erf[η(x−x †)], Ptot = 1 3 ρr,0e−4x −ρ d,0Erf[η(x−x †)] −ρ d,0 2η 3√π e−η2(x−x†)2

ECDM singularities Using Eq. (22) we can compute the total pressure and the total density ρtot =ρr,0e−4x +ρ m,0e−3x +ρ d,0 Erf[η(x−x †)], Ptot = 1 3 ρr,0e−4x −ρ d,0Erf[η(x−x †)] −ρ d,0 2η 3√π e−η2(x−x†)2 . (A5) FIG. 6:Total density and total pressure of the SSCDM and ECDM universes (see Eqs. (A2) and (A5)) with respect to scale factor. For∆x→ ∞andη→ ∞, we...

[6] [6]

(C12) 15

Ladder models (LeΛCDM and LoΛCDM) We will refer to both the even, Λ le(z), and odd, Λ lo(z), cases as Λ L(e−x −1) for simplicity q(x) =−1 + 3Ωm,0ex + 4Ωr,0 2 (Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0) ,(C9) j(x) = 1 + 2Ωr,0 Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0 ,(C10) s(x) = 1− 2Ωr,0(3Ωm,0ex + 4Ωr,0) + 3(Ωd,0ΛL(e−x −1)e 4x + Ωm,0ex + Ωr,0)(3Ωm,0ex + 8Ωr,0) 2(Ωd,0Λ...

[7] [7]

All the results in this section are presented for a spatially flat FLR W universe filled with DE, matter, and radiation

ECDM q(x) =−1 + −Ωd,0 2√π e4x−(x−x†,e)2η2 η+ 3Ω m,0ex + 4Ωr,0 2 (Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)]) ,(C13) j(x) = 1 + e−(x−x†,e)2η2 −Ωd,0e4xη(−3 + 2(x−x †,e)η2) + 2√πΩr,0 √π(Ω m,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)]) ,(C14) s(x) = h 2π Ωm,0ex + Ωr,0 + Ωd,0e4x Erf[η(x−x †,e)] 2i−1 ( e−2(x−x†,e)2η2 −2e8xη2(−3 + 2(x−x †,e)η2)Ω2 d,0 +e4x+(x−x†,e)2η2 √πηΩ...

[8] [8]

Ladder models (LeΛCDM and LoΛCDM) We will refer to both the even, Λ le(z), and odd, Λ lo(z), cases as Λ L(e−x −1) for simplicity S(1) 3 = 1 + 2Ωr,0e−x Ωr,0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x ,(D9) S(1) 4 = 1− Ωr0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x −2 10Ωr,0e−x + 9Ωm,0 + 6Ωd,0ΛL(e−x −1)e −3x Ωr,0e−x , (D10) S(1) 5 = 1 + Ωr,0e−x + Ωm,0 + Ωd,0ΛL(e−x −1)e −3x −2 76...