FAIR k = 5 comparison using SN Ia, BAO (DESI 2024), RSD, KiDS, Planck compressed priors, and optional CMB lensing

Δχ² = −11.31 (FAIR k=5, same parameters as ΛCDM)

5/6 late-time probes improved; BAO penalty negligible (+0.07)

Late-time H(z) only; no recombination / no perturbations / no TT/TE/EE

Robust across 3 independent runs; insensitive to ±10% variations in fixed parameters

Posterior comparison (ΛCDM vs ECTI-P, FAIR k = 5)

Across six independent late-time cosmological probes, ECTI-P yields a net improvement in total χ² relative to ΛCDM, with 5/6 probes improved and a negligible BAO penalty (+0.07), under a strictly FAIR comparison with the same number of free parameters (k = 5).

ECTI-P is a purely phenomenological, late-time, background-only extension of ΛCDM.

The model modifies only the expansion history H(z) at low redshift.

Explicitly, ECTI-P does not modify:

• recombination physics
• perturbation equations
• linear or non-linear growth sector
• neutrinos or radiation content
• sound horizon physics rₛ
• CMB primary anisotropies

Accordingly:

• no Boltzmann solvers (CLASS or CAMB)
• no TT / TE / EE likelihoods
• no CLik

Planck information is included exclusively through compressed distance priors and, optionally, through the Planck 2018 lensing-only constraint.

ECTI-P modifies the late-time expansion history through an effective deformation of the ΛCDM background evolution.

The expansion rate is written as:

E(z)^2 = E_LCDM(z)^2 × F(z; beta, z_t)

with:

F(z; beta, z_t) → 1   for   z >> z_t

The parameters (beta, z_t) control the amplitude and redshift location of a smooth late-time transition in H(z).

the following figures are provided for physical interpretation and visualization. They are derived from the reference MAP parameters and are not part of the automatic reproduction pipeline.

ECTI-P can be reinterpreted as an effective dark-energy fluid producing the same background expansion history.

From the model-implied E(z), we derive:

• the effective equation of state w_eff(z)
• the deceleration parameter q(z)

These diagnostics are purely interpretative and do not modify the likelihood pipeline.

• w_eff(z) deviates smoothly from −1 at low redshift
• the deviation is localized in the z ≈ 0.1–0.6 range
• the acceleration transition redshift shifts slightly relative to ΛCDM
• the model remains background-consistent at higher redshift

At the MAP point used for visualization:

• q₀ (ΛCDM) ≈ −0.55

• q₀ (ECTI-P) ≈ −0.16

• z_acc (ΛCDM) ≈ 0.67

• z_acc (ECTI-P) ≈ 0.63

These values illustrate the modified late-time acceleration behaviour implied by ECTI-P at the background level, while remaining consistent with high-redshift constraints used in the likelihood.

Importantly, ECTI-P modifies only the late-time background expansion history and does not alter early-universe physics, recombination, or perturbation dynamics within the current implementation.

The effective-fluid diagnostics presented here are therefore strictly background-level reinterpretations of the model.

Both models are sampled with the identical parameter vector:

theta = (H0, Omega_m0, sigma8, M, omega_b)

where:

• H0 : Hubble constant

• Omega_m0 : present-day matter density

• sigma8 : amplitude of matter fluctuations

• M : SN absolute magnitude

• omega_b : physical baryon density

• ΛCDM uses the standard background evolution.

• ECTI-P uses the same vector, with its internal late-time parameters (β, zₜ) fixed and not sampled.

No additional degrees of freedom are introduced in ECTI-P.

Pantheon+ (2022) compilation with SH0ES calibration.

DESI 2024 BAO measurements implemented as a Gaussian likelihood.

Curated compilation of published fσ₈(z) measurements.
The data file is included directly in this repository, together with an explicit row-by-row provenance document.

The RSD likelihood assumes standard GR linear growth, computed by solving the linear growth equation self-consistently using the model-specific background expansion H(z). No modified gravity or additional growth parameters are introduced.

KiDS S₈ compressed Gaussian prior (not the full cosmic shear likelihood).

Planck 2018 compressed distance priors (R, ℓ_A, ω_b) with full covariance.

Planck 2018 lensing-only derived one-dimensional constraint, enabled via a notebook flag and consistently included in χ² and BIC.

The results are stable across three independent paired MCMC runs.

All chains satisfy:

• R̂ < 1.02
• high effective sample size
• no stuck walkers or pathological behavior

The ECTI-P specific parameters (β, z_t) are treated as fixed characteristics of the effective late-time modification for the MCMC sampling ($k=5$).

To verify that the observed $\chi^2$ improvement is not the result of hyper-fine-tuning, we performed a grid search in the $(\beta, z_t)$ plane around the reference values (±10%).

Result : The $\chi^2$ landscape is shallow around the reference solution.
Even for ±10% deviations in the fixed parameters, ECTI-P retains a substantial advantage over ΛCDM, with $\Delta\chi^2 \approx -9$ to $-11$.

• Supernova distance modulus residuals

  

  Supernova distance modulus residuals at the reference MAP parameters.

ECTI-P improves the global fit without introducing redshift-dependent systematics or residual structure.

• BAO residuals

• RSD residuals

Both ΛCDM and ECTI-P remain consistent with fσ₈ measurements, indicating that the late-time background modification does not introduce growth-sector tensions.

All figures are generated automatically by the notebook.

Repository structure:

• ECTI_P_vs_LCDM_full_likelihood_Planck_3D_+*.ipynb
  Main analysis notebook (self-contained, generates all outputs)

• data/rsd/
  RSD tables and provenance

• README.md
  Project overview and scientific context

• RUNNING.md
  Reproducibility and execution instructions

• CITATION.cff
  Citation metadata

• LICENSE
  MIT license

  All figures, tables, and diagnostics are generated automatically by the notebook and are not stored in the repository.

For full reproduction instructions, see RUNNING.md

The definitive test of this phenomenological extension requires confrontation with the full Planck TT/TE/EE likelihood, which necessitates a Boltzmann treatment and HPC environment not currently available to the autor.

For questions regarding the methodology, reproducibility, or scope of the results,
feel free to contact the author via GitHub or email.

If you use this work, please cite: Pantheon+ (Brout et al. 2022) SH0ES (Riess et al. 2022) DESI 2024 BAO Planck 2018 results (Planck Legacy Archive) The primary RSD survey papers listed in the RSD provenance file A CITATION.cff file is provided.

Released under the repository license. All data values originate from published literature and must be cited accordingly.