Λcos: Apparent Phantom Crossing as Template Bias

Code and data for the paper:

B. Shatto, "Apparent Phantom Crossing as Template Bias: A Bounded-Clock Deformation of ΛCDM" (2026).

This repository contains the analysis pipeline, figure-generation scripts, and the LaTeX source for the paper itself (under paper/).

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Overview

The Λcos model is a one-parameter deformation of the fiducial flat ΛCDM expansion history using a bounded auxiliary variable. It yields

$$\frac{H^2(z)}{H_0^2} = \alpha,(1+z)^3 ;-; \beta,(1+z) ;+; \Omega_\Lambda$$

with $\alpha$, $\beta$ determined by $s_0$ and a fixed reference $\Omega_\Lambda$. Under the fiducial-matter diagnostic split, the effective residual satisfies $w_\mathrm{eff}(z) > -1$.

Reproducible results in this repository:

• Joint Pantheon+ + DESI DR2 BAO fit for ΛCDM, Λcos, and wCDM (§V.B, §V.G)
• Template-bias mock fits across CPL, BA, JBP, and a three-parameter polynomial (§IV.B)
• CPL threshold scan across $s_0 \in [0.01, 0.40]$ (§IV.C)
• Prior sensitivity for Λcos under flat, $s_0^2$, and $\log_{10}(s_0)$ priors (§V.C)
• Ω_Λ sensitivity scan across 0.680–0.715 (§V.D)
• CMB distance priors for flat ΛCDM, non-flat ΛCDM, Λcos at fixed Ω_Λ, and Λcos with Ω_Λ free (§V.E)
• Savage-Dickey Bayes factor at the $s_0$ prior boundary (§V.E)
• Clock exponent comparison for Models A, B, C, D (Appendix A)
• Linear growth comparison against DESI DR1 ShapeFit+BAO $f\sigma_{s8}$ at six tracer effective redshifts (§VI.C)
• Figure generation for Figs. 1–6

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Follow-up note: the γ-CDM diagnostic check

paper2/gamma_cdm_note.tex (compiled PDF) is a short, standalone two-page note, "A Diagnostic Check on a One-Parameter Matter-Exponent Deformation of ΛCDM." It is not a revision of the paper above and does not depend on it: a separate one-parameter deformation,

$$\frac{H^2(z)}{H_0^2} = \Omega_m,(1+z)^\gamma + \Omega_\Lambda, \qquad \gamma = 3 \text{ recovers } \Lambda\text{CDM},$$

used purely as a diagnostic for how much of the DESI-era preference for dynamical dark energy is driven by the very-low-redshift supernova sample.

On the full Pantheon+ + DESI DR2 BAO vector the deformation prefers $\gamma < 3$ at $\Delta\chi^2 \approx -8.75$. The standard $z > 0.01$ cosmology cut (removing 111 low-$z$ SNe on the usual peculiar-velocity/calibration grounds) reduces this to $\Delta\chi^2 \approx -2.97$, and a $z > 0.1$ stress cut to $-1.89$; a Savage-Dickey comparison favors ΛCDM under both a narrow and a wide prior on $\gamma$. Conclusion: the large apparent preference is substantially a property of the very-low-redshift supernova sample, not evidence for this or any nonstandard late-time expansion history. The note is offered as a small independent data point for the low-redshift supernova-systematics discussion, not as a candidate cosmology.

Supporting analysis: scripts/gamma_gate.py, scripts/diagnose_clock_asymmetry.py, scripts/fit_clock_asymmetry.py, scripts/lcos_z001_robustness.py, with outputs under results/gamma_gate_* and results/clock_asymmetry_*. The note is not yet separately archived with its own DOI; its data-availability statement points to this repository and to the Zenodo deposit of the pipeline above (design commit 343c9d8, results commit 69c4604).

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Citation

If you use this code or data, please cite the paper and the Zenodo archive of this repository.

@article{Shatto2026Lambdacos,
  title  = {Apparent Phantom Crossing as Template Bias: A Bounded-Clock
            Deformation of ΛCDM},
  author = {Shatto, B.},
  year   = {2026}
}

@misc{ShattoLambdacosCode2026,
  author = {Shatto, B.},
  title  = {Λcos: Code and Data for "Apparent Phantom Crossing as Template Bias"},
  year   = {2026},
  doi    = {10.5281/zenodo.19798852}
}

---

Repository contents

.
├── README.md                                  This file
├── LICENSE                                    MIT
├── requirements.txt                           Python dependencies
├── data/
│   ├── pantheon_plus.csv                      Pantheon+ SNe Ia magnitudes
│   ├── pantheon_plus_cov.npy                  1701 × 1701 statistical + systematic covariance
│   ├── desi_dr2_bao.csv                       DESI DR2 BAO observables
│   ├── desi_dr2_bao_cov.npy                   Inter-observable covariance for the 13 BAO points
│   └── desi_dr1_fs_fsigma8.csv                DESI DR1 ShapeFit+BAO compressed fσ_s8 amplitudes at 6 tracer effective redshifts (§VI.C)
├── scripts/
│   ├── fit_lcdm.py                            Flat ΛCDM MCMC fit (§V.B)
│   ├── fit_lcos.py                            Λcos MCMC fit; --omega_lambda VALUE (default 0.685, §V.B)
│   ├── fit_wcdm.py                            wCDM MCMC fit (§V.G)
│   ├── fit_clock_exponents.py                 Clock exponent comparison (Appendix A)
│   ├── fit_lcdm_cmb.py                        ΛCDM + CMB distance priors; --non_flat (§V.E)
│   ├── fit_lcos_cmb.py                        Λcos + CMB distance priors; --free_omega_lambda (§V.E)
│   ├── omega_lambda_scan.py                   Aggregate §V.D Ω_Λ sensitivity table
│   ├── prior_sensitivity.py                   §V.C prior-reweighting of the baseline Λcos chain
│   ├── bayes_factor.py                        §V.E Savage-Dickey Bayes factor at the s₀ prior boundary
│   ├── template_bias.py                       Template-bias mocks + Fig. 1 (§IV.B)
│   ├── threshold_scan.py                      CPL threshold scan + Fig. 2 (§IV.C)
│   ├── make_plots.py                          Λcos corner (Fig. 3) and residuals (Fig. 4)
│   ├── growth.py                              Linear-growth ODE solver for arbitrary E(z); validates against textbook Ω_m(z)^0.55 (§VI.C)
│   ├── compute_rsd_chi2.py                    χ²_RSD comparison vs DESI DR1 FS at the SN+BAO best fit (§VI.C)
│   ├── make_growth_figures.py                 Fig. 5 (fσ_8 trajectories + DR1 FS data) and Fig. 6 (Ω_m(z) diagnostic) (§VI.C)
│   ├── _summary.py                            Shared harmonized summary-JSON schema helper
│   ├── lcos_z001_robustness.py                Paper 1 robustness under the z>0.01 cut (apples-to-apples s0 UL)
│   ├── fit_clock_asymmetry.py                 Continuous clock-asymmetry (ε) fit, two tiers (paper2 precursor)
│   ├── diagnose_clock_asymmetry.py            Post-run diagnostics for the ε fit (audit, Ω_Λ-free scan, SN/BAO split)
│   └── gamma_gate.py                          γ-CDM gate: fits, evidence, and threshold scan for paper2
├── results/                                   MCMC chains, post-burn samples, summaries, generated figures
│   ├── lcdm_chain.npy, lcdm_post.csv, lcdm_summary.json, lcdm_corner.png
│   ├── lcos_chain.npy, lcos_post.csv, lcos_summary.json, lcos_corner.{png,pdf}
│   ├── lcos_omegaL_<v>_*.{npy,csv,json,png}   Λcos at alternative Ω_Λ ∈ {0.680, 0.690, 0.700, 0.715} (§V.D)
│   ├── wcdm_chain.npy, wcdm_post.csv, wcdm_summary.json, wcdm_corner.png
│   ├── lcdm_cmb_*.{npy,csv,json,png}          Flat ΛCDM + CMB priors (§V.E)
│   ├── lcdm_cmb_nonflat_*.{npy,csv,json,png}  Non-flat ΛCDM + CMB priors (§V.E)
│   ├── lcos_cmb_*.{npy,csv,json,png}          Λcos at Ω_Λ = 0.685 + CMB priors (§V.E)
│   ├── lcos_cmb_freeOL_*.{npy,csv,json,png}   Λcos with Ω_Λ free + CMB priors (§V.E)
│   ├── clock_exponent_{A,B,C,D}_chain.npy
│   ├── clock_exponent_{A,B,C,D}_postburn.csv
│   ├── clock_exponent_results.csv             Appendix A summary across all four models
│   ├── prior_sensitivity.csv                  §V.C reweighted-prior medians and 95% upper limits
│   ├── bayes_factor.csv                       §V.E B_01 across bandwidths for stability
│   ├── template_bias.csv, template_bias.{png,pdf}     §IV.B fits and Fig. 1
│   ├── threshold_scan.csv, threshold_scan.{png,pdf}   §IV.C scan and Fig. 2
│   ├── residuals.{png,pdf}                    Fig. 4
│   ├── growth_LCDM_Om0p315.csv                §VI.C growth trajectory at Ω_m = 0.315
│   ├── growth_Lcos_s00p076.csv                §VI.C growth trajectory at the Λcos posterior median
│   ├── growth_Lcos_s00p185.csv                §VI.C growth trajectory at the 95% upper limit
│   ├── rsd_chi2.csv, rsd_residuals.csv        §VI.C χ²_RSD summary and per-tracer pulls
│   ├── fig5_fsigma8.{png,pdf}                 Fig. 5 (fσ_8 + DR1 FS data)
│   ├── fig6_omegam_z.{png,pdf}                Fig. 6 (Ω_m(z) diagnostic)
│   ├── lcos_z001_robustness.{json,log}, lcos_z001_s0_profile.csv, lcos_all_s0_profile.csv   Paper 1 z>0.01 robustness
│   ├── clock_asymmetry_*.{json,csv,log,npy}   ε-fit tiers, gates, and post-run diagnostics (audit, Ω_Λ-free scan, SN/BAO split)
│   └── gamma_gate_*.{json,csv,log}            γ-CDM gate: fits, evidence (primary/robust priors), threshold-scan splits
├── tables/
│   ├── clock_exponent_appendix_A_fits.csv     Curated Appendix A reference values
│   └── omega_lambda_scan.csv                  Aggregated §V.D Ω_Λ sensitivity table
├── figures/
│   ├── fig1_template_bias_overlay.pdf         §IV.B: w(z) overlays for CPL/BA/JBP/Polynomial
│   ├── fig2_threshold_scan.pdf                §IV.C: recovered (w₀, w_a) vs s₀
│   ├── fig3_lcos_corner.pdf                   §V.B: Λcos posterior in (s₀, H₀r_d, M_B)
│   ├── fig4_hubble_residuals.pdf              §V.B: Pantheon+ binned residuals for ΛCDM and Λcos
│   ├── fig5_fsigma8.pdf                       §VI.C: fσ_8(z) trajectories + DESI DR1 FS data
│   └── fig6_omegam_z.pdf                      §VI.C: Ω_m(z) diagnostic out to z = 3
├── paper/                                     LaTeX source for the manuscript
│   ├── paper.tex                              REVTeX 4.2 single-source LaTeX (preamble + body + bibliography)
│   ├── references.bib                         22 entries
│   ├── figures/                               Self-contained copies of ../figures/*.pdf
│   ├── paper.pdf                              Compiled output (single-column, JCAP submission format)
│   ├── Makefile                               Build pipeline (pdflatex + bibtex + pdflatex × 2)
│   └── README.md                              Build instructions
└── paper2/                                    LaTeX source for the follow-up diagnostic note
    ├── gamma_cdm_note.tex                     REVTeX 4.2 two-page note (see "Follow-up note" above)
    └── gamma_cdm_note.pdf                     Compiled output

figures/ holds the paper-facing PDFs at their published filenames. They are stable copies of the corresponding script outputs in results/.

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Installation

Python 3.10 or later recommended.

git clone https://github.com/dmobius3/lambda-cos.git
cd lambda-cos
python -m venv venv
source venv/bin/activate
pip install -r requirements.txt

Dependencies (requirements.txt):

numpy>=1.24
scipy>=1.10
emcee>=3.1
corner>=2.2
matplotlib>=3.7
h5py>=3.8
pandas>=2.0

Total install footprint about 200 MB. The primary SN+BAO MCMC fit takes roughly 5 minutes on a recent laptop (32 walkers, 5000 steps).

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Quickstart: reproducing the primary fit

The headline result is the joint Pantheon+ + DESI DR2 BAO Λcos fit (§V.B). All scripts are written to be run from the scripts/ directory and resolve paths via ../data/ and ../results/.

cd scripts
python fit_lcos.py

Outputs to results/: lcos_chain.npy, lcos_post.csv, lcos_summary.json, lcos_corner.png.

For the ΛCDM baseline:

python fit_lcdm.py

For wCDM (§V.G):

python fit_wcdm.py

Reference summary values from the deposited posteriors:

Λcos:    s0 median ≈ 0.076  (68% CI: 0.023, 0.143)
         s0 95% UL ≈ 0.185  (flat prior)
         H0 r_d    ≈ 10008  km/s
         M_B       ≈ -19.353
         tau_max   ≈ 46.9
ΛCDM:    Ω_m       ≈ 0.312  (68% CI: 0.304, 0.321)
         H0 r_d    ≈ 10043  km/s
         M_B       ≈ -19.355
         tau_max   ≈ 35.9
wCDM:    Ω_m       ≈ 0.297
         w         ≈ -0.855  (68% CI: -0.89, -0.82)
         Δχ²       ≈ -13.05 vs flat ΛCDM
         ΔAIC      ≈ -11.05
         ΔBIC      ≈  -5.61
         tau_max   ≈ 47.4

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Reproducing each figure

cd scripts

# Figure 1 — w(z) recoveries from CPL, BA, JBP, Polynomial
python template_bias.py
# -> results/template_bias.{png,pdf}

# Figure 2 — CPL threshold scan
python threshold_scan.py
# -> results/threshold_scan.{png,pdf}

# Figures 3 and 4 — Λcos corner plot and Pantheon+ residuals
# Requires lcdm_post.csv and lcos_post.csv (run fit_lcdm.py and fit_lcos.py first)
python make_plots.py
# -> results/lcos_corner.{png,pdf}   (Fig. 3)
# -> results/residuals.{png,pdf}     (Fig. 4)

# Figures 5 and 6 — fσ_8(z) with DESI DR1 FS overlay and Ω_m(z) diagnostic
# Requires growth_LCDM_*.csv and growth_Lcos_*.csv (see §VI.C reproduction below)
python make_growth_figures.py
# -> results/fig5_fsigma8.{png,pdf}  (Fig. 5)
# -> results/fig6_omegam_z.{png,pdf} (Fig. 6)

The paper-facing PDFs in figures/ (fig1_template_bias_overlay.pdf, fig2_threshold_scan.pdf, fig3_lcos_corner.pdf, fig4_hubble_residuals.pdf, fig5_fsigma8.pdf, fig6_omegam_z.pdf) are stable copies of the corresponding results/ outputs renamed to match the in-paper figure numbers.

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Reproducing the template-bias scan (§IV.C)

cd scripts
python threshold_scan.py

Iterates CPL fits across $s_0 \in [0.01, 0.40]$ in steps of $0.01$, recording $(w_0, w_a, \chi^2)$ at each value to results/threshold_scan.csv and producing Fig. 2.

The single-$s_0$ mock comparison across all four parameterizations (Table in §IV.B) is produced separately by template_bias.py, which writes results/template_bias.csv and Fig. 1.

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Reproducing Appendix A (clock exponent selection)

cd scripts
python fit_clock_exponents.py

Models A (n = 0), B (n = −1), C (n = +1), D (n = −1/2) are fit with the same MCMC setup as the primary Λcos run. Outputs to results/:

• clock_exponent_{A,B,C,D}_chain.npy — full chains
• clock_exponent_{A,B,C,D}_postburn.csv — post-burn samples
• clock_exponent_results.csv — summary with one row per model: best-fit parameters, χ² split (SN, BAO, total), Δχ² vs ΛCDM, acceptance fraction

The curated paper-facing values are also deposited at tables/clock_exponent_appendix_A_fits.csv.

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Reproducing the Ω_Λ sensitivity scan (§V.D)

cd scripts
python fit_lcos.py --omega_lambda 0.680
python fit_lcos.py --omega_lambda 0.685   # canonical
python fit_lcos.py --omega_lambda 0.690
python fit_lcos.py --omega_lambda 0.700
python fit_lcos.py --omega_lambda 0.715
python omega_lambda_scan.py

Outputs to results/lcos_omegaL_<v>_*.{npy,csv,json,png} for each non-canonical value (the canonical Ω_Λ = 0.685 writes to lcos_* without a suffix). The aggregator reads each summary JSON and produces tables/omega_lambda_scan.csv with one row per Ω_Λ (s₀ median, s₀ 95% UL, χ²_min, χ²_SN, χ²_BAO, Δχ² vs ΛCDM baseline, τ_max, acceptance).

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Reproducing §V.E (CMB distance priors)

§V.E adds compressed Planck 2018 distance priors (R = 1.7502 ± 0.0046, ℓ_A = 301.47 ± 0.09) to the SN+BAO likelihood. Four fits in total:

cd scripts
python fit_lcdm_cmb.py                       # Flat ΛCDM + CMB priors  (3 params)
python fit_lcdm_cmb.py --non_flat            # Non-flat ΛCDM + CMB     (4 params, Ω_k = 1 - Ω_m - Ω_Λ - Ω_r)
python fit_lcos_cmb.py                       # Λcos at Ω_Λ = 0.685 + CMB priors (3 params)
python fit_lcos_cmb.py --free_omega_lambda   # Λcos with Ω_Λ free + CMB priors  (4 params)

Outputs:

• results/lcdm_cmb_* and results/lcdm_cmb_nonflat_* for the two ΛCDM cases
• results/lcos_cmb_* and results/lcos_cmb_freeOL_* for the two Λcos cases

Each summary JSON reports the χ² split (SN, BAO, CMB), the best-fit point from a post-MCMC optimizer pass, the integrated autocorrelation time per parameter, the acceptance fraction, and (for the 4-parameter fits) the Ω_Λ posterior quantiles.

The CMB priors are implemented with the standard compressed-prior forms

R   = sqrt(Ω_m) * ∫₀^z* dz/E(z)
ℓ_A ≈ π c / (H0 r_d) * ∫₀^z* dz/E(z)        (treating r_d ≈ r_s(z*); ~2% offset for standard cosmology)

with z* = 1090 and Ω_r = 9.15 × 10⁻⁵ included in E(z) for the high-z integral.

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Reproducing §VI.C (linear-growth consistency check)

§VI.C compares the linear-growth prediction f σ_8(z) of ΛCDM and Λcos against the DESI DR1 ShapeFit+BAO compressed growth amplitudes (DESI 2024 Paper V, Appendix A, Eqs. A.13–A.24) at six tracer effective redshifts. The Λcos correction enters only through H(z); no perturbation-level parameter is introduced.

cd scripts

# Step 1: validate the growth ODE solver against textbook ΛCDM
python growth.py --validate
# -> max |Δf/f| ≲ 0.5% vs Ω_m(z)^0.55 over z ∈ [0, 2.4]

# Step 2: compute fσ_8(z) trajectories for ΛCDM and Λcos at two s₀ values
python growth.py --model LCDM --Om 0.315
python growth.py --model Lcos --s0 0.076 --OL 0.685
python growth.py --model Lcos --s0 0.185 --OL 0.685
# -> results/growth_LCDM_Om0p315.csv
# -> results/growth_Lcos_s00p076.csv  (posterior median)
# -> results/growth_Lcos_s00p185.csv  (95% upper limit)

# Step 3: χ²_RSD against DESI DR1 FS at the SN+BAO best fit
python compute_rsd_chi2.py
# -> results/rsd_chi2.csv          (per-model χ²_RSD and Δχ²)
# -> results/rsd_residuals.csv     (per-tracer pulls)

# Step 4: figures
python make_growth_figures.py
# -> results/fig5_fsigma8.{png,pdf}   (Fig. 5)
# -> results/fig6_omegam_z.{png,pdf}  (Fig. 6)

Reference values:

χ²_RSD (6 tracer bins, diagonal-error consistency check):
  ΛCDM (Ω_m = 0.315)                4.64
  Λcos (s₀ = 0.076, median)         4.68    Δχ² = +0.04
  Λcos (s₀ = 0.185, 95% UL)         4.90    Δχ² = +0.26

Ω_m(z) split at z = 2.3:
  Λcos – ΛCDM at s₀ = 0.185         −2.9%

Note on data provenance: DESI DR2 (March 2025) released BAO distances only, not a DR2 full-shape product. The growth comparison therefore uses DR1 FS; the combination of DR2 BAO with DR1 FS follows the collaboration's own precedent (Elbers et al., arXiv:2503.14744; Forero-Sánchez et al., arXiv:2602.18761 for the rigorous joint treatment).

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Data sources and provenance

Dataset	Source	Reference
Pantheon+ SNe Ia	pantheonplussh0es.github.io	Brout et al., Astrophys. J. 938, 110 (2022)
DESI DR2 BAO	data.desi.lbl.gov	DESI Collaboration, arXiv:2503.14738 (2025)
DESI DR1 ShapeFit+BAO fσ_s8	DESI 2024 Paper V, Appendix A	DESI Collaboration, J. Cosmol. Astropart. Phys. 2025, 008, arXiv:2411.12021
Planck 2018 distance priors	Compressed (R, ℓ_A) from Planck VI	Planck Collaboration VI, Astron. Astrophys. 641, A6 (2020)

The files under data/ are formatted derivatives of the public sources above, repackaged for direct loading by the fit scripts. No proprietary data is included.

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Reproducibility notes

• Random seeding: fit_clock_exponents.py sets RNG_SEED = 12345, which pins the walker initialization only; the emcee chain evolution itself is not seeded. The other MCMC scripts (fit_lcdm.py, fit_lcos.py, fit_wcdm.py) initialize walkers from np.random.randn without an explicit seed. In all cases run-to-run variations are within the posterior thickness and do not change the reported summary values; for boundary-saturated fits (e.g. clock-exponent models B and D) the best-fit s₀ in the flat-likelihood region near the prior floor can vary between runs, while χ² and Δχ² reproduce to ~0.001.
• MCMC configuration: 32 walkers, 5000 steps, 1000 burn-in across all fit scripts.
• Numerical accuracy: distance integrals use SciPy's cumulative_trapezoid on a 4000-point grid in z ∈ [0, 2.5] (or up to z_max ≈ 2.4 from the BAO range). For the clock-exponent run the grid extends to 1.002 × max(z_data).
• Working directory: scripts are run from the scripts/ subdirectory; paths resolve via ../data/ and ../results/.
• Platform: tested on macOS 14.x and Ubuntu 22.04 with Python 3.11. No GPU required.

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License

This repository is released under the MIT License. See LICENSE for the full text.

The Pantheon+ and DESI DR2 BAO data products are redistributed under the terms of their original publications; refer to the linked sources above for their license terms.

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Contact

• Author: B. Shatto, bshatto.pe@gmail.com
• Issues and questions: please open a GitHub issue.