Λcos: Apparent Phantom Crossing as Template Bias

Code and data for the paper:

B. Shatto, "Apparent Phantom Crossing as Template Bias: A Non-Phantom Test Case with Λcos" (2026).

This repository contains the analysis pipeline and figure-generation scripts needed to reproduce all results in the 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 (§5.2, §5.7)
• Template-bias mock fits across CPL, BA, JBP, and a three-parameter polynomial (§4.2)
• CPL threshold scan across $s_0 \in [0.01, 0.40]$ (§4.3)
• Clock exponent comparison for Models A, B, C, D (Appendix A)
• Figure generation for Figs. 1–4

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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 Non-Phantom
            Test Case with Λcos},
  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}
}

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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
├── scripts/
│   ├── fit_lcdm.py                            Flat ΛCDM MCMC fit (§5.2)
│   ├── fit_lcos.py                            Λcos MCMC fit; --omega_lambda VALUE (default 0.685, §5.2)
│   ├── fit_wcdm.py                            wCDM MCMC fit (§5.7)
│   ├── fit_clock_exponents.py                 Clock exponent comparison (Appendix A)
│   ├── fit_lcdm_cmb.py                        ΛCDM + CMB distance priors; --non_flat (§5.5)
│   ├── fit_lcos_cmb.py                        Λcos + CMB distance priors; --free_omega_lambda (§5.5)
│   ├── omega_lambda_scan.py                   Aggregate §5.4 Ω_Λ sensitivity table
│   ├── template_bias.py                       Template-bias mocks + Fig. 1 (§4.2)
│   ├── threshold_scan.py                      CPL threshold scan + Fig. 2 (§4.3)
│   └── make_plots.py                          Λcos corner (Fig. 3) and residuals (Fig. 4)
├── 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} (§5.4)
│   ├── wcdm_chain.npy, wcdm_post.csv, wcdm_summary.json, wcdm_corner.png
│   ├── lcdm_cmb_*.{npy,csv,json,png}          Flat ΛCDM + CMB priors (§5.5)
│   ├── lcdm_cmb_nonflat_*.{npy,csv,json,png}  Non-flat ΛCDM + CMB priors (§5.5)
│   ├── lcos_cmb_*.{npy,csv,json,png}          Λcos at Ω_Λ = 0.685 + CMB priors (§5.5)
│   ├── lcos_cmb_freeOL_*.{npy,csv,json,png}   Λcos with Ω_Λ free + CMB priors (§5.5)
│   ├── 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
│   ├── template_bias.csv, template_bias.{png,pdf}     §4.2 fits and Fig. 1
│   ├── threshold_scan.csv, threshold_scan.{png,pdf}   §4.3 scan and Fig. 2
│   └── residuals.{png,pdf}                    Fig. 4
├── tables/
│   ├── clock_exponent_appendix_A_fits.csv     Curated Appendix A reference values
│   └── omega_lambda_scan.csv                  Aggregated §5.4 Ω_Λ sensitivity table
└── figures/
    ├── fig1_template_bias_overlay.pdf         §4.2: w(z) overlays for CPL/BA/JBP/Polynomial
    ├── fig2_threshold_scan.pdf                §4.3: recovered (w₀, w_a) vs s₀
    ├── fig3_lcos_corner.pdf                   §5.2: Λcos posterior in (s₀, H₀r_d, M_B)
    └── fig4_hubble_residuals.pdf              §5.2: Pantheon+ binned residuals for ΛCDM and Λcos

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 (§5.2). 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 (§5.7):

python fit_wcdm.py

Reference summary values from the deposited posteriors:

Λcos:    s0        ≈ 0.080 (mean), 0.073 (median)
         s0 95% UL ≈ 0.180  (flat prior)
         H0 r_d    ≈ 10010  km/s
         M_B       ≈ -19.353
         tau_max   ≈ 43.9
ΛCDM:    Ω_m       ≈ 0.312
         H0 r_d    ≈ 10044  km/s
         M_B       ≈ -19.355
         tau_max   ≈ 35.8
wCDM:    Ω_m       ≈ 0.296
         w         ≈ -0.853
         Δχ²       ≈ -13.05 vs flat ΛCDM
         ΔAIC      ≈ -11.05
         ΔBIC      ≈  -5.60
         tau_max   ≈ 44.9

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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)

The paper-facing PDFs in figures/ (fig1_template_bias_overlay.pdf, fig2_threshold_scan.pdf, fig3_lcos_corner.pdf, fig4_hubble_residuals.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 (§4.3)

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 §4.2) 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 (§5.4)

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 §5.5 (CMB distance priors)

§5.5 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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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)
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 uses a fixed seed (RNG_SEED = 12345). The other MCMC scripts (fit_lcdm.py, fit_lcos.py, fit_wcdm.py) initialize walkers from np.random.randn without an explicit seed; small numerical variations between runs are within the posterior thickness and do not change the reported summary values.
• 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.