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/).
The Λcos model is a one-parameter deformation of the fiducial flat ΛCDM expansion history using a bounded auxiliary variable. It yields
with
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
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} }
.
├── README.md This file
├── LICENSE MIT
├── requirements.txt Python dependencies
├── data/
│ ├── pantheon_plus.csv Pantheon+ SNe Ia magnitudes
│ ├── pantheon_plus_cov.npy 1701 ×ばつ 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
├── 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)
├── 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 ×ばつ 2)
└── README.md Build instructions
figures/ holds the paper-facing PDFs at their published filenames. They are stable copies of the corresponding script outputs in results/.
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).
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.pyOutputs 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
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.
cd scripts
python threshold_scan.pyIterates CPL fits across 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.
cd scripts
python fit_clock_exponents.pyModels 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 chainsclock_exponent_{A,B,C,D}_postburn.csv— post-burn samplesclock_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.
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).
§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_*andresults/lcdm_cmb_nonflat_*for the two ΛCDM casesresults/lcos_cmb_*andresults/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 ×ばつ 10−5 included in E(z) for the high-z integral.
§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).
| 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.
- Random seeding:
fit_clock_exponents.pysetsRNG_SEED = 12345, which pins the walker initialization only; theemceechain evolution itself is not seeded. The other MCMC scripts (fit_lcdm.py,fit_lcos.py,fit_wcdm.py) initialize walkers fromnp.random.randnwithout 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_trapezoidon 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 ×ばつ 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.
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.
- Author: B. Shatto, bshatto.pe@gmail.com
- Issues and questions: please open a GitHub issue.