📊 Prior Specification

This page documents the prior distributions used in the ξ Dark Energy inference pipeline. It exists because the posterior constraints, model comparison metrics, and physical interpretation are not reproducible unless the prior volume is explicitly stated.

Where the bounds come from

Every bound on this page is read from the sampling code, not from a results file: log_prior in src/xi_protocol1b_bao_run.py of the xi-dark-energy-bao-mcmc repository, commit 25ed86e (the code used for the final Protocol 1b run). All priors are top-hat (uniform) on an open interval; a sample outside the interval has zero probability.

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Why priors matter

The ξ Dark Energy project compares standard and exploratory cosmological models using Type Ia supernovae and BAO measurements. In this setting, priors affect:

• posterior parameter constraints,
• convergence behavior,
• Bayes-factor-like interpretations,
• AIC / BIC comparisons indirectly through fit stability,
• whether oscillatory extensions are physically meaningful or merely prior-volume artifacts.

The prior specification is therefore part of the scientific result, not a technical detail.

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Model hierarchy

ΛCDM

The baseline model is used as the reference cosmology.

Parameter	Meaning	Prior (uniform, from code)	Notes
H0	Hubble constant in km s⁻¹ Mpc⁻¹	50 < x < 100	Brackets SH0ES-like and Planck-like values
Omega_m	Matter density fraction	0.1 < x < 0.5	Restricted to physical positive values

The code has no supernova nuisance parameter (no absolute magnitude M, no offset): the SH0ES-calibrated distance moduli are used directly in the likelihood.

The baseline model should remain intentionally conservative. It should not be given overly restrictive priors that make the extended models appear better by comparison.

CPL

The CPL model extends the dark-energy equation of state as:

$$w(a) = w_0 + w_a\,(1 - a)$$

Parameter	Meaning	Prior (uniform, from code)	Notes
H0	Hubble constant	Same as ΛCDM	Same prior as baseline
Omega_m	Matter density fraction	Same as ΛCDM	Same prior as baseline
w0	Present-day equation of state	-2 < x < 0	Includes w0 = -1
wa	Evolution parameter	-3 < x < 2	Allows both increasing and decreasing evolution

CPL is useful as a conventional time-varying dark energy comparator. It should not be tuned to disadvantage or advantage the ξ model.

ξ oscillatory

The exploratory ξ model introduces an oscillatory component in the dark-energy sector. In the current documentation, the parameterization is treated as phenomenological rather than as a confirmed physical theory.

Parameter	Meaning	Prior (uniform, from code)	Notes
H0	Hubble constant	Same as ΛCDM	Common across models
Omega_m	Matter density fraction	Same as ΛCDM	Common across models
w0_mean	Mean dark-energy equation-of-state level	−1.5 < x < −0.5	Includes -1
eps (ε)	Oscillation amplitude	0 < x < 0.3	One-sided: positive amplitudes only (see caveat below)
T_Gyr (T)	Oscillation period in Gyr	1 < x < 10 Gyr	Linear-uniform, not logarithmic
phi0 (φ₀)	Phase	−π < x < π	Full phase interval

Important v0.2 caveat

The implementation only allows positive amplitudes (0 < ε < 0.3). A strictly positive amplitude can encode a phase convention or a physical asymmetry depending on the model definition. The next version should test a symmetric prior on epsilon, and the sign convention must be stated explicitly in any result.

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Prior table (from the code)

Each parameter appears once, at the first model that uses it; H₀ and Ωₘ are shared by the three models.

Parameter	First used in	Prior (uniform, open interval)	Unit
H₀	ΛCDM	50 < x < 100	km/s/Mpc
Ωₘ	ΛCDM	0.1 < x < 0.5	—
w₀	CPL	−2 < x < 0	—
wₐ	CPL	−3 < x < 2	—
w₀,mean	Ξosc	−1.5 < x < −0.5	—
ε	Ξosc	0 < x < 0.3	—
T	Ξosc	1 < x < 10	Gyr
φ₀	Ξosc	−π < x < π	rad

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Prior implementation contract

A sample is valid only if all parameters fall inside their prior support and all derived quantities required by the likelihood are finite. Schematically:

def log_prior(theta):
    if not within_bounds(theta):
        return -np.inf
    if not derived_quantities_are_finite(theta):
        return -np.inf
    return 0.0

This corresponds to uniform priors over the allowed region and zero probability outside it.

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Physical constraints

The prior should enforce basic physical sanity checks.

Constraint	Reason
H0 > 0	Expansion rate must be positive
0 < Omega_m < 1	Matter density fraction must remain physical in the explored model (the sampled prior is tighter: see the table above)
Finite luminosity distance	Required for supernova likelihood
Finite BAO distances	Required for DESI BAO likelihood
Full phase coverage for phi0	Avoid phase selection bias
Explicit amplitude convention	Avoid ambiguity in the sign of epsilon

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Known prior-related caveats

1. Oscillation amplitude convention

If epsilon is restricted to positive values, the model must explain whether the sign is absorbed by the phase or whether negative amplitudes represent distinct physical behavior.

2. Period prior sensitivity

The parameter T can be especially sensitive to prior choices. A broad linear prior and a broad log prior do not encode the same scientific assumption.

3. Boundary effects

If posterior mass accumulates near a prior boundary, the result should be treated as prior-limited rather than data-constrained. This is the case here: the marginals of ε and T reach the upper edge of their priors (ε: upper 1σ edge 0.262 against a bound of 0.3; T: multi-featured posterior up to 10 Gyr). See Oscillation Amplitude and Known Limitations.

4. Model comparison caution

AIC and BIC do not fully account for prior volume in the same way Bayesian evidence does. If model comparison is discussed, priors should still be documented to avoid overinterpreting small differences.

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Prior sensitivity checks

Before interpreting ξ as a physically meaningful oscillatory signal, run at least the following checks:

• Repeat the ξ fit with a symmetric amplitude prior.
• Repeat with a wider period prior.
• Repeat with a log-period prior if physically appropriate.
• Check whether phi0 remains broad or becomes constrained.
• Check whether epsilon is away from zero or prior-boundary dominated.
• Compare posterior predictive curves, not only parameter medians.
• Document how AIC/BIC changes under prior variations.

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Reporting template

Use this short block in result pages:

**Priors.** All models were sampled with broad uniform priors over physically allowed ranges. The ΛCDM, CPL, and ξ models share the same priors on common parameters (`H0`, `Omega_m`); no nuisance parameter is sampled. The ξ-specific parameters use broad priors on the mean equation-of-state level, oscillation amplitude, period, and phase. Prior sensitivity tests are planned for v0.2, especially for the amplitude and period parameters.

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Links to related pages

Page	Role
Likelihood	Defines the data likelihood
MCMC Setup	Defines sampler configuration
Convergence	Diagnoses chain quality
Model Comparison	Interprets AIC/BIC and model hierarchy
Known Limitations	Explains v0.1 caveats and v0.2 fixes
Roadmap	Tracks planned prior sensitivity tests