Three Model Families

This page defines the three flat-universe models fitted to Pantheon+SH0ES + DESI DR2 BAO: ΛCDM (2 free parameters), CPL (4) and Ξosc (6). All three share H₀ and Ωₘ and are sampled with the same top-hat priors on those parameters.

Expansion histories

In a flat universe, E(z) = H(z)/H₀. Matter is the only other component, so only the dark-energy density term changes between models.

Model	w	E²(z)
ΛCDM	$-1$	$\Omega_m (1+z)^3 + (1-\Omega_m)$
CPL	$w_0 + w_a\,\dfrac{z}{1+z}$	$\Omega_m (1+z)^3 + (1-\Omega_m)(1+z)^{3(1+w_0+w_a)}\exp\!\left[-\dfrac{3 w_a z}{1+z}\right]$
Ξosc	$w_{0,\mathrm{mean}} + \varepsilon\cos\!\left(\dfrac{2\pi t}{T} + \varphi_0\right)$	$\Omega_m (1+z)^3 + (1-\Omega_m)\,\dfrac{\rho_{\mathrm{DE}}(z)}{\rho_{\mathrm{DE}}(0)}$ with $\dfrac{\rho_{\mathrm{DE}}(z)}{\rho_{\mathrm{DE}}(0)} = \exp\!\left[3\int_0^z \dfrac{1+w}{1+z'}\,dz'\right]$

How Ξosc is implemented

The oscillation is a function of cosmic time t, but the likelihood is evaluated in redshift. The code (XiOscillator) therefore builds, once, a redshift grid and the cosmic age t(z) on that grid from a fixed reference cosmology — flat ΛCDM with H₀ = 70 and Ωₘ = 0.3 (astropy) — then evaluates w on it, integrates the density ratio with a trapezoidal rule and interpolates. The time axis is not recomputed from the cosmology being sampled; this is limitation 2 on Known Limitations.

Two consequences worth knowing

w(z=0) is not w₀,mean. The present-day value is w₀,mean + ε cos(2π t₀/T + φ₀), which depends on the phase. The w0_mean parameter is the average level around which w oscillates.

Ξosc does not nest ΛCDM. Setting ε → 0 gives a constant w = w₀,mean (a wCDM model), which equals ΛCDM only if w₀,mean = −1 as well. Neither is Ξosc nested in CPL. AIC and BIC remain usable, but a likelihood-ratio reading of "ε = 0" is not a test of ΛCDM.

Free parameters and priors

All priors are uniform inside the bounds below and zero outside (open intervals; log_prior returns −∞ outside). They are read from the sampling script — see the Prior Specification page for the discussion of why they matter.

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

Why these three

Property	ΛCDM	CPL	Ξosc
Dark-energy behaviour	constant	smooth, monotonic in z	periodic in cosmic time
Role	baseline	standard dynamical comparator	exploratory probe
Extra parameters vs ΛCDM	—	w₀, wₐ	w₀,mean, ε, T, φ₀
Weakly constrained parameters	—	wₐ	T, φ₀ (and ε near its bounds)