Cosmological Models

Overview

The model functions live in src/xi_protocol1b_bao_run.py, not in xi_cosmo. Each E_* function returns the dimensionless expansion rate E(z) = H(z)/H₀ for a flat universe. Equations and interpretation: Three Model Families.

E_lcdm

# src/xi_protocol1b_bao_run.py
def E_lcdm(z: np.ndarray, Om: float) -> np.ndarray

Parameter	Description
z	redshift array
Om	matter fraction Ωₘ

Returns sqrt(Om (1+z)³ + (1 − Om)).

E_cpl

def E_cpl(z: np.ndarray, Om: float, w0: float, wa: float) -> np.ndarray

Returns E(z) for w(z) = w₀ + wₐ z/(1+z), using the closed-form dark-energy density (1+z)^(3(1+w0+wa)) · exp(−3 wa z/(1+z)).

XiOscillator

class XiOscillator:
    def __init__(self, zmax: float, ngrid: int = 3000, Om_ref: float = 0.3, H0_ref: float = 70.0)
    def w_grid(self, w0_mean, eps, T_gyr, phi0) -> np.ndarray
    def rho_de_ratio_grid(self, w0_mean, eps, T_gyr, phi0) -> np.ndarray
    def E_grid(self, z_grid, Om, w0_mean, eps, T_gyr, phi0) -> np.ndarray
    def w_of_z(self, z, w0_mean, eps, T_gyr, phi0) -> np.ndarray

Constructor argument	Description
zmax	upper end of the internal redshift grid; must cover the largest redshift evaluated
ngrid	grid size (default 3,000)
Om_ref, H0_ref	reference cosmology used once to compute the cosmic age t(z) on the grid (defaults 0.3, 70); astropy.cosmology.FlatLambdaCDM

Method	Returns
w_grid	w(t(z)) = w₀,mean + ε cos(2π t/T + φ₀) on the internal grid; T_gyr in Gyr, phi0 in rad
rho_de_ratio_grid	ρ_DE(z)/ρ_DE(0) = exp(3 ∫(1+w)/(1+z′)dz′), cumulative trapezoid
E_grid	E(z) at the requested redshifts, by linear interpolation of the density ratio
w_of_z	w at the requested redshifts, by linear interpolation

Example

import numpy as np
from xi_protocol1b_bao_run import XiOscillator

xi = XiOscillator(zmax=2.4)
z = np.linspace(0, 2.4, 100)
w = xi.w_of_z(z, w0_mean=-1.0, eps=0.1, T_gyr=5.0, phi0=0.0)   # illustrative values

E_model

def E_model(z, params, model: str, xi_helper: XiOscillator | None = None) -> np.ndarray

Dispatches on model ∈ "Xi_osc". params is ordered as in MODEL_LABELS: [H0, Om], [H0, Om, w0, wa], [H0, Om, w0_mean, eps, T_gyr, phi0]. xi_helper is required for "Xi_osc" (raises ValueError otherwise); any other model name raises ValueError.

Distances

def distance_grids(zmax, params, model, xi_helper=None, ngrid=3000)
    -> (zg, Eg, DC, DH, DL)          # Mpc
def mu_model(z_sn, params, model, xi_helper=None) -> np.ndarray
def bao_predictions(z_bao, types_bao, params, model, xi_helper=None, rd=147.09) -> np.ndarray

• distance_grids integrates 1/E on a uniform grid with a cumulative trapezoidal rule: DC = (c/H0) ∫ dz/E, DH = c/(H0 E), DL = (1+z) DC.
• mu_model interpolates DL at the supernova redshifts and returns 5 log10(DL) + 25.
• bao_predictions returns DM/rd, DH/rd or DV/rd for each entry of types_bao ("DM_over_rs", "DH_over_rs", "DV_over_rs"); any other type raises ValueError. rd defaults to the fixed fiducial value.

Notes

• The grid for each probe ends at the largest redshift of that probe, and XiOscillator must be built with a zmax at least as large as both.
• Results are not validated against an independent distance code in the repository's tests.
• Interpretation of the oscillation and its time variable: Three Model Families.