Expansion history

Models: Expansion history

This page documents expansion_model and its parameter vector expansion_smg, which fix the background evolution of the dark energy component in parametrized runs. It applies only to the parametrized α models; in covariant theories the expansion history follows from solving the scalar-field equations, and expansion_model is not even read by the code.

What it controls

In a parametrized model the α-functions describe the perturbations but say nothing about the background. hi_class therefore needs a separate prescription for ρ_smg(a) and p_smg(a), chosen with

expansion_model = lcdm        # or wowa, wowa_w, wede
expansion_smg = 0.5           # parameter vector; first entry usually overwritten by the closure equation

expansion_model is mandatory whenever a parametrized gravity_model is used — the code stops with expansion_model not read, you should specify one! otherwise. The four options and their aliases:

expansion_model	accepted aliases	expansion_smg
lcdm	—	Ω_smg
wowa	w0wa, cpl	Ω_smg, w₀, wₐ
wowa_w	w0wa_w, cpl_w	Ω_smg, w₀, wₐ
wede	—	Ω_smg, w₀, Ω_ede

lcdm

The dark energy density is constant, exactly like a cosmological constant:

• ρ_smg = Ω_smg H₀², p_smg = −ρ_smg.

The single entry of expansion_smg is Ω_smg. The background is then identical to ΛCDM; all modified-gravity effects come from the α-functions at the perturbation level.

wowa

The CPL (Chevallier–Polarski–Linder) equation of state w(a) = w₀ + wₐ(1 − a), implemented through its analytic density,

• ρ_smg(a) = Ω_smg H₀² · a^(−3(1 + w₀ + wₐ)) · exp[3 wₐ (a − 1)],
• p_smg(a) = w(a) ρ_smg(a).

expansion_smg = Ω_smg, w₀, wₐ, so for example expansion_smg = 0.5, -1., 0. reproduces lcdm.

wowa_w

Documented here for the first time. wowa_w describes the same CPL equation of state w(a) = w₀ + wₐ(1 − a), but obtains the density differently: it sets the internal flag rho_evolution_smg to true, so instead of using the analytic ρ_smg(a) above the code integrates the continuity equation

• ρ′_smg = −3 a H (1 + w) ρ_smg

along with the rest of the background, starting from the analytic CPL value at the initial time. For a pure CPL equation of state the two routes agree to numerical precision, so wowa and wowa_w give the same cosmology.

When does the difference matter? wowa_w is the template to use when the equation of state is not one with a closed-form density: only w(a) enters the equations, and the density stays consistent automatically through the continuity equation. If you want to implement a custom w(a), modify the wowa_w branch (the w(a) assignment in gravity_smg/gravity_models_smg.c and the initial condition in the same file) rather than deriving and coding an analytic ρ(a). It also exercises the rho_evolution_smg code path, which makes the density and its derivative available as evolved quantities.

wede

Early dark energy in the Doran–Robbers parametrization (arXiv:astro-ph/0601544), implemented as in Pettorino, Amendola & Wetterich (arXiv:1301.5279). expansion_smg = Ω_smg, w₀, Ω_ede, where Ω_smg is the dark energy fraction today, w₀ its equation of state at late times, and Ω_ede the constant dark energy fraction at early times. The dark energy fraction evolves as

• Ω_de(a) = [Ω_smg − Ω_ede (1 − a^(−3w₀))] / [Ω_smg + (1 − Ω_smg) a^(3w₀)] + Ω_ede (1 − a^(−3w₀)),

which interpolates between Ω_ede deep in matter/radiation domination and Ω_smg today; the code reconstructs ρ_smg and w(a) from this function and the matter–radiation equality scale. To keep the expressions regular when Ω_ede → 0, a tiny constant (default 10⁻¹⁰, adjustable via the ini parameter wede_Omega_e_regularizer_smg) is added to Ω_ede.

The Ω_smg closure convention

As for every hi_class model, one background parameter is adjusted by shooting to satisfy the budget equation — with Omega_smg = -1 in the ini file the code varies entry tuning_index_smg (default 0, i.e. the Ω_smg entry of expansion_smg) until the density today matches the value required by the other density parameters; see Running hi_class for the details.