Parametrized alphas

Models: Parametrized alphas

This page is the reference for the six parametrized (EFT-based) gravity models in hi_class, in which the α-functions are specified directly as functions of time. It covers the exact gravity_model strings, the meaning and order of every entry of parameters_smg, and the functional form each parametrization assumes.

The α-functions and M*²

At the level of linear perturbations, any Horndeski theory is fully described by its expansion history plus four independent functions of time (Bellini & Sawicki, arXiv:1404.3713):

Symbol	Name	Physical meaning
α_K	kineticity	Kinetic energy of the scalar perturbations. Large α_K lowers the sound speed of dark energy; on its own it has very little effect on observables at sub-horizon scales.
α_B	braiding	Kinetic mixing between the scalar field and the metric. Sources clustering of dark energy and modifies the growth of matter perturbations.
α_M	Planck-mass running	Rate of evolution of the effective Planck mass, α_M = d ln M*²/d ln a. Modifies the growth of structure and gravitational lensing, and makes gravitational-wave amplitudes decay differently.
α_T	tensor speed excess	Deviation of the gravitational-wave speed from the speed of light, c_T² = 1 + α_T. Also modifies the anisotropic stress between the metric potentials.

The fifth function is the effective Planck mass M² itself (in units of the standard Planck mass). It is not independent: hi_class integrates it from the running, d(M² − 1)/d ln a = α_M M*² (eq. 3.3 of arXiv:1404.3713), starting from an initial value that each parametrization prescribes. The exact definitions and sign conventions used by the code are those of the hi_class paper (arXiv:1605.06102).

In these models the α's are functions of time chosen directly, without reference to any scalar-field Lagrangian. The background is not determined by the α's: you must choose the expansion history separately with expansion_model and expansion_smg, described on the expansion models page. Both gravity_model and expansion_model are mandatory for parametrized runs; the code stops with an error if either is missing.

A general run therefore looks like

Omega_smg = -1               # enable smg; see Running hi_class for the closure convention
gravity_model = propto_omega
parameters_smg = 1., 0., 0., 0., 1.
expansion_model = lcdm
expansion_smg = 0.5          # overwritten by the closure equation

The six available parametrizations, with the length of their parameters_smg vector (checked against the parsing in gravity_smg/gravity_models_smg.c):

gravity_model	entries	parameters_smg order
propto_omega	5	c_K, c_B, c_M, c_T, M*²_ini
propto_scale	5	c_K, c_B, c_M, c_T, M*²_ini
constant_alphas	5	c_K, c_B, c_M, c_T, M*²_ini
eft_alphas_power_law	8	δM²₀, c_K, c_B, c_T, δM²_exp, c_K_exp, c_B_exp, c_T_exp
eft_gammas_power_law	8	Ω₀, γ₁₀, γ₂₀, γ₃₀, Ω_exp, γ₁_exp, γ₂_exp, γ₃_exp
eft_gammas_exponential	8	Ω₀, γ₁₀, γ₂₀, γ₃₀, Ω_exp, γ₁_exp, γ₂_exp, γ₃_exp

propto_omega

Each α is proportional to the fractional energy density of the dark energy component,

• α_i(a) = c_i · Ω_smg(a), for i = K, B, M, T,

where Ω_smg(a) = ρ_smg/ρ_tot is set by the chosen expansion model. This is the most common parametrization in the literature: modified gravity switches on only when dark energy becomes dynamically relevant.

gravity_model = propto_omega
parameters_smg = c_K, c_B, c_M, c_T, M*^2_ini

The last entry is the initial value of the effective Planck mass, from which M²(a) is integrated using α_M = c_M Ω_smg(a). Since Ω_smg → 0 at early times, M²_ini is effectively the early-time Planck mass; M*^2_ini = 1 recovers the standard value. The default model in hi_class.ini is propto_omega with parameters_smg = 1., 0., 0., 0., 1., i.e. pure kineticity — a k-essence-like model.

propto_scale

Each α is proportional to the scale factor,

• α_i(a) = c_i · a, for i = K, B, M, T.

gravity_model = propto_scale
parameters_smg = c_K, c_B, c_M, c_T, M*^2_ini

M² is again integrated from α_M = c_M a starting from M²_ini.

constant_alphas

The α's are constant in time,

• α_i(a) = c_i, for i = K, B, M, T.

gravity_model = constant_alphas
parameters_smg = c_K, c_B, c_M, c_T, M*^2_ini

Note that a constant α_M still means an evolving Planck mass: M² grows as a power law of the scale factor, starting from M²_ini at the beginning of the background integration.

eft_alphas_power_law

Here α_K, α_B, α_T and the Planck-mass shift δM² ≡ M*² − 1 are power laws of the scale factor:

• α_K(a) = c_K · a^(c_K_exp)
• α_B(a) = c_B · a^(c_B_exp)
• α_T(a) = c_T · a^(c_T_exp)
• δM²(a) = δM²₀ · a^(δM²_exp), so that α_M(a) = δM²₀ δM²_exp a^(δM²_exp) / (1 + δM²₀ a^(δM²_exp))

gravity_model = eft_alphas_power_law
parameters_smg = delta_M2_0, c_K, c_B, c_T, delta_M2_0_exp, c_K_exp, c_B_exp, c_T_exp

Note the order: the Planck-mass parameters come first and fifth, not in the α_K, α_B, α_M, α_T order of the previous three models. This parametrization generalizes propto_scale, with one difference: there it is α_M that is proportional to the scale factor, while here it is the Planck-mass shift δM² itself that follows the power law (α_M is derived from it). There is no independent M*²_ini: the initial Planck mass is fixed by evaluating δM²(a) at the initial time.

eft_gammas_power_law and eft_gammas_exponential

These two parametrizations work in the (Ω, γ₁, γ₂, γ₃) basis of the EFT of dark energy action rather than directly in the α basis: Ω(a) is the conformal coupling to the Ricci scalar, γ₁ multiplies the (δg⁰⁰)² mass-type operator, γ₂ the δg⁰⁰ δK kinetic-mixing operator and γ₃ the operator that separates the tensor speed from unity — the notation used e.g. in the Planck dark energy and modified gravity analysis (arXiv:1502.01590). Each of the four functions is assumed to be either a power law of the scale factor,

• f(a) = f₀ · a^(f_exp) (eft_gammas_power_law),

or an exponential of one,

• f(a) = exp(f₀ · a^(f_exp)) − 1 (eft_gammas_exponential),

for f ∈ {Ω, γ₁, γ₂, γ₃}. Both take the same eight-parameter vector:

gravity_model = eft_gammas_power_law     # or eft_gammas_exponential
parameters_smg = Omega_0, gamma_1, gamma_2, gamma_3, Omega_0_exp, gamma_1_exp, gamma_2_exp, gamma_3_exp

The code translates these into the α basis (with primes denoting d/da and ρ_tot, p_tot the total density and pressure):

• M*² = 1 + Ω + γ₃ (used as the initial condition; the evolution is then integrated from α_M)
• α_M = a (Ω′ + γ₃′) / (1 + Ω + γ₃)
• α_T = −γ₃ / (1 + Ω + γ₃)
• α_B = −(γ₂ H₀/√ρ_tot + a Ω′) / (1 + Ω + γ₃)
• α_K = 2 (2 γ₁ H₀²/ρ_tot + C/H²) / (1 + Ω + γ₃), where C/H² = [ −a² Ω″ + 3 (Ω + a Ω′/2)(ρ_tot + p_tot)/ρ_tot + 3 (ρ_smg + p_smg)/ρ_tot ] / 2 is the EFT kinetic function fixed by Ω(a) and the chosen expansion history.

(In the code's units √ρ_tot = H for a flat universe, so the γ₁ and γ₂ terms scale as (H₀/H)² and H₀/H respectively.) Note that a nonzero Ω contributes to α_B and α_K as well as to α_M, so the γ parametrizations are not a relabeling of eft_alphas_power_law.

Running an example

A minimal, runnable propto_omega configuration is shown in the code snippet on the website, hiclass-code.net. Remember that one parameter of the background must be adjusted to satisfy the budget (closure) equation — for parametrized models this is by default the Ω_smg entry of expansion_smg — as explained in Running hi_class. The choice of expansion history and its parameters is documented in Models: Expansion history; stability checks that apply to these models are described in Stability and precision.