CosmoFit.HDE

class CosmoFit.HDE(params)

Bases: Cosmology

Holographic dark energy with a future-event-horizon cutoff.

The holographic principle bounds the energy in a region by the area of its boundary, which for a cosmological infrared cutoff L gives rho_DE = 3 c^2 M_p^2 / L^2. Li (2004) showed that the only choice of L that yields an accelerating universe and a sensible equation of state is the future event horizon,

L(a) = a Int_a^inf da’ / (a’^2 H(a’)),

which is what this class implements. The single free parameter is c_hde, the dimensionless constant in that relation.

What makes this model different from every other one here

E(z) has no closed form. Omega_DE obeys

d Omega_DE / d ln a

= Omega_DE (1 - Omega_DE) (1 + 2 sqrt(Omega_DE) / c),

which is solved once per refresh() and splined. The expansion rate then follows algebraically from flatness,

E(z)^2 = Omega_m (1+z)^3 / (1 - Omega_DE(z)),

so this is the library’s first background model that integrates an ODE rather than evaluating a formula. Everything downstream – distances, growth, r_d – only ever asks for E and dEdz, so nothing else changes.

The equation of state is w = -1/3 - (2/3) sqrt(Omega_DE) / c, and its behaviour is the model’s signature: w -> -1/3 at early times, when dark energy is negligible, and w -> -1/3 - 2/(3c) in the far future. So c = 1 approaches a cosmological constant only asymptotically, c > 1 stays quintessence-like, and c < 1 crosses into phantom territory. Fits have generally preferred c somewhat below 1, which is the interesting part – the crossing is not put in by hand.

Flat only

The event-horizon integral, and hence the ODE above, assumes flatness. Curvature changes the causal structure the holographic bound is applied to rather than merely adding a term, so a curved version is a different model and is refused here instead of being silently approximated.

Caveats

The cutoff is the future event horizon, so the present energy density depends on the entire future expansion history. That is unusual as physics and has been argued about at length; it is also what the model is, and the alternative cutoffs (Hubble radius, particle horizon) do not accelerate.

Its perturbations are outside what a standard Boltzmann code solves, so it is refused by CAMBBackend; use the compressed distance priors for its CMB constraint.

References

Li (2004), Phys. Lett. B 603, 1, arXiv:hep-th/0403127.

Wang, Mortsell et al. (2017), Phys. Rept. 696, 1, arXiv:1612.00345 (review).

__init__(params)

Parameters:

params (CosmologyParameters)

Methods

E(z)
H(z)
Omega_de(z)	Dark-energy density in units of today's critical density -- the term as it appears inside E(z)^2, which is what every other model in this library returns here and what Recombination expects.
Omega_matter(z)	Matter density at redshift z, in units of today's critical density -- i.e. the Omega_m (1+z)^3 term as it appears inside E(z)^2, before dividing by E(z)^2.
__init__(params)
dEdz(z)	Analytic, using the ODE rather than differencing E.
mu(a[, k])	Effective-to-Newtonian gravitational coupling ratio, G_eff(a,k)/G_N, entering the linear growth equation solved by GrowthCalculator (see that module's docstring for the equation itself).
omega_de_fraction(z)	The density parameter rho_DE(z) / rho_crit(z), which is what the ODE solves for.
plain_name()	This model's name as plain text: MODEL_NAME if it declares one, else the class name.
plot_label()	This model's name as it should appear in a figure legend or title -- MODEL_LABEL (LaTeX) if it declares one, else plain_name().
refresh()	Re-solve the background ODE, then rebuild everything that depends on it.
w_de(z)	Equation of state, w = -1/3 - (2/3) sqrt(Omega_DE) / c.

Attributes

A_planck
A_s
EXTRA_PARAMS	Extra parameters this model adds beyond CosmologyParameters.
H0
MB
MODEL_LABEL	a LaTeX math string where the plain name is really a set of symbols (LCDM -> $\Lambda$CDM), None where the plain name is already what a reader should see (an acronym like CPL).
MODEL_NAME	Plain-text name for this model, for tables, JSON and log lines.
N_eff
Omega_b
Omega_de0
Omega_k
Omega_m
alpha
c_hde
compute_rd	Whether rd is computed from the physical densities (rd_computed()) rather than read from the free rd parameter.
derive_sigma8	Whether sigma8 is derived from the Boltzmann code rather than read from the free sigma8 parameter.
h	Reduced Hubble constant, H0 / 100.
ln1e10As
m_nu
n_s
omega_b_h2	Physical baryon density, Omega_b h^2 -- the combination BBN and the CMB actually constrain.
omega_cdm_h2	Physical cold dark matter density, Omega_c h^2 -- matter less baryons.
omega_m_h2	Physical matter density, Omega_m h^2.
rd	The sound horizon at the drag epoch [Mpc] that the BAO likelihoods divide by.
sigma8	Present-day sigma_8, the normalization every growth prediction is built on.
tau_reio
w0
wa

Parameters:

params (CosmologyParameters)