3.2. The cosmological models

A few models are already implemented. I give a brief description below, with references for works that discuss some of them in detail and works that analyzed them with this code. The models are objects created from the cosmic_objects.CosmologicalSetup class. This class has a generic module solve_background that calls the Fluid’s module rho_over_rho0 of each fluid to obtain the solution for their energy densities. When a solution cannot be obtained directly (like in some interacting models), a fourth-order Runge-Kutta integration is done using the function generic_runge_kutta from EPIC.utils’s integrators and the fluids` drho_da. There is an intermediate function get_Hubble_Friedmann to calculate the Hubble rate either by just summing the energy densities, when called from the Runge-Kutta integration, or calculating them with rho_over_rho0.

Some new models can be introduced in the code just by editing the model_recipes.ini, available_species.ini and (optionally) default_parameter_values.ini configuration files, without needing to rebuild and install the EPIC’s package. The format of the configuration .ini files is pretty straightforward and the containing information can serve as a guide for what needs to be defined.

The \(\Lambda\text{CDM}\) model

When baryons and radiation are included, the solution to this cosmology will require values for the parameters \(\Omega_{c0}\), \(\Omega_{b0}\), \(T_{\text{CMB}}\), \(H_0\), or \(h\), \(\Omega_{c0} h^2\), \(\Omega_{b0} h^2\), \(T_{\text{CMB}}\), and will find \(\Omega_{\Lambda} = 1 - \left( \Omega_{c0} + \Omega_{b0} + \Omega_{r0} \right)\) or \(\Omega_{\Lambda} h^2 = h^2 - \left( \Omega_{c0} h^2 + \Omega_{b0} h^2 + \Omega_{r0} h^2 \right)\) if physical. [1] The radiation density parameter \(\Omega_{r0}\) is calculated according to the CMB temperature \(T_{\text{CMB}}\), including the contribution of the neutrinos (and antineutrinos) of the standard model. Extending this model to allow curvature is not completely supported yet. The Friedmann equation is

\[\frac{H(z)}{100 \,\, \text{km s$^{-1}$ Mpc$^{-1}$}} = \sqrt{ (\Omega_{b0} h^2 + \Omega_{c0} h^2) (1+z)^3 + \Omega_{r0} h^2 (1+z)^4 + \Omega_d h^2}\]


\[H(z) = H_0 \sqrt{ (\Omega_{b0} + \Omega_{c0}) (1+z)^3 + \Omega_{r0} (1+z)^4 + \Omega_d},\]

\(H_0\) is in units of \(\text{km s$^{-1}$ Mpc$^{-1}$}\). This model is identified in the code by the label lcdm.

The \(w\text{CDM}\) model

Identified by wcdm, this is like the standard model except that the dark energy equation of state can be any constant \(w_d\), thus having the \(\Lambda\text{CDM}\) model as a specific case with \(w_d = -1\). The Friedmann equation is like the above but with the dark energy contribution multiplied by \((1+z)^{3(1+w_d)}\).

The Chevallier-Polarski-Linder parametrization

The CPL parametrization [2] of the dark energy equation of state

\[w(a) = w_0 + w_a \left( 1-a \right)\]

is also available. In this case, the dark energy contribution in the Friedmann equation is multiplied by \(\left(1 + z \right)^{3\left(1 + w_0 + w_a\right)} e^{-3 w_a z /\left(1 + z\right)}\) or \((a/a_0)^{-3\left(1 + w_0 + w_a\right)} e^{-3 w_a \left(1 - a/a_0\right)}\), in terms of the scale factor.

The Barboza-Alcaniz parametrization

The Barboza-Alcaniz dark energy equation of state parametrization [3]

\[w(z) = w_0 + w_1 \frac{z \left(1 + z\right)}{1 + z^2}\]

is implemented. This models gives a dark energy contribution in the Friedmann equation that is multiplied by the term \(x^{3(1+w_0)} \left( x^2 - 2 x + 2 \right)^{-3 w_1/2}\), where \(x \equiv a_0/a\).

Interacting Dark Energy models

A comprehensive review of models that consider a possible interaction between dark energy and dark matter is given by Wang et al. (2016) [4]. In interacting models, the individual conservation equations of the two dark fluids are violated, although still preserving the total energy conservation:

\[\begin{split}\dot\rho_c + 3 H \rho_c &= Q \\ \dot\rho_d + 3 H (1 + w_d) \rho_d &= -Q.\end{split}\]

The shape of \(Q\) is what characterizes each model. Common forms are proportional to \(\rho_c\), to \(\rho_d\) (both supported) or to some combination of both (not supported in this version).

To create an instance of a coupled model (cde) with \(Q \propto \rho_c\), use:

The mandatory species are idm and ide. You can add baryons in the optional_species list keyword argument, but note that matter is not available as a combined species for this model type since dark matter is interacting with another fluid while baryons are not. What is new here is the interaction_setup dictionary. This is where we tell the code which species are interacting (at the moment only an energy exchange within a pair is supported), to which of them (idm) we associate the interaction parameter xi, indicate the second one (ide) as having an interaction term proportional to the other (idm) and specify the sign of the interaction term for each fluid, in this case that means \(Q_c = 3 H \xi \rho_c\) and \(Q_d = - 3 H \xi \rho_c\).


Here, I am exaggerating the value of the interaction parameter so we can see a variation on the dark energy density that is due to the interaction, not the equation-of-state parameter, which is \(-1\). This same cosmology can be realized with the model type cde_lambda without specifying the parameter wd, since the ilambda fluid has fixed \(w_d = -1\). The dark matter interacting term \(Q_c\) is positive with \(\xi\) positive, thus the lowering of the dark energy density as its energy flows towards dark matter.

Fast-varying dark energy equation-of-state models

Models of dark energy with fast-varying equation-of-state parameter have been studied in some works [5]. Three such models were implemented as described in Marcondes and Pan (2017) [6]. We used this code in that work. They have all the density parameters present in the \(\Lambda\text{CDM}\) model besides the dark energy parameters that we describe in the following.

Model 1

This model fv1 has the free parameters \(w_p\), \(w_f\), \(a_t\) and \(\tau\) characterizing the equation of state

\[w_d(a) = w_f + \frac{w_p - w_f}{1 + (a/a_t)^{1/\tau}}.\]

\(w_p\) and \(w_f\) are the asymptotic values of \(w_d\) in the past (\(a \to 0\)) and in the future (\(a \to \infty\)), respectively; \(a_t\) is the scale factor at the transition epoch and \(\tau\) is the transition width. The Friedmann equation is

\[\frac{H(a)^2}{H_0^2} = \frac{\Omega_{r0}}{a^4} + \frac{\Omega_{m0}}{a^3} + \frac{\Omega_{d0}}{a^{3(1+w_p)}} f_1(a),\]


\[f_1(a) = \left( \frac{a^{1/\tau} + a_t^{1/\tau}}{1 + a_t^{1/\tau}} \right)^{3\tau(w_p - w_f)}.\]

Model 2

This model fv2 alters the previous model to allow the dark energy to feature an extremum value of the equation of state:

\[w_d(a) = w_p + (w_0 - w_p) \, a \, \frac{1 - (a/a_t)^{1/\tau}}{1 - (1/a_t)^{1/\tau}},\]

where \(w_0\) is the current value of the equation of state and the other parameters have the interpretation as in the previous model. The Friedmann equation is

\[\frac{H(a)^2}{H_0^2} = \frac{\Omega_{r0}}{a^4} + \frac{\Omega_{m0}}{a^3} + \frac{\Omega_{d0}}{a^{3(1+w_p)}} e^{f_2(a)},\]


\[f_2(a) = 3 (w_0 - w_p) \frac{1+ (1- a_t^{-1/\tau})\tau + a \bigl[\bigl\lbrace (a/a_t)^{1/\tau} - 1 \bigr\rbrace \tau - 1 \bigr]}{(1+\tau)(1 - a_t^{-1/\tau})}.\]

Model 3

Finally, we have a third model fv3 with the same parameters as in Model 2 but with equation of state

\[w_d(a) = w_p + (w_0 - w_p) \, a^{1/\tau} \, \frac{1 - (a/a_t)^{1/\tau}}{1 - (1/a_t)^{1/\tau}}.\]

It has a Friedmann equation identical to Model 2’s except that \(f_2(a)\) is replaced by

\[f_3(a) = 3(w_0 - w_p) \tau \frac{2 - a_t^{-1/\tau} + a_t^{1/\tau} \bigl[(a/a_t)^{1/\tau} - 2 \bigr]}{2 \bigl(1 - a_t^{-1/\tau}\bigr)}.\]


[1]That is, assuming derived=lambda, but we could also have done, for example, physical=False, derived=matter, specify \(\Omega_{\Lambda}\) and the code would get \(\Omega_{m0} = 1 - \left( \Omega_{\Lambda} + \Omega_{r0} \right)\) or, still, without specifying the derived parameter and with physical true, specify all the fluids’ density parameters and get \(h = \sqrt{\Omega_{c0} h^2 + \Omega_{b0} h^2 + \Omega_{r0} h^2 + \Omega_{\Lambda} h^2}\).
[2]Chevallier M. & Polarski D., “Accelerating Universes with scaling dark matter”. International Journal of Modern Physics D 10 (2001) 213-223; Linder E. V., “Exploring the Expansion History of the Universe”. Physical Review Letters 90 (2003) 091301.
[3]Barboza E. M. & Alcaniz J. S., “A parametric model for dark energy”. Physics Letters B 666 (2008) 415-419.
[4]Wang B., Abdalla E., Atrio-Barandela F., Pavón D., “Dark matter and dark energy interactions: theoretial challenges, cosmological implications and observational signatures”. Reports on Progress in Physics 79 (2016) 096901.
[5]Corasaniti P. S. & Copeland E. J., “Constraining the quintessence equation of state with SnIa data and CMB peaks”. Physical Review D 65 (2002) 043004; Basset B. A., Kunz M., Silk J., “A late-time transition in the cosmic dark energy?”. Monthly Notices of the Royal Astronomical Society 336 (2002) 1217-1222; De Felice A., Nesseris S., Tsujikawa S., “Observational constraints on dark energy with a fast varying equation of state”. Journal of Cosmology and Astroparticle Physics 1205, 029 (2012).
[6]Marcondes R. J. F. & Pan S., “Cosmic chronometer constraints on some fast-varying dark energy equations of state”. arXiv:1711.06157 [astro-ph.CO].