4.3. 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. There are some models for which we have analytical solutions for the evolutions of the energy densities and can write the Hubble rate in a closed form: this is done via the function \(h\,E(z)\) (Eh in the module observables) from which \(H(z) = 100\,h\, E(z)\) is obtained in the function H from the same module. Generally, this is implemented using the physical densities \(\Omega_{i0} h^2\) and \(h\) as parameters. For other models, a numerical integral is performed with fourth-order Runge-Kutta. Since this integration is done backwards in time from \(a = 1\), it is easier to deal with the density parameters \(\Omega_{i0}\) and \(H_0\). The result is a \(H(z)\) function that can be interpolated for any redshift within the interval of integration. This is registered in a Cosmology class object as the attribute H_solution, from which \(h\,E(z)\) as well can be obtained when convenient. The code is written so that the integration is done only once given a set of parameters and model. In all cases the dark energy density parameter is obtained, as a derived parameter, from the flatness condition that all the density parameters must add up to 1. At the end of this section, I instruct the user on how to include a new model for use with this program.

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

The standard models accepts the parameters \(h\), \(\Omega_{c0} h^2\), \(\Omega_{b0} h^2\), \(\Omega_{r0} h^2\), and has \(\Omega_{c0}\), \(\Omega_{b0}\), \(\Omega_{r0}\), \(\Omega_{d0}\), as derived parameters. Extending this model to allow curvature is not completely supported yet. The Friedmann equation is

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


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

with \(\Omega_d = 1 - \Omega_{b0} - \Omega_{c0} - \Omega_{r0}\) and \(H_0 = 100 \, h \,\, \text{km s$^{-1}$ Mpc$^{-1}$}\). 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 \((1+z)^{3(1+w_d)}\).

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. [2]. 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 \(\rho_d\) or to some combination of both.

Proportional to \(\rho_d\)

This model uses \(Q = 3 H \xi \rho_d\). We were interested in studying the growth rate of matter perturbations in such a model in an analytic way in a previous work [3]. Some assumptions were made and the growth rate was approximated as \(f \approx \Omega_m^{\gamma}\), with \(\gamma\) determined in terms of the coupling constant \(\xi\), a free parameter of the model. Other parameters involved are \(\sigma_{8,0}\) and \(\Omega_{d0}\), and \(w_0\) and \(w_1\) from

\[w_d(\Omega_d) = w_0 + w_1 \Omega_d + \mathcal{O}(\Omega_d^2).\]

A recurrence relation is used to approximate the evolution of densities. This implementation, dubbed model2 in the code, is meant to be used with the myfsigma8 or growth_rate_sigma_eight functions only, to have its predictions compared with the \(f\sigma_8(z)\) data.

Interacting dark energy in the dark \(SU(2)_R\) model

As per the recent paper by Landim et al [4], This model actually proposes an interaction possibly between more than two components of a dark sector from the decay of the heaviest particle (\(\varphi^{+}\)) of the dark energy doublet (\(\varphi^0, \varphi^{+}\)). The other particles in the sector are a dark matter candidate \(\psi\), a dark matter neutrino \(\nu_d\). The right-hand side of the conservation equations of these particles are obtained in terms of a characteristic decay time scale \(\Gamma\) and mass ratios with respect to the heaviest particle. The model is implemented in two flavours: the more general darkSU2 and the specific case with the neutrino mass set to zero, darkSU2wnu0. The solution to the full set of background density evolutions is obtained numerically.

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 [1]. Three such models were implemented as described in Marcondes and Pan (2017) [5]. 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 fastvarying1 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 fastvarying2 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 fastvarying3 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

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

Including a new model

To define a new model in the program, you will need to implement your model calculation of the observables you want to use. Choose a label for the model and use conditionals on the cosmology.model attribute. In your .ini file, use the same label to describe this new model in the corresponding section and do not forget to give \(\LaTeX\) representations for your parameters. If there are derived parameters that you want to get, include their recipes in derived.py and derived_bf.py. The first one handles the Markov chains, while the second calculates the best-fit estimate point. No other changes are required throughout the code.

[1]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).
[2]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.
[3]Marcondes R. J. F., Landim R. C. G., Costa A. A., Wang B. and Abdalla E., “Analytic study of the effect of dark energy-dark matter interaction on the growth of structures”. Journal of Cosmology and Astroparticle Physics 1612, 009 (2016).
[4]Landim R. C. G., Marcondes R. J. F., Bernardi F. F. and Abdalla E., “Interacting dark energy in the dark SU(2)R model”. arXiv:1711.07282 [astro-ph.CO]
[5]Marcondes R. J. F. & Pan S., “Cosmic chronometer constraints on some fast-varying dark energy equations of state”. arXiv:1711.06157 [astro-ph.CO].