# 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

or

\(H_0\) 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

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 \(a^{-3\left(1 + w_0 + w_a\right)} e^{-3 w_a \left(1 - a\right)}\), in terms of the scale factor.

## The Barboza-Alcaniz parametrization¶

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

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 \(x \equiv 1/a\).

## The Jassal-Bagla-Padmanabhan parametrization¶

Starting with version 1.4, the JBP parametrization [4] of the equation of state

can also be used. In this case, \(a^{-3\left(1 + w_0\right)} e^{- 3 w_1 \left[a\left(1 - a/2\right) - 1/2\right]}\) or \((1+z)^{3\left(1+w_0\right)} e^{3 w_1 z^2/2 \left(1+z\right)^2}\) is the term that goes into the dark energy contribution in the Friedmann equation.

## 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) [5].
In interacting models, the individual conservation equations of the two dark
fluids are violated, although still preserving the total energy conservation:

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 [6]. Three such models were implemented as described in Marcondes and Pan (2017) [7]. 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.

### The Linder-Huterer parametrization (Model 1)¶

The model `lh`

has the free parameters
\(w_p\),
\(w_f\),
\(a_t\) and
\(\tau\) characterizing the equation of state [8]

\(w_p\) 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

where

### The Felice-Nesseris-Tsujikawa parametrization (Model 2)¶

This FNT model `fv2`

alters the previous model to allow the dark energy
to feature an extremum value of the equation of state: [6]

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

with

### The Felice-Nesseris-Tsujikawa parametrization (Model 3)¶

Finally, we have another FNT model `fv3`

with the same parameters as in Model 2 but with equation of state [6]

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

Footnotes

[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] | Jassal H. K., Bagla J. S., Padmanabhan T., “WMAP constraints on low redshift evolution of dark energy”. Monthly Notices of the Royal Astronomical Society 356 (2005) L11-L16; Jassal H. K., Bagla J. S., Padmanabhan T., “Observational constraints on low redshift evolution of dark energy: How consistent are different observations?”. Physical Review D 72 (2005) 103503. |

[5] | 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. |

[6] | (1, 2, 3) 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). |

[7] | Marcondes R. J. F. & Pan S., “Cosmic chronometer constraints on some fast-varying dark energy equations of state”. arXiv:1711.06157 [astro-ph.CO]. |

[8] | Linder E. V. & Huterer D., “How many dark energy parameters?”. Physical Review D 72 (2005) 043509. |