Pycosmicstar

Python Cosmic Star Formation Rate

Theory

The Implemented Cosmological Background

The standard cosmological model, Cold Dark Matter plus Cosmolocical Constan ($\Lambda$CDM), consider that the energetic content of the Universe is dominated by dark energy followed by dark matter and ordinary matter (baryonic). Those components would be in a homogeneous Friedmann-Roberston-Walker metric and that the Universe expansion is determined by the scale factor $a$. The methods associated to this cosmology were implemented in the class lcdmcosmology. This class has the following input attributes:

omegam: ($\Omega_{m}$, default 0.24) - The dark matter parameter;
omegab : ($\Omega_{b}$, default 0.04) - The baryonic parameter;
omegal: ($\Omega_{\Lambda}$, default 0.73) - The dark energy parameter;
h: (default, 0.7) - The h of the Hubble constant ($H_{0} = 100h$ ).

Its public methods are:

dt_dz(z) - time, $t$, and redshift, $z$, relation:

\begin{equation}
\frac{dt}{dz} = \frac{9.78 h^{-1}Gyr}{(1+z)\sqrt{\Omega_{\Lambda} + \Omega_{m}(1 + z)^{3}}};
\end{equation}

H(z) - Hubble parameter as a function of $z$:

\begin{equation}
H(z) = 100 h \sqrt{\Omega_{\Lambda} + \Omega_{m}(1 + z)^{3}};
\end{equation}

dr_dz(z) - Comove distance variation for flat Universe:

\begin{equation}
\frac{dr_{z}(z)}{dz} = \frac{c}{H_{0}} \frac{1}{\sqrt{(1+z)^{2}(1+\Omega_{m}z)- z(2+z)\Omega_{\Lambda}}},
\end{equation}

where $c$ is the speed of light.

dV_dz(z) - Comove volume variation:

\begin{equation}
\frac{dV}{dz}(z) = 4 \pi r_{z}^{2}(z)\left(\frac{dr_{z}(z)}{dz}\right);
\end{equation}

growthFunction(z) - Growth Function of cosmological density perturbations - for the Universe with matter and a cosmological constant, the growth function is well approximated by:

\begin{equation}\label{eq:growth}
D(a)\approx \frac{5 \Omega_{\rm m}(a)\ a}{2[1- \Omega_{\Lambda}(a)+\Omega_{\rm m}^{4/7}+\frac{1}{2}\Omega_{\rm m}(a)]},
\end{equation}

$a = 1 / (1 + z)$ is the scale factor;

dgrowth_dt(z) - derivative of the growth function with respect to time.

dsigma2_dk(k) - derivative of the variance of linear density field, $\sigma$ :

\begin{equation}
\frac{d \sigma}{dk} = \frac{1}{2\pi^{2}} k^{2}P(k)W^{2}(k,M),
\end{equation}

where $P(k)$ is the power spectrum smoothed with a spherical top-hat filter function of radius $R$, which on average encloses a mass $M$ $(R=[3M/4\pi\rho_{\rm B}(z)]^{1/3})$, namely:

\begin{equation}
P(k) = BkT(k),
\end{equation}

where the normalization factor $B$ is determined observationally.

For the transfer function, we consider :

\begin{equation}
T(k) = \frac{1}{\{1+[ak+(bk)^{3/2}+(ck)^{2}]^{\nu}\}^{2/\nu}},
\end{equation}

with $\nu = 1.13$, $a = (6.4 /\Gamma) h^{-1}\rm{Mpc}$, $b= (3.0/ \Gamma)h^{-1}\rm{Mpc}$, $c = (1.7 /\Gamma) h^{-1}\rm{Mpc}$, and $\Gamma = \Omega_{\rm m} h\,\,{\rm e}^{-\Omega_{\rm b}(1+\sqrt{2h}/\Omega_{\rm m})}$ is the so-called shape parameter of the power spectrum.

The top-hat filter is:

\begin{equation}
W(k,M) = \frac{3}{(kR)^{3}}[\sin(kR)-k R\cos(kR)];
\end{equation}

sigma(kmass) - variance of linear density field (\textbf{kmass} is the mass scale):

\begin{equation}\label{eq:sigma}
\sigma^{2}(M) = \int_{0}^{\infty}{\frac{d \sigma}{dk}dk};
\end{equation}

rodm(z) - dark matter density:

\begin{equation}
\rho_{DM}(z) = \frac{\rho_{DM}^{0}}{a^{3}},
\end{equation}

where $\rho_{DM}^{0}$ is the present day dark matter density.

robr(z) - baryonic matterdensity:

\begin{equation}
\rho_{b}(z) = \frac{\rho_{b}^{0}}{a^{3}},
\end{equation}

$\rho_{b}^{0}$ is the present day baryonic matter density.

The like Press-Schechter Formalism

The core of the like Press-Schechter Formalism (PSF) is in define haloes as concentrations of mass that have already left the linear regime by crossing the threshold $\delta_{c}$ for non-linear collapse . In this case, knowing variance of linear density field and how the cosmological density perturbations grow it is possible to have the halo mass function. The functional form to describe the halo abundance is:

\begin{equation}
\frac{dn}{dM} = f(\sigma, z)\frac{\rho_{DM}}{M^{2}}\frac{d \ln(\sigma)}{d\ln(M)},
\end{equation}

being $f(\sigma, z)$ the $\sigma$-weighted distribution function. In the literature is possible to find different forms of $f(\sigma, z)$. In the class structures we have implemented the following distribution (we define $\nu \equiv \sigma(M) D(z)/\delta_{c}$):

Press & Schechter (1974) :

\begin{equation}
f(\sigma, z) = \sqrt{\frac{2}{\pi}} \nu \exp{\left(- \frac{\nu^{2}}{2}\right)};
\end{equation}

Sheth & Tormen (1999):

\begin{equation}
f(\sigma, z) = A \sqrt{\frac{2a}{\pi}} \left[1 + \left(\frac{\nu^{2}}{a}\right)^{p} \right]\exp{\left(- \frac{a\nu^{2}}{2}\right)},
\end{equation}

where $A = 0.3222$, $a= 0.707$ and $p=0.3$;

Jenkis et al. (2001):

\begin{equation}
f(\sigma) = 0.315 \exp{\left( -|ln(\sigma^{-1}) + 0.61|^{3.8}\right)}
\end{equation}

Tinker et al. (2010):

\begin{equation}
f(\sigma, z) = A \left[\left(\frac{b}{\sigma}\right)^{a} + 1\right]\exp(-\frac{c}{\sigma^{2}}),
\end{equation}

being:

\begin{equation}
A = 0.186(1+z)^{-0.14},
\end{equation}

\begin{equation}
a = 1.47(1+z)^{0.06},
\end{equation}

\begin{equation}
b = 2.57(1+z)^{\alpha},
\end{equation}

\begin{equation}
c = 1.19,
\end{equation}

\begin{equation}
\alpha = \exp{\left[-\left(\frac{0.75}{\log_{10}(\Delta_{h}/75)}\right)^{1.2}\right]}
\end{equation}

and $\Delta_{h}$ is a parameter that represent the overdensity within a sphere of radius $R$ with respect to the mean density of the Universe. This parameter is considered as an input parameter of the structures class, called delta_halo.

Warren et al. (2006):

\begin{equation}
f(\sigma) = 0.7234(\sigma^{1.625} + 0.2538)\exp(-1.1982 \sigma^{2})
\end{equation}

The choice of the mass function to be used is passed as an string attribute, massFunctionType, and its possible values are:

The public methods of structures are:

massFunction(lm, z) - Return the dark haloes mass function, lm is the $\log_{10}$ of the mass of the dark halo;
fbstruc(z) - Return the faction of barions into structures. Pereira & Miranda (2010) consider that the baryon distribution traces the dark matter distribution without bias, also that stars can form only in structures with mass larger than a threshold $M_{min}$.Thus, the fraction of baryons at redshift $z$ that are in structures is given by:\begin{equation}\label{eq:fbary}f_{\rm b}(z)=\frac{\int_{M_{\rm min}}^{M_{\rm max}} {\frac{dn(\sigma,z)}{dM} MdM}}{\int_{0}^{\infty} {\frac{dn(\sigma,z)}{dM} MdM}};\end{equation}
halos_n(z) - return the integral of the mass function of dark halos multiplied by mass in the range of $\log(M_{min})$ a $\log(M_{max})$;
numerical_density_halos(z) - Return the numerial density of dark halos within the comove volume;
abt(z) - return the accretion rate which accounts for the increase in the fraction of baryons in structures. From the definition of $f_{\rm b}(z)$ this accretion is given by \citep{pereira10,d1}:\begin{equation}a_{\rm b}(t) = \Omega_{\rm b}\rho_{\rm c}\left(\frac{dt}{dz}\right)^{-1}\left|\frac{df_{\rm b}(z)}{dz}\right|,\label{abaryon}\end{equation}

where $\rho_{\rm c}=3H_{0}^{2}/8\pi G$ is the critical density of the Universe.

The Cosmic Star Formation Rate

In the class cosmicstarformation, that extends the class structures, were implemented the methods:

cosmicStarFormationRate(z) - returns the cosmic star formation rate, $\dot{\rho}_{*}(z)$, in M$_{\odot}$yr$^{-1}$Mpc$^{-3}$;
gasDensityInStructures(z) - returns the density of gas in structures, $\rho_{g}(z)$, in M$_{\odot}$Mpc$^{-3}$;
phi(m) - returns the Initial Mass Function (IMF), $\Phi(m)$, which gives the distribution of the stellar mass. Here was assumed the IMF, namely:

\begin{equation}
\Phi(m) = A m^{-(1+x)},
\end{equation}
where $A$ is a normalization factor and $x$ is an input attribute called eimf (default value eimf=1.35). The constant $A$ is determined by:
\begin{equation}
\int_{m_{\rm inf}}^{m_{\rm sup}}Am^{-(1+x)}mdm = 1,
\label{norimf}
\end{equation}
and it was considered $m_{\rm inf}=0.1{\rm M}_{\odot}$ and $m_{\rm sup}=140{\rm M}_{\odot}$ as limits of the last equation.

Generally speaking, the cosmic star formation rate (CSFR) is obtained by the solution of the equation governing the total gas density in the haloes:

\begin{equation}
\dot\rho_{\rm g}=-\frac{d^{2}M_{\star}}{dVdt}+\frac{d^{2}M_{\rm ej}}{dVdt}+a_{\rm b}(t)\label{rhogas},
\end{equation}

where $a_{b}(t)$ is infall rate of barions into structures. The CSFR is:

\begin{equation}
\frac{d^{2}M_{\star}}{dVdt} \equiv \dot{\rho}_{*}(t) = \frac{\rho_{g}(t)}{\tau},
\end{equation}

being $\tau$ a parameter that represents the time-scale for star formation. In the code, $\tau$ is an input attribute, tau, in units of Gyr.

The term $d^{2}M_{\rm ej}/dVdt$ takes into account the mass ejected from star that returned to the interstellar medium of the system, being:

\begin{equation}
\frac{d^{2} M_{\rm ej}}{dVdt} = \int_{m(t)}^{\rm M_{sup}}{(m-m_{\rm r})\Phi(m)\dot{\rho}_{*}(t-\tau_{m})dm},\label{mej1}
\end{equation}

where $m(t)$ corresponds to the stellar mass whose lifetime is equal to $t$, $m_{r}$ is the mass of the remnant, which depends on the progenitor mass, and the star formation rate is taken at the retarded time $(t-\tau_{m})$, where $\tau_{m}$ is the lifetime of a star of mass $m$. Here it was considered $\tau_{m}$, in yrs, given by the metallicity-independent fit of:

\begin{equation}
\log_{10}(\tau_{\rm m})=10.0-3.6\,\log_{10}\left(\frac{M}{\rm M_{\odot}}\right) +\left[ \log_{10}
\left( \frac{M}{\rm M_{\odot}}\right) \right]^{2},
\end{equation}

The mass of remnant as a function of the progenitor mass are:

Objects with mass lower than 1$M_{\odot}$ have high lifetime, not contribute for M$_{\rm ej}$;
[1$M_{\odot}$ , 8$M_{\odot}$] - Carbon-oxygen- white dwarf - $m_{r} = 0.1156m +0.4551$ $M_{\odot}$
(8$M_{\odot}$,10$M_{\odot}$] - oxygen-neon-magnesium white dwarfs - $m_{r} = 1.35$$M_{\odot}$
(10$M_{\odot}$, 25$M_{\odot}$) - neutron stars - $m_{r}=1.4$$M_{\odot}$
[25$M_{\odot}$,140$M_{\odot}$] - Black Hole - $m_{r} = \frac{13}{24}(m-20{\rm M}_{\odot})$

In hierarchical models for galaxy formation the first star-forming halos are predicted to collapse at redshift $z\gtrsim 20$, having masses $\sim 10^{6}{\rm M}_{\odot}$. These facts are parameterized in the code by choice of the redshift that start the star formation, zmax, and by changes in the value of lower limit of the superior integral of the Equation of fraction of barions into structures lmin.

The input attributes of the class cosmicstarformation are:

tau (default 2.5 ): time scale, in Gyr, of the CSFR.
eimf (default 1.35): exponent of the Initial Mass Function.
nsch (default 1): the normalization factor in the CSFR model.
lmin (default 6.0): log10 of the minal mass of the dark halo where it is possible to have star formation.
zmax (default 20.0): redshift that start the star formation.
omegam (default 0.24): The dark matter parameter.
omegab (default 0.04): The baryonic matter parameter.
omegal (default 0.73): The dark energy parameter.
h (default 0.7): The h of the Hubble constant ($H_{0} = 100h$).
massFunctionType (default "ST"): The type of mass function of dark matter halos used.

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