ReHILAE: Is Re-ionisation of Hydrogen due to Lyman-alpha emitters (LAEs)?

Is Re-ionisation of Hydrogen due to Lyman-alpha emitters (LAEs)?

GitHub Repository: https://github.com/JP-Carr/ReHILAE

Group members and roles:

James Gold – Coordinator/PI

Connor Donovan – Write-up lead/blog

Jack Bowden – Theory lead

James Carr – Code lead

Joe Philips – Administrator

Our group, ReHILAE, is investigating whether or not LAEs are the dominating LyC (ionising radiation) sources during re-ionisation and therefore the main driving force behind the epoch of re-ionisation.

We will be building on the work of Faisst, A. (2016), initially we will be running and improving on existing python scripts which represent the equations on page 7 of this paper. Following on from this we focus on improving the availability of ionising photons, which is currently given by:

$\dot{n}_{ion} = f_{esc}\xi_{ion}\rho_{UV}$

This formula can be improved by considering the work of Sobral, D. and Matthee, J. (2019). Using equation (9) in this paper (page 8) we can replace the $\xi_{ion}\rho_{UV}$ in the equation above with:

$Q_{ion,Ly\alpha}[s^{-1}] = \frac{L_{Ly\alpha}}{c_{H\alpha}(1-f_{esc,LyC})(0.042EW_{0})}$

Week 1 Research Update:

Today, we created our own Python scripts describing all the equations needed for our model of re-ionisation. These equations are shown below:

$z = (\sinh{[\frac{3\Omega^{1/2}_{\Lambda}t}{2H^{-1}_{0}}]} (\frac{\Omega_{\Lambda}}{\Omega_{M}})^{-1/2})^{-2/3} -1$

$f_{esc} = \frac{f_{esc,0}}{100}(\frac{1+z}{3})^{\alpha}$

$\log_{10}(\xi_{ion}) = 24.4 + \log_{10}(1+z)$

Where $f_{esc,0} = 2.3 \pm 0.05$ and $\alpha = 1.17 \pm 0.02$ . The final factor we need to consider before calculating the fraction of ionised hydrogen is the recombination time for ionised hydrogen.

$t_{rec} = [\alpha_{B}n_{H}C(1+Y_{p}/4X_{p})(1+z)^3]^{-1}$

Where $Y_{p}$ and $X_{p}$ are the primordial mass fractions of helium (25%) and hydrogen (75%) respectively. The $\alpha_{B}$ term is the recombination coefficient and the $n_{H}$ term is the hydrogen number density, both of which are described below.

$\alpha_{B} = 2.6 \times 10^{-13}(\frac{T}{10^{4}})^{-0.76}$

$n_H = 1.67 \times 10^{-7}(\frac{\Omega_{b}h^2}{0.02})(\frac{X_p}{0.75})$

Where $T$ is the temperature of the IGM. We then plotted each of these equations to illustrate the relationships between these variables.

We also plotted the UV density luminosity function as a function of redshift:

The main and final equation used in our model is a differential equation that describes the evolution of the fraction of ionised hydrogen as function of redshift or cosmic time, which is shown below.

$\dot{Q}_{HII} = \frac{\dot{n}_{ion}}{n_H} - \frac{Q_{HII}}{t_{rec}}$

The code below shows how this equation was implemented and solved in Python. The odeint function returns an array containing the fraction of ionised hydrogen for each inputted value of time (which is immediately converted to redshift).

def dQ_dt(Q,t):
    z= ((((28./(t))-1.)**(1./2.)-1.))
    
    trec = 1./(C*alpha_B*(1.+Y_p/(4.*X_p))*n_H*(1.+z)**(3.)) 
    nion = f_esc0*xi_ion*p(z)/ (2.938e+73)
    return ((nion)/n_H-Q/trec)*3.1536e+16

ts = np.linspace(0.051,14,10000000) # time in Gyr
zs= ((((28./(ts))-1.)**(1./2.)-1.)) # conversion from Gyr to redshift

Q = odeint(dQ_dt, 0., ts)

This array can be plotted, to visualise the epoch of re-ionisation: