# Methods and Cosmology¶

When using **fruitbat** there are different methods and cosmologies built-in from
which you can choose. Below is a brief description of how each method is calculated
and the parameters for each cosmology.

## Methods¶

Currently there are three built-in dispersion measure-redshift relations. In **fruitbat** these are
referred to as ‘methods’. To calculate the look-up tables **fruitbat** uses the equations described below
for each method to calculate the dispersion measre (\(\rm{DM}\)) for a given redshift (\(z\)).

### F(z) Integral¶

The methods `'Ioka2003'`

, `'Inoue2004'`

, `'Zhang2018'`

all have a common integral \(F(z)\).

Where \(z\) is the redshift, \(\Omega_m\) is the cosmic matter density, \(\Omega_{\Lambda}\) is cosmic dark energy density and \(w\) characterises the dark energy equation of state. This is typically assumed to have a constant value of \(\sim 1.06\) which introduces an error of approximately 6% for \(z < 2\).

The lookup tables in **fruitbat** explicitly solve this integral for each redshift to when calculating
a dispersion measure. See the Figure below for a comparison between the assumed value and the integral in
**fruitbat**.

### Ioka 2003¶

The Ioka (2003) method assumes that all baryons in the Universe are fully ionised and that there is a 1-to-1 relation between eletrons and baryons. i.e. The number is free electrons in the Universe is the same as the number of baryons. The DM at a given redshift using the Ioka method is calculated as follows:

Where \(c\) is the speed of light, \(H_0\) is the Hubble constant, \(\Omega_b\) is the cosmic baryon density, \(G\) is the gravitational constant, \(m_p\) is the mass of the proton and \(F(z)\) is the integral defined earlier.

### Inoue 2004¶

The Inoue (2004) method assumes that hydrogen is fully ionised and helium is singly ionised. THe DM at a given redshift using the Inoue method is calculated as follows.

Where \(c\) is the speed of light, \(H_0\) is the Hubble constant, \(\Omega_b\) is the cosmic baryon density and \(F(z)\) is the integral defined earlier. The factor of \(9.2 \times 10^{-10}\) comes estimating the number of free electrons at high redshifts from models of reionisation.

### Zhang 2018¶

The Zhang (2018) method assumes that all baryons in the Universe are fully ionised and that there is a 0.875-to-1 ratio between eletrons and baryons, and that 85% of baryons are in the intergalactic medium. The DM at a given redshift using the Zhang method is calculated as follows:

Where \(c\) is the speed of light, \(H_0\) is the Hubble constant, \(\Omega_b\) is the cosmic baryon density, \(G\) is the gravitational constant, \(m_p\) is the mass of the proton, \(\chi\) is the free electron per baryon in the Universe, \(f_{igm}\) is the fraction of baryons in the intergalactic medium and \(F(z)\) is the integral defined earlier.

\(\chi\) is calculated as follows:

Where \(\chi_{e, H}\) and \(\chi_{e, He}\) denote the ionisation fraction of hydrogen and helium respectively and \(y_1 \sim y_2 \sim 1\) denote the possible slight deviation from the 3/4 - 1/4 split of hydrogen and helium abundance in the Universe. Assuming that hydrogen and helium are both ionised, then \(\chi(z) \sim 0.875\).

## Cosmology¶

Each method in **fruitbat** has a list of pre-calculated lookup tables with
different cosmologies. The table below lists the parameters that are used for
each cosmology.

Cosmological Parameters | |||||
---|---|---|---|---|---|

Keyword | \(H_0\) | \(\Omega_b\) | \(\Omega_m\) | \(\Omega_\Lambda\) | \(w\) |

`'WMAP5'` |
\(70.2\) | \(0.0459\) | \(0.277\) | \(0.723\) | -1 |

`'WMAP7'` |
\(70.4\) | \(0.0455\) | \(0.272\) | \(0.728\) | -1 |

`'WMAP9'` |
\(69.32\) | \(0.04628\) | \(0.2865\) | \(0.7134\) | -1 |

`'Planck13'` |
\(67.77\) | \(0.0483\) | \(0.3071\) | \(0.6914\) | -1 |

`'Planck15'` |
\(67.74\) | \(0.0486\) | \(0.3075\) | \(0.6910\) | -1 |

`'Planck18'` |
\(67.66\) | \(0.04897\) | \(0.3111\) | \(0.6874\) | -1 |

Below is a figure comparing the different methods and cosmologies in **fruitbat**. The left figure shows how the different methods compare assuming a `'Planck18'`

cosmology..
The right figure shows how the `'Inoue2004'`

method changes with different assumed cosmologies.