pyspk 1.8
Python package to predict the suppression of the total matter power spectrum due to baryonic physics

# py-SP(k) - A hydrodynamical simulation-based model for the impact of baryon physics on the non-linear matter power spectrum
                          _____ ____   ____   _ 
        ____  __  __     / ___// __ \_/_/ /__| |
       / __ \/ / / /_____\__ \/ /_/ / // //_// /
      / /_/ / /_/ /_____/__/ / ____/ // ,<  / / 
     / .___/\__, /     /____/_/   / //_/|_|/_/  
    /_/    /____/                 |_|    /_/    

py-SP(k) [(Salcido et al. 2023)](https://academic.oup.com/mnras/article/523/2/2247/7165765) is a python package aimed at predicting the suppression of the total matter power spectrum due to baryonic physics as a function of the baryon fraction of haloes and redshift.

## Requirements

The module requires the following:

- numpy
- scipy

## Installation

The easiest way to install py-SP(k) is using pip:

```
pip install pyspk [--user]
```

The --user flag may be required if you do not have root privileges.

## Usage

py-SP(k) is not restrictive to a particular shape of the baryon fraction – halo mass relation. In order to provide flexibility to the user, we have implemented 3 different methods to provide py-SP(k) with the required $f_b$ - $M_\mathrm{halo}$ relation. In the following sections we describe these implementations. A jupyter notebook with more detailed examples can be found within this [repository](https://github.com/jemme07/pyspk/blob/main/examples/pySPk_Examples.ipynb). 

### Method 1: Using a power-law fit to the $f_b$ - $M_\mathrm{halo}$ relation

py-SP(k) can be provided with power-law fitted parameters to the $f_b$ - $M_\mathrm{halo}$ relation using the functional form:

$$f_b/(\Omega_b/\Omega_m)=a\left(\frac{M_{SO}}{M_{\mathrm{pivot}}}\right)^{b},$$

where $M_{SO}$ could be either $M_{200c}$ or $M_{500c}$ in $\mathrm{M}_ \odot$, $a$ is the normalisation of the $f_b$ - $M_\mathrm{halo}$ relation at $M_\mathrm{pivot}$, and $b$ is the power-law slope. The power-law can be normalised at any pivot point in units of $\mathrm{M}_ {\odot}$. If a pivot point is not given, `spk.sup_model()` uses a default pivot point of $M_{\mathrm{pivot}} = 1 \mathrm{M}_ \odot$. $a$, $b$ and $M_\mathrm{pivot}$ can be specified at each redshift independently.  

Next, we show a simple example using power-law fit parameters:

```
import pyspk as spk

z = 0.125
fb_a = 0.4
fb_pow = 0.3
fb_pivot = 10 ** 13.5

k, sup = spk.sup_model(SO=200, z=z, fb_a=fb_a, fb_pow=fb_pow, fb_pivot=fb_pivot)
```

### Method 2: Redshift-dependent power-law fit to the $f_b$ - $M_\mathrm{halo}$ relation. 

For the mass range that can be relatively well probed in current X-ray and Sunyaev-Zel'dovich effect observations (roughly $10^{13} \leq M_{500c} [\mathrm{M}_ \odot] \leq 10^{15}$), the total baryon fraction of haloes can be roughly approximated by a power-law with constant slope (e.g. Mulroy et al. 2019; Akino et al. 2022). Akino et al. (2022) determined the of the baryon budget for X-ray-selected galaxy groups and clusters using weak-lensing mass measurements. They provide a parametric redshift-dependent power-law fit to the gas mass - halo mass and stellar mass - halo mass relations, finding very little redshift evolution. 

We implemented a modified version of the functional form presented in Akino et al. (2022), to fit the total $f_b$ - $M_\mathrm{halo}$ relation as follows:

$$f_b/(\Omega_b/\Omega_m)= \left(\frac{e^\alpha}{100}\right) \left(\frac{M_{500c}}{10^{14} \mathrm{M}_ \odot}\right)^{\beta - 1} \left(\frac{E(z)}{E(0.3)}\right)^{\gamma},$$

where $\alpha$ sets the power-law normalisation, $\beta$ sets power-law slope, $\gamma$ provides the redshift dependence and $E(z)$ is the usual dimensionless Hubble parameter. For simplicity, we use the cosmology implementation of `astropy` to specify the cosmological parameters in py-SP(k).

Note that this power-law has a normalisation that is redshift dependent, while the the slope is constant in redshift. While this provides a less flexible approach compared with Methods 1 (simple power-law) and Method 4 (binned data), we find that this parametrisation provides a reasonable agreement with our simulations up to redshift $z=1$, which is the redshift range proved by Akino et al. (2022). For higher redshifts, we find that simulations require a mass-dependent slope, especially at the lower mass range. 

In the following example we use the redshift-dependent power-law fit parameters with a flat LambdaCDM cosmology. Note that any `astropy` cosmology could be used instead.

```
import pyspk.model as spk
from astropy.cosmology import FlatLambdaCDM

H0 = 70 
Omega_m = 0.2793

cosmo = FlatLambdaCDM(H0=H0, Om0=Omega_m) 

alpha = 4.189
beta = 1.273
gamma = 0.298
z = 0.5

k, sup = spk.sup_model(SO=500, z=z, alpha=alpha, beta=beta, gamma=gamma, cosmo=cosmo)
```

### Method 3: Redshift-dependent double power-law fit to the $f_b$ - $M_\mathrm{halo}$ relation. 

We implemented a redshift-dependent double power-law fit to the total $f_b$ - $M_\mathrm{halo}$ relation as follows:

$$f_b/(\Omega_b/\Omega_m)= \frac{1}{2} \epsilon \left[\left(\frac{M_{500c}}{M_{\mathrm{pivot}}}\right)^{\alpha} + \left(\frac{M_{500c}}{M_{\mathrm{pivot}}}\right)^{\beta}\right] \left(\frac{E(z)}{E(0.3)}\right)^{\gamma},$$

where $\epsilon$ sets the normalisation at the pivot point, $M_{\mathrm{pivot}}$, in units of $\mathrm{M}_ {\odot}$, $\alpha$ and $\beta$ are the power-law slopes at low and high mass respectively, $\gamma$ provides the redshift dependence and $E(z)$ is the usual dimensionless Hubble parameter. For simplicity, we use the cosmology implementation of `astropy` to specify the cosmological parameters in py-SP(k).

We find that this double redshift-dependent double power-law form provides a good fit to the whole range of baryon fractions in the ANTILLES simulations. 

In the following example we use the redshift-dependent double power-law fit parameters with a flat LambdaCDM cosmology. Note that any `astropy` cosmology could be used instead.

```
import pyspk.model as spk
from astropy.cosmology import FlatLambdaCDM

H0 = 68.0 
Omega_m = 0.306

cosmo = FlatLambdaCDM(H0=H0, Om0=Omega_m) 

epsilon = 0.410
alpha = -0.385
beta = 0.451
gamma = 0.125
m_pivot = 1e13
z = 0.2

k, sup = spk.sup_model(SO=500, z=z, epsilon=epsilon, alpha=alpha, beta=beta, gamma=gamma, m_pivot=m_pivot, cosmo=cosmo)
```


### Method 4: Binned data for the $f_b$ - $M_\mathrm{halo}$ relation. 

The final, and most flexible method is to provide py-SP(k) with the baryon fraction binned in bins of halo mass. This could be, for example, obtained from observational constraints, measured directly form simulations, or sampled from a predefined distribution or functional form. For an example using data obtained from the BAHAMAS simulations (McCarthy et al. 2017), please refer to the [examples](https://github.com/jemme07/pyspk/blob/main/examples/pySPk_Examples.ipynb) provided. 


## Priors

While py-SP(k) was calibrated using a wide range of sub-grid feedback parameters, some applications may require a more limited range of baryon fractions that encompass current observational constraints. For such applications, we used the gas mass - halo mass and stellar mass - halo mass constraints from the fits in Table 5 in Akino et al. (2022), and find the subset of simulations from our 400 models that agree to within $\pm 2$ or $3 \times \sigma$ of the inferred baryon budget at redshift $z=0.1$. We note that for our simulations, we include all stellar and gas particles within a spherical overdensity radius. Hence, in order to make reasonable comparisons with the fits in Akino et al. (2022), we included an additional 15\% contribution to the total stellar masses from the contribution of blue galaxies, and 30\% additional stellar mass to the brightest cluster galaxies (BCGs) to account for the diffuse intracluster light (ICL, see Akino et al. 2022). 

We utilised the simulations satisfying these restrictions to determine the redshift-dependent power-law parameters for the $f_b$ - $M_\mathrm{halo}$ relation up to redshift $z=1$ (Method 2), and then utilised these parameters to infer suitable priors. We limited the fitting range to $6 \times 10^{12} \leq M_{500c} [\mathrm{M}_ \odot] \leq 10^{14}$. 

Priors inferred from simulations that fall within $\pm 2 \times \sigma$ of the inferred baryon budget:

| Parameter   | Description        | Prior           |
| ----------- | ------------------ | --------------- |
| $\alpha$    | Normaliasation     | $\mathcal{N}$(4.16, 0.07) |
| $\beta$     | Slope              | $\mathcal{N}$(1.20, 0.05) |
| $\gamma$    | Redshift evolution | $\mathcal{N}$(0.39, 0.09) |

where $\mathcal{N}(\mu,\sigma)$ is a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$.

Priors inferred from simulations that fall within $\pm 3 \times \sigma$ of the inferred baryon budget:

| Parameter   | Description        | Prior           |
| ----------- | ------------------ | --------------- |
| $\alpha$    | Normaliasation     | $\mathcal{N}$(4.18, 0.12) |
| $\beta$     | Slope              | $\mathcal{N}$(1.26, 0.08) |
| $\gamma$    | Redshift evolution | $\mathcal{N}$(0.42, 0.10) |

where $\mathcal{N}(\mu,\sigma)$ is a Gaussian distribution with mean $\mu$ and standard deviation $\sigma$.

## Acknowledging the code

Please cite py-SP(k) using:

```
@ARTICLE{SPK_Salcido_2023,
    author = {Salcido, Jaime and McCarthy, Ian G and Kwan, Juliana and Upadhye, Amol and Font, Andreea S},
    title = "{SP(k) – a hydrodynamical simulation-based model for the impact of baryon physics on the non-linear matter power spectrum}",
    journal = {Monthly Notices of the Royal Astronomical Society},
    volume = {523},
    number = {2},
    pages = {2247-2262},
    year = {2023},
    month = {05},
    issn = {0035-8711},
    doi = {10.1093/mnras/stad1474},
    url = {https://doi.org/10.1093/mnras/stad1474},
    eprint = {https://academic.oup.com/mnras/article-pdf/523/2/2247/50512773/stad1474.pdf},
}
```
For any questions and enquires please contact me via email at *j.salcidonegrete@ljmu.ac.uk*