What is the picasso model?

The picasso model is the combination of two different parts:

• A polytropic gas model, mapping intracluster gas thermodynamics to a gravitational potential distribution, given a set of model parameters \(\vartheta_{\rm gas}\);

• A neural network predicting a vector of model parameters \(\vartheta_{\rm gas}\) from a vector of halo properties \(\vartheta_{\rm halo}\).

The gas model

The polytropic gas model is written as:

\[\begin{split}\rho(\phi) = \rho_0 \theta^{\Gamma(r) / (\Gamma(r) - 1)}(\phi), \\[10pt] P(\phi) = P_0 \theta^{1 / (\Gamma(r) - 1)}(\phi),\end{split}\]

where \(\phi\) is the halo’s gravitational potential, and

\[\theta(\phi) = 1 - \theta_0 (\phi - \phi_0).\]

The gas polytropic index, \(\Gamma\), is allowed to vary with radius as:

\[\begin{split}\Gamma(r) = \begin{cases} \begin{aligned} & \; 1 + (\Gamma_0 - 1) \frac{1}{1 + e^{-x}} & c_\Gamma \geqslant 0; \\ & \; \Gamma_0 + (\Gamma_0 - 1) \left(1 - \frac{1}{1 + e^{x}}\right) & c_\Gamma < 0, \\ \end{aligned} \end{cases}\end{split}\]

with \(x \equiv r / (c_\gamma R_{500c})\). The model has five parameters: \((\rho_0, P_0)\) are the central value of gas density and pressure, \(\Gamma_0\) is the asymptotic value of the polytropic index as \(r \rightarrow \infty\), \(c_\gamma\) is the polytropic concentration (\(c_\gamma = 0\) implies \(\Gamma(r) = \Gamma_0\)), and \(\theta_0\) is a shape parameter. In the Ostriker model,

\[\theta_0 = \frac{\Gamma - 1}{\Gamma} \times \frac{\rho_0}{P_0}\]

We further write the fraction of non-thermal pressure as a power-law of radius, plus a constant plateau:

\[f_{\rm nt}(r) = A_{\rm nt} + (B_{\rm nt} - A_{\rm nt}) \left(\frac{r}{2R_{500c}}\right)^{C_{\rm nt}}\]

This adds three parameters to our gas model: \(A_{\rm nt}\) is the central value of non-thermal pressure fraction, \(B_{\rm nt}\) is the non-thermal pressure fraction at \(r=2R_{500c}\), and \(C_{\rm nt}\) is the power law evolution with radius.

Therefore, for a potential distribution \(\phi(r)\), gas thermodynamics are fully specified by a vector \(\vartheta_{\rm gas} = (\rho_0, \, P_0, \, \Gamma_0, \, c_\gamma, \, \theta_0, \, A_{\rm nt}, \, B_{\rm nt}, \, C_{\rm nt})\).

See also:

• Impact of model parameters of gas thermodynamics, for a visual representation of the impact of each parameter of the model on gas thermodynamics;

• picasso.polytrop: Polytropic gas model and picasso.nonthermal: Non-thermal pressure support, for the documentation of the functions providing the numerical implementation of the model above;

• Using the picasso analytical gas model, for code examples.

Neural network predictions

To pick values for the gas model components, picasso uses a fully-connected neural network that is trained to predict the \(\vartheta_{\rm gas}\) vector given a set of halo properties \(\vartheta_{\rm halo}\). The key assumptions are that, for a given halo, the complexity of the gas distribution can be captured by the parameters of \(\vartheta_{\rm gas}\), and that halo properties contain enough information to robustly predict the values of these parameters. The neural network is trained to reproduce a given target data, in general the properties of halos found in hydrodynamic simulations. The components of \(\vartheta_{\rm halo}\) can vary, depending on the properties available during the training and on how widely usable a trained model aims to be.

See also:

• Available pre-trained predictors, for a list of the available pre-trained models;

• picasso.predictors: From halo properties to gas properties, for the documentation of the prediction functions available in picasso;

• Using the picasso trained predictors, for code examples.