Black-to-White hole transition

The code computes the Black-to-White hole transitin amplitude:

$$ W_{\alpha} (j_{\pm}, j_{0}, \zeta_{\pm}, \zeta_0; \Delta l) = \sum_{ j_{ab}^{\pm}, j_{a} } w_{\alpha} \left( \sum_{i_5} d_{i_5} \prod_{\pm} V_{\gamma, n_{f}^{\pm}}^{coh} \left(j_{ab}^{\pm}, j_{a}, i_5; \Delta l \right) \right) $$

and the crossing time:

$$ \tau_{\alpha} \left( m \right) = \int_{0}^{\infty} dT \ T \ P_{\alpha} \left( m | T \right) \ , $$

where:

$$ P_{\alpha} \left( m | T \right) = \frac{\mu_{\alpha} (m, T) \ |W_{\alpha} (m , T )|^2}{\int_{0}^{\infty} dT \ \mu_{\alpha} (m, T) \ |W_{\alpha} (m, T)|^2} \ . $$

The parameters for the computation can be specified within the file parameters.jl.

Structure of computation

The computation is divided into multiple steps:

• bin/vertex_computation.jl computes all the required full vertex amplitude tensors. A supercluster is recommended for high values of $\Delta l$.

• bin/vertex_contraction.jl evaluates the factor $\left( \sum_{i_5} d_{i_5} \prod_{\pm} V_{\gamma, n_{f}^{\pm}}^{coh} \left(j_{ab}^{\pm}, j_{a}, i_5; \Delta l \right) \right)$ for each value of $j_{ab}^{\pm}, j_{a}$.

• bin/weight_factor.jl computes the weight factor $w_{\alpha}$ (whose expression is not explicitly reported here) for each value of $j_{ab}^{\pm}, j_{a}$ and $T_1 \dots T_N$.

• bin/amplitude_assembling.jl puts together all the terms and computes the total amplitude.

The scripts can be run all in sequence with run_all.jl

The Julia codes are parallelized on the available cores. It is therefore advisable for the performance to parallelize the codes keeping into account the number of CPU cores present on the system.

A full list of the employed Julia packages can be found in ./inc/pkgs.jl. Before executing the source codes, all packages must be installed.

The Julia's Just-in-Time compiler is such that the first execution of functions is considerably slower that following ones, and it also allocates much more memory. To avoid this, you can use the DaemonMode package.

Usage

To run the Julia scripts (on a single machine with the synthax below) you can use the following command:

$JULIA_EXECUTABLE_PATH   -p   [N-1]    $JULIA_CODE_PATH   $STORE_FOLDER    $DATA_SL2CFOAM_FOLDER

where [N-1] is the number of workers and:

• STORE_FOLDER: folder where data are saved

• DATA_SL2CFOAM_FOLDER: folder with fastwigxj tables, where booster functions and vertex amplitudes are stored/retrieved.

Data analysis

The data analysis is performed in the analysis.ipynb notebook, inside the folder notebooks/Julia.

The notebooks inside notebooks/Mathematica describe data of the boundary geometry. Specifically, triangulation.nb reconstrutcs the full geometry of one 4-simplex, including the orientation of normals to the tetrahedra.

The folder cluster_scripts contains simple scripts to run the code on a single supercluster host.