On the Generality and Persistence of Cosmological Stasis

Implementation of 2408.00835 by James Halverson and Sneh Pandya.

Abstract

Hierarchical decays of $N$ matter species to radiation may balance against Hubble expansion to yield stasis, a new phase of cosmological evolution with constant matter and radiation abundances. We analyze stasis with various machine learning techniques on the full $2N$-dimensional space of decay rates and abundances, which serve as inputs to the system of Boltzmann equations that governs the dynamics. We construct a differentiable Boltzmann solver to maximize the number of stasis $e$-folds $\mathcal{N}$. High-stasis configurations obtained by gradient ascent motivate log-uniform distributions on rates and abundances to accompany power-law distributions of previous works. We demonstrate that random configurations drawn from these families of distributions regularly exhibit many $e$-folds of stasis. We additionally use them as priors in a Bayesian analysis conditioned on stasis, using stochastic variational inference with normalizing flows to model the posterior. All three numerical analyses demonstrate the generality of stasis and point to a new model in which the rates and abundances are exponential in the species index. We show that the exponential model solves the exact stasis equations, is an attractor, and satisfies $\mathcal{N}\propto N$, exhibiting inflation-level $e$-folding with a relatively low number of species. This is contrasted with the $\mathcal{N}\propto \log(N)$ scaling of power-law models. Finally, we discuss implications for the emergent string conjecture and string axiverse.

Using this code

This project is developed for Python3.9 interpreter on a linux machine. Using an Anaconda virtual environment is recommended. Our implementation hinges on diffrax and numpyro, packages built on jax 🚀.

To install dependencies, you can run

pip install -r requirements.txt

or consult online documentation for appropriate dependencies. It is highly reccomended to follow jax install directions from the source.

Our codebase is structured as follows:

src/scripts contains the main ingredients for our methodology.

• stasis_simulation_differentiable.py is the differentiable stasis simulation, containing the differentiable Boltzmann solver and stasis finder. This is used in SVI and gradient ascent experiments.
• stasis_simulation_non_diff.py is the non-differentiable stasis simulation. This is used for creating plots and features an accurate, sliding-window stasis finder algorithm. This also has parametric functionality built in via the model argument, which allows one to generate power-law and exponential-dependent stasis configurations.
• svi.py contains code for doing stochastic variational inference with normalizing flows. This requires a yaml file specifying experimental parameters, an example of which is given in src/tutorials/example_svi_experiment.yaml.

src/tutorials contains examples which coincide with the experiments in our paper.

• gradient_ascent.ipynb gives a walk-through in how gradients in the simulation can be used to produce stasis with minimal physical biases in the initialization. It also visualizes the optimization trajectories with a gif.
• model_comparison.ipynb does a numerical comparison between power-law and exponential models of stasis. It also shows that both stasis models are attractors.
• random_stasis.ipynb shows that rates and abundances drawn from certain families of distributions can result in persistent epochs of stasis.

Contact

Code author: Sneh Pandya

Issues and questions: @snehjp2, pandya.sne@northeastern.edu