Cosmic-Fields-Lite

Cosmic-Fields-Lite is a lightweight and modular framework for performing field simulations in cosmology. This framework was used for studying free-streaming of wave dark matter; see arXiv:2408.05591 for the study and these youtube videos for visualization. The codebase contains several field equations on both CPU and GPU (CUDA), offering choices for numerical methods and simulation outputs.

Overview

This codebase aims to be:

1. As fast as possible. Users should be able to write code that exhausts hardward potential within this framework.
2. Easily modifiable and extensible. Users should be able to focus on physics-relevant code, such as that for setting initial conditions or the field equation.

To achieve these goals, the framework is written in a modular structure. This allows users to easily switch between different initial conditions, field equations, output methods, and even between using CPUs or GPUs for computation. Users have to and only have to provide the low level implementation for the physics-relevant code. This means users have full control over optimization of core routines, and they are not limited to a specific set of provided features. This flexibility makes it easy for the user to test new ideas.

Sample usage

The following code initializes a homogeneous Klein Gordon field with (initially) unit field strength and zero time derivative. Then the field is evolved from t=0 to t=10. Field and density spectra are saved to disk per unit time.

#include "param.hpp"
#include "initializer.hpp"
#include "equations.hpp"
#include "observer.hpp"

struct MyParam {
  long long int N = 256; // Number of lattice sites (per axis)
  double L = 256.0; // Box size
  double m = 1.0; // Field mass
  double f = 1.0; // The initial (homogeneous) field value
  double dt_f = 0.0; // The initial (homogeneous) field time derivative value
  double t_start = 0.0; // Start time of numerical integration
  double t_end = 10.0; // End time of numerical integration
  double t_interval = 1.0; // Interval between saving outputs
};

int main() {
  using namespace Eigen;
  using namespace boost::numeric::odeint;

  typedef KleinGordonEquation Equation;
  typedef Eigen::VectorXd State;
  typedef WorkspaceGeneric<State> Workspace;
  
  MyParam param;
  
  Workspace workspace(param, homogeneous_field);
  
  Equation eqn(workspace);

  ConstIntervalObserver<Equation> observer("output/sample_equation/", param, eqn);

  auto stepper = runge_kutta4_classic<State>();

  integrate_const(stepper, eqn, workspace.state, param.t_start, param.t_end, 0.1, observer);
}

Here's a break down of the code:

• MyParam is a POD struct specifying parameters for the simulation. You may define your own struct to include new parameters (coupling strength, FRW universe parameters, time step size, etc), as long as it is a POD and contains lattice parameters N and L.
• Workspace is a type containing temporary variables for a simulation (e.g. the field). It is initialized with param and a callback homogeneous_field, which sets the field to homogeneous value param.f and time derivative param.dt_f. You can easily define your own callbacks (using lambdas) to set other sorts of initial conditions.
• Equation is the equation to be solved. Here it is the pre-defined KleinGordonEquation. You can of course write your own equations.
• ConstIntervalObserver<Equation> specifies how to save outputs during simulation. By default it saves spectra for field and density.
• stepper is the RK4 method provided by the boost odeint library. You can choose other methods (e.g. Euler, DOPRI5) in the library, or even write your own. The odeint library is responsible for the main numerical integration loop in this codebase.
• integrate_const is a convenience function in the odeint library. It runs the simulation and saves results to "output/sample_equation", as specified by observer.

How to get and build the project

Compiler requirement: a C++ compiler supporting C++20. (I used g++ 12.2.0.)

Required dependency: fftw3

Optional dependency: CUDA Toolkit

I also included header-only libraries Eigen 3.4.0 and boost 1.84 along with the codebase in the external directory.

Makefile is used for build system. I have tested compilation on Linux and MacOS systems. To compile the project:

• Download the project with (for example) git clone https://github.com/hypermania/Cosmic-Fields-Lite.
• (If default settings don't work:) Modify the Makefile so that it knows where your fftw or CUDA include files / library files are.
• If you have CUDA Toolkit installed, simply run make -j.
• If you don't have CUDA Toolkit, run make -j disable-cuda=true. (I use compiler flags to comment out CUDA-dependent code. e.g. CudaComovingCurvatureEquationInFRW)

Note: If you have a CUDA compatible NVIDIA GPU, using CUDA is highly recommended. In our case, it produced more than 10 times speedup.

Documentation

LaTeX version of documentation is in documentation.pdf. If you have doxygen, you can also build an html version by running doxygen doxygen.config.

Convenience utilities for visualizing output

Two Mathematica notebooks spectra.nb, snapshots.nb and a python script plot_util.py are included for visualizing outputs from the program. By default, running the entire notebook / python script will read sample data from output/Growth_and_FS and produce spectra and snapshots. If you generate new outputs from the program, you just need to change dir or project_dir variables to the new output directory.

Overview of implemented functionalities

Symbol	Description
generate_inhomogeneous_gaussian_random_field	Function for initializing Gaussian random fields with spatially inhomogeneous variances. This procedure is crucial for generating the initial conditions used in the paper.
KleinGordonEquationInFRW and CudaKleinGordonEquationInFRW	Klein Gordon equation that runs on CPU and GPU. Used in section 4.2.1 of paper.
ComovingCurvatureEquationInFRW, CudaComovingCurvatureEquationInFRW and CudaApproximateComovingCurvatureEquationInFRW	A scalar field in the presence of external gravity that is consistent with some set of comoving curvature perturbations. Used in section 4.2.2 of paper.
CudaSqrtPotentialEquationInFRW	A scalar field with monodromy potential. Used in section 4.2.3 of paper.
CudaFixedCurvatureEquationInFRW	A scalar field in a fixed gravitational potential.
CudaLambdaEquationInFRW	A scalar field with lambda phi^4 interaction.

Notes on using CUDA

We do separate compilation of .cpp files and .cu files; .cu files are automatically compiled by nvcc, whereas .cpp files are compiled by the host compiler. We use the thrust library (included with CUDA Toolkit) extensively, with field state vectors having type thrust::device_vector<double>. Initialization procedures usually prepare some profile on the CPU and then copy it to device_vector<double> state in the workspace.

A straightforward way to use CUDA for a simulation is to implement an Equation class with thrust::device_vector<double> as state vector. You will probably need to write your own CUDA kernels for that purpose. See equations_cuda.cu for some examples. Don't worry about adapting CUDA with the numerical integrators (e.g. RK4); the files in src/odeint_thrust will take care of that automatically.