This repository is the companion code to the papers Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Three-field Inflation (I and II), by Vikas Aragam, Sonia Paban, and Robert Rosati.

Quick start

First clone the repo.

Mathematica

Example usage and plots reproducing many of the figures of our work are available in bog_plotting_clean.nb, just clone the repo and get started.

The Bogoliubov coefficients for two and three fields are stored in the *.wl files. The files with *Small* in the name correspond to the branch of solutions with ${\cal M}_{ss,0}, {\cal M}_{bb,0} \leq 9/4$ (cf. Eq (2.11) of paper II), while the other files are the branch with larger constant masses. Each file contains two variables

• regIIIcoeffsSimp: these are the Bogoliubov coefficients of the sharp feature's excited state
• αsols: the WKB exponents during the feature are $\pm \sqrt{\alpha_i}$.

Not discussed in the papers but also included here are the two-field Bogoliubov coefficients including a constant contribution to the entropic mass. These should match the $\tau \rightarrow 0$ limit of the $(\zeta, s)$ 2$\times$2 block of the three-field coefficients.

Julia

Installing Julia

First install Julia, then clone this repo. Open a terminal in the cloned repo's folder.

As a small aside, the Julia workflow is a little different than python -- because Julia uses just-in-time compilation, you'll avoid excessive recompilation if you leave a julia REPL running between editing and rerunning code.

Activating the environment

Next we'll want to activate the environment in this folder

julia> import Pkg; Pkg.activate(".")

(as a shortcut, you can also press the ] key and type activate .).

Instantiating the environment

The first time you run this, you'll also need install the dependencies and precompile them (you only have to do this once):

julia> Pkg.instantiate()

This will take a little while.

Then you'll be able to run the codes here.

How to run

Each time you re-open Julia, you'll need to reactivate the environment in this folder before running any of the codes. Then you can load the codes with

julia> include("bog_plotting_minimal.jl")

to generate a few plots interactively. The code is organized around handling the description of the synthetic background in a SyntheticModelspec, defined as

	@with_kw struct SyntheticModelspec
		d::Int = 3
		feature_at::Float64 =23.0 # e-folds before the end of inflation
		feature_fwhm::Float64=0.25
		efolds::Float64 = 80.0
		ϵ::Float64=0.01
		Mss_ratio::Float64 =-3.0 # Mss/(Ω^2+τ^2)
		Mss_const::Float64=  0.0 # H^2
		Ω::Float64=1.0
		τ::Float64=1.0
		Msb_ratio::Float64= 0.0  # Msb/τ
		Msb_const::Float64= 0.0  # H^2
		Mbb_ratio::Float64= 1.0  # Mbb/τ^2
		Mbb_const::Float64= 0.0 # H^2
		Hinit::Float64 = 1e-4
	end

The functions get_all_bogoliubov_mass(κ,sms::SyntheticModelspec), analyticPz(κ,sms::SyntheticModelspec), etc, all take a SyntheticModelspec and return the quantities of interest. Please see the examples in bog_plotting_minimal.jl. For now, all of the Bogoliubov coefficients in Julia are the branch that allows for $0 < M_0 < 9/4$. They are source-to-source translations of the Mathematica results using https://github.com/rjrosati/mathematica2julia

Alternatively, you can start a notebook-style interface with

julia> import Pluto; Pluto.run()

and selecting bog_plotting_pluto.jl in your browser.

Pluto is a little different than Jupyter notebooks or Mathematica in that it is stateless, so changing a variable in one part of the notebook changes it in all parts of the notebook, and the order of the cells doesn't matter.

See https://github.com/fonsp/Pluto.jl/wiki/%F0%9F%94%8E-Basic-Commands-in-Pluto for some basic how-to's with Pluto notebooks.

Citations

If these results are useful to you, please cite the corresponding papers :)

@article{Aragam:2023adu,
    author = "Aragam, Vikas and Paban, Sonia and Rosati, Robert",
    title = "{Primordial stochastic gravitational wave backgrounds from a sharp feature in three-field inflation. Part~I. The radiation era}",
    eprint = "2304.00065",
    archivePrefix = "arXiv",
    primaryClass = "astro-ph.CO",
    reportNumber = "UTWI-9-2023",
    doi = "10.1088/1475-7516/2023/11/014",
    journal = "JCAP",
    volume = "11",
    pages = "014",
    year = "2023"
}

@article{Aragam:2024nej,
    author = "Aragam, Vikas and Paban, Sonia and Rosati, Robert",
    title = "{Primordial Stochastic Gravitational Wave Backgrounds from a Sharp Feature in Three-field Inflation II: The Inflationary Era}",
    eprint = "2409.09023",
    archivePrefix = "arXiv",
    primaryClass = "astro-ph.CO",
    month = "9",
    year = "2024"
}