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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8" />
<title>Solving the Boltzmann Equation</title>
<!-- Bootstrap core CSS -->
<link
rel="stylesheet"
href="https://stackpath.bootstrapcdn.com/bootstrap/4.4.1/css/bootstrap.min.css"
integrity="sha384-Vkoo8x4CGsO3+Hhxv8T/Q5PaXtkKtu6ug5TOeNV6gBiFeWPGFN9MuhOf23Q9Ifjh"
crossorigin="anonymous"
/>
<script src="//cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-MML-AM_CHTML"></script>
<script src="https://cdn.plot.ly/plotly-2.9.0.min.js"></script>
<style>
.alert {
padding: 20px;
background-color: #f44336;
color: white;
opacity: 1;
transition: opacity 0.6s;
margin-bottom: 15px;
}
.alert.success {
background-color: #04aa6d;
}
.alert.info {
background-color: #2196f3;
}
.alert.warning {
background-color: #ff9800;
}
.closebtn {
margin-left: 15px;
color: white;
font-weight: bold;
float: right;
font-size: 22px;
line-height: 20px;
cursor: pointer;
transition: 0.3s;
}
.closebtn:hover {
color: black;
}
</style>
</head>
<body>
<noscript
>This page contains webassembly and javascript content, please enable
javascript in your browser.</noscript
>
<script src="./bootstrap.js"></script>
<!-- Page Content -->
<div class="container">
<div class="row">
<div class="col-lg-12 text-center">
<h2 class="mt-5 text-left">Solving the Boltzmann Equation</h2>
<p class="text-left">
This page solves the Boltzmann equation for the dark matter comoving
number density and relic density. $$ \frac{dY}{dx} =
-\sqrt{\frac{\pi}{45}}\frac{m_{\chi}M_{pl}}{x^2}g_{*}^{1/2}\langle\sigma
v\rangle \left(Y^2-Y_{\mathrm{eq}}^{2}\right) $$ where: $$
\begin{align} Y &= n_{\chi} / s, & x &= m_{\chi} / T, &
\langle\sigma v\rangle &\sim \langle\sigma v\rangle_{0} x^{-n}
\end{align} $$
</p>
<h4 class="mt-5 text-left">Glossary</h4>
<ul class="text-left">
<li>\(m_{\chi}\): Dark Matter mass</li>
<li>
\(\langle\sigma v\rangle_{0}\): Leading-order coefficient of the
thermally averaged annihilation cross section
</li>
<li>
\(n\): Leading-order power in \(x\) of the thermally averaged
annihilation cross section
</li>
<li>\(T\): Temperature of the Standard Model bath</li>
<li>
\(x\): Scaleless temperature (Dark Matter mass divided by Standard
Model temperature)
</li>
<li>
\(Y\): Comoving number density (number density \(n_{\chi}\)
divided by Standard Model entropy density \(s\))
</li>
<li>\(Y_{\mathrm{eq}}\): Equilibrium value of \(Y\)</li>
<li>
\(M_{\mathrm{pl}}\): Plank mass (taken to be \(1.220910 \times
10^{19}\))
</li>
</ul>
<p class="text-left">
Use the input boxes below to changes the parameter values.
</p>
</div>
</div>
<div class="row">
<div class="col-sm">
<div class="input-group mb-3">
<div class="input-group-prepend">
<span class="input-group-text" id="mass-addon3">
\(m_{\chi} \ [\mathrm{GeV}]\)</span
>
</div>
<input
type="text"
class="form-control"
id="mass"
aria-describedby="mass-addon3"
value="100.0"
/>
</div>
</div>
<div class="col-sm">
<div class="input-group mb-3">
<div class="input-group-prepend">
<span class="input-group-text" id="sigma-addon3"
>\(\langle\sigma v\rangle_{0} \ [\mathrm{GeV}^{-2}]\)</span
>
</div>
<input
type="text"
class="form-control"
id="sigma"
aria-describedby="sigma-addon3"
value="1e-9"
/>
</div>
</div>
<div class="col-sm">
<div class="input-group mb-3">
<div class="input-group-prepend">
<span class="input-group-text" id="n-addon3">n</span>
</div>
<input
type="text"
class="form-control"
id="n"
aria-describedby="n-addon3"
value="0"
/>
</div>
</div>
</div>
<div class="row">
<div class="col-sm">
<div class="input-group mb-3">
<div class="input-group-prepend">
<span class="input-group-text" id="x-start-addon3"
>Start \(\ x = m_{\chi}/T\)</span
>
</div>
<input
type="text"
class="form-control"
id="x-start"
aria-describedby="x-state-addon3"
value="1.0"
/>
</div>
</div>
<div class="col-sm">
<div class="input-group mb-3">
<div class="input-group-prepend">
<span class="input-group-text" id="x-end-addon3"
>End \(\ x = m_{\chi}/T\)</span
>
</div>
<input
type="text"
class="form-control"
id="x-end"
aria-describedby="x-end-addon3"
value="300"
/>
</div>
</div>
</div>
<div class="row">
<div class="col-lg-12 text-center">
<div class="input-group">
<div class="input-group-append">
<button
class="btn btn-outline-secondary"
type="button"
id="btnsolve"
>
Solve
</button>
</div>
</div>
</div>
</div>
<div class="row">
<div class="col-lg-12">
<h2 class="mt-5 text-left">Solution</h2>
</div>
</div>
<div class="row">
<div class="col-lg-12 text-center" id="wasm-support">
<script>
const supported = (() => {
try {
if (
typeof WebAssembly === "object" &&
typeof WebAssembly.instantiate === "function"
) {
const module = new WebAssembly.Module(
Uint8Array.of(0x0, 0x61, 0x73, 0x6d, 0x01, 0x00, 0x00, 0x00)
);
if (module instanceof WebAssembly.Module)
return (
new WebAssembly.Instance(module) instanceof
WebAssembly.Instance
);
}
} catch (e) {}
return false;
})();
if (!supported) {
document.getElementById("wasm-support").innerHTML = `
<div class="alert warning">
<span class="closebtn">×</span>
<strong>Warning!</strong> Indicates a warning that might need attention.
</div>`;
} else {
}
var close = document.getElementsByClassName("closebtn");
var i;
for (i = 0; i < close.length; i++) {
close[i].onclick = function () {
var div = this.parentElement;
div.style.opacity = "0";
setTimeout(function () {
div.style.display = "none";
}, 600);
};
}
</script>
</div>
</div>
<div class="row">
<div class="col-lg-12 text-center">
<div id="plotlydiv">
<!-- Plotly chart will be drawn inside this DIV -->
</div>
</div>
</div>
</div>
<!-- Footer -->
<footer
class="footer bg-dark"
style="height: 30px; line-height: 30px; width: 100%; margin-top: 60px"
>
<div class="container">
<span class="text-muted">Copyright © 2022 Logan A. Morrison</span>
</div>
</footer>
<!-- Footer -->
<!-- Bootstrap core JavaScript -->
<script
src="https://code.jquery.com/jquery-3.4.1.slim.min.js"
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</body>
</html>