{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Finding the size of the universe" ] }, { "cell_type": "markdown", "metadata": { "heading_collapsed": true }, "source": [ "## Theory" ] }, { "cell_type": "markdown", "metadata": { "heading_collapsed": true, "hidden": true }, "source": [ "### Friedman Equations" ] }, { "cell_type": "markdown", "metadata": { "hidden": true }, "source": [ "The Friedman equations describe the evolution of the scale factor R (the size of) the universe.\n", "\n", "They are,\n", "\n", "\\begin{equation}\n", " \\frac{\\ddot{R}}{R} + \\frac{4\\pi G}{3} (1 + \\epsilon) - \\frac{\\Lambda}{3} = 0\n", "\\end{equation}\n", "\n", "and\n", "\n", "\\begin{equation}\n", " \\Big(\\frac{\\dot{R}}{R}\\Big)^2 - \\frac{8\\pi G \\rho}{3} - \\frac{\\Lambda}{3} = - \\frac{k c ^ 2}{R^2}\n", "\\end{equation}\n", "\n", "Where $\\Lambda$ is the cosmological constant, $\\rho$ is the density of the universe (matter and radiation), $k$ is the curvature of the universe {where + is a closed universe, 0 is a flat and infinite universe, and - is an open universe}." ] }, { "cell_type": "markdown", "metadata": { "hidden": true }, "source": [ "It is useful to cast the second equation using the **conformal time**:\n", "\n", "\\begin{equation}\n", " \\eta (t_1) = \\int^{t_1}_{0} \\frac{c}{R(t)} dt\n", "\\end{equation}\n", "\n", "Which lets us know how far apart objects are before taking into account the scale factor (i.e. mapping to a Minkowski universe).\n", "\n", "By the chain rule,\n", "\n", "\\begin{equation}\n", " \\dot{R} = \\frac{c}{R(t)}\\frac{dR}{d\\eta}\n", "\\end{equation}\n", "\n", "Before we rewrite the Friedman equations using these building blocks, we need to find an expression for the density $\\rho$." ] }, { "cell_type": "markdown", "metadata": { "heading_collapsed": true, "hidden": true }, "source": [ "### The continuity equations" ] }, { "cell_type": "markdown", "metadata": { "hidden": true }, "source": [ "We need to find a way to link the density of the universe to the size of the universe. This isn't as straightforward as one may expect. For instance, how dense is a photon gas? How dense is a matter and photon gas? To do so we can approximate by considering the conservation of energy of a sphere undergoing adiabatic expansion,\n", "\n", "\\begin{equation}\n", " \\rho c^2 V = (\\rho + d\\rho)c^2(V + dV) + p dV\n", "\\end{equation}\n", "\n", "We can obtain the result that,\n", "\n", "\\begin{equation}\n", " \\frac{d\\rho}{\\rho} = \\Big(1 + \\frac{p}{\\rho c^2}\\Big) \\frac{dV}{V}\n", "\\end{equation}\n", "\n", "Which we can rewrite,\n", "\n", "\\begin{equation}\n", " \\frac{d\\rho}{\\rho} = \\Big(1 + \\frac{\\epsilon}{3}\\Big) \\frac{dV}{V}\n", "\\end{equation}\n", "\n", "In a matter dominated universe $\\epsilon$ is approximately zero. In a radiation dominated universe, $\\epsilon$ is approximately unity.\n", "\n", "Now, we expect that the volume of the universe varies as $V \\propto R^3$, using the above equations we can write,\n", "\n", "\\begin{equation}\n", " \\rho \\propto R^{-(3 + \\epsilon)}\n", "\\end{equation}\n" ] }, { "cell_type": "markdown", "metadata": { "heading_collapsed": true, "hidden": true }, "source": [ "### The Master Equations" ] }, { "cell_type": "markdown", "metadata": { "hidden": true }, "source": [ "Using the above, we can rewrite the first order Friedman ODE as,\n", "\n", "\\begin{equation}\n", " \\Big(\\frac{1}{a}\\frac{da}{d\\eta}\\Big)^2 - \\frac{8\\pi G \\rho_0}{3a^{1+\\epsilon}}\\frac{R^2_0}{c^2} = \\Big(\\frac{1}{a}\\frac{da}{d\\eta}\\Big)^2 - \\frac{a^{1+\\epsilon}_m}{a^{1+\\epsilon}} = -k\n", "\\end{equation}\n", "\n", "Which gives us an expression that we can integrate.\n", "\n", "$\\rho_0$ and $R_0$ are the density and scale factors as measured today." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Practice" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solving the Master Equation" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following imports have been done for you:" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false, "run_control": { "frozen": false, "read_only": false } }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/home/niall/.local/lib/python3.5/site-packages/matplotlib/backends/backend_gtk3agg.py:18: UserWarning: The Gtk3Agg backend is known to not work on Python 3.x with pycairo. Try installing cairocffi.\n", " \"The Gtk3Agg backend is known to not work on Python 3.x with pycairo. \"\n" ] } ], "source": [ "#NAME: The Size of the Universe\n", "#DESCRIPTION: Determining and plotting the size of the Universe against time.\n", "\n", "import numpy as np\n", "from scipy.integrate import odeint\n", "\n", "import matplotlib\n", "from matplotlib import pyplot as plt\n", "%matplotlib notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Write a function *cosmo_master_eqn* which returns the value of $\\frac{da}{d\\eta}$, given a value of a, a linspace t, $am$, $\\epsilon$, and $k$ (in that order)." ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": true, "run_control": { "frozen": false, "read_only": false } }, "outputs": [], "source": [ "# May be worth setting keyword args\n", "def cosmo_master_eqn(y, eta, am, epsilon, k):\n", " a = y\n", " dydt = (a ** (1 - epsilon) * (am) ** (1 + epsilon) - k*a**2) ** 0.5\n", " return dydt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, set the initial values to pass to the integrator. Make the universe {flat|open|closed}." ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": true, "run_control": { "frozen": false, "read_only": false } }, "outputs": [], "source": [ "# Initial conditions\n", "rho_0 = 7.85846e-31\n", "c = 3 * 10 ** 8\n", "G = 6.67 * 10 ** - 11\n", "epsilon = 0\n", "am = 0.04 ** (1/(1+epsilon))\n", "k = -1\n", "y0 = [0]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Create a *linspace* of the conformal time coordinates." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": true, "run_control": { "frozen": false, "read_only": false } }, "outputs": [], "source": [ "start_eta = 0.0\n", "end_eta = np.pi\n", "steps = 100000\n", "eta = np.linspace(start_eta, end_eta, steps)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Use *odeint* to solve the *cosmo_master_eqn*." ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "collapsed": false, "run_control": { "frozen": false, "read_only": false } }, "outputs": [], "source": [ "solution = odeint(cosmo_master_eqn, y0, eta, args=(am, epsilon, k))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Plot the solution!" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "collapsed": false, "run_control": { "frozen": false, "read_only": false } }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support.' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. 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