Homework3_nathennguyen.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Problem 1\n",
    "(sorry, I've been meaning to learn Latex)\n",
    "\n",
    "Friedmann Eq\n",
    "[a'(t)/a(t)]^2 = 8*pi*G*eps(t)/(3c^2) - k*c^2/a(t)^2 \n",
    "eps being epsilon, energy density \n",
    "\n",
    "Rearrange\n",
    "a'(t)^2 = [8*pi*G*eps(t)/(3c^2)]*a(t)^2 - k*c^2\n",
    "\n",
    "Take the derivative with respect to time\n",
    "2*a'*a\" = [8*pi*G*eps(t)/(3c^2)]* [eps'*a^2 + 2*eps*a*a'] \n",
    "\n",
    "divide through by 2*a*a'\n",
    "a\"/a = [4*pi*G*eps(t)/(3c^2)]*[eps'*(a/a') + 2eps]  This equation is {A} \n",
    "\n",
    "\n",
    "From the fluid eq. \n",
    "eps'(t) + [3*a'(t)/a(t)]*[eps+P] = 0\n",
    "\n",
    "we get\n",
    "eps'(t) = -[3*a'(t)/a(t)]*[eps+P]\n",
    "\n",
    "further rearranging \n",
    "eps'(t)*[a(t)/a'(t)] = -3[eps+P]\n",
    "\n",
    "\n",
    "We can plug this relation into {A}\n",
    "a\"/a = [4*pi*G*eps(t)/(3c^2)]*[{-3eps-3P} + 2eps]\n",
    "\n",
    "which yeilds \n",
    "a\"/a = [4*pi*G*eps(t)/(3c^2)]*[-eps - 3P]\n",
    "\n",
    "Finnaly, we get the 3rd equation\n",
    "a\"/a = -[4*pi*G*eps(t)/(3c^2)]*[eps + 3P]\n",
    "Thus we have shown that this equation, the acceleartion of the universe, is derived from the Friedman and Fluid Equation "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Problem 2\n",
    "Both a & b concern a flat universe, thus (1-Omega0) goes to zero. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.integrate import quad\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "For proper distance we have the integral \n",
    "r = Integral[(c*dt)/a] = Integral{0 to z}[(c*dz)/H(z)]\n",
    "\n",
    "We are given that \n",
    "(H(z)/H0)^2 = Omega{r}(1+z)^4 + Omega{m}(1+z)^3 + Omega{lambda} + (1-Omega{o}(1+z)^2 \n",
    "\n",
    "We know that the universe is flat for this problem, so the Omega for curviture goes to 0, and in problem a Omega of radiation is very small, so we can have that go to zero.\n",
    "Solving for H(z) we get \n",
    "H(z) = H0*sqrt[Omega{m}(1+z)^3 + Omega{lambda}]\n",
    "\n",
    "Plugging into the integral for comoving distance we get \n",
    "\n",
    "r = integral{0 to z}[(c*dz)/(H0*sqrt[Omega{m}(1+z)^3 + Omega{lambda}])] \n",
    "\n",
    "For a we are given that \n",
    "Omega{m} = 0.3\n",
    "Omega{lambda} = 0.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 122,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[   0.           42.28160742   82.50190391  120.6194699   156.64270868\n",
      "  190.61963202  222.62720439  252.76162204  281.13022204  307.84520738\n",
      "  333.0190667   356.76142266  379.17700491  400.36446401  420.41578812\n",
      "  439.41613491  457.44393754  474.57118246  490.86378666  506.3820249\n",
      "  521.18097368  535.31095069  548.81793655  561.74397134  574.12752206\n",
      "  586.00381982  597.40516702  608.36121576  618.89921923  629.04425819\n",
      "  638.81944469  648.24610505  657.34394425  666.13119354  674.62474307\n",
      "  682.84026102  690.79230081  698.49439755  705.95915501  713.19832396\n",
      "  720.22287299  727.04305249  733.66845256  740.10805546  746.37028329\n",
      "  752.46304116  758.39375657  764.16941525  769.79659377  775.28148941\n",
      "  780.6299474   785.84748585  790.93931863  795.91037629  800.76532538\n",
      "  805.50858608  810.14434851  814.67658778  819.10907778  823.445404\n",
      "  827.68897531  831.84303494  835.91067056  839.89482369  843.79829842\n",
      "  847.62376945  851.37378968  855.05079714  858.65712155  862.19499037\n",
      "  865.66653448  869.07379345  872.41872049  875.7031871   878.92898733\n",
      "  882.09784188  885.21140188  888.27125245  891.27891604  894.23585555\n",
      "  897.14347731  900.00313383  902.81612642  905.58370762  908.30708356\n",
      "  910.9874161   913.62582492  916.22338945  918.7811507   921.30011299\n",
      "  923.78124562  926.2254844   928.63373311  931.00686491  933.34572364\n",
      "  935.6511251   937.92385819  940.1646861   942.37434728  944.55355658]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fbf21943780>"
      ]
     },
     "execution_count": 122,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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BBQUF1QxRSk2YEJqjevWCkSNhR83QEclqhYWFFBYWVvv+RNaeMrMJwMXuXmxmA4H60be+\ncPdBZtYfaODu/aOO8GFAJ0Kz1HjgwC07MNSnkVolJTBoENx9NwweDCeemHREIpIOVe3TqGw/jTGE\nGkVFD3N3P60a8ZXqCzxlZjsSajEXAXWAEdHIrAVAz+iNisxsBFAEbAIuV3ZIr5Ur4cILYfVq+Ogj\naNIk6YhEJFNstaZhZisITUnDCUNuoSyBuLu/lf7w4lNNIzU++gjOPjssMnjLLdp2VSTXVbWmUVnS\n2B7oDvQCWgMvAcPdfVYqAk01JY2a+9e/oH//MJT2rLOSjkZEakPKksYWD92JkDzuIMyRuLf6IaaH\nkkb1ffcdXHUVvPUWPPcctGiRdEQiUltS1qcRPawuYRn084D9CDO5n69JgJJZli8PzVENG8LEiZrZ\nLSKV2+rkPjN7EngPaAf81d07uvvN7l7hcFfJPlOnQqdO0K0bPP+8EoaIbFtlfRolwNqt3OfunlG/\nYtQ8VTUvvAAXXwz33Qc9eyYdjYgkJWXNU+6uRQFzkDvceWcoL78MHTsmHZGIZJM4M8IlR2zcCH37\nwnvvheVAfv7zpCMSkWxTWZ/G5G3dHOcayQxr1sDpp8PChfDOO0oYIlI9ldU0WpjZjG3cv3sqg5H0\nWL4cTj45bJj04IOasCci1Vdp0ohx/6ZUBSLp8ckn0KNHWHDwL38Bi93dJSLyY4ksWJgOGj31Y9Om\nwUknwY03wmWXJR2NiGSilE7uk+z13ntw5plw771hHSkRkVRQ0shBY8fCBRfA0KFwwglJRyMiuSTW\nXAwz28/MjouO65tZRk3skzKjRoVlzUeNUsIQkdTbZtIws0uBZ4CHolNN0PpTGemZZ0LfxSuvQJcu\nSUcjIrkoTk3jCuBIYBWAuxcDP0tnUFJ1w4aFlWrHjoUOHZKORkRyVZw+jW/d/VuLxmpG+2xomFIG\nGToUbrgBxo+HVq2SjkZEclmcmsZbZvZfQH0z605oqhqT3rAkrqefhuuvV8IQkdqxzXkaZlYH6AMc\nH50aCzyaaZMi8nGexsiRcMUVMG4ctG6ddDQiko1SvnOfme0MbHD3zdHrOsBO7r6uRpGmWL4ljdGj\n4ZJLQh9G27ZJRyMi2aqqSSNO89QbQL1yr+sD46samKTO66+HvTBefFEJQ0RqV5yksZO7ryl94e6r\nCYlDEjBxIpx3Xhheq70wRKS2xUkaa83s+0GcZnYosD59IcnWzJoVljcfPDhs0SoiUtviDLm9Ghhh\nZv+JXu8NnJu+kKQiCxaE1Wr/8Q845ZSkoxGRfBVrlVsz2xE4iDA/Y667b0x3YFWVyx3hX34ZZnhf\ndhn065d0NCKSS1I+eip6aGdgf0LNxAHcfUh1g0yHXE0aGzZA9+5w2GFwxx1JRyMiuSYdQ26HAgcA\nU4HNpefdvW91g0yHXEwaJSXQs2fYae+pp2C7WMtLiojEl479NDoALXPuN3IWuPZaWLkyzMVQwhCR\nTBAnacwkdH4vSXMsUs7DD8NLL8EHH8BOOyUdjYhIECdp/BQoMrNJwLfROXf309IXVn574w3485/h\n7behYcOkoxERKRMnaQxMdxBSprgYevWCf/8bmjVLOhoRkR+KNXoqG+RCR/jXX4dRUtddF5YJERFJ\nt5SvPWVmR5jZh2a2xsw2mlmJma2qWZiypZISOP/8sEWrEoaIZKo4Y3LuBX4NzAPqEpZJvz+dQeWj\ngQNhzZow41tEJFPFGsjp7vOAOu6+2d0HAz3SG1Z+GTUKHn8cRowIczJERDJVnI7wtWa2EzDNzG4H\nlgKx27+kcnPmwKWXhuG1jRolHY2ISOXi1DQujK67ElgHNAHOqukbm1kdM5tiZmOi13uY2TgzKzaz\n18ysQblrB5jZPDObY2bHb/2p2WXdOjj7bPj737XMuYhkhzhJ4wx3X+/u37j7QHf/E3ByCt67H1BE\ntJYV0B8Y5+7Ngdej15hZS8Kqui0JzWL3m1lOzI++4gpo104d3yKSPeL88v1tBecuqsmbmlkT4CTg\nUcqauk4DnoiOnwDOiI5PB4a7+0Z3XwDMBzrV5P0zweDBYUOlBx4AU2OfiGSJrfZpmFkvwqip/Uub\nkCK7Al/U8H3vAq4Ddit3rpG7L4uOlwGlLfz7AB+Uu24R0LiG75+omTPh+uuhsBB22SXpaERE4qus\nI/w94D+EZUTuoKxGsAqYXt03NLNTgOXuPsXMCiq6xt3dzCqbqZe1s/jWrQsr195xB7RqlXQ0IiJV\ns9Wk4e4LgYVmdhyw3t03m9lBhM2YZtTgPTsDp5nZSYR5H7uZ2ZPAMjPby92XmtnewPLo+sVA03L3\nN4nO/cjAgQO/Py4oKKCgoKAGYabHNdeEfozevZOORETyUWFhIYWFhdW+P85+Gh8DXYGGwLvAh8B3\n7n5+td+17NndgGvd/dRoOO8X7j7IzPoDDdy9f9QRPozQj9EYGA8cuOWaIdmwjMgLL8DVV8PUqbD7\n7klHIyKSnv00tnP3dWbWB7jf3W83s2nVD/FHSn/T30bYi7wPsADoCeDuRWY2gjDSahNwecZnhwos\nWQK//z2MHKmEISLZK05NYwpwOaHzuo+7zzKzGe7eujYCjCuTaxolJdCjR9jn+6abko5GRKRMyhcs\nBK4GBgDPRwnjF8Cb1Q0wH913H6xeDf/1X0lHIiJSM1oaPc3mzYMjjoD33oPmzZOORkTkh1LWp2Fm\nd7t7vy3maJTSzn0xbN4Mv/0t3HijEoaI5IbKOsKHRF8rWqw78/6kz0B33RVWre3bN+lIRERSI1bz\nlJn9FMDdV6Q9omrKtOapoiI46iiYNAkOOCDpaEREKpayjnALBprZSqAYKDazlWam8T/bsHkz9OkD\nN9+shCEiuaWy0VN/BLoAHd29obs3JEyw62Jmf6qV6LLUAw/A9tuHeRkiIrlkq81TZjYV6L5lk1TU\nVDXO3dvWQnyxZUrz1OefQ/v2MGECtGiRdDQiIpVL5TyN7Svqw4jOxZlJnnfc4fLLQ8e3EoaI5KLK\nfvlvrOb38tYzz8Cnn4alQkREclFlzVObCdu7VqSeu2dUbSPp5qlvvgm1i2efhc6dEwtDRKRKqto8\npRnhKXL11bB2LTzySGIhiIhUWTpWuZVtmDYNhg+HWbOSjkREJL3iLFgolXCHK64IczL23DPpaERE\n0ktJo4aGDIHvvguT+UREcp36NGrg669D5/fo0dCxY62+tYhISqgjvBZdcw2sWqXObxHJXkoataR0\nn4xZs6BRo1p7WxGRlErHzn1Sgeuvh+uuU8IQkfyiIbfV8OabZcNsRUTyiWoaVbR5M/zxjzBoENSt\nm3Q0IiK1S0mjip54AnbdFc4+O+lIRERqnzrCq2DdurDX93PPQadOaX0rEZFaoY7wNLrnHjj8cCUM\nEclfqmnE9NVXoZbxzjtw0EFpexsRkVqlmkaaDBoEZ56phCEi+U01jRgWL4Y2bWD6dGjcOC1vISKS\nCM0IT4NLL4WGDUNtQ0QklyhppNj8+aHze968kDhERHKJ+jRS7G9/g759lTBEREDLiFRq3jx46aVQ\n2xAREdU0KnXzzXDVVbD77klHIiKSGdSnsRXFxdClS6hlKGmISK5Sn0aK3Hwz9OunhCEiUp5qGhWY\nOxeOPBI++QR22y0ljxQRyUiqaaTArbeGvgwlDBGRH6r1pGFmTc3sTTObZWYzzeyq6PweZjbOzIrN\n7DUza1DungFmNs/M5pjZ8emM77PPYMwYuPLKdL6LiEh2qvXmKTPbC9jL3aea2S7Ax8AZwEXASne/\n3cxuABq6e38zawkMAzoCjYHxQHN3L9niuSlpnrrqKqhXT7O/RSQ/VLV5qtbnabj7UmBpdLzGzGYT\nksFpQLfosieAQqA/cDow3N03AgvMbD7QCfgg1bEtXw5Dh0JRUaqfLCKSGxLt0zCz/YB2wESgkbsv\ni761DGgUHe8DLCp32yJCkkm5u++Gc8+FvfZKx9NFRLJfYjPCo6apkUA/d19tVlY7cnc3s8ramir8\n3sCBA78/LigooKCgIHY8q1bBQw/BpEmxbxERyTqFhYUUFhZW+/5Ehtya2Q7Ai8Ar7v7P6NwcoMDd\nl5rZ3sCb7n6wmfUHcPfbouteBW5y94lbPLNGfRqDBsGMGaF5SkQkX2T8kFsLVYp/AUWlCSMyGugd\nHfcGRpU7f56Z7Whm+wPNgJTWB779NjRN3XBDKp8qIpJ7kmie6gJcAEw3synRuQHAbcAIM+sDLAB6\nArh7kZmNAIqATcDlqV4D/emnoXXrUEREZOvyfka4O7RtC7ffDieckIbAREQyWMY3T2WaN96ATZvg\n+LROGRQRyQ15nzTuvBP+9Cew2HlWRCR/5XXz1OzZcPTRsGAB1K2bnrhERDKZmqeq4K674A9/UMIQ\nEYkrb2saK1ZA8+ZhGfSf/SyNgYmIZDDVNGJ69FE480wlDBGRqsjLmsamTfCLX8Dzz0P79mkOTEQk\ng6mmEcOLL0LjxkoYIiJVlZdJ4957tcmSiEh15F3zVOkw24ULYaedaiEwEZEMpuapbbj/frjkEiUM\nEZHqyKuaxurVsO++MH06NGlSS4GJiGQw1TQq8eSTcMwxShgiItWVN0nDHR58MMwAFxGR6smbpDFp\nEqxbFzrBRUSkevImaTz8cOgA3y5vfmIRkdTLi47wVatCB/icOdCoUS0HJiKSwdQRXoFhw+C445Qw\nRERqKueThjs89FBomhIRkZrJ+aTx8cfw9dehpiEiIjWT80lDHeAiIqmT0x3ha9eGiXyzZsE++yQU\nmIhIBlNHeDkjR0KXLkoYIiKpktNJY/BguOiipKMQEckdOds89emn0KkTLFqkFW1FRLZGzVORIUOg\nVy8lDBGRVMrJmkZJSdgDfORIbekqIlIZ1TSACRNgt92gXbukIxERyS05mTQGD4bf/hYsdu4UEZE4\ncq55avVqaNoUiovhZz9LOioRkcyW981TI0dCt25KGCIi6ZBzSWPoULjwwqSjEBHJTTnVPLVokdO6\nNSxZAnXrJh2RiEjmy+vmqaefhjPPVMIQEUmXnEoaQ4fCBRckHYWISO7KmqRhZj3MbI6ZzTOzGyq6\nZuXK0AkuIiLpkRVJw8zqAPcCPYCWQC8za7Hldb16ad8MgMLCwqRDyBj6LMrosyijz6L6suVXbCdg\nvrsvcPeNwNPA6VtepKapQP8hyuizKKPPoow+i+rLlqTRGPi83OtF0bkfaNOm1uIREclL2ZI0cmNc\nsIhIlsuKeRpmdjgw0N17RK8HACXuPqjcNZn/g4iIZKCqzNPIlqSxPTAXOBZYAkwCern77EQDExHJ\nM9snHUAc7r7JzK4ExgJ1gH8pYYiI1L6sqGmIiEhmyJaO8K2KM+kvH5hZUzN708xmmdlMM7sq6ZiS\nZmZ1zGyKmY1JOpYkmVkDM3vWzGabWVHUR5iXzGxA9H9khpkNM7O82RDazB4zs2VmNqPcuT3MbJyZ\nFZvZa2bWYFvPyeqkEXfSX57YCPzR3VsBhwNX5PFnUaofUIRG390NvOzuLYA2QF427ZrZfsAlQHt3\nb01o6j4vyZhq2WDC78ry+gPj3L058Hr0ulJZnTSIOekvH7j7UnefGh2vIfxi2CfZqJJjZk2Ak4BH\ngbzdw9HMdge6uvtjEPoH3f2bhMNKyirCH1f1o8E19YHFyYZUe9z9beCrLU6fBjwRHT8BnLGt52R7\n0og16S/fRH9RtQMmJhtJou4CrgNKkg4kYfsDK8xssJlNNrNHzKx+0kElwd2/BP4BfEYYhfm1u49P\nNqrENXL3ZdHxMqDRtm7I9qSR780OP2JmuwDPAv2iGkfeMbNTgOXuPoU8rmVEtgfaA/e7e3tgLTGa\nIHKRmf0CuBrYj1AL38XMzk80qAziYVTUNn+nZnvSWAw0Lfe6KaG2kZfMbAdgJDDU3UclHU+COgOn\nmdmnwHDgGDMbknBMSVkELHL3D6PXzxKSSD46FHjP3b9w903Ac4R/K/lsmZntBWBmewPLt3VDtieN\nj4BmZrafme0InAuMTjimRJiZAf8Citz9n0nHkyR3/3/u3tTd9yd0dL7h7r9JOq4kuPtS4HMzax6d\nOg6YlWBISZoDHG5m9aL/L8cRBkrks9FA7+i4N7DNPzazYnLf1mjS3w90AS4AppvZlOjcAHd/NcGY\nMkW+N2PvhngBAAADO0lEQVT2BZ6K/rD6BLgo4XgS4e7TohrnR4S+rsnAw8lGVXvMbDjQDdjTzD4H\n/gzcBowwsz7AAqDnNp+jyX0iIhJXtjdPiYhILVLSEBGR2JQ0REQkNiUNERGJTUlDRERiU9IQEZHY\nlDRERCQ2JQ3JO2a2OdpnY7qZPRet11WV+ytd02tr3zez35vZhdHxwWY21cw+NrMDzKzXVu4pMLNv\nzOzF6PUVUeylZYaZlZjZQWbWNdovY0ZFzxJJBSUNyUfr3L2du7chLJf9+yrev60ZsRV+390fcvcn\no5dnAM+4ewfg58CvK3neBHc/JXrGfVHs7dy9HTCGsNbY3Gjp6xOr9JOIVFFWLyMikgLvA4fA96ug\n3gv8FFgHXOLuc81sf2AYsDPl1jaLFnj7N7Ar4f/SZe7+bvS9vwGnAOuB0919uZkNBFYT1jvqB2w2\ns2OBekCLaPmXx9397jiBm9lRwDmEZfC/P12dD0EkLtU0JG9FOz8eD8yMTj0M9HX3Qwl7cdwfnb8b\nuC+qmSwp94hfA69Gf/EfAkyLzu8MvO/ubYEJhN3iINRA3N1fAR4E7nT3YwhLlb8d1R7iJowGhJ3Y\nfpOvS+BLMlTTkHxUL/qrvjFhkbYHo36NI4BnwgKoAOwYfe0MnBkdDwUGRceTgMeiJelHuXtp0vjO\n3V+Kjj8Gupd7b6vguDq1gweBIe7+fjXuFak21TQkH62Pagf7AhsIWwQbYSe3duVKq8oeEvUhdCXs\n6/J4aSc3YUvRUiWk+I8zM+tN2Dvm5lQ+VyQOJQ3JW+6+HrgK+DuwBvjUzM6GsD+JmbWJLn2XsC8H\nwPc7vZnZz4EV7v4oYS+T8n0LFamolgGhn2PXODGb2QFRvBe4e75vZSsJUNKQfPT96CZ3nwrMJ+wj\ncD7Qx8ymEvo5Tosu6wdcYWbTCduElt5/NDDVzCYTOqRL+yPKj54qv4Xm1o6nETrFp5pZv23Efj2h\n4/y5LYbedon3o4vUjPbTEMlgZlYAXOPup8a8fj9gjLu3TmNYksdU0xDJbN8Cvyyd3FcZM+tKGBK8\nIu1RSd5STUNERGJTTUNERGJT0hARkdiUNEREJDYlDRERiU1JQ0REYvv/6iBw8KcaOR4AAAAASUVO\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbf21cd4fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "H0= 2.2683e-18 #seconds\n",
    "H0gyr= 0.07154 #gigayears\n",
    "c = 30000000 #m*s^-1\n",
    "\n",
    "def integrand(z):\n",
    "    return (c/(H0*(0.3*(1+z)**3 + 0.7)**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "values = [] \n",
    "a = np.linspace(0,10,100) #redshift runs from 0 to 10 redshift\n",
    "for i in a: \n",
    "    intergral0,err = quad(integrand,0,i)\n",
    "    \n",
    "    values.append(intergral0)\n",
    "    \n",
    "values=np.asarray(values)*3.24e-23 #convert meters to Mpc\n",
    "\n",
    "\n",
    "print (values)\n",
    "plt.plot(a,values)\n",
    "plt.ylabel('Distance [Mpc]')\n",
    "plt.xlabel('Redshift [Z]')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Then to solve for time, we can use the integral\n",
    "Integral{0 to z}[(c*dz)/H(z)]\n",
    "\n",
    "rearranging the integral with dz/dt = H(z)/a(t) we can get \n",
    "{Delta}*t= Integral{0 to z} [dz/((1+z)*H0*sqrt[Omega{m}(1+z)^3 + Omega{lambda}]]\n",
    "\n",
    "{Delta}t can also be thought of as the lookback age\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.           0.27610248   0.54435023   0.80501617   1.05836074\n",
      "   1.30463285   1.54407061   1.77690209   2.00334593   2.22361195\n",
      "   2.43790168   2.64640879   2.84931958   3.04681334   3.23906272\n",
      "   3.42623405   3.60848767   3.78597819   3.95885473   4.1272612\n",
      "   4.29133649   4.45121471   4.60702534   4.75889344   4.90693984\n",
      "   5.05128125   5.19203047   5.32929649   5.46318465   5.59379677\n",
      "   5.72123129   5.84558335   5.96694494   6.08540501   6.20104955\n",
      "   6.31396171   6.42422192   6.53190792   6.63709493   6.73985567\n",
      "   6.84026051   6.93837747   7.03427238   7.1280089    7.21964864\n",
      "   7.30925118   7.39687417   7.48257341   7.56640286   7.64841478\n",
      "   7.7286597    7.80718657   7.88404273   7.95927404   8.03292487\n",
      "   8.10503819   8.17565559   8.24481738   8.31256255   8.37892889\n",
      "   8.44395301   8.50767034   8.57011524   8.63132098   8.69131981\n",
      "   8.75014297   8.80782076   8.86438253   8.91985675   8.97427099\n",
      "   9.02765202   9.08002577   9.13141743   9.18185138   9.23135131\n",
      "   9.2799402    9.32764032   9.37447332   9.42046019   9.4656213\n",
      "   9.50997645   9.55354484   9.59634512   9.63839542   9.67971331\n",
      "   9.7203159    9.7602198    9.79944113   9.83799557   9.87589837\n",
      "   9.91316433   9.94980786   9.98584296  10.02128323  10.05614193\n",
      "  10.09043192  10.12416573  10.15735555  10.19001323  10.22215032\n",
      "  10.25377805  10.28490734  10.31554883  10.3457129   10.37540961\n",
      "  10.40464881  10.43344005  10.46179265  10.4897157   10.51721804\n",
      "  10.54430829  10.57099486  10.59728592  10.62318946  10.64871327\n",
      "  10.67386492  10.69865181  10.72308117  10.74716002  10.77089523\n",
      "  10.79429349  10.81736134  10.84010515  10.86253114  10.88464539\n",
      "  10.90645382  10.92796222  10.94917623  10.97010138  10.99074304\n",
      "  11.01110648  11.03119684  11.05101913  11.07057826  11.08987902\n",
      "  11.10892608  11.12772404  11.14627735  11.16459038  11.18266742\n",
      "  11.20051263  11.2181301   11.23552383  11.25269771  11.26965557\n",
      "  11.28640114  11.30293809  11.31926997  11.3354003   11.35133249\n",
      "  11.3670699   11.3826158   11.39797341  11.41314585  11.42813622\n",
      "  11.44294751  11.45758267  11.4720446   11.48633612  11.50045999\n",
      "  11.51441894  11.5282156   11.5418526   11.55533247  11.56865771\n",
      "  11.58183078  11.59485406  11.60772992  11.62046065  11.63304852\n",
      "  11.64549573  11.65780446  11.66997684  11.68201496  11.69392085\n",
      "  11.70569653  11.71734397  11.72886509  11.74026179  11.75153593\n",
      "  11.76268933  11.77372378  11.78464103  11.79544281  11.8061308\n",
      "  11.81670667  11.82717203  11.8375285   11.84777763  11.85792097\n",
      "  11.86796003  11.87789629  11.88773121  11.89746623  11.90710274\n",
      "  11.91664213  11.92608575  11.93543494  11.944691    11.95385523\n",
      "  11.96292887  11.97191318  11.98080937  11.98961864  11.99834216\n",
      "  12.0069811   12.01553659  12.02400975  12.03240168  12.04071346\n",
      "  12.04894614  12.05710078  12.06517839  12.07318     12.08110658\n",
      "  12.08895912  12.09673858  12.1044459   12.11208201  12.11964781\n",
      "  12.12714422  12.13457211  12.14193235  12.14922581  12.1564533\n",
      "  12.16361568  12.17071375  12.17774831  12.18472016  12.19163006\n",
      "  12.19847879  12.20526709  12.21199571  12.21866537  12.2252768\n",
      "  12.2318307   12.23832776  12.24476866  12.2511541   12.25748472\n",
      "  12.26376118  12.26998412  12.27615419  12.282272    12.28833816\n",
      "  12.2943533   12.30031799  12.30623283  12.3120984   12.31791527\n",
      "  12.32368401  12.32940515  12.33507926  12.34070687  12.3462885\n",
      "  12.35182469  12.35731594  12.36276277  12.36816566  12.37352512\n",
      "  12.37884164  12.38411568  12.38934772  12.39453824  12.39968767\n",
      "  12.40479649  12.40986513  12.41489403  12.41988362  12.42483435\n",
      "  12.42974661  12.43462084  12.43945744  12.44425681  12.44901936\n",
      "  12.45374548  12.45843555  12.46308995  12.46770908  12.47229329\n",
      "  12.47684295  12.48135843  12.48584009  12.49028827  12.49470333\n",
      "  12.49908561  12.50343545  12.50775317  12.51203912  12.51629362\n",
      "  12.52051698  12.52470954  12.52887159  12.53300345  12.53710542\n",
      "  12.54117781  12.5452209   12.54923501  12.5532204   12.55717738\n",
      "  12.56110622  12.5650072   12.5688806   12.57272669  12.57654573\n",
      "  12.58033799  12.58410374  12.58784323  12.59155671  12.59524444\n",
      "  12.59890667  12.60254364  12.60615559  12.60974278  12.61330542\n",
      "  12.61684376  12.62035802  12.62384844  12.62731524  12.63075864\n",
      "  12.63417887  12.63757614  12.64095066  12.64430265  12.64763231\n",
      "  12.65093986  12.6542255   12.65748943  12.66073186  12.66395297\n",
      "  12.66715296  12.67033204  12.67349038  12.67662817  12.67974561\n",
      "  12.68284287  12.68592015  12.68897761  12.69201544  12.69503381\n",
      "  12.69803289  12.70101286  12.70397389  12.70691615  12.7098398   12.712745\n",
      "  12.71563192  12.71850071  12.72135155  12.72418457  12.72699995\n",
      "  12.72979782  12.73257835  12.73534169  12.73808797  12.74081735\n",
      "  12.74352998  12.74622599  12.74890554  12.75156875  12.75421577\n",
      "  12.75684673  12.75946178  12.76206104  12.76464464  12.76721273\n",
      "  12.76976543  12.77230287  12.77482517  12.77733246  12.77982487\n",
      "  12.78230252  12.78476553  12.78721402  12.78964811  12.79206793\n",
      "  12.79447358  12.79686518  12.79924284  12.80160669  12.80395683\n",
      "  12.80629337  12.80861642  12.81092609  12.81322249  12.81550573\n",
      "  12.81777591  12.82003313  12.8222775   12.82450911  12.82672808\n",
      "  12.8289345   12.83112847  12.83331009  12.83547945  12.83763665\n",
      "  12.83978179  12.84191496  12.84403625  12.84614575  12.84824357\n",
      "  12.85032978  12.85240447  12.85446775  12.85651968  12.85856037\n",
      "  12.86058989  12.86260834  12.86461579  12.86661233  12.86859804\n",
      "  12.870573    12.8725373   12.87449102  12.87643423  12.87836702\n",
      "  12.88028946  12.88220162  12.8841036   12.88599545  12.88787727\n",
      "  12.88974911  12.89161106  12.89346319  12.89530557  12.89713827\n",
      "  12.89896136  12.90077492  12.902579    12.9043737   12.90615906\n",
      "  12.90793516  12.90970206  12.91145984  12.91320855  12.91494827\n",
      "  12.91667905  12.91840097  12.92011408  12.92181844  12.92351413\n",
      "  12.9252012   12.92687971  12.92854972  12.9302113   12.9318645\n",
      "  12.93350938  12.935146    12.93677442  12.9383947   12.94000689\n",
      "  12.94161104  12.94320722  12.94479549  12.94637588  12.94794847\n",
      "  12.9495133   12.95107042  12.95261989  12.95416177  12.9556961\n",
      "  12.95722294  12.95874233  12.96025433  12.96175898  12.96325635\n",
      "  12.96474647  12.96622939  12.96770518  12.96917386  12.97063549\n",
      "  12.97209012  12.97353779  12.97497856  12.97641246  12.97783954\n",
      "  12.97925985  12.98067343  12.98208033  12.98348059  12.98487425\n",
      "  12.98626136  12.98764197  12.9890161   12.99038382  12.99174515\n",
      "  12.99310014  12.99444883  12.99579126  12.99712747  12.99845751\n",
      "  12.99978141  13.00109921  13.00241095  13.00371668  13.00501642\n",
      "  13.00631021  13.00759811  13.00888013  13.01015632]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fbf21cc72b0>"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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Cb7gsNrO/mtm+ecS4EDjSzDpHE8ycCKSyb82oUfDjH6vwF5HylEsbwB3AIuDBaPsCoB8w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\nkQr1/9D3Lpdjby9SAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbf22070e80>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def timefunc1(z):\n",
    "    return 1/((1+z)*(H0gyr*(0.3*(1+z)**3 + 0.7)**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "valuest1 = [] \n",
    "a = np.linspace(0,10,500)\n",
    "for i in a:\n",
    "    intergral1,err = quad(timefunc1,0,i)\n",
    "    valuest1.append(intergral1)\n",
    "    \n",
    "valuest1=np.asarray(valuest1)\n",
    "\n",
    "plt.plot(a,valuest1)\n",
    "\n",
    "print (valuest1)\n",
    "plt.ylabel('Look Back Age [Gyr]')\n",
    "plt.xlabel('Redshift [Z]')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Part B\n",
    "\n",
    "e are given that \n",
    "(H(z)/H0)^2 = Omega{r}(1+z)^4 + Omega{m}(1+z)^3 + Omega{lambda} + (1-Omega{o}(1+z)^2 \n",
    "\n",
    "In part B, Omega{r} does not disappear, so our equation changes a bit\n",
    "H(z) = H0*sqrt[Omega{r}(1+z)^4 Omega{m}(1+z)^3 ]\n",
    "Omega{lambda} is zero, so it disappears, but we pick up a (1+z)^4 along with the Omega{r}\n",
    "\n",
    "Omega{r} = Omega{m} = 0.5 \n",
    "\n",
    "Omega{lambda} = 0 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[   0.           39.77476657   73.59941573  102.73328684  128.10039504\n",
      "  150.39486354  170.14871569  187.77688889  203.60796234  217.90564346\n",
      "  230.88411339  242.71919381  253.55660964  263.51819551  272.70662102\n",
      "  281.20903296  289.09989422  296.44321959  303.29435298  309.70139281\n",
      "  315.70634419  321.34605738  326.65299753  331.65588009  336.3801985\n",
      "  340.84866511  345.08158128  349.0971499   352.91174038  356.54011432\n",
      "  359.9956187   363.2903517   366.43530575  369.44049126  372.31504417\n",
      "  375.06731953  377.70497334  380.23503433  382.66396698  384.99772717\n",
      "  387.24181127  389.40129971  391.48089565  393.48495943  395.41753924\n",
      "  397.28239868  399.08304129  400.82273273  402.50452059  404.13125238\n",
      "  405.70559166  407.23003271  408.70691375  410.13842898  411.52663952\n",
      "  412.87348338  414.18078448  415.45026095  416.68353273  417.88212843\n",
      "  419.04749176  420.18098726  421.28390574  422.35746916  423.40283514\n",
      "  424.42110121  425.41330859  426.3804458   427.32345195  428.2432198\n",
      "  429.14059853  430.01639647  430.87138343  431.70629303  432.52182477\n",
      "  433.31864603  434.09739384  434.85867665  435.60307587  436.33114735\n",
      "  437.04342282  437.74041113  438.42259953  439.09045472  439.74442401\n",
      "  440.38493625  441.01240283  441.62721851  442.22976228  442.82039812\n",
      "  443.39947578  443.96733141  444.52428827  445.07065728  445.60673767\n",
      "  446.13281749  446.64917412  447.15607478  447.65377698  448.14252896]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fbf21a19240>"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbf21c3b550>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def integrandb(z):\n",
    "    return (c/(H0*((0.50*(1+z)**4) + (0.50*(1+z)**3) )**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "valuesb = [] \n",
    "a = np.linspace(0,10,100)\n",
    "for i in a: \n",
    "    intergralb,err = quad(integrandb,0,i)\n",
    "    \n",
    "    valuesb.append(intergralb)\n",
    "    \n",
    "valuesb=np.asarray(valuesb)*3.24e-23 #convert meters to Mpc\n",
    "\n",
    "\n",
    "print (valuesb)\n",
    "plt.plot(a,valuesb)\n",
    "plt.ylabel('Distance [Mpc]')\n",
    "plt.xlabel('Redshift [Z]')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.          0.27259192  0.53083255  0.77572113  1.00817076  1.22901723\n",
      "  1.43902682  1.63890312  1.82929312  2.01079267  2.18395122  2.34927615\n",
      "  2.5072366   2.65826687  2.80276952  2.94111811  3.07365969  3.20071707\n",
      "  3.32259076  3.43956088  3.55188876  3.65981844  3.76357808  3.86338111\n",
      "  3.95942743  4.05190441  4.1409878   4.22684264  4.30962397  4.38947761\n",
      "  4.4665408   4.54094277  4.61280533  4.68224337  4.74936532  4.81427359\n",
      "  4.87706495  4.93783095  4.9966582   5.05362871  5.10882017  5.16230627\n",
      "  5.21415686  5.26443827  5.31321347  5.36054229  5.40648163  5.45108558\n",
      "  5.49440562  5.53649077  5.57738772  5.61714098  5.65579296  5.69338413\n",
      "  5.72995312  5.7655368   5.80017039  5.83388754  5.86672042  5.89869981\n",
      "  5.92985514  5.96021458  5.98980511  6.01865258  6.04678173  6.07421631\n",
      "  6.10097908  6.12709186  6.1525756   6.17745042  6.20173562  6.22544975\n",
      "  6.24861063  6.2712354   6.29334052  6.31494184  6.3360546   6.35669346\n",
      "  6.37687254  6.39660544  6.41590525  6.4347846   6.45325564  6.47133009\n",
      "  6.48901926  6.50633406  6.52328499  6.53988222  6.55613553  6.5720544\n",
      "  6.58764795  6.60292503  6.61789414  6.63256354  6.64694119  6.66103481\n",
      "  6.67485183  6.68839947  6.7016847   6.71471427  6.72749471  6.74003236\n",
      "  6.75233333  6.76440355  6.77624879  6.78787459  6.79928637  6.81048936\n",
      "  6.82148863  6.8322891   6.84289554  6.85331258  6.86354472  6.8735963\n",
      "  6.88347158  6.89317464  6.90270949  6.91207999  6.92128991  6.9303429\n",
      "  6.93924252  6.94799222  6.95659534  6.96505516  6.97337484  6.98155747\n",
      "  6.98960605  6.99752349  7.00531264  7.01297625  7.02051701  7.02793754\n",
      "  7.03524039  7.04242804  7.04950289  7.05646731  7.06332357  7.07007392\n",
      "  7.07672053  7.08326552  7.08971094  7.09605882  7.10231111  7.10846973\n",
      "  7.11453654  7.12051336  7.12640196  7.13220408  7.1379214   7.14355556\n",
      "  7.14910819  7.15458083  7.15997504  7.16529229  7.17053405  7.17570174\n",
      "  7.18079676  7.18582045  7.19077416  7.19565917  7.20047674  7.20522813\n",
      "  7.20991453  7.21453713  7.21909708  7.22359551  7.22803353  7.2324122\n",
      "  7.2367326   7.24099574  7.24520263  7.24935427  7.25345161  7.2574956\n",
      "  7.26148717  7.26542722  7.26931662  7.27315626  7.27694696  7.28068957\n",
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      "  7.6005069   7.60103169  7.60155303  7.60207095  7.60258547  7.60309663\n",
      "  7.60360446  7.60410898  7.60461022  7.60510822  7.605603    7.60609459\n",
      "  7.60658301  7.60706829  7.60755047  7.60802956  7.60850559  7.60897859\n",
      "  7.60944859  7.6099156   7.61037966  7.61084078  7.611299    7.61175434\n",
      "  7.61220683  7.61265647  7.61310331  7.61354736  7.61398865  7.6144272\n",
      "  7.61486303  7.61529616  7.61572662  7.61615443  7.61657961  7.61700218\n",
      "  7.61742217  7.61783958  7.61825446  7.61866681  7.61907665  7.61948401\n",
      "  7.61988891  7.62029137  7.6206914   7.62108902  7.62148426  7.62187714\n",
      "  7.62226766  7.62265586  7.62304175  7.62342534  7.62380666  7.62418573\n",
      "  7.62456256  7.62493716  7.62530957  7.62567978  7.62604783  7.62641373\n",
      "  7.62677749  7.62713913  7.62749867  7.62785612  7.62821151  7.62856484\n",
      "  7.62891613  7.6292654   7.62961266  7.62995793  7.63030123  7.63064256\n",
      "  7.63098195  7.63131941  7.63165494  7.63198858  7.63232033  7.6326502\n",
      "  7.63297822  7.63330439  7.63362872  7.63395124  7.63427195  7.63459087\n",
      "  7.63490802  7.6352234   7.63553703  7.63584891  7.63615908  7.63646753\n",
      "  7.63677427  7.63707933  7.63738272  7.63768444  7.63798451  7.63828294\n",
      "  7.63857974  7.63887493  7.63916851  7.63946051  7.63975092  7.64003976\n",
      "  7.64032704  7.64061278]\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7fbf21957828>"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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YObLLhxMRqWglH3UTLSV4apuTfCFaO7bokC+lV1+F/v0V8iIi7Yk1vBLAzEYB5xAuxq4A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\nIiI1TkEvIlLj/j9f/9RF1bnG3AAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7fbf2193ed30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "def timefunc2(z):\n",
    "    return 1/((1+z)*(H0gyr*((0.50*(1+z)**4) + (0.50*(1+z)**3) )**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "valuest2 = [] \n",
    "a = np.linspace(0,10,500)\n",
    "for i in a:\n",
    "    intergral2,err = quad(timefunc2,0,i)\n",
    "    valuest2.append(intergral2)\n",
    "    \n",
    "valuest2=np.asarray(valuest2)\n",
    "\n",
    "plt.plot(a,valuest2)\n",
    "\n",
    "print (valuest2)\n",
    "plt.ylabel('Look Back Age [Gyr]')\n",
    "plt.xlabel('Redshift [Z]')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#Problem 3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To find the age of the benchmark universe, we can intgrate the age of the universe over a large redshift which gives an approximate age of 13.4596 Gigeyears\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.           2.43790168   4.29133649   5.72123129   6.84026051\n",
      "   7.7286597    8.44395301   9.02765202   9.50997645   9.91316433\n",
      "  10.25377805  10.54430829  10.79429349  11.01110648  11.20051263\n",
      "  11.3670699   11.51441894  11.64549573  11.76268933  11.86796003\n",
      "  11.96292887  12.04894614  12.12714422  12.19847879  12.26376118\n",
      "  12.32368401  12.37884164  12.42974661  12.47684295  12.52051698\n",
      "  12.56110622  12.59890667  12.63417887  12.66715296  12.69803289\n",
      "  12.72699995  12.75421577  12.77982487  12.80395683  12.82672808\n",
      "  12.84824357  12.86859804  12.88787727  12.90615906  12.92351413\n",
      "  12.94000689  12.9556961   12.97063549  12.98487425  12.99845751\n",
      "  13.01142673  13.02382004  13.03567262  13.04701691  13.05788289\n",
      "  13.06829832  13.07828891  13.08787853  13.09708933  13.10594194\n",
      "  13.11445555  13.12264806  13.13053615  13.13813543  13.14546049\n",
      "  13.15252498  13.15934169  13.16592261  13.17227901  13.17842145\n",
      "  13.1843599   13.19010369  13.19566166  13.20104211  13.20625286\n",
      "  13.21130133  13.2161945   13.22093898  13.225541    13.2300065\n",
      "  13.23434106  13.23855002  13.24263839  13.24661096  13.25047227\n",
      "  13.25422664  13.25787817  13.26143075  13.2648881   13.26825375\n",
      "  13.27153108  13.27472329  13.27783345  13.28086447  13.28381916\n",
      "  13.28670017  13.28951005  13.29225123  13.29492605  13.29753673\n",
      "  13.3000854   13.30257411  13.3050048   13.30737937  13.3096996\n",
      "  13.31196722  13.31418389  13.31635118  13.31847064  13.32054372\n",
      "  13.32257184  13.32455634  13.32649853  13.32839966  13.33026093\n",
      "  13.33208351  13.33386851  13.33561701  13.33733005  13.33900861\n",
      "  13.34065368  13.34226617  13.34384698  13.34539697  13.34691699\n",
      "  13.34840783  13.34987027  13.35130507  13.35271295  13.35409461\n",
      "  13.35545073  13.35678197  13.35808897  13.35937232  13.36063264\n",
      "  13.36187049  13.36308643  13.36428101  13.36545473  13.36660811\n",
      "  13.36774165  13.3688558   13.36995104  13.37102781  13.37208654\n",
      "  13.37312766  13.37415156  13.37515866  13.37614932  13.37712393\n",
      "  13.37808285  13.37902642  13.37995499  13.38086889  13.38176844\n",
      "  13.38265396  13.38352575  13.38438411  13.38522932  13.38606167\n",
      "  13.38688143  13.38768885  13.38848421  13.38926775  13.39003972\n",
      "  13.39080035  13.39154988  13.39228852  13.39301651  13.39373406\n",
      "  13.39444136  13.39513864  13.39582608  13.39650388  13.39717222\n",
      "  13.39783129  13.39848128  13.39912235  13.39975467  13.40037841\n",
      "  13.40099373  13.4016008   13.40219976  13.40279076  13.40337396\n",
      "  13.40394949  13.4045175   13.40507813  13.4056315   13.40617775\n",
      "  13.406717    13.40724938  13.40777502  13.40829403  13.40880653\n",
      "  13.40931262  13.40981243  13.41030607  13.41079363  13.41127522\n",
      "  13.41175095  13.41222091  13.4126852   13.41314392  13.41359715\n",
      "  13.414045    13.41448755  13.41492488  13.41535709  13.41578425\n",
      "  13.41620645  13.41662378  13.41703629  13.41744409  13.41784723\n",
      "  13.4182458   13.41863986  13.41902949  13.41941476  13.41979573\n",
      "  13.42017248  13.42054506  13.42091354  13.42127799  13.42163846\n",
      "  13.42199502  13.42234772  13.42269662  13.42304179  13.42338327\n",
      "  13.42372113  13.4240554   13.42438616  13.42471345  13.42503731\n",
      "  13.4253578   13.42567498  13.42598888  13.42629955  13.42660705\n",
      "  13.42691141  13.42721268  13.4275109   13.42780612  13.42809838\n",
      "  13.42838772  13.42867418  13.4289578   13.42923862  13.42951668\n",
      "  13.42979201  13.43006466  13.43033466  13.43060205  13.43086686\n",
      "  13.43112912  13.43138888  13.43164615  13.43190099  13.43215341\n",
      "  13.43240346  13.43265115  13.43289653  13.43313962  13.43338046\n",
      "  13.43361907  13.43385547  13.43408971  13.4343218   13.43455178\n",
      "  13.43477967  13.4350055   13.43522929  13.43545107  13.43567086\n",
      "  13.4358887   13.43610459  13.43631858  13.43653068  13.43674091\n",
      "  13.43694929  13.43715586  13.43736063  13.43756363  13.43776487\n",
      "  13.43796437  13.43816217  13.43835827  13.4385527   13.43874548\n",
      "  13.43893662  13.43912615  13.43931409  13.43950046  13.43968526\n",
      "  13.43986853  13.44005028  13.44023052  13.44040928  13.44058656\n",
      "  13.4407624   13.4409368   13.44110978  13.44128136  13.44145155\n",
      "  13.44162037  13.44178784  13.44195396  13.44211876  13.44228224\n",
      "  13.44244443  13.44260534  13.44276498  13.44292337  13.44308051\n",
      "  13.44323643  13.44339114  13.44354464  13.44369696  13.4438481\n",
      "  13.44399808  13.44414691  13.44429461  13.44444118  13.44458664\n",
      "  13.44473099  13.44487426  13.44501645  13.44515757  13.44529764\n",
      "  13.44543666  13.44557465  13.44571161  13.44584757  13.44598252\n",
      "  13.44611648  13.44624945  13.44638146  13.44651251  13.4466426\n",
      "  13.44677175  13.44689997  13.44702727  13.44715365  13.44727913\n",
      "  13.44740371  13.4475274   13.44765022  13.44777217  13.44789325\n",
      "  13.44801349  13.44813288  13.44825144  13.44836917  13.44848607\n",
      "  13.44860217  13.44871747  13.44883197  13.44894568  13.44905861\n",
      "  13.44917076  13.44928215  13.44939279  13.44950267  13.44961181\n",
      "  13.44972021  13.44982788  13.44993483  13.45004106  13.45014658\n",
      "  13.4502514   13.45035552  13.45045896  13.45056171  13.45066378\n",
      "  13.45076518  13.45086591  13.45096599  13.45106542  13.45116419\n",
      "  13.45126233  13.45135983  13.4514567   13.45155295  13.45164858\n",
      "  13.45174359  13.451838    13.45193181  13.45202502  13.45211764\n",
      "  13.45220968  13.45230113  13.45239201  13.45248232  13.45257206\n",
      "  13.45266124  13.45274987  13.45283794  13.45292547  13.45301246\n",
      "  13.45309891  13.45318483  13.45327022  13.45335509  13.45343944\n",
      "  13.45352328  13.45360661  13.45368943  13.45377175  13.45385357\n",
      "  13.4539349   13.45401575  13.4540961   13.45417598  13.45425538\n",
      "  13.45433431  13.45441277  13.45449077  13.45456831  13.45464538\n",
      "  13.45472201  13.45479819  13.45487392  13.4549492   13.45502405\n",
      "  13.45509847  13.45517245  13.45524601  13.45531914  13.45539185\n",
      "  13.45546414  13.45553602  13.45560748  13.45567854  13.4557492\n",
      "  13.45581945  13.45588931  13.45595877  13.45602784  13.45609652\n",
      "  13.45616481  13.45623272  13.45630026  13.45636741  13.45643419\n",
      "  13.45650061  13.45656665  13.45663233  13.45669764  13.4567626\n",
      "  13.4568272   13.45689144  13.45695534  13.45701889  13.45708209\n",
      "  13.45714494  13.45720746  13.45726964  13.45733148  13.45739299\n",
      "  13.45745417  13.45751502  13.45757554  13.45763574  13.45769562\n",
      "  13.45775519  13.45781443  13.45787337  13.45793199  13.4579903\n",
      "  13.45804831  13.45810601  13.45816341  13.45822051  13.45827731\n",
      "  13.45833382  13.45839003  13.45844595  13.45850158  13.45855692\n",
      "  13.45861198  13.45866676  13.45872125  13.45877547  13.45882941\n",
      "  13.45888307  13.45893646  13.45898958  13.45904243  13.45909502\n",
      "  13.45914733  13.45919939  13.45925118  13.45930271  13.45935399\n",
      "  13.45940501  13.45945577  13.45950628  13.45955654  13.45960655]\n"
     ]
    }
   ],
   "source": [
    "def timefunc1(z):\n",
    "    return 1/((1+z)*(H0gyr*(0.3*(1+z)**3 + 0.7)**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "valuest1 = [] \n",
    "a = np.linspace(0,100,500)\n",
    "for i in a:\n",
    "    intergral1,err = quad(timefunc1,0,i)\n",
    "    valuest1.append(intergral1)\n",
    "    \n",
    "valuest1=np.asarray(valuest1)\n",
    "\n",
    "print (valuest1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this matter only universe, we do not know that curviture is 0, therefore Omega{o} = Omega{m}\n",
    "\n",
    "that means that \n",
    "H(z) = H0*sqrt[Omega{m}(1+z)^2 + (1-Omega)(1+z)^2] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  0.           2.33150043   3.99272775   5.23576314   6.20043398\n",
      "   6.97052521   7.59929156   8.12219801   8.56377268   8.9415191\n",
      "   9.26826166   9.55361031   9.80490768  10.02786004  10.22696876\n",
      "  10.40583209  10.56736068  10.71393415  10.84751681  10.96974436\n",
      "  11.08198979  11.185414    11.2810052   11.36960982  11.451957\n",
      "  11.52867824  11.60032315  11.6673723   11.73024772  11.78932152\n",
      "  11.84492305  11.89734488  11.94684776  11.99366482  12.03800515\n",
      "  12.08005679  12.11998932  12.15795608  12.19409603  12.22853542\n",
      "  12.26138919  12.29276222  12.32275041  12.35144159  12.37891637\n",
      "  12.40524886  12.43050733  12.45475472  12.47804921  12.50044462\n",
      "  12.52199083  12.54273412  12.56271748  12.58198094  12.60056177\n",
      "  12.61849473  12.63581228  12.65254478  12.66872062  12.68436642\n",
      "  12.69950713  12.71416617  12.72836556  12.74212601  12.75546699\n",
      "  12.76840689  12.78096301  12.79315171  12.80498841  12.81648772\n",
      "  12.82766345  12.83852866  12.84909573  12.85937642  12.86938185\n",
      "  12.87912261  12.88860874  12.89784979  12.90685484  12.91563254\n",
      "  12.92419112  12.93253842  12.94068193  12.94862878  12.95638577\n",
      "  12.96395942  12.97135592  12.97858122  12.985641    12.99254069\n",
      "  12.99928549  13.00588038  13.01233014  13.01863933  13.02481236\n",
      "  13.03085341  13.03676655  13.04255563  13.04822439  13.05377641\n",
      "  13.05921513  13.06454386  13.06976577  13.07488393  13.0799013\n",
      "  13.08482071  13.08964489  13.09437649  13.09901803  13.10357198\n",
      "  13.1080407   13.11242646  13.11673148  13.12095786  13.12510767\n",
      "  13.12918288  13.13318542  13.13711714  13.14097981  13.14477518\n",
      "  13.14850493  13.15217066  13.15577396  13.15931633  13.16279925\n",
      "  13.16622414  13.16959238  13.17290532  13.17616424  13.1793704\n",
      "  13.18252502  13.18562928  13.18868434  13.19169129  13.19465123\n",
      "  13.1975652   13.20043421  13.20325926  13.20604131  13.20878129\n",
      "  13.21148011  13.21413864  13.21675775  13.21933826  13.221881\n",
      "  13.22438674  13.22685625  13.22929029  13.23168957  13.2340548\n",
      "  13.23638667  13.23868585  13.24095299  13.24318873  13.24539369\n",
      "  13.24756847  13.24971365  13.25182982  13.25391753  13.25597732\n",
      "  13.25800973  13.26001527  13.26199445  13.26394777  13.26587569\n",
      "  13.2677787   13.26965725  13.27151178  13.27334274  13.27515055\n",
      "  13.27693562  13.27869837  13.28043918  13.28215845  13.28385656\n",
      "  13.28553387  13.28719076  13.28882756  13.29044463  13.29204231\n",
      "  13.29362092  13.29518079  13.29672224  13.29824557  13.29975108\n",
      "  13.30123908  13.30270985  13.30416368  13.30560084  13.3070216\n",
      "  13.30842623  13.30981499  13.31118813  13.31254591  13.31388856\n",
      "  13.31521633  13.31652945  13.31782815  13.31911265  13.32038318\n",
      "  13.32163994  13.32288316  13.32411304  13.32532977  13.32653357\n",
      "  13.32772463  13.32890313  13.33006927  13.33122322  13.33236518\n",
      "  13.33349531  13.33461379  13.33572079  13.33681648  13.33790102\n",
      "  13.33897457  13.34003729  13.34108933  13.34213085  13.343162\n",
      "  13.34418292  13.34519375  13.34619464  13.34718573  13.34816714\n",
      "  13.34913903  13.3501015   13.35105471  13.35199876  13.35293379\n",
      "  13.35385992  13.35477727  13.35568596  13.35658609  13.35747779\n",
      "  13.35836117  13.35923634  13.36010341  13.36096247  13.36181364\n",
      "  13.36265702  13.36349272  13.36432082  13.36514142  13.36595463\n",
      "  13.36676054  13.36755924  13.36835082  13.36913537  13.36991298\n",
      "  13.37068373  13.37144772  13.37220502  13.37295572  13.37369991\n",
      "  13.37443765  13.37516903  13.37589412  13.37661301  13.37732577\n",
      "  13.37803247  13.37873318  13.37942798  13.38011694  13.38080013\n",
      "  13.3814776   13.38214944  13.38281571  13.38347648  13.3841318\n",
      "  13.38478174  13.38542637  13.38606574  13.38669992  13.38732896\n",
      "  13.38795293  13.38857188  13.38918587  13.38979495  13.39039919\n",
      "  13.39099863  13.39159333  13.39218334  13.39276871  13.3933495\n",
      "  13.39392576  13.39449754  13.39506487  13.39562783  13.39618644\n",
      "  13.39674077  13.39729085  13.39783673  13.39837846  13.39891609\n",
      "  13.39944964  13.39997918  13.40050474  13.40102636  13.40154409\n",
      "  13.40205796  13.40256803  13.40307431  13.40357687  13.40407573\n",
      "  13.40457094  13.40506252  13.40555053  13.40603499  13.40651594\n",
      "  13.40699342  13.40746747  13.40793811  13.40840538  13.40886932\n",
      "  13.40932997  13.40978734  13.41024148  13.41069242  13.41114019\n",
      "  13.41158482  13.41202634  13.41246479  13.41290019  13.41333258\n",
      "  13.41376197  13.41418841  13.41461192  13.41503253  13.41545026\n",
      "  13.41586515  13.41627723  13.41668651  13.41709302  13.4174968\n",
      "  13.41789787  13.41829625  13.41869197  13.41908506  13.41947553\n",
      "  13.41986342  13.42024874  13.42063153  13.4210118   13.42138958\n",
      "  13.42176489  13.42213776  13.4225082   13.42287624  13.4232419\n",
      "  13.4236052   13.42396617  13.42432482  13.42468118  13.42503527\n",
      "  13.4253871   13.4257367   13.42608409  13.42642928  13.4267723\n",
      "  13.42711317  13.4274519   13.42778851  13.42812303  13.42845547\n",
      "  13.42878584  13.42911418  13.42944048  13.42976479  13.4300871\n",
      "  13.43040743  13.43072582  13.43104226  13.43135678  13.43166939\n",
      "  13.43198012  13.43228897  13.43259596  13.43290111  13.43320444\n",
      "  13.43350595  13.43380567  13.4341036   13.43439977  13.43469419\n",
      "  13.43498688  13.43527784  13.43556709  13.43585465  13.43614053\n",
      "  13.43642474  13.43670731  13.43698823  13.43726753  13.43754521\n",
      "  13.4378213   13.4380958   13.43836873  13.43864009  13.43890991\n",
      "  13.43917819  13.43944495  13.4397102   13.43997395  13.44023622\n",
      "  13.440497    13.44075633  13.4410142   13.44127063  13.44152564\n",
      "  13.44177922  13.4420314   13.44228218  13.44253158  13.4427796\n",
      "  13.44302626  13.44327156  13.44351553  13.44375816  13.44399947\n",
      "  13.44423946  13.44447816  13.44471556  13.44495168  13.44518653\n",
      "  13.44542012  13.44565245  13.44588354  13.44611339  13.44634202\n",
      "  13.44656944  13.44679565  13.44702066  13.44724448  13.44746712\n",
      "  13.44768859  13.44790889  13.44812805  13.44834606  13.44856293\n",
      "  13.44877867  13.44899329  13.4492068   13.44941921  13.44963052\n",
      "  13.44984074  13.45004988  13.45025795  13.45046495  13.4506709\n",
      "  13.45087579  13.45107965  13.45128247  13.45148427  13.45168504\n",
      "  13.4518848   13.45208356  13.45228132  13.45247808  13.45267387\n",
      "  13.45286867  13.45306251  13.45325538  13.45344729  13.45363826\n",
      "  13.45382828  13.45401736  13.45420552  13.45439275  13.45457906\n",
      "  13.45476446  13.45494895  13.45513255  13.45531525  13.45549707\n",
      "  13.45567801  13.45585807  13.45603726  13.45621559  13.45639307\n",
      "  13.45656969  13.45674547  13.45692041  13.45709451  13.45726779\n",
      "  13.45744024  13.45761187  13.4577827   13.45795271  13.45812193\n",
      "  13.45829035  13.45845798  13.45862482  13.45879088  13.45895617\n",
      "  13.45912069  13.45928444  13.45944743  13.45960967  13.45977116]\n"
     ]
    }
   ],
   "source": [
    "massdensity = 0.01905\n",
    "def timefuncm(z):\n",
    "    return 1/((1+z)*(H0gyr*(massdensity*(1+z)**3 + (1-massdensity)*(1+z)**2)**(1/2)))\n",
    "\n",
    "step = 0.01 \n",
    "valuestm = [] \n",
    "a = np.linspace(0,100,500)\n",
    "for i in a:\n",
    "    intergralm,err = quad(timefuncm,0,i)\n",
    "    valuestm.append(intergralm)\n",
    "    \n",
    "valuestm=np.asarray(valuestm)\n",
    "\n",
    "print (valuestm)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By varrying the matter density parameter, we can find a close age of the universe for the given matter density parameter\n",
    "\n",
    "We find that the matter density= **0.01905** is the optimal mass density\n",
    "\n",
    "As shown above, the age of this matter only universe is **13.45977 Gyr**\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
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