diff --git a/example_notebooks/Schramm.ipynb b/example_notebooks/Schramm.ipynb index 88af0e5..e9efb17 100644 --- a/example_notebooks/Schramm.ipynb +++ b/example_notebooks/Schramm.ipynb @@ -1191,7 +1191,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.9.18" + "version": "3.12.11" } }, "nbformat": 4, diff --git a/example_notebooks/background_evolution.ipynb b/example_notebooks/background_evolution.ipynb index b00dee0..c1877cd 100644 --- a/example_notebooks/background_evolution.ipynb +++ b/example_notebooks/background_evolution.ipynb @@ -324,7 +324,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.12.3" + "version": "3.12.11" } }, "nbformat": 4, diff --git a/example_notebooks/scratch.ipynb b/example_notebooks/scratch.ipynb new file mode 100644 index 0000000..3e59281 --- /dev/null +++ b/example_notebooks/scratch.ipynb @@ -0,0 +1,518 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 718, + "id": "da68cf81-6422-4fbb-9310-0a972a6e37e3", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The autoreload extension is already loaded. To reload it, use:\n", + " %reload_ext autoreload\n" + ] + } + ], + "source": [ + "%load_ext autoreload\n", + "%autoreload\n", + "import jax.numpy as jnp\n", + "import jax\n", + "from jax import jit, vmap\n", + "import sys\n", + "\n", + "sys.path.append(\"../\")\n", + "import linx.const as const \n", + "from linx.nuclear import NuclearRates\n", + "from linx.background import BackgroundModel\n", + "from linx.abundances import AbundanceModel\n", + "import linx.thermo as thermo\n", + "\n", + "import matplotlib\n", + "import matplotlib.pyplot as plt\n", + "import matplotlib.pylab as pylab\n", + "from matplotlib.lines import Line2D\n", + "import matplotlib.patches as mpatches\n", + "from matplotlib.patches import Patch\n", + "from matplotlib.legend_handler import HandlerLine2D\n", + "\n", + "%matplotlib inline\n", + "\n", + "from plot_params import params\n", + "matplotlib.pylab.rcParams.update(params)\n", + "cols_default = plt.rcParams['axes.prop_cycle'].by_key()['color']\n", + "\n", + "from scipy.integrate import quad\n", + "from scipy.interpolate import interp1d\n", + "\n", + "import time\n" + ] + }, + { + "cell_type": "code", + "execution_count": 661, + "id": "61a0b61d-9532-4757-a56e-3dd37673bdfe", + "metadata": {}, + "outputs": [], + "source": [ + "import linx.P_QED as P_QED" + ] + }, + { + "cell_type": "code", + "execution_count": 662, + "id": "a4b1a2c6-f5aa-4e6a-8118-cc3000473fe6", + "metadata": {}, + "outputs": [], + "source": [ + "T = 40" + ] + }, + { + "cell_type": "code", + "execution_count": 517, + "id": "30da5a29-1a43-4545-85a9-97d1f862f880", + "metadata": {}, + "outputs": [], + "source": [ + "e = jnp.sqrt(const.aFS * 4 * jnp.pi)\n", + "prefac = e**3 * T/(12 * jnp.pi**4)\n", + "\n", + "p = jnp.linspace(0,50*T,num=2000) # this integral also peaks at p close to zero, integrating to 50 is fine as long as resolution is good\n", + "Ep = jnp.sqrt(p**2 + me**2)\n", + "integrand = (p**2 + Ep**2)/Ep * 2/(jnp.exp(Ep/T) + 1)\n", + "\n", + "res = jnp.trapezoid(integrand,p)" + ] + }, + { + "cell_type": "code", + "execution_count": 518, + "id": "fea14aac-0f8c-4c63-b199-e162c6e78566", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 518, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(p,integrand)" + ] + }, + { + "cell_type": "code", + "execution_count": 519, + "id": "2e00051c-b5c2-406c-ac7b-f261587a8aa9", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "5263.633435509192\n", + "5263.65845568496\n" + ] + } + ], + "source": [ + "print(jnp.trapezoid(integrand,p))\n", + "print(quad(lambda p: (p**2 + jnp.sqrt(p**2 + me**2)**2)/jnp.sqrt(p**2 + me**2) * 2/(jnp.exp(jnp.sqrt(p**2 + me**2)/T) + 1) , 0, jnp.inf)[0])" + ] + }, + { + "cell_type": "code", + "execution_count": 495, + "id": "06c6e881-45bf-49ef-b4f0-655dd0ed189c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "380969.1889396603" + ] + }, + "execution_count": 495, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "T = 40\n", + "me = 0.511\n", + "quad(lambda p: p**2 * jnp.log( (1 + jnp.exp(-jnp.sqrt(p**2 + me**2)/T))**2 / (1 - jnp.exp(-p/T)) ) , 0, jnp.inf)[0]" + ] + }, + { + "cell_type": "code", + "execution_count": 496, + "id": "ca9ecfdb-c325-4d9b-819a-90ccb11229e3", + "metadata": {}, + "outputs": [], + "source": [ + "p = jnp.linspace(0,50*T,num=3000)\n", + "Ee = jnp.sqrt(p**2 + me**2)\n", + "integrand = p**2 * jnp.log( (1 + jnp.exp(-Ee/T))**2 / (1 - jnp.exp(-p/T)) ) " + ] + }, + { + "cell_type": "code", + "execution_count": 497, + "id": "cb7558da-0af5-4f7e-9a09-a1f49330ebd1", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 497, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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jDxia47DWLoVea+L6ZemzqakpPVfko+OAUbc9a6/Dah+sJ8VGSgCGld4no+6f8/PzWrw50KVBjDHapaR/YkflIsJdUa3K+y5YFiSOLqm6FgYjpQBkTK6XoGCM0eG0jYI75dUmXUt6V11qozx9o6TqlgchYADIll5bGEG+wuOPbpoJuqmMMVEtBc15rLdTPshZ3t719JjDGJmY8Fa81W4uTXoDgKstjLrWQDh3EXJe51aEXl+KmKh3oYPyga0jpXI9dknVLw9CDgOAQy0MPxehk/Z+W4e6+iOgXgvN5v5u6Iavd8rr4aU9dIitlvtDb707aSflgxBn0lvlp2dk99YtWhgAnJvpra0InVdxsUm53hX3vcG3CgBJrB3VtEsqhoARjJQi6Q0ga5xfrTbOpHd4pBRJbwBZQ8AIDavtdbVaVZid9Z5Ld+96CxsCQFbkXN/T24ZaGGa0+z29GxcgFLFS2mw6/wUAhgZ7eneT9C70Xh350Gzv4sZGrcUBAMOKPb077JIy+byYkZGef19h5kGAKJHHAJAhzgeMoEuq14UHoxcgXIvldwLAMHA+YASjpEwMCW/FAoQAsoqA4ecwcmPxtDBGDh70urcUS5wDyBLnA0awWm1cXVK6pnzhkD95j4ABIEOcDxhxd0mp/Gw1j8HyIACyhIDhLw0SV5dUeC4GLQwAWZLKgBHvxL2gSyrGFoY/UoqkN4A0YOJep/MwYpjl/dAS5xsbYisVMblUxmUAjjjLxL32VHb3YtsLI1A4XN06xJZL3ppSAJAFzgeMBxP3YgwYsw/2miquDmx7cgDoK+cDRrDjXhx7YQRGjxyuHRfvEDAAZIPzASPYQEn34o5LYe5BC2OPFgaAjHA+YARJ71i7pEIBgy4pAFlBwAi6pGIMGBp88lNT3jEBA0BWOB0wbLlc2xUvju1Zwwpz1TxG8c6dWH8vACTF6YBRKT7YQjX+gFHtltpbZYlzANng9EzvYJZ33KOk1Kg/F4MuKQDDjpneHSS8+9LCOHy4tgChdnsFS54DwLBhpnenASPGpLcK9vK2tuItEQIAaed0wAj2wlAmxqVB1KjfwlAkvgFkgdMBI9gLoz9J72oLQxVJfAPIALcDhr/woMrFuFptOIeh9mhhAMgApwNG/SipmHMY09NiTLV6i2u0MACkn9MBo59Jbx0VFWykRA4DQBY4HTCChQf7MQ8jPHmPuRgAssDpgBGsI9WPpLca9RPfeyxxDiADnJ7p3c+Je+HEd3GVLikAw4uZ3p12SRX61yVV2tz0glM/ghIA9IqZ3h23MOJNeqvRI0dqxwytBZB2BIw+jZJSo0cfBIziba+pBwCp1bJLyhijY0PPisiT1tpzEeVaFmR156y1r8ZZ3k/Wn+ltRvJ9WRywroVx+3bsvx8AhqaFYYw5JSIviMh6k3K92S9Zay/rQ2/8/nuxlKdxe9awwpGjteO9ajIJALIZMKy114IbeZOPnLPWXgl9Xj97LsbygQSMuGd5B0bGx2pbtRIwADibw/C7qhYiik5pWa/lMsDVanN9GCHV2C1FlxQAl5Pep5u0PNb9QNBr+cBWqzV96pKqCxi0MAA4HDBmmuQ2NAjMxVDe1ObmprZwIh+Li4sdr1abG4t3pdqw0WPHvGdaGACGhd4no+6fKysrWlztR8/KsNqpqSnNd0Q+OgkYwWq1/ZiD0djCKG9vS2lrq2/nAYB26X0y6v45Pz+vxZvNfq6XsaTaOojKNcyFupp6LR9Ql1T/chiFI/VzMfIHDvTtXADQT720MK426TrSILAUQ3nfBUuD9LWFcZShtQCyoesWhrV23RgT1RLQeRVebqLX8n6r7O32dR5G42xvAgYAV1oYUa2BSxET8S7EWD6YiXt9THrrAoTBznskvgFkeab3gjHmZRE5rzO+jTEXjDE689tjrb3of07LXvLfezWu8n6r+F1S/dg8KZDL56Uw6++LQQsDQFa7pKy1mkvQm/rFfT6z7w2+1/JBrCXVzy6pII+xt3qHFgaAVEvlsNrYlwbpe8BgtjeA9HM6YNgB5DDCI6W0S8qWy309FwD0i7NbtNpKRSqlYFhtf3fCGz1ene1tSyUprrK/N4DhwhatbeYvBtElNXb8kdrx7q1bdXMzACBpbNHa5izvQbQwxh45XjvevXmrr+cCgH5xN2D0eXvWqAUI1d4tAgaAdMq5nvDu9zyMoAVTmJ2rdUkBQBo5GzDqWhhj/Q0Yamy+mscgYABIK4cDRiiH0cfVagNjx6t5jN2Vm30/FwD0g7MBI9gLo597ekcFjOLaal3rBgDSwtmAUZ/07n8LY9QPGGrv44/7fj4AiJu7AcPfnnUQw2ofGlpLHgNACrk70zs8cW8QSe9QC2P3JnkMAMODmd4tVIqDm4dR2xcjn/eWB9m7RZcUgOHBTO82t2ft957etXOMjMjY0eoEPloYANLI2YAx6HkYauzEvPe8s7wykPMBQJwIGP6ueIMwduKE97y7vOytlgsAaeJswKhbGqTP+2EExk+eqOVP9j65M5BzAkBccq6vVmtMzssvDML4yZO1493lGwM5JwDExd2AEWzPOjoqxpiBB4ydGwQMAOnibMCobc86gGVBAoXDhyU3Wu3+2rmxPLDzAoCzASOOiXtBC2MQy4IETC4nY/PztcQ3AAwDJu510CU1SOOPnpT7H7xPCwPA0GDiXpvLmw9qDkZg3B9aqzvvVUqlgZ4bAHqRc31580F2SYXnYlhbYW8MAKmSc72FMYi9MBq7pAIMrQWQJs4GjAejpJLpklI7HxEwAKSHswGjsrc7sJVqw/KHDklhesY7vv83fzPQcwNAL9wNGMFMb39exCBNfPrT3vP9Dz4Y+LkBoFs513fcG/QoKTX+qce8550PPxJbLg/8/ADQDWcDRrDj3qC7pNTE45+udYvtsr83gJTIuT7T2yQRMPwuKXX/fbqlACSLmd7tJr0HtLR52MRj1S4ptaN5jC/+0sCvAQACzPTeh25epHtrJzGsVo0cOCCjR496xyS+AaRFzuX8RVJdUnUjpRhaCyAleu6SMsa8JCIvisgF/y19fc1aeyX0mbMisuq/nLPW1iUeWpVnYT/vRhOf+pRsXL3qLUKoa0oNaptYAEiyhTEnIqdF5LqIvCYi6xHBYslae1kfGhj899oq7+ccjCTWkmpsYdhySXY+/DCRawCAQQeMVWvtc9ZaY619MqJ1cC4cQPygcK6D8thZfw5Gkl1SkwsLtePt6xprAcDhHIYxRtfAeHBnfOCUlrUq79d1VfyVapPsktJFCIPd97bfI2AAcCRgGGNeCD1eDhWdDuUmwtb9QNGqvP85jARGSSmTz8vkE094xwQMAK4EjCU/B3HF71paMsYECfAZ/+bfaNXPfbQq7wvrL22uTEI5DDX51JPe8/Zf/3VtmC8AZDZgWGt1RNRSQw4i3MqI3ebmprZqIh+Li4uddUkNeD+MqIChkwhZ6hzAoOh9Mur+ubKyosVTA89hGGMW/NZDVC5izm9FtCqPNDU1pYEp8tFWwNitzvJOsktKTT71VO146733ErsOAG5ZXFyMvH/Oz89r8WZfAoYGBWNMVAd80M10tUnXkgaJpTbK+98llWDAGH/0Ucnlqy0cRkoBGHZxtDCCfIXHH900o91U1tr1Ji0Fr6xVufRJfZdUcgFDJ+tNLPiJ77+ihQEgwwEjnLsIOd8wj+JSxES9Cx2UZ3KUVODA0097z1vX36ubUAgAWWxhfFdv8v5Dk93Xw5P3rLUX9dkfcvuS/17b5VkeJaUO/txnvGcdJbVNHgPAEOt5ASO/62jfG3yrANDvALF/CyO5UVLq4GeqAUPd+39v1b0GgGHi5Gq1dQEjoaVBAqNHjtSWOr/31luJXgsA7MfJgGGD3fZG8t6M66QFrYp7b73tDW0DgGHk5BatQXI56dZFY8Ao3d2Q3eXlpC8HgGNeZYvWNvbzTjh/0Zj4Vps/+78yfvJkotcDwC1n2aK1dZdU0kNqw3tj5A9WZ+Nv/vSnSV8OAERyMmAELYykNk9qZEZGZOqzv+gd3/3pm96e4wAwbJwOGEkuC9Lo0Gc/W8tj3P/gg6QvBwAe4nTAyI1VNzAaBlPPVgOGuvuTv0z0WgAgipsBw1+tdpgCxtgjj8jYsePe8d2/JI8BYPgQMIaErkUftDLu/exnUt7ZSfqSAKAOAWOIzHzh87XVdO/+5CdJXw4A1HE8YAxP0lsdevbZWhBb/z9/nvTlAED6A0bPM72HtIWh13Poc6e8442/eJ19vgEM1UzvXJpneutDZydmJWCo2V963nsu3dv0Vq8FgH7T+6jeT48cOaIvmekd0MX9hjlgTH/+tJjciHe8+qMfJX05AOBwwAgvbT6EASN/6JAc+tyz3vHan/6QXfgADA3nAkbQuhjWgKEO/9qvec+lrXuy0cMiiwAQJwLGEJp+/nkZmajmne788Q+SvhwA8BAwhtDI+JjM/vIXveONq69LcW0t6UsCABcDxnDnMAKHX3zRe9ahtbe//z+TvhwAcDFgDH8LI9hUaXJhwTu+/UfflwpzMgAkjIAxpHRtqWNf/g3vuLi2Kms/+rOkLwmA45yb6Z2WgKHmfvVXvWG26ubly2ysBKAvmOmdgYChW8ge/83f9I51U6W1P6OVASB+zPTOQMBQx37j12utjOX/9t/FlstJXxIARzkXMML7TKQhYIxMTMgjv/Vb3vHORx/K7e9/P+lLAuAo5wJG5f792vHI5ISkgbYygt34bvyX/yrFjY2kLwmAg5wLGGU/YJh8XnKFgqSBtoQe+9o/9I7L21vy0Xf+c9KXBMBB7gWM7Wo+Z2Sy6UCAoTT9/Bdk+nR1R747P/iBrP7pD5O+JACOcbZLSnMDaaLzMj79j/+R5KemvNcf/MF/lJ2VlaQvC4BDnO2SChb3S5PRuTl5/J/+k1rX1HuvfEtKd+8mfVkAHOHcxL3ydjVg5FKS8G408/zzMn/mjHe8s7Is7/2rfy3lra2kLwtAijFxL2NdUmEnfucfyNyXvuQd33v7bXn3X74ipc3NpC8LQEoxca9V0jvFAcPkcvL47/0zmfl8NQm+9e678tbvfV22338/6UsDkGHuBQy/hZFLYQ4jTIcEL/yL87WWxu7NFXn769+QW3/4PWaDA+gL9wLGdrW/f+RAugOGyuXz8sQ3vi4nv/pVbXd4y558+J3vyFv//Buyce2aWGuTvkQAGZKXIWCM0UTEqv9yzlrbWSa7TeWd3dpaUsH6TGmn3VPzZ/6eHHjmGfngP/yB19LYvv6e/NUri3Lg6Wfk6N/9OzL7pV/xdvEDgFQHDD9YLFlrr/ivX9L3+hE0wkNQsxIwAoc++4vyc//+33nLoH/8h9+T8s592Xr3He/x4aX/JIc+96xMf+HzMvULvyCjx4558zoAIFUBQ0TOWWufC15Yay8bY97QkV5xnyjLAUNpK+Lk7/6OHPvyl+Xj731PPvlfr0lxfc0LHms//rH3UPlD03Lg6adl4lOPydj8fPVx7JjkZ2dkZHw86X8GgCGVaMAwxsyISHUf0nqntMxaux7n+Yqrd2rHhVk9dTYVpg/Jya/+rsz//a/Ixuuvy/r//rFsXH1DSveqQ29Ldzdk4+rr3iNq3arCzIwXUHPjE14Qyo2PV48nxsUUCt46XGZkREx+RMyIf+y9rh6L33rxWjHew3vh5VlqLZvg/eqLB5+tvnz488AwMJIKB3/+5yV/8GDmWhinQ7mLsHU/kFyL82S7N5Zrx/pXddZpUnz2i1/0Hjpyauu992TrnXe9Ybjb16/L7s2bYhv2Ctccz+6tW94DQDp95vd/X/LPPJ25gKF/5ke1IjSIzDX7oc3NzaZ/eb7yyiuyuLgYWVYplSR/4KBYW8lkl9R+9C//g8884z0CGkT2bt+WneUVr/VVWt/wlk4vrq1JeXPT2zukoo/7O9Xj+/elUio+FGQApIveI7/1rW81K64uWBfBJDn0UhPcInI+nMPw37/u5za8RHhD2fL8/Pz88vKD1kKndCmNkQMHuv5513nfmUrFCzgaPHSv8fCzlonV//nfLf188PBfVj8QPEKfrft8Mv8+IIpN0TD18UdPdpWPPHHihKysrKxYa08MYwtDWxdRyYS5Jl1VsSBY9MZr3fl5CxkdTfpyADgyce9qk64nDSJLCVwPAGAYA4Y/CiqqJbEU9wgpAEC6Wxjqkj95z+MfX+jnCZslxbE/6q071Ft3qLfhq7dEk96Ns739rqh9lwaJI+mtffDD8O9OG+qtO9Rbd6i3wdfbsCe9Pf1aOwoAkK0uKQBAChAwOtTplrDD9PO9nrsX1Fsy53a13no9v8v1ti/t60rTQ0SWp6en7XPPPec9Ll26ZDtV/Wd3R8/ZiyR/vtdzU2/dod4GX2+9nt+1ert06ZJ33kKhoD+/3uz+OxRJ704YY4q5XC5//Pjxrn/HysqKzHe5lpRuku7ve5u6n+/13NRbd6i3wddbr+d3td5u3bollUqlZK0tRJWnMWDoptz6j7ndw6/RtVKqS7d2bnK/TdKH/Od7PTf11h3qbfD11uv5Xa23o7qwt7V2MhMBAwCQDJLeAIC2EDAAAG0hYAAA2kLAAAC0hYABAGjLUKwlNeBFDoPl1Pdd5NAV/q6HL4ZWCNbX18K7HbaqNxfq1RijC2Pqv/NJa+25iPKe6iirdbhfvfHda1lvZ/wFWZ/0J9N9M+nvnDMtjGBFXGvtZX1oRYWXVXeYbmB1WkR0W9zX/C/mlXbrzYV6NcacEpEXmuw/33MdZbUOW9Ub3719ndEbuLX2YhBojTFaR8l+55Je6mOAS4q80c57rj30r7pe6s2levX/Ar4Udx1lvQ73qTe+ezby370gIi83vKctDT2YiaNuuq07J1oYfvNO/09odMovQxf1Rr32XkfUYTTqTc6HX4R2IF1I8jvnSg7jdJOtYNf9irsmDjPGaLdB4JQ2g9usN+1ScL1ee60jp+uQ797DrLW6mdxs+D1jjHeDt9Ze8+sske+cKwFjpkk/6qpfeS5b8vuOl4K/7IwxF/wEW6t6o157ryOX65DvXvu0Ti4m/Z1zoksKzelfLMF/sP5rTYC9nOxVwQV89zoaPLDQOEoqCa60MDSaRvXNNWuaOc9vArdTb67Xaxx15Hod1uG795Dz1lodfpz4d86VFsbVJk0trbTaXzgu/odpjNEhjY3W26w36rX3OnKyDvnutUe76ETka8PynXOihaEjDIwxUZFTxyE3GyPuimDSlMcfJTET6lfet95cr9d2vlvUYVN89/ZhjNHuuW+H/r0z/gS7paS+c660MNSliIkrdV9Y14T7jxuG853roN5cq9eov8x6rSMX6rCu3vju7c8fCXW54QauM79Xk/zOObWBUjC70W96ZWYZgZiWIJBg9ESTJQSa1lvW69XvU9fJZ7/tDzvUf99rUbOSu62jLNZhq3rju7dvvUV211lrZ5P8zjkVMAAA3XOpSwoA0AMCBgCgLQQMAEBbCBgAgLYQMAAAbSFgAADaQsAAALSFgAEAaAsBAwAg7fj/ba4Ewx/XUV4AAAAASUVORK5CYII=", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.plot(p,integrand)" + ] + }, + { + "cell_type": "code", + "execution_count": 498, + "id": "6bc7a539-f89f-4828-9a05-a692ebfc8d0a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Array(380969.17992295, dtype=float64)" + ] + }, + "execution_count": 498, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "jnp.trapezoid(jnp.nan_to_num(integrand,nan=0),p)" + ] + }, + { + "cell_type": "code", + "execution_count": 254, + "id": "90c1bd7b-a6fe-437f-9d30-f05fef9f3640", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Array([ nan, 0.00193672, 0.00665202, ..., 0. , 0. ,\n", + " 0. ], dtype=float64)" + ] + }, + "execution_count": 254, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "integrand" + ] + }, + { + "cell_type": "code", + "execution_count": 525, + "id": "a1350bb3-4b88-476d-9643-7f447d444b1b", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "0.584374419983926 -0.0013354808639761487 0.00013392287473705834 0.5831728619946869\n" + ] + } + ], + "source": [ + "me = .511\n", + "T = 1.000534\n", + "print(P_QED.explicit_P0(T,me), P_QED.explicit_P2(T,me), P_QED.explicit_P3(T, me), P_QED.P_QED(T, me))" + ] + }, + { + "cell_type": "code", + "execution_count": 529, + "id": "0dd1e876-ec88-4405-9321-ad0dd56276cd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Array(-407.48356157, dtype=float64, weak_type=True)" + ] + }, + "execution_count": 529, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_QED.dPdTQED_2(40.,.511)" + ] + }, + { + "cell_type": "code", + "execution_count": 531, + "id": "10c012ad-207f-4175-954b-00e48e6fcadd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Array(-30.56433696, dtype=float64, weak_type=True)" + ] + }, + "execution_count": 531, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "P_QED.d2PdT2QED_2(40.,.511)" + ] + }, + { + "cell_type": "code", + "execution_count": 641, + "id": "36afc6fe-274f-4667-97e1-803c34838b6a", + "metadata": {}, + "outputs": [], + "source": [ + "# Initialize the class. \n", + "default = BackgroundModel() \n", + "\n", + "# Call the class. \n", + "t_vec1, a_vec1, rho_g_vec1, rho_nu_vec1, _, _, Neff_vec = default(jnp.asarray(0.),me=jnp.asarray(0.5109989)) \n", + "t_vec2, a_vec, rho_g_vec2, rho_nu_vec, _, _, Neff_vec = default(jnp.asarray(0.),me=jnp.asarray(const.me)) " + ] + }, + { + "cell_type": "code", + "execution_count": 635, + "id": "af8485d4-5256-4976-9122-2b28d491895d", + "metadata": {}, + "outputs": [], + "source": [ + "rho_g_vec1_interp = interp1d(t_vec1,rho_g_vec1,bounds_error=False)\n", + "rho_g_vec2_interp = interp1d(t_vec2,rho_g_vec2,bounds_error=False)" + ] + }, + { + "cell_type": "code", + "execution_count": 636, + "id": "537c54c6-a484-4965-b0d4-a90f4e27b5e6", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(0.2, 0.6)" + ] + }, + "execution_count": 636, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.loglog(t_vec1,rho_g_vec1)\n", + "plt.loglog(t_vec2,rho_g_vec2)\n", + "plt.xlim([0.5,2])\n", + "plt.ylim([2e-1,6e-1])" + ] + }, + { + "cell_type": "code", + "execution_count": 637, + "id": "b96087a0-ab8e-4f0d-8de6-6765b2ab642a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[]" + ] + }, + "execution_count": 637, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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", + "text/plain": [ + "
" + ] + }, + "metadata": { + "needs_background": "light" + }, + "output_type": "display_data" + } + ], + "source": [ + "plt.loglog(t_vec1,jnp.abs(rho_g_vec1_interp(t_vec1)-rho_g_vec2_interp(t_vec1))/rho_g_vec2_interp(t_vec1))" + ] + }, + { + "cell_type": "code", + "execution_count": 638, + "id": "67708145-0b93-4193-b023-d6394f2c8a55", + "metadata": {}, + "outputs": [], + "source": [ + "network = 'key_PRIMAT_2023'\n", + "# network = 'key_PRIMAT_2018'\n", + "# network = 'key_PArthENoPE'\n", + "# network = 'key_YOF'\n", + "abundance_model = AbundanceModel(NuclearRates(nuclear_net=network))" + ] + }, + { + "cell_type": "code", + "execution_count": 723, + "id": "4029768d-fc72-482a-a4d3-7e8e69c08f5e", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "The abundances (n_i/n_b) are: \n", + "n: 6.79894161965725e-11\n", + "0.1366720199584961\n", + "The abundances (n_i/n_b) are: \n", + "n: 6.79894161965725e-11\n", + "0.13133788108825684\n", + "The abundances (n_i/n_b) are: \n", + "n: 6.79894161965725e-11\n", + "0.12949514389038086\n", + "The abundances (n_i/n_b) are: \n", + "n: 6.79894161965725e-11\n", + "0.1299757957458496\n", + "The abundances (n_i/n_b) are: \n", + "n: 6.79894161965725e-11\n", + "0.12741518020629883\n" + ] + } + ], + "source": [ + "me = jnp.asarray(0.6)\n", + "\n", + "t_vec1, a_vec1, rho_g_vec1, rho_nu_vec1, _, _, Neff_vec = default(jnp.asarray(0.),me=me) \n", + "\n", + "for i in range(5):\n", + " start = time.time()\n", + " Planck_omega_b_res = abundance_model(\n", + " rho_g_vec1, # photon energy density\n", + " rho_nu_vec1, # neutrino energy density\n", + " rho_NP_vec, # energy density of extra species\n", + " P_NP_vec, # pressure of extra species\n", + " t_vec=t_vec1, # vector of times at which quantities are given\n", + " a_vec=a_vec1, # vector of scale factor at corresponding times\n", + " eta_fac=jnp.asarray(1.), # factor by which to scale baryon-to-photon ratio\n", + " me = me\n", + " )\n", + " \n", + " print('The abundances (n_i/n_b) are: ')\n", + " print('n: ', Planck_omega_b_res[0])\n", + " # print('p: ', Planck_omega_b_res[1])\n", + " # print('d: ', Planck_omega_b_res[2])\n", + " # print('t: ', Planck_omega_b_res[3])\n", + " # print('He3: ', Planck_omega_b_res[4])\n", + " # print('He4: ', Planck_omega_b_res[5])\n", + " # print('')\n", + " # print('More standard abundances are: ')\n", + " # print('D/H: ', Planck_omega_b_res[2]/Planck_omega_b_res[1])\n", + " # print('T/H: ', Planck_omega_b_res[3]/Planck_omega_b_res[1])\n", + " # print('He3/H: ', Planck_omega_b_res[5]/Planck_omega_b_res[1])\n", + " # print('Yp: ', 4*Planck_omega_b_res[5])\n", + " # print('Li7/H: ', Planck_omega_b_res[6]/Planck_omega_b_res[1])\n", + " print(time.time() -start)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7ef75243-ac77-4a8c-b3a0-fb407634c15a", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3 (ipykernel)", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.11" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/linx/P_QED.py b/linx/P_QED.py new file mode 100644 index 0000000..c65c344 --- /dev/null +++ b/linx/P_QED.py @@ -0,0 +1,71 @@ +import os + +import numpy as np + +import jax.numpy as jnp +import jax.lax as lax +from jax import grad, vmap + +import linx.const as const +import equinox as eqx + + +# high temperature behavior is not correct, probably because bounds need to +# change with T. +# Low temperature behavior is perfect. + + +def explicit_P0(T, me): # not needed, computed in thermo + # 4.5 of https://arxiv.org/pdf/1911.04504 + + prefac = T/jnp.pi**2 + + p = jnp.linspace(0,50*T,num=3000) # this integral peaks at p close to T, integrating to 50*T is fine as long as resolution is very good + Ee = jnp.sqrt(p**2 + me**2) + integrand = p**2 * jnp.log( (1 + jnp.exp(-Ee/T))**2 / (1 - jnp.exp(-p/T)) ) + + res = jnp.trapezoid(jnp.nan_to_num(integrand,nan=0),p) # p = 0 gives a nan, should be 0 + + return prefac * res + + +def explicit_P2(T, me): + # first compute 4.7 of https://arxiv.org/pdf/1911.04504, but ignoring the + # last term because it's itty bitty according to them + + e = jnp.sqrt(const.aFS * 4 * jnp.pi) + prefac1 = - e**2 * T**2 / (12 * jnp.pi**2) + prefac2 = - e**2/(8 * jnp.pi**4) + + p = jnp.linspace(0,50*T,num=2000) # this integral peaks at p close to T, integrating to 50*T is fine as long as resolution is good + Ep = jnp.sqrt(p**2 + me**2) + integrand = p**2/Ep * 2/(jnp.exp(Ep/T) + 1) + + res = jnp.trapezoid(integrand,p) + + return prefac1 * res + prefac2 * res**2 + +def explicit_P3(T, me): + # compute 4.24 of https://arxiv.org/pdf/1911.04504 + + e = jnp.sqrt(const.aFS * 4 * jnp.pi) + prefac = e**3 * T/(12 * jnp.pi**4) + + p = jnp.linspace(0,50*T,num=2000) # this integral also peaks at p close to T, integrating to 50 is fine as long as resolution is good + Ep = jnp.sqrt(p**2 + me**2) + integrand = (p**2 + Ep**2)/Ep * 2/(jnp.exp(Ep/T) + 1) + + res = jnp.trapezoid(integrand,p) + + return prefac * res**(3./2) + +def P_QED(T,me): # not needed, sums in thermo + return explicit_P0(T, me) + explicit_P2(T, me) + explicit_P3(T, me) + + +dPdTQED_2 = grad(explicit_P2,argnums=0) +dPdTQED_3 = grad(explicit_P3,argnums=0) + +# we don't need these computed explicitly, actually +d2PdT2QED_2 = grad(dPdTQED_2,argnums=0) +d2PdT2QED_3 = grad(dPdTQED_3,argnums=0) diff --git a/linx/Untitled.ipynb b/linx/Untitled.ipynb new file mode 100644 index 0000000..125abfb --- /dev/null +++ b/linx/Untitled.ipynb @@ -0,0 +1,23 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "id": "a0074e73-6ca5-47ee-a7bc-87a25e8fb3a5", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "", + "name": "" + }, + "language_info": { + "name": "" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/linx/abundances.py b/linx/abundances.py index ed3ac0b..5f96d1a 100644 --- a/linx/abundances.py +++ b/linx/abundances.py @@ -14,6 +14,7 @@ import linx.thermo as thermo from linx.thermo import rho_EM_std_v, p_EM_std_v, nB from linx.special_funcs import zeta_3 +from linx.tau_n_vary_me import tau_n_fac_vary_me class AbundanceModel(eqx.Module): """ @@ -39,8 +40,6 @@ class AbundanceModel(eqx.Module): Spin of each species. species_binding_energy : list Binding energy of each species. - species_mass : list - Mass of each species. throw : bool Whether to raise exceptions on solver failure. """ @@ -53,7 +52,6 @@ class AbundanceModel(eqx.Module): species_excess_mass : dict species_spin : list species_binding_energy : list - species_mass : list throw : bool def __init__(self, nuclear_net, weak_rates=wr.WeakRates(), throw=True): @@ -115,16 +113,17 @@ def __init__(self, nuclear_net, weak_rates=wr.WeakRates(), throw=True): ) # in MeV - self.species_mass = ( - self.species_A * ma + self.species_excess_mass - self.species_Z * me - ) + # requires recompilation for each me--moved to YNSE method + # self.species_mass = ( + # self.species_A * ma + self.species_excess_mass - self.species_Z * me + # ) @eqx.filter_jit def __call__( self, rho_g_vec, rho_nu_vec, rho_NP_vec, P_NP_vec, a_vec=None, t_vec=None, eta_fac=jnp.asarray(1.), tau_n_fac = jnp.asarray(1.), - nuclear_rates_q=None, + nuclear_rates_q=None, me = const.me, Y_i=None, T_start=None, T_end=None, sampling_nTOp=150, rtol=1e-6, atol=1e-9, solver=Kvaerno3(), max_steps=4096, @@ -153,10 +152,12 @@ def __call__( in `const.eta0` (or `const.Omegabh2`). tau_n_fac : float, optional Rescaling factor for neutron decay lifetime, 1 for fiducial value - in `const.eta0` (or `const.Omegabh2`). + in `const.tau_n`. nuclear_rates_q : array, optional q ~ N(0,1) specifies the nuclear rate in its log-normal distribution. If not specified, will be taken to be `q = 0`. + me : float, optional + Electron mass in MeV. Defaults to `const.me`. Y_i : tuple of float, optional Initial abundances :math:`n_i/n_b` for species. Length must be equal to `self.nuclear_net.max_i_species`. Must specify `T_start` and `T_end` if not `None`. @@ -217,6 +218,11 @@ def __call__( T_end = const.T_end + # check if the user has varied me, and adjust the neutron lifetime if so + diff = jnp.abs(me - const.me)/const.me + tau_n_fac = jnp.where(diff > 1e-5, tau_n_fac_vary_me(me), 1.) * tau_n_fac + + # These are in MeV T_g_vec = thermo.T_g(rho_g_vec) T_nu_vec = thermo.T_nu(rho_nu_vec) @@ -266,7 +272,7 @@ def __call__( T_interval_nTOp, nTOp_frwrd, nTOp_bkwrd = self.weak_rates( jnp.array([T_g_vec, T_nu_vec]), - T_start=T_start, T_end=T_end, sampling_nTOp=sampling_nTOp + T_start=T_start, T_end=T_end, sampling_nTOp=sampling_nTOp, me=me ) ################################## @@ -285,7 +291,7 @@ def __call__( n_CMB_start = thermo.n_massless_BE(T_start, 0., 2.) eta_T_start = nB(a_start, eta_fac=eta_fac) / n_CMB_start - Y_YNSE = self.YNSE(Yn_i, Yp_i, const.T_start, eta_T_start) + Y_YNSE = self.YNSE(Yn_i, Yp_i, const.T_start, eta_T_start, me) Y_others_i = Y_YNSE[2:self.nuclear_net.max_i_species] @@ -527,7 +533,7 @@ def Y_prime(self, t, Y, args): return dY - def YNSE(self, Yn, Yp, T, eta): + def YNSE(self, Yn, Yp, T, eta, me=const.me): """ Nuclear statistical equilibrium yields for all species. @@ -541,6 +547,8 @@ def YNSE(self, Yn, Yp, T, eta): The temperature of the baryons in MeV. eta : float The baryon-to-photon ratio. + me: float, optional + Electron mass in MeV. Defaults to const.me Returns ------- @@ -548,8 +556,12 @@ def YNSE(self, Yn, Yp, T, eta): Yields for all species considered in LINX (13 of them). """ + species_mass = ( + self.species_A * ma + self.species_excess_mass - self.species_Z * me + ) + A32Overmn = ( - self.species_mass / ( + species_mass / ( mn**(self.species_A - self.species_Z) * mp**self.species_Z ) diff --git a/linx/background.py b/linx/background.py index 4a37529..404fce5 100644 --- a/linx/background.py +++ b/linx/background.py @@ -41,9 +41,11 @@ class BackgroundModel(eqx.Module): collision_me : bool LO : bool NLO : bool + max_steps : int throw : bool - def __init__(self, decoupled=False, use_FD=True, collision_me=True, LO=True, NLO=True, throw=True): + def __init__(self, decoupled=False, use_FD=True, collision_me=True, LO=True, NLO = True, throw=True, max_steps=512): + """ Initialize the BackgroundModel with thermodynamic options. @@ -70,13 +72,14 @@ def __init__(self, decoupled=False, use_FD=True, collision_me=True, LO=True, NLO self.collision_me = collision_me self.LO = LO self.NLO = NLO + self.max_steps = max_steps self.throw = throw @eqx.filter_jit def __call__( self, Delt_Neff_init, T_start=const.T_start, - T_end=const.T_end, rtol=1e-8, atol=1e-10, - solver=Tsit5(), max_steps=512 + T_end=const.T_end, me=const.me, rtol=1e-8, atol=1e-10, + solver=Tsit5(), ): """ Calculate thermodynamics given an initial :math:`\\Delta N_\\mathrm{eff}`. @@ -84,11 +87,13 @@ def __call__( ---------- Delt_Neff_init : float Initial :math:`\\Delta N_\\mathrm{eff}`. Can be positive or negative. - T_EM_init : float + T_EM_init : float, optional Initial EM (and neutrino) temperature. Default is `const.T_start`. - T_EM_end : float + T_EM_end : float, optional Final EM temperature to terminate integration at. Default is `const.T_end`. + me : float, optional + Electron mass in MeV. Defaults to `const.me`. rtol : float, optional Relative tolerance of the abundance solver. Default is `1e-8`. atol : float, optional @@ -133,18 +138,23 @@ def __call__( ) * Delt_Neff_init Y0 = (lna_init, T_EM_init, T_nu_init) + + # use parametric form to estimate correct start + # time given T_start, assuming T ~ t^(-1/2) and + # initial g_* is 10.75 + t0 = (1.5/T_start * 10.75**(-1./4))**2 def T_EM_check(t, y, args, **kwargs): return y[1] < T_end sol = diffeqsolve( - ODETerm(self.dY), solver, args=(lna_init, rho_extra_init), - t0=0., t1=jnp.inf, dt0=None, y0=Y0, + ODETerm(self.dY), solver, args=(lna_init, rho_extra_init, me), + t0 = t0, t1=jnp.inf, dt0=None, y0=Y0, saveat=SaveAt(steps=True), event=Event(T_EM_check), stepsize_controller=PIDController( rtol=rtol, atol=atol - ), - max_steps=max_steps, + ), + max_steps=self.max_steps, throw=self.throw ) @@ -159,7 +169,7 @@ def T_EM_check(t, y, args, **kwargs): last_step_ind = jnp.max( jnp.argwhere( sol.ys[1] < T_end, - size=512 + size=self.max_steps )[:,0] ) @@ -218,11 +228,11 @@ def dY(self, t, Y, args): """ lna, T_g, T_nu = Y - lna_init, rho_extra_init = args + lna_init, rho_extra_init, me = args - rho_EM = thermo.rho_EM_std(T_g, LO=self.LO, NLO=self.NLO) - rho_plus_p_EM = thermo.rho_plus_p_EM_std(T_g, LO=self.LO, NLO=self.NLO) - drho_EM_dT_g = thermo.drho_EM_dT_g_std(T_g, LO=self.LO, NLO=self.NLO) + rho_EM = thermo.rho_EM_std(T_g, me=me, LO=self.LO, NLO=self.NLO) + rho_plus_p_EM = thermo.rho_plus_p_EM_std(T_g, me=me, LO=self.LO, NLO=self.NLO) + drho_EM_dT_g = thermo.drho_EM_dT_g_std(T_g, me=me, LO=self.LO, NLO=self.NLO) rho_nu = 3*thermo.rho_nue_std(T_nu) rho_plus_p_nu = (4/3) * rho_nu @@ -233,7 +243,7 @@ def dY(self, t, Y, args): H = thermo.Hubble(rho_EM + rho_nu + rho_extra) C_rho_nue, C_rho_numu, _, _ = thermo.collision_terms_std( - T_g, T_nu, T_nu, decoupled=self.decoupled, use_FD=self.use_FD, collision_me=self.collision_me + T_g, T_nu, T_nu, me=me, decoupled=self.decoupled, use_FD=self.use_FD, collision_me=self.collision_me ) drho_EM_dt = -3 * H * rho_plus_p_EM - C_rho_nue - 2*C_rho_numu diff --git a/linx/data/background/MB_coefficients.txt b/linx/data/background/MB_coefficients.txt new file mode 100644 index 0000000..9156165 --- /dev/null +++ b/linx/data/background/MB_coefficients.txt @@ -0,0 +1,503 @@ +# Provided by Miguel Escudero, Greg Jackson, Stefan Sandner and Mikko Laine, to appear +# columns: +# m_e/T_ga, fa1, fa2, fa3, fa4, fs1, fs2, fs3, fs4, fn1, fn2, fn3, fn4 + 0.0000e+00 +8.8411e-01 -0.0000e+00 +3.1837e-02 +0.0000e+00 +8.2874e-01 -0.0000e+00 +0.0000e+00 +0.0000e+00 +8.5227e-01 -0.0000e+00 +4.1088e-02 +0.0000e+00 + 1.0000e-01 +8.8358e-01 +1.1518e-03 +3.1942e-02 -2.3151e-04 +8.2757e-01 -5.0208e-04 +0.0000e+00 +0.0000e+00 +8.5163e-01 +1.3833e-03 +4.1222e-02 -2.9850e-04 + 2.0000e-01 +8.8199e-01 +4.5959e-03 +3.2261e-02 -9.1907e-04 +8.2406e-01 -1.9883e-03 +0.0000e+00 +0.0000e+00 +8.4972e-01 +5.5151e-03 +4.1637e-02 -1.1806e-03 + 3.0000e-01 +8.7933e-01 +1.0298e-02 +3.2793e-02 -2.0401e-03 +8.1827e-01 -4.4013e-03 +0.0000e+00 +0.0000e+00 +8.4656e-01 +1.2339e-02 +4.2356e-02 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a/linx/tau_n_vary_me.py b/linx/tau_n_vary_me.py new file mode 100644 index 0000000..f1c8de6 --- /dev/null +++ b/linx/tau_n_vary_me.py @@ -0,0 +1,34 @@ +import jax.numpy as jnp +import linx.const as const +from jax import jit + + +def tau_n_fac_vary_me(me): + """ Returns tau_n_fac during BBN given + electron mass. + + Parameters + ---------- + me : float + Electron mass during BBN in MeV. + + Returns + ------- + float + The factor by which the neutron lifetime is scaled during + BBN. + """ + + delta = const.mn - const.mp + deltabar = delta/const.me + deltabar_vary_me = delta/me + + # See https://arxiv.org/pdf/1801.08023 Eq (91) + def f_int(deltabar): + return jnp.sqrt(deltabar**2 - 1) * (-8 - 9*deltabar**2 + 2*deltabar**4)/60. + deltabar/4 * jnp.arccosh(deltabar) + + f_int_0 = f_int(deltabar) + f_int_BBN = f_int(deltabar_vary_me) + + # new tau_n_fac = tau_n_BBN / const.tau_n + return const.me**5/me**5 * f_int_0/f_int_BBN \ No newline at end of file diff --git a/linx/thermo.py b/linx/thermo.py index 6758d5f..ffb94e6 100644 --- a/linx/thermo.py +++ b/linx/thermo.py @@ -9,6 +9,7 @@ import linx.const as const from linx.special_funcs import Li, K1, K2 +from linx.P_QED import explicit_P2, explicit_P3, dPdTQED_2,dPdTQED_3 ########################################### # Cosmology # @@ -634,19 +635,23 @@ def p_massive_MB(T, mu, m, g): file_dir = os.path.dirname(__file__) -# QED Corrections - flip to ensure monotonically increasing T for interpax.interp1d +# QED Corrections - flip to ensure monotonically increasing T for interpax.interp1d (assume me = 0.511 MeV) P_QED_tab = np.flip(np.loadtxt(file_dir+"/data/background/"+"QED_P_int.txt"), axis=0) dPdT_QED_tab = np.flip(np.loadtxt(file_dir+"/data/background/"+"QED_dP_intdT.txt"), axis=0) -d2PdT2_QED_tab = np.flip(np.loadtxt(file_dir+"/data/background/"+"QED_d2P_intdT2.txt"), axis=0) +# d2PdT2_QED_tab = np.flip(np.loadtxt(file_dir+"/data/background/"+"QED_d2P_intdT2.txt"), axis=0) # CG: JAX grad obviates this import... -# Effect of standard value of electron mass in scattering matrix elements +# Effect of standard value of electron mass in scattering matrix elements (assume me = 0.511 MeV) f_nue_scat_tab = np.loadtxt(file_dir+"/data/background/"+"nue_scatt.txt") f_numu_scat_tab = np.loadtxt(file_dir+"/data/background/"+"numu_scatt.txt") -# Effect of standard value of electron mass in annihilation matrix elements +# Effect of standard value of electron mass in annihilation matrix elements (assume me = 0.511 MeV) f_nue_ann_tab = np.loadtxt(file_dir+"/data/background/"+"nue_ann.txt") f_numu_ann_tab = np.loadtxt(file_dir+"/data/background/"+"numu_ann.txt") +# Use scattering coefficients provided by Miguel Escudero, Greg Jackson, Stefan Sandner and Mikko Laine, to appear +# no assumption that me = 0.511 MeV +f_coeffs = np.loadtxt(file_dir+"/data/background/"+"MB_coefficients.txt") + try: gpus = devices('gpu') P_QED_tab = device_put( @@ -655,9 +660,9 @@ def p_massive_MB(T, mu, m, g): dPdT_QED_tab = device_put( dPdT_QED_tab, device=gpus[0] ) - d2PdT2_QED_tab = device_put( - d2PdT2_QED_tab , device=gpus[0] - ) + # d2PdT2_QED_tab = device_put( + # d2PdT2_QED_tab , device=gpus[0] + # ) f_nue_scat_tab = device_put( f_nue_scat_tab, device=gpus[0] @@ -672,6 +677,10 @@ def p_massive_MB(T, mu, m, g): f_numu_ann_tab = device_put( f_numu_ann_tab, device=gpus[0] ) + + f_coeffs = device_put( + f_coeffs, device=gpus[0] + ) except (RuntimeError, IndexError): # No GPU available or no GPU devices found - data stays on CPU pass @@ -681,7 +690,7 @@ def p_massive_MB(T, mu, m, g): # Standard EM Sector # ###################### -def rho_EM_std(T_g, mu=0, LO=True, NLO=True): +def rho_EM_std(T_g, mu=0, me=const.me, LO=True, NLO=True): """ Total energy density of EM-coupled SM fluids. @@ -692,6 +701,8 @@ def rho_EM_std(T_g, mu=0, LO=True, NLO=True): mu : float, optional Parameter added for syntax consistency--does not impact function behavior. Defaults to 0. + me : float, optional + Electron mass in MeV. Defaults to const.me. LO : bool True includes leading order QED corrections to the energy density. Defaults to 'True'. @@ -705,25 +716,28 @@ def rho_EM_std(T_g, mu=0, LO=True, NLO=True): Units of MeV^4. """ - corr_QED = ( - -interpax.interp1d( - T_g, P_QED_tab[:,0], - LO*P_QED_tab[:,1]+NLO*P_QED_tab[:,2] - ) - + T_g*interpax.interp1d( - T_g, dPdT_QED_tab[:,0], - LO*dPdT_QED_tab[:,1]+NLO*dPdT_QED_tab[:,2] - ) + corr_QED = jnp.where(jnp.abs(me/const.me - 1) > 1e-8, # if input me is sufficiently different from const.me, + -(LO*explicit_P2(T_g, me) + NLO*explicit_P3(T_g, me)) + T_g*(LO*dPdTQED_2(T_g, me) + NLO*dPdTQED_3(T_g, me)), # compute the QED correction + ( + -interpax.interp1d( + T_g, P_QED_tab[:,0], + LO*P_QED_tab[:,1]+NLO*P_QED_tab[:,2] + ) + + T_g*interpax.interp1d( + T_g, dPdT_QED_tab[:,0], + LO*dPdT_QED_tab[:,1]+NLO*dPdT_QED_tab[:,2] + ) + ) # otherwise just use pretabulated values ) return ( - rho_massless_BE(T_g, 0., 2) + rho_massive_FD(T_g, 0., const.me, 4) + rho_massless_BE(T_g, 0., 2) + rho_massive_FD(T_g, 0., me, 4) + corr_QED ) rho_EM_std_v = vmap(rho_EM_std, in_axes=0) -def p_EM_std(T_g, mu=0, LO=True, NLO=True): +def p_EM_std(T_g, mu=0, me=const.me, LO=True, NLO=True): """ Total pressure of EM-coupled SM fluids. @@ -734,6 +748,8 @@ def p_EM_std(T_g, mu=0, LO=True, NLO=True): mu : float, optional Parameter added for syntax consistency--does not impact function behavior. Defaults to 0. + me : float, optional + Electron mass in MeV. Defaults to const.me. LO : bool True includes leading order QED corrections to the pressure. Defaults to 'True'. @@ -747,19 +763,23 @@ def p_EM_std(T_g, mu=0, LO=True, NLO=True): Units of MeV^4. """ - corr_QED = interpax.interp1d( - T_g, P_QED_tab[:,0], - LO*P_QED_tab[:,1] + NLO*P_QED_tab[:,2] + corr_QED = jnp.where(jnp.abs(me/const.me - 1) > 1e-8, # if input me is sufficiently different from const.me, + LO*explicit_P2(T_g, me) + NLO*explicit_P3(T_g, me), # compute the QED correction + interpax.interp1d( + T_g, P_QED_tab[:,0], + LO*P_QED_tab[:,1] + NLO*P_QED_tab[:,2] + ) # otherwise just use pretabulated values ) + return ( - p_massless_BE(T_g, 0., 2) + p_massive_FD(T_g, 0., const.me, 4) + p_massless_BE(T_g, 0., 2) + p_massive_FD(T_g, 0., me, 4) + corr_QED ) p_EM_std_v = vmap(p_EM_std, in_axes=0) -def rho_plus_p_EM_std(T_g, mu=0, LO=True, NLO=True): +def rho_plus_p_EM_std(T_g, mu=0, me=const.me, LO=True, NLO=True): """ Sum of energy densities and pressures of all EM-coupled SM fluids. @@ -770,6 +790,8 @@ def rho_plus_p_EM_std(T_g, mu=0, LO=True, NLO=True): mu : float, optional Parameter added for syntax consistency--does not impact function behavior. Defaults to 0. + me : float, optional + Electron mass in MeV. Defaults to const.me. LO : bool True includes leading order QED corrections to the energy density and pressure. Defaults to 'True'. @@ -782,15 +804,18 @@ def rho_plus_p_EM_std(T_g, mu=0, LO=True, NLO=True): float Units of MeV^4. """ - - corr_QED = T_g * interpax.interp1d( - T_g, dPdT_QED_tab[:,0], - LO*dPdT_QED_tab[:,1] + NLO*dPdT_QED_tab[:,2] + + corr_QED = jnp.where(jnp.abs(me/const.me - 1) > 1e-8, # if input me is sufficiently different from const.me, + T_g*(LO*dPdTQED_2(T_g, me) + NLO*dPdTQED_3(T_g, me)), # compute the QED correction + T_g * interpax.interp1d( + T_g, dPdT_QED_tab[:,0], + LO*dPdT_QED_tab[:,1] + NLO*dPdT_QED_tab[:,2] + ) # otherwise just use pretabulated values ) return ( - 4/3 * rho_massless_BE(T_g, 0., 2) + rho_massive_FD(T_g, 0., 0.511, 4) - + p_massive_FD(T_g, 0., 0.511, 4) + corr_QED + 4/3 * rho_massless_BE(T_g, 0., 2) + rho_massive_FD(T_g, 0., me, 4) + + p_massive_FD(T_g, 0., me, 4) + corr_QED ) def T_g(rho_g): @@ -957,7 +982,7 @@ def T_nu(rho_nu): def collision_terms_std( - T_g, T_nue, T_numt, mu_nue=0., mu_numt=0., + T_g, T_nue, T_numt, me=const.me, mu_nue=0., mu_numt=0., decoupled=False, use_FD=True, collision_me=True ): """ @@ -972,6 +997,8 @@ def collision_terms_std( Electron neutrino temperature in MeV. T_numt : array_like Mu, tau neutrino temperature in MeV. + me : float, optional + Electron mass in MeV. Defaults to const.me mu_nue : float, optional Chemical potential of electron neutrinos in MeV. Defaults to 0. @@ -1006,6 +1033,11 @@ def collision_terms_std( 0. ) + geL = const.geL + geR = const.geR + gmuL = const.gmuL + gmuR = const.gmuR + def G(T_1, mu_1, T_2, mu_2): return ( @@ -1018,72 +1050,177 @@ def G(T_1, mu_1, T_2, mu_2): ) ) - def G_nue_with_me(T_1, mu_1, T_2, mu_2): + def G_nue_with_me(T_1, mu_1, T_2, mu_2, me): + # CG: update to use interp1d + def interp_fa1(f_tab): + index = 1 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] + ) + + def interp_fa2(f_tab): + index = 2 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] + ) - def interp_f(f_tab): - # Tables have boundary values 0.0 (low T) and 1.0 (high T) - return interpax.interp1d( - T_1, f_tab[:,0], f_tab[:,1], extrap=(0.0, 1.0) + def interp_fs1(f_tab): + index = 5 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] ) - f_nue_ann = lax.cond( - collision_me, interp_f, lambda _: 1., f_nue_ann_tab + def interp_fs2(f_tab): + index = 6 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] + ) +# def interp_f(f_tab): +# # Tables have boundary values 0.0 (low T) and 1.0 (high T) +# return interpax.interp1d( +# T_1, f_tab[:,0], f_tab[:,1], extrap=(0.0, 1.0) +# ) + + + # def interp_f(f_tab): + + # return jnp.interp( + # T_1, f_tab[:,0], f_tab[:,1], left=f_tab[0,1], right=f_tab[-1,1] + # ) + + # f_nue_ann = lax.cond( + # collision_me, interp_f, lambda _: 1., f_nue_ann_tab + # ) + # f_nue_scat = lax.cond( + # collision_me, interp_f, lambda _: 1., f_nue_scat_tab + # ) + + f_ann_1 = lax.cond( + collision_me, interp_fa1, lambda _: 1., f_coeffs + ) + f_scat_1 = lax.cond( + collision_me, interp_fs1, lambda _: 1., f_coeffs ) - f_nue_scat = lax.cond( - collision_me, interp_f, lambda _: 1., f_nue_scat_tab + + f_ann_2 = lax.cond( + collision_me, interp_fa2, lambda _: 1., f_coeffs + ) + f_scat_2 = lax.cond( + collision_me, interp_fs2, lambda _: 1., f_coeffs ) - return ( - 32 * f_a * f_nue_ann * ( + return ( # note f_a and f_s are now folded into f_nue_ann/scat + 4 * (geL**2 + geR**2) * (32 * f_ann_1 * ( T_1**9 * jnp.exp(2 * mu_1 / T_1) - T_2**9 * jnp.exp(2 * mu_2 / T_2) - ) - + 56 * f_s * f_nue_scat * ( - jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) - * T_1**4 * T_2**4 * (T_1 - T_2) + ) + + 56 * f_scat_1 * ( + jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) + * T_1**4 * T_2**4 * (T_1 - T_2) + ) + ) + # new terms (previously baked into tabulated rates) + + 4 * geL*geR * (f_ann_2 * 32 * ( + T_1**9 * jnp.exp(2 * mu_1 / T_1) + - T_2**9 * jnp.exp(2 * mu_2 / T_2) + ) + + 56 * f_scat_2 * ( + jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) + * T_1**4 * T_2**4 * (T_1 - T_2) + ) ) ) + # CG: update to use interp1d + def G_numt_with_me(T_1, mu_1, T_2, mu_2, me): - def G_numt_with_me(T_1, mu_1, T_2, mu_2): + def interp_fa1(f_tab): + index = 1 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] + ) - def interp_f(f_tab): - # Tables have boundary values 0.0 (low T) and 1.0 (high T) - return interpax.interp1d( - T_1, f_tab[:,0], f_tab[:,1], extrap=(0.0, 1.0) + def interp_fa2(f_tab): + index = 2 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] ) - f_numt_ann = lax.cond( - collision_me, interp_f, lambda _: 1., f_numu_ann_tab + def interp_fs1(f_tab): + index = 5 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index] + ) + + def interp_fs2(f_tab): + index = 6 + return jnp.interp( + me/T_1, f_tab[:,0], f_tab[:,index], left=f_tab[0,index], right=f_tab[-1,index]) +# def G_numt_with_me(T_1, mu_1, T_2, mu_2): + +# def interp_f(f_tab): +# # Tables have boundary values 0.0 (low T) and 1.0 (high T) +# return interpax.interp1d( +# T_1, f_tab[:,0], f_tab[:,1], extrap=(0.0, 1.0) +# ) + + # def interp_f(f_tab): + + # return jnp.interp( + # T_1, f_tab[:,0], f_tab[:,1], left=f_tab[0,1], right=f_tab[-1,1] + # ) + + # f_numt_ann = lax.cond( + # collision_me, interp_f, lambda _: 1., f_numu_ann_tab + # ) + # f_numt_scat = lax.cond( + # collision_me, interp_f, lambda _: 1., f_numu_scat_tab + # ) + + f_ann_1 = lax.cond( + collision_me, interp_fa1, lambda _: 1., f_coeffs + ) + f_scat_1 = lax.cond( + collision_me, interp_fs1, lambda _: 1., f_coeffs ) - f_numt_scat = lax.cond( - collision_me, interp_f, lambda _: 1., f_numu_scat_tab + + f_ann_2 = lax.cond( + collision_me, interp_fa2, lambda _: 1., f_coeffs + ) + f_scat_2 = lax.cond( + collision_me, interp_fs2, lambda _: 1., f_coeffs ) - return ( - 32 * f_a * f_numt_ann * ( + return ( # f_s, f_a now folded into f_ann and f_scat + 4 * (gmuL**2 + gmuR**2) * (32 * f_ann_1 * ( T_1**9 * jnp.exp(2 * mu_1 / T_1) - T_2**9 * jnp.exp(2 * mu_2 / T_2) - ) - + 56 * f_s * f_numt_scat * ( - jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) - * T_1**4 * T_2**4 * (T_1 - T_2) + ) + + 56 * f_scat_1 * ( + jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) + * T_1**4 * T_2**4 * (T_1 - T_2) + ) + ) + # new terms (previously baked into tabulated rates) + + 4 * gmuL*gmuR * (f_ann_2 * 32 * ( + T_1**9 * jnp.exp(2 * mu_1 / T_1) + - T_2**9 * jnp.exp(2 * mu_2 / T_2) + ) + + 56 * f_scat_2 * ( + jnp.exp(2 * mu_1 / T_1) * jnp.exp(2 * mu_2 / T_2) + * T_1**4 * T_2**4 * (T_1 - T_2) + ) ) ) - geL = const.geL - geR = const.geR - gmuL = const.gmuL - gmuR = const.gmuR - # Units MeV^4 s^-1 - C_rho_nue = const.GF**2 / jnp.pi**5 * ( - 4 * (geL**2 + geR**2) * G_nue_with_me(T_g, 0., T_nue, mu_nue) + C_rho_nue = const.GF**2 / jnp.pi**5 * ( # prev coeff now in G def + G_nue_with_me(T_g, 0., T_nue, mu_nue, me) + 2 * G(T_numt, mu_numt, T_nue, mu_nue) ) / const.hbar # Units MeV^4 s^-1 C_rho_numu = const.GF**2 / jnp.pi**5 * ( - 4 * (gmuL**2 + gmuR**2) * G_numt_with_me(T_g, 0., T_numt, mu_numt) + G_numt_with_me(T_g, 0., T_numt, mu_numt, me) - G(T_numt, mu_numt, T_nue, mu_nue) ) / const.hbar @@ -1106,5 +1243,5 @@ def interp_f(f_tab): - T_nue**8 * jnp.exp(2 * mu_nue / T_nue) ) ) / const.hbar - + return C_rho_nue, C_rho_numu, C_n_nue, C_n_numu \ No newline at end of file diff --git a/linx/weak_rates.py b/linx/weak_rates.py index e187ce1..38b2759 100644 --- a/linx/weak_rates.py +++ b/linx/weak_rates.py @@ -22,7 +22,7 @@ file_dir = os.path.dirname(__file__) # Particle masses -from linx.const import me, mn, mp +from linx.const import mn, mp Q = mn - mp # Mass difference between neutrons and protons class WeakRates(eqx.Module): @@ -62,8 +62,6 @@ class WeakRates(eqx.Module): L_nTOpCCRTh_res : list L_pTOnCCRTh_res : list - lambda_0 : float - def __init__(self, RC_corr=True, FM_corr=True, weak_mag_corr=True, thermal_corr=True @@ -131,24 +129,11 @@ def __init__(self, self.L_nTOpCCRTh_res = [] self.L_pTOnCCRTh_res = [] - self.lambda_0 = 0. - - # Slight shift in limits to avoid unimportant singularities. - en_vals = jnp.linspace(1.+.1e-6, Q/me-1e-6, 1000) - dlambda_den_vals = self.dlambda_den_RC(en_vals) - self.lambda_0 += trapz(dlambda_den_vals, en_vals) - - if self.FM_corr: - - pe_vals = jnp.linspace( - 0.+1e-4, jnp.sqrt((Q/me)**2 - 1.)-1e-5, 1000 - ) - y_vals = self.dlambda_dp_FM(pe_vals) - self.lambda_0 += trapz(y_vals, pe_vals) + @eqx.filter_jit def __call__( - self, T_vec_ref, T_start, T_end, sampling_nTOp + self, T_vec_ref, T_start, T_end, sampling_nTOp, me=const.me, ): """ Evaluate n <-> p rates over range of EM temperatures. @@ -163,6 +148,8 @@ def __call__( Lowest photon temperature to evaluate rates at. In MeV. sampling_nTOp : int Number of points between T_start and T_end to evaluate at. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -178,12 +165,12 @@ def __call__( jnp.log10(T_start), jnp.log10(T_end), sampling_nTOp ) - nTOp_rates = self.nTOp_rates(T_interval, T_vec_ref) + nTOp_rates = self.nTOp_rates(T_interval, T_vec_ref, me) return (T_interval, ) + nTOp_rates - @eqx.filter_vmap(in_axes=(None, 0, None)) - def nTOp_rates(self, Tg, T_vec_ref): + @eqx.filter_vmap(in_axes=(None, 0, None, None)) + def nTOp_rates(self, Tg, T_vec_ref, me=const.me): """ Dimensionless n <-> p rates, normalized to neutron decay width. @@ -194,6 +181,8 @@ def nTOp_rates(self, Tg, T_vec_ref): T_vec_ref : tuple (EM temperature array, neutrino temperature array) in MeV, used for computing the weak rates. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -202,6 +191,21 @@ def nTOp_rates(self, Tg, T_vec_ref): dimensionless, normalized to the neutron decay width. """ + lambda_0 = 0. + + # Slight shift in limits to avoid unimportant singularities. + en_vals = jnp.linspace(1.+.1e-6, Q/me-1e-6, 1000) + dlambda_den_vals = self.dlambda_den_RC(en_vals, me) + lambda_0 += trapz(dlambda_den_vals, en_vals) + + if self.FM_corr: + + pe_vals = jnp.linspace( + 0.+1e-4, jnp.sqrt((Q/me)**2 - 1.)-1e-5, 1000 + ) + y_vals = self.dlambda_dp_FM(pe_vals, me) + lambda_0 += trapz(y_vals, pe_vals) + Tg_vec_ref, Tnu_vec_ref = T_vec_ref Tnu_of_Tg_ref = Tnu_vec_ref / Tg_vec_ref @@ -229,22 +233,22 @@ def nTOp_rates(self, Tg, T_vec_ref): pTOn_rate = 0. y_CCR_vals = jnp.array([ - self.dGamma_nTOp_dp(p_vals, x, xnu), - self.dGamma_pTOn_dp(p_vals, x, xnu) + self.dGamma_nTOp_dp(p_vals, x, xnu, me), + self.dGamma_pTOn_dp(p_vals, x, xnu, me) ]) CCR_rates = trapz(y_CCR_vals, p_vals) - nTOp_rate += CCR_rates[0] / self.lambda_0 - pTOn_rate += CCR_rates[1] / self.lambda_0 + nTOp_rate += CCR_rates[0] / lambda_0 + pTOn_rate += CCR_rates[1] / lambda_0 if self.FM_corr: y_FMCCR_vals = jnp.array([ - self.ddelt_Gamma_nTOp_FM_dp(p_vals, x, xnu), - self.ddelt_Gamma_pTOn_FM_dp(p_vals, x, xnu) + self.ddelt_Gamma_nTOp_FM_dp(p_vals, x, xnu, me), + self.ddelt_Gamma_pTOn_FM_dp(p_vals, x, xnu, me) ]) FMCCR_rates = trapz(y_FMCCR_vals, p_vals) - nTOp_rate += FMCCR_rates[0] / self.lambda_0 - pTOn_rate += FMCCR_rates[1] / self.lambda_0 + nTOp_rate += FMCCR_rates[0] / lambda_0 + pTOn_rate += FMCCR_rates[1] / lambda_0 if self.thermal_corr: @@ -264,13 +268,13 @@ def nTOp_rates(self, Tg, T_vec_ref): right=self.L_pTOnCCRTh_res[-1] ) ]) - nTOp_rate += thermal_rates[0] / self.lambda_0 - pTOn_rate += thermal_rates[1] / self.lambda_0 + nTOp_rate += thermal_rates[0] / lambda_0 + pTOn_rate += thermal_rates[1] / lambda_0 return (nTOp_rate, pTOn_rate) - def Sirlin_G(self, kmax, en): + def Sirlin_G(self, kmax, en, me=const.me): """ Sirlin's universal function. @@ -281,6 +285,8 @@ def Sirlin_G(self, kmax, en): radiative correction, normalized to electron mass. en : float Dimensionless electron energy, normalized to electron mass) + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -304,7 +310,7 @@ def Sirlin_G(self, kmax, en): + (4./b)*(-spence(1. - 2.*b/(1. + b))) ) - def R_RC(self, kmax, en): + def R_RC(self, kmax, en, me=const.me): """ Resummed radiative correction term at T = 0. @@ -315,6 +321,8 @@ def R_RC(self, kmax, en): radiative correction, normalized to electron mass. en : float Dimensionless electron energy, normalized to electron mass) + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -343,14 +351,14 @@ def R_RC(self, kmax, en): return ( ( 1. + const.aFS / (2.*jnp.pi) * ( - self.Sirlin_G(kmax, en) - 3.*jnp.log(mp / (2*Q)) + self.Sirlin_G(kmax, en, me) - 3.*jnp.log(mp / (2*Q)) ) ) * (L + (const.aFS/jnp.pi)*C + delta_factor) * ( S + 1./(134.*2.*jnp.pi)*(jnp.log(mp/mA) + A_g) + NLL ) ) - def Fermi(self, b): + def Fermi(self, b, me=const.me): """ Fermi function in the relativistic limit. @@ -358,6 +366,8 @@ def Fermi(self, b): ---------- b : float Speed of the electron. + me: float, optional + Electron mass in MeV. Defaults to const.me Notes ----- @@ -378,7 +388,7 @@ def Fermi(self, b): * jnp.abs(gamma(gamma1 + const.aFS/b*1j))**2 ) - def bFermi(self, b): + def bFermi(self, b, me=const.me): """ Fermi function in the relativistic limit, times speed of electron. @@ -386,6 +396,8 @@ def bFermi(self, b): ---------- b : float Speed of the electron. + me: float, optional + Electron mass in MeV. Defaults to const.me Notes ----- @@ -394,10 +406,10 @@ def bFermi(self, b): """ - return b*self.Fermi(b) + return b*self.Fermi(b, me) - @eqx.filter_vmap(in_axes=(None, 0)) - def dlambda_den_RC(self, en): + @eqx.filter_vmap(in_axes=(None, 0, None)) + def dlambda_den_RC(self, en, me=const.me): """ Derivative of lambda with respect to energy of electron, including radiative corrections at T = 0. @@ -407,6 +419,8 @@ def dlambda_den_RC(self, en): en : float Energy of the electron produced in neutron decay, normalized to the electron mass. + me: float, optional + Electron mass in MeV. Defaults to const.me Returns ------- @@ -423,8 +437,8 @@ def dlambda_den_RC(self, en): if self.RC_corr: R_term = ( - self.Fermi(b) - * self.R_RC(q-en, en) + self.Fermi(b, me) + * self.R_RC(q-en, en, me) ) else: R_term = 1. @@ -433,7 +447,7 @@ def dlambda_den_RC(self, en): en**2 * (q-en)**2 * b * R_term ) - def chi_n_decay(self, pe): + def chi_n_decay(self, pe, me=const.me): """ Finite nucleon mass correction term for neutron decay. @@ -442,6 +456,8 @@ def chi_n_decay(self, pe): pe : float Dimensionless momentum of the electron, normalized to electron mass. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -485,8 +501,8 @@ def chi_n_decay(self, pe): + f3 * (q - en)**2 * (pe**2) / (M*en) ) - @eqx.filter_vmap(in_axes=(None, 0)) - def dlambda_dp_FM(self, pe): + @eqx.filter_vmap(in_axes=(None, 0, None)) + def dlambda_dp_FM(self, pe, me=const.me): """ Derivative of the correction to lambda with respect to momentum, due to finite mass effects. @@ -496,6 +512,8 @@ def dlambda_dp_FM(self, pe): pe : float Dimensionless momentum of the electron, normalized to the electron mass. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -514,18 +532,18 @@ def dlambda_dp_FM(self, pe): if self.RC_corr: - R_rad = self.R_RC(Q/me - en, en) + R_rad = self.R_RC(Q/me - en, en, me) else: R_rad = 1. return ( - pe**2 * self.chi_n_decay(pe) - * R_rad * self.Fermi(b) + pe**2 * self.chi_n_decay(pe, me) + * R_rad * self.Fermi(b, me) ) - def chi_Born(self, en, x, x_nu, sgnq): + def chi_Born(self, en, x, x_nu, sgnq, me=const.me): """ Integrand in momentum integral for Born weak rate. @@ -539,6 +557,8 @@ def chi_Born(self, en, x, x_nu, sgnq): Dimensionless inverse nu temperature, normalized to electron mass. sqnq : int Should have value +1 or -1, to switch between chi_+ and chi_-. + me : float, optional + Electron mass in MeV. Defaults to const.me. Notes ----- @@ -555,7 +575,7 @@ def chi_Born(self, en, x, x_nu, sgnq): return e_nu**2 * g_nu * g_e - def Fermi_sgn(self, sgnq, sgnE, b): + def Fermi_sgn(self, sgnq, sgnE, b, me=const.me): """ Fermi function, with a check for proton and electron final state. @@ -567,6 +587,9 @@ def Fermi_sgn(self, sgnq, sgnE, b): +1 or -1 to choose between positive or negative arguments of F. b : float The speed of the electron. + me : float, optional + Electron mass in MeV. Defaults to const.me. + Returns ------- @@ -577,14 +600,14 @@ def Fermi_sgn(self, sgnq, sgnE, b): See Pitrou+ 1801.08023 Eq. (102). """ - result = jnp.where(sgnq*sgnE > 0, self.Fermi(b), 1.) + result = jnp.where(sgnq*sgnE > 0, self.Fermi(b, me), 1.) return result ############################### # n<->p Rates # ############################### - def dGamma_dp(self, p, x, x_nu, sgnq): + def dGamma_dp(self, p, x, x_nu, sgnq, me=const.me): """ Integrand over momentum for n <-> p rate. including radiative corrections. @@ -602,6 +625,8 @@ def dGamma_dp(self, p, x, x_nu, sgnq): electron mass. sgnq : int +1 or -1, to select between n -> p or p -> n. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -618,12 +643,12 @@ def dGamma_dp(self, p, x, x_nu, sgnq): if self.RC_corr: RC_term_plus = ( - self.R_RC(jnp.abs(sgnq*Q/me - en), en) - * self.Fermi_sgn(sgnq, 1, p/en) + self.R_RC(jnp.abs(sgnq*Q/me - en), en, me) + * self.Fermi_sgn(sgnq, 1, p/en, me) ) RC_term_minus = ( - self.R_RC(jnp.abs(sgnq*Q/me + en), en) - * self.Fermi_sgn(sgnq, -1, p/en) + self.R_RC(jnp.abs(sgnq*Q/me + en), en, me) + * self.Fermi_sgn(sgnq, -1, p/en, me) ) else: RC_term_plus = 1. @@ -631,16 +656,16 @@ def dGamma_dp(self, p, x, x_nu, sgnq): return p**2 * ( ( - self.chi_Born(en, x, x_nu, sgnq) + self.chi_Born(en, x, x_nu, sgnq, me) * RC_term_plus ) + ( - self.chi_Born(-en, x, x_nu, sgnq) + self.chi_Born(-en, x, x_nu, sgnq, me) * RC_term_minus ) ) - @eqx.filter_vmap(in_axes=(None, 0, None, None)) - def dGamma_nTOp_dp(self, p, x, xnu): + @eqx.filter_vmap(in_axes=(None, 0, None, None, None)) + def dGamma_nTOp_dp(self, p, x, xnu, me=const.me): """ Integrand over momentum for n -> p rate including radiative corrections. @@ -656,6 +681,8 @@ def dGamma_nTOp_dp(self, p, x, xnu): x_nu : float Dimensionless neutrino inverse temperature, normalized to the electron mass. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -666,10 +693,10 @@ def dGamma_nTOp_dp(self, p, x, xnu): See Pitrou+ 1801.08023 Eq. (101). """ - return self.dGamma_dp(p, x, xnu, 1) + return self.dGamma_dp(p, x, xnu, 1, me) - @eqx.filter_vmap(in_axes=(None, 0, None, None)) - def dGamma_pTOn_dp(self, p, x, xnu): + @eqx.filter_vmap(in_axes=(None, 0, None, None, None)) + def dGamma_pTOn_dp(self, p, x, xnu, me=const.me): """ Integrand over momentum for p -> n rate including radiative corrections. @@ -687,6 +714,8 @@ def dGamma_pTOn_dp(self, p, x, xnu): electron mass. sgnq : int +1 or -1, to select between n -> p or p -> n. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -697,7 +726,7 @@ def dGamma_pTOn_dp(self, p, x, xnu): See Pitrou+ 1801.08023 Eq. (104). """ - return self.dGamma_dp(p, x, xnu, -1) + return self.dGamma_dp(p, x, xnu, -1, me) ########################### @@ -705,7 +734,7 @@ def dGamma_pTOn_dp(self, p, x, xnu): ########################### - def chi_FM(self, en, x, x_nu, sgnq): + def chi_FM(self, en, x, x_nu, sgnq, me=const.me): """ Integrand over momentum for finite mass correction to n <-> p rate. @@ -722,6 +751,8 @@ def chi_FM(self, en, x, x_nu, sgnq): electron mass. sgnq : int +1 or -1 corresponding to chi_+ or chi_-. + me : float, optional + Electron mass in MeV. Defaults to const.me. Notes ----- @@ -824,7 +855,7 @@ def chi_FM(self, en, x, x_nu, sgnq): ) return result - def ddelt_Gamma_FM_dp(self, p, x, znu, sgnq): + def ddelt_Gamma_FM_dp(self, p, x, znu, sgnq, me=const.me): """ Integrand over momentum for finite mass corrections to the n <-> p rate. @@ -842,6 +873,8 @@ def ddelt_Gamma_FM_dp(self, p, x, znu, sgnq): electron mass. sgnq : int +1 or -1, to select between n -> p or p -> n. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -858,11 +891,11 @@ def ddelt_Gamma_FM_dp(self, p, x, znu, sgnq): if self.RC_corr: RC_term_plus = self.R_RC( - jnp.abs(sgnq*Q/me - en), en - ) * self.Fermi_sgn(sgnq, 1, b) + jnp.abs(sgnq*Q/me - en), en, me + ) * self.Fermi_sgn(sgnq, 1, b, me) RC_term_minus = self.R_RC( - jnp.abs(sgnq*Q/me + en), en - ) * self.Fermi_sgn(sgnq, -1, b) + jnp.abs(sgnq*Q/me + en), en, me + ) * self.Fermi_sgn(sgnq, -1, b, me) else: RC_term_plus = 1. @@ -870,17 +903,17 @@ def ddelt_Gamma_FM_dp(self, p, x, znu, sgnq): result = p**2 * ( ( - self.chi_FM(en, x, znu, sgnq) + self.chi_FM(en, x, znu, sgnq, me) * RC_term_plus ) + ( - self.chi_FM(-en, x, znu, sgnq) + self.chi_FM(-en, x, znu, sgnq, me) * RC_term_minus ) ) return result - @eqx.filter_vmap(in_axes=(None, 0, None, None)) - def ddelt_Gamma_nTOp_FM_dp(self, p, x, xnu): + @eqx.filter_vmap(in_axes=(None, 0, None, None, None)) + def ddelt_Gamma_nTOp_FM_dp(self, p, x, xnu, me=const.me): """ Integrand over momentum for finite mass corrections to the n -> p rate. @@ -896,6 +929,8 @@ def ddelt_Gamma_nTOp_FM_dp(self, p, x, xnu): x_nu : float Dimensionless neutrino inverse temperature, normalized to the electron mass. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -906,10 +941,10 @@ def ddelt_Gamma_nTOp_FM_dp(self, p, x, xnu): See Pitrou+ 1801.08023 Eq. (115). """ - return self.ddelt_Gamma_FM_dp(p, x, xnu, 1) + return self.ddelt_Gamma_FM_dp(p, x, xnu, 1, me) - @eqx.filter_vmap(in_axes=(None, 0, None, None)) - def ddelt_Gamma_pTOn_FM_dp(self, p, x, xnu): + @eqx.filter_vmap(in_axes=(None, 0, None, None, None)) + def ddelt_Gamma_pTOn_FM_dp(self, p, x, xnu, me=const.me): """ Integrand over momentum for finite mass corrections to the p -> n rate. @@ -925,6 +960,8 @@ def ddelt_Gamma_pTOn_FM_dp(self, p, x, xnu): x_nu : float Dimensionless neutrino inverse temperature, normalized to the electron mass. + me : float, optional + Electron mass in MeV. Defaults to const.me. Returns ------- @@ -935,4 +972,4 @@ def ddelt_Gamma_pTOn_FM_dp(self, p, x, xnu): See Pitrou+ 1801.08023 Eq. (115). """ - return self.ddelt_Gamma_FM_dp(p, x, xnu, -1) \ No newline at end of file + return self.ddelt_Gamma_FM_dp(p, x, xnu, -1, me) \ No newline at end of file diff --git a/scripts/test_REMOVE.py b/scripts/test_REMOVE.py new file mode 100644 index 0000000..0d9ff4f --- /dev/null +++ b/scripts/test_REMOVE.py @@ -0,0 +1,33 @@ +import jax.numpy as jnp +import jax +from jax import jit, vmap +import sys + +sys.path.append("../") +import linx.const as const +from linx.nuclear import NuclearRates +from linx.background import BackgroundModel +from linx.abundances import AbundanceModel + +thermo_model_DNeff = BackgroundModel() + +( + t_vec_ref, a_vec_ref, rho_g_vec, rho_nu_vec, rho_NP_vec, P_NP_vec, Neff_vec +) = thermo_model_DNeff(jnp.asarray(0.)) + +network = 'key_PRIMAT_2023' +# network = 'key_PRIMAT_2018' +# network = 'key_PArthENoPE' +# network = 'key_YOF' +abundance_model = AbundanceModel(NuclearRates(nuclear_net=network)) + +Planck_omega_b_res = abundance_model( + rho_g_vec, # photon energy density + rho_nu_vec, # neutrino energy density + rho_NP_vec, # energy density of extra species + P_NP_vec, # pressure of extra species + t_vec=t_vec_ref, # vector of times at which quantities are given + a_vec=a_vec_ref # vector of scale factor at corresponding times +) + +print('n: ', Planck_omega_b_res[0]) \ No newline at end of file