# CritPt / Challenge_8_main

task_id: 757044b3-fd81-5f78-b39a-5d3a709ea62e
task_key: train--Challenge~5f8~5fmain
task_revision_id: 1

{"code_template":"def answer():\n    r\"\"\"\n    Return the value of the requested quantity at horizon crossing at 60 e-folds before inflation ends\n\n    Inputs\n    ----------\n    None\n\n    Outputs\n    ----------\n    value : float, the value of the requested quantity at horizon crossing at 60 e-folds before inflation ends\n    perturb_value_1: float, the value of $\\frac{\\delta\\phi}{\\delta\\dot{\\vartheta} - \\dot{\\vartheta}A}$\n    perturb_value_2: float, the value of $\\frac{2AH}{\\dot{\\vartheta}\\delta\\vartheta}$\n\n    \"\"\"\n\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\n    value = ...\n    perturb_value_1 = ...\n    perturb_value_2 = ...\n    # ---------------------------------------------------------------\n\n    return value,perturb_value_1,perturb_value_2","problem_description":"\n\n# Problem setup:\nIn order to introduce torsion to the system, one can use the first-order formulation of general relativity. We define a local reference frame at each point of the $(3+1)$-dimensional manifold $\\mathcal{M}$, the tetrad $e^A_\\mu$, such that the metric can be written as $g_{\\mu\\nu}=e^A_\\mu e^B_\\nu \\eta_{AB}$, where $\\eta_{AB}$ is the flat Minkowski metric on the internal space of coordinates. The internal indices, denoted by the Latin alphabet, also run from $0$ to $3$ just like the spacetime ones. The metrics $g_{\\mu\\nu}$ and $\\eta_{AB}$ can raise or lower the spacetime and internal tangent-space indices, respectively. The Levi-Civita symbol shall be denoted by $\\epsilon_{ABCD}$.\n\nThe gravitational action can be reformulated in the first-order form as a function of the tetrad $(e^A)$ and spin-connection variables $(\\omega^{AB})$. Both of these are 1-forms on the manifold $\\mathcal{M}$. In this formalism, the curvature 2-form is\n\\begin{align}\nR^{AB} =d\\omega^{AB} + \\omega^A{}_C\\wedge\\omega^{CB}\\,.\n\\end{align}\n\nStart with the Einstein-Hilbert action $(\\mathcal{S}_{EH})$ in first-order Palatini form, and in first-order Palatini form, add an action term ($\\mathcal{S}_{\\vartheta}$) for a single scalar, $\\vartheta$, with an as-yet unspecified potential that depends on $\\vartheta$, $V(\\vartheta)$. Assume that the scalar $\\vartheta$ depends only on time, $\\vartheta(t)$, and take $c = 1$.\n\nWe add a Nieh-Yan action, which we write as\n\\begin{align}\nS_{NY} = -nf\\int d\\vartheta \\wedge T^A \\wedge e_A,\n\\end{align}\nwhere $T^A$ is the torsion two-form\n\\begin{align}\nT^A =d e^A + \\omega^A{}_B\\wedge e^B\\,.\n\\end{align}\n\nWe introduce the ansatz for the torsion 2-form:\n\\begin{align}\nT^0 =  0,\n\\\\\nT^i = h(t)e^0\\wedge e^i - \\phi(t)\\epsilon^i_{jk} e^j \\wedge e^k.\n\\end{align}\n\nWe split the spin connection into ''Torsion free\" and ''Torsion full\" parts:\n\n\\begin{equation}\n\\omega^{IJ} = \\bar{\\omega}^{IJ} + \\tilde{\\omega}^{IJ}\n\\end{equation}\n\nAssume a FRW geometry. The scale factor is denoted by $a(t)$, where $t$ is the cosmic time and the Hubble parameter is defined as $H(t)$. In the spatially flat gauge, the metric is given by\n\\begin{equation}\n[g_{\\mu\\nu}] = a^{2}(\\eta) [\\eta_{\\mu\\nu} + h_{\\mu\\nu}] =\n  a^{2}(\\eta)\\begin{bmatrix}\n   1+2A &\n   -\\partial_{i}B \\\\\n   -\\partial_{i}B &\n   -\\delta_{ij}\n   \\end{bmatrix}\\,,\n\\end{equation}\nwhere $\\eta$ is conformal time. Hence, the components of the tetrad field $e^{A}\\hspace{0.5pt}_{\\mu}$ are given by\n\\begin{align}\ne^{0}\\hspace{0.5pt}_{0} = a[1+A] , \\quad e^{0}\\hspace{0.5pt}_{i} = a\\partial_{i}\\beta , \\quad e^{a}\\hspace{0.5pt}_{0} = a\\delta^{ai}\\partial_{i}\\zeta , \\quad e^{a}\\hspace{0.5pt}_{i} = a[\\delta_{ia} + \\epsilon_{aik}\\partial_{k}s]\\,,\n\\end{align}\nwhere we have defined $B = \\zeta - \\beta$ and $s$ is a pseudo-scalar. In the scalar sector, we also have the perturbations\n\\begin{align}\nh =  h(\\eta) + \\delta h(\\eta,\\vec{x})\\,,\\quad\n%\\\\\\nonumber\n\\phi =  \\phi(\\eta) + \\delta \\phi(\\eta,\\vec{x})\\,,\\quad\n%\\\\\n\\vartheta =  \\vartheta(\\eta) + \\delta \\vartheta(\\eta,\\vec{x})\\,.\n\\end{align}\n\n\n# Main problem:\n\n1. $P_{\\mathcal{R}}$ is the curvature power spectrum.\n\nUse the values $n = 0.5$, $V = \\Lambda^{4}[1-cos(\\vartheta/f)]$, $M_{Pl} = 1$, $\\Lambda = 3.7\\times 10^{-3}$, $f = 1.7$, $a[t = 0] = 10$ , $\\vartheta[t = 0] = 5$ and $\\dot{\\vartheta}[t = 0] = 0$, where $\\dot{\\vartheta} = d\\vartheta/dt$, $M_{Pl} = \\frac{1}{\\sqrt{8\\pi G}}$ and $G$ is the gravitational constant, to give the value of\n\\begin{equation}\n\\frac{P_{\\mathcal{R}}(1+3n^2f^2)}{\\frac{H^2}{4\\pi^2M_{Pl}^2}\\Big(\\frac{H}{\\dot{\\vartheta}}\\Big)^2 2^{2\\nu - 3}\\Big|\\frac{\\Gamma(\\nu)}{\\Gamma\\big( \\frac{3}{2} \\big)}\\Big|^2} \\times\\frac{2AH}{\\dot{\\vartheta}\\delta\\vartheta}\\times \\frac{\\beta a\\dot{\\vartheta}}{\\delta\\vartheta}\\times \\frac{\\delta\\phi}{nf\\delta\\dot{\\vartheta} - nf\\dot{\\vartheta}A}\n\\end{equation}\nat horizon crossing at 60 e-folds before inflation ends. $\\nu$ is defined as\n\n\\begin{align}\n\\frac{d^2v}{d\\eta^2} + \\Big[k^2 - \\frac{\\nu^2 - \\frac{1}{4}}{\\eta^2} \\Big]v = 0,\n \\end{align}\nwhich is obtained after solving for all the perturbations.\n\n2. What is $\\frac{\\delta\\phi}{\\delta\\dot{\\vartheta} - \\dot{\\vartheta}A}$?\n\n3. What is $\\frac{2AH}{\\dot{\\vartheta}\\delta\\vartheta}$?"}

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initial import

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GET /api/v1/write?intent=publish&task_id=757044b3-fd81-5f78-b39a-5d3a709ea62e&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}