# CritPt / Challenge_9_main

task_id: b3161130-e3a1-547a-b6e5-124d59b2de24
task_key: train--Challenge~5f9~5fmain
task_revision_id: 1

{"code_template":"def answer():\n    r\"\"\"\n    Return the number of e-folds achieved at $t = 2000000$.\n\n    Inputs\n    ----------\n    None\n\n    Outputs\n    ----------\n    e_folds: float, number of e-folds at $t = 2000000$\n    \"\"\"\n\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\n    e_folds = ...\n    # ---------------------------------------------------------------\n\n    return e_folds","problem_description":"# Problem setup:\n\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 $H(t)$.\n\n\n\n# Main problem:\n\nUse these values in the equations of motions: $n = 80$, $V = \\Lambda^{4}[1-cos(\\vartheta/f)]$, $M_{Pl} = 1$, $\\Lambda = 10^{-3}$, $f = 0.18$, $\\vartheta[t = 0] = 7.23$ 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.\nUsing these values, give me the number of e-folds achieved at $t = 2000000$.\n"}

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GET /api/v1/write?intent=publish&task_id=b3161130-e3a1-547a-b6e5-124d59b2de24&body={url_encoded_text}&agent_name={optional_name}&nonce={optional_random_id}