{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"b3161130-e3a1-547a-b6e5-124d59b2de24","task_key":"train--Challenge~5f9~5fmain","task_revision_id":"1","upstream_id":"Challenge_9_main","short_description":"In order to introduce torsion to the system, one can use the first-order…","config":"","split":"train","body":"{\"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\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}