{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"研究水準の物理問題で、科学的理解と多段階の推論・計算を評価するベンチマークです。公開データには70の課題があり、問題文とコード雛形を組み合わせて解答を構成します。\n\nCritPt 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"},"task_id":"757044b3-fd81-5f78-b39a-5d3a709ea62e","task_key":"train--Challenge~5f8~5fmain","task_revision_id":"1","upstream_id":"Challenge_8_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 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 $\\\\frac{2AH}{\\\\dot{\\\\vartheta}\\\\delta\\\\vartheta}$?\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}