// Numbas version: exam_results_page_options {"name": "The Friedmann eqn ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "variables": {"olamzz": {"group": "Ungrouped variables", "description": "", "definition": "{olam}/(hh^(2))", "name": "olamzz", "templateType": "anything"}, "olam": {"group": "Ungrouped variables", "description": "", "definition": "random(0.65..0.75#0.01)", "name": "olam", "templateType": "randrange"}, "hh": {"group": "Ungrouped variables", "description": "", "definition": "(({om}*(1+{zz})^(3)) + {olam} + ({orad}*(1+{zz})^(4)) - ({kk}*(1 + {zz})^(2)))^(0.5) ", "name": "hh", "templateType": "anything"}, "om": {"group": "Ungrouped variables", "description": "", "definition": "random(0.25..0.35#0.01)", "name": "om", "templateType": "randrange"}, "omzz": {"group": "Ungrouped variables", "description": "", "definition": "({om}*(1+{zz})^(3))/(hh^(2))", "name": "omzz", "templateType": "anything"}, "zz": {"group": "Ungrouped variables", "description": "", "definition": "random(1.5..5#0.1)", "name": "zz", "templateType": "randrange"}, "orad": {"group": "Ungrouped variables", "description": "", "definition": "random(0.00001..0.000001#0.000001)", "name": "orad", "templateType": "randrange"}, "oradzz": {"group": "Ungrouped variables", "description": "", "definition": "{orad}*((1+{zz})^(4))/(hh^(2))", "name": "oradzz", "templateType": "anything"}, "kk": {"group": "Ungrouped variables", "description": "", "definition": "om + olam + orad - 1", "name": "kk", "templateType": "anything"}}, "variable_groups": [], "ungrouped_variables": ["om", "olam", "orad", "kk", "zz", "hh", "omzz", "olamzz", "oradzz"], "extensions": [], "parts": [{"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "{kk}", "correctAnswerFraction": false, "precision": "3", "strictPrecision": true, "mustBeReduced": false, "correctAnswerStyle": "plain", "marks": 1, "precisionPartialCredit": 0, "showFeedbackIcon": true, "scripts": {}, "type": "numberentry", "mustBeReducedPC": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "precisionType": "sigfig", "showPrecisionHint": true, "variableReplacementStrategy": "originalfirst", "minValue": "{kk}"}], "marks": 0, "prompt": "

Calculate the value of the constant $k c^{2}$ as a multiple of $H_{0}^{2}$.

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$kc^{2}/H_{0}^{2}  =$ [[0]] 

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Calculate the value of the Hubble parameter, $H$ at a redshift $z = \\var{zz}$ as a multiple of its current value $H_{0}$.

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$H(z=\\var{zz})/H_{0} =$ [[0]]

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Calculate the density parameters in the 3 components at redshift $z=\\var{zz}$.

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$\\Omega_{M} (z=\\var{zz}) = $ [[0]]

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$\\Omega_{\\Lambda} (z=\\var{zz}) = $ [[1]]

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$\\Omega_{\\gamma} (z=\\var{zz}) = $ [[2]]

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Consider a hypothetical Universe described by the Friedmann equation,

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$ \\left( \\frac{\\dot{a}}{a} \\right)^{2} = \\frac{8  \\pi G \\rho}{3} - \\frac{k c^{2}}{a^{2}},$

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where all parameters have their usual meanings.

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The mass-energy density budget comprises contributions from matter, cosmological constant and relativistic particles (the latter including the background radiation), with present-day density parameters shown below:

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
Matter$\\Omega_{M,0} = \\var{Om}$
Cosmological Constant$\\Omega_{\\Lambda,0} = \\var{Olam}$
Radiation$\\Omega_{\\gamma,0} = \\var{Orad}$
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", "name": "The Friedmann eqn ", "type": "question", "contributors": [{"name": "Richard Wilman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1618/"}]}]}], "contributors": [{"name": "Richard Wilman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1618/"}]}