Consider a hypothetical Universe described by the Friedmann equation,
\n$ \\left( \\frac{\\dot{a}}{a} \\right)^{2} = \\frac{8 \\pi G \\rho}{3} - \\frac{k c^{2}}{a^{2}},$
\nwhere all parameters have their usual meanings. Matter is the only contributor to the mass-energy density, with a present-day density parameter $\\Omega_{M,0} = \\var{Om}$.
", "parts": [{"marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "prompt": "Calculate the value of the constant $k c^{2}$ as a multiple of $H_{0}^{2}$.
\n$kc^{2}/H_{0}^{2} =$ [[0]]
\n", "variableReplacements": [], "gaps": [{"marks": 1, "scripts": {}, "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": true, "variableReplacements": [], "maxValue": "{kk}", "precisionType": "sigfig", "showFeedbackIcon": true, "strictPrecision": true, "showCorrectAnswer": true, "minValue": "{kk}", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "correctAnswerStyle": "plain", "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true}, {"marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "prompt": "Calculate the value of the Hubble parameter, $H$ at a redshift $z = \\var{zz}$ as a multiple of its current value $H_{0}$.
\n$H(z=\\var{zz})/H_{0} =$ [[0]]
\n", "variableReplacements": [], "gaps": [{"marks": 1, "scripts": {}, "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": true, "variableReplacements": [], "maxValue": "{hh}", "precisionType": "sigfig", "showFeedbackIcon": true, "strictPrecision": true, "showCorrectAnswer": true, "minValue": "{hh}", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "correctAnswerStyle": "plain", "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true}, {"marks": 0, "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "prompt": "Calculate the matter density parameter at redshift $z=\\var{zz}$.
\n$\\Omega_{M} (z=\\var{zz}) = $ [[0]]
", "variableReplacements": [], "gaps": [{"marks": 1, "scripts": {}, "precisionPartialCredit": 0, "mustBeReducedPC": 0, "type": "numberentry", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": true, "variableReplacements": [], "maxValue": "{omzz}", "precisionType": "sigfig", "showFeedbackIcon": true, "strictPrecision": true, "showCorrectAnswer": true, "minValue": "{omzz}", "variableReplacementStrategy": "originalfirst", "allowFractions": false, "correctAnswerFraction": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "3", "correctAnswerStyle": "plain", "mustBeReduced": false}], "showFeedbackIcon": true, "showCorrectAnswer": true}], "advice": "", "rulesets": {}, "ungrouped_variables": ["om", "kk", "zz", "hh", "omzz"], "variables": {"omzz": {"name": "omzz", "definition": "({om}*(1+{zz})^(3))/(hh^(2))", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "om": {"name": "om", "definition": "random(0.25..1.1#0.01)", "templateType": "randrange", "description": "", "group": "Ungrouped variables"}, "kk": {"name": "kk", "definition": "om - 1", "templateType": "anything", "description": "", "group": "Ungrouped variables"}, "zz": {"name": "zz", "definition": "random(1.5..5#0.1)", "templateType": "randrange", "description": "", "group": "Ungrouped variables"}, "hh": {"name": "hh", "definition": "(({om}*(1+{zz})^(3)) - ({kk}*(1 + {zz})^(2)))^(0.5) ", "templateType": "anything", "description": "", "group": "Ungrouped variables"}}, "functions": {}, "metadata": {"licence": "None specified", "description": ""}, "variable_groups": [], "tags": [], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question"}, {"name": "Redshift and cosmic expansion", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Wilman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1618/"}], "functions": {}, "metadata": {"licence": "None specified", "description": "This question tests the ability to relate observed redshifts in cosmology to the expansion factors of the Universe at the times of emission and observation.
"}, "tags": [], "ungrouped_variables": ["za", "zb", "z1", "sqrtaem", "z2"], "variables": {"z1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "((1+{za})/(1 + {zb})) - 1", "description": "", "name": "z1"}, "z2": {"templateType": "anything", "group": "Ungrouped variables", "definition": "{sqrtaem}^(-2) -1", "description": "", "name": "z2"}, "za": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(2..4#0.1)", "description": "", "name": "za"}, "zb": {"templateType": "randrange", "group": "Ungrouped variables", "definition": "random(0.5..1.5#0.1)", "description": "", "name": "zb"}, "sqrtaem": {"templateType": "anything", "group": "Ungrouped variables", "definition": "1 + (1/(1+{za}))^(0.5) - (1/(1 + {zb}))^(0.5)", "description": "", "name": "sqrtaem"}}, "statement": "As observed from Earth at the present time, Galaxy A has an observed redshift of $z=\\var{za}$. Galaxy B, along the same light of sight, has an observed redshift of $z = \\var{zb}$. Calculate the redshift of Galaxy A, as seen from an observer in a Galaxy B, at the following cosmic epochs:
", "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "", "parts": [{"gaps": [{"precisionPartialCredit": 0, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{z1}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showCorrectAnswer": true, "strictPrecision": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "precisionType": "sigfig", "showPrecisionHint": true, "correctAnswerFraction": false, "mustBeReduced": false, "scripts": {}, "precision": "2", "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{z1}", "variableReplacements": []}], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "At the time when the light we (on Earth) currently observe from galaxy B was emitted.
\n[[0]]
", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"gaps": [{"precisionPartialCredit": 0, "type": "numberentry", "showFeedbackIcon": true, "minValue": "{z2}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showCorrectAnswer": true, "strictPrecision": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "precisionType": "sigfig", "showPrecisionHint": true, "correctAnswerFraction": false, "mustBeReduced": false, "scripts": {}, "precision": "2", "precisionMessage": "You have not given your answer to the correct precision.", "maxValue": "{z2}", "variableReplacements": []}], "showCorrectAnswer": true, "showFeedbackIcon": true, "prompt": "At the current cosmic epoch.
\n\nFor this part, you may assume a matter-only Universe at the critical density, in which the comoving separation between a galaxy which emits light at $a=a_{1}$ and a galaxy which receives it at $a=a_{2}$ is given by
\n$r_{cm} = \\frac{2c}{H_{0}} [ \\sqrt{a_{2}} - \\sqrt{a_{1}} ].$
\n[[0]]
", "variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "scripts": {}, "variableReplacements": []}], "preamble": {"js": "", "css": ""}, "variable_groups": [], "type": "question"}, {"name": "The Friedmann eqn ", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Wilman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1618/"}], "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"], "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 $H_{0}^{2}$.
\n$kc^{2}/H_{0}^{2} =$ [[0]]
\n", "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "{hh}", "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": "{hh}"}], "marks": 0, "prompt": "Calculate the value of the Hubble parameter, $H$ at a redshift $z = \\var{zz}$ as a multiple of its current value $H_{0}$.
\n$H(z=\\var{zz})/H_{0} =$ [[0]]
\n", "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "{omzz}", "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": "{omzz}"}, {"notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "{olamzz}", "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": "{olamzz}"}, {"notationStyles": ["plain", "en", "si-en"], "showCorrectAnswer": true, "maxValue": "{oradzz}", "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": "{oradzz}"}], "marks": 0, "prompt": "Calculate the density parameters in the 3 components at redshift $z=\\var{zz}$.
\n$\\Omega_{M} (z=\\var{zz}) = $ [[0]]
\n$\\Omega_{\\Lambda} (z=\\var{zz}) = $ [[1]]
\n$\\Omega_{\\gamma} (z=\\var{zz}) = $ [[2]]
\n", "showFeedbackIcon": true}], "tags": [], "advice": "", "metadata": {"description": "", "licence": "None specified"}, "statement": "Consider a hypothetical Universe described by the Friedmann equation,
\n$ \\left( \\frac{\\dot{a}}{a} \\right)^{2} = \\frac{8 \\pi G \\rho}{3} - \\frac{k c^{2}}{a^{2}},$
\nwhere all parameters have their usual meanings.
\nThe 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| Matter | \n$\\Omega_{M,0} = \\var{Om}$ | \n
| Cosmological Constant | \n$\\Omega_{\\Lambda,0} = \\var{Olam}$ | \n
| Radiation | \n$\\Omega_{\\gamma,0} = \\var{Orad}$ | \n
Given this range in density, calculate the maximum possible age of the Universe, in Gyr:[[0]]
\nState the density parameter of the Universe for the latter case:[[1]]
\nCalculate the minimum possible age of the Universe, also in Gyr: [[2]]
\n", "marks": 0, "variableReplacements": [], "scripts": {}}, {"variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "gaps": [{"precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "precisionPartialCredit": 0, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "maxValue": "{v1}", "allowFractions": false, "strictPrecision": true, "precisionType": "sigfig", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "minValue": "{v1}", "notationStyles": ["plain", "en", "si-en"], "precision": "3", "type": "numberentry", "scripts": {}, "showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "precisionPartialCredit": 0, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "maxValue": "{z1}", "allowFractions": false, "strictPrecision": true, "precisionType": "sigfig", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "minValue": "{z1}", "notationStyles": ["plain", "en", "si-en"], "precision": "3", "type": "numberentry", "scripts": {}, "showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false}, {"precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "precisionPartialCredit": 0, "mustBeReduced": false, "showFeedbackIcon": true, "marks": 1, "maxValue": "{w1}", "allowFractions": false, "strictPrecision": true, "precisionType": "dp", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "minValue": "{w1}", "notationStyles": ["plain", "en", "si-en"], "precision": "2", "type": "numberentry", "scripts": {}, "showPrecisionHint": true, "correctAnswerStyle": "plain", "variableReplacements": [], "correctAnswerFraction": false}], "type": "gapfill", "prompt": "A galaxy at a distance of $\\var{d}$ Mpc emits an emission line of rest-frame wavelength 500.00 nm.
\nThe recession velocity of this galaxy in km s$^{-1}$ is: [[0]].
\nThe redshift of the galaxy is: [[1]].
\nThe observed wavelength of the emission line in nm is [[2]].
\n", "marks": 0, "variableReplacements": [], "scripts": {}}], "preamble": {"css": "", "js": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"s2": {"definition": "13.98*(2/3)*(70/h)", "group": "Ungrouped variables", "description": "", "name": "s2", "templateType": "anything"}, "z1": {"definition": "{v1}/(3*10^(5))", "group": "Ungrouped variables", "description": "", "name": "z1", "templateType": "anything"}, "s1": {"definition": "13.98*(70/h)", "group": "Ungrouped variables", "description": "", "name": "s1", "templateType": "anything"}, "h": {"definition": "random(65..75#1)", "group": "Ungrouped variables", "description": "", "name": "h", "templateType": "randrange"}, "w1": {"definition": "500*(1+{z1})", "group": "Ungrouped variables", "description": "", "name": "w1", "templateType": "anything"}, "v1": {"definition": "{h}*{d}", "group": "Ungrouped variables", "description": "", "name": "v1", "templateType": "anything"}, "d": {"definition": "random(10..30#0.5)", "group": "Ungrouped variables", "description": "", "name": "d", "templateType": "randrange"}}, "ungrouped_variables": ["h", "s1", "s2", "d", "v1", "z1", "w1"], "functions": {}, "rulesets": {}, "statement": "Consider a hypothetical Universe with a Hubble constant of $H_{0} = \\var{h}$ km s$^{-1}$ Mpc$^{-1}$. The Universe contains only matter with an unknown density which may range anywhere from zero to the critical density.
", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "variable_groups": [], "advice": "", "type": "question"}]}], "showQuestionGroupNames": false, "metadata": {"licence": "None specified", "description": ""}, "showstudentname": true, "navigation": {"onleave": {"action": "none", "message": ""}, "showresultspage": "oncompletion", "reverse": true, "showfrontpage": true, "allowregen": true, "browse": true, "preventleave": true}, "feedback": {"feedbackmessages": [], "advicethreshold": 0, "showtotalmark": true, "allowrevealanswer": true, "showactualmark": true, "showanswerstate": true, "intro": ""}, "percentPass": "40", "type": "exam", "contributors": [{"name": "Richard Wilman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1618/"}], "extensions": [], "custom_part_types": [], "resources": []}