2018年IOAA理论第11题-宇宙的热历史 - 版本历史

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CHN-5：\ddot typo, change to \dot

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\ddot typo, change to \dot

2019年8月6日 (二) 07:07 Quan787

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 ==英文题目==

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 $$(\frac{\dot a}{a})^2=\frac{8\pi G}{3}(\rho_m+\rho_r)+\frac{\Lambda c^2}{3}-\frac{kc^2}{a^2}$$,
-We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scale factor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant h.
+We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scale factor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant $$h$$.
-(a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time t<sub>H</sub>) using the Hubble parameter. Calculate the present-day Hubble time t<sub>H0</sub>.
+(a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time $$t_H$$) using the Hubble parameter. Calculate the present-day Hubble time $$t_{H0}$$.
 (b) (5 points) Let us define the critical density ρ<sub>c</sub> as the matter density required to explain the expansion of a flat universe without any radiation or dark energy. Find an expression of the critical density, in terms H and G. Calculate the present critical density ρ<sub>c0</sub>.
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 Use this information to find expression for Ω<sub>Λ</sub> and Ω<sub>k</sub>, in terms H, c, Λ, k and a.
-(d) (7 points) Another equation which is valid for matter, radiation and dark energy is often called the Fluid Equation: $$\dot\rho+3\frac{\dot a}{a}(\rho+\frac{p}{c^2})=0$$, where  p is the pressure of some component, ρ is the density and $$\ddot\rho$$ is the rate of change of density over time. Radiation contains photons and massless neutrinos, and they both travel atthe speed of light. The pressure exerted by these particles is 1/3 of their energy density. Show that the density of radiation ρ<sub>r</sub> ∝ (1 + z)<sup>4</sup>, where 𝑧 is cosmological redshift. You may note that if $$\frac{\dot\rho}{\rho}=n\frac{\dot a}{a}$$, then ρ ∝ a<sup>n</sup>
+(d) (7 points) Another equation which is valid for matter, radiation and dark energy is often called the Fluid Equation: $$\dot\rho+3\frac{\dot a}{a}(\rho+\frac{p}{c^2})=0$$, where p is the pressure of some component, ρ is the density and $$\dot\rho$$ is the rate of change of density over time. Radiation contains photons and massless neutrinos, and they both travel atthe speed of light. The pressure exerted by these particles is 1/3 of their energy density. Show that the density of radiation ρ<sub>r</sub> ∝ (1 + z)<sup>4</sup>, where z is cosmological redshift. You may note that if $$\frac{\dot\rho}{\rho}=n\frac{\dot a}{a}$$, then ρ ∝ a<sup>n</sup>
 (e) (4 points) We know that the value of the cosmological constant Λ doesn’t evolve. Its equation of state hasa form p = wρ<sub>Λ</sub>c<sup>2</sup>, where w is an integer. Find the value of w.

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 $$(\frac{\dot a}{a})^2=\frac{8\pi G}{3}(\rho_m+\rho_r)+\frac{\Lambda c^2}{3}-\frac{kc^2}{a^2}$$,
-We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scalefactor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant h.
+We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scale factor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant h.
 (a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time t<sub>H</sub>) using the Hubble parameter. Calculate the present-day Hubble time t<sub>H0</sub>.
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 (j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe was around 1 MeV. At this time, the radiation density in the universe was much more than all other components. Estimate the time ($$t=\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.

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 '''(T11) Thermal History of the Universe (75 points)'''
-Based on Einstein’s general relativity, Russian physicist Alexander Friedmann derived the FriedmannEquation by which the dynamics of a homogeneous and isotropic universe can be well described. TheFriedmann Equation is usually written as follows:
+Based on Einstein’s general relativity, Russian physicist Alexander Friedmann derived the Friedmann Equation by which the dynamics of a homogeneous and isotropic universe can be well described. TheFriedmann Equation is usually written as follows:
 $$(\frac{\dot a}{a})^2=\frac{8\pi G}{3}(\rho_m+\rho_r)+\frac{\Lambda c^2}{3}-\frac{kc^2}{a^2}$$,
-We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scalefactor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmologicalconstant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter,we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant h.
+We define the Hubble parameter as $$=\frac{\dot a}{a}$$, where a is the scale factor and $$\dot a$$ is the rate of change of scalefactor with time. Thus, the Hubble parameter is a function of cosmic time. In the Friedmann Equation, ρ<sub>m</sub> isthe density of matter, including dark matter and baryons, ρ<sub>r</sub> is the density of radiation, Λ is the cosmological constant, and k is the curvature of space. Subscript 0 indicates the value of a physical quantity at present day,e.g. H<sub>0</sub> is the present value Hubble parameter. Also, to avoid confusion with the reduced Hubble parameter, we use the reduced Planck Constant $$\hbar=h/2\pi$$ instead of the Planck constant h.
-(a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time t<sub>H</sub>) using the Hubble parameter. Calculate the present-dayHubble time t<sub>H0</sub>.
+(a) (5 points) What are the dimensions of Hubble parameter? One can define a characteristic timescale forthe expansion of the Universe (i.e. Hubble time t<sub>H</sub>) using the Hubble parameter. Calculate the present-day Hubble time t<sub>H0</sub>.
-(b) (5 points) Let us define the critical density ρ<sub>c</sub> as the matter density required to explain the expansion of aflat universe without any radiation or dark energy. Find an expression of the critical density, in terms H and G. Calculate the present critical density ρ<sub>c0</sub>.
+(b) (5 points) Let us define the critical density ρ<sub>c</sub> as the matter density required to explain the expansion of a flat universe without any radiation or dark energy. Find an expression of the critical density, in terms H and G. Calculate the present critical density ρ<sub>c0</sub>.
-(c) (6 points) It is convenient to define all density parameters in a dimensionless manner like $$\Omega_i=\frac{\rho_i}{\rho_c}$$, i.e. theratio of density to critical density. The Friedmann Equation can be rewritten using these dimensionlessdensity parameters simply as, Ω<sub>m</sub> + Ω<sub>r</sub> + Ω<sub>Λ</sub> + Ω<sub>k</sub> = 1.
+(c) (6 points) It is convenient to define all density parameters in a dimensionless manner like $$\Omega_i=\frac{\rho_i}{\rho_c}$$, i.e. the ratio of density to critical density. The Friedmann Equation can be rewritten using these dimensionless density parameters simply as, Ω<sub>m</sub> + Ω<sub>r</sub> + Ω<sub>Λ</sub> + Ω<sub>k</sub> = 1.
 Use this information to find expression for Ω<sub>Λ</sub> and Ω<sub>k</sub>, in terms H, c, Λ, k and a.
-(d) (7 points) Another equation which is valid for matter, radiation and dark energy is often called the FluidEquation: $$\dot\rho+3\frac{\dot a}{a}(\rho+\frac{p}{c^2})=0$$, where  p is the pressure of some component, ρ is the density and $$\ddot\rho$$ is the rateof change of density over time. Radiation contains photons and massless neutrinos, and they both travel atthe speed of light. The pressure exerted by these particles is 1/3 of their energy density. Show that the densityof radiation ρ<sub>r</sub> ∝ (1 + z)<sup>4</sup>, where 𝑧 is cosmological redshift. You may note that if $$\frac{\dot\rho}{\rho}=n\frac{\dot a}{a}$$, then ρ ∝ a<sup>n</sup>
+(d) (7 points) Another equation which is valid for matter, radiation and dark energy is often called the Fluid Equation: $$\dot\rho+3\frac{\dot a}{a}(\rho+\frac{p}{c^2})=0$$, where  p is the pressure of some component, ρ is the density and $$\ddot\rho$$ is the rate of change of density over time. Radiation contains photons and massless neutrinos, and they both travel atthe speed of light. The pressure exerted by these particles is 1/3 of their energy density. Show that the density of radiation ρ<sub>r</sub> ∝ (1 + z)<sup>4</sup>, where 𝑧 is cosmological redshift. You may note that if $$\frac{\dot\rho}{\rho}=n\frac{\dot a}{a}$$, then ρ ∝ a<sup>n</sup>
 (e) (4 points) We know that the value of the cosmological constant Λ doesn’t evolve. Its equation of state hasa form p = wρ<sub>Λ</sub>c<sup>2</sup>, where w is an integer. Find the value of w.
-(f) (13 points) Planck time, defines a characteristic timescale before which our present physical laws are nolonger valid, and where quantum gravity is needed. The expression for Planck time can be written in terms of $$\hbar$$, G and c and non-dimensional coefficient of this expression in SI units is of the order of unity. Usingdimensional analysis, find expression for Planck time and estimate its value.
+(f) (13 points) Planck time, defines a characteristic timescale before which our present physical laws are nolonger valid, and where quantum gravity is needed. The expression for Planck time can be written in terms of $$\hbar$$, G and c and non-dimensional coefficient of this expression in SI units is of the order of unity. Using dimensional analysis, find expression for Planck time and estimate its value.
-(g) (7 points) Planck length defines the length scale associated with Planck time is given by l<sub>p</sub>=ct<sub>p</sub>. Theminimal mass of a black hole, also called Planck mass, is defined as the mass of a black hole whoseSchwarzschild radius is two times the Planck length.
+(g) (7 points) Planck length defines the length scale associated with Planck time is given by l<sub>p</sub>=ct<sub>p</sub>. Theminimal mass of a black hole, also called Planck mass, is defined as the mass of a black hole whose Schwarzschild radius is two times the Planck length.
-Derive the Planck mass M<sub>p</sub> and calculate M<sub>p</sub>c<sup>2</sup>in GeV. This mass is considered to be an upper threshold forelementary particles, beyond which they will collapse to a black hole.
+Derive the Planck mass M<sub>p</sub> and calculate M<sub>p</sub>c<sup>2</sup> in GeV. This mass is considered to be an upper threshold for elementary particles, beyond which they will collapse to a black hole.
-(h) (4 points) At the very beginning (soon after the Planck time), all the particles were in thermal equilibriumin a primordial soup. As temperature decreased, different particles then decoupled from the primordial soupone by one and could travel freely in the Universe. Photons decoupled at ~300000 years after the Big Bang.These photons emitted at that time are what constitutes the cosmic microwave background (CMB), whichfollows the Stefan-Boltzmann law for blackbody radiation.
+(h) (4 points) At the very beginning (soon after the Planck time), all the particles were in thermal equilibriumin a primordial soup. As temperature decreased, different particles then decoupled from the primordial soup one by one and could travel freely in the Universe. Photons decoupled at ~300000 years after the Big Bang.These photons emitted at that time are what constitutes the cosmic microwave background (CMB), which follows the Stefan-Boltzmann law for blackbody radiation.
 $$\epsilon_r=\frac{\pi^2}{15\hbar^3c^3}(k_BT)^4$$,
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 Show that the temperature of the CMB follows T/(1 + z) = constant.
-(i) (16 points) With the expansion of the Universe, radiation density dropped more quickly than matterdensity, and at some epoch the matter density was equal to the radiation density. Radiation contains bothphotons and neutrinos. Apart from photons, neutrinos additionally contribute to the radiation energy densityby 68% (i.e.  = 1.68Ω<sub>γ0</sub>, where γ indicates photons). Estimate the redshift of matter-radiation equality z<sub>eq</sub> in terms of Ω<sub>m0</sub> and reduced Hubble parameter $$h=\frac{H_0}{100kms^{-1}Mpc^{-1}}$$ You may use the currenttemperature of the CMB: T<sub>0</sub> = 2.73K.
+(i) (16 points) With the expansion of the Universe, radiation density dropped more quickly than matter density, and at some epoch the matter density was equal to the radiation density. Radiation contains both photons and neutrinos. Apart from photons, neutrinos additionally contribute to the radiation energy densityby 68% (i.e.  = 1.68Ω<sub>γ0</sub>, where γ indicates photons). Estimate the redshift of matter-radiation equality z<sub>eq</sub> in terms of Ω<sub>m0</sub> and reduced Hubble parameter $$h=\frac{H_0}{100kms^{-1}Mpc^{-1}}$$ You may use the current temperature of the CMB: T<sub>0</sub> = 2.73K.
-(j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe wasaround 1 MeV. At this time, the radiation density in the universe was much more than all other components.Estimate the time ($$t=\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.
+(j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe was around 1 MeV. At this time, the radiation density in the universe was much more than all other components. Estimate the time ($$t=\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.

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	(i) (16 points) With the expansion of the Universe, radiation density dropped more quickly than matterdensity, and at some epoch the matter density was equal to the radiation density. Radiation contains bothphotons and neutrinos. Apart from photons, neutrinos additionally contribute to the radiation energy densityby 68% (i.e. = 1.68Ω<sub>γ0</sub>, where γ indicates photons). Estimate the redshift of matter-radiation equality z<sub>eq</sub> in terms of Ω<sub>m0</sub> and reduced Hubble parameter $$h=\frac{H_0}{100kms^{-1}Mpc^{-1}}$$ You may use the currenttemperature of the CMB: T<sub>0</sub> = 2.73K.		(i) (16 points) With the expansion of the Universe, radiation density dropped more quickly than matterdensity, and at some epoch the matter density was equal to the radiation density. Radiation contains bothphotons and neutrinos. Apart from photons, neutrinos additionally contribute to the radiation energy densityby 68% (i.e. = 1.68Ω<sub>γ0</sub>, where γ indicates photons). Estimate the redshift of matter-radiation equality z<sub>eq</sub> in terms of Ω<sub>m0</sub> and reduced Hubble parameter $$h=\frac{H_0}{100kms^{-1}Mpc^{-1}}$$ You may use the currenttemperature of the CMB: T<sub>0</sub> = 2.73K.

−	(j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe wasaround 1 MeV. At this time, the radiation density in the universe was much more than all other components.Estimate the time ($$t-\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.	+	(j) (8 points) The neutrinos decoupled from the primordial soup when the temperature of the universe wasaround 1 MeV. At this time, the radiation density in the universe was much more than all other components.Estimate the time ($$t=\frac{1}{2H}$$) when neutrinos decoupled, and express it in seconds since the big bang.