2020年3月7日 (六) 10:08 Jingsong Guo

2020-03-07T10:08:24Z

Jingsong Guo：创建页面，内容为“==英文题目== 8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor aptq. G “ G0fpaq (5) Consider th…”

2020-03-07T09:25:33Z

创建页面，内容为“==英文题目== 8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor aptq. G “ G0fpaq (5) Consider th…”

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==英文题目==

8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor aptq. G “ G0fpaq (5) Consider the model fpaq “ e bpa´1q where b “ 2.09. a) Assuming that the universe is flat, dark energy is absent, and the only constituent is matter, estimate the present age of this weird universe according to this model. Assume that the Friedmann equation: Hpaq 2 “ H2 0 pΩm ` Ωr ` Ωk ` ΩΛq (6) still holds in this setting. b) What is the behaviour of the age of the universe t as the scale factor aptq Ñ 8 ? Note that all parameters with subscript 0 indicate their present value. Take the value of Hubble’s constant as H0 “ 67.8 kms´1Mpc´1 Hint: You might need the following integrals ż 8 0 x 2 e ´x 2 dx “ ? π 4 ż 1 0 x 2 e ´x 2 dx « 0.189471 (7) 9. (15 points) a) Find the shortest distance from Boston (42.36010 N, 71.05890 W) to Beijing (39.90420 N, 116.40740 E)traveling along the Earth’s surface. Assume that the Earth is a uniform sphere of radius 6371 km. b) What fraction of the path lies within the Arctic circle (north of 66.56080 N)?

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	==英文题目==		==英文题目==

−	8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor ~~aptq~~. G ~~“ G0fpaq~~ (5) Consider the model ~~fpaq “~~ e ~~bpa´1q~~ where b “ 2.09. a) Assuming that the universe is flat, dark energy is absent, and the only constituent is matter, estimate the present age of this weird universe according to this model. Assume that the Friedmann equation: ~~Hpaq~~ 2 ~~“ H2 0 pΩm ` Ωr ` Ωk ` ΩΛq~~ (6) still holds in this setting. b) What is the behaviour of the age of the universe t as the scale factor ~~aptq Ñ 8~~ ? Note that all parameters with subscript 0 indicate their present value. Take the value of Hubble’s constant as H0 “ 67.8 ~~kms´1Mpc´1~~ Hint: You might need the following integrals ~~ż 8~~ 0 x 2 ~~e ´x 2~~ dx ~~“ ? π~~ 4 ż 1 0 x 2 ~~e ´x 2~~ dx « 0.189471 (7) 9. (15 points) a) Find the shortest distance from Boston (42.36010 N, 71.05890 W) to Beijing (39.90420 N, 116.40740 E)traveling along the Earth’s surface. Assume that the Earth is a uniform sphere of radius 6371 km. b) What fraction of the path lies within the Arctic circle (north of 66.56080 N)?	+	8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor $$a(0)$$.
		+
		+	$$G = G_0f(a)$$
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		+	(5) Consider the model $$f(a)=e^{b(a-1)}$$ where $$b=2.09$$.
		+
		+	a) Assuming that the universe is flat, dark energy is absent, and the only constituent is matter, estimate the present age of this weird universe according to this model. Assume that the Friedmann equation: $$H(a)^2=H^2_0 (Ω_m+Ω_r+Ω_k+Ω_Λ)$$ still holds in this setting.
		+
		+	b) What is the behaviour of the age of the universe t as the scale factor $$a\rightarrow \infty$$ ?
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		+	Note that all parameters with subscript 0 indicate their present value. Take the value of Hubble’s constant as $$H0=67.8 kms^{-1}Mpc^{-1}$$
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		+	Hint: You might need the following integrals:
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		+	$$\int_{0}^{\infty}x^2e^{-x^{2}}dx=\dfrac{\sqrt{\pi}}{4}$$
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		+	$$\int_{0}^{1}x^2e^{-x^{2}}dx\approx 0.189471$$

2019年USAAAO决赛第8题 - 版本历史

2020年3月7日 (六) 10:08 Jingsong Guo

Jingsong Guo：创建页面，内容为“==英文题目== 8. (15 points) In a rather weird universe, the gravitational constant G varies as a function of the scale factor aptq. G “ G0fpaq (5) Consider th…”