2019年USAAAO决赛第8题

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)?