# 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 $$a(0)$$.

$$G = G_0f(a)$$

(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 $$a\rightarrow \infty$$ ?

Note that all parameters with subscript 0 indicate their present value. Take the value of Hubble’s constant as $$H0=67.8 kms^{-1}Mpc^{-1}$$

Hint: You might need the following integrals:

$$\int_{0}^{\infty}x^2e^{-x^{2}}dx=\dfrac{\sqrt{\pi}}{4}$$

$$\int_{0}^{1}x^2e^{-x^{2}}dx\approx 0.189471$$