Book: An Introduction to Modern Astrophysics, 2nd Edition, by Carroll and Ostlie.???????

(b) The Hubble constant has units of s. For a universe filled with just enough matter to stop ex- pansion way in the future, you can use the Hubble constant to estimate the age of the Universe. The formula is simply to = 3 Ho . Calculate the age of the Universe in Gyr using the value you derived from Hubble's data. Note that you have to convert the Hubble constant into mks units of s-1. It was an extraordinary discovery that the Universe had a finite age and apparently a beginning. +1000 KM 2. The Hubble Law (which was first discovered by Georges Lemaitre in 1927) states that the appar- O. ent expansion velocity of the Universe between two points is directly proportional to the distance between the two points. That is, v = Hor where v is the "velocity" (actually the stretching of the Uni- 500KM verse) between the two points and r is the distance. The value of the Hubble parameter at present is called the Hubble constant, denoted by Ho. Dating back to the first measurements of Ho in 1927, VELOCITY we use the units of kms- Mpc-1. (a) The following figure shows the original plot of the expansion of the Universe from Hubble's 1929 paper. Use this plot to calculate the Hubble constant in units of kms Mpc. DISTANCE 10 PARSECS 2 x 10 PARSECS The attentive student will note that I have switched from h to h. This is called the "reduced" Compton wavelength, the wavelength over one radian. Without the bar, it is the official formula for Ac over a full 27 radians. As a professor in astrophysical hydrodynamics told my graduate class "27 = 1." I am still scratching my head over that. Clearly, my professor should not be an engineer. Figure 1: Hubble's 1929 measurement of the expansion of the Universe. The y-axis is the velocity in units of, and the x-axis in units of parsecs. (c) The critical density needed to close the Universe - that is, to make it come to a stop in its expansion way in the future is given by the formula 3H. Pc = 87. G (4) Assuming the present Hubble constant is Ho = 74kms- Mpc", calculate the critical density of the Universe in units of kg m- (d) In BOB, the effects of the Cosmological Constant (Dark Energy) lead to a law of repulsive acceleration of FA = -Amort (5) where r is the distance, m is a test mass, and f is the unit vector. My team first measured the value of the cosmological constant A as 1.1 x 10-52m-2. The force of gravity Fg decreases as r-2 but the force of Dark Energy FA increases as a function of r. At some point, they must balance. Show that they balance at (3GM) 1/3 r = Ac2 (6)