Question

Using these data, calculate the following quantities. Express your answers in SI units (unless the quantity is dimensionless) and to an appropriate number of significant figures.

• a.The critical density at the current time.
• b.The density of matter at the current time.
• c.The energy density (u_m) due to matter at the current time.
• d.The energy density (u_Λ) due to the cosmological constant.
  

Quantity	Value*
Hubble parameter at current time	72 km s–1 Mpc–1
Curvature parameter	0
Density parameter of matter at current time	0.32
Density parameter of cosmological constant at current time	0.68

^* Note that the uncertainties associated with these measurements are not given.

Answer 1

given :

Hubble parameter (H₀ ) = 72 km s^–1 Mpc^–1 = 72 x 10³/ 3.09 x10¹⁶ x 10⁶ s^–1 = 2.33 x 10^-18 s^–1

curvature parameter (k) = 0

density parameter of matter at current time = 0.32

Density parameter of cosmological constant at current time = 0.68

gravitational constant = 6.67408 × 10^-11 m³ kg^-1 s^-2

a) the critical density is given by _c = 3 H₀² / 8 G = 3 * ( 2.33 x 10^-18) ² / (8 x 6.67408 × 10^-11)= 9.7104 x  10^-27 kg/m³

b) density of matter at current time =

= / _c

= _c x = 9.7104 x  10^-27 x 0.32 = 3.1073 x  10^-27 kg/m³

c) energy density of matter at current time (U_m) = c² = (3.1073 x  10^-27 ) x (3 x 10⁸)² = 2.7966 x 10^-10 J/m³

d) energy density (u_Λ) due to the cosmological constant

⁼ u_Λ / _c

therefore, u_{Λ =} x _c =0.68  x 9.7104 x  10^-27 = 6.6031  x  10^-27 J/m³