Question

In a flat Universe that contains only a fluid with equation of state P = wpc, where w = -1/6, the evolution of the scale factor, R, with cosmic time, t, is given by R(t) = R, (1)", where R = R(to), and to is the present time: to = 3H, t = 0 corresponding to the Big Bang. The Hubble constant is H. = 67 kms-'Mpc-!. In this model of the Universe: 1. How does the Hubble parameter scale with the cosmic time, t? 2. What is the redshift of a galaxy that emitted its light attem = 10/2? 3. Use the relation between the redshift, z, and the scale factor, R(t), to show that the Hubble parameter can be written as H(z) = H (1 + z)5/4.

Answer 1

Answers Given data, . A flat universe that contains only a fluid with excecution of state P- wpc? where, I woollo The evolution of the scale factor & R Cosmic time = t. Therefore TR CES - R. ()YS 0 where, RowR (20) then to is the present time when, 750 Boig Bang is corresponding The Hubble constant Ho = 67 km 5' mpe!

The hubbie parameter scale Cosmic time =HLI ARCL) R(E) at u15. 1) :HCP) ACP) a pri tom urs (4445 HC) - pres la desconto cos 45 pick) a prese nting ( 9 ) Capas HCLRO ()) HCl) Antes Cat) (18 R . Pole" 1464) Š The Hubble parametey Scale with the cosmic time [H (t) = 1 4 quation ) 1. that emitted The redshift its light at of a galaxy tem = to/2 .

tem= tola We know that , 1+2 = femit observed Hz I'lto (to .. . . . . (to) 1 +2 = 1+2 (+6) (33) t2 =2 z 2-1 alote - we know femit a emitted frequency = 1/2 of observed - observed frequency = 40 z = redsheft

(1+2) = Ro (iv) R(E) equation () from the from equation en R(H) = Ro (345 Red Chus from substitute t value from equation. ult) 4647 3 ) 574 from equation ci) to a

then 464) = TECH) ( here ) 574 464) = He inheit from eglection (ov) The = 142 4lx) #. (122)504] Hence, we can prove that the relation beficen the red shift, a and the scale factor, R(E) of hubble parameter H(3) - Ho litz)514