12
The standard model of cosmology
The first evidence suggesting this came from measurements of the redshifts
of type la supernovae. Such supernovae arise as remnants of the explosion of
white dwarfs which accrete matter from neighbouring stars. Eventually the white
dwarf mass exceeds the Chandrasekhar limit and the supernova is born after the
explosion. The intrinsic luminosity of such supernovae is considered to be a
constant. That is, they are taken as standard candles and any variation in their
apparent luminosity as measured on earth must be explicable in terms of their
differing distances from the earth. In a Euclidean space, the apparent luminosity 1
of a source with intrinsic luminosity L at a distance D from the observer is given
by
1= _L
(1.8 I)
41rD2·
We may, therefore, define the 'luminosity distance' DL of a source from the
observer by
(1.82)
DL == J4~"
In GR we must be more careful. So consider the circular mirror, area A, of a
telescope at the origin, nonnal to the line of sight to a source at r I. Light emitted
from the source at time 11 and arriving at the mirror at time 10 is bounded by a
cone with solid angle
A
w=---:"""":"
(1.83)
41r R(lo)2rl
as measured in the locally inertial frame at the source. The emitted photons have
their energy redshifted by a factor
R(tl) =
(1.84)
R(IO)
1+ z
as explained in section 1.2, (see (I. I 8». Also, photons emitted at time intervals of
1511 reach the mirror at time intervals 1510 = 1511 R(IO)/ R(II). Thus, the total power
P received at the mirror is given by
P = L (R(II»)2
(1.85)
R(to) w
and the apparent luminosity by
P
1 = A.
(1.86)
Then, using (1.27), the luminosity distance defined in (1.82) is
DL = Ha l (1 + z) [z - ~(l +QO)z2 + ... ]
(1.87)
= Ho I [I z + 2(1 - qo)z 2] +... .
(1.88)