Primordial Black Holes and Cosmological Phase Transitions Report ...

PBHs and Cosmological Phase Transitions 6
where the + sign corresponds to a receding light ray and the − sign to an
aproaching light ray. Consider a light ray emanating from a galaxy P with
world line r = r1, at coordinate t1, and received by O at coordinate time t0.
Using equation (20) we have (e.g. d’Inverno, 1993)
t0
t1
0
dt
dr
= −
. (21)
R(t) r1 (1 − kr) 1/2
Next, consider a second light ray emanating from P at time t1 +dt1 and received
at O at time t0 + dt0. Thus, we have
t0+dt0
t1+dt1
0
dt
dr
= −
. (22)
R(t) r1 (1 − kr) 1/2
Comparing equations (21) and (22) it turns out that
t0+dt0
t1+dt1
dt
R(t) =
t0
dt
. (23)
t1 R(t)
Assuming that R(t) does not vary greatly over the intervals dt1 and dt0 we can
take it outside the integral, yielding (e.g. d’Inverno, 1993)
dt0 dt1
= . (24)
R(t0) R(t1)
All fundamental particles of the substractum have world lines on which the spatial
coordinates are constant and, hence, the metric reduces to ds 2 = dt 2 . Here
t measures the proper time along the substractum world lines. The intervals dt1
and dt0 are, respectively, the proper time intervals between the rays as measured
at the source and observer. In an expanding Universe we have that t0 >t1 and
so R(t0) >R(t1) which means that the observer O will experience a redshift z
given by (e.g. d’Inverno, 1993)
1+z = ν0
ν1
= R(t0) dt0
=
R(t1) dt1
(25)
where ν1 and ν0 are the frequencies measured by the emitter and the receiver,
respectively. In a contracting Universe O will detect instead a blue shift.
The Hubble parameter H is defined as (e.g. d’Inverno, 1993)
H(t) = ˙ R(t)
R(t)
and the Hubble time is defined as (e.g. Boyanovsky et al., 2006)
tH(t) = 1
H(t)
(26)
R(t)
= . (27)
˙R(t)