196
Inflationary cosmology
which were causally disconnected up to the time of recombination of electrons
and photons after which the photons we now observe as the cosmic microwave
background underwent no further scattering. The puzzle is how these causally
disconnected regions could have ended up with the same microwave background
temperature.
We can estimate how many causally disconnected regions there were at the
time of recombination. From (3.117), the proper distance at time t from any point
to the particle horizon is
dH(t) = 2t
(7.1)
where we have taken n = ! for a radiation-dominated universe. In particular, at
the recombination time tn
dH(tr ) = 2tr •
(7.2)
(We are assuming that for most of the time from t = 0 up until tr the universe was
radiation dominated. This is a reasonable approximation because the temperature
at which the transition from radiation dominance to matter dominance occurs and
the temperature at which recombination of electrons and protons occurs are weB
within an order of magnitude of each other, respectively 0.37 eV and 0.26 eV.)
We need to know how many horizon volumes at time tr have expanded to fiB the
presently observable region of the universe (the present horizon volume). Thus,
we need to know the radius of the region at time tr that has expanded to the radius
of the presently observable universe. Let the volume of the observable universe
at the present time to be Vo(to) and let the horizon volume at the recombination
time be Vr (t r). Since RT is constant, because of conservation of entropy,
Vo(tr) = Vo(to) R: (t,) = Vo(to) (To)3
(7.3)
Vr(tr)
Vr(t,) R (to)
Vr(tr ) Tr
In view of (3.117),
Vo(tr) = (~)3 (To)3
(7.4)
Vr(tr )
tr
Tr
With the universe matter dominated from approximately the time of
recombination to the present time, so that
R(t) ex t 2/3
(7.5)
we have
tex T- 3/2 •
(7.6)
Thus,
Vo(t r ) ~ (Tr
To
y/2 ~ 3.6 x )04
(7.7)
Vr(tr )
for Tr ~ 3.0 x HP K and To ~ 2.73 K. This is the number of horizon volumes
at the recombination time that expanded to fiB the presently observable universe
(the present horizon volume.) As advertised, this is a large number.
Inflationary cosmology
which were causally disconnected up to the time of recombination of electrons
and photons after which the photons we now observe as the cosmic microwave
background underwent no further scattering. The puzzle is how these causally
disconnected regions could have ended up with the same microwave background
temperature.
We can estimate how many causally disconnected regions there were at the
time of recombination. From (3.117), the proper distance at time t from any point
to the particle horizon is
dH(t) = 2t
(7.1)
where we have taken n = ! for a radiation-dominated universe. In particular, at
the recombination time tn
dH(tr ) = 2tr •
(7.2)
(We are assuming that for most of the time from t = 0 up until tr the universe was
radiation dominated. This is a reasonable approximation because the temperature
at which the transition from radiation dominance to matter dominance occurs and
the temperature at which recombination of electrons and protons occurs are weB
within an order of magnitude of each other, respectively 0.37 eV and 0.26 eV.)
We need to know how many horizon volumes at time tr have expanded to fiB the
presently observable region of the universe (the present horizon volume). Thus,
we need to know the radius of the region at time tr that has expanded to the radius
of the presently observable universe. Let the volume of the observable universe
at the present time to be Vo(to) and let the horizon volume at the recombination
time be Vr (t r). Since RT is constant, because of conservation of entropy,
Vo(tr) = Vo(to) R: (t,) = Vo(to) (To)3
(7.3)
Vr(tr)
Vr(t,) R (to)
Vr(tr ) Tr
In view of (3.117),
Vo(tr) = (~)3 (To)3
(7.4)
Vr(tr )
tr
Tr
With the universe matter dominated from approximately the time of
recombination to the present time, so that
R(t) ex t 2/3
(7.5)
we have
tex T- 3/2 •
(7.6)
Thus,
Vo(t r ) ~ (Tr
To
y/2 ~ 3.6 x )04
(7.7)
Vr(tr )
for Tr ~ 3.0 x HP K and To ~ 2.73 K. This is the number of horizon volumes
at the recombination time that expanded to fiB the presently observable universe
(the present horizon volume.) As advertised, this is a large number.