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```Cosmology
3
1
Topics

The Hot Big Bang

The Flatness Problem

The Horizon Problem

Summary
2
The Hot Big Bang
If the space between distant galaxies is
growing, this suggests that galaxies were
closer together in the past.
t
Indeed, assuming
this continues
into the distant
past, all matter must
have been squeezed
together very tightly.
le
t0 = now
d
d0
d (t )  a(t )d0
lo
3
The Hot Big Bang
Therefore, a basic assumption of the big
bang theory of the universe is that the early
universe was dense and hot. If so, it must
have been filled with
t
le
high energy
t0 = now
photons in
d
thermal equilibrium
d0
with matter
d (t )  a(t )d0
lo
4
The Hot Big Bang
The integral over all photon energies of the
number of photons per unit volume n(E)dE/V
times the photon energy, E, gives the photon
energy per unit volume (T)

 (T )  0
En(E)dE
V
3

8
(
E
/
kT
)
4

(kT )  E / kT
dE / kT
3
0 e
(hc)
1
4
8

4
4
4
3

(kT )
 RadT  4732  T eV/m
3
(hc)
15
5
The Hot Big Bang
The number density of photons scales like
1/a3. But the energy, E, of each photon scales
like E = E0/a, where E0 is its energy today.
Therefore, the photon energy density goes
4
like
   /a
R
But since
 (T )  4732 T 4eV/m3
the photon temperature scales like
T  T0 / a where T0 is the current
photon temperature
6
The Hot Big Bang
The fact that T = T0 / a implies that the
black body spectrum of photons from the
early universe is preserved through
time, but with
t
le
the temperature
t0 = now
reduced by the
d
scale factor a
to a predicted
temperature of
d (t )  a(t )d0
T0 ~ 3 K
lo
d0
7
The Hot Big Bang
An important milestone came in 1965, with
the (accidental) discovery of the predicted
cosmic microwave background (CMB)
radiation by Arno Penzias and Robert Wilson
They were
awarded the
Nobel prize in
1978
8
The Hot Big Bang – The CMB
The CMB is the
most perfect
black body
spectrum known.
No other
convincing way
has been found
to explain the
perfection of this spectrum
9
The Hot Big Bang
Another important prediction is that when
the universe was about a second old the
number of protons increased relative to the
number of neutrons
t
le
to a ratio of
t0 = now
7 to 1.
d
3 minutes later,
1H and 4He nuclei
formed in the ratio
3 to 1
d0
d (t )  a(t )d0
lo
10
The Hot Big Bang - Abundances
11
Burles, Nollett & Turner, 1999
The Hot Big Bang - Problems


The Flatness Problem
 Why is the geometry of space so incredibly
flat?
The Horizon Problem
 Why is the temperature of the CMB so
incredibly uniform across the entire sky?
12
The Flatness Problem
The Friedman
equation can be
written as
8 Ga
2
a 
 Kc
3
2
2
H 1(t )  Kc / a
2
where
2
2
H (t )  a(t ) / a(t ), (t )   (t ) / c (t )
2
3H (t ) is the critical density at
and c (t ) 
8 G cosmic time t
13
The Flatness Problem
H 1(t )  Kc / a
2
2
2
Since K is constant in time, we can find its value
by evaluating this equation at our epoch:
H 1  0   Kc
2
0
2
We can then re-write the Friedman equation
as
2
0
2
H
(t )  1
10 
H
14
The Flatness Problem
We now consider the density parameter (t)
in the very early universe. The energy
density of photons exceeded that of matter
when distance scales were about 13,000
times smaller
t = now
0
totally dominated the
energy density of
d (t )  a(t )d0
the universe
d0
lo
15
The Hot Big Bang
In this case, the energy density is given
by  = R / a4, where R is the photon
energy density at the present epoch.
The solution of the Friedman equation is then
a(t )  bt
1/ 2
where b is a constant.
Therefore, the Hubble parameter in the
early universe goes like
a 1
H 
a 2t
16
The Hot Big Bang
When we put H(t) back into the equation for
(t) we get
2
 t 
(t )  1  4 
 1  0 
 1/ H0 
which is valid for very early times. Putting in
the measured value of H0 we find
(t )  1 10 t 1  0 
34 2
17
The Hot Big Bang
(t )  1 1034 t 2 1  0 
This expression exhibits the flatness problem:
In order to have an 0 ≈ 1, as observed, the
density must have been equal to the critical
value to better than 1 part in 1034 when
the universe was about a second old.
This fantastically fined-tuned adjustment of the
density cannot be explained by the big bang
18
The Horizon Problem
Portrait of the universe at t = 300,000 y
CMB temperature fluctuations DT/T ~ 10-5
19
The Horizon Problem
On the following slide, the spatial regions
BC and DE are causally disconnected; that is,
nothing in region BC can affect the region DE,
and vice versa, because the regions do not
overlap. Nonetheless, the temperature at G and
H is measured to be the same to one part in 105
Before t ≈ 300,000 y, all matter was ionized.
Consequently, photons could not travel far
without scattering off charged particles.
Thereafter, neutral atoms formed and
20
photons could then travel freely.
The Horizon Problem
t = 14 Gy
time
neutral atoms exist
and universe is
transparent
to light
G
H
ionized
light cannot
matter exists travel far
A B
C
space
D
t = 300,000 y
E
F
21
Summary

The big bang theory is successful in
predicting the CMB and the abundances of
the light elements.

However, it cannot explain why space is flat

Nor can it explain why the CMB is so isotropic
22
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