The Hubble constant (H) will change with time and with 1 z.

1
H tells us the slope of the R-t curve,
therefore A less steep curve means a lower
expansion rate
The Hubble constant (H) will change with time
and with 1z.
2
Mini quiz
De-acceleration
-1
k ?
0
?
W
R
1
k ?
gt1
?
W
t
3
W 0 gt k -1 and r 0gt (R/R)2R2 -kc2 or R
constant slope of R vs t, R H0t (as in y
a bx with a 0, x R b H0
.
.
4
Einstein-de Sitter is the name for a k 0
universe, Wt 1 universe
See table 11.1, page 305, but we dont know what
q0 is yet!
Physical concepts needed to derive q0 and what it
mean to follow
5
Adiabatic Expansion(dQ 0)needed to calculate
acceleration (change of expansion H with time)
Also need/use the first law of thermodynamics
is dU dQ-pdV, U total
internal energy Q heat p pressure, V
volume. Energy mass x c2 plus internal
kinetic energy (negligible in our case) of the
gas of the Universe
6
U is internal gt dont include total bulk KE
(expansion of Universe) or PE (gravity term) of
Universe in U.
7
Adiabatic expansion (dQ 0) first law of
thermodynamics equation for pressure (p) gt
derive the relationship between R and t Just
before brick wall and smaller 1z (matter
dominated era), p 0 !
Cant assume p 0 when in radiation dominated
era or if have a cosmological constant Wont
derive the equations for these case.
8
Adiabatic Exp. (dQ 0)
gt dU -pdV, needed to derive how the
Hubble constant changes with time. U Mc2.
Remember M r4pR3 /3 combine dU equation with
escape equation to derive how H changes with
time. dU c2d(r4pR3)/3, both r and R depend on
t, dU 0 gt can derive De-acceleration parameter
(q0 today) W0/2
Where R with 2 dots over it acceleration
9
The sign of q tells us if universe
de-accelerating or accelerating. Since for here
2q0 W0, we know the answer (see next slide)
De-accelerating
10
From
.
From R0 gt 0 , R02 gt 0 and 2q0 W0 gt 0 gt R lt 0
Matter dominated, L 0 universe, were
de-accelerating
..
11
Going back in time
Why dont the curves in the book look funny at
low t where t is low enough (100,000 yrs after
the BB) for radiation to be important?
Because this is at at time when the universe was
about 100,000 years old R(t) is too small to be
seen on the scale of the plots
12
One Final detail
Today and back as far as we can see (brick
wall) The matter dominated era, p 0, and W0
2q0. Old books use W0 and 2q0 interchangeably,
but these are physically different things W0
density ratio q0 de-acceleration parameter
13

• GR/surface geometry
• one universe
• self gravitating
• totally isolated universe
• matter dominated

• measure parameters of our model curvature (k)
  critical density rc, W0, H0 q0.
• gt predict the fate of the universe expand
  forever etc. see table 11.1

14
Summary

• Start from geometry and demand that light travel
  a geodesic on the surface

Use a metric with curvature (k)
Require that gravity (and GR) apply relate the
concept of an initial Bang to future assume
matter dominated (M (4/3)pR3 x r )
gt rc, H0 , W0 , and k
15
pressure 0 (part of matter dominated model)
Adiabactic expansion first law of
thermodynamics derive de-acceleration parameter
(q) relationship to W
16
H0 the Hubble constant about 50 km/(sec- Mpc)
Mpc 3 x 1024 cm 3 x 1019 km
Now to back fill with FAQ
If the universe started from a point
Why dont we see a direction for the explosion?
Why didnt all the matter end up in one or many
black holes?
17
Answers
1. Because everywhere in the universe had its own
big bang. There was no preferred spot!
2. (a) Each place in the universe did start from
the DECAY of black hole (b) Matter didnt come
into existence until the density was less than
needed to make a black hole
R 2GM/c2 radius of black hole
18
Looking at the escape equation we see wed like
to measure density and the Hubble constant.
19
Looking at the Robertson-Walker metric we can see
how to relate the Hubble constant to the
expansion rate. This is a little fudge but
gives the concept.
20
Change c to v, dt to t and set k 0 because
were so close, and space isnt curved, set rR
D, divide by t and set R/t to R and then multiply
top and bottom by R gt RrR/R v or DH v !
.
.
21
Entropy, Heat death, big crunch and the like...
22
Entropy, Heat Death, and the like
Book says universe is cooling off and were
losing energy No, because the volume is
expanding as well, the Universe is not losing
energy. Rather the energy/unit density is what
is decreasing.
23
Entropy, Heat Death, and the like
The book also says were headed for more and more
disorder (or increased entropy). On a local
scale this is correct. There is clearly and arrow
to time. (None of us is getting any younger)
Absurd to think if the universe started to
contract ( k 1 case, L0), wed get younger
and eventually turn into babies.
24
Entropy, Heat Death, and the like
The but on the scale of the Universe, weve
assumed for our calculations there is no change
in (total) entropy the cognesetti would
recognize this as dS dQ /T where S the
entropy and T the temperature. gt Unless T 0,
dQ 0 (what we assumed in deriving 2q0 W0
implies dS 0! Or, entropy remains constant! So
weve fudged, but it works!
25
What about the big crunch? Will we bounce or
not?
26
Entropy, Heat Death, and the like
The answer is we dont know. The book makes a
good point that we probably wont bounce
because we will have gained entropy (disorder)
therefore be so disordered, the universe wont
start afresh. But maybe at high temperatures
and density, our normal definition of entropy
fails! So really, we dont know!