Download The homogeneous and isotropic universe: Cosmology

The homogeneous
and
isotropic universe:
Cosmology
Dr. Naylor
Cosmological Principle
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Our place in the universe is not special
Cosmology discusses physics on the very largest scales
Megaparsec scales! (1 pc = 3.261 lyr )
Cosmological principle
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The universe is smooth (homogeneous)
The universe looks the same in all directions (isotropic)
Note homogeneity does not imply isotropy and vice versa:
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Homogeneous magnetic field is not isotropic
Isotropic Spherical shell is not homogeneous
However, assuming isotropy at all given points in space does imply
homogeneity!!
The Metric?
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Assume (i) spacetime can be sliced into constant time hypersurfaces
which are isotropic & homogeneous (ii) mean rest frame of galaxies
agrees definition of simultaneity
2
2
i
j
ds
=
a
(t)h
dx
dx
Consider the 3D line element (index i,j) 3
ij
2
2
i
2
i
j
ds
=
−dt
+
g
dtdx
+
a
(t)h
dx
dx
Including time we have
0i
ij
where g00=-1 because dt is the propertime on a constant surface
dxi=0
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Also, g0i=0, because if t=constant and local Lorentz frame of
galaxies are to agree, then ē0 and ēi must be orthogonal
ds2 = −dt2 + a2 (t)hij dxi dxj
The most general isotropic 3D line element is the spacial spherically
symmetric [exterior] Schwarzschild metric:
ds23 = eλ(r) dr2 + r2 [dθ2 + sin2 θdφ]
Homogeneity
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Einstein equations
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1 αβ
− g R = Gαβ = κT αβ
2
Spacial part Gij has following solutions
Grr
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R
αβ
e2Λ
Gθθ = −re−2Λ Λ"
= − 2 (1 − e−2Λ )
r
2
ij
R = −2Gij g = 2 [r(1 − e−2Λ )]" = k
r
Gφφ = sin2 θGθθ
Homogeneity implies k = constant and therefore integrating the
1
above equation implies that g = e2Λ =
rr
Local flatness (grr =1 @ r=0)
Thus we find that
ds23
A=0
1 − kr2 /6 − A/r
1
2
2
2
2
=
dr
+
r
[dθ
+
sin
θdφ]
2
1 − kr
Full Friedman-Lematre-Robertson-Walker
(FLRW) metric
is
!
"
1
2
2
2
2
ds = −dt + a (t)
dr
+
r
(dθ
+
sin
θdφ)
2
1 − kr
2
2
2
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Spatial curvature
The FLRW metric allows for three values
of k=-1,0,1
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We can always rescale the radial
coordinate r and scale factor a(t) such
that the only special cases are k=-1,0,1
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E.g. for k=-3 if then r→√3 r and a(t)
→1/√3 a(t) then line element becomes
that with k=-1
k=1 implies a closed (spherical) universe
k=-1 implies an open (hyperbolic) universe
[horse-saddle]
k=0 corresponds to a flat (Euclidean)
universe.
From WMAP
FLRW equations
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We will assume that the universe is a perfect fluid with stress-energy
momentum tensor T µ = diag(−ρc2 , p, p, p)
ν
The Einstein G00 equation for the FLRW metric leads to the
Friedman equation
! "
2
G00
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8πG
kc
Λ
⇒H =
ρ− 2 +
3
a
3
=
µ
Tν,µ
+
Γµαµ Tνα
−
µ
Γα
T
νµ α
=0
If we include CC
For μ=ν=0 the relevant Christoffel symbols are
Γ000
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H=
ȧ
a
While conservation of energy-momentum implies
µ
Tν;µ
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where
2
= 0;
Γ101
=
Γ202
=
Γ202
ȧ
=
a
This all implies the following cosmological fluid equation
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p"
ρ̇ + 3H ρ + 2 = 0
c
Hubble’s constant
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Edwin Hubble showed in 1929 that all the galaxies are moving away
from us with a velocity v = H0r
Not really a constant!
This does not break the cosmological principle [imagine baking a cake
with raisins in it]
˙
d!r
!r|
|
ȧ
!v =
=
!r = !r
dt
|!r |
a
⇒
ȧ
H=
a
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The expansion of the universe implies that in the past there must
have been some Big Bang from which the universe was created
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Depending on the sign of k the universe will collapse to a big crunch,
or expand forever!!
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Where and when did the big bang
happen?
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Everywhere & nowhere
Imagine an expanding sphere
http://en.wikipedia.org/wiki/Hubble's_law
Cosmological Doppler-shift
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In special relativity the redshift factor z satisfies 1 + z = !1 + v/c ≈ 1 + v
c
1 − v/c
but some galaxies with z~6 have been detected!
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Does this mean faster than light travel? NO
Wrong
Space itself is expanding and there is no violation of causality because no
signals can be sent between such galaxies
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Imagine ants living on the surface of a balloon all limited to speed c. As
the balloon expands the red-shift might appear as v>c, but only due to
expansion!
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Cosmological red-shift: dv = Hdr = dr where for 2 galaxies nearby is
a
dλ
dv
=
where dλ ≡ λr − λe & time between emission and reception
λe
c
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is dt=dr/c.
Correct
λr
a(tr )
dλ
da
=
This all implies that
=
⇒ λ ∝ a and hence 1 + z =
λe
a(te )
λe
a
Using the FLRW metric this result is also valid for large distances
CMB & space travel?
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The cosmic microwave background (CMB) was detected by Penzias &
Wilson in 1965 (Nobel Prize)
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This originates in the early universe when photons decouple from free
electron & H+ ions and can travel in the universe
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This happened when H+ ions recombined with electrons at a
temperature of about 50 000K
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Why is this important in SR?
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What happens if we try to travel near to the speed of light?
Well microwave radiation has a
wavelength of the order 10cm
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Hint: special-relativistic Doppler-shift formula
This is a major obstacle to relativistic travel ...
Matter & radiation in the universe?
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!
p"
ρ̇ + 3H ρ + 2 = 0
c
ρ 1/a3
! "2
8πG
ȧ
=
ρ
a
3
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Matter (non-relativistic) is that
which exerts no pressure
p=0
a(t) =
!
t
t0
"2/3
Radiation (relativistic particles)
such as light has a pressure
p = σc2/3
Fluid equation
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ρ 1/a4
Friedman equation with k=0
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a(t) =
!
t
t0
"1/2
• Solutions with k=-1,1 also lead to similar results, but are not analytic
• Recent results also suggest we should include cosmological constant!!
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The age of the universe?
Hubble’s constant H0 = 100h kms-1Mpc-1 where h =0.72 ± 0.08
The units are a little unusual so on converting kilometres into
parsecs and converting seconds to years we find the Hubble time
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As is well known from radioactive decay of Uranium isotopes the
Earth is about 5 billion years old and stars & galaxies are even older
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H0-1= 9.77h-1×109 yrs
A good check is that the Universe must be older than the oldest
star! Chemical evolution of stars leads to 10~13 billion years
In matter dominated universe (k=0, flat)
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a(t) =
!
so t0 = 2H0/3 = 6.51h-1×109 yrs (even smaller!)
t
t0
"
ȧ
2
⇒H ≡ =
a
3t0
If we consider k=1 (closed) situation is worse, but near to open
(k=-1) improves the situation. Results a Λ>0 term is needed and
this improves the situation
Final Test Comments
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Final Test (30pts of your mark) will be next week!
You need more than 15pts (50%) in final test to pass course!
Approximately 60 minutes from 9:00AM~10:00AM
About 5/6 questions on SR and GR.
Possible questions:
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Postulates of SR, Michelson-Morley, relativistic Dopplershift, Equivalence Principle, spherically symmetric +
homogeneous isotropic spacetimes, cosmological red-shift
from FLRW metric, Schwarzschild metric
Good Luck!