Inflation, Dark Energy, and the Cosmological Constant
Intro Cosmology Short Course
Lecture 5
Paul Stankus, ORNL
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˙ a (t)( )2
=8πGN
3c 2ρ(t)EnergyDensity
{ a(t)( )2
− k c 2
SpatialCurvature
{
The Friedmann Equation0
GR
Recall from Lecture 4:
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2 ˙ a (t)˙ ̇ a (t) =8πGN
3c 2ρ (t)2a(t) ˙ a (t) + ˙ ρ (t) a(t)( )
2
( )
€
˙ ̇ a (t)
a(t)=
4πGN
3c 22ρ (t)+
dρ(a)
daNeed an
Equation of State
1 2 3 a(t)
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
yadda, yadda, yadda…
€
dE = TdS − PdVBasic
Thermodynamics
Sudden expansion, fluid fills empty space without loss of energy.
dE 0 PdV 0 therefore dS 0
Gradual expansion (equilibrium maintained), fluid loses energy through PdV work.
dE PdV if and only if dS 0Isentropic Adiabatic
Perfect Fluid
Hot
Hot
Hot
Hot
Cool
€
˙ ̇ a (t)
a(t)= −
4πGN
3c 2ρ(t)+ 3P(t)[ ]The second/dynamical
Friedmann Equation€
˙ a (t)
a(t)
⎛
⎝ ⎜
⎞
⎠ ⎟2
=8πGN
3c 2ρ (t)−
k c 2
a(t)( )2
The Friedmann Equation
0
Together with the perfect fluid assumption dEPdV gives
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ρ > 0 , P ≥ 0
⇒ ˙ ̇ a (t)< 0 for all tNo static
Universes!
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˙ ̇ a (t) < 0 , ˙ a (t) > 0 for early t
⇒ a(t) =0 at finite tBig
Bang!QuickTime™ and a
TIFF (Uncompressed) decompressorare needed to see this picture.
Over-
€
Something
about space- time
curvature
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=
Something
about
mass- energy
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
€
Metric Tensor gμν
Riemann Tensor Rαβγδ
Ricci Tensor Rμν = Rαμαν
Ricci Scalar R = Rαα
€
Stress- Energy Tensor Tμν
€
Rμν −1
2Rgμν
For continuity
≡ Gμν Einstein Tensor
1 2 4 4 3 4 4 − Λ
Unknown{ gμν = 8πGN
To match Newton
1 2 3 Tμν
€
Gμν = 8πTμν Einstein Field Equation(s)B. Riemann GermanFormalized non-Euclidean geometry (1854)
G. Ricci-Curbastro Italian Tensor calculus (1888)
Column A Column B
0
“The Heavens endure from everlasting to everlasting.”
€
˙ ̇ a (t)
a(t)= −
4πGN
3c 2ρ(t)+ 3P(t)[ ] +
Λ
3€
˙ a (t)
a(t)
⎛
⎝ ⎜
⎞
⎠ ⎟2
=8πGN
3c 2ρ (t)−
k c 2
a(t)( )2 +
Λ
3 The full Friedmann equations with non-zero “cosmological
constant”
Albert Einstein German
General Theory of Relativity (1915);
Static, closed universe (1917)
€
∃ρ > 0 , P = 0, Λ > 0, k > 0
⇒ ˙ ̇ a (t) = ˙ a (t) =0 for all t
Einstein closed, static Universe
Cosmological “Leftists” vs “Rightists”
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Gμν − Λgμν = 8π Tμν
€
Gμν = 8π Tμν + Λgμν
Left: as part of geometry
Right: as part of energy&pressure
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Gμν = 8π
ρ 0 0 0
0 P 0 0
0 0 P 0
0 0 0 P
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
+ Λ
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
⎡
⎣
⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥
€
=8π
ρ + Λ8π 0 0 0
0 P − Λ8π 0 0
0 0 P − Λ8π 0
0 0 0 P − Λ8π
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥
T for matter and radiation
in local rest frame
Effective T in the presence of
a cosmological constant
Cosmological constant as part of General Relativity
Space filled by energy of uniform density and Pρ
Indistinguishable!
Inflationary Universes
W. de Sitter Dutch
Vacuum-energy-filled universes “de Sitter space” (1917)
€
˙ a (t)
a(t)
⎛
⎝ ⎜
⎞
⎠ ⎟2
= H(t)( )2
=Λ
3
What if a flat Universe had no matter or radiation but only cosmological constant (and/or vacuum energy)?
€
a(t)∝ exp +Λ
3t
⎛
⎝ ⎜
⎞
⎠ ⎟
Exponentially expanding!
…but yet oddly static: No beginning, no ending, H(t) never changes
“de Sitter Space”
€
a(t)∝ exp +Λ
3t
⎛
⎝ ⎜
⎞
⎠ ⎟ "de Sitter"
€
a(t)∝ exp −Λ
3t
⎛
⎝ ⎜
⎞
⎠ ⎟ "de Sitter"
€
a(t) = Constant "Minkowski"
Con
tinuu
m
Negative pressure: not so exotic….
€
dE = ρ dV = −PdV
∴ P = −ρConstant energy density Negative pressure
E
€
ρ ~r E
2
E
Unchanged!
€
ρ ~r E
2
Light Cones in De Sitter Space
t (1/H0)
Now
BB
Radiation Dominated
Inflationary De Sitter
(c/H0)
Robertson-Walker Coordinates
€
a(t) = tt0
⎛ ⎝ ⎜ ⎞
⎠ ⎟
1 2
H0 =1
2t0
χ γ (t) = ±c
a(t)dt + Const∫
= ±2c t t0( )1 2
+ Const
€
a(t) = K exp Ht( ) H0 = H
χ γ (t) = ±c
a(t)dt + Const∫
= ±c
Kexp −Ht( ) + Const
Radiation-filled
Vacuum-energy-filled
Us
Inflation solves (I): Uniformity/Entropy
t (1/H0)Now
BB no inflation
Radiation Dominated
Inflationary De Sitter
(c/H0)
Robertson-Walker Coordinates
Us
BB with inflation
CMB decouples In
flatio
n
Alan Guth American
Cosmological inflation (1981)
Inflation Solves (II): Early Flatness
(Yet another) Friedmann Equation
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d
dtΩM(t) −1[ ]Non -Flatness
1 2 4 3 4
ΩM(t) −1[ ]Non -Flatness
1 2 4 3 4
= H(t)ΩM(t) =˙ a (t)
a(t)ΩM(t)
0
Non-Flatness grows ~a(t) Huge increase!?
€
˙ a (t)
a(t)
⎛
⎝ ⎜
⎞
⎠ ⎟2
=8πGN
3c 2ρ (t)
Energy Density∝1/a 3 or 1/ a 4
1 2 4 3 4 −
k c 2
a(t)( )2
Curvature∝1/ a 2
1 2 3
+Λ
3Cosmological
Constant
{Dominates with increasing a(t), leading to faster increase
Flat (Generic)
0
OpenClosed
Our lonely future?
Horizon
Matter/radiation Dark energy
The New Standard Cosmology in Four Easy Steps
Inflation, dominated by “inflaton field” vacuum energy
Radiation-dominated thermal equilibrium
Matter-dominated, non-uniformities grow (structure)
Start of acceleration in a(t), return to domination by cosmological constant and/or vacuum energy.
w=P/
-1/3
+1/3
-1
aeHt at1/2 at2/3
t
now
acc
dec
Points to take home
• For full behavior of a(t), need an equation of state (P/ρ)
• Matter/radiation universes: evolving, and finite age
• Cosmological constant a legal but unneeded piece of GR; is indistinguishable from constant-density vacuum energy
• All >0 evolve to inflationary (de Sitter) universes; Do we
live at a special time? Part 2
• Early inflation solves flatness and uniformity puzzles
• Life in inflationary spaces is very lonely