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

( )

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˙ ̇ 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

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˙ ̇ 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-

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Something

about space- time

curvature

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

Something

about

mass- energy

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

€

Metric Tensor gμν

Riemann Tensor Rαβγδ

Ricci Tensor Rμν = Rαμαν

Ricci Scalar R = Rαα

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Stress- Energy Tensor Tμν

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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

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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)

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∃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

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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)

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˙ 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)?

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a(t)∝ exp +Λ

3t

⎛

⎝ ⎜

⎞

⎠ ⎟

Exponentially expanding!

…but yet oddly static: No beginning, no ending, H(t) never changes

“de Sitter Space”

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a(t)∝ exp +Λ

3t

⎛

⎝ ⎜

⎞

⎠ ⎟ "de Sitter"

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a(t)∝ exp −Λ

3t

⎛

⎝ ⎜

⎞

⎠ ⎟ "de Sitter"

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a(t) = Constant "Minkowski"

Con

tinuu

m

Negative pressure: not so exotic….

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dE = ρ dV = −PdV

∴ P = −ρConstant energy density Negative pressure

E

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ρ ~r E

2

E

Unchanged!

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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

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a(t) = tt0

⎛ ⎝ ⎜ ⎞

⎠ ⎟

1 2

H0 =1

2t0

χ γ (t) = ±c

a(t)dt + Const∫

= ±2c t t0( )1 2

+ Const

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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