IX. The Cosmological Constant - Macquarie...

IX. The Cosmological Constant

ASTR378 Cosmology : IX. The Cosmological Constant 96

Einstein’s “Greatest Blunder”

•  At the time (~1915), Universe believed static, supported by observational data

•  Universe containing only matter must be expanding or contracting – the only static Universe is an empty one introduce a “fudge factor”, Λ, with constant density and pressure

•  Friedmann Equation with Λ:


�

˙ a a

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

+ kc 2

a2 − c 2Λ3

= 8πG3

ρ

Problems with Λ

•  Acceleration Equation with Λ – acts as a repulsive force:

•  If Λ has constant P, ρ then infinitesimal expansion / contraction of the Universe would lead to runaway expansion / contraction unstable

•  (Inelegant – took away from “formal beauty of theory”) •  1929: Hubble paper on redshift-distance relation Universe

expanding, no need for Λ (although sometimes invoked when the age of the Universe appeared less than the ages of astronomical objects...)


�

˙ ̇ a a

= − 4πG3

ρ + 3Pc 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ +

Λc 2

3

1998: The Rehabilitation ofΛ

•  Type Ia Supernovae (SNe Ia): thermonuclear explosion of a white dwarf > ~ 1.4 M (single/ double degenerate scenarios)

•  Standard Candles: SNe Ia have ~same luminosity can get distances to host galaxies, calibration improved by taking shape of light curve into account

•  In 1998, two competing teams observing SNe Ia found that for z 1, SNe Ia were too faint


4 April

SN

28 April

How to Find Supernovae


Two Competing Teams, Same Result


S. Perlmutter, G. Aldering, S. Deustua, S. Fabbro, G. Goldhaber, D. Groom,A. Kim, M. Kim, R. Knop, P. Nugent, (LBL & CfPA)N. Walton (Isaac Newton Group)A. Fruchter, N. Panagia (STSci)A. Goobar (Univ of Stockholm)R. Pain (IN2P3, Paris)I. Hook, C. Lidman (ESO)M. DellaValle (Univ of Padova)R. Ellis (CalTech)R. McMahon (IofA, Cambridge)B. Schaefer (Yale)P. Ruiz-Lapuente (Univ of Barcelona)H. Newberg (Fermilab)C. Pennypacker

• Brian Schmidt (ANU)• Nick Suntzeff, Bob Schommer, Chris Smith (CTIO)• Mark Phillips (Carnegie)• Bruno Leibundgut and Jason Spyromilio (ESO)• Bob Kirshner, Peter Challis, Tom Matheson (Harvard)• Alex Filippenko, Weidong Li, Saurabh Jha (Berkeley)• Peter Garnavich, Stephen Holland (Notre Dame)• Chris Stubbs (UW)• John Tonry, Brian Barris (University of Hawaii)• Adam Reiss (Space Telescope)• Alejandro Clocchiatti (Catolica Chile)• Jesper Sollerman (Stockholm)

The Accelerating Universe

•  What else could make the SNe Ia at high z too faint? –  Dust? –  Are today’s SNe the same as those at

z~1? –  Are the various observational

corrections (e.g., K-correction) causing systematic errors?

–  ...?

•  Eleven years later, however, the general consensus is that the SN Ia data are best fit by an accelerating expansion, parameterised as Λ


Adding Λ to the Mix

•  Can define a density parameter for Λ, ΩΛ:

•  So, to have a flat Universe: open Universe: closed Universe:


�

ΩΛ = Λc 2

3H 2 ; Ω+ ΩΛ −1= kc 2

a2H 2

Ω+ ΩΛ =10 < Ω+ ΩΛ <1Ω+ ΩΛ >1

ΩΛ vs ΩΜ


Λ as a Fluid

•  We can define Λ as a fluid with energy density ρΛ and pressure PΛ. Rewriting the Friedmann Equation:

•  Rewriting the Fluid Equation:

since ρΛ is constant by definition. Λ has a negative effective pressure: as the Universe expands, work is done on the Λ fluid, so energy density can remain constant even though a increasing

ASTR378 Cosmology : IX. The Cosmological Constant 105 �

ρΛ ≡ Λ8πG

⇒H 2 = 8πG3

ρ + ρΛ( ) − kc 2

a2

˙ ρ Λ + 3 ˙ a a

ρΛ + PΛ

c 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0 ⇒ PΛ = −ρΛc 2

Equations of State Revisited

•  Friedmann Equation, Fluid Equation, Acceleration Equation:

•  Equation of State: P = P(ρ), for cosmological components can write in the form P = wρ(w dimensionless)

•  For non-relativistic gases, w ≈ <v2>/3c2 << 1 •  For photons and relativistic gases, Prel = 1/3(ρrel) ; “mildly”

relativistic gases have 0 < w < 1/3 •  A cosmological constant has w < 0, e.g., w=-1/3 (“dark

energy”; experimentally w appears to be ~ -1


�

˙ a a

⎛ ⎝ ⎜

⎞ ⎠ ⎟

2

+ kc 2

a2 = 8πG3

ρ ; ˙ ρ + 3 ˙ a a

ρ + Pc 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = 0 ; ˙ ̇ a

a= − 4πG

3ρ + 3P

c 2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Living with Λ

•  Current observational constraints on Ωmat and ΩΛ from three different sources: –  Type Ia supernovae (SNe, blue) –  Sound waves in the Cosmic Microwave

Background (CMB, orange) –  Large-scale structure in the distribution of

galaxies (Baryon Acoustic Oscillations=BAO, green)

•  They all overlap at the same location (!) •  They all overlap where Ωmat + ΩΛ≈ 1

(!); best guess Ωmat ~ 0.3, ΩΛ~ 0.7


From Kowalski et al. 2008

Contours:68.3%, 95.4% and 99.7% confidence levels

k > 0

k < 0

So What is Λ?

•  What has an energy density ρΛ that remains constant as the Universe expands or contracts?

•  One idea: vacuum energy – the energy density of virtual particle pairs that appear and annihilate each other in empty vacuum

•  Vacuum energy would not be affected by the expansion of the Universe, so ρvac would be constant with time and independent of the expansion / contraction of the Universe

•  However, the expected vacuum energy density has not been successfully calculated – one guess is the Planck energy EP per Planck volume (lP3), but this yields ρvac~3×10133 eV m-3, roughly 124 orders of magnitude > ρcrit ! ...?


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