Density Perturbations in an Inflationary Universe

1
Density Perturbations in an Inflationary Universe

• Guth Pi (1982)
• Bardeen, Steinhardt Turner (1983)
• Katie Mack

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Density Perturbations in an Inflationary Universe

• Guth Pi (1982)
• Bardeen, Steinhardt Turner (1983)
• Katie Mack

1982 Michael Jackson releases Thriller
3
Outline of Talk

• Basics of inflation (summary of Sudeeps talk)
• Varieties of inflationary models
• Guths original model
• New inflation
• Chaotic inflation
• Density Perturbations
• Inflationary Timeline
• Conclusion

4
Inflation

• Proposed by Guth in 1981 to solve
• Horizon problem
• Flatness problem
• Basic idea universe undergoes exponential
  expansion in early history

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Inflation

• Horizon Problem
• Problem correlated regions in CMB are outside
  each others horizons
  
• Solution regions were expanded out of each
  others horizons

(from Sudeeps talk)
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Inflation

• Flatness Problem
• Problem Total density parameter O improbably
  close to 1 (universe very close to flat)
  
• Solution Exponential expansion flattens universe

(from Sudeeps talk)
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Varieties of Inflationary Models

• Guths original scenario
• Phase transition Bubbles of true vacuum form in
  false vacuum background
  
• Bubbles coalesce to reheat universe
• Problem
• Space between bubbles expands exponentially - no
  coalescence
  
• Universe inflates forever (graceful exit
  problem)

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Varieties of Inflationary Models

• New Inflation (this scenario is used by Guth Pi
  and B,ST)
  
• Coleman-Weinberg potential (chosen for
  super-symmetry breaking in GUT)
  
• Scalar field starts at local minimum (false
  vacuum)
  
• Local minimum is separated from global minimum
  (true vacuum) by temperature-dependent barrier
  
• Unstable equilibrium as T - 0 field slowly
  rolls toward true vacuum state
  
• Whole universe can be contained in one bubble
• Reheating occurs when field oscillates about true
  minimum
  

V(f)
T 0
f
V(f)
T 0
f
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Varieties of Inflationary Models

• New Inflation (this scenario is used by Guth Pi
  and B,ST)
  
• Coleman-Weinberg potential (chosen for
  super-symmetry breaking in GUT)
  
• Scalar field starts at local minimum (false
  vacuum)
  
• Local minimum is separated from global minimum
  (true vacuum) by temperature-dependent barrier
  
• Unstable equilibrium as T - 0 field slowly
  rolls toward true vacuum state
  
• Whole universe can be contained in one bubble
• Reheating occurs when field oscillates about true
  minimum

V(f)
T 0
Problem Gives incorrect magnitude of density
perturbations
f
V(f)
T 0
f
10
Coleman-Weinberg Potential
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Varieties of Inflationary Models

• Chaotic Inflation (currently favored model)
• Slow-roll of potential achieved with drag term
  in equation of motion
  
• Powerlaw potential
• Production of particles (reheating) occurs as
  field oscillates about its minimum

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Varieties of Inflationary Models

• Chaotic Inflation (currently favored model)
• Slow-roll of potential achieved with drag term
  in equation of motion
  
• Powerlaw potential
• Production of particles (reheating) occurs as
  field oscillates about its minimum

Problem Fine-tuning required for correct
magnitude of density perturbations
13
Other Models

• Hybrid inflation
• Multiple scalar fields, not necessarily all with
  consequences to dynamics
  
• Eternal inflation
• Universe infinitely reproduces new universes
• True vacuum is achieved in many different parts
  of the inflating universe that are not causally
  connected this is a self-perpetuating process

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Inflation and Density Perturbations

• Inflation - nearly homogeneous universe
• but if exactly homogeneous, no structure or CMB
  anisotropies
  
• Must be mechanism for density perturbations in
  inflation
  
• CMB anisotropy, LSS
• Superhorizon scales
• Nearly scale-free

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Inflation and Density Perturbations

• The short version
• Quantum fluctuations before/during inflation
• Small, subhorizon fluctuations frozen in when
  universe expands and they cross the horizon
  
• Post-inflation, standard growth of perturbations
• (more discussion later)

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Observing Density Perturbations

• How do we measure primordial density
  perturbations?
  
• CMB anisotropy
• Sachs-Wolfe effect
• ?T/T result of redshifting of photons coming out
  of gravitational potential wells
  
• Contrast in redshift - depth of potential wells
  - measure of d?/?

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Sachs-Wolfe Effect
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Observing Density Perturbations

• 1982 only upper limit on CMB anisotropy
  (pre-COBE)
  
• d?/?
• COBE - d?/?10-5
• Harrison-Zeldovich spectrum scale-independent
  P(k) k (or P(k) kn where n 1)

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Observing Density Perturbations
Turnover in spectrum indicates horizon size at
matter-radiation equality
Growth the same on all scales before horizon
crossing
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Not Quite Scale-Independent

• Inflation predicts nearly (but not quite)
  scale-independent fluctuations
  
• P(k) kn , n?1
• (more on this later)

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Timeline of Density Perturbation Production

• Inflation begins - exponential expansion
• Quantum fluctuations frozen in
• Fluctuations in scalar field - time delay
• Post inflation scales re-enter horizon, normal
  evolution of perturbations

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1) Inflation begins

• Scalar field f slowly rolling down potential -
  exponential expansion
  
• Quantum fluctuations df in scalar field on
  horizon scale
  
• Analogy to Hawking radiation

Zero-point fluctuations in the scalar field with
wavelengths of order the Hubble radius
Attributed to Hawking temperature (H/2p)
associated with event horizon
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1) Inflation begins

• Scalar field f slowly rolling down potential -
  exponential expansion
  
• Quantum fluctuations df in scalar field on
  horizon scale
  
• Analogy to Hawking radiation

Zero-point fluctuations in the scalar field with
wavelengths of order the Hubble radius
Attributed to Hawking temperature (H/2p)
associated with event horizon
horizon shrinks, isolating particles
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2) Fluctuations frozen in

• Fluctuations produced when proper length scale
  comparable to Hubble Radius 1/H (horizon scale)
  
• Fluctuations frozen in by expansion as they are
  carried outside the horizon

horizon scale
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2) Fluctuations frozen in

• Fluctuations produced when proper length scale
  comparable to Hubble Radius 1/H (horizon scale)
  
• Fluctuations frozen in by expansion as they are
  carried outside the horizon

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2) Fluctuations frozen in

• Fluctuations produced when proper length scale
  comparable to Hubble Radius 1/H (horizon scale)
  
• Fluctuations frozen in by expansion as they are
  carried outside the horizon

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2) Fluctuations frozen in

• Fluctuations produced when proper length scale
  comparable to Hubble Radius 1/H (horizon scale)
  
• Fluctuations frozen in by expansion as they are
  carried outside the horizon

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2) Fluctuations frozen in

• Fluctuations produced when proper length scale
  comparable to Hubble Radius 1/H (horizon scale)
  
• Fluctuations frozen in by expansion as they are
  carried outside the horizon

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3) Time delay

• Perturbations df - different end times for
  inflation in different locations
  
• Regions in which inflation ends later become
  larger density is diluted
  
• Early end to inflation underdensity
• Late end to inflation overdensity

f
f
df
dt
t
t
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4) Inflation Ends

• Horizon begins to grow again (with respect to
  space)
  
• Scales re-enter horizon
• Normal evolution of perturbations occurs as
  perturbations become causally connected

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horizon
Horizon crossing in comoving coordinates
smooth patch on CMB
now
end
Horizon crossing in physical coordinates
start
start
horizon crossing
end
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Scale near-independence

• Why are the primordial perturbations nearly
  scale independent?
  
• d?/? H fractional perturbation amplitude on a
  given scale upon re-entering the horizon
  
• d?/? H H?f/(df/dt)
• (df/dt) will be different at different times as
  the field rolls down the potential (it increases
  with time)
  
• ?f H, and while H is nearly constant during
  inflation, it does grow slowly with time
  
• Slight scale dependence

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Density Perturbation Magnitude Estimate

• Using the Coleman-Weinberg potential of New
  Inflation, both authors find d?/? 10 on scale
  of current horizon
  
• What went wrong?
• Choice of potential (was chosen for supersymmetry
  breaking) could be fixed with implausible
  amount of fine-tuning of parameters (level of
  10-12)
• Current method choose functional form of
  potential, normalize with d?/? 10-5
  
• Generally phenomenological potential is not
  determined from first principles

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Conclusions and Current Work

• Inflationary models succeed in creating nearly
  scale-free density perturbations which can grow
  to create structure in the universe
  
• But getting the magnitude of the perturbations
  right requires scaling the potential accordingly
  
• No conclusion yet on correct functional form of
  inflationary potential
  
• However, scale-independence is well constrained
  with CMB, Lyman-alpha forest measurements