PortsmouthPortsmouth

Models of inflation and primordial perturbations

Models of inflation and primordial perturbations

David Wands

Institute of Cosmology and GravitationUniversity of Portsmouth

Cosmo – 02 Chicago, September 2002

• period of accelerated expansion in the very early universe

• requires negative pressure

e.g. self-interacting scalar field

• speculative and uncertain physics

Cosmological inflation:Cosmological inflation: Starobinsky (1980)

Guth (1981)

V()

•just the kind of peculiar cosmological behaviour we observe today

• inflation in the very early universe

• observed through primordial perturbation spectra

•gravitational waves (we hope)

•matter/radiation perturbations

Models of inflation:Models of inflation:

gravitational

instability

new observational data offers precision tests of cosmological parameters and the nature of the

primordial perturbations

Perturbations in FRW universe:Perturbations in FRW universe:

Characteristic timescales for comoving wavemode k

– oscillation period/wavelength a / k– Hubble damping time-scale H -1

• small-scales k > aH under-damped oscillator

• large-scales k < aH over-damped oscillator

03 2 Hwave equation

H-1 / a

conformal time

comoving wavelength, k-1

radiation or matter era

decelerated expansion

frozen-in

oscillates

comoving Hubble length

inflation

accelerated expansion (or contraction)

vacuum

Vacuum fluctuationsVacuum fluctuations

V()

• small-scale/underdamped zero-point fluctuations

• large-scale/overdamped perturbations in growing mode

linear evolution Gaussian random field

k

e ik

k2

2

3

23

2

2)2(

4

Hk k

aHk

fluctuations of any light fields (m<3H/2) `frozen-in’ on large scales

Hawking ’82, Starobinsky ’82, Guth & Pi ‘82

*** assumes Bunch-Davies vacuum on small scales ***

probe of trans-Planckian effects for k/a > MPl Brandenberger & Martin (2000)

effect likely to be small for H << MPl Starobinsky; Niemeyer; Easther et al (2002)

Inflation -> matter perturbationsInflation -> matter perturbations

HR

for adiabatic perturbations on super-horizon scales0R

during inflation scalar field fluctuation,

scalar curvature on uniform-field hypersurfaces

HR

during matter+radiation era

density perturbation,

scalar curvature on uniform-density

hypersurfaces

time

high energy / not-so-slow roll

1. large field ( MPl )

e.g. chaotic inflation

not-so-high energy / very slow roll

2. small field

e.g. new or natural inflation

3. hybrid inflation

e.g., susy or sugra models

Single-field models:Single-field models:

slow-roll solution for potential-dominated, over-damped evolution

gives useful approximation to growing mode for { , || } << 1

2

22

16 H

H

V

VM P

2

22

8 H

m

V

VM P

Kinney, Melchiorri & Riotto (2001)

0

0

0

see, e.g., Lyth & Riotto

can be distinguished by observationscan be distinguished by observations• slow time-dependence during inflation

-> weak scale-dependence of spectra

• tensor/scalar ratio suppressed at low energies/slow-roll

261 n

162

2

R

T

• Microwave background +2dF + BBN + Sn1A + HST

Lewis & Bridle (2002)

Observational constraintsObservational constraints

08.0scalar

tensor

spectral index

261 n

Harrison-Zel’dovich

n 1, 0

• additional fields may play an important role

• initial state new inflation

• ending inflation hybrid inflation

• enhancing inflation assisted inflation

warm inflation

brane-world inflation

• may yield additional information

• non-gaussianity

• isocurvature (non-adiabatic) modes

digging deeper:digging deeper:

Inflation -> primordial perturbations (II)Inflation -> primordial perturbations (II)scalar field fluctuations density perturbationstwo fields (,) matter and radiation (m,)curvature of uniform-field slices curvature of uniform-density

slices

HR *

HR

aHkSS

RS

aHkS

R

T

T

S

R

inflation*

*

primordial 0

1

model-dependent transfer functions

HSS

HSR

,

HS *

n

n

n

nS

m

m

isocurvature

initial power spectra:initial power spectra:

uncorrelated at horizon-crossing

2

*

2*

2* 2

H

2

*

22

*2

* 2

H

SR

0**** SR

primordial power spectra:primordial power spectra:

correlated! Langlois (1999)2

*

2*

22

2*

22*

2

STTSR

STS

STRR

SSRS

SS

RS

correlation angle measures contribution of non-adiabatic modes to primordial curvature perturbation

22/122/12 1cos

RS

RS

T

T

SR

RS

can reconstruct curvature perturbation at horizon-crossing

222* sin RR

consistency conditions:consistency conditions:

single-field inflation: Liddle and Lyth (1992)

TnR

T

R

TRR 8

2*

2*

2

22

*2

gravitational waves:

2

1632

2

2*

2

2*

2

T

Pl

n

RM

HTT

two-field inflation: Wands, Bartolo, Matarrese & Riotto (2002) 22

2*

2*

2

2

sin8sin TnR

T

R

T

multi-field inflation: Sasaki & Stewart (1996)

TnR

TRR 8

2

22

*2

microwave background predictionsmicrowave background predictions

adiabatic isocurvature correlation

Cl = A2 + B2 + 2 A B cos

Bucher, Moodley & Turok ’00 nS = 1

Trotta, Riazuelo & Durrer ’01

Amendola, Gordon, Wands & Sasaki ’01

best-fit to Boomerang, Maxima & DASI

B/A = 0.3, cos = +1, nS = 0.8

b = 0.02, cdm = 0.1, = 0.7

}

curvaton scenario:curvaton scenario:

large-scale density perturbation generated entirely by non-adiabatic modes after inflation

assume negligible curvature perturbation during inflation

02* R

Lyth & Wands, Moroi & Takahashi, Enqvist & Sloth (2002)

22,

2*

22 )/( decayRS STR

late-decay, hence energy density non-negligible at decaydecayRST ,

light during inflation, hence acquires isocurvature spectrum

2*S

•negligible gravitational waves

•100%correlated residual isocurvature modes

•detectable non-Gaussianity if ,decay<<1

primordial non-gaussianityprimordial non-gaussianity

simple definition in terms of conformal Newtonian potential =3R/5

where fNL = 0 for linear evolution

22GaussGaussNLGauss f Komatsu & Spergel (2001)

also Wang & Kamiokowski (2000)

significant constraints on fNL from all-sky microwave background surveys

• COBE fNL < 2000

• MAP fNL < 20

• Planck fNL < 5theoretical predictions

• non-linear inflaton dynamics constrained fNL < 1 Gangui et al ‘94

• dominant effect from second-order metric perturbations? Acquaviva et al

(2002) • but non-linear isocurvature modes could allow fNL >> 1 Bartolo et al (2002)

• e.g., curvaton fNL 1 / ,decay Lyth, Ungarelli & Wands (2002)

Conclusions:Conclusions:

1. Observations of tilt of density perturbations (n1) and gravitational waves (>0) can distinguish between slow-roll models

2. Isocurvature perturbations and/or non-Gaussianity may provide valuable info

3. Non-adiabatic perturbations (off-trajectory) in multi-field models are an additional source of curvature perturbations on large scales

4. Consistency relations remain an important test in multi-field models - can falsify slow-roll inflation