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