Transcript Document
Observational constraints on
primordial perturbations
Antony Lewis
CITA, Toronto
http://cosmologist.info
Primordial fluid at redshift < 109
• Photons
• Nearly massless neutrinos
Free-streaming (no scattering) after neutrino decoupling at z ~ 109
• Baryons
tightly coupled to photons by Thomson scattering
• Dark Matter
Assume cold. Coupled only via gravity.
• Dark energy
probably negligible early on
Perturbations O(10-5) => linear evolution
• Scalar, vector, tensor modes evolve independently
• Each Fourier k mode evolves independently
General regular linear primordial perturbation
General perturbation
Scalar
Adiabatic
(observed)
Matter density
Cancelling matter density
(unobservable)
Neutrino density
(contrived)
Neutrino velocity
(very contrived)
Vector
Neutrino vorticity
(very contrived)
Tensor
Gravitational waves
+ irregular modes, neutrino n-pole modes, n-Tensor modes Rebhan and Schwarz: gr-qc/9403032
+ other possible components, e.g. defects, magnetic fields, exotic stuff…
Irregular (decaying) modes
• Generally ~ a-1, a-2 or a-1/2
• E.g. decaying vector modes unobservable at late times
unless ridiculously large early on
Adiabatic decay ~ a-1/2 after
neutrino decoupling.
possibly observable if generated
around or after neutrino
decoupling
Otherwise have to be very large
(non-linear?) at early times
Amendola, Finelli: astro-ph/0411273
WMAP + other CMB data
Redhead et al: astro-ph/0402359
+ Galaxy surveys, galaxy weak lensing, Hubble Space Telescope, supernovae, etc...
Constraints from data
• Can compute P( {ө} | data) using e.g. assumption of
Gaussianity of CMB field and priors on parameters
• Often want marginalized constraints. e.g.
1 | data 1 P(1 2 3 ... n | data )d1d 2 ..d n
• BUT: Large n-integrals very hard to compute!
• If we instead sample from P( {ө} | data) then it is easy:
1
1 | data
N
1(i )
i
Use Markov Chain Monte Carlo to sample
MCMC sampling for parameter estimation
• Number density of samples
proportional to probability
density
• At its best scales linearly
with number of parameters
(as opposed to exponentially
for brute integration)
•
For CMB:
P( {ө} | data) ~ P(Cl(ө)|data)
Theoretical Cl numerically
computed using linearised GR
+ Boltzmann equations
(CAMB)
CosmoMC code at http://cosmologist.info/cosmomc
Lewis, Bridle: astro-ph/0205436
Adiabatic modes
What is the primordial power spectrum?
Reconstruct in bins
by sampling posterior
using MCMC with
current data
On most scales P(k) ~ 2.3 x 10-9
Close to scale invariant
Bridle, Lewis, Weller, Efstathiou:
astro-ph/0302306
WMAP TT power spectrum at low l
compared to theoretical power law model (mean over realizations)
data from http://lambda.gsfc.nasa.gov/
Low quadrupole
Indication of less power on very large scales?
•
Any physical model cannot give
sharper cut in power than a step
function with zero power for k< kc
•
k cut model favoured by data,
but only by ~1 sigma
•
No physical model will be
favoured by the data by any
more than this
e.g. Contaldi et al: astro-ph/0303636
•
Allowing for foreground
uncertainties etc, evidence is
even weaker
astro-ph/0302306
Matter isocurvature modes
• Possible in two-field inflation models, e.g. ‘curvaton’ scenario
• Curvaton model gives adiabatic + correlated CDM or baryon
isocurvature, no tensors
• CDM, baryon isocurvature indistinguishable – differ only by
cancelling matter mode
Constrain B = ratio of matter isocurvature
to adiabatic
No evidence, though still allowed.
Not very well constrained.
Gordon, Lewis: astro-ph/0212248
General isocurvature models
• General mixtures currently
poorly constrained
Bucher et al: astro-ph/0401417
Polarization can break degeneracies
Bucher et al. astro-ph/0012141
The future: CMB Polarization
Stokes’ Parameters
-
Q
U
Q → -Q, U → -U under 90 degree rotation
Spin-2 field Q + i U
or Rank 2 trace free symmetric tensor
θ
θ = ½ tan-1 U/Q
sqrt(Q2 + U2)
E and B polarization
Trace free gradient:
E polarization
e.g.
Curl:
B polarization
Why polarization?
• E polarization from scalar, vector and tensor modes
(constrain parameters, break degeneracies)
• B polarization only from vector and tensor modes (curl grad = 0)
+ non-linear scalars
Primordial Gravitational Waves
• Well motivated by some inflationary models
- Amplitude measures inflaton potential at horizon crossing
- distinguish models of inflation
• Observation would rule out other models
- ekpyrotic scenario predicts exponentially small amplitude
- small also in many models of inflation, esp. two field e.g. curvaton
• Weakly constrained from CMB temperature anisotropy
- significant power only at l<100, cosmic variance limited to 10%
- degenerate with other parameters (tilt, reionization, etc)
Look at CMB polarization: ‘B-mode’ smoking gun
CMB polarization from primordial
gravitational waves (tensors)
Tensor B-mode
Tensor E-mode
Adiabatic E-mode
Weak lensing
Planck noise
(optimistic)
• Amplitude of tensors unknown
• Clear signal from B modes – there are none from scalar modes
• Tensor B is always small compared to adiabatic E
Seljak, Zaldarriaga: astro-ph/9609169
Regular vector mode: ‘neutrino vorticity mode’
logical possibility but unmotivated (contrived). Spectrum unknown.
B-modes
Similar to gravitational wave spectrum on large scales: distinctive small scale
Lewis: astro-ph/0403583
Other B-modes?
•Topological defects
Seljak, Pen, Turok: astro-ph/9704231
Non-Gaussian signals
global defects:
10% local strings from
brane inflation:
r=0.1
Pogosian, Tye, Wasserman, Wyman:
hep-th/0304188
lensing
Conclusions
•
Currently only very weak evidence for any deviations from standard near
scale-invariant purely adiabatic primordial spectrum
•
Precision E polarization
- Much improved constraints on isocurvature modes
•
Large scale Gaussian B-mode CMB polarization from primordial
gravitational waves:
- energy scale of inflation
- rule out most ekpyrotic and pure curvaton/
inhomogeneous reheating models and others
•
Small scale B-modes:
- Strong signal from any vector vorticity modes
(+strong magnetic fields, topological defects, lensing, etc)
http://CosmoCoffee.info
arXiv paper discussion and comments