Microwave Background Polarization: Theoretical …cmbpol.uchicago.edu/workshops/technology2008/depot/kosowsky-arth… · Tensor Perturbations and Microwave Polarization Current Cosmology

Tensor Perturbations and Microwave PolarizationCurrent Cosmology

B-mode ScienceConclusions

Microwave Background Polarization: TheoreticalPerspectives

Arthur Kosowsky

Department of Physics and AstronomyUniversity of Pittsburgh

CMBpol Technology Workshop

Arthur Kosowsky Gravity Waves and the CMB



Outline

Tensor Perturbations and Microwave Polarization

Current Cosmology

B-mode ScienceGravitational Waves from InflationGravitational LensingOptical ActivityReionization

Conclusions




Outline


Current Cosmology


Conclusions




Microwave Temperature and Polarization Anisotropies

Microwave background radiation is described by its temperatureand polarization as a function of sky direction: T (n) and Pab(n)

Temperature is a scalar function; polarization is a symmetric,traceless, rank-2 tensor function (2 degrees of freedom)

Common polarization Stokes Parameters Q and U depend ontensor basis




Harmonic Decompositions

Temperature:

T (n) =∞∑l=2

l∑m=−l

a(lm)Y(lm)(n)

Polarization:

Pab(n) =∞∑l=2

l∑m=−l

[aG(lm)Y

G(lm)ab(n) + aC

(lm)YC(lm)ab(n)

]

Polarization has two degrees of freedom, so two sets of basisfunctions are required to span the space of polarization fields




Tensor Spherical Harmonics

One convenient basis is the ”gradient/curl” basis (Kamionkowski,Kosowsky, Stebbins 1997):

Y G(lm)ab = Nl

[Y(lm):ab +

1

2gabY(lm):c

:c

]

Y C(lm)ab =

Nl

2

[Y(lm):acε

cb + Y(lm):bcε

ca

]where gab and εab are the metric and antisymmetric tensors on thesphere, and ”:” indicates a covariant derivative

This looks like the familiar gradient/curl decomposition of a vectorfield. Independent of polarization basis




G and C Polarization Modes

Random C/B (left) and G/E (right) polarization modes (Bunn2003)




Geometry of Scalar and Tensor Modes

A physically useful basis:

One Fourier mode of a scalar perturbation has spatial dependencee ik·x: for fixed k depends only on angle between k and x, so isaxially symmetric. Therefore it cannot have any curl component:aC(lm) ≡ 0 for scalar perturbations!

One Fourier mode of a tensor perturbation (a spin-2 field) hasaxial spatial dependence like cos(2φ) so aC

(lm) is nonzero in general.




Curl Polarization

Curl polarization of the microwave background radiation providesphysics other than dominant primordial scalar perturbations.

Primordial tensor perturbations generated by inflation

Gravitational lensing of primordial non-scalar primordialperturbations

Optical activity from magnetic fields or nonstandard photoninteractions




Partial Sky Effects

Unique E-B decomposition only on the full sky. On regions withboundaries, some modes are ambiguous! (Bunn et al. 2003; Lewis,Challinor, and Turok 2002)

In all models, B-modes have a much smaller amplitude thanE-modes, making their extraction technically challenging (e.g. K.Smith 2005)




Outline


Current Cosmology


Conclusions




Current Parameters




Extra Parameters




Inflation Parameters




Gravitational Waves from InflationGravitational LensingOptical ActivityReionization

Outline


Current Cosmology


Conclusions





Inflation Basics

Inflation is a period in the early universe of accelerating expansion,which means that the stress-energy tensor must satisfyw ≡ p/ρ < −1/3. Solves flatness, horizon, and monopole problemsof standard cosmology, generates scalar and tensor fluctuations.

The universe must undergo enough inflation to increase the scalefactor by e60, or 60 e-foldings. Then inflation must end.

Scalar density perturbations generated by inflation 60 e-foldingsbefore the end must have amplitude of 10−5 to match fluctuationsobserved in the microwave background.





Fundamental Relations

These conditions imply that:

Energy scale of inflation M ≡ ρ1/4 ≈ 10−5/2(1 + w)1/4mPl

Tensor-scalar ratio r ≡ 14ρ/(ρ + p) ≈ 14(1 + w)

To determine the tensor amplitude on current horizon scales, wemust estimate 1 + w at 60 e-foldings before the end of inflation.





Effective Scalar Field Description

Usually consider scalar field with

L = (∂tφ)2 − V (φ)

H2 =1

3m2Pl

(1

2φ2 + V (φ)

)φ + 3Hφ + V ′(φ) = 0

where H = a/a





Slow-Roll Inflation

For inflation to occur, w < −1/3; this implies V � φ2

For inflation to sustain, need φ� V ′

Slow-roll parameters ε = mPl2

(V ′

V

)2and η = m2

PlV ′′

V must both be

small





Fluctuations

Resulting fluctuations have power-law spectra described by

ns − 1 = 2η − 6ε

nt = −2ε, r = 16ε

Note a consistency relation r = 8nt between the tensor-scalar ratioand the tensor power spectrum





Models for V (φ)

Many models have been constructed of both types (chaoticinflation, natural inflation, hybrid inflation, etc.)





Current WMAP 5-year Limits

Planck may be able to reach r = 10−2.





Future Possibilities





Less Simple Possibilities

If more than one dynamical field occur during inflation, additionaleffects can arise:

I Non-gaussian perturbations

I Isocurvature perturbations

I Topological defects

These all would show up in temperature and polarization maps





A Probe of Gravity at Very High Energies

Inflation properties can depend on the “ultraviolet completion” ofgravitation – in other words, on certain properties of quantumgravity at the Planck scale.

The effective scalar field potential must have quantum correctionssuppressed. This can put constraints on any effective field theoryof quantum gravity at very high energies.

See Douglas and Kachru 2007





Inflation and String Theory

Preliminary work seems to indicate that either class of inflationmodels (large or small-field potentials) can be realized in astring-theory context, by different physical mechanisms.

If true, even a non-detection of tensor modes would put anobservational constraint on string theory.

See McAllister and Silverstein 2008





A Unique Probe of New Physics

The B-polarization signal in the cosmic microwave backgroundprovides unique constraints on physics at energy scales which willnever be probed by direct means.





Lensing Displacement

Gravitational lensing creates a vector displacement field d(n) onthe sky

Pab(n)→ Pab(n) + d(n))

Dominated by mass perturbations at z = 2 to 3, tail to higherredshift





B-polarization Power Spectra

K. Smith (theoryworkshop)





Lensing Power Spectrum Issues

Science content: are there strong science drivers for sub-degreepolarization measurements from lensing?

Lensing as a contaminant to primordial B-mode tensor signal:lensing limit on r of roughly 10−4 if reionization bump at low lmeasurable; 10−3 if only recombination signal (mode counting:K. Smith, theory workshop)





B-mode Lensing Map Science

Temperature mapby S. Das

Science beyondpower spectrum?





Polarization Rotation

If all linear polarization rotated by angle α,

C′ BBl = CBB

l cos2 2α + CEEl sin2 2α

C′ TBl = CTE

l sin 2α

C′ EBl =

1

2(CEE

l − CBBl ) sin 4α

(Lue, Wang, Kamionkowski 1999)

WMAP5 α < 10◦ (Cabella, Natoli, and Silk 2008)





Faraday Rotation

Rotation due tomagnetic fields,proportional to ν−2

Kosowsky et al.2005





Chern-Simons Photon Interaction

Lint = gεµναβAµTνFαβ

for some external field Tν : can represent effect of spacetimetorsion or other nonstandard effects. Results in rotation,independent of frequency (Carroll, Fields, and Jackiw 1990)

Current limits on rotation give gT � H0, best limit on thisinteraction (Alexander, Ochoa, and Kosowsky 2008; Kosteleckyand Mewes 2007)

[Also generates circular polarization! But negligible compared torotation]





Lyman-Alpha Absorption in Quasars

QSO spectra at z ' 6from SDSS

Note: Ly-α absorptionsaturates at z ≈ 6 forxi = 10−4

Fan et al. 2006





Reionization Simulations

Lidz et al., from theoryworkshop





Reionization Histories

Various filling factorsand efficiencies

Furlanetto and Loeb2005





Does CMBpol Help?

CMBpol can probereionization abovez = 15, no othercompetitive

Mortonson and Hu 2007




Outline


Current Cosmology


Conclusions




Selling Points

B-mode polarization signal is currently the only known probe ofGUT-scale physics. Information about the UV completion ofquantum gravity.

A potentially clean measure of gravitational lensing, differentsystematics than weak lensing or microwave temperature lensing

Probe of early reionization




Open Theory Issues

Further development of quantum gravity implications of inflationconstraints

Does B-mode lensing provide unique cosmological constraints?Lensing signal separation simulations needed

What fundamental interactions can be constrained by opticalactivity?

How valuable is the galactic polarization signal at CMBfrequencies?




Foregrounds!

Measurements will be foreground dominated

More data and modeling needed: how many frequency channelsrequired to extract B-polarization science from large foregroundsignal?


Documents

Microwave Background Polarization: Theoretical …cmbpol.uchicago.edu/workshops/technology2008/depot/kosowsky-arth… · Tensor Perturbations and Microwave Polarization Current Cosmology