Microwave Background Polarization: Theoretical Perspectives

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1 Microwave Background Polarization: Theoretical Perspectives Department of Physics and Astronomy University of Pittsburgh CMBpol Technology Workshop

2 Outline Tensor Perturbations and Microwave Polarization

3 Outline Tensor Perturbations and Microwave Polarization

4 Microwave Temperature and Polarization Anisotropies Microwave background radiation is described by its temperature and polarization as a function of sky direction: T (ˆn) and P ab (ˆn) 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 on tensor basis

5 Harmonic Decompositions Temperature: Polarization: T (ˆn) = l a (lm) Y (lm) (ˆn) l=2 m= l P ab (ˆn) = l l=2 m= l [ a G (lm) Y G (lm)ab (ˆn) + ac (lm) Y C (lm)ab (ˆn) ] Polarization has two degrees of freedom, so two sets of basis functions are required to span the space of polarization fields

6 Tensor Spherical Harmonics One convenient basis is the gradient/curl basis (Kamionkowski, Kosowsky, Stebbins 1997): [ Y(lm)ab G = N l Y (lm):ab + 1 ] 2 g :c aby (lm):c Y(lm)ab C = N l [ Y(lm):ac ɛ c b + Y 2 (lm):bc ɛ c a] where g ab and ɛ ab are the metric and antisymmetric tensors on the sphere, and : indicates a covariant derivative This looks like the familiar gradient/curl decomposition of a vector field. Independent of polarization basis

7 G and C Polarization Modes Random C/B (left) and G/E (right) polarization modes (Bunn 2003)

8 Geometry of Scalar and Tensor Modes A physically useful basis: One Fourier mode of a scalar perturbation has spatial dependence e ik x : for fixed k depends only on angle between k and x, so is axially symmetric. Therefore it cannot have any curl component: a(lm) C 0 for scalar perturbations! One Fourier mode of a tensor perturbation (a spin-2 field) has axial spatial dependence like cos(2φ) so a(lm) C is nonzero in general.

9 Curl Polarization Curl polarization of the microwave background radiation provides physics other than dominant primordial scalar perturbations. Primordial tensor perturbations generated by inflation Gravitational lensing of primordial non-scalar primordial perturbations Optical activity from magnetic fields or nonstandard photon interactions

10 Partial Sky Effects Unique E-B decomposition only on the full sky. On regions with boundaries, some modes are ambiguous! (Bunn et al. 2003; Lewis, Challinor, and Turok 2002) In all models, B-modes have a much smaller amplitude than E-modes, making their extraction technically challenging (e.g. K. Smith 2005)

11 Outline Tensor Perturbations and Microwave Polarization

12 Current Parameters

13 Extra Parameters

14 Inflation Parameters

15 Outline Tensor Perturbations and Microwave Polarization

16 Inflation Basics Inflation is a period in the early universe of accelerating expansion, which means that the stress-energy tensor must satisfy w p/ρ < 1/3. Solves flatness, horizon, and monopole problems of standard cosmology, generates scalar and tensor fluctuations. The universe must undergo enough inflation to increase the scale factor by e 60, or 60 e-foldings. Then inflation must end. Scalar density perturbations generated by inflation 60 e-foldings before the end must have amplitude of 10 5 to match fluctuations observed in the microwave background.

17 Fundamental Relations These conditions imply that: Energy scale of inflation M ρ 1/4 10 5/2 (1 + w) 1/4 m Pl Tensor-scalar ratio r 14ρ/(ρ + p) 14(1 + w) To determine the tensor amplitude on current horizon scales, we must estimate 1 + w at 60 e-foldings before the end of inflation.

18 Effective Scalar Field Description Usually consider scalar field with L = ( t φ) 2 V (φ) H 2 = 1 ( ) 1 3mPl 2 2 φ 2 + V (φ) φ + 3H φ + V (φ) = 0 where H = ȧ/a

19 Slow-Roll Inflation For inflation to occur, w < 1/3; this implies V φ 2 For inflation to sustain, need φ V Slow-roll parameters ɛ = m Pl 2 small ( ) V 2 V and η = m 2 V Pl V must both be

20 Fluctuations Resulting fluctuations have power-law spectra described by n s 1 = 2η 6ɛ n t = 2ɛ, r = 16ɛ Note a consistency relation r = 8n t between the tensor-scalar ratio and the tensor power spectrum

21 Models for V (φ) Many models have been constructed of both types (chaotic inflation, natural inflation, hybrid inflation, etc.)

22 Current WMAP 5-year Limits Planck may be able to reach r = 10 2.

23 Future Possibilities

24 Less Simple Possibilities If more than one dynamical field occur during inflation, additional effects can arise: Non-gaussian perturbations Isocurvature perturbations Topological defects These all would show up in temperature and polarization maps

25 A Probe of Gravity at Very High Energies Inflation properties can depend on the ultraviolet completion of gravitation in other words, on certain properties of quantum gravity at the Planck scale. The effective scalar field potential must have quantum corrections suppressed. This can put constraints on any effective field theory of quantum gravity at very high energies. See Douglas and Kachru 2007

26 Inflation and String Theory Preliminary work seems to indicate that either class of inflation models (large or small-field potentials) can be realized in a string-theory context, by different physical mechanisms. If true, even a non-detection of tensor modes would put an observational constraint on string theory. See McAllister and Silverstein 2008

27 A Unique Probe of New Physics The B-polarization signal in the cosmic microwave background provides unique constraints on physics at energy scales which will never be probed by direct means.

28 Lensing Displacement Gravitational lensing creates a vector displacement field d(ˆn) on the sky P ab (ˆn) P ab (ˆn) + d(ˆn)) Dominated by mass perturbations at z = 2 to 3, tail to higher redshift

29 B-polarization Power Spectra K. Smith (theory workshop)

30 Lensing Power Spectrum Issues Science content: are there strong science drivers for sub-degree polarization 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 l measurable; 10 3 if only recombination signal (mode counting: K. Smith, theory workshop)

31 B-mode Lensing Map Science Temperature map by S. Das Science beyond power spectrum?

32 Polarization Rotation If all linear polarization rotated by angle α, C BB l = C BB l C TB l cos 2 2α + C EE l = C TE l sin 2α sin 2 2α C EB l = 1 EE (Cl 2 Cl BB ) sin 4α (Lue, Wang, Kamionkowski 1999) WMAP5 α < 10 (Cabella, Natoli, and Silk 2008)

33 Faraday Rotation Rotation due to magnetic fields, proportional to ν 2 Kosowsky et al. 2005

34 Chern-Simons Photon Interaction L int = gɛ µναβ A µ T ν F αβ for some external field T ν : can represent effect of spacetime torsion or other nonstandard effects. Results in rotation, independent of frequency (Carroll, Fields, and Jackiw 1990) Current limits on rotation give gt H 0, best limit on this interaction (Alexander, Ochoa, and Kosowsky 2008; Kostelecky and Mewes 2007) [Also generates circular polarization! But negligible compared to rotation]

35 Lyman-Alpha Absorption in Quasars QSO spectra at z 6 from SDSS Note: Ly-α absorption saturates at z 6 for x i = 10 4 Fan et al. 2006

36 Simulations Lidz et al., from theory workshop

37 Histories Various filling factors and efficiencies Furlanetto and Loeb 2005

38 Does CMBpol Help? CMBpol can probe reionization above z = 15, no other competitive Mortonson and Hu 2007

39 Outline Tensor Perturbations and Microwave Polarization

40 Selling Points B-mode polarization signal is currently the only known probe of GUT-scale physics. Information about the UV completion of quantum gravity. A potentially clean measure of gravitational lensing, different systematics than weak lensing or microwave temperature lensing Probe of early reionization

41 Open Theory Issues Further development of quantum gravity implications of inflation constraints Does B-mode lensing provide unique cosmological constraints? Lensing signal separation simulations needed What fundamental interactions can be constrained by optical activity? How valuable is the galactic polarization signal at CMB frequencies?

42 Foregrounds! Measurements will be foreground dominated More data and modeling needed: how many frequency channels required to extract B-polarization science from large foreground signal?

Gravitational Waves and the Microwave Background

Gravitational Waves and the Microwave Background Department of Physics and Astronomy University of Pittsburgh KICP Inaugural Symposium, December 11, 2005 Outline Tensor Perturbations and Microwave Polarization