Rotation of Linear Polarization Plane from Cosmological - PowerPoint PPT Presentation

Rotation of Linear Polarization Plane from Cosmological Pseudoscalar Fields Matteo Galaverni based on a work with: Fabio Finelli University of Ferrara INAF Physics Departm ent I talian National I nstitute for Astrophysics - Bologna

Overview � Pseudoscalar – photon coupling. � Main effects on CMB polarization. � Modified Einstein – Boltzmann equations for a time dependent linear polarization rotation angle. � Fixed DM (or DE ) model: -full linear polarization angular power spectra; -comparison with constant rotation angle approximation. Work based based on: on: Work -F. Finelli and MG, “ Rotation of Linear Polarization Plane and Circular Polarization from Cosmological Pseudoscalar Fields ”, arXiv:0802.4210 [astro-ph], accepted in Phys. Rev. D. -F. Finelli and MG, “CMB Cosmological Birefringence and Ultralight Pseudo Nambu-Goldstone Bosons”, in preparation . GGI, 11-02-2009 2

Pseudoscalar – photon coupling Pseudoscalar fields are invoked to solve the strong CP- problem of QCD [R. Peccei and H.Quinn PRL 38 (1977)] g 2 L QCD = L PERT + 1 φ 2 ∇ μ φ ∇ μ φ + G a μν ˜ G μν a 32 π 2 f a They are also good candidates for cold dark matter (misalignment axion production). Pseudoscalar particles interact with ordinary matter : photons, nucleons, [electrons]. The coupling with photons play a key role for most of the searches: L φγ = g φ E · B φ = − g φ 4 F μν ˜ F μν φ F μν ≡ 1 F μν ≡ ∇ μ A ν − ∇ ν A μ where: ˜ 2 ² μνρσ F ρσ and GGI, 11-02-2009 3

Pseudoscalar – photon coupling Most of this searches make use of the Primakoff effect , by which pseudo- scalars convert into photons in presence of an external electromagnetic field. γ φ •Dichroism in laser experiments •Solar axions (e.g. CAST) φ γ γ •Birefringence in laser experiments •Light shining through walls _experiments GGI, 11-02-2009 4

Current Constraints [Battesti et al., arXiv:0705.0615] GGI, 11-02-2009 5

Cosmological background γ γ Photon propagation in a time dependent background of pseudoscalar particles acting as DM (e.g. axion-like particles) or DE (e.g. ultralight pseudo Nambu- φ Goldstone bosons) We want to evaluate the effect on CMB polarization of a coupling of this kind between pseudoscalar field and photon , improving the estimate obtained by D.Harari and P. Sikivie in 1992 [Phys. Lett. B 289 67] for linear polarization: L = − 1 4 F μν F μν − 1 2 ∇ μ φ ∇ μ φ − V ( φ ) − g φ 4 φ F μν ˜ F μν GGI, 11-02-2009 6

Pseudoscalar – photon coupling - Assume a spatially flat Roberson-Walker universe: £ − d η 2 + d x 2 ¤ ds 2 = − dt 2 + a 2 ( t ) d x 2 = a 2 ( η ) - Neglect the spatial variations of the pseudoscalar field: φ = φ ( η ) φ is homogeneous throughout our universe (inflation occurs after the PQ-symmetry breaking): PQ scale is much higher than 10 11÷12 GeV, case motivated by anthropic considerations [Linde, Phys. Lett. B 201 (1988), M. Tegmark, A. Aguirre, M. Rees, F. Wilczek Phys. Rev. D 73 (2006), M.P. Hertzberg, M. Tegmark, F. Wilczek Phys. Rev. D 78 (2008) ] For a plane wave propagating along z-axis, the equation for Fourier transform of the vector potential (in the Coulomb Gauge ) : ∇ · A = 0 ∙ ¸ k 2 + g φ k d φ A 00 ˜ ˜ + ( η , k ) + A + ( η , k ) = 0 d η ∙ ¸ k 2 − g φ k d φ A 00 ˜ ˜ − ( η , k ) + A − ( η , k ) = 0 d η GGI, 11-02-2009 7

Adiabatic solution It is possible to search a solution in this form: r p R 1 ± g φ 1 k φ 0 ≡ k ˜ e ± i ω s d η where: ω s ( η ) = k 1 ± ∆ ( η ) √ 2 ω s A s = 3 ω 0 2 ω 00 It is a good approximation of the solution when: s s ¿ 1 and ¿ 1 . 4 ω 4 2 ω 3 s s If also : ∆ ( η ) ¿ 1 ∙ µ ¶¸ Z 1 η ± 1 ˜ p A ± ' exp ± ik ∆ ( η ) d η 2 2 k (1 ± ∆ / 4) 1 p = exp [ ± i ( k η ± g φ φ / 2)] . 2 k (1 ± g φ φ 0 k/ 4) GGI, 11-02-2009 8

Adiabatic solution The two main effects on the propagation of the wave are: • a k- independent shift between the two polarized waves, which corresponds to rotation of the plane of linear polarization of an angle: θ ( η ) = g φ 2 [ φ ( η ) − φ ( η rec )] • production of a certain degree of circular polarization (dependent on k): ¯ ¯ ¯ ¯ 2 2 ¯ ¯ ¯ ¯ ¯ ˜ ¯ ˜ A 0 A 0 ¯ − ¯ = g φ 0 ( η ) Π V ( η ) ≡ V 2 ' ∆ ( η ) + − ˜ T = ¯ ¯ ¯ ¯ 2 ¯ ¯ ¯ ¯ 2 2 k ¯ ˜ ¯ ˜ A 0 A 0 ¯ + ¯ + − GGI, 11-02-2009 9

CMB Polarization •Linear polarization of CMB was predicted soon after CMB discovery in 1968 by Martin Rees TT [ Rees, ApJ 153 1968 ] (Thomson scattering of anisotropic radiation at last scattering give rise to linear TE polarization). EE •The first detection of CMB BB polarization was made by the Degree Angular Scale lensing Interferometer (DASI, Kovac et al. , Nature 420, 2002). Plot of signal for TT, TE, EE, BB for the best fit model. •First full-sky polarization map [Page et al ., 2006] released from WMAP in 2006. GGI, 11-02-2009 10

E and B linear polarization Potential sources of B polarization: E-mode - “gradient-like” • Cosmological gravitational waves (tensor perturbation of the metric) • Gravitational lensing of E-mode polarization • Faraday Rotation of E-mode polarization (magnetic fields) • Coupling of CMB photons with a B-mode - “curl-like” pseudoscalar field (e.g. axion). … [Zaldarriaga, astro-ph/0305272] GGI, 11-02-2009 11

Polarization Boltzmann equation One of the main effects of coupling between photons and pseudoscalar fields is cosmological birefringence : θ ( η ) = g φ 2 [ φ ( η ) − φ ( η rec )] Including the time dependent rotation angle contribution in the Boltzmann equation for polarization [Liu et al ., PRL 97, 161303 (2006)] : ∆ 0 Q ± iU ( k, η ) + ik μ ∆ Q ± iU ( k, η ) = − n e σ T a ( η ) [ ∆ Q ± iU ( k, η ) # r X 6 π 2 S ( m ) 5 ± 2 Y m + ( k, η ) P m ∓ i 2 θ 0 ( η ) ∆ Q ± iU ( k, η ) . GGI, 11-02-2009 12

Polarization Boltzmann equation Following the line of sight strategy for scalar perturbations, we have an additional term in polarization sources: Z η 0 ∆ T ( k, η ) = d η g ( η ) S T ( k, η ) j ` ( k η 0 − k η ) , 0 Z η 0 P ( k, η ) j ` ( k η 0 − k η ) d η g ( η ) S (0) ∆ E ( k, η ) = ( k η 0 − k η ) 2 cos [2 θ ( η )] , 0 Z η 0 P ( k, η ) j ` ( k η 0 − k η ) d η g ( η ) S (0) ∆ B ( k, η ) = ( k η 0 − k η ) 2 sin [2 θ ( η )] . 0 If θ is constant in time the new terms exit from the time integrals and: ∆ E ( θ = 0) cos(2¯ ∆ E = θ ) , ∆ E ( θ = 0) sin(2¯ ∆ B = θ ) . GGI, 11-02-2009 13

Constant rotation angle In the constant rotation angle approximation new polarization power spectra are given by [A. Lue, L. Wang, M. Kamionkowski PRL 83 , 1506 (1999)]: C EE,obs cos 2 (2¯ C EE = θ ) , ` ` C BB,obs sin 2 (2¯ C EE = θ ) , ` ` 1 C EB,obs sin(4¯ 2 C EE = θ ) , ` ` C TE,obs cos(2¯ C TE = θ ) , ` ` C TB,obs sin(2¯ C TE = θ ) . ` ` Where are the primordial power spectra produced by scalar C XY l fluctuations in absence of parity violation, while are what we C XY,obs l would observe in the presence of anfor an isotropic, k-independent rotation θ of the plane of liner polarization. GGI, 11-02-2009 14

Constraints on the rotation angle - analyzing a subset of WMAP3 and BOOMERANG data [B. Feng, et al., PRL 96 221302 (2006)] − 13 . 7 deg < ¯ θ < 1 . 9 deg (2 σ ) - analyzing WMAP three years polarization data [P.Cabella, et al., PRD 76 123014 (2007)] − 8 . 5 deg < ¯ θ < 3 . 5 deg (2 σ ) - analyzing WMAP five years polarization data [E. Komatsu, et al ., arXiv:0803.0547] − 5 . 9 deg < ¯ θ < 2 . 4 deg (2 σ ) - analyzing QUaD experiment second and third season observations [QUaD Collaboration, arXiv:0811.0618 ] − 1 . 2 deg < ¯ θ < 3 . 9 deg (2 σ ) GGI, 11-02-2009 15

Cosine-type potential Assuming that dark matter is given by massive pseudoscalar particles (e.g. axions), we consider the potential: µ ¶ V ( φ ) = m 2 f 2 1 − cos φ N ' 1 a 2 m 2 φ 2 N 2 f a the evolution of Ф is given by the equation: φ + 3 H ˙ ¨ φ + m 2 ( T ) φ = 0 If the solution simply is: φ ' φ i m ¿ 3 H If the field begins to oscillate and the solution, in a matter m > 3 H dominated universe ( ), is: a/a = 2 / 3 t ˙ φ 0 m φ t À 1 φ ( t ) ' mt sin( mt ) GGI, 11-02-2009 16

Cosine-type potential (µ η 0 " ¶ 3 # r ¶ 3 µ η θ ( η ) 3 g φ M pl m η 0 θ ( η ) = sin θ = θ 0 π 2 m η 0 η 3 η 0 " ¶ 3 #) µ η 0 ¶ 3 µ η rec m η 0 − sin η rec 3 η 0 z m = 10 − 22 eV , g φ = 10 − 20 eV − 1 θ 0 ∼ 0 . 506 rad EE θ = 0 θ = θ 0 θ = θ ( η ) θ = θ ( η ) GGI, 11-02-2009 17

Cosine-type potential θ 0 ∼ 0 . 506 rad BB θ = θ ( η ) θ = θ 0 r = 0 . 1 θ = θ ( η ) lensing TE θ = 0 θ = θ 0 θ = θ ( η ) θ = θ ( η ) GGI, 11-02-2009 18

WMAP 2008 - TE Cosine-type potential [note vertical axis] Cosine-type potential with: WMAP collaboration m = 10 − 22 eV , g φ = 10 − 20 eV − 1 [ arXiv:0803.0593 ] GGI, 11-02-2009 19