Magnetic Fields in the Early Universe - D. Grasso and H.R. Rubinstein

2.4. Effects on the CMBR polarization

Thomson scattering is a natural polarizing mechanism for the CMBR. It is enough if the photon distribution function seen by the electrons has a quadrupole anisotropy to obtain polarization. At early times, the tight coupling between the photons and the electron-baryon fluid prevents the development of any photon anisotropy in the baryon's rest frame, hence the polarization vanishes. As decoupling proceeds, the photons begin to free-stream and temperature quadrupole anisotropies can source a space dependent polarization. For this reason temperature and polarization anisotropies are expected to be correlated (for a comprehensive review on the subject see [79]).

The expected polarization anisotropy is not large, perhaps about 10^-6. Currently the best polarization limit comes from the Saskatoon experiment [80], with a 95% confidence level upper limit of 25 µK at angular scales of about a degree, corresponding to 9 × 10^-6 of the mean temperature. Future balloons and satellites observations, like e.g. the PLANCK [67] mission to be launched in 2007, are expected to have a good chance to measure the CMBR polarization power spectrum.

Kosowsky and Loeb [81] first observed that the possible presence of magnetic fields at the decoupling time may induce a sizeable Faraday rotation in the CMBR. Since the rotation angle depend on the wavelength, it is possible to estimate this effect by comparing the polarization vector on a given direction at two different frequencies. The basic formula is [15]

(2.49)

where phi is the amount by which the plane of polarization of linearly polarized radiation has been rotated, after traversing a distance L in a homogeneous magnetic field B in a direction qhat . x_e is the ionized fraction of the total electron density n_e and m is the electron mass. Finally lambda is the radiation wavelength.

Although the magnetic field strength is expected to be larger at early times, the induced Faraday rotation depends also on the free electron density (see Eq. (2.49)) which drops to negligible values as recombination ends. Therefore, rotation is generated during the brief period of time when the free electron density has dropped enough to end the tight coupling but not so much that Faraday rotation ceases. A detailed computation requires the solution of the radiative transport equations in comoving coordinates [81]

	(2.50)
	(2.51)
	(2.52)

In the above Delta _T, Delta _Q and Delta _U respectively represent the fluctuations of temperature and of the of the Stokes parameters Q and U [15]. The linear polarization is Delta _P = sqrt[ Delta _Q² + Delta _U²]. The numerical subscripts on the radiation brightnesses Delta _X indicate moments defined by an expansion of the directional dependence in Legendre polynomials P(µ):

(2.53)

Therefore, the subscripts 0, 1, 2 label respectively monopole, dipole and quadrupole moments. kappadot = x_e n_e sigma _T adot / a is the differential optical depth and the quantities V_b and R have been defined in the previous section. Derivatives are respect to conformal time. Finally, omega _B is the Faraday rotation rate

(2.54)

From Eqs. (2.51, 2.52) wee see that Faraday rotation mixes Q and U Stokes parameters. The polarization brightness Delta _Q is induced by the function S_P = - Delta _T(2) - Delta _Q(2) + Delta _Q(0) and Delta _U is generated as Delta _Q and Delta _U are rotated into each other. In the absence of magnetic fields, Delta _U retains its tight-coupling value of zero. The set of Eqs. (2.50-2.52) is not easily solved. A convenient approximation is the tight coupling approximation which is an expansion in powers of k tau _C with tau _C = kappadot ^-1. This parameter measures the average conformal time between collisions. At decoupling the photon mean free path grows rapidly and the approximation breaksdown, except for long wavelength as measured with respect to the thickness of the last scattering surface. For these frequencies the approximation is still accurate.

Kosowsky and Loeb assumed a uniform magnetic field on the scale of the width of the last scattering surface, a comoving scale of about 5 Mpc. This assumption is natural if the coherent magnetic field observed in galaxies comes from a primordial origin, since galaxies were assembled from a comoving scale of a few Mpc. The mean results was obtained by averaging over the entire sky. Therefore the equations still depend only on k and µ = cos( kbhat · qhat ) and not on the line-of-sight vector qhat and the perturbation wave-vector k separately. The evolution of the polarization brightnesses, for given values of k and µ is represented in Fig. 2.2 as a function of the redshift.

Figure 2.2. The evolution of the polarization brightnesses, for k = 0.16 Mpc^-1 and µ = 0.5 (in arbitrary units). Also plotted as a dotted line is the differential visibility function taudot e^- in units of Mpc^-1. From Ref. [81].

By following the approach described in the above Kosowsky and Loeb estimated the polarization angle produced by a magnetic field on CMB photons with frequency ₁ < ₂ to be

(2.55)

A 10% correction may apply to this expression to account for the effects _b h² and ₀ h² (in the range _b h² > 0.007 and ₀ h² < 0.3).

For a primordial field of B₀ ~ 10^-9 G which could result in the observed galactic field without dynamo amplification, one can therefore expect a rotation measure of order 1.6 deg cm^-2 = 280 rad m^-2. This rotation is considerable by astrophysical standards and could in principle be measured.

We noticed at the beginning of this section that temperature and polarization anisotropies of the CMBR are generally expected to be correlated. The statistical properties of such correlation may be affected by the presence of a magnetic field at the decoupling time in a peculiar way. In fact, it was showed by Scannapieco and Ferreira [82] that such a field may induce an observable parity odd cross correlation between polarization and temperature anisotropies. Any polarization pattern on the sky can be separated into "electric" (E) and "magnetic" (B) components. The nomenclature reflects the global parity property. Like multipole radiation, the harmonics of an E-mode have (-1) parity on the sphere, whereas those of a B-mode have (-1)⁺¹ parity. Indeed, given a measurement of the Stokes parameters Q and U, this data can be decomposed into a sum over spin ± 2 spherical harmonics

(2.56)

Under parity inversion, _sY_m (-1) _sY_m so that ₂Y_m ± _-2Y_m are parity eigenstates. It is then convenient to define the coefficients

(2.57)

so that the E-mode remains unchanged under parity inversion for even , whereas the B-mode changes sign.

In an isotropic Universe, cross correlation between the B and E polarizations is forbidden as this would imply parity violation. Magnetic fields, however, are maximally parity violating and therefore they may reveal their presence by producing such a cross correlation, Faraday rotation being the physical process which is responsible for this effect. The authors of Ref. [82] determined the expected cross correlation between temperature and E and B polarization modes. On the basis of such result they concluded that magnetic fields strengths as low as 10^-9 (present time value obtained assuming adiabatic scaling) could be detectable by the PLANCK satellite mission. It is worthwhile to note that Scannapieco and Ferreira only considered homogeneous magnetic fields. We note, however, that most of their considerations should apply also to the case of magnetic fields with a finite coherence length. In this case measurements taken in different patches of the sky should present different temperature-polarization cross correlation depending on the magnetic field and the line-of-sight direction angle.

The consequences of Faraday rotation may go beyond the effect they produce on the CMBR polarization. Indeed, Harari, Hayward and Zaldarriaga [83] observed that Faraday rotation may also perturb the temperature power spectrum of CMBR. The effect mainly comes as a back-reaction of the radiation depolarization which induces a larger photon diffusion length reducing the viscous damping of temperature anisotropies.

In the absence of the magnetic field (_B = 0), to the first order in the tight-coupling approximation one finds

(2.58)

and

(2.59)

₀ = _T(0) + . Obviously, all multipoles with l > 3 vanish to this order. Replacing all quantities in terms of ₀ one obtains [84]:

(2.60)

that can be interpreted as the equation of a forced oscillator in the presence of damping.

In the presence of the magnetic field _B 0. The depolarization depends upon two angles: a) the angle between the magnetic field and wave propagation and b) the angle of the field with the wave vector k. Since we assume that the vector k is determined by stochastic Gaussian fluctuations, its spectrum will have no preferred direction. Therefore this dependence will average out when integrated. It is also assumed that for evolution purposes, the magnetic field has no component perpendicular to k. This imposed axial symmetry is compatible with the derivation of the above written Boltzmann equations. Under these assumptions Harari et al. found [83]

(2.61)

where the coefficient F was defined by

(2.62)

which gives

(2.63)

Physically, F represents the average Faraday rotation between two photon-electron scattering. Note that assuming perfect conductivity

(2.64)

and therefore F is a time independent quantity. Faraday rotation between collisions becomes considerably large at frequencies around and below _d. This quantity is implicitly defined by

(2.65)

which gives

(2.66)

From Eqs. (2.61) and, the definition of S_P given in the first part of this section, one can extract

	(2.67)
	(2.68)

and, from the equation for _T in the tight coupling,

(2.69)

In the above the coefficients are defined so that d_i 1 + O(F²) for small F, i.e. small Faraday rotation, while d_i O(1 / F) as F (for the exact definition see Ref. [83]). Equations (2.61, 2.68) and (2.69) condense the main effects of a magnetic field upon polarization. When there is no magnetic field (F = 0, d = 1) _U = 0 and _Q = -15/8 _T₂ sin² . A magnetic field generates _U, through Faraday rotation, and reduces _Q. In the limit of very large F (large Faraday rotation between collisions) the polarization vanishes. The quadrupole anisotropy _T₂ is also reduced by the depolarizing effect of the magnetic field, by a factor 5/6 in the large F limit, because of the feedback of _Q upon the anisotropy or, in other words, because of the polarization dependence of Thomson scattering. The dipole _T₁ and monopole _T₀ are affected by the magnetic field only through its incidence upon the damping mechanism due to photon diffusion for small wavelengths. Indeed, the equation for ₀ = _T₀ + , neglecting O(R²) contributions, now reads

(2.70)

which is the equation of a damped harmonic oscillator.

The damping of the temperature anisotropies on small angular scales can be determined by solving the radiative transfer equation to second order in the tight-coupling approximation. By assuming solutions of the form

(2.71)

for X = T, Q, and U, and similarly for the baryon velocity V_b, Harari et al. [83] found the following solution for Eq. (2.70)

(2.72)

where the photon-diffusion damping length-scale is

(2.73)

The damping affects the multipole coefficients of the anisotropy power spectrum which are defined by

(2.74)

The average damping factor due to photon diffusion upon the C_l's is given by an integral of e^-2 times the visibility function across the last scattering surface [84, 85]. It depends upon cosmological parameters, notably R, and upon the recombination history.

In the Fig. 2.3 it is represented the correction to the temperature power spectrum expected for several values of the parameter F. We see from that figure that on small angular scales the effect of the magnetic field is to increase the temperature anisotropies. The magnitude of this effect was estimated to be up to 7.5% in a CDM Universe on small angular scales (l approx 1000) at a level that should be reachable from future CMBR satellite experiments like MAP [66] and PLANCK [67]. The frequency at which the effect should be detectable will, however, depend on the strength and coherence length of the magnetic field at the recombination time. Both experiment should be sensitive to magnetic fields around B_{z = 1000} = 0.1 G or, equivalently, B₀ = 10^-7 G a level that is comparable to be BBN limit (see Chap. 3).

Figure 2.3. Numerical integration for the multipoles of the anisotropy correlation function in a standard CDM model without a primordial magnetic field (F = 0), and with F = 1, 4, 9, which correspond to ₀ = _d, _d/2, _d/3 respectively, with _d approx 27 GHz (B_* / 0.01 Gauss)^1/2. From Ref. [83].