Cosmic Birefringence Fluctuations and Cosmic Microwave Background $B$-mode Polarization

Recently, BICEP2 measurements of the cosmic microwave background (CMB) $B$-mode polarization has indicated the presence of primordial gravitational waves at degree angular scales, inferring the tensor-to-scalar ratio of $r=0.2$ and a running scalar spectral index. In this {\em Letter}, we show that the existence of the fluctuations of cosmological birefringence can give rise to CMB $B$-mode polarization that fits BICEP2 data with $r<0.11$ and no running of the scalar spectral index. Thus, it might be too hasty to conclude that many inflation models with small $r$ are ruled out based on BICEP2 result.

The spatial flatness and homogeneity of the present Universe strongly suggest that a period of de Sitter expansion or inflation had occurred in the early Universe [1]. During inflation, quantum fluctuations of the inflaton field may give rise to energy density perturbations (scalar modes) [2], which can serve as the seeds for the formation of large-scale structures of the Universe. In addition, a spectrum of gravitational waves (tensor modes) is produced from the de Sitter vacuum [3].
Gravitational waves are very weakly coupled to matter, so once produced, they remain as a stochastic background until today, and thus provide a potentially important probe of the inflationary epoch. Detection of these primordial waves by using terrestrial wave detectors or the timing of millisecond pulsars [4] would indeed require an experimental sensitivity of several orders of magnitude beyond the current reach. However, like scalar perturbation, horizon-sized tensor perturbation induces largescale temperature anisotropy of the cosmic microwave background (CMB) via the Sachs-Wolfe effect [5]. In addition, the tensor modes uniquely induce CMB B-mode polarization that is the primary goal of ongoing and future CMB experiments [6].
Recently, WMAP+SPT CMB data has placed an upper limit on the contribution of tensor modes to the CMB anisotropy, in terms of the tensor-to-scalar ratio, which is r < 0.18 at 95% confidence level, tightening to r < 0.11 when also including measurements of the Hubble constant and baryon acoustic oscillations (BAO) [7]. Planck Collaboration XVI has quoted r < 0.11 using a combination of Planck, SPT, and ACT anisotropy data, plus WMAP polarization; however, the constraint relaxes to r < 0.26 (95% confidence) when running of the scalar spectral index is allowed with dn s /d ln k = −0.022±0.010 (68%) [8]. More recently, BICEP2 CMB experiment has found an excess of B-mode power at degree angular scales, indicating the presence of tensor modes with r = 0.20 +0.07 −0.05 and dn s /d ln k = −0.028 ± 0.009 [9]. If this result is confirmed, it would give a very strong support to inflation model and open a new window for probing the inflationary dynamics.
In this Letter, we investigate an another source for generating CMB B-mode polarization. The generated B-mode power spectrum can explain the BICEP2 excess B-mode power, while complying to the limit r < 0.11 and dn s /dlnk = 0. Here we consider a nearly massless pseudoscalar Φ ≡ M φ that couples to the electromagnetic field strength via (−β/4)φF µνF µν , where β is a coupling constant and M is the reduced Planck mass. The effect of this coupling to CMB polarization has been previously studied [10][11][12][13]. It is well known that the above φ-photon interaction may lead to cosmic birefringence [14] that induces rotation of the polarization plane of the CMB, thus converting E-mode into B-mode polarization [15,16]. For such a pseudoscalar, we consider the contribution of φ perturbation to cosmic birefringence.
We assume a conformally flat metric, ds 2 = a 2 (η)(dη 2 − d x 2 ), where a(η) is the cosmic scale factor and η is the conformal time defined by dt = a(η)dη. The φFF term leads to a rotational velocity of the polarization plane of a photon propagating in the directionn [14], Thomson scatterings of anisotropic CMB photons by free electrons give rise to linear polarization, which can be described by the Stokes parameters Q(η, x) and U (η, x). The time evolution of the linear polarization is governed by the collisional Boltzmann equation, which would be modified due to the rotational velocity of the polarization plane (1) by including a temporal rate of change of the Stokes parameters: where the dot denotes d/dη. This gives a convolution of the Fourier modes of the Stokes parameters with the spectral rotation that can be easily incorporated into the Boltzmann code. Now we consider the time evolution of φ. Decompose φ into the vacuum expectation value and the perturbation: φ(η, x) =φ(η) + δφ(η, x). For the metric perturbation, we adopt the synchronous gauge: Neglecting the back reaction of the interaction, we obtain the mean field evolution as where H ≡ȧ/a and V (φ) is the scalar potential. The equation of motion for the Fourier mode δφ k is given bÿ where h k is the Fourier transform of the trace of h ij . If φ is nearly massless or its effective mass is less than the present Hubble parameter, the mass term and the source term in Eq. (4) can be neglected. In this case, V (φ) can be either null or behaves just like a cosmological constant withφ = 0. However, its perturbation is dispersive and can be cast into δφ k (η) = δφ k,i f (kη), where δφ k,i is the initial perturbation amplitude and f (kη) is a dispersion factor. For a super-horizon mode with kη ≪ 1, f (kη) = 1; the factor then oscillates with a decaying envelope once the mode enter the horizon. Let us define the initial power spectrum P δφ (k) by δφ k,i δφ k ′ ,i = (2π 2 /k 3 )P δφ (k) δ( k − k ′ ). We solve for f (kη) numerically using Eq. (4) withφ = 0 and the initial power spectrum P δφ (k) = Ak n−1 , where A is a constant amplitude squared and n is the spectral index. The space-time background has no difference from that of the Lambda Cold Dark Matter model. Assuming a scale-invariant spectrum (n = 1) and a combined constant parameter Aβ 2 , the induced B-mode polarization is computed using our full Boltzmann code based on the CMBFast [17]. Note that both C T B l and C EB l power spectra vanish due to the fact that δφ = 0. We have tuned the value of Aβ 2 to best fit the BICEP2 data as shown in Fig. 1. The likelihood plot in Fig. 2 shows the maximum likelihood value of Aβ 2 = 0.0072±0.0032, with 1-sigma error. We have also produced the rotation power spectrum [11,18], where η s denotes the time when the primary CMB polarization is generated on the last scattering surface or the rescattering surface. The rotation power spectra for the recombination and the reionization with Aβ 2 = 0.0072 are shown in Fig. 3. Recently, constraints on directiondependent cosmological birefringence from WMAP 7year data have been derived, with an upper limit on the quadrupole of a scale-invariant rotation power spectrum, C α 2 < 3.8 × 10 −3 [19]. Our quadrupole is within this limit. In fact, the limit should become weaker for our case because our C α l scales as l −4 for l > 100. Recently, the gravitational lensing B-mode polarization has been detected by cross correlating B modes measured by the SPTpol experiment with lensing B modes inferred from cosmic infrared background fluctuations measured by Herschel and E modes measured by SPTpol [20]. Another CMB experiment called POLARBEAR has also confirmed this cross correlation [21]. However, we note that this detection has no constraint on the rotation-induced B-mode polarization because the rotation power spectrum and the lensing power spectrum are uncorrelated.
There have been physical constraints on A and β. Let us assume that inflation generates the initial condition for dark energy perturbation. Then, n ≃ 1 and A ≃ (H/2π) 2 /M 2 , where H is the Hubble scale of inflation. The recent CMB anisotropy measured by the Planck mission has put an upper limit on A < 3.4 × 10 −11 [8]. This implies that the present spectral energy density of dark energy perturbation relative to the critical energy density, Ω δφ < 10 −15 , which is negligible compared to that of radiation. The most stringent limit on β comes from the absence of a γ-ray burst in coincidence with Supernova 1987A neutrinos, which would have been converted in the galactic magnetic field from a burst of axion-like particles due to the Primakoff production in the supernova core: β < 2.4 × 10 7 for m φ < 10 −9 eV [22]. Hence the combined limit is Aβ 2 < 2×10 4 , which is much bigger than the value that we have used here.
Cosmological birefringence perturbation can generate a rotation-induced B-mode power spectrum. The BI-CEP2 experiment may have firstly detected cosmological birefringence B modes at degree angular scales as proposed in this paper. It would be very important to make direct measurements of B-mode polarization at sub-degree scales where birefringence B modes can be mixed with lensing B modes. It thus poses a big challenge to do the separation of different B-mode signals. It is apparent that the rotation-induced B-mode has acoustic oscillations but to detect them will require next-generation experiments. In principle, one may use de-lensing methods [23] or lensing contributions to CMB bi-spectra [24] to single out the lensing B modes. Furthermore, de-rotation techniques can be used to remove the rotation-induced B modes [25]. More investigations along this line should be done before we can confirm the detection of the genuine B modes. Thus, it might be too hasty to conclude that many inflation models with small r are ruled out based on BICEP2 result. With the proper mechanisms like birefringence to induce the largescale B-mode polarization, many inflation models can be still compatible with BICEP2 result.