The local B-polarization of the CMB: a very sensitive probe of cosmic defects

We present a new and especially powerful signature of cosmic strings and other topological or non-topological defects in the polarization of the cosmic microwave background (CMB). We show that even if defects contribute 1% or less in the CMB temperature anisotropy spectrum, their signature in the local $\tilde{B}$-polarization correlation function at angular scales of tens of arc minutes is much larger than that due to gravitational waves from inflation, even if the latter contribute with a ratio as big as $r\simeq 0.1$ to the temperature anisotropies. We show that when going from non-local to local $\tilde{B}$-polarization, the ratio of the defect signal-to-noise with respect to the inflationary value increases by about an order of magnitude. Proposed B-polarization experiments, with a good sensitivity on arcminute scales, may either detect a contribution from topological defects produced after inflation or place stringent limits on them. Even Planck should be able to improve present constraints on defect models by at least an order of magnitude, to the level of $\ep<10^{-7}$. A future full-sky experiment like CMBpol, with polarization sensitivities of the order of $1\mu$K-arcmin, will be able to constrain the defect parameter $\ep=Gv^2$ to a few $\times10^{-9}$, depending on the defect model.


I. INTRODUCTION
Many inflationary models terminate with a phase transition which often also leads to the formation of cosmic strings and other topological defects [1]. Furthermore, we have recently argued [2] that the end of hybrid inflation may involve the self-ordering of a N -component scalar field. Even though for N > 4 it does not lead to the formation of topological defects, the self-ordering dynamics leads to a scale-invariant spectrum of fluctuations which leaves a signature on the CMB [3,4]. It has been shown long ago that topological defects do not generate acoustic peaks [5] and therefore they cannot provide the main contribution to the CMB anisotropies. However, they still may provide a fraction of about 10%, similar to a possible gravitational wave contribution [6] in the temperature anisotropies of the CMB.
The perturbations from cosmic strings and other topological defects are proportional to the dimensionless variable = Gv 2 where v is the symmetry breaking scale. For cosmic strings µ = v 2 is the energy per unit length of the string [7]. Present CMB data limit the contribution from defects [6] such that < 7 × 10 −7 . Stronger limits on have been derived from the gravitational waves emitted from cosmic string loops [8][9][10], but these are quite model dependent and will not be discussed here.
In this Letter we show that measuring the localBpolarization correlation function of the CMB provides stringent limits on defects or, alternatively, detects them. The physical reason for this is twofold. First, defects lead not only to tensor but also to even larger vector perturbations [4]. What is more important, vector modes gen-erate much stronger B-polarization than tensor modes with the same amplitude, see e.g. [11]. B-polarization is not only a 'smoking gun' for gravitational waves from inflation, but it is also extremely sensitive to the presence of vector perturbations (vorticity). Furthermore, the Bpolarization of the angular power spectrum of topological defects, especially of cosmic strings, peaks on somewhat smaller scales than the one from tensors due to inflation. The localB-correlation function, which is obtained from the polarization by two additional derivatives, enhances fluctuations on small angular scales. As we shall see, measuring the localB instead of the usual non-local B correlation function results in an enhancement of the signal to noise ratio from defects with respect to the inflationary one by about a factor 10.

II. THE LOCALB-POLARIZATION CORRELATION FUNCTION
Since Thomson scattering is direction dependent, a non-vanishing quadrupole anisotropy on the surface of last scattering leads to a slight polarization of the CMB [11]. This polarization is described as a rank-2 tensor field P ab on the sphere, the CMB sky. It is usually decomposed into Stokes parameters, P ab = (Iσ ab )/2 = Iδ ab /2 + P ab , where σ (µ) are the Pauli matrices [11], and I corresponds to the intensity of the radiation and contains the temperature anisotropies. Thomson scattering does not induce circular polarization so we expect V = 0 for the CMB polarization, and hence P ab to be real. We define an or-arXiv:1003.0299v3 [astro-ph.CO] 22 Dec 2010 thonormal frame (e 1 , e 2 , n) and the circular polarization vectors e ± = 1 √ 2 (e 1 ± ie 2 ), which allows us to introduce the components P ±± = 2e a ± e b ± P ab = Q ± iU and P +− ∼ V = 0. The second derivatives of this polarization tensor are related to the localẼ-andB-polarizations, Here ∇ ± are the derivatives in the directions e ± and cd is the 2-dimensional totally anti-symmetric tensor. These functions are defined locally. The usual E-and B-modes can be obtained by applying the inverse Laplacian to the localẼ-andB-polarizations. Such inversions of differential operators depend on boundary conditions which can affect the result for local observations. TheB-correlation function, CB(θ) ≡ B (n)B(n ) n·n =cos θ , is measurable locally. It is related to the B-polarization power spectrum C B by [11] Here P (x) are the Legendre polynomials. Analogous formulae also hold for CẼ. Note the additional factor n = ( + 2)!/( − 2)! = ( 2 − 1)( + 2) ∼ 4 as compared to the usual non-local E-and B-polarization correlation functions. At first sight one might argue that whether one expresses a result in terms of C B 's or CB = n C B should really not make a difference since both contain the same information. For an ideal full sky experiment which directly measures the C B with only instrumental errors this is true. But a CMB experiment usually measures a polarization direction and amplitude with a given resolution over a patch of sky and with a significant noise level, and this makes a big difference as we shall show.

III. RESULTS
In Fig. 1 we show the localB-polarization power spectra for tensor perturbations from inflation, cosmic strings, textures and the large-N limit of the non-linear sigma-model. All spectra are normalized such that they make up 10% of the temperature anisotropy at = 10. Details of how these calculations are done can be found in [4] for global defects and the large-N limit and in [12] for cosmic strings. A comparison of the non-local Bpolarization power spectra for cosmic strings and inflation can be found in [13].
It had already been noted in Refs. [15] and [7] that the B-polarization power spectra for defects are larger than those from inflation for the same temperature anisotropy. Defects peak at somewhat higher 's than inflationary perturbations, since B-modes from defects are dominated by their vector (vorticity) modes. This contribution is maximal on scales that are somewhat smaller than the horizon scale, while gravitational waves truly peak at the Hubble horizon at decoupling, which corresponds to ∼ 100. As a consequence, the localB-polarization spectra for defects are even larger than those from inflation because of the factor n 4 . This is most pronounced for cosmic strings, which have considerable power on small scales, but it is also true for other defects.
Due to the extra factor n , in the localB-power spectra shown in Fig. 1, power at small scales (high ) counts significantly more than power at larger scales (low ). This is the reason why defect models dominate over the inflationary B-modes of the same amplitude. This is seen very prominently in the 2-point angular correlation function shown in Fig. 2 where we can compare the defect peaks coming from cosmic strings, textures and large-N . Note the decreasing height but increasing width of the peak as we go from cosmic strings to large-N models.
For 0.2 < θ < 1 o , where the inflationaryB-polarization is about −2 mK 2 , that from cosmic strings is −150 mK 2 , about a factor 100 larger. For textures and the large-N model, the difference is somewhat smaller, roughly a factor of 50 and 10 respectively. The very pronounced peak on very small scales is not visible due to the noise.
Even though constructed ad hoc, coherent causal seed models (but not topological defects) can have acoustic peaks, see Ref. [18], which thus cannot be used as a differentiating signature from inflation. But the fact that polarization is generated at the last scattering surface implies that it cannot have power on scales larger than the horizon at decoupling, corresponding to about ∼ 100, or angles θ > 2 o , unless something like inflation has taken place [16]. This can only be circumvented if one allows for acausality, i.e. superluminal motion, of the seeds [19], however improbable. In Ref. [20] the authors have shown that this superhorizon signature appears not only in the TE-cross correlation spectrum, but also in the localBpolarization spectrum. We find that this is somewhat weakened by re-ionization, which adds power on large scales to the B-polarization from defects, see Fig. 1.

IV. OBSERVATIONAL PROSPECTS
It is clear from Fig. 2 that cosmic defects with equal amplitude as the tensor component from inflation (note = 7 × 10 −7 is equivalent to r = 0.1) would have a significant peak in the two-point correlation function of the localB-polarization, on angular scales of order tens of arc-minutes. A relevant issue is whether this peak could be measured with full-sky probes like Planck [21] or CMBpol [22], or even with small-area experiments. This is difficult because, although CMB experiments typically have a flat (white) noise power spectrum for the Stokes parameters, the local n ∼ 4 factor induces a very blue spectrum for the noise in the localB-modes, which erases the significance of the broad defect peak at ∼ 500 in the CB power spectrum. Moreover, in order to extract the cosmologicalB-polarization signal it is necessary first to clean the map from the contribution coming from gravitationally lensedẼ-modes. This induces an extra 'lensing noise' ∆ P,eff ∼ 4.5 µK·arcmin for uncleaned maps that can be reduced to ∼ (0.1 − 0.7) µK·arcmin by iterative cleaning or a simple quadratic estimator respectively [14]. Furthermore, CMB experiments have an angular resolution determined by the microwave horn beam width, θ FWHM , which induces an uncertainty in the C 's that can be described by an exponential factor exp[ ( + 1)σ 2 b ], with σ b = θ FWHM / √ 8 log 2. Resolutions of order 10 arcminutes, like those of the Planck HFI experiment, correspond to multipoles b = 1/σ b ∼ 800.
Adding the steep polarization noise, with typical amplitude ∆ P,eff = (0.5 − 12) µK·arcmin, would make the signal disappear under the small-scale noise. In order to regulate this divergence, we smooth both the signal and the noise with a Gaussian smoothing of width σ s , corresponding to a smoothing scale s < b . We choose s = 400 in our analysis. In order to compute the signal-to-noise ratio S/N for detection of the defect peak in the localB-correlation function, we split the interval θ ∈ [0, 1 o ] in 10 equal bins [23]. We then evaluate the theoretical correlation function at the center of those bins, S i = CB(θ i ), and write the covariance matrix of the correlated bins as where the covariance matrix in -space is assumed to be diagonal, cov[CB , CB ] = 2(CB ) 2 δ /(2 + 1)f sky , with CB = (CB + N ) exp[− ( + 1)/ 2 s ]. Here f sky is the fraction of the observed sky which we set to 0.7 for satellite probes. The signal-to-noise ratio for the defect model is Fig. 3 we show this ratio as a function of the normalized polarization sensitivity for all types of defects as well as for inflation (where 7 × 10 −7 / has to be replaced by 0.1/r). The horizontal lines correspond to 3, 5 and 10-σ respectively. To show why the choice of θ max = 1 o is optimal we also plot (dashed lines) the S/N for θ max = 4 o , at fixed resolution (6 ). For the latter, the noise level allowed for a 3-σ detection increases by more than a factor of 2 for inflation while it does not change much for defects. This behaviour is a telltale sign for defects, and shows that their signal is strongly localised in the angular correlation function, which distinguishes them e.g. from inflationary tensor perturbations and lensed E-modes: the S/N curve from defects does = Gv 2 , of various defects, at 3-σ in the range θ ∈ [0, 1 o ], for Planck (∆ P,eff = 11.2 µK·arcmin), CMBpol-like exp. (∆ P,eff = 0.7 µK·arcmin) and a dedicated CMB experiment with (∆ P,eff = 0.01 µK·arcmin). We set f sky = 0.7. S/N = 3 Strings Semi-local Textures Large-N Planck 1.2 · 10 −7 1.1 · 10 −7 1.0 · 10 −7 1.6 · 10 −7 CMBpol 7.7 · 10 −9 6.9 · 10 −9 6.3 · 10 −9 1.0 · 10 −8 B exp 1.1 · 10 −10 1.0 · 10 −10 0.9 · 10 −10 1.4 · 10 −10 not change much for angles above ∼ 1 o , while the one from inflation increases significantly.
In Table I we give the values of which are measured at 3σ by Planck (assuming ∆ P,eff = 11.2 µK·arcmin [20], where the de-lensing error is added in quadrature), a CMBpol-like experiment with polarization sensitivity ∆ P,eff = 0.7 µK·arcmin, and a dedicated CMB experiment with ∆ P,eff = 0.01 µK·arcmin. Note, however, that it is not clear how to perform the de-lensing of the Bmodes to the level of precision needed for the last case.
In Fig. 4 we show the ratio of S/N from defects to the one from inflation for non-local (dashed) and local B-modes (solid curves). Clearly, in the local polarization the defect signal is substantially enhanced. It is interesting to note that actually textures fare better than cosmic strings even though they have less power on small scales.

V. CONCLUSIONS
In this Letter we have shown that measuring the lo-calB-polarization correlation function on small scales, θ < ∼ 1 o is a superb way to detect topological and nontopological defects, or alternatively to constrain their contribution to the CMB. For simple inflationary models which lead to defect formation at the end of inflation, a value of 10 −7 ÷ 10 −8 seems rather natural, hence the achieved limits include the relevant regime. The fact that the localB-polarization from defects is dominated by the vector mode, which peaks on scales smaller than the horizon, is responsible for a significant enhancement of the localB-polarization correlation function on tens of arc-minute scales.
Even though the Planck satellite is not the ideal probe for constraining these models, if it finally reaches down to r ≤ 0.025, see Ref. [17], it will either lead to the detection of a defect contribution, or it will constrain it to = Gv 2 < ∼ 10 −7 , depending on the defect model (textures being the most constrained and Large-N nontopological defects the least). Future CMB experiments, with 0.1 arc-minute resolution and sensitivities at the level of 0.1 µK in polarization, could in principle reach the bound < 10 −10 for most defect types, which would rule out a large fraction of present models.