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<ul><li><p>B-modes from cosmic strings</p><p>Levon Pogosian*Department of Physics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada</p><p>Mark Wyman</p><p>Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, ON, L6H 2A4, Canada(Received 14 November 2007; published 15 April 2008)</p><p>Detecting the parity-odd, or B-mode, polarization pattern in the cosmic microwave backgroundradiation due to primordial gravity waves is considered to be the final observational key to confirmingthe inflationary paradigm. The search for viable models of inflation from particle physics and string theoryhas (re)discovered another source for B-modes: cosmic strings. Strings naturally generate as much vector-mode perturbation as they do scalar, producing B-mode polarization with a spectrum distinct from thatexpected from inflation itself. In a large set of models, B-modes arising from cosmic strings are moreprominent than those expected from primordial gravity waves. In light of this, we study the physicalunderpinnings of string-sourced B-modes and the model dependence of the amplitude and shape of theCBBl power spectrum. Observational detection of a string-sourced B-mode spectrum would be a directprobe of post-inflationary physics near the grand unified theory (GUT) scale. Conversely, nondetectionwould put an upper limit on a possible cosmic string tension of G & 107 within the next three years.</p><p>DOI: 10.1103/PhysRevD.77.083509 PACS numbers: 98.80.Cq</p><p>I. INTRODUCTION</p><p>In the past few years, observations of the cosmic micro-wave background (CMB) and large-scale structure datahave largely confirmed the predictions of the inflationaryparadigm. Now, observational cosmologists have begun toturn their instruments towards the final unconfirmed sig-nature of inflation: primordial B-mode, or odd-parity, po-larization in the CMB [1]. Meanwhile, string theorists haveworked diligently towards realizing observationally viablemodels of inflation within reasonably well-understoodstring compactifications (see, e.g., [24] for reviews). Inparallel, viable models of hybrid inflation continue to beexplored in the context of particle physics (e.g., [58]).Brane and hybrid inflation models often predict the pro-duction of cosmic strings [2,6,7,919], linear defects aris-ing from the breaking of U(1) symmetries towards the endof the inflationary epoch. These relics of the inflationaryperiod provide a rich phenomenology, but here we focus ontheir contribution to the polarization of the CMB.</p><p>The production of cosmic strings in such models seemsat first unimportant, since they cannot source the majorityof the CMB anisotropy [20]. However, it now appears thatstrings may be the most prominent source of observable B-mode polarization in many of these braneworld or hybridinflationary models [21,22]. Strings, as extended objects,source vector mode perturbations in the CMB as readily asthey do scalar mode perturbations. This is in contrast withthe inflationary picture, where gravity waves generated byinflation source tensor mode perturbations that are gener-ally much smaller than the adiabatic scalar mode perturba-tions. Defect-sourced B-mode polarization was first</p><p>calculated in [23] for global defects, and for local stringsin [2426]. Recently, B-mode predictions from field theo-retical simulations of local strings were also reported in[27]. It is true that cosmic strings are subdominant sourcesof CMB perturbation, limited to producing less than about10% of the primordial anisotropy [2834].1 But since theinflationary tensor-to-scalar ratiothe parameter that gov-erns the strength of inflationary gravity wavesis ex-pected also to be below 10%, cosmic strings andprimordial gravity waves may well be observables of simi-lar detectability.</p><p>Models of hybrid and brane inflation [35], tend to pro-duce a low tensor-to-scalar ratio [36]. Hybrid inflationarymodels from particle physics, for instance, typically haveunobservably small (r & 104) tensor-to-scalar ratios[37,38]. Most models of inflation that have been realizedin string theory are of the hybrid type, and also have smallscalar-to-tensor ratios, typically r 1010 (e.g. [3941]).On the other hand, many of these models produce cosmicstrings near the grand unified theory (GUT) scale [4245],which implies a cosmic string tension (G) close to theobservational upper bound of G & 2 107 [3234].</p><p>Among models of inflation motivated by string theory,brane inflation (reviewed recently in Refs. [2,4]) can createthe widest range of cosmic string tensions, with 1013 100 requires an accurate calculation of the amplitude of theB-mode polarization that comes from the conversion of E-mode polarization to B-mode polarization by gravitationallensing. This is because the peak BB contribution fromstrings falls at 1000, overlapping heavily with thespectrum expected from lensing of E. Hence, the presenceof a string-induced B-mode would induce an excess inpower throughout this region of values above what isexpected from lensing. Accurately characterizing this ex-cess will require very good predictions of the expectedlensing signal. If all goes well, polarization experimentscould eventually see strings with tensions as low as G109 [21]. Because strings can source B-modes in theaftermath of a hybrid inflationary epoch that cannot di-rectly source B-modes, the class of inflationary modelsthat B-mode experiments can test is broadened.</p><p>Detecting neither string or primordial gravity wave-sourced B-modes would strongly constrain many inflation-ary models. These constraints would be different from, andperhaps stronger than, the constraints derived from mea-surement of other cosmological parameters (e.g., the spec-tral index ns).</p><p>In this paper, we outline the physics behind cosmicstrings generation of B-mode polarization, placing a heavyemphasis on how much variability might be expected fromvarious models of string networks. This gives us an idea ofhow much a successful detection of cosmic string B-modesmight teach us about these networks. We also review someof the currently operating, soon to be launched, and futureplanned experiments to measure the CMBs B-mode po-larization to give estimates for what range of string ten-sions they will be able to detect.</p><p>II. HOW STRINGS POLARIZE THE CMB</p><p>The string-sourced perturbations are fundamentally dif-ferent from those produced by inflation. Inflation sets theinitial conditions for the perturbations (i.e. their initialspectrum) which then evolve in time without additionaldisturbances being produced. In this sense, inflationaryperturbations can be classified as passive. Inhomo-geneities produced by cosmic strings are activestringnetworks are thought to persist throughout the history ofthe universe and actively source metric perturbations at alltimes. It is well known [49] that, in the absence of a sourceterm, vector modes decay quickly, scalar modes have agrowing and a decaying solution, and tensor modes aresustained on superhorizon scales and decay after enteringthe horizon. For this reason, vector modes are not relevantin passive scenarios, and are rarely considered in theliterature. Cosmic strings, on the other hand, activelysource scalar, vector, and tensor perturbations. For them,scalar and vector-mode perturbations are typically of simi-lar magnitude. The string-generated tensor modes are alsoat a comparable level, but their observational impact isgenerally lower because of the oscillatory nature of gravitywaves [50].</p><p>The nonzero CMB temperature quadrupole at the timeof recombination leads to linear polarization of CMBthrough Compton scattering of photons on baryons.Before recombination, the photons and baryons form atightly coupled fluid with the anisotropy characterizedonly by monopole (density contrast) and dipole (massflow) components. After last scattering, photons propagatefreely. Hence polarization is produced during a narrowtime window in the last scattering epoch, when scatteringbecomes sufficiently rare for a temperature quadrupole todevelop. Essentially, this window corresponds to the timebetween the last scattering and the next-to-last scattering.Polarization can also be produced at more recent epochsduring reionization.</p><p>2This conversion covers only the region of 2< < 100, andwas calculated by comparing the total BB power in stringsversus the total BB power from tensor modes in this range.This particular relation is dependent on the parameters of thestring model, and a more complete conversion is suggested inSec. IV.</p><p>3More complicated setups involving wrapped branes mightloosen such bounds [46,47].</p><p>LEVON POGOSIAN AND MARK WYMAN PHYSICAL REVIEW D 77, 083509 (2008)</p><p>083509-2</p></li><li><p>At any point on the sky, CMB polarization can berepresented by a headless vector with a magnitude jPj Q2 U2</p><p>pand an orientation angle 1=2</p><p>arctanU=Q, where Q and U are two of the four Stokesparameters [51].4 The pattern of the P-vectors on the skycan be separated into parity-even and parity-odd contribu-tions, or the so-called E and B-modes. While intensitygradients automatically generate E-mode patterns, the B-mode is not produced unless there are metric perturbationsthat can locally have a nonvanishing handedness, i.e. havea parity-odd component. Local departures from zero hand-edness can be due to tensor modes, i.e. gravity waves,which can be represented as linear combinations of left-and right-handed polarizations [52]. The B-mode can arisefrom nonzero vector modes, or vorticity.</p><p>Cosmic strings induce both vector and tensor compo-nents in the metric by actively generating anisotropicstress, with the vector contribution being the most impor-tant. This implies a nonvanishing vorticity in the photon-baryon plasma at the surface of last scattering, whichgenerates B-mode polarization in the CMB. The vectormetric potential V in Fourier space, in the generalizedNewtonian gauge, can be defined as [53]</p><p> h0jk; eikx VQ1; (1)where h is the linear perturbation to the metric and Q1</p><p>is a divergenceless harmonic eigenmode of the Laplacian.The reader is referred to [53,54] for a more detailed de-scription of the formalism. At last scattering, V is given by[53],</p><p> V; k 8Ga2</p><p>Z </p><p>0</p><p>da4k1pf1f 1s ;(2)</p><p>where f represents the fluid and s the seed contributions tothe anisotropic stress, a subscript indicates the values ofthe conformal time and scale factor a at last scattering.The vector anisotropic stress of the fluid, included here, isindirectly sourced by the strings through the metric.However, the direct coupling of the string anisotropic stressto the vector metric potential dominates over the (indirectlysourced) fluid contribution. Finally, one can see from thisexpression that the vector potential will decay as a2 if it isnot actively seeded.</p><p>It is instructive to consider the vector-mode contributionfrom one straight cosmic string segment. Without loss ofgenerality, one can choose k zk [55], in which case theT13 and T23 components of the Fourier transform of theenergy-momentum tensor of the string network become</p><p>pure vector modes, i.e. they do not contribute to the scalaror tensor modes. The two components are statisticallyequivalent, and it is sufficient to work with just one ofthem. e.g. T13. The Fourier components of the energy-momentum tensor of the string are given by [20,55,56]</p><p> T00 1 v2</p><p>p sinkX03l=2</p><p>kX03=2cos ~k ~x0 k _X3v;</p><p>Tij v2 _Xi _Xj </p><p>1 v22</p><p>X0iX0j</p><p>T00;</p><p>(3)</p><p>where X0i is the component of the strings orientation in thedirection i and _Xi is the strings velocity in the direction i,v is the velocity, is the wiggliness parameter that re-normalizes the strings tension based on its small-scalestructure, is the tension, and ~xo is the strings locationin space. We chose k zk. In our notation, the anisotropicstress is related to T13 via [54]</p><p> 1s k; 22</p><p>pT13k; : (4)</p><p>The key observation one can make from these equations isthat the string contribution to anisotropic stress is alwayspresent, even for a stationary string. The strings orienta-tion inherently violates isotropy.</p><p>For a single string, the strength of the scalar, vector, andtensor contributions depend on the orientation of the stringwith respect to k. A network of strings can be thought of asa collection of many string segments at random positions,each with a random orientation and velocity. The B-modepower spectrum is determined by the two-point correlationhPm1smPn1sni, where the sums are taken over all seg-ments present at last scattering. If the segments are statis-tically independentwhich is approximately true sincethey are taken to be comparable to the horizon sizethen the ensemble average reduces to a single sum overcontributions of separate strings, i.e. hPnj1snj2i. In fact, ifall strings are statistically equivalent, this reduces toNhj1s1j2i, where N is the number of segments and1s 1 is the contribution from one string. One can analyti-cally evaluate the average hj1s1j2i for a straight stringusing Eqs. (3) and (4) by averaging over uniformly distrib-uted velocity directions, string positions, and orientations.The dependence of the B-mode spectrum on the networkparameters can be inferred from how hj1s1j2i varies withv, l, and . We will return to this when we interpret ournumerical results, but we sketch the main dependenciesnow.</p><p>Two important network parametersthe correlationlength, l, and the strings root-mean square (rms) velocity,venter T00 (see Eq. (3)) in a sine and cosine function,respectively. Scaling l and v up and down, then, amounts tochanging the Fourier space dependence of the string con-tribution. The dominant Fourier modes will determine the</p><p>4The other two parameters are the intensity I, and the onequantifying the circular polarization, V. Circular polarization isnot produced at last scattering and is expected to be zero oncosmological scales in the absence of large-scale magnetic fieldsor other exotic physics.</p><p>B-MODES FROM COSMIC STRINGS PHYSICAL REVIEW D 77, 083509 (2008)</p><p>083509-3</p></li><li><p>dominant angular modes of the B-mode spectrum and setthe location of the angular peaks. The sine function thatdepends on l sets the location of the peak in string-sourcedpower. The cosine function that depends on v also takes acontribution from a random phase. Varying v changes thenumber of cosine-peaks that contribute to the shape of theoverall peak. Higher velocity strings will contribute morecosine-peaks and lead to broader B-mode spectra.Changing the rms velocity also changes the ratiohjTijj2i=hjT00j2i, i.e. the relative contribution of the aniso-tropic stress versus the isotropic stress. Namely, increasingv reduces this ratio. This can be seen by computing thisratio from Eq. (3),</p><p>hjTijj2ihjT00j2i</p><p> hjX0iX0jj2i 2v2hjX0iX0jj2i hX0iX0j _Xl _Xki</p><p>Ov4;so that any nonzero velocity always decreases the aniso-tropic contribution. String wiggliness similarly suppressesthe anisotropic stress relative to the isotropic stress. All ofthese effects can be seen in Sec. III B and Figs. 24, whichpresent the results from our numerical experiments.</p><p>III. DEPENDENCE OF THE B-MODE SP...</p></li></ul>