Cosmic microwave anisotropies from BPS semilocal strings
Jon Urrestilla, Neil Bevis, Mark Hindmarsh, Martin Kunz, Andrew R. Liddle
aa r X i v : . [ a s t r o - ph ] J u l Cosmic microwave anisotropies from BPS semilocal strings
Jon Urrestilla, ∗ Neil Bevis,
2, 1, † Mark Hindmarsh, ‡ Martin Kunz,
3, 1, § and Andrew R. Liddle ¶ Department of Physics & Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 1211 Gen`eve 4, Switzerland (Dated: 11/07/2008)We present the first ever calculation of cosmic microwave background (
CMB ) anisotropy powerspectra from semilocal cosmic strings, obtained via simulations of a classical field theory. Semilo-cal strings are a type of non-topological defect arising in some models of inflation motivated byfundamental physics, and are thought to relax the constraints on the symmetry breaking scale ascompared to models with (topological) cosmic strings. We derive constraints on the model param-eters, including the string tension parameter µ , from fits to cosmological data, and find that in thisregard BPS semilocal strings resemble global textures more than topological strings. The observedmicrowave anisotropy at ℓ = 10 is reproduced if Gµ = 5 . × − ( G is Newton’s constant). Howeveras with other defects the spectral shape does not match observations, and in models with inflationaryperturbations plus semilocal strings the 95% confidence level upper bound is Gµ < . × − when CMB data, Hubble Key Project and Big Bang Nucleosynthesis data are used (c.f.
Gµ < . × − for cosmic strings). We additionally carry out a Bayesian model comparison of several models withand without defects, showing models with defects are neither conclusively favoured nor disfavouredat present. I. INTRODUCTION
Recent observational data are tightly constraining cos-mological models, in many cases rendering them non-viable. One of the constraints comes from the fact thatmany potentially successful high-energy physics moti-vated models of inflation predict a cosmic string network[1, 2] in the post-inflationary era: this occurs in super-symmetric hybrid inflation [3], in grand unified theories[4] and in some string theory inflation models [5, 6, 7, 8].These strings and other type of defects contribute tothe cosmic microwave background (
CMB ) anisotropiesin addition to the primordial inflationary fluctuations,allowing
CMB experiments to provide upper limits onthe contribution from defects [9, 10, 11, 12, 13]. In thenear future, observations will either reveal the presence ofthese defects or will significantly tighten the constraintsupon the models from which they are predicted (see forexample Refs. [14, 15, 16, 17, 18]).However, a simple and elegant method for evading theconstraints coming from cosmic strings in these mod-els was proposed in Refs. [19, 20]. A duplication ofthe matter fields responsible for the string formation,which arises quite naturally in several models [20, 21, 22],changes the nature of the strings from topological andtherefore long-lived strings, to non-topological semilocalstrings [23]. All the desired inflationary properties of themodel persist, and semilocal strings are believed to be ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] cosmologically less harmful than their topological coun-terparts, in that their energy density may be lower for agiven symmetry breaking scale. In this paper we confirmthese expectations by performing the first
CMB calcula-tions for semilocal strings and determine the constraintsupon semilocal strings imposed by current data.Calculating the additional contribution that cosmicstrings would make to the
CMB temperature and po-larization power spectra is not an easy task. They arehighly non-linear entities, and the range of scales in-volved in the problem is enormous. However, as thewidth of the string is much smaller than the string length,one possibility is to neglect the finite width and performNambu–Goto type simulations [24, 25, 26, 27]. Unfor-tunately, these simulations ignore the string decay intogravitational waves or particles and the correspondingback-reaction onto the network. Their representation ofsmall-scale loop production in the network is also underdebate [28, 29, 30, 31, 32]. Yet another level of simplifi-cation often used in
CMB calculations for local strings isto simulate merely a stochastic ensemble of unconnectedstring segments [10, 11, 14, 16, 33, 34]. Lacking any dy-namical equations, in this case the sub-horizon decay ofstrings must be taken into account by the random re-moval of segments, at a rate chosen to match that seenin network simulations, but qualitatively accurate resultsare obtained with much less computational cost. In theglobal defect case, the energy is not localized into thestring cores and their resolution is not essential. This en-ables the calculation of
CMB contributions from globaldefects using field theory simulations [35, 36].None of these three approaches is useful for the studyof semilocal strings, since they are not well-modelled byNambu–Goto strings while, unlike the global case, thestring cores are potentially important. Fortunately, ina recent breakthrough we performed the first
CMB cal-culations for local cosmic strings to derive from field-theoretical simulations [37]. We achieved this by the useof a parametrized approximation to reduce the range ofscales required in the simulations, but which had no re-solvable effect on the
CMB predictions. That approachhas opened the door for the possibility of simulating de-fects with no known approximating models.We have therefore employed the machinery developedfor the local string case to the simplest model containingsemilocal strings, with the parameters chosen to give theBogomol’nyi–Prasad–Sommerfield (BPS) limit (see Sec-tion II). The BPS limit arises naturally in D-term SUSYinflation and D /D CMB calculations using the linear σ model. We leavea discussion of our texture calculations to an Appendix.We have also performed a multi-parameter compari-son with current data, yielding results similar to those ofRef. [13] where it was seen that local cosmic strings aremildly favoured by current
CMB data. We find that thepreference for semilocal strings is slightly greater and, asin Ref. [13], we explore this preference using Bayesianevidence calculations, as well as considering the implica-tions of non-
CMB data.This paper is organized as follows: in the next Sectionwe discuss the semilocal string model, before giving abrief overview of the simulation and
CMB calculationmethod in Section III. We will then present our
CMB results and consistency checks in Section IV, includingthe comparison with local strings and global textures.Section V then presents the comparison with data.
II. SEMILOCAL STRING MODEL
The simplest model giving rise to semilocal strings isgiven by the Lagrangian density [23] L = | D µ φ | + | D µ φ | − e F µν F µν − λ (cid:16) | φ | + | φ | − η (cid:17) , (1)where D µ = ∂ µ + iA µ and F µν = ∂ µ A ν − ∂ ν A µ . The fields φ and φ are complex scalar fields and A µ is an Abeliangauge field. The parameter η gives the symmetry break-ing scale. This Lagrangian possesses all the necessaryingredients to give rise to semilocal strings and so allows See for instance Ref. [41] for a comparison of the dynamics ofthe linear and non-linear σ models. insight into more complicated models, such as D-terminflation in N = 1 or N = 2 supersymmetric (SUSY)models [42, 43] and D /D S , which is simply connected,and the existence and stability of the strings depend onenergetic and dynamical causes, not on the topology ofthe vacuum. There is only one parameter governing thestability of the strings, namely, β ≡ λ/ e : for β < β > β = 1) is the transi-tional case, where the strings are neutrally stable [45, 46].There is a zero mode concerning the string width in theBPS case: all strings with widths that range from thewidth of an Abelian Higgs string to infinity are degen-erate in energy. But even though the string can havea very large (infinite) core, the magnetic flux is never-theless quantized. It is also worth noting that when twosemilocal strings of any width collide, they intercommute[47, 48, 49] and their widths may revert to the minimumwidth (i.e., the width of an Abelian Higgs string) [48]. Asmentioned earlier, the BPS case arises naturally in, forexample, D-term inflationary scenarios, In more genericcases, non-BPS cases will arise. We can anticipate somequalitative behaviour for the non-BPS case. For β > β <
1, the lower the valueof β , the more semilocal string networks resemble AbelianHiggs string networks [40], and their CMB predictionswill tend to that of Abelian Higgs strings.One important difference between Abelian Higgs cos-mic strings and semilocal strings is that the latter arenon-topological, so they can have ends. We therefore donot expect infinite semilocal strings to be formed by theKibble mechanism in a cosmological phase transition. In-stead, a collection of segments forms. The evolution ofthe segments is very complicated: some will shrink tozero, others will join both ends to form loops, while somewill join to nearby segments to make longer segments,and eventually form virtually infinite strings [40]. At theBPS limit, the formation of longer segments is stronglysuppressed, which led the authors of Refs. [19, 20] to con-jecture that no strings would be formed in such a modelafter inflation, and to claim that the D-term inflationwithout cosmic strings could take place.While the semilocal model can be reduced to theAbelian Higgs model by removing one complex scalarfield, it can also be reduced to a global texture model byremoving the gauge field (see Appendix). A key questionis how those removals modify the predicted anisotropy,keeping fundamental Lagrangian parameters fixed. Animportant derived parameter is the string mass per unitlength in the Abelian Higgs and semilocal cases [44, 50]: µ ≡ πη , (2)and the CMB anisotropies are proportional ( Gµ ) , where G is the gravitational constant. We continue to use thismeasure even when talking about the texture model, butin that case it has no direct interpretation and is merelya measure of the energy scale of the model. It is knownthat textures give significantly lower anisotropies thanstrings for a given Gµ , since textures decay much morequickly once inside the horizon. Our original precon-ception was that the anisotropies from semilocal stringswould be of the same order as, but a little smaller than,those of textures. This expectation is because the gaugefield cancels some of the field gradients present in thetexture case and contributes little to the energy density[46], while in the BPS limit there are no long strings andhence no expectation of large anisotropies from string-like sources. Indeed, we will find that this expectation isconfirmed by our detailed analysis. III. CMB CALCULATION METHODA. Overview
As discussed in the introduction, the method used tocalculate
CMB power spectra employed here is exactlythat used for Abelian Higgs strings in Ref. [37] and we re-fer the interested reader to that article for greater depth.However we will briefly describe the fundamentals of theapproach in this Section.Since observations reveal that the cosmological per-turbations are small (except on short scales in the cur-rent epoch) the defects can be taken to evolve in anexpanding flat Friedmann–Lemaˆıtre–Robertson–Walkeruniverse. Inflation is taken to set up small primor-dial fluctuations, which then evolve passively under theEinstein–Boltzmann equations. However, the defectsperturb the metric, via their energy–momentum tensor,and these perturbations also have to be taken into ac-count when computing
CMB power spectra. But sincethe perturbations introduced by either inflation or defectsare very small, any coupling between them is insignificantand the two perturbation sets may be calculated sepa-rately. The result is two statistically independent setsof
CMB anisotropies, the power spectra of which simplyadd to give the total.Fundamentally the
CMB power spectra are two-pointcorrelation functions of the
CMB temperature (or po-larization) fluctuations, and in the defect case they canbe determined from the two-point unequal-time correla-tion functions ( uetc s) of the defect energy–momentumtensor ˜ T µν [35, 51, 52, 53]:˜ U κλµν ( k , τ, τ ′ ) = D ˜ T κλ ( k , τ ) ˜ T ∗ µν ( k , τ ′ ) E . (3)Here k is the comoving wavevector, τ and τ ′ are twoconformal times and ∼ denotes a Fourier space quantity,as in the notation of Ref. [37].However, the above 55 complex functions of 3 variablesmay be reduced to just 5 real functions of two variables, via the use of statistical isotropy, energy–momentum con-servation, and a property of defect networks known as scaling . The last property derives from the causal na-ture of the network: that field smoothing is limited bythe horizon and, as such, defect networks often tend toan attractor regime in which statistical measures of theirdistribution are a function of a single quantity: the hori-zon size τ . Assuming scaling and statistical isotropy, the uetc s can be written as [53, 37]˜ U κλµν ( k, τ, τ ) = η √ τ τ ′ V ˜ C κλµν ( kτ, kτ ′ ) (4)where V is the fiducial simulation volume and ˜ C is thescaling function of the uetc . Further, the functions ˜ C decay for large or small τ /τ ′ or for large k √ τ τ ′ and hencetheir measurement is only required over a fairly limitedparameter space.Of the 5 independent scaling functions, 3 ofthem represent scalar degrees of freedom ( ˜ C S11 ( kτ, kτ ′ ),˜ C S12 ( kτ, kτ ′ ), ˜ C S22 ( kτ, kτ ′ )) while the remaining two repre-sent the vector and tensor degrees of freedom respectively( ˜ C V ( kτ, kτ ′ ), ˜ C T ( kτ, kτ ′ )) [37, 52]. Unlike the inflation-ary case, the amplitude of the tensor perturbations isfixed and directly obtained from the simulations, and itis not possible to neglect the vector perturbations as theyare continuously sourced by the presence of the defectnetwork.The uetc scaling functions are then fed into a modi-fied version of the CMBeasy
Boltzmann code [54] whichcomputes the
CMB temperature and polarization powerspectra. However, in order for
CMBeasy to use the uetc data, the functions ˜ C ( kτ, kτ ′ ) must be first de-composed as:˜ C ( kτ, kτ ′ ) = X n λ n ˜ c n ( kτ ) ˜ c n ( kτ ′ ) , (5)which in the numerical case is tantamount to deter-mining the eigenvalues and eigenvectors of a real, sym-metric matrix. Note that ˜ C S12 is not symmetric, but˜ C S21 ( kτ, kτ ′ ) = ˜ C S12 ( kτ ′ , kτ ) and so the M × M matricesrepresenting the scalar degress of freedom ( ˜ C S11 , ˜ C S12 , ˜ C S21 and ˜ C S22 ) are tiled together to yield a symmetric 2 M × M matrix which is then decomposed this way.The modified CMBeasy then takes a single eigenvec-tor ˜ c n as a source function and determines the corre-sponding contribution to the CMB power spectra, withsummation over all such contributions yielding the to-tal. We found that for the semilocal case, the number ofeigenvectors that should be included in the computationof the power spectra to achieve convergence was lowerthan that needed in the Abelian Higgs case; nevertheless,we use the same number of eigenvectors as in the AbelianHiggs case (see Table I) to minimize the differences in thecomputation. Also there are some choices about how todecompose the uetc s into the form of Eq. (5), and againwe follow exactly the method of Ref. [37].Note that scaling is broken at the radiation–mattertransition and therefore the scaling uetc functions mustbe determined for the defects under both matter andradiation-domination. These are decomposed into eigen-vectors and fed into the modified
CMBeasy code, whichthen interpolates between the radiation and matter re-sults to model the transition. For the transition into theΛ–dominated epoch, the sources simply decay and simu-lation is not required. In any case, perturbations sourcedat such late times affect only the very largest scales.
B. Semilocal simulations
In order to calculate the scaling uetc functions forthe semilocal string model above, we perform fieldtheory simulations under both radiation- and matter-domination. However, since at times of importance for
CMB calculations the string width is many orders ofmagnitude smaller than the horizon size, it is not pos-sible to perform simulations for such times while stillresolving the string cores. Fortunately, scaling impliesthat this is not required and that the scaling uetc func-tions may be obtained from simulations of early times,with τ ∼ η − , and then used to describe the late-time behaviour of the system. Note that although weperform simulations in a matter-dominated backgroundfor τ ∼ η − , when the real universe was radiation-dominated, scaling implies that the network propertiesare the same at 100 η − as they are in the true late-timematter era.However, performing the simulations required withcurrent or near future computational technology is a con-siderable challenge. The (minimum) string width ( r min )is fixed in physical coordinates and, in the comoving co-ordinates required for the simulations, it varies as: r min = 1 a √ λη = 12 aeη , (6)where a is the cosmic scale factor. It therefore shrinksas τ − under a radiation- or as τ − under a matter-dominated universe. If the strings are resolved at theend of the simulation, then they are enormous and largerthan the horizon size for times only a little earlier, par-ticularly under matter domination. The system will notscale, or even contain significant regions where the fieldslie on the vacuum manifold, if τ is comparable to r min .Thus, the uetc data can only be collected over a limitedregion of the required parameter space.A simple means of circumventing the above problemis to promote the parameters λ and e in the action totime-dependent variables [37] as: λ = λ a − − s ) e = e a − (1 − s ) . (7)Now the scale r min varies as: r min = r min , a − s , (8) and variation of the action yields equations of motion as:¨ φ + 2 ˙ aa ˙ φ − D j D j φ = − a s λ (cid:0) | φ | + | φ | − η (cid:1) φ ¨ φ + 2 ˙ aa ˙ φ − D j D j φ = − a s λ (cid:0) | φ | + | φ | − η (cid:1) φ ˙ F j + 2(1 − s ) ˙ aaF j − ∂ i F ij = − a s e I m [ φ ∗ D j φ + φ ∗ D j φ ] . (9)For the case of s = 1, the true dynamics are obtained,while for s = 0 the factors of a responsible for the shrink-age of the (minimum) string width are removed from theequations of motion and r min becomes comoving. This issimilar to methods of Refs. [55, 56], where the undesir-able factors of a were directly removed from the dynami-cal equations, but here the parameter s can be varied andits effects measured. This is particularly important be-cause s = 1 means that energy–momentum conservationis violated and it is T µν that sources the perturbations.For the simulations under radiation-domination we canactually simulate the s = 1 case directly and comparein detail our intermediate results for s < CMB results for Abelian Higgs strings are insensitive to s , and here we extend that result to semilocal strings.There is good evidence from previous simulations ofboth Abelian Higgs and semilocal strings that the net-work results are independent of the initial conditions oncescaling is reached [39, 56, 57, 58, 37]. The goal is there-fore to produce an initial configuration in a numericallyfeasible way that obeys the analogue of Gauss’ law inthe model and that will yield scaling fairly fast, in or-der to maximize the dynamical range of the simulation.Following again Ref. [37] we set all temporal derivativesand gauge fields to zero, and the fields φ and φ areset in such a way that they lie in the vacuum manifold (cid:0) | φ | + | φ | = η (cid:1) with random phases in S . The sys-tem is then evolved using the equations of motion ob-tained after discretizing the action via the standard Mo-riarty et al. [59] scheme and encoded in C++ using theLATfield Library [60]. This scheme preserves the Gaussconstraint during the evolution, even in the case of s < IV. SIMULATION AND CMB POWERSPECTRUM RESULTSA. Numerical aspects and tests for scaling
The numerical simulations were performed on the UKNational Cosmology Supercomputer [61], where we sim-ulated five different realizations for both matter- andradiation-dominated epochs and for s = 0 . s = 0 . s = 1 for radiation alone. Of those, only thesimulations with s = 0 . CMB r m i n / r m i n ,
1 100 10 000 λ / λ e / e τ FIG. 1: Variation of r min and the parameters e and λ for the s = 0 . N x [ η − ] 0.5Time-step ∆ t [ η − ] 0.1Scalar coupling λ e τ i [ η − ] 1.0Final conformal time τ end [ η − ] 128Initial s − . s η − ] 32 τ sim range of ξ fit [ η − ] 50 - 128Dynamic range ( τ /τ ′ ) max ∼ uetc sample matrix size 512 uetc matrix sampling linearNo. eigenvectors 128TABLE I: Parameters used in production runs. The firstpart of the table lists the parameters of the simulations, withthe second part listing parameters of the uetc method forcalculating the CMB power spectrum. Note that the valuesare identical to those of Ref. [37], except for the time rangeof the simulations. power spectra; the others were used to test the validityof our approach. We begin and end the simulation withthe same value for r min ; that is, we initially set s to anegative value so that r min increases until a time τ s , af-ter which s is positive so that r min shrinks during theprimary part of the simulation (see Fig.1). This processaccelerates the formation of vacuum regions.Table I shows the values of the parameters used for theproduction runs. Most of the parameters match those ofRef. [37]; furthermore, the coupling parameters were alsovaried as shown in Fig. 1 to replicate the behaviour ofRef. [37]. Because of the existence of zero modes in thesemilocal system it is dangerous to run the simulation for τ s = . τ s = . τ s = . FIG. 2: Average value of ξ , as defined in Eq. (10), for s = 0 . , . , .
0, in the radiation era. The average is overfive different realizations for each value of s , and the shadedregions correspond to 1- σ and 2- σ deviations. The best-fitline for times τ >
50 is also shown (dashed line). Note thatthe best-fit line is an excellent approximation, and that thedifferences between different values of s are minimal. longer than the half-crossing-time. In order to keep thedynamic range of Ref. [37], here we had to begin taking uetc data at an earlier time ( τ = 50 rather than τ = 64).One might wonder whether the system is already into thescaling regime by those early times, but we will show inthis Section that in fact this is the case.One means to test for scaling is via the string lengthdensity, which should be proportional to τ − , since theaverage length of string per horizon volume should beproportional to τ and the horizon volume varies as τ .The detection of (topological) Abelian Higgs strings ina lattice field simulation is fairly straightforward, sinceone can track, for instance, zeros of the Higgs field orthe winding of its phase around them. Alternatively onecould use the average Lagrangian density (as in Ref. [37])or the T component of the energy–momentum tensor,which should also vary as τ − .In fact, for (non-topological) semilocal strings thosealternative methods must be employed: the zeros of theHiggs fields are not a good measure, since there is no needfor the string core to be a zero of the Higgs field. Besides,semilocal strings in the Bogomol’nyi limit ( β = 1) are ex-pected to form short segments that either collapse veryquickly or become fatter, diluting away the core of thestring. That is why we used a different measure to quan-tify the scaling of the semilocal BPS string system.In this work, we use the T component of the energy–momentum tensor to define our length measure ξ as: ξ = η √ T , (10)and use it to check the scaling of the system. Figure 2
10 20 30 40 50 600102030405060 C S ( k τ , k τ ) k τ
10 20 30 40 50 60−50−40−30−20−10010 C S ( k τ , k τ ) k τ
10 20 30 40 50 600102030405060 C S ( k τ , k τ ) k τ −3 −2 −1 C V ( k τ , k τ ) k τ
10 20 30 40 50 6000.511.522.533.544.55 C T ( k τ , k τ ) k τ FIG. 3: The dependence of the 5 etc s upon s in the radiation era at τ sim = 128 η − . The shaded regions show the 1- σ and2- σ uncertainties for the s = 0 . s = 1 . s = 0 . shows the result of the simulations for the radiation erafor s = 0 . , . , .
0, as well as the best fit for times τ >
50. First of all, note that the differences with respectto varying the value of s are minimal. It is also clear fromthese figures that after the initial non-scaling period, ξ follows a linear behaviour ξ ∝ ( τ − τ ξ =0 ) . (11)There is nothing fundamental about the offset τ ξ =0 , asargued for Abelian Higgs case in Ref. [37]; it is a con-sequence of the non-scaling initial period of the simula-tions and is negligible in the late-time limit of interestfor CMB calculations. What is fundamental is the valueof the slope of ξ , which shows little dependence upon s : 0 . ± .
01 for s = 0 .
0, 0 . ± .
02 for s = 0 . . ± .
02 for s = 1 .
0. The difference between the threeplots in Figure 2 is due almost entirely to differing valuesof τ ξ =0 .However, given that the system scales not with τ butwith τ − τ ξ =0 , then following Ref. [37] we perform a rescal-ing of the measured uetc functions ˜ C ( kτ, kτ ′ ) in accor-dance with: τ → τ − τ ξ =0 . (12)These functions then provide a more important test ofscaling for the present work: after the rescaling, do theyshow negligible absolute time dependence? Figures 3 and4 show the equal-time correlator ( etc ) scaling functions,that is when τ = τ ′ . These are the most important part
10 20 30 40 50 600102030405060 k τ C S ( k τ , k τ ) FIG. 4: The equal-time scaling function ˜ C S11 ( kτ, kτ ) averagedover five realizations for s = 0 . τ valuesin the range 50 η − < τ < η − . The shaded regions showthe 1- σ and 2- σ uncertainties in the mean indicated for thelatest time (solid). For visualization purposes, the mean offsetacross the five realizations is used, whereas the actual CMB calculations use independent offsets for each realization. of the uetc s, and the most suitable for visualization.The different lines in Fig. 4 correspond to the etc func-tion averaged over five realizations, for different times;the shaded regions correspond to the 1- σ and 2- σ un-certainties in the mean at the last time step (when the l ( l + ) C l / ( G µ ) FIG. 5:
CMB temperature power spectrum from semilocalstring simulations. The solid black line is the total prediction,which is obtained by adding the contribution of scalar (dash-dot red), vector (thin blue) and tensor (dashed green) modes. system is closest to scaling). Clearly, the different lineslie within the statistical uncertainties of the simulation,and there is no absolute dependence of the etc on time.As the uetc s are the only quantities entering the
CMB power spectrum calculations, we are confident that theearlier sampling does not affect the results.We can also make use of the etc s to show that the de-pendence of the simulations with s is quite mild. Figure 3shows the etc s for the last time step in the simulationfor s = 0 . , . , . uetc s need forthe CMB calculation. In all cases the dependence on s lies well within the statistical uncertainties. B. Temperature power spectra from semilocalstrings
After checking that our numerical approach for simu-lating semilocal strings is satisfactory by monitoring theinfluence of the parameter s in the simulations and con-firming the early onset of scaling, the resulting uetc swere fed into the modified CMBeasy
Boltzmann codeas explained in Section III. The perturbations obtainedfrom the strings and from inflation are calculated sep-arately and added in quadrature – the tiny interactionbetween the two contributions can be safely neglected.Besides, it is numerically quite slow to recompute thestring spectra for many different values of the cosmologi-cal parameters. For these reasons, the string spectra werecomputed for a fixed value of the cosmological parameters l ( l + ) C l / ( G µ ) l ( l + ) C l / ( G µ ) FIG. 6:
CMB temperature power spectrum from AbelianHiggs strings (left) and texture (right) simulations. The solidblack line is the total prediction, and its constituents: scalar(dash-dot red), vector (thin blue) and tensor (dashed green).Note the difference in the scale between the two graphs. (the central values given by cosmological experiments ).We then tested at the end that the string spectra do notchange significantly when the cosmological parametersare varied within the limits imposed by the CMB data.Note that this is the only point where the cosmologicalparameters are involved in the calculation of the
CMB spectra.Figure 5 shows the
CMB temperature power spectrumfrom semilocal cosmic strings, divided into scalar, vectorand tensor components. For comparison, we plot the cor-responding power spectra from Abelian Higgs strings andtextures in Fig. 6. The power spectra for Abelian Higgs The calculation of the string spectra was done for h =0 .
72 [62],Ω b h = 0 . Λ = 0 .
75 [64]. We also assumed spatialflatness and the optical depth to the last-scattering surface τ =0 . l ( l + ) C l TT / ( G µ ) FIG. 7: The temperature power spectra from semilocal strings(thick) and textures (thin), showing the scalar (dash-dot red),vector (dashed blue) and tensor (lower solid green) compo-nents. strings are taken from Ref. [37], whereas the texturespectra have been calculated by using new simulations ofthe texture linear- σ model obtained by setting to zero thegauge fields in the semilocal model Eq. (1) (see Appendixfor details on the texture calculations).The first thing to notice from these graphs is that,at fixed Gµ , the Abelian Higgs strings give significantlyhigher anisotropies than the other two by a factor ofabout 5. Semilocal strings and textures are quite sim-ilar, with the former being slightly lower. This is in goodaccord with the intuition that semilocal strings ought tobe similar to textures, but with some of the anisotropiescancelled by the gauge field. Ultimately, this differencein anisotropy amplitude will result in a weakening of ob-servational constraints on Gµ for semilocal and textureas compared to Abelian Higgs strings.Going beyond the amplitude, it is clear that the semilo-cal temperature power spectra shape shares propertieswith both the Abelian Higgs and the texture prediction.It is close to the texture prediction in, e.g., the level ofthe power of the anisotropies at fixed Gµ , although itdoes not have the small oscillations. On the other hand,it peaks for larger values of ℓ than textures, more likeAbelian Higgs strings.Figure 7 shows a direct comparison between the semilo- Refs. [15, 37] employed a code in which a bug has been discov-ered, and this had a small effect in Ref. [13] since it used theirresults directly (see the respective errata). Here we have usedthe corrected power spectra from Refs. [15, 37] and quote thecorrected results from Ref. [13]. Gµ Abelian Higgs string 2 . × − Semilocal string 5 . × − Textures 4 . × − TABLE II: Normalization of different defects to match theobserved ℓ = 10 multipole value. l ( l + ) C l / π [ µ K ] WMAPAbelian Higgs stringsSemilocal stringsTextures
FIG. 8: The temperature power spectra for Abelian Higgsstrings, semilocal strings, textures and the best fit inflation-ary model. The points represent the three-year
WMAP ex-perimental data [65]. The curves for defects are normalizedto match the data at ℓ = 10. cal strings prediction and the texture prediction for thetemperature power spectrum. As discussed before, thetexture model can be obtained by setting to zero thegauge coupling constant in the semilocal model, so thedifferences appearing in the curves arise from couplingthe scalar fields to the gauge fields. The overall shape ofthe curves for each model is roughly the same, with thatfor textures being somewhat higher and peaking earlierthan that for the semilocal strings. If the Abelian Higgsmodel prediction was plotted in the same graph, it wouldlie way above this curve, since the overall scale of thespectrum is much higher.However, bear in mind that the normalization of thespectra is a free parameter, and one would like to com-pare the relative shapes of the predictions to understandtheir detailed observational implications. For example, itis customary to encode the information about the overallpower coming from the defects by Gµ . The normal-ization of the strings is proportional to ( Gµ ) . Thus, Gµ is the value of the normalization by which the de-fect spectra matches the WMAP data [65] at multipole ℓ = 10, shown in Table II for each type of defect. Thisis merely a tool to compare different predictions. Fig-ure 8 shows pictorially the different defect predictionswith these normalizations, together with the experimen-tal data and the inflationary best-fit model. Note thatfor textures it is not natural to talk about “energy perunit length”, but we adopt the same normalization strat-egy Eq. (2) by analogy with the strings, in order to easethe comparison between models. Different normalizationschemes are discussed in the Appendix. C. Polarization power spectra from semilocalstrings
The polarization power spectra are shown in Fig. 9, de-composed as usual into the ee , te and bb components[66]. The curves are normalized so they have equal tt power at ℓ = 10 (and match the WMAP measurementon this scale). Note that, as in Ref. [15], we do not in-clude the coupling between inflation and defect perturba-tions as a result of gravitational lensing at late times. Wedo not, however, expect this effect to be significant: thegravitational lensing of the inflationary scalar ee spec-trum by the defect perturbations is likely to be insignifi-cant relative to that produced by the inflationary pertur-bations, since defects are sub-dominant. It is also likelyto be negligible relative to the B-mode from defects sincedefects still produce a strong B-mode signal even whentheir sub-dominance is taken into account. The lensing ofthe defect ee and bb spectra by the inflationary pertur-bations should be negligible also since defects contributeequally strongly to these two spectra.With the temperature spectra all normalized to thesame value at ℓ = 10, the polarization spectra are allof similar magnitude and shape (if normalized to thesame Gµ , the Abelian Higgs contribution would be muchhigher). In the ee case, all three defect models peakat the same value of ℓ ∼ ℓ coming from reionizationfollows the following trend: it peaks at the lowest valueof ℓ for textures, next for semilocal strings, and highestfor Abelian Higgs strings. The same trend can be seenfor the low- ℓ features of the te spectra. The high- ℓ be-haviour of the te is roughly the same for all three cases,showing large oscillations that eventually die out. Note,however, that the oscillations for the semilocal case arethe smallest, and there is no alternation between anticor-relation and correlation due to the combined effect of thescalar–vector–tensor contributions which have differentphases.The bb power spectrum is arguably the most inter-esting prediction from defects [14, 15, 18, 35]: inflationcontributes weakly to the bb polarization [67], and thesignal predicted from defects might be detected in thenear future, providing a smoking gun for the existenceof defects in cosmology. In Fig. 9 it can be seen that See Fig. 12 in Section V for a figure with the normalizationof each case set to match the respective 95% upper limits fromcurrent data. −1 −2 −1 l ( l + ) C l / π [ µ K ] −2 −1 l TEEEBB
FIG. 9: Polarization power spectra for semilocal strings (solidred), compared to Abelian Higgs strings (dashed black) andtextures (dot-dash blue). The top figure is the te powerspectra, the ee is in the middle, and bb at the bottom. Allthe curves are normalized by making the temperature spec-tra match the WMAP ℓ = 10 value, i.e., using the values of Gµ = Gµ as in table II. In the te figure, the thick linescorrespond to correlations and the thin lines to anticorrela-tions. The results plotted here correspond to a value τ = 0 . the shape of the bb power spectra is similar for semilocalstrings, textures and Abelian Higgs string. Both peaksfollow the aforementioned trend, with textures peakingfor lowest ℓ ∼ ℓ ∼ ℓ ∼ bb power spectrum for textures decays first, then that forsemilocal strings, and the Abelian Higgs string bb powerspectrum decays at the highest ℓ . V. CONSTRAINTS FROM CURRENT CMBDATA
Clearly, a string component at the Gµ level shownin Table II is ruled out by the experiments. There-0 Parameter PL PL+SL PL+AH PL+TX f . ± .
06 0 . ± .
05 0 . ± . n s . ± .
02 1 . ± .
04 1 . ± .
03 1 . ± . h . ± .
03 0 . ± .
06 0 . ± .
06 0 . ± . b h . ± . . ± . . ± . . ± . m h . ± .
007 0 . ± .
007 0 . ± .
007 0 . ± . A s ) 3 . ± .
06 2 . ± .
08 2 . ± .
08 2 . ± . τ . ± .
03 0 . ± .
04 0 . ± .
04 0 . ± . CMB data only. s Ω b h f C M B f C M B + H K + BB N FIG. 10: The 2-dimensional constraints from the
MCMC chains using
CMB only data (bottom), and using
CMB + BBN + HKP data (top). fore, in order to quantify the allowed level for a stringcomponent, we introduce a parameter f [13]: the frac-tional contribution of strings to the temperature powerspectra at multipole ℓ = 10. We then use a modifiedversion of CosmoMC [68] to perform a multi-parameterMarkov Chain Monte Carlo ( MCMC ) likelihood analysisfor
CMB data (
WMAP , ACBAR, BOOMERANG, CBIand VSA projects [65, 69, 70, 71, 72, 73]) when semilocalstrings or textures are included. We also reproduce theresults for Abelian Higgs strings from [13] for compari-son. The likelihood analysis is performed for the usualsix-parameter Power-Law (PL) model (Ω b h , h , n s , τ ,Ω m h , A s ), plus the new extra f . We vary only thenormalization of the string (or texture) power spectraand keep the form of them fixed, following Refs. [9, 13].This approach is justified since the strings (or textures)contribute only a small fraction to the total power, andthe changes in the cosmological parameters are not large P r obab ili t y den s i t y FIG. 11: The 1-dimensional marginalized likelihood plots forthe fraction of semilocal strings f allowed at multipole ℓ =10 from the MCMC chains using
CMB only data (dark), andusing
CMB + BBN + HKP data (light). enough that the changes in the form of the string or tex-ture spectra are relevant. The best-fit values of the pa-rameters can be found in Table III for PL alone, andfor PL with semilocal (SL) strings, Abelian Higgs (AH)strings and textures (TX).Table III shows that, as in previous results [9, 13], thedegeneracies of the cosmological parameters with respectto the
CMB data allow for a rather high value of f =0 . ± .
06, which corresponds to Gµ = [1 . ± . × − .In order to accommodate the defect contribution, otherparameters (most notably Ω b h , h , and n s ) get shiftedto higher values, as was the case for the Abelian Higgsstrings [13].Figure 10 shows the marginalized 2D likelihood distri-butions for the parameters Ω b h , h , and n s versus f .It is clear in the figure that there is ample space for anoticable value of f . In fact, it can be seen in the 1-D marginalized likelihood plot (Fig. 11) that the CMB data alone prefers to have semilocal strings at a 2- σ level.However, as mentioned before (see Table III), the best-fit values of Ω b h and h are high compared to the con-1cordance model: Non- CMB experiments for those twoquantities yield h = 0 . ± . HKP [62]) and Ω b h = 0 . ± . BBN [63]). If weinclude these two non-
CMB experimental values into our
MCMC scheme (as gaussian likelihoods), we confirmthat the new set of data is more constraining also forsemilocal strings (as seen in Figs. 10 and 11). However,the 95% confidence level upper bound on f for semilocalstrings remains rather high, going from Gµ < . × − ( f < .
25) for the
CMB only case to
Gµ < . × − ( f < . f and Gµ we obtain forour analysis of semilocal strings and textures, togetherwith that for Abelian Higgs strings from Ref. [13]. Theparameter f depends mainly on the form of the de-fect power spectra, and we see that the data allow for ahigher contribution from semilocal strings (and textures)than from Abelian Higgs strings. Besides, the valuesof Gµ are also higher for semilocal strings (and tex-tures) than for the Abelian Higgs strings. Therefore, theexpectation that the constraints on the Abelian Higgsstring tensions would be alleviated by semilocal stringsis confirmed: the 95% upper bound limit for semilo-cal strings is Gµ < . × − for CMB + BBN + HKP ( Gµ < . × − for CMB only) whereas for AbelianHiggs string it is
Gµ < . × − for CMB + BBN + HKP ( Gµ < . × − for CMB only).Figure 12 shows the temperature power spectrumand the bb polarization spectrum for semilocal strings,Abelian Higgs strings and textures, normalized by usingthe upper bounds obtained from performing the MCMC likelihood analysis for both
CMB and non-
CMB ( BBN and
HKP ) data. In the temperature power spectrum fig-ure it can be seen that due to the different shapes of thecurves, even though the textures are allowed the high-est level of power at ℓ = 10, the semilocal and AbelianHiggs strings go up to higher peaks. The signal is high-est for textures for ℓ .
60, for semilocal strings for60 . ℓ .
250 and then for Abelian Higgs for 250 . ℓ . Onthe other hand, for the bb polarization, the normaliza-tion makes the trend of the power coming from the threecases just the reverse of the temperature case: textureshave the highest peak, followed by semilocal strings andthen Abelian Higgs strings. Needless to say, the normal-ization does not change the value of ℓ at which each casepeaks, and we find that textures peak first, then semilo-cal strings, and then Abelian Higgs strings.It is worth mentioning that, in all cases studied, thevalue for the spectral index n s is higher than the one ob-tained for cases without defects; and n s = 1 is not ruled −4 −3 −2 −1 l TTBB l ( l + ) C l / π [ µ K ] FIG. 12: Temperature power spectra (top) and bb polariza-tion power spectra (bottom) for semilocal strings (solid red),compared to Abelian Higgs strings (dashed black) and tex-tures (dot-dash blue). The curves are normalized using the95% confidence level upper bound from a MCMC analysisusing
CMB , BBN and
HKP data.
CMB only
CMB + BBN + HKP
Model f Gµ n s f Gµ n s SL 0 .
25 2 . × − .
09 0 .
14 2 . × − . .
33 2 . × − .
14 0 .
16 1 . × − . .
17 1 . × − .
06 0 .
10 0 . × − . f , Gµ and n s for PL+X,using CMB only, and using
CMB + BBN + HKP . out (see table IV). Therefore, when comparing differ-ent cosmological models, one should also consider the(scale-invariant) Harrison–Zel’dovich (HZ) model with 5parameters (Ω b h , h , τ , Ω m h , A s ) and a fixed n s = 1.Table V shows the goodness of fit for the concordancePL model, as well as for the HZ model, when AbelianHiggs strings, semilocal strings and textures are added;for both CMB only and
CMB + BBN + HKP data. Inmost cases, the inclusion of the defects makes the fit bet-ter, with semilocal strings obtaining marginally the bestfit. This is not too surprising for the cases where defects See Ref. [74] for a discussion about n s in light of WMAP3 froma Bayesian model selection point of view. CMB only
CMB + BBN + HKP
Model ∆ χ ln( E ) ∆ χ ln( E )HZ 6 . − . ± . . − . ± . − . . ± . − . . ± . − . . ± . − . − . ± . − . . ± . . − . ± . − . − . ± . − . − . ± . − . . ± . − . . ± . − . . ± . − . − . ± . are added to the PL model, because we are adding an ex-tra parameter to the PL model, and of course obtaininga better fit. But the models with HZ + defects have thesame number of parameters as the PL model (6 parame-ters), and surprisingly, the fit is better when considering CMB data only.In order to compare different models, we use Bayesianevidence values [75]. The evidences are calculated us-ing the Savage–Dickey method [76, 77] with flat priorsof 0 < f < . < n s < .
25 (Table V). Inthe case of
CMB data only, adding a defect contributionto the PL model makes little change to the evidence,whereas HZ+defects models have a significant higher ev-idence (and semilocal strings are marginally better thantextures or Abelian Higgs strings). Therefore, taken atface value, HZ+SL strings is the six-parameter modelwhich fits the data best, though the others are not con-vincingly worse. With the non-
CMB data added, theevidences get reduced, but not to the level to concludethat a string contribution is ruled out.This type of comparison will have some dependenceon the choice of prior used for the defect component (theother priors are less important as they are shared by themodels). Our imposition of a prior on f is just oneout of several plausible options. A prior on µ might bemore appropriate but is hard to formulate; for instancea logarithmic (Jeffreys) prior on µ would put most of theprior weight on models with negligible defect contribu-tion and hence leave the data unable to distinguish them(the same situation occurs with the inflationary tensorcomponent – see Ref. [74]). In any event, we can safelysay that present data are not good enough to exclude adefect contribution, and leave open the tantalising possi-bility that such a contribution may be a significant one. VI. DISCUSSION
In this article we report on the first ever semilocalstring simulations to predict
CMB power spectra. Thishas been possible because of recent improvements in sim-ulating field theoretical defects, as there is no Nambu–Goto type approximation for semilocal strings. The case of Abelian Higgs model strings is much better known,so we compared the semilocal string predictions to theAbelian Higgs string ones, trying to minimize the dif-ferences in the approaches. Since semilocal strings wereexpected to share properties with both an Abelian Higgsstring prediction and a texture prediction, we also pro-duced a set of simulations for texture-type defects to com-pare all three defect contributions.Our first conclusion is that the amplitude of the spectrafor the semilocal strings is indeed lower than in AbelianHiggs models (for a given energy scale); therefore, thenormalization at ℓ = 10 to match the WMAP value givesa higher value for semilocal strings Gµ = 5 . × − than for Abelian Higgs strings Gµ = 2 . × − . More-over, the form of the power spectra for semilocal stringsallows for their fractional contribution to the tempera-ture power spectra at multipole ℓ = 10 to be higher thanthe one for Abelian Higgs strings: the 95% confidenceupper bounds are f < .
25 ( f < .
14) for semilo-cal strings for
CMB only (
CMB + BBN + HKP ); whereas f < .
17 ( f < .
10) for Abelian Higgs strings.These two ingredients contribute to allow a highervalue of Gµ for semilocal strings when CMB data (andalso when
HKP and
BBN data) are taking into ac-count:
Gµ < . × − ( Gµ < . × − ) for semilo-cal strings for CMB only (
CMB + BBN + HKP ); whereas
Gµ < . × − ( Gµ < . × − ) for Abelian Higgsstrings. Thus, if a promising high-energy physics infla-tion model faces a problem of creating Abelian Higgscosmic strings at a scale that it is too high for the CMB ,a possible resolution could be to extend the model topredict semilocal strings instead. This weakens the Gµ constraint by a factor of nearly three, which may give thetheory extra breathing space.As in the case of textures [9] and local strings [11,13], the best fit for the cosmological parameters whentaking into account semilocal strings gives a rather highspectral index n s , comparing to the concordance model.For CMB only data, the preferred value of n s is higherthan 1 ( n s = 1 . ± . BBN and
HKP dataare included n s = 0 . ± .
02. This is in contrast to theAbelian Higgs data, where the
CMB data alone predict n s = 1; but the result is very similar when BBN and
HKP data are included [13]. This result also alleviatespressure on high–energy physics models (such as D-termand F-term hybrid inflation) which predict defects andan n s close to unity.As n s = 1 is close to our best-fit values, we studiedhow well a combination of Harrison–Zel’dovich plus de-fects would fit the data, comparing it to the concordancemodel. Using goodness of fit and Bayesian Evidence cri-teria, we find that models with defects are not ruledout (even when BBN and
HKP data are included), andshould be considered as competitive models. In fact, tak-ing into account only
CMB data, there is significant ev-idence in favour of a Harrison–Zel’dovich plus semilocalstrings model.We have not included large-scale structure datasets3into our analysis for two reasons. Firstly, perturbationsfrom defects are non-linear and non-Gaussian as soonas they are created [9], and so it is not clear that thecommonly-used model [78] for non-linear corrections tothe galaxy power spectrum applies. Secondly, recentanalyses [79, 80] show scale-dependent bias even on rela-tively large scales (0 . < k/h Mpc − < . bb polar-ization peak (normalized using the 95% confidence levelupper bound on f ) is of the same order of magnitudefor Abelian Higgs strings, semilocal strings and textures,though the position at which the maximum occurs varyfrom case to case. A detection of a bb signal on thesescales could be an indicator of the existence of cosmicdefects [15].In the present work we have studied only BPS semilo-cal strings, but we will in a future work extend the anal-ysis to lower values of β , which would arise in F-term su-persymmetric models (or even in P-term models [81, 82]).The behaviour of semilocal strings does depend stronglyof β [40]: the lower the value of β the more the semilocalstrings behave as Abelian Higgs strings. Tracking howthe CMB predictions change with respect to β will giveus insight into how the different components in a stringnetwork contribute to the creation of anisotropies.There is also a whole new class of strings coming fromstring inflation: ( p, q )-strings [6, 7, 8]. Even thoughthere is some work about the networks of these defects[83, 84, 85, 86], basic properties, such as scaling, are stillnot really understood. Provided some sound CMB pre-dictions from these strings are calculated, it would beinteresting to compare the results with those of AbelianHiggs or semilocal strings.Ultimately, one would like to be able to discriminatebetween all the different defect types, be it Abelian Higgs,semilocal, textures of ( p, q )-strings. As we have shownhere, the predictions do look similar, but there are differ-ences, e.g., the aforementioned position of the bb peak.The question is whether these differences are strongenough, or whether future data will be good enough, topinpoint whether defects do appear and if so, which type.In a future publication we will address this issue. Acknowledgments
We thank M. Nitta for pointing out Ref. [49] andK. Sousa for discussion. We acknowledge support fromPPARC/STFC (N.B., M.H, A.R.L.), the Swiss NSF(M.K.), Marie Curie Intra-European Fellowship MEIF-CT-2005-009628 (J.U.). This work was partially sup-ported by Basque Government (IT-357-07), the SpanishConsolider-Ingenio 2010 Programme CPAN (CSD2007- 00042) and FPA2005-04823 (J.U.). The simulationswere performed on cosmos , the UK National Cosmol-ogy Supercomputer, supported by SGI, Intel, HEFCEand PPARC.
APPENDIX A: TEXTURE SIMULATIONS
In this work we have compared the
CMB predictionsto those from the Abelian Higgs model and texture mod-els. For the latter we have performed a new set ofsimulations for the linear- σ global texture model com-paring the results additionally with previous ones fromRefs. [9, 35, 36, 87].We exploited the fact that a texture model can be ob-tained directly from Lagrangian Eq. (1) by setting thegauge field to zero, obtaining L = | ∂ µ φ | + | ∂ µ φ | − λ (cid:16) | φ | + | φ | − η (cid:17) = X i =1 ( ∂ µ ψ i ) − λ X i =1 ψ i − η ! (A1)where ψ i , ( i = 1 ...
4) are real scalar fields and φ = ψ + iψ , φ = ψ + iψ . l ( l + ) C l TT / ( G µ ) FIG. 13: The temperature power spectra from linear σ tex-tures (thick) obtained and used in this article; together withthe spectra from a non-linear σ model (thin) from [37]. Thedifferent components are scalar (dash-dot red), vector (dashedblue) and tensor (lower solid green). This is the Lagrangian we have simulated to obtain thefigures used in the main body if this article. Note thatit is not canonically normalized with respect to the realscalar fields, which becomes clear when compared to thenon-linear σ model: L = 12 X i =1 ( ∂ µ ˜ ψ i ) − ˜ λ X i =1 ˜ ψ i − κ ! (A2)4where in this case ˜ λ is a Lagrange multiplier fixing thefields onto the vacuum manifold.The normalization of the textures is usually given forthe Lagrangian canonically normalized for scalar fields as[53] ε = 4 πGκ . (A3)Comparing both Lagrangians (A1) and (A2), it can beseen that κ = 2 η , so the normalization scheme for tex-tures used throughout this work Eq. (2) relates to thatof Ref. [53] by ε = 4 Gµ (A4)Note that Ref. [35] uses another definition of ε , bigger bya factor 2 π . The procedure followed to obtain the CMB predictionsfrom textures was exactly the same as for the semilocalcase (and for the Abelian Higgs case). The results can beseen in Fig. 6. Earlier work on global textures [41, 88, 89]represented important steps towards the formalism weare using here. However, they neglected decoherence andso overemphasized the presence of the acoustic peaks.We also compared the texture predictions we obtainin our simulations with those of a non-linear σ model bysome of the authors of this article [37] in Fig. 13. Thosesimulations were performed using different parameters;e.g., smaller lattices (256 ) and smaller number of eigen-vectors used to obtain the CMB power spectra. In anycase, the agreement with our linear- σ model is very good,which makes our comparison of texture with semilocaland Abelian Higgs strings more widely valid. [1] A. Vilenkin and E. P. S. Shellard, Cosmic Strings andOther Topological Defects (Cambridge University Press,Cambridge, U.K., 1994).[2] M. B. Hindmarsh and T. W. B. Kibble, Rept. Prog. Phys. , 477 (1995), hep-ph/9411342.[3] D. H. Lyth and A. Riotto, Phys. Rept. , 1 (1999),hep-ph/9807278.[4] R. Jeannerot, J. Rocher, and M. Sakellariadou, Phys.Rev. D68 , 103514 (2003), hep-ph/0308134.[5] M. Majumdar and A. Christine-Davis, JHEP , 056(2002), hep-th/0202148.[6] S. Sarangi and S. H. H. Tye, Phys. Lett. B536 , 185(2002), hep-th/0204074.[7] E. J. Copeland, R. C. Myers, and J. Polchinski, JHEP , 013 (2004), hep-th/0312067.[8] G. Dvali and A. Vilenkin, JCAP , 010 (2004), hep-th/0312007.[9] N. Bevis, M. Hindmarsh, and M. Kunz, Phys. Rev. D70 ,043508 (2004), astro-ph/0403029.[10] M. Wyman, L. Pogosian, and I. Wasserman, Phys. Rev.
D72 , 023513 (2005), astro-ph/0503364 [Erratum-ibid. D (2006) 089905].[11] R. A. Battye, B. Garbrecht, and A. Moss, JCAP ,007 (2006), astro-ph/0607339.[12] A. A. Fraisse, JCAP , 008 (2007), astro-ph/0603589.[13] N. Bevis, M. Hindmarsh, M. Kunz, and J. Urrestilla,Phys. Rev. Lett. , 021301 (2008), astro-ph/0702223[Erratum in preparation].[14] U. Seljak and A. Slosar, Phys. Rev. D74 , 063523 (2006),astro-ph/0604143.[15] N. Bevis, M. Hindmarsh, M. Kunz, and J. Urrestilla,Phys. Rev.
D76 , 043005 (2007), arXiv:0704.3800 [astro-ph] [Erratum in preparation].[16] R. A. Battye, B. Garbrecht, A. Moss, and H. Stoica,JCAP , 020 (2008), 0710.1541.[17] A. A. Fraisse, C. Ringeval, D. N. Spergel, and F. R.Bouchet (2007), arXiv:0708.1162 [astro-ph].[18] L. Pogosian and M. Wyman (2007), arXiv:0711.0747[astro-ph].[19] J. Urrestilla, A. Achucarro, and A. C. Davis, Phys. Rev. Lett. , 251302 (2004), hep-th/0402032.[20] K. Dasgupta, J. P. Hsu, R. Kallosh, A. Linde, and M. Za-germann, JHEP , 030 (2004), hep-th/0405247.[21] A. Achucarro, A. Celi, M. Esole, J. Van den Bergh, andA. Van Proeyen, JHEP , 102 (2006), hep-th/0511001.[22] K. Dasgupta, H. Firouzjahi, and R. Gwyn, JHEP , 093(2007), hep-th/0702193.[23] T. Vachaspati and A. Achucarro, Phys. Rev. D44 , 3067(1991).[24] B. Allen, R. R. Caldwell, E. P. S. Shellard, A. Stebbins,and S. Veeraraghavan, Phys. Rev. Lett. , 3061 (1996),astro-ph/9609038.[25] B. Allen, R. R. Caldwell, S. Dodelson, L. Knox, E. P. S.Shellard, and A. Stebbins, Phys. Rev. Lett. , 2624(1997), astro-ph/9704160.[26] C. Contaldi, M. Hindmarsh, and J. Magueijo, Phys. Rev.Lett. , 679 (1999), astro-ph/9808201.[27] M. Landriau and E. P. S. Shellard, Phys. Rev. D69 ,023003 (2004), astro-ph/0302166.[28] G. R. Vincent, M. Hindmarsh, and M. Sakellariadou,Phys. Rev.
D56 , 637 (1997), astro-ph/9612135.[29] V. Vanchurin, K. D. Olum, and A. Vilenkin, Phys. Rev.
D74 , 063527 (2006), gr-qc/0511159.[30] V. Vanchurin, K. Olum, and A. Vilenkin, Phys. Rev.
D72 , 063514 (2005), gr-qc/0501040.[31] C. Ringeval, M. Sakellariadou, and F. Bouchet, JCAP , 023 (2007), astro-ph/0511646.[32] C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev.
D73 , 043515 (2006), astro-ph/0511792.[33] A. Albrecht, R. A. Battye, and J. Robinson, Phys. Rev.Lett. , 4736 (1997), astro-ph/9707129.[34] L. Pogosian and T. Vachaspati, Phys. Rev. D60 , 083504(1999), astro-ph/9903361.[35] U.-L. Pen, U. Seljak, and N. Turok, Phys. Rev. Lett. ,1611 (1997), astro-ph/9704165.[36] R. Durrer, M. Kunz, and A. Melchiorri, Phys. Rev. D59 ,123005 (1999), astro-ph/9811174.[37] N. Bevis, M. Hindmarsh, M. Kunz, and J. Urrestilla,Phys. Rev.
D75 , 065015 (2007), astro-ph/0605018 [Er-ratum in preparation].[38] A. Achucarro, J. Borrill, and A. R. Liddle, Phys. Rev. D57 , 3742 (1998), hep-ph/9702368.[39] A. Achucarro, J. Borrill, and A. R. Liddle, Phys. Rev.Lett. , 3742 (1999), hep-ph/9802306.[40] A. Achucarro, P. Salmi, and J. Urrestilla, Phys. Rev. D75 , 121703 (2007), astro-ph/0512487.[41] R. Durrer and Z. H. Zhou, Phys. Rev.
D53 , 5394 (1996),astro-ph/9508016.[42] A. Achucarro, A. C. Davis, M. Pickles, and J. Urrestilla,Phys. Rev.
D66 , 105013 (2002), hep-th/0109097.[43] M. Pickles and J. Urrestilla, JHEP , 052 (2003), hep-th/0211240.[44] A. Achucarro and T. Vachaspati, Phys. Rept. , 347(2000), hep-ph/9904229.[45] M. Hindmarsh, Phys. Rev. Lett. , 1263 (1992).[46] M. Hindmarsh, Nucl. Phys. B392 , 461 (1993), hep-ph/9206229.[47] R. A. Leese and T. M. Samols, Nucl. Phys.
B396 , 639(1993).[48] P. Laguna, V. Natchu, R. A. Matzner, and T. Vachaspati,Phys. Rev. Lett. , 041602 (2007), hep-th/0604177.[49] M. Eto et al., Phys. Rev. Lett. , 091602 (2007), hep-th/0609214.[50] E. Bogomol’nyi, Sov. J. Nucl. Phys. , 449 (1976).[51] N. Turok, Phys. Rev. D54 , 3686 (1996), astro-ph/9604172.[52] R. Durrer and M. Kunz, Phys. Rev.
D57 , 3199 (1998),astro-ph/9711133.[53] R. Durrer, M. Kunz, and A. Melchiorri, Phys. Rept. ,1 (2002), astro-ph/0110348.[54] M. Doran, JCAP , 011 (2005), astro-ph/0302138,URL .[55] W. H. Press, B. S. Ryden, and D. N. Spergel, Astrophys.J. , 590 (1989).[56] J. N. Moore, E. P. S. Shellard, and C. J. A. P. Martins,Phys. Rev.
D65 , 023503 (2002), hep-ph/0107171.[57] G. Vincent, N. D. Antunes, and M. Hindmarsh, Phys.Rev. Lett. , 2277 (1998), hep-ph/9708427.[58] J. Urrestilla, A. Achucarro, J. Borrill, and A. R. Liddle,JHEP , 033 (2002), hep-ph/0106282.[59] K. J. M. Moriarty, E. Myers, and C. Rebbi, Phys. Lett. B207 , 411 (1988).[60] N. Bevis and M. Hindmarsh,
LATfield Web page × .[62] W. L. Freedman et al., Astrophys. J. , 47 (2001),astro-ph/0012376.[63] D. Kirkman, D. Tytler, N. Suzuki, J. M. O’Meara, andD. Lubin, Astrophys. J. Suppl. , 1 (2003), astro-ph/0302006.[64] R. A. Knop et al. (The Supernova Cosmology Project), Astrophys. J. , 102 (2003), astro-ph/0309368.[65] G. Hinshaw et al. (WMAP), Astrophys. J. Suppl. ,288 (2007), astro-ph/0603451.[66] W. Hu and M. J. White, Phys. Rev. D56 , 596 (1997),astro-ph/9702170.[67] M. Kamionkowski, A. Kosowsky, and A. Stebbins, Phys.Rev.