Constraining topological defects with temperature and polarization anisotropies
Joanes Lizarraga, Jon Urrestilla, David Daverio, Mark Hindmarsh, Martin Kunz, Andrew R. Liddle
aa r X i v : . [ a s t r o - ph . C O ] N ov Constraining topological defects with temperature and polarization anisotropies
Joanes Lizarraga, Jon Urrestilla, David Daverio, Mark Hindmarsh,
3, 4
Martin Kunz,
2, 5 and Andrew R. Liddle Department of Theoretical Physics, University of the Basque Country UPV/EHU, 48080 Bilbao, Spain D´epartement de Physique Th´eorique & Center for Astroparticle Physics,Universit´e de Gen`eve, Quai E. Ansermet 24, CH-1211 Gen`eve 4, Switzerland Department of Physics & Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom Department of Physics and Helsinki Institute of Physics, PL 64, FI-00014 University of Helsinki, Finland African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, 7945, South Africa Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, United Kingdom (Dated: November 22, 2018)We analyse the possible contribution of topological defects to cosmic microwave anisotropies, bothtemperature and polarisation. We allow for the presence of both inflationary scalars and tensors,and of polarised dust foregrounds that may contribute to or dominate the B-mode polarisationsignal. We confirm and quantify our previous statements that topological defects on their own area poor fit to the B-mode signal. However, adding topological defects to a models with a tensorcomponent or a dust component improves the fit around ℓ = 200. Fitting simultaneously to bothtemperature and polarisation data, we find that textures fit almost as well as tensors (∆ χ = 2 . ℓ . The 95%confidence upper limits on models combining defects and dust are Gµ < . × − (Abelian Higgsstrings), Gµ < . × − (semilocal strings) and Gµ < . × − (textures), a small reduction onthe Planck bounds. The most economical fit overall is obtained by the standard ΛCDM model witha polarised dust component.
I. INTRODUCTION
The recent detection of B-mode polarisation [1–3]opens a new avenue for constraining models of the earlyUniverse at very high energy. The claim by the BICEP2collaboration that the B-mode polarisation on large an-gular scales [3] is caused by primordial inflationary tensormodes has generated great excitement, and stimulatedthe search for other possible B-mode polarization sourcessignalling new physics, such as cosmic defects [4, 5], self-ordering scalar fields [6] or primordial magnetic fields [7].However, it has subsequently become apparent that con-ventional astrophysics can plausibly account for the en-tire observed B-mode signal with polarised dust emission[8, 9]. At the very least it is clear that dust contamina-tion must be explored alongside any proposed primordialcontribution.In a previous paper [4] we showed that the predictedspectra from defects had the wrong shape to entirelyexplain the observed B-mode signal at low multipoles,although a good fit could be obtained in combinationwith inflationary tensors. However, we did not considerthe possibility of foreground contributions to the polar-isation, nor did we analyse the BICEP2 data in combi-nation with other cosmic microwave background (CMB)datasets. In this article we complement our previous pa-per by providing a comprehensive analysis of the defectcontribution to the microwave anisotropies, both in tem-perature and polarisation, allowing all three of the abovesignal sources, i.e., inflationary gravitational waves, dust,and cosmic defects.Cosmic defects produce B-mode polarization throughboth tensor and vector modes (see e.g. Refs. [10–15] forreviews). The relative proportions of scalar, vector and tensor perturbations are essentially fixed for a given typeof defect, so a constraint on one of the modes will implyconstraints on the others. It is worth noting that eventhough defects are highly constrained via the CMB tem-perature anisotropies [16–22], they can still contributeimportantly to the B-mode polarization.In our analysis we study three types of cosmic defects:Abelian Higgs strings [23], O(4) global textures [24], andsemilocal strings [25–28]. Other defect models exist, suchas self-ordering scalar fields, global monopoles, and globalstrings. However, with the three types of defects underconsideration we are able to obtain a global view of theinterplay between cosmic defects and the other signals,and can also study where the differences between the dif-ferent defect predictions are important. The imprints ofdefects on the temperature and polarization power spec-tra are qualitatively similar [12, 29–35], though there areimportant quantitative differences.In the next section we describe the defect models wehave considered. In Section III we describe the methodol-ogy, cosmological models and datasets that we have used.The results of the analysis are reported in Section IV, andin the final section we present conclusions and discussion.
II. CMB SPECTRA FROM DEFECTS
The cosmic defects that are most strongly motivatedby particle physics, as they arise from spontaneously-broken gauge symmetries, are cosmic strings [10, 11, 14,36]. They are predicted to form in many high-energyinflationary models [37–45]. Our two other examples ofcosmic defects, textures and semilocal strings, arise whenspontaneously-broken global symmetries are present.The perturbation power spectra of topological defectscan be calculated from numerical simulations of an un-derlying field theory in an expanding cosmological model[29, 32–34]. Cosmic string spectra have been calculatedfor the Abelian Higgs model [29–31], textures in an O(4)non-linear [12, 32, 33] or linear [34] σ -model, and semilo-cal strings in a U(1) theory with an extra SU(2) globalsymmetry [34].Another approach to model cosmic string networks, in-cluding those arising from superstring models, is basedon simulating directly the evolution of string-like objectsbased on the Nambu-Goto action [46–53]. There has beenvery intense work into understanding the loop generationand dynamics in this model [54–60]. However, there isno numerical simulation of the Nambu-Goto model cal-culating the full CMB temperature or polarization spec-tra, although there has been some work in that direction[61, 62]. An alternative to full Nambu-Goto type sim-ulations is afforded by the unconnected segment model(USM) [63–65], which introduces an extra layer of mod-eling and can be tuned to mimic not only Nambu-Gotostrings but also the behaviour of Abelian-Higgs stringnetworks, in which case it gives a good approximation tothe power spectra of the CMB anisotropies [18].There are also other approaches for the other defectsconsidered in this work. For textures, there is an analyticapproximation in the large N limit of the O( N ) non-linear σ -model [66]. In Ref. [34] a comparison betweenthe linear and non-linear σ -model can be found, showingthat they are very close. There is also a model describingthe evolution of semilocal strings [67, 68].The defect spectra used in this paper were calculated inRefs. [31, 34] using a modified version of CMBeasy [69],with the best-fit parameters of the WMAP 7-year analy-sis [70]. We do not vary the cosmological model used forcomputing the defect spectra, as the spectra change littlefor the allowed range of cosmological parameters. Sincethe defect contribution is sub-dominant in the temper-ature power spectrum, the resulting inaccuracies in theparameter posteriors are insignificant.Figure 1 shows the power spectra obtained from thefield theoretical simulations of Abelian Higgs strings(AH) [31], semilocal strings (SL), and textures (TX) [34],for the temperature and B-mode polarization spectra,normalised to the
Planck temperature power spectrumat ℓ = 10. There are important differences between thepower spectra obtained from defects or from inflation.Defects produce scalar, vector and tensor perturbationsin proportions which are fixed for a given defect model,while in inflationary models vector modes are absent andthe tensor contribution can vary almost independentlyof the scalar, apart from the inflationary consistency re-lation [71]. In addition, defect-induced polarization issuppressed on large angular scales, as causality requirestheir fluctuations to be uncorrelated beyond the horizondistance at decoupling [6].The amplitude of the perturbations produced by de-fects is usually parametrised by the dimensionless num- ber Gµ , where G is Newton’s constant and µ = 2 πv ,where v is the expectation value of the canonically-normalised symmetry-breaking field, assumed complex. A parameter often used to quantify the contribution ofdefects to the power spectrum is f , which is the frac-tional contribution of defects to the total model tem-perature power spectrum at multipole ℓ = 10. Withthese definitions, and for small contributions from de-fects, f ∝ ( Gµ ) . The values of Gµ needed to fit the Planck data at ℓ = 10 (i.e. the value for which f = 1),and the Planck
95% upper bounds for Gµ and f , canbe found in Table I. Note that Gµ is calculated as thenormalization of strings needed to match the observed power at ℓ = 10, whereas the limit on f is the upperbound on the ratio of the power in strings to the totalpower in the best-fit model at ℓ = 10.CMB data already put strong constraints on the de-fect contribution in models which combine it with a pri-mordial inflationary power spectrum, mainly through theincreasingly accurate measurement of the temperature l ( l + ) C TT l / π l10 −3 −2 −1 l ( l + ) C BB l / π l FIG. 1: Temperature (TT) and B-mode polarization (BB)defect spectra, normalized to make the temperature spectramatch
Planck data at ℓ = 10. Different lines correspond totextures (solid red line), semilocal strings (dashed black line),and Abelian Higgs strings (dot-dashed blue line). For Abelian Higgs strings at critical coupling, µ is the energy perunit length. For a theory with a canonically-normalized scalarfield with expectation value κ , we have Gµ = πκ . The textureliterature uses a parameter ǫ , defined as ǫ = 4 πGκ [33]. Fordetails see the appendix in Ref. [34]. Gµ Gµ ( < f ( < × − . × − × − × − × − × − Gµ is the normalization of different defects tomatch the observed ℓ = 10 multipole value (i.e. to explain thefull temperature signal at that multipole, f = 1). The lasttwo columns show the 95% confidence upper limit obtained bythe Planck collaboration [22] for the
Planck + WP + High- ℓ dataset. l ( l + ) C TT l / π [ µ K ] l10 −3 −2 −1 l ( l + ) C BB l / π [ µ K ] l FIG. 2: Defect spectra normalised to the 95% upper limitsobtained using
Planck + WP + High- ℓ . Different lines cor-respond to textures (solid red line), semilocal strings (dashedblack line), and Abelian Higgs strings (dot-dashed blue line). power spectrum to higher and higher multipoles [16–22].Figure 2 shows the temperature and B-mode spectra forAH, SL and TX at the upper 95% level for defects ob-tained in Ref. [22]. Even though the shapes of the spectraare similar (see Fig. 1) the peaks are not exactly at thesame ℓ , and they fall off at different rates at high ℓ . Thecosmic string model (AH) has the slowest fall-off, and itsamplitude is the most tightly constrained by the temper-ature data. As a result, the possible B-mode contributionis the smallest at low ℓ (lower panel).As explained in Ref. [4], and as is also clear fromFig. 2, the shape of the power spectrum of the defectsis qualitatively wrong, and cannot give a good fit toBICEP2 data. Including the constraint from the tem-perature power spectrum is likely to make the fit evenworse. A similar conclusion was obtained in Ref. [6] for self-ordering scalar fields, which is understandable sinceself-ordering scalar fields are closely related to the O(4)model under study here.An apparently contradictory conclusion was obtainedin Ref. [5], where the BICEP2 data was fitted to theUSM, allowing the inter-string distance parameter tovary. They found that a string-only model differed in χ by only 2.65 from the best-fit model with primordialtensor modes, albeit for inter-string distance values largerthan the causal horizon at decoupling. It was suggestedthe model spectrum was representative of global stringsor textures. Numerical simulations of textures do showthat they have a larger correlation length than the otherdefects, and hence a B-mode peak at lower ℓ , but it is alsoapparent from a comparison of Fig. 2 to Fig. 1 of Ref. [5]that the shape of the texture power spectrum at low ℓ is not accurately modelled by the USM. For example, at ℓ = 70, the best-fit USM spectrum is approximately 50%higher than the texture spectrum, and twice as high at ℓ = 40, which will tend to make the texture spectrum aworse fit to the data.We will see (Table II) that, when fitting BICEP2 dataonly, the ∆ χ between the O(4) texture model and pri-mordial tensor modes is 5.0, significantly larger than theUSM best-fit value. We will also see that textures cancombine with primordial gravitational waves to improvethe fit to the BICEP2 data, as they help with the pointsat ℓ &
200 which are above the lensing signal [4–6].Note that while a super-horizon inter-string distance isphysically questionable, it was argued [5] that one couldappear in models where the string-forming phase transi-tion happens during inflation, leading to the delayed on-set of scaling in the string network. However, an analysisof the delayed scaling model which takes into account thedynamics of the inter-string distance finds that a string-only model described by the USM is not a good fit [72].
III. MODELS AND METHODOLOGY
We perform a set of parameter estimations for modelswhere defects coexist with other sources of B-mode po-larisation, namely inflationary gravitational waves, dustand lensing. The lensing signal is always present, andwas recently detected by POLARBEAR [1, 2]. However,the extra constraining power of the POLARBEAR datais weak, and for simplicity we do not include it in ouranalysis.In order to reliably explore the parameter space weperform Markov Chain Monte-Carlo (MCMC) runs withthe publicly-available
Monte Python code [73, 74], whichuses
Class [75, 76] as its Boltzmann equation solver forthe inflationary component of the power spectrum. Wecompare our predictions to the following CMB datasets: • Planck +WP:
Planck ℓ andhigh- ℓ ), including WMAP 9 year [78] low- ℓ polari-sation data. • High- ℓ : SPT [79, 80] and ACT [81]. • BICEP2: BICEP2 BB polarization data [3].The likelihoods are the official codes provided by eachexperiment. The basic inflationary ΛCDM model, the “Power-Law”( PL ) model, is represented by the following set of param-eters: { Ω c h , Ω b h , τ, H , A s , n s } (1)where Ω c h is the physical cold dark matter density, Ω b h is the baryon density, τ is the optical depth to reioniza-tion, H is the Hubble constant, A s is the amplitude ofthe scalar spectrum, and n s its spectral index.We add a number of extra ingredients to the PL model, sometimes by themselves, sometimes in combi-nations. Our main extra ingredient is given by topolog-ical defects parametrized as ( Gµ ) , for each of the threemodels explained in the previous section (Abelian Higgscosmic strings AH, textures TX, or semilocal stringsSL). Another parameter describing an extra ingredientis r which parametrises the amount of inflationary grav-itational waves through the tensor-to-scalar ratio (at k = 0 .
002 Mpc − ). Scalar perturbation quantities arealso specified at a pivot scale k = 0 .
002 Mpc − .The BICEP2 collaboration included also the runningof the scalar spectral index α s in order to improve theagreement between the BICEP2 and Planck data [3]. Al-though several papers [82–84], showed that there is noworrying tension between BICEP2 and
Planck data, wenevertheless also study the impact of α s here.As mentioned above, the observed B-mode polariza-tion signal may have a contribution from polarised dustemission [8, 9]. We characterized this B-mode channelby A dust 3 , using the dust model proposed by the Planck collaboration [85]: C BB, dust ℓ = A dust ℓ − . (2)Our models are constructed using those building blocks,starting from the models with just one extra ingredient,and moving to more complex models where several addi-tional ingredients are present simultaneously. Although there may be minor differences between the likelihoodused by the BICEP2 collaboration in their publication [3] andthe public version [82], we expect that this is not important forour main conclusions. Another parametrisation of dust is used in the literature, givenby ∆ BB , which is related to ours via∆ BB, dust ,ℓ = ℓ π C ℓ = A dust π ℓ − . IV. RESULTS
In this section we present the results from fitting dif-ferent combinations of datasets with various cosmologicalmodels.First, in subsection IV A we fit CMB data with our ba-sic model ( PL ) with defects Gµ , with inflationary tensormodes r , and with both r and Gµ . The CMB data chosenare the BICEP2 data alone (for which we only fit for theB-mode spectrum), or all the CMB data. We also con-sidered the case where the data used did not include theHigh- ℓ data, but the results for both these two choicesof data (with and without High- ℓ ) were identical, so weonly show the parameter constraints with all the CMBdata. In subsection IV B we consider a model where therunning of the scalar spectral index α s is also free. Thelast case, subsection IV C, corresponds to models whichinclude a dust contribution as described above.The results showed in the Tables in the subsequentsections state only the relevant parameters for the givencase. In all cases flat parameter priors were used,in the case of defects the prior being flat in ( Gµ ) which is proportional to the fractional defect contribu-tion to the power spectra f . The prior ranges were0 < ( Gµ ) < < A dust / ( µK ) < .
75. Allother parameters, including foreground parameters withthe exception of the new polarised dust amplitude, weremodelled as in the
Planck collaboration papers [77, 86].
A. Primordial tensor modes and defects
We begin the analysis by extending the results of ourprevious paper [4] with quantitative statements. All re-sults, using only BICEP2 BB data and using the fullCMB set, can be found in Tables II and III respectively.The structure of the tables is the following: on the left weshow results from chains containing defects; on the right-hand side, we show results from models without defects,included here as reference values.The values in Table II, especially best-fit likelihoods,show that the fit is rather poor, as suggested in Ref. [4].Actually, for a model with only one extra component,none of the defect models ( PL + Gµ ) provides a fit thatis comparable to the model including only inflationarygravitational waves ( PL + r ), although the texture fit isonly moderately worse.We then fit the BICEP2 data with a model which con-tains both defects and gravitational waves ( PL + r + Gµ ),in order to assess whether defects could assist tensormodes. As mentioned before, at low ℓ defects cannotexplain the power measured. Nevertheless, defects peakat higher ℓ , which might help to fit those points that lieabove the lensing curve. The fit is improved (the like-lihood is better), although it should be noted that thislast model has 2 extra ingredients.As a next step we use the full CMB dataset ( Planck +WP + High- ℓ + BICEP2) and include the contributions Dataset BICEP2 (only BB )Model PL + Gµ PL + r + Gµ PL + r Param AH SL TX AH SL TX - r - - - 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( Gµ ) . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . - − ln L max . . . . . . . Planck + WP + High- ℓ + BICEP2Model PL + Gµ PL + r + Gµ PL + r Param AH SL TX AH SL TX - n s . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . r - - - 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( Gµ ) . +0 . − . . +0 . − . . +0 . − . < . < . < .
74 - f . +0 . − . . +0 . − . . +0 . − . < . < . < .
035 - − ln L max . . . . . . . to temperature and polarization (both E- and B-modes)from the different ingredients. If we compare models withonly one extra ingredient, we find that PL + Gµ [AH] fitsthe data quite poorly, whereas PL + r , PL + Gµ [TX] and PL + Gµ [SL] fit the data at roughly the same level, with r being the best model followed closely by TX.The Gµ constraint obtained from the full set of CMBdata is tighter than that from only BICEP2, especiallyfor the AH case. For this case, Planck bounds are strongenough to push the corresponding BB spectrum far be-low the BICEP2 data, in other words, the BICEP2 datado not constrain further the AH model in the com-bined
Planck + BICEP2 case. By contrast, temperaturebounds for SL and TX [22] leave their BB power spectraaround the values of BICEP2 (for high ℓ ), such that BI-CEP2 alone is able to put comparable constraints on thelevel of allowed defects. Our results are consistent withthe observation that accurately-determined B-modes candistinguish between different types of defects [87].The final possibility is the mixture of inflationary grav-itational waves and cosmic defects, PL + r + Gµ . Weobserve that there is less room for defects, and we onlyobtain upper 95% bounds. Roughly, the mean values inthe previous cases become 95% values now. Here againSL and TX do marginally better than AH strings, whichare not able to lower r . The reason has already beenmentioned: their contribution is so suppressed by theconstraints from the Planck data that the effect on ten-sor modes is negligible.
B. Running of the scalar index
Here we do not consider the BICEP2 data alone, sincethe running of the scalar spectral index affects mainlythe temperature channel. The results from a fit to thefull CMB dataset can be found in Table IV.The model which contains gravitational waves plusrunning is slightly preferred over PL + r + Gµ , possiblybecause if one also allows for running, r could take highervalues (e.g. see PL + r + α s of Table IV) and therefore abetter fit of B-modes. Another typical effect of including α s is that the scalar spectral index is pushed up, whichin principle implies more room for a defect contribution,though in this case it only affects AH strings. Runningalso changes the tilt of the temperature spectrum, caus-ing an unexpected anticorrelation between Gµ and f .It is worth noting that for the TX and SL cases, allow-ing for the running of the scalar index does not increasethe value of r ; it remains around the same values as forcases without α s . At the same time, allowing for defectsdoes not reduce the magnitude of the running. C. Dust
As discussed in Section III, we consider a dust modelproposed by the
Planck collaboration [85], given by C BB, dust ℓ = A dust ℓ − . (3)A similar model has been used by Mortonson and Sel-jak [8] and Flauger et al. [9] to examine the robustnessof the BICEP2 result’s interpretation as primordial, andwe follow their approach. Dataset
Planck + WP + High- ℓ + BICEP2Model PL + r + Gµ + α s PL + r + α s Param AH SL TX - n s . +0 . − . . +0 . − . . +0 . − . . +0 . − . α s − . +0 . − . − . +0 . − . − . +0 . − . − . +0 . − . r . +0 . − . . +0 . − . . +0 . − . . +0 . − . ( Gµ ) < . < . < .
71 - f < . < . < .
038 - − ln L max . . . . In Fig. 3 we show the contributions to the B-modepower spectrum from inflationary tensors, AH strings,textures, and dust, together with the data points fromBICEP2. The normalization is the one obtained fromfitting only the BICEP2 data to a model PL plus oneextra ingredient (see Tables II and V). Note that thelensing spectrum is added in all cases. In the figure it canbe seen that dust and r have more importance for lower ℓ ;therefore, in B-modes dust is in more direct competitionwith r than with defects.As in the previous section, we start analyzing BICEP2B-mode data. The first thing we notice is that a model in-cluding just dust as an extra ingredient is able to improvethe fit of PL + r . Moreover, if we consider a compositemodel ( PL + r + A dust ), the gravitational wave detectiondisappears and the fit does not improve (as previouslyfound in Ref. [8]). The best-fit value for r is at r = 0, i.e.a model with dust alone provides the best fit.Dust combined with defects gives better results thanwith gravitational waves. Note that in all cases (be itwith r or with any Gµ ) the dust contribution is at thesame level. However, dust lowers the amount of defectsto about half the one obtained using PL + Gµ , and moreor less at the level of PL + r + Gµ . This last set of
50 100 150 200 250 300 3500.010.020.030.040.050.06 l ( l + ) C BB l / π [ µ K ] l FIG. 3: B-mode spectra, including the lensing contribution,using best-fit normalization values given in Tables II and V,for tensors (thick solid black line), dust (thick dashed greenline), AH strings (dot-dashed blue) and textures (solid redline). BICEP2 data points are also shown. models does not improve the best-fit likelihood. Finally,in a model with all ingredients ( PL + r + Gµ + A dust ), wefind that a model with no dust is possible at one-sigma,and thus we quote an upper 95% confidence limit. Thisis due to the fact that dust and inflationary tensors canboth account for the low ℓ part of the spectrum, whereasdefects account for the higher ℓ .Considering the full CMB dataset, the picture isroughly the same. Dust does a very good job on its own,and any other combination improves only marginally thebest-fit log-likelihood. Once again, since the tempera-ture power spectrum is also constraining the defect con-tribution, we find only 95% upper bounds for defects.The bounds for SL and especially TX are tighter thanthose from Planck (see Table I). In all cases, the combi-nation PL + Gµ + A dust does better than the equivalent PL + r + Gµ .Note that the different mean values of A dust are dueto the differences in the lensing spectra due to differentcosmologies used in Tables V and VI. V. DISCUSSION AND CONCLUSIONS
In this paper we investigate quantitatively the impactof the recent detection of B-mode polarisation by the BI-CEP2 collaboration [3] on models containing topologicaldefects, extending the results of our earlier, more qual-itative, study [4]. In accordance with our earlier paper,we find that topological defects on their own are a poorfit to the signal and that we need an additional contribu-tion, either inflationary gravitational waves or polariseddust emission.When considering only the BICEP2 B-mode polariza-tion data, we find that the combination of topological de-fects and inflationary gravitational waves ( PL + r + Gµ )or topological defects and dust ( PL + Gµ + A dust ) slightlyimproves the fit over inflationary gravitational wavesalone ( PL + r ) or dust alone ( PL + A dust ). This is be-cause topological defects help to fit the BICEP2 datapoints at ℓ & Dataset BICEP2 (only BB )Model PL + Gµ + A dust PL + r + Gµ + A dust PL + r + A dust PL + A dust Param AH SL TX AH SL TX - - r - - - < . < . < . < .
22 -10 ( Gµ ) . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . - - A dust [ µK ] 0 . +0 . − . . +0 . − . . +0 . − . < . < . < .
25 0 . +0 . − . . +0 . − . − ln L max . . . . . . . . Planck + WP + High- ℓ + BICEP2Model PL + Gµ + A dust PL + r + Gµ + A dust PL + r + A dust PL + A dust Param AH SL TX AH SL TX - - r - - - < . < . < . < .
11 -10 ( Gµ ) < . < . < . < . < . < .
56 - - f < . < . < . < . < . < .
027 - - A dust [ µK ] 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . . +0 . − . − ln L max . . . . . . . . improve the fit over either contribution alone. We notethat there are hints in the cross-correlation between BI-CEP2 and Keck array data [3] that the central values ofthe B-mode power spectrum will decrease in the future,which will have the effect of more strongly constrainingthe defect contribution.The situation changes slightly when we consider thefull CMB dataset, consisting of Planck + WP + High- ℓ + BICEP2. In this case the texture model on its own( PL + Gµ [TX]) is only slightly worse than inflationarygravitational waves ( PL + r ), while cosmic strings ( PL + Gµ [AH]) are ruled out as the sole source of B-modes.Dust on the other hand is much better, so that PL + A dust is the globally-preferred model, and neither defects norinflationary gravitational waves are able to improve thegoodness of fit significantly.When considering parameter constraints on Gµ , wefind that BICEP2 on its own constrains the SL and TXmodels to roughly the same level as Planck data do. Inother words, the constraints obtained from BICEP2 alonefor SL and TX are as strong as the ones obtained from
Planck data. On the other hand,
Planck data constrainsAH strings more strongly than BICEP2. The reason isthat the combined temperature anisotropy dataset con-straints on defects come from ℓ & Gµ .The constraints become tighter when including a con-tribution from inflationary gravitational waves or dust.For the full CMB dataset and for a model with defectsand dust (see Table VII), we find Gµ < . × − for AH, Gµ at < f at < . × − . × − . × − Gµ and f obtainedusing PL + Gµ + A dust and the full CMB dataset. Gµ < . × − for TX and Gµ < . × − for SL (all at95%). These constraints are tighter than ones found bythe Planck collaboration for the temperature data alone[22] (especially for texture, see Table I), which shows theimportance that even the current B-mode polarisationdata has for constraining topological defects.
Note added -
While this paper was being refereed the
Planck collaboration submitted a paper [88] where theyupdate their dust model to C BBℓ ∝ ℓ − . (to be com-pared with equation (2)). We do not expect our results tochange significantly with this new power law. We testedthe case of PL + Gµ [AH] + A dust fitted to the full CMBdataset, and found that the upper 95% confidence limitin Gµ moves from 2 . × − to 2 . × − (see Table VII),which supports our expectations. Acknowledgments
This work has been possible thanks to the comput-ing infrastructure of the i2Basque academic network,the COSMOS Consortium supercomputer (within theDiRAC Facility jointly funded by STFC and the LargeFacilities Capital Fund of BIS), and the Andromeda clus-ter of the University of Geneva. JL and JU acknowledgesupport from the Basque Government (IT-559-10), theUniversity of the Basque Country UPV/EHU (EHUA12/11), MINECO (FPA2012-34456) and Consolider In-genio (CPAN CSD2007-00042 and EPI CSD2010-00064). DD and MK acknowledge financial support from theSwiss National Science Foundation. MH and ARLacknowledge support from the Science and Technol-ogy Facilities Council (grant numbers ST/J000477/1,ST/K006606/1, and ST/L000644/1). [1] P. Ade et al. (POLARBEAR Collaboration),Phys.Rev.Lett. , 021301 (2014), 1312.6646.[2] P. Ade et al. (POLARBEAR Collaboration), Astro-phys.J. , 171 (2014), 1403.2369.[3] P. Ade et al. (BICEP2 Collaboration), Phys. Rev. Lett. , 241101 (2014), 1403.3985.[4] J. Lizarraga, J. Urrestilla, D. Daverio, M. Hindmarsh,M. Kunz, and A. R. Liddle, Phys.Rev.Lett. , 171301(2014), 1403.4924.[5] A. Moss and L. Pogosian, Phys.Rev.Lett. , 171302(2014), 1403.6105.[6] R. Durrer, D. G. Figueroa, and M. Kunz, JCAP ,029 (2014), 1404.3855.[7] C. Bonvin, R. Durrer, and R. Maartens, Phys.Rev.Lett. , 191303 (2014), 1403.6768.[8] M. J. Mortonson and U. Seljak (2014), 1405.5857.[9] R. Flauger, J. C. Hill, and D. N. Spergel, JCAP ,039 (2014), 1405.7351.[10] A. Vilenkin and E. Shellard,
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