Detecting and distinguishing topological defects in future data from the CMBPol satellite
Pia Mukherjee, Jon Urrestilla, Martin Kunz, Andrew R. Liddle, Neil Bevis, Mark Hindmarsh
aa r X i v : . [ a s t r o - ph . C O ] N ov Detecting and distinguishing topological defects in future data from the CMBPolsatellite
Pia Mukherjee, ∗ Jon Urrestilla,
2, 1, † Martin Kunz,
3, 1, ‡ Andrew R. Liddle, § Neil Bevis, ¶ and Mark Hindmarsh ∗∗ Department of Physics & Astronomy, University of Sussex, Brighton, BN1 9QH, United Kingdom Department of Theoretical Physics, University of the Basque Country UPV-EHU, 48040 Bilbao, Spain D´epartement de Physique Th´eorique, Universit´e de Gen`eve, 1211 Gen`eve 4, Switzerland Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom (Dated: October 30, 2018)The proposed CMBPol mission will be able to detect the imprint of topological defects on thecosmic microwave background (CMB) provided the contribution is sufficiently strong. We quantifythe detection threshold for cosmic strings and for textures, and analyse the satellite’s ability todistinguish between these different types of defects. We also assess the level of danger of misidentifi-cation of a defect signature as from the wrong defect type or as an effect of primordial gravitationalwaves. A 0 .
002 fractional contribution of cosmic strings to the CMB temperature spectrum at mul-tipole ten, and similarly a 0 .
001 fractional contribution of textures, can be detected and correctlyidentified at the 3 σ level. We also confirm that a tensor contribution of r = 0 . σ , in agreement with the CMBpol mission concept study. These results are supported bya model selection analysis. I. INTRODUCTION
Cosmological probes are reaching a sensitivity wherethey are able to meaningfully constrain models of theearly Universe. Data compilations including Wilkin-son Microwave Anisotropy Probe (WMAP) data [1–5]already indicate that the observed inhomogeneities aremostly due to primordial adiabatic scalar perturbations[6, 7]. However, there remains room for low-level contri-butions from other sources such as cosmic defects [8–11]and primordial tensor perturbations, believed to be gen-erated by inflation alongside the scalars.These will be detected primarily from the signal theyproduce in cosmic microwave background (CMB) polar-ization, in particular the B-modes which have yet to bedetected and are a target for future probes. The possibledetection of B-modes produced by primordial gravita-tional waves (tensor modes) is often referred to as the‘smoking gun’ of inflation. The amplitude of the primor-dial gravitational wave background would provide strongconstraints on high-energy physics models of inflation,including some appealing models coming from string the-ory/brane inflation. String/M-theory may thus be con-strained by cosmological data: even though there existmodels [12] that give rise to a measurable tensor-to-scalarratio r , a fairly general prediction from string cosmol-ogy seems to be that the level of primordial gravitationalwaves, given by r , is very low ( r ≪ − , in some cases ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] ¶ Electronic address: [email protected] ∗∗ Electronic address: [email protected] even r ∼ − ). As emphasized by Kallosh et al. [13]it is hard to obtain an inflationary model coming fromstring theory which predicts measurably high primordialtensor modes. Thus, a future detection of r in the ac-cessible range r & − –10 − would present importantimplications for string cosmology.Another typical prediction of string cosmology is theproduction of cosmic (super)strings [14–16]. Indeed, cos-mic strings are a quite general prediction from high-energy inflationary models within the Grand UnifiedTheory (GUT) framework [17]. Strings produced afterinflation will also generate CMB anisotropies [18–20].Cosmic strings are not the only possible cosmic defectsin high-energy inflationary models: global monopoles,semilocal strings and textures are all examples of cosmicdefects that could be created after inflation and remainconsistent with the Universe we observe. Determiningthe nature of cosmic defects would provide invaluable in-formation on high-energy symmetry breaking.Defects produce scalar, vector and tensor perturba-tions. In contrast to the standard inflationary model,their vector perturbation modes do not die out since theyare seeded continuously by the defects. Moreover, thereare no free parameters that quantify the relative amountof scalar, vector and tensor perturbations independently,only an overall normalization factor; the relative amountof those perturbations is fixed for a given model. As de-fects produce vector and tensor modes, they create polar-ization B-modes directly [21–23]. It is interesting to notethat even though cosmic defects can contribute at mosta small fraction of the temperature perturbations, whichmust be mostly created by inflationary scalar modes tomatch the temperature anisotropy data, they can still bedominant in the B-mode spectra. Urrestilla et al. [24]have shown that Planck satellite [25] data would notsuffer from significant degeneracy between tensors andstrings. Thus, if Planck detects extra ingredients in theB-mode polarization spectra, its accuracy will be enoughto say whether the source of the spectra are primordialtensor modes or cosmic defects.CMBPol [26] is a proposed space mission that hashigher sensitivity than Planck and is specifically designedto target the polarization anisotropies. Here we performan analysis, partly along similar lines to Ref. [24], to de-termine both the detection threshold for different typesof signal in CMBpol data and the ability of the satelliteto distinguish between different defect types as well asprimordial tensors. We use both parameter estimationand Bayesian model selection tools to achieve this. Thedetection thresholds we find improve on those expectedfrom Planck over an order of magnitude, under realis-tic assumptions about foreground residuals and withoutassuming any level of delensing. II. DIFFERENT COSMIC DEFECTS.
High-energy physics models of inflation often give riseto cosmic defects after inflation ends. The most stud-ied ones are cosmic strings. These are one-dimensionalobjects that are extremely long (cosmic size) and yet mi-croscopic in width, which generate CMB perturbations.They can arise in field theories (for example, they areexpected in SUSY GUT models [17]) and can also bepresent as cosmic superstrings arising in fundamentalstring theories [14–16].Other kinds of defects are also possible; global defectscan be formed, such as global monopoles or textures [27–30]. Global monopoles do not present a problem in acosmological setup (contrary to their local counterparts,the ‘usual’ magnetic monopoles), because their scalingproperties are such that their energy density remains afixed small fraction of the total. Textures are also per-missable byproducts of cosmological symmetry breakingprocesses; indeed textures have been invoked in the CMBcontext as a possible explanation of the cold spot [31].In previous papers by some of us [23, 29, 30, 32] wecalculated the CMB power spectra (temperature and po-larization spectra) of cosmic strings, semilocal strings[33, 34] and textures from field theoretical simulations.There we showed that the spectra from all these de-fects are very different from those of primordial infla-tionary models (including tensors). We also showed thateven though there are similarities amongst them, thereare also differences, with the semilocal predictions lyingsomewhere between textures and strings.The zoo of possible defects is richer than that describedhere, but rather than performing an extensive compar-ison, we choose to focus on just two of them: cosmicstrings and textures. An exhaustive analysis would notgenerate further insight at this stage. Besides, the exactprediction for each kind of defect has its subtleties, andoften different calculational approaches result in slightlydiffering spectra [35]. Our aim is to verify and quantifyat what level the spectra created by two different kinds l ( l + ) C l / π [ µ K ] TT −4 −2 l ( l + ) C l / π [ µ K ] l BB FIG. 1: The CMB temperature and B-mode polarizationspectra for different components. The figure shows thetemperature spectra from inflationary scalar modes (blackdashed), inflationary tensor modes (black thin), cosmicstrings (gray dot-dashed), and textures (black thick). Thenormalizations shown are at the threshold detection levelsidentified later in this paper assuming the true model isknown: for the tensors this is at the r = 0 . f st10 = 0 . f tex10 = 0 . of defects, such as the ones shown in Fig. 1, can be dis-tinguished by CMBpol.We use the latest, more accurate, spectra derived fromfield theoretical simulations of cosmic strings and tex-tures [36]. Those spectra have been obtained by makingthe minimal possible computational changes in order tocapture the differences between those two defect types.The Abelian Higgs model used to model cosmic stringsand the linear σ -model for the textures were evolved us-ing the same discretization algorithms, the same typeof initial conditions, and the same procedure to calcu-late the power spectra (more details can be found inRefs. [30, 32]). As in our previous papers, we quantifythe amount of defects by f , which is the fractional con-tribution to the total TT power spectrum at l = 10.Observational data sets an upper limit on defects of afew percent: for strings f ∼ . f ∼ .
16 [30]. In turn, this parameter f can betranslated into a value of Gµ , with G the gravitationalconstant, and µ the string tension. For the Abelian Higgsmodel simulations used in this paper, f = 0 . Gµ ≃ × − .For textures, it is not natural to talk about a “string”tension, but we will use µ defined as 2 πφ , where φ isthe symmetry-breaking scale, to ease comparison (see theappendix of Ref. [30] for more details). III. METHODS
We simulate CMBPol data as described in the CMBPolmission concept study [26] in its high-resolution version.The treatment there follows the approach of Ref. [37]in modelling residuals from foreground subtraction andpropagating their effects into uncertainties in cosmologi-cal parameters.We consider a flat ΛCDM model with the same setof fiducial parameters as used in Ref. [26]: H =72 km s − Mpc − , Ω b h = 0 . c h = 0 . τ =0 . A s = 2 . × − , n s = 0 . r and/or the level of strings and textures (for whichwe quote the level of these defects relative to the totalTT power spectrum at multipole l = 10, which we la-bel as f st , tex10 ). We assume the inflationary consistencyrelation n t = − r/ k ∗ = 0 .
05 Mpc − . We assume that 80% of the sky canbe used for cosmological analysis.The effect of lensing in the inflationary spectra is in-cluded in the prediction of the signal. We work in theGaussian limit (ignoring mode correlations due to lensingor defects) where the likelihood takes its usual form [38],and ignore lensing due to defects. The task of detectingdefects through B-modes primarily amounts to detect-ing the excess variance in the C l from defects againstthis lensing contribution, which is more or less fixed bythe other spectra. Accordingly its recovery is approxi-mately limited by the cosmic variance of the lensing sig-nal, but this is fully modeled through the likelihood. Weuse the pessimistic dust model (third column of Table 10of Ref. [26]), and use an intermediate value for the levelof foreground residuals (not 1% or 10%, but 5%).We simulate instrumental data with the input sourcesbeing adiabatic primordial scalars plus either cosmicstrings or textures at different contribution levels. Wethen analyze that data in two ways, the first being aparameter estimation exercise and the second a modelcomparison.We use CosmoMC [39] to obtain parameter confidencecontours. Our fiducial model consists of a flat ΛCDMmodel with the parameters quoted above, and we includesome defects (one case with cosmic strings, the other withtextures). Then we try to fit that simulated data usingall the different possibilities that can be assembled fromthe different components: a model with strings; a modelwith tensors; a model with textures; a model with stringsand textures; with strings and tensors; with textures andtensors; and with strings, tensors and textures. This exercise allows us to infer the level of defects needed toclearly distinguish one from the other.For our model-level analysis, we compute the Bayesfactors of the set of models mentioned above, that is,models with one extra ingredient (strings, textures ortensors) and models with combinations of two of those orall three extra ingredients. In order to obtain the Bayesfactors we use the Savage–Dickey ratio [40, 41], and weconsider two sets of priors for these extra ingredients:flat linear priors and flat logarithmic priors. The relativeBayes factors of all these models will pinpoint which ofthose models is favoured and at which level.We have also analysed the data under different assump-tions than those described above. Ignoring foregrounduncertainties reduces parameter error bars by about afactor of two. If we turn off lensing (i.e. assume that theCMB can be perfectly delensed) then error bars decreaseby a factor of about seven. Thus, an order of magnitudeimprovement in f is still possible over the uncertaintiesdescribed in the rest of this paper.Previous works that considered constraints on stringsin CMBPol-like experiments include Refs. [21] and [42].However, the assumptions made there are different tothose considered here regarding, for example, polariza-tion sensitivities and lensing residuals. In addition,Ref. [21] uses the Unconnected Segment Model [43] forstring perturbations, which has a different Gµ for a given f unless special parameters are chosen [44]. IV. PARAMETER ESTIMATION ANALYSIS
We carry out two kinds of parameter estimation anal-yses. In the first, we fit assuming we already know thefiducial model used to generate the data. This enablesus to determine the sensitivity of CMBpol to the differ-ent defect signals, under the best possible circumstances.We then carry out an analysis in the presence of modeluncertainty, to assess the possible effect of mistaken as-sumptions.
A. Fitting with the correct model
If we assume we already know the correct model theanalysis is particularly straightforward. We first usethe fiducial model described above together with cos-mic strings only, with f st10 = 0 . Gµ ≃ × − ). Fitting for the same parameters aswent into the model, strings are detected at high signifi-cance, with f st10 = 0 . ± . Gµ ≃ × − ) would qualify for a 3 σ detec-tion, and hence this is the detection threshold for stringswith this CMBpol configuration.Repeating the analysis with textures, using a fiducialvalue of f tex10 = 0 . Gµ ≃ . × − ), we find that infitting for textures f tex10 = 0 . ± . ω c h f s t −3 H f s t −3 n s f s t −3 log(10 A s ) f s t −3 FIG. 2: The correlation between strings and cold dark matterdensity, the hubble parameter, the scalar spectral index, andthe amplitude of primordial perturbations. These are the pa-rameters with which strings are most correlated, at the -37%,34%, 28% and -24% levels respectively. Textures are less cor-related with cosmological parameters, the levels being -10%,20%, 9%, and -8% respectively.
The 3 σ detection threshold for textures is therefore a0.0005 fractional contribution to the CMB TT powerspectrum at l = 10 ( Gµ ≃ . × − ).The power spectra shown in Fig. 1 were normalized toindicate these detection thresholds. The distinct shapesof the spectra in both TT and BB are evident, with thedefect spectra more resembling each other than the ten-sors. B. Fitting with a range of models
In reality we do not know a priori which model is cor-rect, and indeed our primary interest is likely to be in de-termining the correct model. One should be concernedabout whether one might be able to draw conclusionsbased on the wrong model assumption, e.g. in the actualpresence of strings, instead fitting primordial tensors andapparently detecting r at some significance. We wish alsoto know whether or not such data fits are able to drawus towards the correct model conclusion. The parame-ter estimation approach of this section is complementedby the more robust model-level analysis we provide inSection V. −3 r 0 1 2 3x 10 −3 f −4 f −3 r 0 1 2x 10 −3 f −3 f FIG. 3: 1D marginalized likelihood for tensors, strings, andtextures, for the fiducial model with strings f st10 = 0 .
004 (toprow) and for the fiducial model with textures f tex10 = 0 . σ detection.
1. True model is strings
If we fit for tensors instead of strings, we do get a milddetection of r with r = 0 . ± . σ to 0.1103),the scalar spectral index (which goes down more than1 σ to 0.961) and H (goes down more than 1 σ to 71.8).Parameter correlations are shown in Fig. 2. If insteadwe wrongly fit for textures we get a strong detection oftextures with f tex10 = 0 . ± . H shift to 0.1102, 0.962 and 71.8 respectively) There istherefore a danger of being led astray through assump-tion of the incorrect cosmological model. The bias in thevalues for other parameters is a potential signal but theremay be no independent means of estimating them to thesame accuracy. It is therefore important to test differentmodel assumptions, and this motivates attempts to fitmultiple components.When fitting for strings and textures together, or forall of strings, textures and tensors, results are very simi-lar. Figure 3 shows the marginalized likelihoods for eachcomponent from a fit where all parameters are simultane- f f t e x −3 −4 f f t e x −3 −3 FIG. 4: The correlation between strings and textures in sim-ulated CMBPol data with strings (left panel) and textures(right panel), showing 68% and 95% confidence contours. ously varied, the upper panels showing a fiducial stringmodel. From these likelihoods we get f st10 = 0 . ± . f tex10 < . , . r < . , . ≈ l as textures, and at low l strings have a littlepeak where the tensors dip, making strings a little moredissimilar to tensors than textures). Table I summarizesthe uncertainties found under various assumptions.The correct component can also be sought based onthe quality of the fits, i.e. by looking at the best-fitand mean likelihoods achieved in the MCMC analysis.Strings+textures+tensors and strings alone lead to sim-ilar best-fit or mean log-likelihood, in each case beingat least 3 better than with textures alone and 10 bet-ter than tensors alone. We conclude that at this fiducialstring contribution, the model of tensors alone could bediscounted, and strings favoured over textures, thoughnot convincingly.We find that a slightly higher fiducial value of f st10 =0 .
003 gives a 4 σ detection ( f st10 = 0 . ± . σ threshold for identi-fying strings correctly in favour of these alternatives istherefore f st10 ≃ .
002 ( Gµ ≃ × − ).
2. True model is textures
If we fit for tensors instead of textures, we get a falsedetection r = 0 . ± . Model has δf st10 δf tex10 δr String 0 . − − String − . − String − − . . . ∗ − String 0 . . ∗ . ∗ Texture − . − Texture 0 . − − Texture − − . . ∗ . − Texture 0 . ∗ . ∗ . ∗ TABLE I: Standard deviation achieved when trying to fit thedata with a model with one, two or three extra components.In the string case the fiducial value is f st10 = 0 . f tex10 = 0 . ases are smaller because textures are less correlated withcosmological parameters (see caption of Fig. 2). If weperform the fit for strings instead of textures, strings re-ceive a false detection with f st10 = 0 . ± . f tex10 < . , . f st10 < . , . r < . , . l = 10 is not clearly de-tectable when fitting for these two additional parameters.Study of best-fit and mean likelihoods will again enablea ranking of models considered, with models involvingtextures preferred by about 3 (6) as compared to modelswith just strings (tensors).Further simulations show that f tex10 = 0 . σ detection of textures ( f tex10 = 0 . ± . f tex10 = 0 . σ detection of tex-tures ( f tex10 = 0 . ± . f tex10 ≃ .
001 ( Gµ ≃ . × − ). V. MODEL SELECTION ANALYSIS
Until we have uncovered the presence of defects, weare less interested in constraints on the defect parametersand more in the fundamental question whether there areany defects in the Universe, and if yes, which kind. Thisis a question of model selection rather than parameterestimation, and should be dealt with by computing Bayesfactors between the different models, including a modelwith no extra ingredients. In this section we will onlyconsider the fiducial string model.Before embarking on a model selection analysis, weneed to consider the priors that we want to place on theparameters. Here we will look at two different priors.In the first one, we assume that the prior is uniform in f in the interval [0 ,
1] for all extra contributions. Thisinterval is much wider than the precision of CMBPol,which will lead to a significant “Occam’s razor” factorthat the defect models need to overcome in order to befavoured against the no-defect baseline model. While thisprior makes sense when looking for a signal that may bepresent at some level in the C ℓ , it appears at least asnatural to impose a prior which assumes that the phasetransition in which the defects were generated happenswith equal probability at an arbitrary energy scales withsome cut-off. This leads to a prior that is uniform inlog Gµ for cosmic strings, or more generally uniform inthe logarithm of the amplitude. As limits we choose thatthe energy scales ( ∼ √ µ ) would range from the GUTscale ( ∼ GeV) to the SUSY breaking scale ( ∼ GeV). This in turns translates into values of log f ranging from −
52 (SUSY scale) to 0 (GUT scale). Ac-tually, log f = 0 corresponds to a situation where allthe CMB signal is coming directly from strings, and itis the absolute maximum number possible according tothe definition of f . As we show in the Appendix, ourBayes factors are only slightly changed when choosingother physical scales for our lower cut-off for the prior.For r there is not quite an equivalent to the symmetrybreaking scales of the universe since it is a ratio of ten-sors to scalars rather than on an absolute scale, so wejust used the same range as for f , i.e., r ranges from 1to 10 − in our logarithmic prior analysis.Having specified the priors, we are left with the tech-nical question of how to compute the Bayes factors. Onepossibility is to compute the model probabilities directlyusing for example nested sampling [45–47]. Here weinstead employ the Savage–Dickey (SD) density ratio[40, 41], since a model with a given kind of defects isnested within the simpler model without defects at thepoint f = 0. The Bayes factor in favour of the simplermodel is then just the value of the (marginalized and In this section we will refer to the model with no extra ingredientsas the ‘no defect’ model, meaning a model without strings ortextures, but also without tensors. normalized) posterior at f = 0 divided by the prior atthe same point. For our linear prior in the defect ampli-tude, the prior is always just equal to 1. We include anAppendix in which we describe the techniques employedin order to accurately obtain the normalized posteriorvalues and subsequent Bayes factors.For our analysis, we ran chains with a fiducial cos-mic string fractional amplitude of f st10 = 0 . → ‘s’ → ‘st’ as well as through‘no defects’ → ‘t’ → ‘st’. The Bayes factor from bothsequences must agree within the error bars. Indeed thisis the case for all results quoted in the paper. A. Analysis for linear prior
We first performed the analysis described above usinglinear priors for all the three extra components. Figure 5ashows the results from this exercise, in the shape of a cubewith each axis denoting the presence of strings, texturesand tensors respectively. The model in the lower left cor-ner of the cube is the ‘no defect’ model while the diago-nally opposite corner corresponds to ‘str’. The numbersgiven denote ln B , with positive values for models thatare favoured over the ‘no defect’ case. The fiducial stringamplitude was chosen so that there is strong evidence forthe presence of strings, which means that it is difficultto evaluate the SD ratio far out in the tail of the distri-bution in the ‘s’ case. The Appendix discusses how weevaluated SD ratios for these cases. We notice that theonly other model with a positive evidence is ‘t’, which isdue to the partial degeneracy of the strings and textures.The ‘r’, ‘rt’ and ‘rts’ models are significantly disfavoured.Given these results we would conclude that there isstrong evidence in favour of defects – indeed, this is aboutthe minimal string contribution for which CMBPol wouldbe able to make such a statement. We notice that in aparameter estimation context, the significance is 5 σ if weonly fit for the correct component (supposing we knowwhich that component is), and it is a borderline detectionwhen fitting for all three components (of order 3 σ ).As in the parameter estimation case, we would still notbe able to distinguish decisively between strings and tex-ture, although strings are favoured by a factor of roughly40 (∆ ln B = 3 . FIG. 5: Pictorial representation of the logarithm of the Bayes factors for different models, relative to a model with ‘no defect’.The lower left corner of the cube corresponds to the ‘no defect’ model, and the axes of the cube correspond to adding strings(s), textures (t) or tensors (r). Thus, the diagonally opposite corner correspons to a model with strings+textures+tensors.Figure a) depicts a model comparison with linear priors in f , whereas in figure b) we have used a logarithmic prior instead. B. Analysis for logarithmic prior
We also performed a model selection analysis usinglogarithmic priors for the extra parameters. ObtainingSD ratios for this case is even harder than for the lin-ear case, since it presents the additional difficulty thatthe models are not actually nested, since f = 0 isnot attainable (recall that the range for our priors islog f ∈ [ − , B ≈ B ≈ VI. CONCLUSIONS
Simulating data as per the CMBPol mission conceptstudy, propagating uncertainties due to foreground resid-uals, we find that the level of cosmic strings and tex-tures that can be detected and correctly identified (at3 σ ) by CMBPol is 0.002 and 0.001 of the total TT powerspectrum at multipole 10 respectively (correspondingly, Gµ ≃ × − for strings and Gµ ≃ . × − fortextures). Similarly a tensor fraction of 0.0018 shouldbe discernible. Contributions from strings and texturesare highly correlated with each other, so at lower lev-els the signal would be harder to attribute to one or theother conclusively. Tensors are not much correlated withstrings but are somewhat correlated with textures.We also performed a model selection analysis for a fidu-cial model containing cosmic strings. Using a flat prioron f , we found that a model with only strings is favoredover all models but the texture-only model is also betterthan a model without any defects. Models with severaltypes of defects are all strongly disfavored because of alarge “Occam’s razor” factor. This changes when takinga prior that is flat in log ( Gµ ). In this case, it is notpossible to rule out the presence of defects, and all mod-els containing strings are strongly favoured (with modelscontaining textures but no strings having an intermediateprobability).A CMBPol-like experiment, as has been proposed bothin the US and Europe, has the ability to illuminate uson important issues regarding high-energy physics in theearly universe, that we can only speculate about at thistime. It is roughly two orders of magnitude better (in f )than what we can achieve at this point with WMAP, andover an order of magnitude better than what the Planckmission will achieve. Appendix A: Posterior calculation for theSavage–Dickey density ratio
In this appendix we summarise the different techniquesused in this work to perform the model selection analy-sis. In both prior choices (linear and logarithmic) for thedefect contribution, we encounter challenges in obtainingan accurate normalized posterior. We will explain thosechallenges and describe how we overcame them.For both prior choices, the interval for the parametercharacterizing the string contribution is much wider thanthe precision of CMBPol, which leads to a significant“Occam’s razor” factor. For example, if the posteriorwas Gaussian with a variance of σ and the prior flat witha width of 1, then its normalization alone contributes afactor 1 √ πσ ∼ >
300 (A1)in favour of the simpler no-strings model, where we usedthe variance of the string contribution discussed earlier.In order to strongly support the presence of strings, weneed to overcome this factor as well as reach down toat least exp( − ≈ /
150 with the posterior. We there-fore need to have an accurate estimate of the posteriorover four to five orders of magnitude. A normal MCMCchain would need to be exceedingly long to reach that farout; we would effectively be counting only every 50,000-th sample! This problem can be alleviated by runningMC chains at higher temperatures (we used T = 2, and T = 4 where necessary) that probe the tails much bet-ter. This increases the computational cost of using theSD ratio, but not prohibitively, especially since we foundthat scaled versions of the T = 1 covariance matrix weresufficient to use for the proposal densities of the higher-temperature chains. For T = 1 we are not samplingfrom the desired probability density but from exp( − λ/T )where λ = − ln L with L is the likelihood. If we scaledthe Gaussian proposal distribution the same way, thenthe covariance matrix C at temperature T should bechanged to C = C ( T /T ). What we did in practicewas to increase the proposal scale in CosmoMC from 2 . . . √ T = 1 for model selec-tion and parameter estimation, we need to correct forthe temperature. This can be done through importancesampling, by adjusting the sample weights w i , w i ( T = 1) = w i ( T ) L ( T = 1) L ( T ) = w i ( T ) e − λ e − λ/T . (A2)Figure 6 shows four chains for different temperatures thatuse the correction given above. The resulting probabil-ity densities agree well in the high-probability peak, butthe high- T chains probe the low-probability regions muchbetter. (cid:0) (cid:1) (cid:2) (cid:3) (cid:4) (cid:5) (cid:6) (cid:7) log f st10 -7 -6 -5 -4 -3 -2 -1 FIG. 6: The marginalized probability distribution functionfor f st10 in the logarithmic prior case for four different tem-peratures: T = 1 (blue), T = 2 (cyan), T = 4 (magenta)and T = 8 (red) [the curves with lower temperatures end athigher f st10 ]. All chains agree in the high-probability regionnear log f st10 = − .
7, but only the two highest temperaturechains can probe the low-probability tail which is reached for f st10 → The logarithmic prior presents the additional difficultythat the models are not actually nested since f = 0is not attainable. However, we know from the previousdiscussion about parameter constraints that, e.g., a de-fect fraction of f = 10 − is completely undetectableby CMBPol and corresponds for all practical (althoughmaybe not for philosophical) purposes to a model withoutdefects. For this reason, the Bayes factor of the modelwith f = 10 − relative to the more general model witharbitrary defect contribution is the same as the one of amodel with f = 0. But the former model is nested inthe general model and allows us to compute the Bayesfactor with the help of the SD ratio.Additionally we do not want to sample all the waydown to very big negative values of log f in the chains(in our case log f = − f (in the example described in the main text, we used f = 10 − ). This cut-off is arbitrary as long as it isin the asymptotic region where the posterior has becomeflat since defects are no longer detected. In addition, it isbetter to choose it slightly lower than the value at whichwe evaluate the SD ratio in order to avoid edge effects.We proceed as follows: first we evaluate the posteriorat the chosen point (in our example, log f = − units,the width is ǫ = 6). Since we are in the region where theposterior is already flat, we add a stretch of width ∆ (inour example ∆ = 46 to reach log f = − q the value measured at the cut-off point inside the asymptotic region (e.g. q = 1 . × − in Fig. 6). Then, the normalized posterior actually has avalue of p = q/ (1 + q ∆). The prior is flat over the wholerange so that its value is 1 / (∆ + ǫ ) and hence the Bayesfactor is B = 1 + q ∆ q (∆ + ǫ ) . (A3)It is worth checking how much the Bayes factor changesfor different values ∆. Recall that ∆ basically gives usthe lower energy scale taken into account in the priorfor log f . For example, ∆ = 46 corresponds to SUSYbreaking scale, and ∆ = 50 would correspond to the elec-troweak scale. It is easy to verify that ln B (∆ = 46) ∼ .
09 and ln B (∆ = 50) ∼ .
01, so a wide range of lowercutoffs would give virtually the same results.Unfortunately there is one additional hurdle that needsto be overcome. In the model analysis we have per-formed, we often need to fit for several kinds of com-ponents: strings (s), textures (t) and tensors (r); andcombinations of them: ‘st’, ‘sr’, ‘tr’ and ‘str’ models. Inorder to compute SD ratios for models with more thanone component, we marginalize over all but one compo-nent. However, this marginalization should be done notover the limited range of f that we actually sample from,but over the full range (taking account the stretch ∆ inall extra components).We illustrate how this was performed in the concreteexample ‘st’ → ‘s’, where the simulated range of the pa-rameters is smaller than the full range both in f st10 andin f tex10 . First we get the weight of the full simulatedchain W simst , and the (normalized) pdf p simst of the ‘st’chain by marginalizing over all parameters except f tex10 .That would be enough if our model was nicely nestedand we did not have to add the stretch ∆. To accountfor the stretch, we consider the interval of unit width log f st10 ∈ [ − , −
5] of the ‘st’ chain. We calculate bothits weight W (1)st and its normalized pdf p (1)st , once againby marginalizing over all parameters except f tex10 . Notethat p (1)st should be the same as the pdf obtained from the‘t’ chains, since p (1)st is calculated virtually in the regionwith only textures (no strings). We verified that this wasthe case.We now have all the ingredients that are necessary tomarginalize over the full range: Let W ∆st = ∆ × W (1)st bethe estimated weight over all the ∆ stretch. Then thenormalized marginalized pdf for the full range is givenby p fullst = p (1)st W ∆st + p simst W simst W ∆st + W simst (A4)We still have to account for the ∆ stretch in f tex10 , butwe are just in the case tackled earlier in this section, andone just needs to apply Eq. (A3) to get the Bayes factor. Acknowledgments
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