Can topological defects mimic the BICEP2 B-mode signal?
We show that the B-mode polarization signal detected at low multipoles by BICEP2 cannot be entirely due to topological defects. This would be incompatible with the high-multipole B-mode polarization data and also with existing temperature anisotropy data. Adding cosmic strings to a model with tensors, we find that B-modes on their own provide a comparable limit on the defects to that already coming from Planck satellite temperature data. We note that strings at this limit give a modest improvement to the best-fit of the B-mode data, at a somewhat lower tensor-to-scalar ratio of .
The detection of low-multipole B-mode polarization anisotropies by the BICEP2 project Ade et al. (2014) opens a new observational window on models that generate the primordial perturbations leading to structure formation. The leading candidate to explain such a B-mode signal is primordial gravitational wave (tensor) perturbations generated by the inflationary cosmology. For a tensor-to-scalar ratio of around , these give a good match to the spectral shape in the region – , while falling some way short of the observed signal at higher multipoles for reasons yet to be uncovered.
An alternative mechanism of generating primordial B-modes is the presence of an admixture of topological defects (see e.g. Refs. Vilenkin and Shellard (1994); Hindmarsh and Kibble (1995); Durrer et al. (2002); Copeland and Kibble (2010); Hindmarsh (2011) for reviews). Many inflation scenarios, particularly of hybrid inflation type, end with a phase transition. Defect production at such a transition is natural and plausibly a sub-dominant contributor to the total temperature anisotropy. Many papers have used recent data to impose constraints on the fraction of defects, typically obtaining limits of a few percent contribution to the large-angle temperature anisotropies Wyman et al. (2005); Bevis et al. (2008); Battye and Moss (2010); Dunkley et al. (2011); Urrestilla et al. (2011); Avgoustidis et al. (2011); Ade et al. (2013a). The tensor and defect spectra were previously compared in Refs. Urrestilla et al. (2008a); Mukherjee et al. (2011).
An important question then arises: does the observed B-mode polarization confirm the existence of a primordial gravitational wave background due to inflationary dynamics in the early Universe, or could it instead be entirely due to the presence of topological defects? In this Letter we show that topological defects alone cannot explain the BICEP2 data points.
Ii B-mode constraints from BICEP2
As with inflationary tensors, a distinctive signature of topological defects lies in the B-mode polarization, where the signal is not masked by a dominant contribution from inflationary scalars. Figure 1 shows a comparison of cosmic microwave background (CMB) spectra predicted from inflation with those of cosmic strings as computed via field theory simulations111Strings can also be studied in the Nambu–Goto approximation, most recently in Ref. Blanco-Pillado et al. (2014). However, the shapes of the cosmic string CMB spectra are reasonably generic and can be understood from simple modelling Pogosian and Vachaspati (1999); Martins et al. (2004); Battye and Moss (2010). There are significant differences in other observational constraints: for a review see Ref. Hindmarsh (2011). by Bevis et al. Bevis et al. (2007, 2010), for a particular value of near the Planck upper limit Ade et al. (2013a) (where is the fractional contribution of defects to the temperature anisotropies at ). The scalar B-mode spectrum is the one inevitably produced by lensing of the scalar E-modes. In the B-mode channel the string spectrum has a quite different shape to the inflationary tensors, peaking towards smaller scales. Figure 2 shows the B-mode polarization spectra for several classes of defects (textures, semilocal strings, and Abelian Higgs strings Urrestilla et al. (2008b)), showing that they share the same general shape in the multipole range of interest. We focus on cosmic strings (using the Abelian Higgs model) as a specific example for the remainder of this work.
We first attempt to match the cosmic string B-mode spectrum to the BICEP2 data, showing the result in the lower panel of Figure 3. It is clear that the defect spectrum has the wrong shape, and could only match the low-multipole data at by substantially over-predicting the high multipole data (). In detail, we see that we need to generate the necessary power at , which in turn leads to a B-mode amplitude which is a factor of about 5 too large at higher .
In addition, matching the low-multipole data requires a fractional contribution to the total TT power spectrum at far larger than the maximum allowed by Planck Ade et al. (2013a), as shown in the upper panel of Figure 3. We show the defect contributions to the temperature spectrum as the blue-dotted curves, with the required contributions to match the B-mode polarization amplitude at as the highest blue-dotted curve (which corresponds to ). The solid back line is the best-fit CDM model, while the grey dashed line shows the sum of the string prediction with the Planck best-fit CDM model Ade et al. (2013b). The model in which strings match the B-mode polarization amplitude at is clearly incompatible with the temperature data. Allowing the parameters of the CDM model to vary does not help: the 95% upper limit from Planck is around 0.03 to 0.055 depending on the type of defect Ade et al. (2013a).
We can therefore immediately conclude that defects do not provide an alternative to inflationary tensors in explaining the observed data.
We can also use the B-mode data to constrain the contribution of defects to the total anisotropy in a scenario where both strings and inflationary gravitational waves contribute significantly, as anticipated in Refs. Seljak and Slosar (2006); Pogosian and Wyman (2008). In fact, because the strings contribute more substantially at higher multipoles than inflationary tensors do, a modest admixture of defects improves the fit to the BICEP2 data; as seen in Fig. 4 a string fraction of around 0.04 would explain the excess signal at (as an alternative to the more prosaic possible explanations of a foreground contribution or undiscovered systematic), while a fraction above about 0.06 is disfavoured. It is noteworthy that the first detection of the B-modes already gives a limit on defects which is competitive with that from the temperature spectrum. This conclusion can of course only strengthen if some or all of the BICEP2 signal turns out not to be cosmological.
If this detection of B-mode polarization is confirmed, then primordial gravitational waves appear to be a necessary addition to the standard cosmological model. However, the BICEP2 data points do not agree well with expectations at higher . It is intriguing that an admixture of topological defects appears able to improve the fit, while reducing the tensor-to-scalar ratio to . But precise quantitative statements for such a model, which would simultaneously include primordial tensors, defects, and perhaps also a running of the scalar spectral index, require a more careful numerical analysis.
In conclusion, we have shown that topological defects alone cannot explain the BICEP2 data points, and that B-modes already give a constraint on defects competitive with that from temperature anisotropies.
Acknowledgements.JL and JU acknowledge support from the University of the Basque Country UPV/EHU (EHUA 12/11), the Basque Government (IT-559-10), the Spanish Ministry (FPA2012-34456) and the Consolider-Ingenio Programme CPAN (CSD2007-00042), EPI (CSD2010-00064). DD and MK acknowledge financial support from the Swiss NSF. MH and ARL acknowledge support from the Science and Technology Facilities Council (grant numbers ST/J000477/1 and ST/K006606/1).
Shortly after our article was posted on arxiv.org, a related paper Moss and Pogosian (2014) was posted investigating similar ideas.
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