Steps to Reconcile Inflationary Tensor and Scalar Spectra
Abstract
The recent BICEP2 B-mode polarization determination of an inflationary tensor-scalar ratio is in tension with simple scale-free models of inflation due to a lack of a corresponding low multipole excess in the temperature power spectrum which places a limit of (95% CL) on such models. Single-field inflationary models that reconcile these two observations, even those where the tilt runs substantially, introduce a scale into the scalar power spectrum. To cancel the tensor excess, and simultaneously explain the excess already present in CDM, ideally the model should introduce this scale as a relatively sharp transition in the tensor-scalar ratio around the horizon at recombination. We consider models which generate such a step in this quantity and find that they can improve the joint fit to the temperature and polarization data by up to without changing cosmological parameters. Precision E-mode polarization measurements should be able to test this explanation.
I Introduction
The recent BICEP2 measurement of a tensor-scalar ratio from degree scale B-mode polarization of the cosmic-microwave background (CMB) Ade et al. (2014) is in “moderately-strong” tension with slow-roll inflation models that predict scale-free, albeit slightly tilted () power-law power spectra. This tension is due to the implied excess in the temperature spectrum at low multipoles which is not observed and restricts (95% CL) in this context Ade et al. (2013a).
These findings can be reconciled in the single-field inflationary paradigm by introducing a scale into the scalar power spectra to suppress power on these large-angular scales. For example a large running of tilt, , is possible as a compromise Ade et al. (2014). Here the scale introduced is associated with the scalar spectrum transiently passing through a scale-invariant slope near observed scales. However, such a large running is uncomfortable in the simplest models of inflation which typically produce running of order . Moreover, a large running also requires further additional parameters in order that inflation does not end too quickly after the observed scales leave the horizon Easther and Peiris (2006).
The temperature anisotropy excess implied by tensors is also not a smooth function of scale, but rather cut off at the horizon at recombination. To counter this excess, a transition in the scalar power spectrum that occurs more sharply, though coincidentally near these scales, would be preferred. Such a transition can occur without affecting the tensor spectrum if there is a slow-roll violating step in the tensor-scalar ratio while the Hubble rate is left nearly fixed. In this work we consider the effects of placing such a feature near scales associated with the horizon at recombination, thereby suppressing the scalar spectrum on large scales.
This slow-roll violating behavior also produces oscillations in the power spectrum Adams et al. (2001); Peiris et al. (2003); Park and Sorbo (2012); Miranda et al. (2012) and generates enhanced non-Gaussianity Chen et al. (2007, 2008) if this transition occurs in much less than an efold. For transitions that alleviate the tensor-scalar tension, these oscillations would violate tight constraints on the acoustic peaks and hence only transitions that occur over at least an efold are allowed. The resulting non-Gaussianity is then undetectable Adshead et al. (2011); Adshead and Hu (2012). Throughout, we work in natural units where the reduced Planck mass as well as .
Ii Step Solutions
In slow roll inflation, the tensor power spectrum in each gravitational wave polarization state is directly related to the Hubble scale during inflation
(1) |
whereas the scalar or curvature power spectrum is given by
(2) |
where and is the sound speed, yielding a tensor-scalar ratio . The addition of a nearly scale invariant tensor spectrum to the CMB temperature anisotropy produces excess power below which at is difficult to accommodate in slow roll inflation where the scalar spectrum is, to a good approximation, a scale-free power law (see Fig. 1).
The scalar power spectrum can be changed largely without affecting the tensors if the quantity changes while remains small. As shown in Fig. 1, the excess power resembles a step in this quantity on scales near the horizon at recombination. Hence to alleviate the tension between the tensor inference from the BICEP2 experiment, , and the upper limits from the combined CMB temperature power spectrum (95% CL), we examine models where there is a step in this quantity. In this paper we quote at the scalar pivot of Mpc where it is unaffected by changes to the scalar power spectrum that we introduce whereas the upper limit is quoted at Mpc.
As an example, we consider a step in the warp
(3) |
of Dirac-Born-Infeld (DBI) inflation^{1}^{1}1Of course, we are well outside the region of validity of UV complete versions of DBI inflation. However, this is merely a phenomenological proof of principle rather than a working construction. Silverstein and Tong (2004); Alishahiha et al. (2004) with the Lagrangian
(4) |
where the kinetic term , the sound speed
(5) |
Here parameterize the height, field position and field width of the step while the underlying parameters and the inflaton potential are set to to fix and Adshead et al. (2013). In Ref. Miranda et al. (2012), we showed that such a model produces a step in the quantity that controls the tensor-scalar ratio. To keep this discussion model independent, we follow Ref. Miranda and Hu (2013) and quantify the amplitude of the step by the change in this quantity
(6) |
where “” and “” denote the quantities before and after the step on the slow roll attractor. For definiteness, we take . In place of we quote the sound horizon
(7) |
at the step and in place of the width in field space , we take the inverse of the number of efolds the inflaton takes in traversing the step
(8) |
See Ref. Adshead et al. (2012); Miranda and Hu (2013) for details of this description. We utilize the generalized slow roll technique Stewart (2002); Dvorkin and Hu (2010); Hu (2011) to calculate the power spectra of these models since at the step the slow roll approximation is transiently violated.
(Mpc) | ||||||||
---|---|---|---|---|---|---|---|---|
0 | 0 | - | - | 2.1972 | 0.961 | 9802.7 | 89.1 | 40.1 |
0 | -0.15 | 337.1 | 1.58 | 2.2003 | 0.957 | 9798.6 | 89.2 | 36.1 |
0.1 | 0 | - | - | 2.1961 | 0.962 | 9806.5 | 47.9 | 2.7 |
0.1 | -0.22 | 339.2 | 1.60 | 2.2000 | 0.958 | 9797.8 | 48.2 | -5.7 |
0.2 | 0 | - | - | 2.1939 | 0.963 | 9812.3 | 39.4 | 0 |
0.2 | -0.31 | 351.8 | 1.47 | 2.2002 | 0.959 | 9798.1 | 39.9 | -13.7 |
Iii Joint Fit
We jointly fit the Planck CMB temperature results, WMAP9 polarization results, and BICEP2 to models with and without steps in the tensor-scalar ratio parameter . We use the MIGRAD variable metric algorithm from the CERN Minuit2 code James and Roos (1975) and a modified version of CAMB Lewis et al. (2000); Howlett et al. (2012) for model comparisons. The Planck likelihood includes the Planck low- spectrum (Commander, ) and the high- spectrum (CAMspec, ), whereas the BICEP2 likelihood^{2}^{2}2http://bicepkeck.org/ includes both its and contributions.
We begin with the baseline best fit 6 parameter slow-roll flat CDM model with . This model sets the non-inflationary cosmological parameters to , , , and the inflationary scalar amplitude at Mpc, , and spectral tilt, . When considering alternate models we fix the non-inflationary parameters to these values while allowing the inflationary parameters, including and to vary.
As shown in Tab. 1, this model is strongly penalized by the BICEP2 data. Moving to the model with the same parameters removes this penalty at the expense of making the Planck likelihood worse by due to the excess in the temperature power spectrum shown in Fig. 1.
Next we fit for a step with parameters , , controlling the amplitude, location and width of the step. The best fit model at more than removes the penalty from the temperature excess for Planck while fitting the BICEP2 results equally well. The net result is a preference for a step feature at the level of over no feature. The inclusion of BICEP2 results slightly degrades the fit to due to changes in the spectrum (see below). The model with a step is very close to the global maximum with further optimization in allowing only an improvement of . With the addition of the step, there remains a small high- change in the vicinity of the first acoustic peak in Fig. 1 which is interestingly marginally favored by the data. Note that we have fixed the non-inflationary parameters to their values without the step, for example . Thus the likelihood may in fact increase in a full fit (see Fig. 3). Conversely, we do not consider any compromise solutions where non-inflationary cosmological parameters ameliorate the tension without a step. We leave these considerations to a future work.
The best fit step also predicts changes to the polarization. Like the spectrum, the excess power from the tensor contribution is partially compensated by the reduction in the scalar spectrum for . This is a signature of the step model which requires only a moderate increase in data to test as witnessed by the change in the BICEP2 likelihood of it induces. Differences at , shown here at fixed , are largely degenerate with changes in the ionization history Mortonson et al. (2009)
Due to potential contributions from foregrounds in the BICEP2 data which may imply a shift to Ade et al. (2014), we also test models at which would formally be in tension with the BICEP2 likelihood without foreground subtraction. Even in this case, the Planck portion of the likelihood improves with the inclusion of a step though the preference is weakened to versus no step. At , the Planck data still prefers a step to remove power at a reduced improvement of , a fact that was already evident in the Planck collaboration analysis of anticorrelated isocurvature perturbations Ade et al. (2013b). Such an explanation should also help resolve the tensor-scalar tension albeit outside of the context of single-field inflation. Interestingly, the addition of tensors at both and in fact further helps step models fit the Planck data due to the changes shown in Fig. 1 independent of the BICEP2 result.
Iv Discussion
A transient violation of slow-roll which generates a step in the scalar power spectrum at scales near to the horizon size at recombination can alleviate problems of predicted excess power in the temperature spectrum, present already in the best fit CDM spectrum, and greatly exacerbated by tensor contributions implied by the BICEP2 measurement. Such a step may be generated by a sharp change in the speed of the rolling of the inflaton or by a sharp change in the speed of sound over a period of around an efolding which combine to form the tensor-scalar ratio. Preference for a step from the temperature power spectrum is at a level of if and is still at , the lowest plausible value that would fit the BICEP2 data.
Such an explanation makes several concrete predictions. Since slow-roll is transiently violated in this scenario, there will be an enhancement in the associated three-point correlation function. However, we do not expect this signal to be observable as it impacts only a small number of modes Adshead et al. (2011); Adshead and Hu (2012). -mode fluctuations on similar scales would be predicted to have a smaller enhancement then with tensors alone. This prediction should soon be testable; in the BICEP2 data it brings down the total likelihood improvement to with a step at .
While we have used a DBI type Lagrangian to illustrate the impact of a change in the tensor-scalar ratio parameter due to a step in the sound speed, we do not expect that our results require this form, though precise details of the fit may change. Transient shifts in the speed of sound have been found to occur in inflationary models where additional heavy degrees of freedom have been integrated out Achucarro et al. (2012). We leave investigation of specific constructions to future work.
Acknowledgements.
While this work was in preparation, the work Contaldi et al. (2014) appeared which has some overlap with the work presented here. We thank Maurício Calvão, Cora Dvorkin, Dan Grin, Chris Sheehy and Ioav Waga for useful conversations. This work was supported in part by the Kavli Institute for Cosmological Physics at the University of Chicago through grants NSF PHY-1125897 and an endowment from the Kavli Foundation and its founder Fred Kavli. WH was additionally supported by U.S. Dept. of Energy contract DE-FG02-13ER41958 and VM by the Brazilian Research Agency CAPES Foundation and by U.S. Fulbright Organization.References
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