Searching for features of a string-inspired inflationary model with cosmological observations

Searching for features of a string-inspired inflationary model with cosmological observations

Yi-Fu Cai Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy, University of Science and Technology of China, Chinese Academy of Science, Hefei, Anhui 230026, China Department of Physics, McGill University, Montréal, Quebec H3A 2T8, Canada    Elisa G. M. Ferreira Department of Physics, McGill University, Montréal, Quebec H3A 2T8, Canada    Bin Hu Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands    Jerome Quintin Department of Physics, McGill University, Montréal, Quebec H3A 2T8, Canada

The latest Planck results show a power deficit in the temperature anisotropies near in the cosmic microwave background (CMB). This observation can hardly be explained within the standard inflationary -cold-dark-matter (CDM) scenario. In this paper we consider a string theory inspired inflationary model (axion monodromy inflation) with a step-like modulation in the potential which gives rise to observable signatures in the primordial perturbations. One interesting phenomenon is that the primordial scalar modes experience a sudden suppression at a critical scale when the modulation occurs. By fitting to the CMB data, we find that the model can nicely explain the power deficit anomaly as well as predict specific patterns in the temperature-polarization correlation and polarization autocorrelation spectra. Though the significance of the result is not sufficient to claim a detection, our analysis reveals that fundamental physics at extremely high energy scales, namely, some effects inspired by string theory, may be observationally testable in forthcoming cosmological experiments.

98.80.Es, 11.25.Mj, 98.70.Vc, 98.80.Cq

I Introduction

With accumulated high-precision measurements of the cosmic microwave background (CMB) radiation Hinshaw:2012aka (); Ade:2013zuv (); Ade:2015xua (), it is believed that our universe has experienced a dramatic phase of expansion at very early times as described by the inflationary CDM cosmology from which a power law spectrum of primordial perturbations is obtained Mukhanov:1990me (). The recently released Planck data, however, reported deviations from a power law connected to a deficit at multipoles in the temperature power spectrum Ade:2015lrj (). Although this anomaly does not have sufficient statistical significance due to the cosmic variance at these multipoles, it is of enough interest to ask whether such an experimental signature provides a hint to new physics beyond CDM.

In order to understand cosmological data from fundamental physics, we consider an important class of large field inflationary cosmology realized by string theory, which is dubbed axion monodromy inflation (AMI) Silverstein:2008sg (); McAllister:2008hb (). In the corresponding stringy setup, a number of axion fields coupled to fluxes can realize a super-Planckian field variation with a soft shift symmetry breaking along the axion potential due to couplings or D-branes. This is how a class of monomial potentials has been achieved McAllister:2014mpa (). This model and its extensions have drawn a lot of attention in the literature (e.g., see Flaugeretal (); Cai:2014vua () and references therein). In particular, it was observed in Cai:2014vua () that axion monodromies can be obtained in terms of D3- and D5-branes with torsional cycles in which there exists at least one scalar mode that is free from the dangerous moduli stabilization effects from the Kähler potential. This scalar field, when applied to the early universe, can drive a sufficiently long phase of inflation where the potential parameters are sensitive to flux couplings, namely, the potential experiences a modulation at a critical value of the inflaton field. As a result, it provides an interesting implementation of step inflation Starobinsky:1992ts (); Adams:1997de () from the perspective of string theory.

In this paper we aim at examining observational signatures of this type of inflationary model in cosmological surveys. Specifically, we present an estimate of the power spectrum of primordial curvature perturbations and analytically find that it possesses a suppression feature at a critical length scale. Using the Planck 2013 data, we perform a numerical analysis and find that this feature can nicely interpret the anomaly at low multipoles as observed in the recent CMB observations which cannot be explained by the standard model, i.e. CDM. Afterwards, we compute the E-mode polarization spectra, and interestingly, we find that our model also predicts nontrivial patterns on these spectra. We take the convention of the Planck team throughout the paper.

Ii Model

We work with the string-inspired inflationary model of axion monodromy in the presence of torsional cycles and flux couplings Cai:2014vua (). In this framework, the 4-dimensional effective action is described by a canonical scalar field with a polynomial potential coming from either the Chern-Simons term or flux couplings. Due to a variation of the flux coupling, the inflaton’s potential can be expressed as




where is the reduced Planck mass (see Cai:2014vua () for the string theory implementation with ).

The coefficient of the potential takes the form in the ultraviolet (UV) regime () but becomes in the infrared (IR) regime (), where is a dimensionless parameter. The other dimensionless parameter is associated with the smooth transition between the UV and IR regimes of the potential near the critical scale characterized by . As was observed in Cai:2014vua (), the inflaton field undergoes standard slow roll dynamics in both the UV and IR regimes, but near the potential modulation, the slow roll condition is briefly broken.

In this case it is convenient to introduce the generalized slow roll (GSR) parameters Stewartetal (), which up to -th order are given by , , where a dot denotes a cosmic time derivative. Then one can follow the formalism of the GSR approximation to derive an analytic expression of the power spectrum. Here we skip the detailed procedure (which will be presented in a follow-up paper) but directly write down the power spectrum of primordial curvature perturbation as


where is the featureless power spectrum from the standard model and represents the modifications introduced by the GSR approximation. Similar to the standard case, the power spectrum can be parametrized by , where and are the amplitude and the spectral index, respectively, and is the pivot scale. For a specific power law potential, we have


and the power spectrum quantities and can be approximately written as Planck:2013jfk (); Ade:2015lrj ()


To apply these relations it is convenient to introduce at where the feature is located. The factor grasps the signatures brought by the potential modulation, and therefore, its value equals unity away from but exhibits the GSR feature near . For , these features have an oscillatory behavior near , analogous to Dvorkin:2009ne () where a quadratic model with a different modulation form was considered.

In this paper we focus our interest in the case of . Accordingly, the GSR feature is very smooth and is only located in the vicinity of the potential modulation and mainly depends on the evolution of . To leading order one has , where with being the Euler-Masheroni constant. This factor can be further parametrized as follows,

Figure 1: Power spectra of primordial curvature perturbations as a function of the comoving wave number . The blue curve shows the standard CDM model and the black curve presents an example of the AMI with feature given by (3).

The parametrized form (3) together with Eqs. (2) and (6) gives the primordial power spectrum of the AMI with feature, which is described by three extra parameters on top of and : , , and (or equivalently), as demonstrated in Fig. 1. However, one extra parameter remains undetermined, i.e. the e-folding number () at the pivot scale, since it involves detailed knowledge of the reheating theory following it Liddle:2003as (); Martin:2010kz (). A different value of leads to a different , altering the position of the feature. This shows that is highly degenerate with, at least, . In the literature, is sometimes fixed for simplicity Ade:2015lrj (); Benetti:2012wu (), but this leads to preferred values of , as seen from Eq. (5). For the sake of generality, we investigate both possibilities: having fixed and having it as a free parameter. Additionally, the parameter space can be further reduced by imposing theoretical constraints, namely, the power spectrum must always be positive, i.e. . We impose this constraint in all numerical calculations below.

Iii Methodology

We wish to compare the theoretical predictions of our model and to test the validity of our parametrization of the power spectrum with the current data. We perform a global fitting by running the CosmoMC package Lewis:2002ah (), a Markov Chain Monte Carlo (MCMC) parameter sampler. We modify the publicly available CAMB Lewis:1999bs () Boltzmann code with our parametrization (3), which gives the power spectrum with a feature described by , , and . Those are combined with the “vanilla” parameters of Planck’s one-parameter extension of the baseline CDM model Ade:2015lrj (); Planck:2013jfk () consisting of 7 parameters (, , , , , , ) where and are the baryon and cold dark matter densities, is the optical depth to reionization, is the ratio of the sound horizon at decoupling to the angular diameter distance to the last scattering surface (multiplied by 100), is the spectral index, is the primordial amplitude, and is the tensor-to-scalar ratio. As usual, we assume adiabatic initial conditions and a flat background universe.

Due to the limited knowledge of reheating, there remains uncertainty in determining the exact -folding number when the pivot mode () crosses the horizon. In order to take this into account, we first take the Planck convention Planck:2013jfk () by setting and vary when we perform the global fitting. Then, in order to have a closer look at the effect of fixing on the parameter estimation, we perform three extra runs. Specifically, we fix at , , and , which correspond to , and according to the modeling of entropy generation at the end of inflation given by Eq. (24) in Planck:2013jfk (). In this case, when the details of the entropy generation process are fixed, the relation between and is known. Thus, fixing fixes and vice-versa. redHere we emphasize again, in the case of fixing , we only investigate a specific formula for reheating, the one that is adopted by the Planck collaboration Planck:2013jfk (). As will be demonstrated below, the sensitivity of the feature in different multipole ranges is expected to depend on the choice of , so it is crucial to set the relationship between and consistently. This is important in its own right, but the exploration of different reheating processes is beyond the scope of this paper.

When considering as a free parameter, we do not assume any specific reheating scenario. This means that the relation between and is not known. In this case, the pivot scale is fixed at the value , which is equivalent to “marginalizing” over the uncertainty of the reheating mechanism. However, it is important to know since it changes the position of the possible feature, and this is determined by the MCMC analysis.

We use the Planck 2013 TT power spectrum, both the low- () and high- () multipole data. In order to break the degeneracy between reionization optical depth and the primordial amplitude, we include the WMAP Bennett:2012zja () low- polarization power spectrum (). Besides that, we adopt the baryon acoustic oscillation data from BOSS “LOWZ” and CMASS-DR11 surveys Anderson:2013zyy () in order to obtain a better estimation on and , etc. We name the above data compilation Planck13 in the following analysis. Moreover, we implement a logarithmic prior for and , namely and , to be able to span a large parameter space. The theoretical constraint of restricts the value of to be close to unity. Given this observation, we choose a flat prior for such that .

Iv Results

After performing the global fitting of all parameters, we find that the best-fit values of the vanilla parameters are in agreement with the Planck values Ade:2015xua (). Then we present the posterior of and in Fig. 2. Since the current data cannot provide any significant constraint on , we do not report any bound on it. We specifically consider the AMI with feature for the , and free cases. The improvement in the maximum likelihood obtained in our runs with respect to CDM is given by the quantity , which is found to be , respectively.

Figure 2: One-dimensional posteriors of the feature parameters and for different e-folding numbers as a result of the fitting to the Planck13 data.

We find that is the best-fit value of the scale at which the feature in the primordial power spectrum is located. This roughly corresponds to the angular scale with multipole . Indeed, we see from Fig. 3 that for the and free cases, the suppression feature becomes manifest near , and accordingly, it leads to a slightly better fit to the Planck 2015 low- data Aghanim:2015xee (). Hence, our model can naturally explain the suppression anomaly near , and the interpretation is that this observed anomaly is due to a short phase of slow-roll violation during inflation Cai:2014vua ().

Figure 3: The CMB temperature-temperature power spectra for the best-fit CDM and AMI with feature models. Additionally, we superimpose the Planck 2015 data. The vertical axis of the top panel is defined as . For the bottom panel, we plot the power spectrum residual () with respect to the CDM one.

We notice from Fig. 3 that in the high- regime our best-fit curves from Planck13 data have a slight discrepancy with the Planck 2015 results. This is due to the fact that there is roughly a shift in between the 2013 Ade:2013zuv () and 2015 Ade:2015xua () data induced by the absolute calibration parameter correction of the 143 GHz channel in the 2015 pipeline Adam:2015vua (). Moreover, our analysis reveals that the posterior of for the and cases can have a second peak away from the anomaly scale, hence they do not reproduce the suppression in Fig. 3. The anticorrelation between and (the larger is, the farther away from the end of inflation Hubble crossing occurs, and hence the smaller ) demonstrates that in the case of small (or large ) one can fit the small scale structures in the primordial power spectrum better and vice versa.

The current CMB data can only report an upper bound for the parameter, at C.L., and it is insensitive to the parameter compared to its theoretical prior. As for the degeneracy between feature and CDM parameters, no significant correlation is obtained. The most interesting correlation is found between and the tensor-to-scalar ratio (). In the localized feature regime (), the tensor-to-scalar ratio is dragged to a greater value at C.L. This is because the potential modulation leads to a power deficit in the scalar spectrum without affecting as much the tensor spectrum (see Cai:2014vua ()).

Several previous studies of CMB features reported . However, these signals were not statistically conclusive Benetti:2012wu (); Planck:2013jfk (); Ade:2015lrj (); Benetti:2013cja (); Hazra:2014goa (); Achucarro:2013cva (); Achucarro:2014msa (); Hu:2014hra (); Meerburg:2013cla (); Meerburg:2013dla (); Chen:2012ja (); Chen:2014cwa (). The improvement obtained here is also not statistically significant, but yet, it shows that the localized feature is mildly favored by the data. As pointed out in Planck:2013jfk (), it is difficult to prove that these results are not coming from statistics overfitting of noisy data. Moreover, one needs to be careful with the values found since the best-fit value provided by CosmoMC in its Metropolis-Hastings mode might not be fully trustworthy.

Observational signatures of the new model can also be analyzed within the CMB polarization data, namely, the E-mode spectra. On the one hand, since oscillations in the temperature maps are strongly washed out on small scales Mortonson:2009qv (), the inclusion of the polarization maps will help us capture information of early universe models at relevant scales. On the other hand, if the origin of the low- anomaly in the TT spectrum comes from the inflationary phase, a similar signal should also appear in the low- regime of the TE and EE spectra.

Figure 4: Comparison between the theoretical CMB polarization power spectra (TE and EE) and the Planck 2015 data. In the left panel, we show and the residuals with respect to the CDM best-fit. In the right panel, we show and . Note that the scales differ for and as shown by the different vertical axis scales (the same way as Fig. 48 of Aghanim:2015xee ()).

The Planck collaboration happened to release the polarization data very recently Aghanim:2015xee (). In Fig. 4, we show the TE and EE power spectra for the best-fit CDM and AMI model with feature (obtained from the Planck13 data) and we make a first comparison with the Planck 2015 data. Looking at the residues of the TE and EE power spectra, one can see that our model is distinct from CDM both at low- and high-. At low-, our model predicts power deficits around in the TE and EE spectra (similar to the TT spectrum), of which the amplitudes ( and ) are of order and , respectively; while at high- the relative powers oscillate around CDM in both the TE and EE spectra. The oscillations at are due to small differences in the best-fit CDM parameters via the mild correlations with the feature parameters, but the suppressions in the TE and EE spectra are explicit predictions of the model compared to CDM. Although these suppression signals in the E-mode spectra are difficult to be tested by the present observations since the differences are currently much smaller than the overall scattering in the data, they may provide a promising window for future CMB surveys which are expected to greatly improve their accuracy of polarization measurements.

V Conclusions

Evidence of fundamental theory (e.g. string theory) occurs at extremely high energy scales, and hence, is difficult to be found directly by experiments. To search for evidence, it is important to investigate the associated cosmological implications, in particular, applications to the very early universe and their observational consequences. In this sense, various theoretical models have been proposed to explain certain CMB anomalies such as step potentials Adams:2001vc (); Benetti:2012wu (); Adshead:2011jq (); Miranda:2013wxa (); Benetti:2013cja (); Hazra:2014goa (), transient sound speed reduction Achucarro:2010da (); Achucarro:2013cva (); Achucarro:2014msa (); Hu:2014hra (), massive fields Chen:2011zf (); Chen:2012ja (); Chen:2014cwa (), varying Planck mass models Ashoorioon:2014yua (), pre-inflationary fast roll models Gruppuso:2015xqa (), linear oscillations Jackson:2013mka (); Meerburg:2013cla (); Meerburg:2013dla (); Ashoorioonetal (), logarithmic oscillations Martin:2000xs (); Danielsson:2002kx (); Bozza:2003pr (), and cutoff models Contaldi:2003zv (); Sinha:2005mn (); Iqbal:2015tta ().

In the present paper, we studied the primordial power spectra of one class of inflationary model inspired by string theory and examined their patterns in the CMB. We interestingly found a suppression feature in the power spectrum of curvature perturbations in the vicinity of a critical scale, which could explain the power deficit anomaly of temperature anisotropies as indicated by the recent Planck data.

We further studied the angular power spectra of CMB polarization modes and performed a preliminary comparison with the Planck 2015 data. We showed that some patterns in these spectra are manifestly different from CDM and hence the model is observationally distinguishable if experimental accuracy is largely improved in the future. While these polarization signals could be important, it is also necessary to explore observational evidences from other avenues. For instance, we expect that the modulation in the inflaton’s potential can give rise to specific non-Gaussianity signals (e.g., see Achucarro:2012fd (); Palma:2014hra (); Novaes:2015uza (); Mooij:2015cxa ()) as well as signals in the matter power spectrum that may be sensitive to the large scale structure experiments. The study of these topics will be important along with this present work.

We thank Ana Achucarro, Frederico Arroja, Fang Chen, Chunshan Lin, Daan Meerburg, Shinji Mukohyama, Misao Sasaki, Yi Wang, and Scott Watson for valuable discussions. Y.F.C. is supported in part by the Chinese National Youth Thousand Talents Program and by the USTC start-up funding under Grant No. KY2030000049. E.F. thanks CNPq (Science without Borders) for financial support. B.H. is supported by the Dutch Foundation for Fundamental Research on Matter (FOM). J.Q. acknowledges the Fonds de recherche du Québec - Nature et technologies (FRQNT) and the Walter C. Sumner Foundation for financial support. Computations were made in part on the supercomputer Guillimin from McGill University, managed by Calcul Québec and Compute Canada. The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), NanoQuébec, RMGA, and FRQNT.


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