New model of axion monodromy inflation and its cosmological implications

We propose a new realization of axion monodromy inflation in which axion monodromy arises from torsional cycles in a type IIB compactification. A class of monomial potentials is obtained with specific values for the power index. Moreover, the inflaton mass changes profile due to the couplings between various fields after compactification. Consequently, the potential obtains a step-like profile at some critical scale. We study the cosmological implications of one concrete realization of this model. At the background level, it realizes a sufficiently long inflationary stage, which allows for the violation of the slow-roll conditions for a short period of time when the inflaton is close to the critical scale. Accordingly, the Hubble horizon is perturbed and affects the dynamics of primordial cosmological perturbations. In particular, we analyze the angular power spectrum of B-mode polarization and find a boost on very large scales. We also find that the amplitude of scalar perturbations is suppressed near the critical scale. Thus our model provides an interpretation for the low-$\ell$ suppression of temperature anisotropies in the CMB power spectrum. We examine these effects and confront the model to observations.


I. INTRODUCTION
Large field inflationary models with super-Planckian field ranges are observationally allowed by the latest cosmological experiments [1,2]. The recent detection of primordial B-mode polarization reported by the BICEP2 experiment implies that the tensor-to-scalar ratio, r, which is the ratio between the amplitude of the spectra of tensor and scalar perturbations, is likely to be nonzero [3]. This favors models of large field inflation via the Lyth bound [4]. While the understanding of the BICEP2 data in terms of primordial tensor fluctuations becomes less clear [5,6] because of a potential contamination by the dust foreground detected by the Planck group [7], it remains of theoretical interests to study the realization of large field inflation within the framework of fundamental particle physics.
Considering the fact that inflation could occur near the Grand Unified Theory (GUT) scale, it is natural to ask whether large field inflation can be realized within string theory. In the literature, this topic has received extensive attention (for instance, see [8][9][10] for recent reviews). Recently, it was put forward and analyzed in [11][12][13] that a string theory model of large field inflation can be achieved under the axion monodromy construction. In the corresponding stringy setup, a number of axion fields coupled to fluxes can yield a super-Planckian field variation while softly breaking the shift symmetry along the axion potential due to the coupling or due to Dbranes. As a result, a class of axion monodromy inflation * Electronic address: yifucai@physics.mcgill.ca † Electronic address: fchen@kitp.ucsb.edu ‡ Electronic address: elisafenu@physics.mcgill.ca § Electronic address: jquintin@physics.mcgill.ca models with monomial potentials was obtained in [14]. Various extended analyses of axion monodromy inflation were performed in the literature, for instance, in the context of super-gravity (SUGRA) realizations [15][16][17], by involving extra moduli fields [18,19], or by making connections to natural inflation [20][21][22][23]. Also, a potential back-reaction issue against the moduli stabilization was recently discussed in [24][25][26] from different perspectives.
In the present paper, we propose a new class of axion monodromy inflation models in terms of a set of D3-and D5-branes with torsional cycles. In our concrete construction, the potential for the inflaton field approximately takes the form of a power law function in which the exponent (p) varies depending on the detailed integral over the internal manifold as well as on the contribution from a Chern-Simons term. In general, these functions can appear as a linear combination in a single potential, which can be expressed as where the c i 's are to be determined by the specific constructions. Therefore, our model can easily give rise to a sufficiently long period of inflation that is required by cosmological observations. More interestingly, we take a closer look at the shape of the inflaton's potential and we find that it may receive a modulation due to the uncertainty in the integration of the internal space. As a result, we expect this nontrivial modulation in the potential to leave important signals for cosmological experiments.
As a representative example, we study a specific inflation model derived from our construction. In particular, we analyze a model of chaotic inflation with a potential of the form m 2 ϕ 2 in which the inflaton mass obtains a small modulation due to the variation of the internal space. Therefore, the inflaton potential possesses a steplike feature at a critical scale ϕ c . We study in detail the cosmological implications of this model and we find that the mass modulation leads to a short phase of slow-roll violation during inflation when the inflaton field approaches ϕ c . The Hubble horizon near the corresponding scale is also perturbed due to the violation of the slow-roll conditions. Accordingly, for the primordial fluctuations exiting the Hubble horizon at the same moment, the process of getting squeezed can be dramatically affected. In general, the spectrum of these fluctuation modes can become wiggly, but if the modulation is smooth enough, the spectrum can present a suppression behavior near the critical scale and then grow back to a larger amplitude at even larger length scales. We analyze these effects in the primordial angular power spectrum of temperature anisotropies and confront them with the Planck 2013 data. Our results show that, for certain values of the model parameters, an appropriate suppression of the power can be achieved to explain the cosmic microwave background (CMB) anomaly at large angular scales. We further study the angular power spectrum of B-mode polarization that arises from primordial gravitational waves and obtain a small enhancement on very large scales. While being aware of the fact that the amplitude of primordial tensor fluctuations can always be modified by tuning the inflationary model parameters, the small amplification feature in the B-mode spectrum is a key prediction made by our model. This particular prediction may shed light on probing new physics by virtue of accumulating high-precision CMB polarization data in the near future.
The paper is organized as follows. In Sec. II, we first briefly review the general picture of axion monodromy inflation and then propose a new framework for axion monodromy inflation in string theory. Afterwards, we select a specific realization of our new axion monodromy model in Sec. III and we study its cosmological implications in detail. We first study the background inflationary solution and analyze the dynamics of the slow-roll parameters. Then we numerically calculate the primordial power spectra of scalar and tensor perturbations generated during inflation. We then confront the theoretical predictions of our specific model with the latest CMB data by analyzing the temperature and B-mode polarization angular power spectra, and we perform Markov Chain Monte Carlo (MCMC) simulations to constrain the model. We conclude with a discussion in Sec. IV. Throughout this paper, we adopt the convention M 2 p ≡ 1/8πG.

II. A MODEL OF AXION MONODROMY INFLATION
In this section we first present a brief review of the regular version of axion monodromy inflation and then propose a new model in the presence of torsional cycles.

A. A brief review of existing axion monodromy models
The idea of axion monodromy inflation originally proposed in [11] has provided an interesting interpretation that a sufficiently long period of inflation can persist through many cycles of the axion period while the inflaton's potential is well controlled by the axion shift symmetry. This model, and related extensions, were widely studied since then (see, e.g., [12,13,15,27,28] and references therein).
In most of these models, the axion field comes from the reduction of a NS-NS two-form B 2 on a two-cycle which has a continuous shift symmetry due to the higher dimensional gauge symmetry of the two-form, and the monodromy is induced by branes or fluxes. As a simple example, one can consider a D5-brane wrapping an internal two-cycle Σ 2 in type IIB string theory. The axion, which is given by b = 1 α Σ2 B 2 , has a shift symmetry b → b + const. However, the presence of the D5-brane gently breaks such a shift symmetry and generates the monodromy. To be explicit, we can consider the Dirac-Born-Infeld (DBI) action for the D5-brane, which is an effective field description of string theory at low energy scales. Here, M 4 denotes the 3 + 1 spacetime, and G ab and B ab , where a, b = 0, ..., 5, are the induced metric and NS-NS two-form on the D5-brane world volume, respectively.
After integrating over the two-cycle Σ 2 , one obtains the potential of the axion in the four-dimensional effective theory, where is the size of the two-cycle Σ 2 in string units and ρ is a dimensionless coefficient determined by the warp factor. For large values of b, the potential is linear, V (b) ∝ b. Correspondingly, the Lagrangian of the canonically normalized inflaton field ϕ is given by, where f is the axion decay constant. In addition, a similar scenario can be achieved when the D5-brane is replaced by an NS5-brane.
It has been interestingly observed that there are many variants of the axion's potential that may arise from axion monodromy. Thus, it is convenient to parameterize these theories in terms of an exponent p: For example, a model with p = 2/3 was obtained in [11] based on a construction of Nil manifolds; the case of p = 2 was found after taking into account an appropriate coupling of the axion to a four-form field [29]; more examples with p = 2/3, 4/3, 2 and 3 were obtained in [14] by considering effective couplings of the NS-NS B 2 field to the RR F 1 field through |F 5 | 2 and |F 3 | 2 , respectively, after reducing type IIB on a circle with specific background fluxes and moduli stabilization. Another interesting model of axion monodromy inflation was constructed in [15], where the axion arises from the KK compactification of higher dimensional gauge fields or p-form potentials in the presence of fluxes and/or torsion homology, and the axion potential and monodromy arise from an F-term. In the same work the authors argued that this scenario could naturally avoid the η-problem widely existing in the inflationary paradigm [30] since those axions did not appear in the Kähler potential as they did in other models.

B. New axion monodromy inflation from warped torsion classes
We propose a new class of axion monodromy inflation models in the present subsection. Specifically, we consider a type IIB string theory compactification and assume that there are N D3-branes along the 3 + 1 space-time. Thus the metric naturally has a warp factor e −2A ∼ 1/N and takes the form where µ = 1, ..., 4 represents the coordinates of the physical space-time, and i = 1, ..., 6 runs over the coordinates of the internal manifold M 6 . It is well known that in (conformal) Calabi-Yau (CY) manifold compactification, the massless modes are in oneto-one correspondence with the cohomology groups of the manifold. However, the situation becomes slightly different if there are torsional cycles since in this case, the massless modes are not sensitive to D-branes wrapping torsional cycles. In the following we use X 6 to denote the unwarped internal manifold and distinguish it from M 6 . In general, there are two independent torsion classes in a six-dimensional manifold, i.e., torsion one-cycles Σ 1 and torsion two-cycles Σ 2 . They can be related to torsion four-cycles Σ 4 and torsion three-cycles Σ 3 via for a d-dimensional internal manifold and any positive integer r. For simplicity we take where p and k are positive integers and in general not equal. Now let us consider a set of non-harmonic Laplacian eigenforms of X 6 , where γ 1 ,ρ 4 , η 2 , andω 3 are associated with the generators of TorH i (X 6 , Z) with i = 1, 4, 2, and 3, respectively. We note that, ρ 2 ,γ 5 , ω 3 , andη 4 are trivial in de Rham cohomology but are non-trivial generators of H i (X 6 , Z), i = 2, 5, 3, 4. Additionally, we have the following integral constraints: Then we can expand type IIB NS-NS and RR forms in terms of these eigenforms as and they are the most general expansions regardless of the orientation. The corresponding ten-dimensional field strengths are given by Next, we perform the dimensional reduction of the type IIB SUGRA action on M 6 . The type IIB SUGRA action including local sources in ten-dimension is then given by in Einstein frame, where τ = C 0 + ie −φ is the axiondilaton field with C 0 being the string axion (one should be aware that this is not the same axion field as the one we will discuss below in the four-dimensional effective theory) and φ being the dilation field. Moreover, which is the five-form flux, and with F 3 being the RR three-form flux and H 3 the NS-NS three-form flux. Also, G is the determinant of the tendimensional metric g M N , where M, N = 0, ..., 9. Finally, S loc is the action for localized sources in the system, i.e. D3-branes in our case.
Plugging Eq. (11) and Eq. (12) into Eq. (13) gives rise to the four-dimensional effective action that we are interested in. In particular, we are mostly interested in the contributions from the scalar fields b, c and c, and hence, the effective action in four dimensions can be simplified as where Also, g 4 is the determinant of the unwarped space-time metric and the hodge star 6 is associated with the unwarped internal manifold X 6 . In the above, we have chosen C 0 to be zero and the string coupling to be unity for simplicity. The ellipsis at the end of the action denotes the contributions from the ten-dimensional Chern-Simons term and other relevant terms, which are all linear in the b, c, and c fields.
Thus the b-and c-related terms in the action can be written as wherec = κ 4 c/(2 √ 3κ 10 ). The above three scalar fields b, c (orc equivalently), and c can be candidates for the inflation field. On one hand, as has been argued in [15], in the (conformal) CY compactification the b field is generally involved in the Kähler potential. Thus if one takes the b field as the inflaton field, upon the moduli stabilization, the corresponding potential could achieve a correction that renders a possibly large value of the η parameter (η ∼ 1), and therefore may invalidate the occurrence of inflation. On the other hand, the c field gives rise to standard chaotic inflation. Therefore, we focus on the cosmological implications of thec field.
We assume that the moduli fields have stabilized from now on and that the T i 's are all fixed and of the same order. Before we study the inflaton field, we also have to stabilize the b field. We notice that the b field couples to the c field in a non-standard way and that the value b min where its potential is the lowest depends on c. Since we know that e −4A ∼ 1/N 2 , the c dependence is mild if c < N 2 and strong if c > N 2 . Particularly, for c < N 2 we ignore its dependence in the b potential and b min is fixed; for c > N 2 we will see that it can give rise to cosmological inflation. Thus, we can set c = N 2 temporarily in order to find b min in this region. Now that b min is fixed, b min can also be determined. Thus we can normalize the c field and write down the four-dimensional effective action as follows, where Moreover, we have included the contributions from the Chern-Simons term and other relevant terms, which appears in the linear function gϕ in Eq. (20). One could do another field transformation in Eq. (20) to absorb the linear term and leave the mass m 2 unchanged. The resulting cosmological solution would lead to simple chaotic inflation. Instead, we would like to explore another possibility in which the mass of the inflaton field varies along the inflaton value, which occurs at To see this more explicitly, we rewrite the mass term as follows, where m 2 a and m 2 b are the inflaton masses when the inflaton field is smaller and larger than the critical value ϕ c , respectively. We have used the θ function here to describe the most extremal case. However, as we discussed previously, b min (c) is smooth, so instead of the step function θ, it is better to use the following expression, The parameter C H depends on the smoothness of b min (c) and thus on the potential of the b field. From now on we will treat it as a free parameter.

III. INFLATIONARY DYNAMICS AND COSMOLOGICAL IMPLICATIONS
In this section, we select one specific model of the new axion monodromy model as a demonstration to study its cosmological implications. Specifically, we consider the effective action in (20) and assume a negligible Chern-Simons term. Then the Lagrangian derived at the end of the previous section can be rewritten as which is viewed as an effective Lagrangian in 4dimensional spacetime. The potential of the inflaton takes the form of m 2 b ϕ 2 /2 when |ϕ| > ϕ c (which corresponds to the UV regime) while taking the form of m 2 a ϕ 2 /2 when |ϕ| ≤ ϕ c (which belongs to the IR regime). We note that m b > m a is required for the model to be consistent with theoretical constraints. For convenience, we would like to parameterize the potential in the following form, where we have introduced one dimensionless coefficient C H in order to smoothly connect the UV and IR regimes of the potential near the critical scale characterized by ϕ c . This newly introduced parameter ought to be determined by the explicit physics of the mass transition in the corresponding string theory construction. However, due to the lack of specific stringy derivations, we argue that C H can be treated as a free parameter in our model that can be constrained by cosmological observations as will be analyzed in the following subsections.
We provide a sketch of the parameterized potential V (ϕ) given by Eq. (26) in Fig. 1. This figure is helpful to gain a semi-analytic understanding of the background dynamics, which is the topic we turn to in the following subsection. We note that features in the inflationary potential such as above have a long history, namely see [31] for a pioneer study and [32,33] for phenomenological constructions. Also, we note that a similarly featured potential was implemented phenomenologically by a model of multifield inflation in [34][35][36].

A. Background dynamics
In this subsection, we perform the detailed analysis of the background inflationary dynamics of the present model. It is well known that a model of ϕ 2 inflation possesses a local attractor solution regardless of the initial condition for the background scalar and that this is a slow-roll solution [37]. Accordingly, in both the UV and IR regimes of our model, the inflaton field can enter a period of slow-roll dynamics. Thus, in order to analyze the background dynamics semi-analytically, it is convenient to introduce a series of slow-roll parameters as follows, where a dot denotes a derivative with respect to cosmic time (t) and where the subscript ,ϕ denotes a derivative with respect to ϕ. During inflation, the size of the universe is growing nearly exponentially while the Hubble rate is slowly varying. Thus, instead of cosmic time, it is very efficient to apply the inflationary e-folding number to characterize the time duration of the process. Accordingly, the inflationary e-folding number is defined by where the end of inflation is determined by the condition (t end ) = 1. When the single field ϕ evolves stably along the slowroll trajectory, it is not difficult to observe that the value of approximately equals the value of ϕ . One can easily derive the following approximate relations, when the inflaton field is far away from the critical scale. When ϕ evolves near the critical value ϕ c , the potential experiences a sudden decrease and its profile becomes very steep. In response to this change in the potential, and ϕ can increase significantly before returning to their regular value after the critical phase. The slow-roll parameter |η ϕ | can also obtain a dramatic increase around the critical scale but the shape of the amplification is fairly asymmetric since this parameter characterizes the second derivative of the potential. It is interesting to note that |ξ ϕ | in standard quadratic inflation is always zero, but in our model, it can become very large when the inflaton evolves through ϕ c .
In the following, we numerically compute the background dynamics of our model. In Fig. 2, we plot the evolution of the Hubble rate H, the inflaton field ϕ, as well as the slow-roll parameters , ϕ , η ϕ , and ξ ϕ with respect to the e-folding number N (solid black curves). Our model has four parameters, which are chosen to be, in Planck units, for this specific example. In order to make a comparison, we also compute the background dynamics of quadratic inflation with a mass m a (dotted red curves). We note that we set N = 0 to be the end of inflation where (t end ) = 1. Thus, in these figures, inflation begins on the right where all the slow-roll parameters are small and proceeds to the left where both the Hubble rate and the inflaton field decrease to smaller values.
From the plots in Fig. 2, one can clearly see that the behavior of the slow-roll parameters is in agreement with the arguments given above. It is interesting to note that when the inflaton field approaches ϕ c , its evolution presents a small step feature, and correspondingly, the Hubble rate experiences a sudden decrease at the same moment. Even though this variation is very small, it still leads to a series of changes in the slow-roll parameters. In particular, ξ ϕ , which characterizes the third derivative of the potential with respect to ϕ, is vanishingly small for standard inflationary models. Here, we notice that it can be amplified up to O(10).
We can see from Fig. 2 that the deviations between our model and the regular m 2 ϕ 2 model are still very small. Firstly, this is consistent with the nature of inflationary cosmology in which any deviation from the attractor solution damps out quickly. Secondly, even though these differences are very small, they can affect the process during which the primordial perturbation modes exit the Hubble radius near the pivot scale. We note that in our numerical computations, the pivot scale, which is taken to be k * = 0.002 Mpc −1 , corresponds to the e-folding number N 55. Accordingly, this can leave an imprint in the primordial power spectra. This is the subject that we explore in the following subsection.

B. Perturbation analysis
We already see from the previous section that, at the background level, a step-like feature in the potential induces a short period of time where the slow-roll conditions are violated. We therefore expect this to have implications in the perturbative regime. In fact, the near scaleinvariant power spectra typically obtained from quadratic inflation are likely to be modified, so in this section, we explore the quantitative consequences of the model introduced above.
The power spectra for metric perturbations are usually found by solving the Einstein field equations expanded about the background solution to linear order. For scalar modes, a gauge-invariant perturbation quantity is the curvature perturbation R, and for tensor modes, it is the usual gravitational wave polarization state h (see [38] for a review of cosmological perturbations). The respective power spectra are then defined as P R (k) ≡ k 3 |R k | 2 /(2π 2 ) and P h (k) ≡ k 3 |h k | 2 /(2π 2 ). Given the complexity of the potential energy for the scalar field and of the background dynamics, we solve the perturbations equations for R and h numerically in order to obtain their power spectra. More specifically, we use the recently devel-oped MultiModeCode 1 [39][40][41]. This code is optimized for multi-field inflation, but its implementation makes it easy to evolve single field inflation with the potential defined in Eq. (26).
The resulting power spectra are shown in Fig. 3. We plotted the standard near scale-invariant power spectra for quadratic inflation in black, and the color curves show the modified potential with different parameter values. We first notice that for k 10 −2 Mpc −1 , the standard m 2 ϕ 2 potential and the modified potential yield near identical power spectra. Moreover, for the parameter values shown in Fig. 3, our model predicts a similar tilt and tensor-to-scalar ratio compared to quadratic inflation with n s ≈ 0.96 and r ≈ 0.14 at the pivot scale of k * = 0.05 Mpc −1 . It is only on larger length scales that the spectra deviate from near scale invariance due to the short phase of slow-roll violation during the inflationary period. Overall, the modified potential tends to lower the amplitude of the curvature perturbations and enhance the amount of gravitational waves produced. The effect is more radical when the drop in the potential depicted in Fig. 1 is very steep. In our parametrization of the potential, this is equivalent to having a large value of C H . This is shown by the orange curve which has small damped non-harmonic oscillations for tensor modes but very large ones in the curvature power spectrum. However, such a large value for C H is not expected from string theory (see Sec. II B).
The effect is less drastic when the drop in the potential is smooth and this can be seen with the curves in red, blue, and green. In particular, the smallest effect is found when the large mass scale m b is close to m a (red curve). In this case, the jump in the potential is small and the feature in the power spectra occurs only in the far infrared. Finally, we note that changing the critical field value ϕ c at which the jump in the potential occurs affects the scale at which the power spectra leave near scale-invariance. More specifically, the lower the energy scale of the drop in the potential, the lower the length scale of the feature in the power spectra.
Let us provide some further physical intuition for the above description of the power spectra. The small k modes (large wavelength) exit the Hubble radius at early times when the inflaton is in its first slow-roll phase, i.e. when the potential looks like m 2 b ϕ 2 /2. At this point, the power spectra are given by their usual forms [42] with an amplitude proportional to H 2 /( M 2 p ) for curvature perturbations and proportional to H 2 /M 2 p for tensor modes. Thus, the amplitudes depend on m 2 b (the larger mass scale). Similarly, the large k modes (small wavelength) exit the Hubble radius at late times when the inflaton is in its second slow-roll phase, i.e. when the potential looks like m 2 a ϕ 2 /2, so the amplitude of the spectra depends on m 2 a (the smaller mass scale). In particular, the amplitude To further connect with observations, we evolve the resulting power spectra after inflation to the angular power spectra that can be probed observationally at the time of the CMB with telescopes such as Planck. To do this, we use CAMB 2 [43] where we input our power spectrum for curvature perturbations and tensor modes as shown in Fig. 3 instead of a parameterized power spectrum as it is usually done for the standard ΛCDM model. Furthermore, we use the best fit to the cosmological parameters found by Planck [1] in the code.
First, the temperature (TT) angular power spectrum is shown in Fig. 4. The grey squares show the measured data by Planck 2013 and the black curve shows the standard ΛCDM curve for a parameterized power spectrum after inflation. The color curves show the angular power spectrum for the modified potential, and similar to our conclusion from the power spectra after inflation, the different angular power spectra are very similar on small scales ( 300).
The differences with ΛCDM are more distinct around the first acoustic peak and at low 's. In particular, we notice that the amplitude of the green curve is much smaller than the other ones for 300. This was expected from the fact that the curvature power spectrum for the green curve is the one that deviates from near scaleinvariance at the smallest scale (see lower panel of Fig. 3) since it has the smallest ϕ c . Although the low amplitude of the angular power spectrum around the first acoustic peak may imply larger differences in the cosmological parameters compared to the Planck 2013 results, the green curve has the advantage of explaining the lowsuppression of the spectrum. Such a suppression is also obtained in the case of the blue and red curves, although at a smaller extent than for the case of the green curve, but these curves have the advantage of closely matching ΛCDM around the first acoustic peak. To summarize, the larger the k-mode at which the feature appears in the curvature power spectrum, the larger the deviation from ΛCDM, but the larger the low-suppression is.
The situation is somewhat different for the orange curve in the case of a steep drop in the potential (large C H ). Indeed, this case follows ΛCDM very closely at all angular scales except at = 2 where the amplitude for the modified potential is larger and this seems to be in greater conflict with the observation. This reinforces our intuition that a steep drop in the potential is not favored.
Second, we show the unlensed B-mode polarization angular power spectrum in Fig. 5, i.e. the primordial contribution to the total B-mode power spectrum. The figure shows that the axion monodromy model with different parameter values agrees with the quadratic potential at almost all angular scales. In fact, for 7, the spectra are basically identical. This is in agreement with the top panel of Fig. 3 which showed very similar tensor power spectra for large k.
It is only at very large angular scales ( 7) that one can notice a small deviation. This deviation is only significant when the critical scale ϕ c is small since this is when the deviation from near scale invariance in the primordial power spectrum occurs at larger k. We notice that the amplification at small in the BB power spectrum is very small compared to the suppression in the TT power spectrum. This is in accordance with the power spectra after inflation shown in Fig. 3 where the suppression in P R can be of more than one order of magnitude, whereas the amplification in P h is only of order one. Clearly, the suppression in the TT spectrum and the amplification in the BB spectrum are correlated, so one needs to compare both spectra with observations to constrain the model. Still, the amplification at large angular scales in the BB spectrum is an interesting feature of this model and one hopes to explore this region as well as the entire spectrum with future CMB polarization data.

C. Fitting to cosmological data
After analyzing the temperature angular power spectrum, one would hope to be able to determine which model best fits the observed data. In Fig. 4, the cosmological parameters are fixed, but when one allows them to be free, it is possible to fit all the cosmological parameters to the data given the model that is studied. This is what we explore with MCMC simulations. We perform a model comparison with CosmoMC 3 [44] by running the MCMC code for the red, blue, and green curves of Fig. 4. The usual cosmological parameters (except the ones related to a parameterized power spectrum such as n s and A s , which are ignored) are given the same prior range as Planck [1]. The runs reached convergency as we can see from the Gelman-Rubin statistic R −1 value [45,46] of 0.01325, 0.01276, and 0.01103 in the case of the red, blue, and green curves, respectively. The resulting likelihoods can then be compared to determine which of the three cases best fits the Planck 2013 data, but we find that the likelihoods are all identical within the precision of the results. Furthermore, running the action 2 from CosmoMC leads to the same conclusion: the respective χ 2 are near identical and it is therefore not possible to tell which case is best.
For this reason, we leave model comparison aside and simply compare the CosmoMC results for our model with the standard ΛCDM model from Planck 2013. In Fig. 6, we show constraints on pairs of cosmological parameters and their respective probability distributions for the CosmoMC runs described above and for the results from Planck 2013. We first notice that the colored curve which represent our model are almost perfectly superposed, in agreement with our likelihood analysis. Also, we notice that the resulting best fits to the cosmological parameters are all very close to one another even when comparing with the Planck 2013 results. They differ by fractions of the order of 1% at most. This suggests that the model tested here is in good agreement with the observations. We stress though that the above analysis cannot indicate which of the axion monodromy model and the standard model is best. Indeed, the standard model assumes a parameterized power spectrum of the form P R (k) = A s (k/k * ) ns−1 and thus fits two additional parameters, i.e. A s and n s . This in part explains why the cosmological parameters in Fig. 6 are better constrained by our model than by the standard model and why more correlation seems present (especially for Ω b h 2 ) in the standard model than in the axion monodromy model.

IV. CONCLUSIONS
In the present paper, we proposed a new model of axion monodromy inflation. Analogously to regular models of axion monodromy inflation in which the effective theory is derived from the reduction of a higher dimensional theory on a two-cycle, our model is constructed in the presence of torsional cycles. In particular, our model possesses at least three scalar fields that can be candidates to the inflaton. Amongst them, two fields are free of the η problem due to mass corrections of the moduli stabilization in the Kähler , the cold dark matter density today (Ωch 2 ), the Hubble parameter today (H0), and the dark energy density divided by the critical density today (ΩΛ). The color scheme is the same as for Fig. 4.
potential, and hence, the theory is well established from the perspective of string theory. As a particular realization of this model, we studied quadratic inflation where the potential is effectively transformed to have two mass scales separated by a step-like feature at some critical scale. At the background level, one finds a short phase in which the slow-roll conditions are violated. Perturbatively, the power spectrum of curvature perturbations is either suppressed or oscillates near the critical scale depending on the details of the potential, and the power spectrum of tensor modes is boosted on large scales.
In light of recent and forthcoming CMB experiments, a lot of analyses have aimed at searching for features of inflation or its alternatives (see, e.g., [35,36,[47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65] for extensive discussions). In our model, we have two interesting predictions: one is that there exits a moderate boost in the CMB B-mode polarization spectrum at small 's; the other is a defect of temperature anisotropies in the same regime, which provides a natural interpretation for the low-suppression anomaly in the CMB data. It will be interesting to perform a full global fitting analysis with the Planck 2014 data to confront these predictions with cosmological observations, which will be presented in a follow-up letter [66].
We point out that while this paper was being prepared for submission, the preprint of [67] appeared, which explores features of standard axion monodromy inflation with drifting oscillations.