Probing the Weak Gravity Conjecture in the Cosmic Microwave Background

The weak gravity conjecture imposes severe constraints on natural inflation. A transPlanckian axion decay constant can only be realized if the potential exhibits an additional (subdominant) modulation with sub-Planckian periodicity. The resulting wiggles in the axion potential generate a characteristic modulation in the scalar power spectrum of inflation which is logarithmic in the angular scale. The compatibility of this modulation is tested against the most recent Cosmic Microwave Background (CMB) data by Planck and BICEP/Keck. Intriguingly, we find that the modulation completely resolves the tension of natural inflation with the CMB. A Bayesian model comparison reveals that natural inflation with modulations describes all existing data equally well as the cosmological standard model ΛCDM. In addition, the bound of a tensor-to-scalar ratio r > 0.002 correlated with a striking small-scale suppression of the scalar power spectrum occurs. Future CMB experiments could directly probe the modulation through their improved sensitivity to smaller angular scales and possibly the measurement of spectral distortions. They could, thus, verify a key prediction of the weak gravity conjecture and provide dramatic new insights into the theory of quantum gravity.


I. INTRODUCTION
The cosmic microwave background (CMB) provides a window to physics at very high energy scales. It probes the era of cosmic inflation and could even contain imprints from the theory of quantum gravity. Within recent years, major theoretical advances have been made in understanding how such signatures could possibly look like. The progress roots in the observation that theories of quantum gravity have to fulfill certain self-consistency conditions. A particular intriguing example is the weak gravity conjecture (WGC) which -in its original form -constrains the strength of gauge forces relative to gravity [ ]. The conjecture is motivated by the absence of an infinite tower arXiv:1911.11148v1 [astro-ph.CO] 25 Nov 2019 of stable black hole remnants which would otherwise plague the theory.
Possibly even more important is the application of the WGC to non-perturbative axion physics [ -]. This is because axions are the prime candidates to realize large-field inflation within a consistent theory of quantum gravity -for which string theory is the leading candidate. String axions, descending from higher-dimensional p-form gauge fields, possess continuous shift symmetries which hold at all orders in perturbation theory and survive quantum gravity effects [ , ]. Non-perturbative instanton terms break the shift symmetries down to discrete remnants in a controlled way. In the simplest case, the resulting axion potential features the familiar cosine shape of natural inflation [ ]. Its periodicity is determined by the axion decay constant f which -in natural inflation -has to be trans-Planckian since a too red spectrum of CMB perturbations would otherwise arise. While f > 1 for a single fundamental axion does not arise in a controllable regime of string theory [ , ], an effective trans-Planckian f can consistently be realized via the alignment of two or more axions with sub-Planckian decay constants [ ].
The WGC imposes that any axion must be subject to a (possibly subdominant) modulation with sub-Planckian periodicity f mod . Natural inflation (with f > 1) can still be realized through the axion alignment mechanism, but it necessarily comes with subdominant 'wiggles' on top of the leading potential [ , -]. The resulting scheme was dubbed 'modulated natural inflation' [ ]. It was realized that the modulations find a simple explanation in terms of higher instanton corrections.
The non-perturbative breaking of the axionic shift symmetries in many cases comes from instantons which are described by modular functions (see e.g. [ -]). Therefore, the axion potential exhibits the desired cosine shape with additional wiggles which result from the higher harmonics in the modular functions. It is, in fact, not surprising that modular functions play a crucial role since -as the WGC itself -they are deeply connected to the duality symmetries of string theory. The explicit shape of the inflaton potential has been derived in [ ], where δ mod is the amplitude of the modulation term with sub-Planckian periodicity f mod < 1. It is believed that this form of the potential applies to a wide class of large-field inflation models consistent with the WGC.
The purpose of the present work is to derive the signatures of the weak gravity conjecture in the CMB. The wiggles in the axion potential, which it requires, seed a characteristic modulation While cosmological data do presently not require trans-Planckian excursions of the inflaton field, a natural choice of initial conditions and an observable tensor mode signal make large-field inflation a particular attractive candidate to describe the early expansion of the universe.
in the primordial power spectrum of scalar perturbations which is logarithmic in the angular scale.
We will derive analytic expressions for the primordial power spectra and implement them in the Code for Anisotropies in the Microwave Background (CAMB) [ ]. This will allow us to solve the Einstein-Boltzmann equations for cosmological perturbations and compute the CMB temperature and polarization power spectra. We will then directly test the modulation against the most recent CMB data by Planck [ , ] and BICEP/Keck [ ]. Finally, we will make exciting predictions for spectral distortions in the CMB which can be tested with future satellite missions.

II. THE WEAK GRAVITY CONJECTURE
In this section, we review the WGC [ ] and its implications for large-field inflation in more detail. Originally, the WGC was formulated to constrain the strength of gauge forces relative to gravity. In its so-called electric version it states that any U( ) gauge theory coupled to gravity should contain a particle with charge-to-mass ratio q m > 1 . ( ) In the absence of a lower bound on q, peculiar consequences would arise: for vanishing coupling strength, the gauge boson kinetic term becomes infinite. A non-propagating gauge boson implies that the gauge symmetry effectively behaves as a global symmetry. This appears problematic in the light of strong arguments against the existence of exact global symmetries [ -]. Violation of the weak gravity conjecture would, furthermore, imply that extremal black holes cannot decay.
The theory would be plagued by an infinite tower of stable gravitational bound states. There also exists a magnetic version of the WGC which states that any U( ) gauge theory with coupling g breaks down at a cutoff scale Λ < 1/g [ ]. This condition ensures that the minimally charged magnetic object of the gauge theory is not a black hole. The application of the magnetic WGC to axion systems is less straight-forward compared to the electric version. In particular, it is still open, to which extent the magnetic WGC constrains effective trans-Planckian axion decay constants [ -]. More precisely, q in this expression stands for the product of charge and gauge coupling.  In this work, the we are mainly interested in the application of the WGC to axion systems [ -].
We consider an axion whose shift symmetry is broken to a discrete remnant via instanton terms.
In the simplest case, the resulting axion potential takes the form where S denotes the instanton action, and we use Planck units (M P = 1) throughout this work. The string theory duality symmetries provide an inherent link between U( ) gauge theories and non-perturbative axion physics. In simple terms, the U( ) gauge charge translates to the inverse axion decay constant and the mass translates to the instanton action. For a single axion, the WGC then requires (f S) −1 > 1 for at least one instanton of the theory. It is straight-forward to generalize the WGC to multi-axion systems with Lagrangian where the index α runs over the axions and i runs over the instanton terms. The WGC is satisfied if the convex hull, spanned by the vectors ±(c i,α /S i ), contains a ball with radius unity. More The convex hull condition has originally been formulated for the gauge theory version of the WGC in [ ].
precisely, the required minimal radius is of O(1), but its exact value depends on the type of axion under consideration. Some examples are discussed in [ ]. In a system with many axions of different mass, the convex hull condition ensures that the one-axion WGC is satisfied after all heavier axions have been integrated out. The strong WGC imposes that the leading instanton(s) fulfill(s) the WGC.
One immediately realizes that the WGC has dramatic implications for large-field inflation which are illustrated in Fig. . The strong WGC entirely excludes natural inflation. This is because it requires a periodicity f < 1 of the leading instanton term, i.e. of the one which dominates the potential. Given that the strong WGC is most likely overrestrictive, the more interesting scenario is, however, that only the 'standard' WGC applies which is theoretically on much firmer grounds.
In this case, the condition (f S) −1 > 1 can be satisfied by a subleading instanton. Large-field inflation with f > 1 can now be realized, but it necessarily comes with a subdominant highfrequency modulation on the potential.
We expect that the amplitude of this modulation cannot be arbitrarily small. In the next section, we will discuss the explicit realization of trans-Planckian f through the alignment mechanism. We will see that the modulation term increases rapidly for trans-Planckian f and leads to an effective cutoff scale Λ 4 < e −c f with an O(1) number c. In the observationally most interesting regime f = 2 − 10, the modulation can still be controlled, but it affects cosmological observables. This raises the exciting prospect of testing the WGC through CMB data. Before we discuss this in detail, we turn to the concrete realization of natural inflation with modulations.

III. MODULATED NATURAL INFLATION
Natural inflation (NI) in its simplest form assigns a cosine potential to the inflaton (cf. ( )).
Let us briefly describe the axion alignment mechanism for realizing an effective trans-Planckian decay constant for the axion [ ]. One considers two canonically normalized axion fields φ 1,2 with potential V = Λ 4 a e −Sa 1 − cos where all individual axion decay constants f 1,2 , g 1,2 are taken to be sub-Planckian. For simplicity, we consider the case Λ 4 b e −S b Λ 4 a e −Sa such that we can integrate out the heavy axion directioñ φ ∝ φ 1 g 1 + φ 2 g 2 . The potential for the light axion φ ∝ φ 1 g 2 − φ 2 g 1 is ( ) The effective axion decay constant f is strongly enhanced compared to the individual axion decay constants in the the alignment limit f 1 /f 2 g 1 /g 2 , allowing in particular for f > 1.
However, we have seen in the previous section, that the pure cosine potential in combination with a trans-Planckian axion decay constant is in conflict with the WGC. Indeed, one easily verifies that the convex hull for the potential ( ) becomes increasingly narrow in the alignment limit and does not contain the unit ball. Hence, the convex hull condition is violated. As described earlier, a subdominant modulation with sub-Planckian periodicity can reconcile NI with the WGC [ , - ]. From the bottom-up perspective, introducing the modulation may seem ad hoc. However, subleading instantons appear quite naturally in string theory. Non-perturbative terms often contain a series of higher harmonics. This is well-known for toroidal string compactifications [ -, ], but also established in more general string theory setups [ , , ]. The possible impact of higher instantons on inflation models has also been noted in [ , ].
The subleading instantons manifest in form of η-or θ-functions which replace the exponential shape of a single instanton. As an example, we consider toroidal string compactifications with superpotential and Kahler potential [ , ] with the chiral superfields ψ 1,2 and the Kahler moduli T 1,2 which parameterize the volume of two internal sub-tori. The imaginary parts of the T i are identified with the two axions participating in the alignment mechanism. The parameters A 1,2 , B 1,2 , which are taken to be constants, descend from integrating out heavy chiral fields with non-vanishing vacuum expectation values. The coefficients n 1,2 , m 1,2 are determined by the localization properties of the chiral fields. The model above can e.g. be realized in heterotic orbifold compactifications, where the non-perturbative terms are identified with world-sheet instantons [ ]. By expanding the Dedekind η-function, one can verify that the superpotential contains an infinite number of subleading instantons, Including the full series of higher harmonics, it can be shown that the convex hull condition is satisfied even in the alignment case (see [ ]).
The relevant potential of the light axion direction φ follows after setting the remaining fields to their vacuum expectation values and integrating out the heavy axion direction. We choose the indices 1, 2 such that B 2 > B 1 and T 1,0 > T 2,0 , where T i,0 denotes the expectation value of T i . This leads to the approximate form of the potential [ ] Up to the last bracket, this is simply the potential of natural inflation with ( ) The relative modulation amplitude δ mod increases towards the alignment limit. This is because the leading instantons partly cancel for alignment, whereas the higher orders are not affected by cancellation. Note that f mod is of the size of the original axion decay constants, i.e. it is generically sub-Planckian. The ratio f /f mod is a measure for the alignment -the more the axion decay constants are aligned, the bigger this factor becomes.
A very interesting property of the model is that it exhibits an intrinsic relation between the alignment factor and the cutoff-scale. Among all possible parameter choices we find where the factor π/36 in the exponent is related to the expansion of the η-function and may somewhat differ for other modular functions. Since Λ effectively determines the scale of inflation, It is sufficient to include the next-to-leading terms in the η-expansion in order to arrive at the potential ( ).
Contributions beyond this order are suppressed even in the alignment limit. We required T1,2 > 1 and A1,2, B1,2 < 1 in order to be in the controllable regime of the theory.
it can be constrained observationally. The correct normalization of the CMB power spectrum The scheme described by the potential ( ) will in the following be called 'modulated natural inflation' (MNI). We point out that the shape of the potential does not depend on the concrete compactification, but merely on the fact that the higher harmonics exist. Presumably, it is applies universally to natural inflation models consistent with the weak gravity conjecture.
In Tab. I, we provide an exemplary parameter choice for the MNI model. We will later show that this benchmark point provides a very good fit to existing CMB data.

IV. PRIMORDIAL POWER SPECTRUM OF MODULATED NATURAL INFLATION
In this section we derive analytic expressions for the scalar and tensor power spectra of MNI.
It is convenient to start with the simpler case of natural inflation, i.e. to set δ mod = 0 for the moment. In a second step, we will later derive how the expressions are modified for non-vanishing δ mod . We introduce Furthermore, we define the (potential) slow roll parameters where the index 0 again indicates that we are neglecting the modulation for the moment. The scalar and tensor power spectra are well-approximated by a power law form determined by the slow roll expressions, where k * denotes the pivot scale. The normalization A 0 is given as ( ) The scalar and tensor spectral indices are determined as In the above expressions, we have introduced N * which denotes the number of e-foldings when the CMB scales crossed the horizon. One typically finds N * 50 − 60 with the exact value depending on the post-inflationary evolution of the universe. We have also employed that the corresponding field value φ * is determined by with φ end marking the end of inflation (where the slow roll condition is violated).
We now turn to the power spectrum of MNI including the modulation. The slow roll parameters ε and η of the full model are introduced as in ( ), with V 0 replaced by V (defined in ( )). Notice that each derivative acting on the modulated part of the potential pulls out a factor f −1 mod > 1. Therefore, the modulations have a much stronger impact on η than on ε. In order to remain in the slow-roll regime, we need to require |η| 1 which translates to If this condition is violated, a strong scale-dependence of the power spectrum arises which is inconsistent with observation. The only caveat consists in very small f mod , for which the power spectrum oscillations are so rapid that they are not individually traced in the CMB. This situation can, for example, arise in axion monodromy inflation models [ , ]. However, in MNI such highfrequency oscillations are inaccessible due to the constraint on the alignment factor ( ) which prevents too small f mod . We expect not more than a few oscillations over the range of scales observable in the CMB. Therefore, we can safely require slow roll and impose ( ).
The condition ( ) ensures that we can treat the modulation as a perturbation in ε. Including terms up to first order in δ mod , we find Notice that we have replaced ε by ε 0 * (= ε 0 evaluated at the pivot scale) in the definition of δ.
This is justified since it merely amounts to neglecting a second order correction (we fully keep track of the scale dependence of ε 0 in the leading term). The modulation in ε causes a modulation in the scalar power spectrum As a final step, we need to relate the field value to a physical scale in the CMB. The modulation does (mildly) affect the relation φ(k). However, this is negligible compared to the direct impact of the modulation term in ( ). We will, therefore, employ the 'unperturbed' relation ( ) and perform an expansion around φ * . Neglecting the scale-dependence of ε in the vicinity of the pivot with φ * determined from ( ). Our final expression for the scalar and tensor power spectra of MNI reads where we expressed ε 0 * in terms of the tensor spectral index n t and introduced Notice that we neglected modulations in the tensor power spectrum. The latter is set by the energy scale of inflation V , while the scalar power spectrum scales with V /ε. Therefore, the modulation in P t is suppressed by δ mod /δ ∼ (f mod ε 0 * ) −1 1 compared to the one in P R . Therefore, in the regime, where P R is consistent with observation, the modulations in P t play no role.
In order to test the validity of our analytic approximation, we also performed exact numerical calculations of the primordial power spectra for a number of benchmark points. This was done by The expressions ( ) and ( ) seem to suggest a (slight) violation of the inflationary consistency relation r = −8nt in MNI. In reality, the consistency relation is, however, satisfied due to a small modulation in nt. The latter is neglected in ( ) since its effect on Pt resides at the per mill level in the relevant parameter space of MNI. Only the phase ∆ is not very accurately determined by our approximation ( ). This is expected since the phase is extremely sensitive to changes of the remaining parameters. We can safely ignore this issue since the phase is essentially a free parameter. Through very minor change of f mod one can realize any value of ∆ while virtually not affecting the power spectrum otherwise.
The deviation of P R from the standard power-law form results in an intriguing deviation from ΛCDM cosmology. If the modulations of the CMB power spectrum could be experimentally proven, this would yield strong indication that the WGC holds in nature. The prospect of testing general laws of quantum gravity by the spectra of cosmological perturbations is extremely exciting.

V. CMB ANALYSIS
The analytic expressions of the primordial power spectra ( ) and ( ) can directly be implemented in standard tools for cosmological analysis and the corresponding predictions can be derived. We are particularly interested in the consistency of MNI with present CMB data and in the most striking observable differences compared to ΛCDM. In the following, we describe the analysis method adopted in this work. The discussion of the results is given in Sec. VI.

A. Monte Carlo analysis
We perform a Monte Carlo Markov Chain (MCMC) analysis to explore the parameter space of MNI and to derive constraints on parameters from a combination of the latest cosmological data, employing the publicly available sampler CosmoMC [ ]. In order to solve the Einstein-Boltzmann equations for cosmological perturbations and compute the theoretical predictions , such as the CMB temperature and polarization power spectra, we modified the current version of the Code for Anisotropies in the Microwave Background (CAMB) [ ], so that the primordial spectra are now given as ( ) -( ). Since it is of particular interest, whether CMB data favor a non-vanishing modulation, we will treat the case δ = 0 separately. It corresponds to standard natural inflation and will, hence, be dubbed 'NI' in the following. For comparison with the cosmological standard model, we will also perform the CMB analysis for the ΛCDM scenario (as reference model) with one parameter extension, i.e. leaving the tensor-to-scalar ratio as free parameter (= ΛCDM + r).
The cosmological models are fully specified by the following set of parameters: the physical densities of cold dark matter Ω c h 2 and baryons Ω b h 2 , the angular size of the sound horizon at recombination θ, the reionization optical depth τ , the primordial amplitude ln(10 10 A s ) and the remaining inflationary parameters. For ΛCDM + r, we vary the spectral index n s of scalar perturbations and the tensor-to-scalar ratio r.
In the standard CAMB code, the primordial power spectrum is parameterized, at first order, as the power-law PR = As k k * (ns−1) , which we retain when we focus on constraints on the ΛCDM model. For the MNI model, the choice of input parameters is ambiguous: we could either express the model in terms of the potential parameters in ( ), or directly by the derived power spectrum parameters. In order to allow for a more meaningful statistical comparison with ΛCDM + r, the input primordial power spectra should share a similar structure, i.e. we should treat n 0 on the same footing as n s in ΛCDM + r. Therefore we decided in favor of the second option and to sample over n 0 , δ and f mod , ∆ complemented with the constraint n 0 < 1 − 2 N * (n 0 < 0.967 for N * = 60). Notice that n t is not treated as an independent parameter. At each sampling step, we can invert ( ) to get f and then obtain n t from ( ). While ( ) cannot be inverted analytically, a very precise estimate is found by employing the Warner approximation for the inverse Langevin function. We obtain Due to the increased number of parameters and their mutual degeneracies, the MCMC analysis for MNI is extremely time-consuming. Therefore, we decided to fix the e-fold number N * = 60 which would correspond to the case of instantaneous reheating after inflation. We have run a test case to verify that our results are qualitatively unchanged if we allow N * to float in the range N * = 50 − 60. In the special case of NI (δ = 0), the parameters f mod and ∆ become irrelevant.
In the analysis, we assume purely adiabatic initial conditions. For the concrete realization of modulated natural inflation described in Sec. III this amounts to neglecting quantum fluctuations in field directions orthogonal to the inflaton. This is a valid approximation in a large fraction of The upper bound on n0 is approached in the limit f → ∞.
For the test case, the final distribution is somewhat peaked towards the boundary N * = 60, with negligible spread in the other parameters with respect to the case of fixed N * .
We also tested a logarithmic sampling over δ. The logarithmic prior tends to give more statistical weight to smaller values of δ, therefore making the results less distinguishable from NI.
the model parameter space, where an effective single field description arises. We also include a contribution of N eff = 3.046 active neutrinos, with a total mass of m ν = 0.06 eV.
We recall that sampling over the phenomenological parameters describing the primordial power spectra P R and P t is only one of the possible choices for deriving constraints on inflationary parameters. For example, the Planck collaboration has extensively used the method of sampling the Hubble Flow Functions (HFF) [ -]. Here, we opt for the phenomenological approach of sampling over the power spectra parameters for two reasons: first, all the models under scrutiny in this work obey slow-roll conditions and therefore we do not need a method that allows for a more general description beyond slow-roll approximations. Secondly, it has been shown that the two methods (phenomenological and HFF) agree at a level that is acceptable to answer the question of whether the MNI model reasonably describes current cosmological data [ ].
We choose to build our data set combining measurements of CMB temperature and polarization anisotropies as well as measurements of CMB lensing signal from the latest Planck satellite data release (the baseline combination labeled as "TTTEEE+lowE+lensing" in the Planck pa- We present the results of the analysis in Sec. VI.

B. Model comparison
We also perform a model-comparison analysis of MNI with respect to the standard ΛCDM + r model and NI (i.e. MNI with fixed δ = 0). Indeed, in addition to providing constraints on the model parameters, we also want to know to what extent MNI describes the cosmological data as well as the standard ΛCDM + r scenario and whether a non-vanishing modulation is preferred (comparison with NI). To answer these questions, we employ the Deviance Information Criterion In particular, we include one massive neutrino with mass 0.06 eV and 2.046 massless species. This choice approximates the case of minimal mass in the normal ordering scenario for massive neutrinos, and has been adopted  model selection criterion. With this caveat, we follow previous works in literature and consider ∆DIC = 10/5/1 to provide, respectively, strong/moderate/null preference for the reference model.

VI. RESULTS OF THE CMB ANALYSIS
The main results of our analysis are presented in Tab. III, where the constraints on the inflationary parameters are presented. Noteworthy, the ones on other cosmological parameter do not present significant differences with those of the standard model, so it was decided not to show them. To facilitate comparison, we also show the derived inflationary parameters, calculated at the pivot scale k * = 0.05 Mpc −1 . We employ the standard definitions A s = P R (k * ) and ( ) Turning first to natural inflation with the pure cosine potential, we confirm a mild tension with CMB data (in agreement with Planck results). In the parameter region, where NI reproduces the preferred spectral index, the tensor-to-scalar ratio is slightly to high. The best fit point features an axion decay constant of f = 6.8 which minimizes the tension. It exhibits a smaller n s , but larger r compared to ΛCDM.
We observe that MNI is able to resolve the tension faced by NI. This possibility is enabled by the modulation term in ( ) whose presence is enforced by the weak gravity conjecture. As stated in Tab. III, the data prefer a non vanishing modulation amplitude of δ = 0.05 − 0.23 (at the 1 σ level). The key feature in MNI is the possibility to allow for smaller value of n 0 , while keeping n s in the preferred window of 0.96 − 0.97. Indeed, there is an inverse degeneracy between n 0 and δ as shown in Fig. : when larger values of δ can be sampled, smaller values of n 0 are allowed. The modulation term adjusts such that it 'compensates' the otherwise too strong scale dependence of the scalar power spectrum which would derive from the small n 0 (remember that in NI n 0 simply corresponds to n s ). A satisfactory description of CMB data is, however, only obtained within a limited range of n 0 0.93. For smaller values of n 0 , one can still obtain n s in the desired range through a relatively large modulation term. However, the resulting power spectrum would deviate too strongly from the power-law form leading to a degraded fit.
Another interesting observation is that CMB data pin down the axion decay constant of MNI in a relatively narrow window f 4.1 − 5.4 (at the 1 σ level). This implies that no large parametric enhancement of the axion decay constant is required to match CMB observations. Indeed, the Other possibilities to resolve the tension of natural inflation with CMB data have e.g. been discussed in [ , ].  A more striking result concerns the tensor-to-scalar ratio. In the ΛCDM + r model, there is currently only an upper limit, r < 0.065 at % C.L.. The latter was obtained in the context of power-law primordial spectra with the inflation consistency relation n t = −8r imposed.

ΛCDM
The situation changes drastically for MNI. As can be seen in Fig If MNI is realized in nature, we hence expect a discovery of tensor modes by the next generation of CMB experiments. This is easy to understand: CMB constraints on the spectral index (and its scale dependence) exclude too small values of n 0 (see Fig. ). The lower bound on n 0 translates to a lower bound on r through ( ) and ( ). We emphasize that this lower limit is not driven by a preference of r > 0 in the existing CMB data. Rather, it is forced by the model (in combination with the experimental constraints on n 0 ). We can conclude that the observation of a non-vanishing tensor mode signal is crucial for probing the weak gravity conjecture. If future experiments exclude r > 0.002, there would be little hope for observing any signature of the WGC in the CMB. If, on the other hand, a tensor mode signal is detected, experimental tests of the WGC may become feasible (see next section). These will be related to the scale-dependence of the spectral index which is visible in the right panel of Fig. , where we depict the probability contours for the running n run and running-of-the-running n runrun of the spectral index. Notice a preference for negative running in MNI.
We can now present the results of the model comparison analysis. We take ΛCDM + r to be the reference model, so that the DIC M ref ≡ DIC ΛCDM and ∆DIC M ≡ DIC M − DIC ΛCDM . We obtain the following: • ∆DIC MNI = 0.5, that is a no preference of the current cosmological data between ΛCDM +r and modulated natural inflation. Both models describe the existing data equally well; • ∆DIC NI = 3.8, that is a mild preference of both, ΛCDM+r and modulated natural inflation with respect to natural inflation; A deeper understanding can be obtained if we break down the DIC values in their components, according to ( ). The three models show the same Bayesian complexity p d 30, with differences at the level of the first decimal figure. Therefore, they share the same number of effective parameters needed to describe the data. The best fit and average likelihoods are also similar between ΛCDM+r and MNI, with −2 lnL 3517 and −2ln L 3547. Natural inflation is penalized with respect to both ΛCDM + r and MNI by significantly higher values of the best fit likelihood −2 lnL = 3523 and of the average likelihood −2ln L = 3552.
We should, hence, emphasize that the modulation term in the MNI potential fulfills two purposes: it renders the inflation model consistent with quantum gravity constraints (in the form of the weak gravity conjecture) and resolves the (mild) tension of NI with CMB data.

VII. FUTURE PROSPECTS
In the previous section, we have seen that MNI provides as good as a fit to all current cosmological data as the cosmological standard model (ΛCDM + r). In this section, we briefly discuss how future cosmological surveys can distinguish between the two models.
From the discussion in the previous section, we have seen that current data, when interpreted in the context of MNI, prefer non-zero values of r > 0.002, see Fig. . Future CMB experiments will strongly improve the sensitivity to B-modes, and therefore will be beneficial to our understanding of the viability of the MNI model. For the reader's convenience, we briefly report the expected sensitivity σ(r) on the tensor-to-scalar ratio r, or the expected exclusion limit at % C.L.: • the ground-based Simons Observatory (first light in ) [ , ] will probe σ(r) = 0.003.
• CMB-S (proposed project completion in ) [ , ] will set a limit r < 0.001 in absence of a tensor signal or clearly detect tensor modes if r > 0.003.
• the satellite mission LiteBIRD (selected for launch in ) [ ] will reach σ(r) ∼ 0.001 (exact value depends on the specific noise model) • the proposed satellite mission PICO [ ] would discover tensor modes at 5σ significance if r > 5 × 10 −4 .
With future CMB surveys, we envisage that we could face two different situations: one possibility is that the overall amplitude of the primordial gravitational wave spectrum is so low that only an upper bound on r will be put by future experiments. In this case, the probability contours for MNI in the (n s , r) plane will shrink, but they will be still centered at non-zero r given the improved sensitivity of future cosmological surveys on n 0 (see the discussion in the previous section about the relation between n 0 and r in MNI). Via the statistical tools employed in this work, one would find strong statistical preference for ΛCDM + r compared to MNI once the exclusions approach the lower bound set by data when interpreted in the context of MNI. At this point it would be clear that MNI is not the correct model of inflation and there is no hope of detecting signatures of the weak gravity conjecture in the CMB.
The much more exciting possibility is that future CMB experiments discover tensor modes at high statistical significance. Not only is the detection of non-vanishing r a scientific milestone, it would also be an important step towards testing the MNI model and, thus, the possible signatures of the WGC. In Fig. , we show the BB power spectra for both, ΛCDM + r and MNI, at the corresponding best fit points. The expected sensitivity from future surveys is represented with the error bars forecasted from LiteBIRD as the pink errorbars on top of the tensor BB in ΛCDM + r [ , ]. For comparison, the current error bars from BK are also shown [ ]. Notice that the expected BB signal is larger in MNI compared to ΛCDM+r due to the higher value of r at the best fit point (see Tab. III). The magnitude of a detected tensor signal, in combination with sensitivity improvements on the spectral index, could thus already lead to a slight statistical preference for either MNI or ΛCDM + r. We defer to future works for a thorough forecast analysis.
In addition to observable imprints on the tensor spectrum, a key feature of MNI is a scaledependent modulation term in the spectrum of scalar perturbations. In Fig. , we show the comparison between the best fit scalar power spectra obtained within ΛCDM + r (black line) and MNI (red line). We also show the power spectrum in the MNI benchmark model (see Tab. I)   proposed that can measure spectral distortions in the CMB frequency spectrum (see e.g. the PIXIE proposal [ , ]). These surveys would be able to probe much smaller scales (k 10 3 Mpc −1 ) than those accessible to experiments targeted to CMB anisotropies, and have the potential to test deviations from a standard power-law behaviour of P R [ , ]. It is clear that the possibility to access such a wide range of scales would either reduce the region of parameter space available to MNI to mimic a power-law behavior or allow to identify deviations from a standard power-law that manifest only at the smallest scales. In essence, we have identified a promising path to discover modulated natural inflation and, thus, to directly verify a key prediction of the weak gravity conjecture (namely the modulations in the power spectrum). Specifically, we have predicted a tensor mode signal correlated with a small scale suppression in the scalar power spectrum. In Fig. , we provide the posterior distribution in the two key observables: the tensor-to-scalar ratio and the amplitude of the scalar power spectrum at small (k = 10 3 MPc −1 ) scales. A clear separation between the preferred regions is observed between MNI and ΛCDM + r.

VIII. SUMMARY AND CONCLUSIONS
The weak gravity conjecture imposes stiking constraints on quantum field theories coupled to gravity. While the conjecture is supported by strong arguments related to black hole remnants, efforts towards its direct theoretical proof are still ongoing. In this work, we followed a different avenue and asked the question, whether predictions of the WGC can be tested experimentally.
Specifically, we concentrated on signatures in the cosmic microwave background.
Our working assumption was that inflation is realized through an axion field and that its potential is generated by non-perturbative instanton effects. This framework is particularly appealing since the flatness of the inflaton potential carries a strong protection from the underlying shift symmetry which is preserved at the perturbative level. In the simplest case, the resulting potential would display a cosine shape as in the prominent model of natural inflation. The observed nearly scale invariant spectrum of inflationary perturbations requires the corresponding axion decay constant to be trans-Planckian.
The WGC provides concrete constraints on trans-Planckian axions. It states that a trans-Planckian axion decay constant can only be realized if the potential exhibits an additional (possibly subdominant) modulation with sub-Planckian periodicity which manifests in the form of 'wiggles' on the leading potential. We provided a concrete realization of the modulation within string theory, where instanton effects arise in the form of modular functions. These contain a series of higher harmonics which can naturally be identified with the sub-Planckian modulation imposed by the WGC. The explicit inflaton potential, which is expected to hold rather generically in natural inflation models consistent with the WGC, is given in ( ). The model was dubbed 'modulated natural inflation'.
We then derived analytic expressions for the scalar and tensor power spectra (see ( ) and ( )).
The wiggles in the inflaton potential imposed by the WGC translate to modulations in the scalar power spectrum. The latter exhibit a frequency which changes logarithmically on angular scales.
The power spectrum is distinct from the cosmological standard model (ΛCDM + r) in which it is restricted to a plain power law behavior.
We implemented the primordial spectra of modulated natural inflation in the CAMB code and computed cosmological predictions such as the temperature and polarization power spectra.
These were tested against the latest CMB data from Planck, BICEP/Keck and BAO data from SDSS-BOSS. Parameter constraints were derived in an MCMC analysis employing the sampler CosmoMC.
A first important result was that the modulations improve the consistency of natural inflation with the CMB. We then performed dedicated statistical comparison of modulated natural inflation against ΛCDM + r. The outcome was that both models describe all existing cosmological data equally well. The Deviance Information Criterion yielded absolutely no statistical preference for any of the models. They also share very similar posterior distributions in the familiar n s -r-plane (see Fig. ).
A striking difference between both models is, however, that modulated natural inflation requires a non-vanishing tensor mode signal r > 0.002 at % C.L.. This lower limit is within reach of near-future ground-based CMB experiments such as the Simons Observatory. Furthermore, the scalar power spectrum of modulated natural inflation looks strikingly different from a power law outside the regime of scales presently probed by the CMB. In particular, a significant suppression of power at k 100 Mpc −1 is predicted (see Fig. ). Such small angular scales (large k) are accessible to proposed future CMB missions such as PIXIE via the measurement of spectral distortions.
Although the expected signal is a factor of a few below the forecast sensitivity of PIXIE, we argue that the possibility to test the weak gravity conjecture via spectral distortions deserves further attention. Additional signatures related to the power suppression may arise e.g. in structure formation.
We can, hence, envision two situations: either CMB experiments will exclude r > 0.002 and rule out modulated natural inflation. This would blow all hopes of observing a signature of the WGC in the cosmic microwave background. If, on the other hand, a tensor mode signal is discovered, the weak gravity conjecture predicts that it must be intrinsically linked to a small scale suppression of the scalar power spectrum.
The prospect of finding evidence for the weak gravity conjecture in future CMB data is extremely exciting. It would provide invaluable insights into the theory of quantum gravity. Provided that a tensor mode signal in the CMB is measured by near future CMB experiments, our analysis makes a poweful case for a dedicated satelite mission devoted to spectral distortion as a probe of signals generated by inflation.

ACKNOWLEDGMENTS
We would like to thank Katherine Freese, Massimiliano Lattanzi, Hans Peter Nilles and Luca