We apply the Effective Field Theory of Inflation to study the case where the continuous shift symmetry of the Goldstone boson π is softly broken to a discrete subgroup. This case includes and generalizes recently proposed String Theory inspired models of Inflation based on Axion Monodromy. The models we study have the property that the 2-point function oscillates as a function of the wavenumber, leading to oscillations in the CMB power spectrum. The non-linear realization of time diffeomorphisms induces some self-interactions for the Goldstone boson that lead to a peculiar nonGaussianity whose shape oscillates as a function of the wavenumber. We find that in the regime of validity of the effective theory, the oscillatory signal contained in the n−point correlation functions, with n > 2, is smaller than the one contained in the 2-point function, implying that the signature of oscillations, if ever detected, will be easier to find first in the 2-point function, and only then in the higher order correlation functions. Still the signal contained in higher-order correlation functions, that we study here in generality, could be detected at a subleading level, providing a very compelling consistency check for an approximate discrete shift symmetry being realized during inflation.

Peaking consistently in June for nearly eleven years, the annual modulation signal reported by DAMA/NaI and DAMA/LIBRA offers strong evidence for the identity of dark matter. DAMA's signal strongly suggest that dark matter inelastically scatters into an excited state split by O(100 keV). We propose that DAMA is observing hyperfine transitions of a composite dark matter particle. As an example, we consider a meson of a QCD-like sector, built out of constituent fermions whose spin-spin interactions break the degeneracy of the ground state. An axially coupled U (1) gauge boson that mixes kinetically with hypercharge induces inelastic hyperfine transitions of the meson dark matter that can explain the DAMA signal.This letter proposes a new class of inelastic dark matter (iDM) models that can explain the annual modulation reported by DAMA/NaI and DAMA/LIBRA [1]. DAMA's signal peaks in early June, consistent with dark matter scattering, and has remained in phase for nearly eleven years. Moreover, the fractional modulation of the signal appears anomalously large, and the nuclear recoil spectrum has a peak near E R O(30 keV).The hypothesis that dark matter scatters inelastically off nuclei into a O(100 keV) excited state elegantly explains the salient features of the DAMA signal [2]. IDM models predict nuclear recoil spectra with a characteristic peak and an O(1) modulation fraction [3]. The large dark matter velocity threshold required by inelastic kinematics also implies that heavier nuclei targets like 127 I in DAMA provide enhanced signal sensitivity relative to lighter targets such as 74 Ge in CDMS.In composite inelastic dark matter models (CiDM), DAMA's observed signal arises from inelastic hyperfine transitions of a composite dark matter particle. (For other examples of composite dark matter, see [4].) We illustrate this mechanism with a simple model where the majority of dark matter is a meson of a strongly coupled SU (N c ) gauge sector that confines near Λ GeV. These mesons are comprised of constituent fermions whose hyperfine interactions split the ground state by O(100 keV). When one constituent quark is non-relativistic, a hierarchy between the hyperfine scale and the dark matter mass follows inevitably from an enhanced spin symmetry. The dark matter couples to a new U (1) A vector boson that kinetically mixes with the Standard Model's hypercharge [5]. Another version of iDM with kinetic mixing is given in [6]. Axial couplings of the U (1) A to the constituent fermions mediate inelastic hyperfine transitions that dominate low-energy nuclear scattering.The model considered here has two Dirac fermions, Ψ H and Ψ L , transforming in the fundamental representation of the SU (N c ) gauge group. The new U (1) A couples axially to Ψ H,L , each of which have equal and opposite unit charge. We introduce a charge-2 Higgs φ, whose vacuum expectation value generates a mass for Ψ H,L and the A . The U (1) A is non-anomalous for this particle content. The dark matter candidate is aΨ L Ψ H bound state, and its stability ca...