The most simple superrenormalizable model of quantum gravity is based on the general local covariant six-derivative action. In addition to graviton such a theory has massive scalar and tensor modes. It was shown recently that in the case when the massive poles emerge in complex conjugate pairs, the theory has also unitary S-matrix and hence can be seen as a candidate to be a consistent quantum gravity theory. In the present work we construct the modified Newton potential and explore the gravitational light bending in a general six-derivative theory, including the most interesting case of complex massive poles. In the case of the light deflection the results are obtained within classical and semiclassical approaches.MSC: 53B50, 83D05, 81T20

Recently there has been a growing interest in quantum gravity theories with more than four derivatives, including both their quantum and classical aspects. In this work we extend the recent results concerning the non-singularity of the modified Newtonian potential to the most relevant case in which the propagator has complex poles. The model we consider is Einstein-Hilbert action augmented by curvature-squared higher-derivative terms which contain polynomials on the d'Alembert operator. We show that the classical potential of these theories is a real quantity and it is regular at the origin despite the (complex or real) nature or the multiplicity of the massive poles. The expression for the potential is explicitly derived for some interesting particular cases. Finally, the issue of the mechanism behind the cancellation of the singularity is discussed; specifically we argue that the regularity of the potential can hold even if the number of massive tensor modes and scalar ones is not the same.

Pluto and Eris are icy dwarf planets with nearly identical sizes, comparable densities and similar surface compositions as revealed by spectroscopic studies 1,2 . Pluto possesses an atmosphere whereas Eris does not; the difference probably arises from their differing distances from the Sun, and explains their different albedos 3 . Makemake is another icy dwarf planet with a spectrum similar to Eris and Pluto 4 , and is currently at a distance to the Sun intermediate between the two. Although Makemake's size (1,420 6 60 km) and albedo are roughly known 5,6 , there has been no constraint on its density and there were expectations that it could have a Plutolike atmosphere 4,7,8 . Here we report the results from a stellar occultation by Makemake on 2011 April 23. Our preferred solution that fits the occultation chords corresponds to a body with projected axes of 1,430 6 9 km (1s) and 1,502 6 45 km, implying a V-band geometric albedo p V 5 0.77 6 0.03. This albedo is larger than that of Pluto, but smaller than that of Eris. The disappearances and reappearances of the star were abrupt, showing that Makemake has no global Pluto-like atmosphere at an upper limit of 4-12 nanobar (1s) for the surface pressure, although a localized atmosphere is possible. A density of 1.7 6 0.3 g cm 23 is inferred from the data. Stellar occultations allow detection of very tenuous atmospheres and can provide accurate sizes and albedos 9,10,11,3,12 , so we embarked on a programme of predicting and observing occultations by (136472) Makemake, also known as 2005 FY 9 . The occultation of the faint star NOMAD 1181-0235723 (with magnitude m R 5 18.22, where NOMAD is the Naval Observatory Merged Astronomic Dataset) was predicted in 2010 by methods similar to those used to predict occultations by several large bodies 13 , but refined as shown in Supplementary Information section 1. We arranged a campaign involving 16 telescopes, listed in Supplementary Table 1. The occultation was successfully recorded from seven telescopes, listed in Table 1, at five sites. From the images obtained, we made photometric measurements as a function of time (light curves).The light curves of the occultation are shown in Fig. 1. Fitting synthetic square-well models to the light curves yielded the disappearance and reappearance times of the star (Table 1), from which we calculate one chord in the plane of the sky for each site (see Supplementary Information section 3). On the basis of analyses of the light curves, taking into account the cycle time between the images and the dispersion of the data, we deduce that there were no secondary occultations, so we can reject the existence of a satellite larger than about 200 km in diameter in the areas sampled by the chords. The result is consistent with a deep-image survey that did not find any satellites 16 . The chords can be fitted with two shape models (Fig. 2). Our preferred shape, which is compatible with our own and other observations (see Supplementary Information section 8), corresponds to an elliptical object ...

It is shown that polynomial gravity theories with more than four derivatives in each scalar and tensor sectors have a regular weak-field limit, without curvature singularities. This is achieved by proving that in these models the effect of the higher derivatives can be regarded as a complete regularization of the delta-source. We also show how this result implies that a wide class of non-local ghost-free gravities has a regular Newtonian limit too, and discuss the applicability of this approach to the case of weakly non-local models.1 We use the same sign conventions as [10]. Also, we set c = = 1. 2 That is, we shall consider non-local gravity models which are extensions of GR in the UV-limit, which means that for large momentum the propagator decays faster than in GR. Specifically, we require that f 0 (z) and f 2 (z) (defined in (3) and (4)) are constant or diverge at least linearly as z −→ ∞, and that f s (0) = 1. Owed to this improved behaviour in the UV, sometimes these models are called non-local HDG. The situation is quite different from non-local IR modifications of GR, such as those defined by form factors of the type F i ∝ ✷ −1 and F i ∝ ✷ −2 [24-26], or the logarithmic ones, F i ∝ ln ✷, which come from the integration of quantum matter fields in curved space-time [27][28][29][30][31].

In the present work we show that, in the linear regime, gravity theories with more than four derivatives can have remarkable regularity properties if compared to their fourth-order counterpart. To this end, we derive the expressions for the metric potentials associated to a pointlike mass in a general higher-order gravity model in the Newtonian limit. It is shown that any polynomial model with at least six derivatives in both spin-2 and spin-0 sectors has regular curvature invariants. We also discuss the dynamical problem of the collapse of a small mass, considered as a spherical superposition of nonspinning gyratons. Similarly to the static case, for models with more than four derivatives the Kretschmann invariant is regular during the collapse of a thick null shell. We also verify the existence of the mass gap for the formation of mini black holes even if complex and/or degenerate poles are allowed, generalizing previous considerations on the subject and covering the case of Lee-Wick gravity. These interesting regularity properties of sixth-and higher-derivative models at the linear level reinforce the question of whether there can be nonsingular black holes in the full nonlinear model. MSC: 53B50, 83D05, PACS: 04.20.-q, 04.50.Kd