## Page under construction.

You have taken the blue pill. Very well.

**Present research**: **String field theory (SFT), nonlocal quantum gravity and nonlocal cosmology – Dimensional flow and quantum gravity – Multifractional spacetimes**

**Past research**: **Braneworld and noncommutative cosmology – Higher-order gravity – Loop quantum gravity (LQG) and cosmology (LQC) – BCS condensation in quantum gravity and cosmology – Hořava–Lifshitz gravity – Miscellaneous topics in cosmology**

### Distribution of papers per topic

- String field theory [12,17,20,26,30,61,67] and nonlocal quantum gravity [16,19,27,33,57,84,87] (16%).
- Multiscale and multifractional spacetimes [31,32,36,38,39,41,42,45,47,48,51–53,55,56,58,60,70–78,80,82,83,86] (34%).
- Other theories of quantum gravity, related cosmology and dimensional flow [14,18,25,28,29,44,46,51,54,59,63–65,68,69,73,81,85] (21%).
- Loop quantum cosmology and loop quantum gravity [15,21–24,34,35,37,40,43,49,50,63,66,69,79] (18%).
- Braneworld and noncommutative cosmology and effective string cosmologies [1–11,13,62,77,78] (17% of the total production).

**String field theory (SFT), nonlocal quantum gravity and nonlocal cosmology**

The tachyonic effective action of **open string field theory** may be relevant for the early universe. **(i)** I have started an analytic study of this action [12] which is important also as a mathematical example of nonlocal theories. Beside cosmological applications [16,20,27,33], **nonperturbative solutions of bosonic and susy open SFT and boundary SFT** in Minkowski spacetime were found and compared, whenever possible, with numerical or perturbative results in the literature [17,20,26,30]. **(ii)** It was shown that the **diffusion-equation method** we developed is founded on the gauge symmetry (reparametrization) structure of SFT; in particular, the diffusion equation is an implementation of a large gauge transformation at the spacetime level [30]. **(iii)** For the first time, we constructed nonperturbative analytic solutions in SFT describing the full decay of an unstable brane, confronting them with previous numerical results limited to marginal deformations [12,17,20,26]. Sen’s descent relations are reproduced to a very high level of accuracy. **(iv)** Our diffusion-equation method also helped us to find solutions of generic nonlocal systems with exponential operators [16,27,30,33,57] and to clarify long-standing conceptual issues such as the initial-condition problem [19,30] and other interpretational issues which were still not well understood [12,16,17,19,20,26,27,33]. In fact, the phenomenology of nonlocal theories, even on Minkowski background, was unknown and usually considered to be inaccessible nonperturbatively. Thanks also to our results, the topic has been experiencing a remarkable boost of interest since 2007.

Similar techniques are relevant also in the context of quantum gravity, where nonlocality can favour good renormalization properties. **(v)** In 2010, we formulated a gravity/matter effective action which is nonlocal in all sectors [33] and admits a wide class of cosmological solutions which have interesting applications in the early universe. **(vi)** A nonlocal model of gravity with connections with string theory [61] and very promising cosmological applications has been formulated recently and is under further study [67]. Covariant operators with an infinite number of derivatives are inserted into a gravitational action according to a set of strict requirements (absence of ghosts, unitarity, compatibility with SFT and local supergravity, and so on) [19,57,67]. These operators, which are exponentials of the second-order Laplace–Beltrami operator, arise in effective actions of SFT when massive modes are neglected. When employed in perturbative quantum-gravity models, exponential operators **improve the renormalization **properties in the ultraviolet (UV). There is also reiterated evidence that the big bang is replaced by a classical **bounce**, but so far the exact cosmological dynamics and the inflationary spectra have not been computed rigorously due to well-known difficulties in handling nonlocal operators. L. Modesto and I have proposed a ten-dimensional field theory for gravity which describes simultaneously the nonlinear (in the metric) extension of the closed-SFT effective nonlocal action for the graviton and the nonlocal extension of ten-dimensional supergravity. The model is further extended to 11 dimensions and interpreted as a nonlocal-field-theory limit of M-theory, the mother theory supposed to include all strings [67]. This is the first proposal that unifies the low-energy limit of M-theory (11D local supergravity) and the nonlocal characteristics of SFT actions.

**Dimensional flow and quantum gravity**

As soon as gravity goes quantum, the very concept of smooth continuous geometry can break down at microscopic scales in favour of more abstract but more fundamental degrees of freedom. Fortunately, whenever this happens it is possible to find approximations such that the geometry retains at least some of its characteristics, *in primis* the concept of spacetime dimension. Then, usually one is able to track the behaviour of these features down to ultraviolet scales, for instance through the study of the spectral and the Hausdorff dimension. It is found that, in virtually all known approaches to **quantum gravity**, one of (or both) these dimensions runs from 4 in the infrared to some value ≤ 2 in the ultraviolet (see, e.g., [31]). The transition between the two regimes varies depending on the model but it is continuous in general. Examples include causal dynamical triangulations, noncommutative geometry and κ-Minkowski spacetime, nonlocal quantum gravity, causal sets and others. **(i)** I have contributed to the study of dimensional flow in asymptotically safe quantum gravity [54], loop quantum gravity and spin foams [49,63,69], Hořava–Lifshitz gravity [54], Stelle’s gravity [68], spacetimes with black holes [59], and **string theory** [61]. **(ii)** In the latter case (with L. Modesto), we calculated the spectral dimension of the open-string worldsheet and target spacetime, thus describing analytically how, at large distances (low resolution), a string looks like a particle. **(iii)** A reinterpretation of the spectral dimension in the context of perturbative quantum field theories (including of gravity) has also been offered [68].

**(iv)** With M. Arzano, we studied a model of quantum gravity that, near the horizon of a black hole, gives rise to a nonlocal dispersion relation. We verified that the spacetime dimension is variable in this case [59] and, using the results of the gravitational waves of event GW150914, we have also found experimental bounds on the fundamental scale of quantum gravity in this and other models [73,82].

**(v)** With D. Oriti and our former PhD student J. Thürigen, we have studied the spectral dimension of graphs with the purpose of calculating it for spacetimes arising from a fundamentally discrete geometry, as in loop quantum gravity, Regge calculus, spin foams, and group field theory [49,63,69]. We have performed a detailed exact and numerical analysis starting from very simple configurations, i.e., regular unlabelled lattices or random lattices with fixed vertex valence, progressively including details mimicking actual spin foam graphs. The purpose was to acquire full and unprecedented control of the details of the spectral dimension, and discriminate between lattice artifacts, regularity artifacts, and genuinely physical features. We did find evidence of a dimensional flow in a class of states describing quantum geometries. Contrary to expectations, the spectral dimension varies with the observation scale only in superpositions of states defined on regular complexes.

**(vi)** Finally, to dispel some confusion about what people mean with **multiscale** and **multifractal** spacetimes, I provided a first **rigorous definition** of both concepts that draws a landscape of theories according to the properties of their dimensional flow [72].

The above developments in quantum gravity indicate that theories on effective spacetimes with certain **fractal-like features** have interesting UV and cosmological properties, with possibly observable consequences. To verify this point and in an attempt to realize dimensional flow as a direct, controllable feature, I have developed a class of field theories where the action displays an exotic measure with anomalous scaling. I constructed a class of nontrivial multiscale geometries (multifractional spacetimes) via **fractional calculus**. The latter strikes an extremely rich compromise between full-fledged fractal geometry (to which it is related by precise approximation schemes) and phenomenological continuum scenarios.

**(i)** At first, in [31,32,36] I outlined a model for a power-counting renormalizable **field theory** living in a fractal spacetime and with applications to quantum gravity. **(ii)** This model admits a rigorous formalization outlined in the short paper [38] and in [39,41]. The idea is that, replacing ordinary calculus with fractional calculus, one can obtain a continuum geometry sharing the main properties enjoyed by multifractal sets, including invariance under a set of similarity or affinity symmetries and noninteger and scale-dependent Hausdorff and spectral dimensions. One can realize this via a complex fractional measure whose form is dictated by quite general dynamics-independent arguments in fractal geometry and quantum gravity [83]. These models are endowed with a surprising wealth of novel calculable features. Different scales correspond to different geometric regimes and at ultra-small scales **discrete symmetries **appear. This is a feature unobservable in the original proposal and it can have applications in discrete quantum gravity settings, where such a transition from discreteness to a continuum is only conjectured. Discrete symmetries arise because fractal measures display logarithmic oscillations and the system is invariant under dilatation transformations with fixed ratios; such features are present in a number of chaotic systems, as in models of earthquake patterns or in major financial-market crashes.

**(iii)** At ultra-microscopic scales, unsuspected connections with other models such a noncommutative geometry begin to emerge, while at larger distances anomalous scaling reproduces the behaviour of field theories of gravity based on the renormalization group [42,51,75]. When conditions allow it, multiscale spacetimes can be regarded not only as stand-alone proposals for exotic (but not necessarily quantum) gravity, but also as descriptions of certain effective regimes of other quantum-gravity theories.

Since the first works in 2010, this class of scenarios has been developed to a considerable level of sophistication. **Three independent theories** have been formulated and two of them (with weighted or “*q*-” derivatives) have been constructed in full, including the Standard Model and cosmology. The observability of signatures of these theories and their discrimination from the standard general-relativistic picture or from other quantum-gravity models are hot issues that have been confronted with the most recent experiments, especially those involving the observation of the cosmic microwave background (e.g., WMAP and PLANCK), primordial gravitational waves (**LIGO**), extragalactic supernovæ, large-scale structures, and particle physics (**LHC**). One of the main goals of my research has been to address the following question: What can collider, astrophysical cosmological experiments say about multiscale physics? Unambiguous physical evidences for phenomena be yond the Standard Model would open up a new season for our modern view of the high-energy and geometrical structure of spacetime. Therefore, I have capitalized my efforts in the inspection of these models. **(iv)** First of all, while clarifying the basic definition and properties of multifractional spacetimes [47,53], I have constructed with my collaborators field theories for a scalar and for the complete Standard Model, studying their renormalization properties [55,56,70,71]. I have also studied the dynamics of relativistic and nonrelativistic bodies and particles [58,72]. **(v)** In [60], I have built the cosmology of three of the four multiscale theories. In this framework, early-universe inflation is characterized by an oscillatory phase that can leave an imprint in the CMB spectrum. **(vi)** In 2016, the theory has received a dramatic boost. For the first time, **phenomenological constraints** from observations have been found that place strong bounds on the characteristic scales of the geometry, including from **electroweak interaction**s and data that are unable to constrain traditional quantum-gravity models efficiently (e.g., **gravitational waves** from the GW150914 event) [70,71,82,83]. The imprint of the log oscillations of geometry on the CMB has been tested using PLANCK data and we placed a surprising **upper bound < 3 on the UV dimension of spacetime in the early universe [74].**

All these achievements stem from the application of the tools of fractal geometry and fractional calculus to a field-theory and quantum-gravity context. The theory received a positive welcome and has influenced the general approach towards, and terminology of, the study of fractal properties in quantum gravity.

**Braneworld and noncommutative cosmology**

In some applications of string theory to cosmological models, large or noncompact extra spatial dimensions are allowed, so that the visible universe is confined into a four-dimensional variety (a brane) embedded in a larger spacetime. In this context, during my PhD I considered an inflationary period started by a scalar field, with either a Klein–Gordon or Dirac–Born–Infeld (DBI) tachyonic action, slowly rolling down its potential and driving the accelerated expansion. Quantum fluctuations of this scalar field generate the perturbation structure explaining the small anisotropies of the cosmic microwave background (CMB). Models with a variety of different high-energy ingredients were confronted, using either a **modified effective Friedmann** on the brane and/or a maximally symmetric realization of **noncommutative** spacetime. I showed that **(i)** the slow-roll consistency relations describing the inflationary spectra are broken in the presence of extra dimensions and can discriminate between standard and braneworld scenarios. This point was unclear in older literature due to an accidental degeneracy between the Randall–Sundrum and 4D scenario [1-6]. **(ii)** A very compact formalism was provided which treats different models (4D, Randall–Sundrum, Gauss–Bonnet, etc.) in a simple, unified way, valid within finite time intervals or in particular energy regimes (or “patches”) experienced by the inflaton field during the early cosmological evolution. This formalism permitted the simultaneous consideration of standard and tachyonic inflation, without performing separate analyses of the dynamics, thus generalizing many previous results regarding inflationary attractor, exact solutions, perturbation spectra, non-Gaussianities, Hamilton–Jacobi formulation, dualities and holographic interpretation. The predictions of this class of models were later confirmed by other groups independently [3,4,6-8,10,11]. **(iii)** I also developed the noncommutative cosmological model by Brandenberger and Ho from both the theoretical and experimental point of view, showing that new features arise when considering the presence of a noncommutative scale. Their observational consequences in the light of WMAP [4-6,8,10] and PLANCK [62] data have been fully analyzed and strong constraints have been placed upon these models. With S. Tsujikawa, J. Ohashi and S. Kuroyanagi, we have shown that the majority of these models are not particularly viable and, after almost 15 years since their birth, we have managed to rule them out at a good confidence level.

I investigated low-energy effective **Gauss–Bonnet gravitational theories** with a nonminimally coupled scalar field, which can be regarded as either the dilaton or a compactification modulus in the context of string theory. **(i)** At first, we were interested in a phase space analysis and the classical stability of asymptotic solutions, where a barotropic perfect fluid coupled to the scalar field was allowed. Analytic solutions were confronted with observational constraints for the deceleration parameter and it was shown that Gauss–Bonnet gravity alone may not explain the current acceleration of the universe. We also studied the structure of future big-rip singularities in this class of models. A nonminimal coupling between the fluid and the modulus field opens up the general possibility of avoiding big rips [9]. **(ii)** However, stability against inhomogeneous perturbations is not guaranteed and, in general, there appear ghost modes even for attractors in the phase space. Through cosmological linear perturbations, we extracted no-ghost and sub-luminal constraint equations, written in terms of background quantities, which must be satisfied for consistency. These results, which had a positive welcome in the community and influenced later works by other research groups, were then generalized to a two-field configuration. Single-field models as candidates for dark energy were explored numerically and severe bounds on the parameter space of initial conditions were placed. A number of cases proposed in the literature were tested and most of them found to be unstable or observationally inadequate to explain dark energy [13].

**Loop quantum gravity (LQG) and cosmology (LQC)**

In the context of the nonperturbative, background independent theory of loop quantum gravity, a minisuperspace reduction to a cosmological background was performed and inflationary observables were computed. **(i)** A preliminary analysis under simplified conditions indicated that it is possible to generate a scale-dependent scalar spectrum [15]. **(ii)** The inflationary spectrum of tensor and scalar perturbations in the presence of inverse-volume quantum corrections has been computed [23,34]. **Observational constraints** on the set of **cosmological observables** and on the size of the quantum corrections have been placed upon the model [37,40], and different parametrizations theoretically discriminated; surprisingly, the typical minisuperspace parametrizations turns out to be inconsistent with anomaly cancellation in this inhomogeneous setting, while inhomogeneities are naturally accommodated in a “lattice-refinement” picture. More in detail, I was able to derive the Mukhanov–Sasaki equation for scalar perturbations for the first time, together with a novel conservation equation for curvature perturbation in the presence of inverse-volume quantum corrections. Solutions of the Mukhanov–Sasaki equation then allowed us to find the full set of inflationary observables, including spectral indices, their running, and the tensor-to-scalar ratio. In doing so, unexpected theoretical constraints on the parametrizations of the model emerged, pointing towards an inconsistency between the popular minisuperspace or improved dynamics parametrization and inhomogeneous perturbations. All these results were new and of conspicuous relevance for later research directions in the LQC community. In two papers in collaboration with M. Bojowald and S. Tsujikawa [37,40], observational bounds on the free parameters of the theory were placed using WMAP7 data. Observations determined upper bounds on the quantum corrections, which can be considerably greater than what naively expected in other quantum-gravity settings. **(iii)** The role of the Barbero–Immirzi parameter in classical general relativity was clarified after promoting it to a scalar field. This field turns out to be a pseudo-scalar coupled with a topological term. The resulting canonical structure of gravity was studied at the classical and quantum level [24]. **(iv)** The relation between spin-foam-type **path integrals and loop quantum cosmology** was studied with particular emphasis on Green functions and the analogy with the quantum mechanical relativistic particle [35]. In collaboration with D. Oriti and S. Gielen, we investigated the structure of the minisuperspace path integral formulation of (loop) quantum cosmology, in particular regarding the mutual relations between (a) different representations of the physical Hilbert space, (b) two-point functions and the composition laws governing the evolution of the quantum universe, (c) different time formalisms. Thanks to a complete comparison with the single particle, the combination of all these aspects in a self-consistent framework allow us to acquire a precise dictionary and novel physical insights. **(v)** Invited reviews of LQC have appeared in a special issue of SIGMA [43] and in Annalen der Physik [50] and they have collected a sizable number of citations. **(vi)** A “manifesto” of the effective dynamics approach in LQC, which also contains a comprehensive comparison with other LQC approaches and several conceptual clarifications, has been redacted in collaboration with some of the most prominent researchers in the field [66]. **(vii)** Group field theory is a candidate theory of everything based on an action on a group manifold. Loop quantum gravity and spin foams are special limits of this theory. I studied some cosmological aspects of this framework, both at a very preliminary stage [44] and at a later stage where some fundamental advancement showed how to obtain the so-called “improved dynamics” of LQC directly from the full theory via **condensate quantum states** [65].

**BCS condensation in quantum gravity and cosmology**

In collaboration with S. Alexander ([21,22] and work in progress), I have studied a formulation of loop quantum gravity in vacuum which is dual (in a precise way) to a **Fermi liquid**. Deformation of the topological sector breaks invariance under **large gauge transformations**, leading to the emergence of nonlocal degrees of freedom at the quantum level. Connection components define a degenerate geometry (Jacobson’s sector) and are rearranged into a fermionic field, which obeys a massive Dirac equation. A four-fermion interaction is responsible for the creation of **Cooper pairs**, semiclassically corresponding to wormholes living on the horizon of a de Sitter spacetime. This scenario is thought to have important consequences regarding the measurement theory (providing a natural decoherence process), the area-entropy law (emerging from the counting of Cooper pairs), and the **cosmological constant**. The latter is exponentially suppressed as a nonperturbative, large-scale effect of pure gravity. At small scales gravity admits a perturbative description in terms of a false ground state made of free fermions (Fermi sea), in analogy with the QCD confinement mechanism. A different application of BCS condensates to late-time cosmology and **dark energy** was presented in [29].

Soon after Hořava’s proposal for a field theory model of quantum gravity, I considered its cosmology for the first time [25]. Cosmological equations of motion were derived, the existence of a **bounce **resolving the big-bang singularity was argued, and scalar matter was introduced extending the original proposal for pure gravity. A correspondence between this model and causal dynamical triangulations (CDT) and asymptotic-safety scenarios was supported at the level of minisuperspace; tensor and scalar perturbations were computed. One of the main purposes was to clarify the role of the so-called detailed balance condition. I showed that if the latter is enforced, then the scalar sector of the theory has intrinsic instabilities and a bad infrared limit where Lorentz invariance is not restored [28].

**Miscellaneous topics in cosmology**

**(i)** *Multifield cosmology.* We explored the classical stability properties of multifield solutions of assisted inflation type, where several fields collectively evolve to the same configuration. We found very simple stability conditions, less restrictive than that required for tracking in quintessence models. Our results do not rely on the slow-roll approximation and are relevant also in the context of dark energy [18].

**(ii)** *Cosmological tachyon.* The DBI tachyon has been considered in literature also as a late-time dark energy model. Despite its original motivation from string theory, this case is purely phenomenological. In a work with A. Liddle [14], we showed that there are a number of fine tunings on the parameters and/or initial conditions of this dynamical system, which make it less effective than canonical quintessence.

**(iii)** *Quantum tunneling.* With C. Kiefer and C.S. Steinwachs, we have determined the observational viability of tunneling models of natural inflation in Wheeler–DeWitt quantum gravity. In these models of geometrodynamics, the universe is represented by the quantum-tunneling cosmological state formulated by Vilenkin [64,81].