**Stephen Adler**, Institute for Advanced Study

*Implications of a Frame-Dependent Dark Energy Action*

I review motivations for a frame-dependent dark energy action proportional to $\int d^4x (-g)^{1/2}/g_{00}^2$, and discuss implications for the black hole horizon and for perturbations on the Robertson-Walker line element.

**Julian Barbour**, University of Oxford

*Shape Dynamics: Perspectives and Problems*

Shape Dynamics(SD) can be derived from principles that differ in significant respects from Einstein's derivation of GR. It requires a spatially closed universe and allows a smaller set of solutions than GR does for this case. There are indications that its solution space can be fully characterized and endowed with a measure. These architectonic features suggest that SD can explain the arrows of time as direct consequences of the law of the universe. They do not require special initial conditions. I will discuss these and other major issues on which SD may cast light. I will also discuss the problems that face SD.

**Yuri Bonder, **UNAM

*Relationalism and the speed of light: Are we in a relationship?*

Most practical studies in Shape Dynamics involve an N-body Newtonian interaction which is described by a homogeneous potential. This property allows one to proof several interesting features like the emergence of an arrow of time. However, more generic interactions are not described by these kind of potentials and introduce additional dimensionful coupling constants. Thus, it is an open question whether more generic interactions can be written in a fully relational manner. By studying the concrete example of the gravitational Weber interaction which is, in a sense, a more realistic theory of gravity, we show that it is possible translate non-Newtonian interactions, which have inhomogeneous potentials and additional coupling constants, into a relational language. This opens the door to study other interactions and may shed light into the relationalization of gravity as described by general relativity.

**Beatrice Bonga, **Pennsylvania State University

Inflationary and pre-inflationary dynamics with the Starobinsky potential

Inflation is the leading paradigm of the early Universe, according to which the tiny temperature fluctuations observed in the cosmic microwave background (CMB) originate from quantum vacuum fluctuations at very early times. Recent observations show that the Starobinsky potential is favored among the single field inflationary models. However, the calculations that match the data exclude the Planck era. I will explain why this era is important and how using techniques from loop quantum gravity, the effects of this period can be studied. In particular, we find that for a large part of the initial data surface predictions for the power spectrum of the CMB are indistinguishable from predictions neglecting the pre-inflationary era. However, initial conditions exist for which the quantum gravitational corrections to the power spectrum are potentially observable.

**Erik Curiel,** Harvard University

*A Weyl-Type Theorem in Geometrized Newtonian Gravity, and How It May Bear on Shape Dynamics*

The Weyl Theorem states that the conformal structure and the projective structure jointly suffice to fix the metric up to a global constant. This is a powerful interpretive tool in general relativity: it says in effect that if I know the paths of light rays in vacuo and I know the images of the paths of freely falling particles (i.e., the spacetime curves they follow with no preferred parametrization), then I know the metric. It has particular relevance to Shape Dynamics, where the conformal structure is taken as the object of primitive geometrical interest, and one does not generally want a preferred parametrization of timelike geodesics. The spacetimes of geometrized Newtonian gravity share many important features with the way that Shape Dynamics approaches the construction of relativistic spacetimes, in particular the fixing of a preferred foliation of spacetime by spacelike slices. Studying the way that geometrized Newtonian gravity does and does not allow one to recover a Weyl-type Theorem may, therefore, shed light on the ways that Shape Dynamics may allow one to recover the structure of relativistic spacetimes. I show that, in geometrized Newtonian gravity, the conformal structure of the spatial and temporal metrics, in conjunction with the projective structure of timelike curves, allows one to recover the full metrical structure of a geometrized Newtonian gravity only if one also fixes an affine parametreization of at least one timelike geodesic. This suggests that, in far as the analogy between geometrized Newtonian gravity and Shape Dynamics is a physically significant one, Shape Dynamics may be right not to demand preferred parametrizations of timelike geodesics from the start. The tools I develop to prove the theorem may also be applicable to problems in Shape Dynamics itself. But I make no promises.

**Astrid Eichhorn**, University of Heidelberg

*Renormalization Group flows of spacetime*

I will discuss the use of (functional) Renormalization Group in models of quantum gravity. I will highlight the challenges that occur in continuum approaches to quantum gravity, such as asymptotically safe gravity, as well as challenges in discrete approaches, such as tensor models.

**Henrique Gomes**, Perimeter Institute

*Timeless cosmology with records*

On the path towards quantum gravity we find friction between temporal relations in quantum mechanics (QM) (where they are fixed and field-independent), and in general relativity (where they are field-dependent and dynamic). In this talk, I will erase that distinction. I encode gravity, along with other types of interactions, in the timeless configuration space of spatial fields, with dynamics obtained through a path integral formulation. The framework demands that boundary conditions for this path integral be uniquely given. Such uniqueness arises if a reduced configuration space can be defined and if it has a profoundly asymmetric fundamental structure. These requirements place strong restrictions on the field and symmetry content of theories encompassed here. When these constraints are met, the emerging theory has no non-unitary measurement process; the Born rule is given merely by a particular volume element built from the path integral in (reduced) configuration space. Time, including space-time, emerges as an effective concept; valid for certain curves in configuration space but not assumed from the start. When some notion of time becomes available, conservation of (positive) probability currents ensues. I will show that, in the appropriate limits, a Schroedinger equation dictates the evolution of weakly coupled source fields on a classical gravitational background. Due to the asymmetry of reduced configuration space, these probabilities and currents avoid a known difficulty of standard WKB approximations for Wheeler DeWitt in minisuperspace: the selection of a unique Hamilton-Jacobi solution to serve as background. I illustrate these constructions with a simple example of a quantum gravitational theory for which the formalism is applicable, and give a formula for calculating gravitational semi-classical relative probabilities in it. Although this simple model gives the same likelihood for the evolution of all TT gravitational modes, there is evidence that a slightly more complicated model would favor modes with the smallest eigenvalues of the Laplacian and thus drive towards homogeneity.

**Sean Gryb, **University of Bristol

*Quantum singularity resolution in homogeneous cosmology and the implications for shape dynamics*

I will present results on the quantization of an FRLW model that utilises a Schrodinger-type evolution equation. In contrast to standard Wheeler--DeWitt-type quantisations, the quantum model resolves the classical singularity, exhibits a quantum bounce, and displays novel early-universe phenomenology. A global scale emerges because of a scale anomaly, and suggests an interesting scenario for quantum shape dynamics. I will give the details of the quantization procedure and show how these techniques can be used more generally for anisotropic models. I will end by speculating about how these techniques might be applicable to a genuine quantum shape model of the universe.

**Daniel Guariento**, Perimeter Institute

*Self-gravitating fluid solutions of Shape Dynamics*

Shape Dynamics possesses a large set of solutions in common with General Relativity. Upon close inspection, these solutions behave in surprising ways, so in order to probe the fitness of Shape Dynamics as a viable alternative to General Relativity one must understand increasingly complex solutions, on which to base perturbative studies and numerical analyses. We show that a class of time-dependent exact solutions of Shape Dynamics exists from first principles, representing a central inhomogeneity in an evolving cosmological environment. By assuming only a perfect fluid source in a spherically symmetric geometry, we show that this solution satisfies in all generality the Hamiltonian structure of Shape Dynamics. The solutions are characterized by shear-free flow of the fluid and admit an interpretation as cosmological black holes.

**James Isenberg**, University of Oregon

*What we know and don’t know about solutions to the Einstein Constraint Equations*

**Tim Koslowski**, UNAM

*The quantum equation of state of the universe produces a small cosmological constant*

Relationalism is the strict disentanglement of physical law from the definition of physical object. This can be formalized in the shape dynamcis postulate that the objective evolution of the universe is described by an "equation of state of a curve in relational configuration space." The application of this postulate to General Relativity implies that gravity is described by an equation of state of a curve on conformal superspace. It turns out that the naive quantization of these equations of state introduces an undesired preferred time parametrization. However, it turns out that one can still describe the quantum evolution of the system as an equation of state of the Bohmian trajectory which remains manifestly parametrization independent. These quantum systems generically develop quasi-isolated bound states (atoms) that can be used as reference systems. It turns out that the system as a whole expands if described in units defined by these atoms. This produces phenomenological effects that are usually ascribed to the presence of a cosmological constant. This "effective cosmological constant" is however unaffected by vacuum energy. I pesent the formal argument for this statement and show this explicitly by remormalizing a scalar field coupled to shape dynamics.

**Flavio Mercati, **Sapienza University of Rome

*Compact spherically symmetric solutions and gravitational collapse in SD*

I will review the current status of our understanding of spherically symmetric compact solutions of Shape Dynamics, which have nontrivial degrees of freedom when matter is present. I will show some new solutions of GR in a CMC foliation: a single thin spherical shell of matter in equilibrium in a compact foliation of de Sitter, and the simplest possible model of a black hole or compact star. This is provided by a universe with the topology of a 3-sphere with two thin spherical shells of dust. One of the shells models the `fixed stars’, or the `rest of the universe’, while the other shell models collapsing matter. Both are needed for a truly relational description of gravitational collapse. It turns out that such a solution of GR cannot be evolved past a point at which the foliationceases to be admissible, but it still makes sense past that point as a solution of Shape Dynamics, because the shape degrees of freedom seem to be unaffected. My conjecture is that we have found another example of departure between GR and SD, and this departure happens whenever ordinary matter undergoes gravitational collapse.

**Roberto Percacci, **SISSA

*Weyl invariance and quantum gravity*

I will discuss various ways of realizing the Weyl group in a theory of gravity, and the presence or absence of anomalies.

**David Sloan**, University of Oxford

*Through the Big Bang*

I will show how the intrinsic definition of observables in relativity through dynamical similiarity (known as Shape Dynamics) leads to the continuation of Einstein's equations classically through the big bang singularity in simple cosmological scenarios. By appealing to general principles I argue that this is a generic feature, and that the singularity can be viewed as an artifact of the redundant description imposed by absolute length scales. I will then lay out some other welcome features of intrinsic relational systems, and discuss the broader questions raised by a theory of physics that is independent of physical dimensions such as mass and length.

**Lee Smolin, **Perimeter Institute

*Shape dynamics in terms of connection variables*

**Claes Uggla**, Karlstad University

*Dynamical systems approaches and methods in cosmology*

I will with simple examples from spatially homogeneous and isotropic cosmology illustrate the importance of respecting the global features of a state space for a given model when reformulating field equations to useful dynamical systems. In particular I will use examples from f(R) gravity and GR with a minimally coupled scalar field. In this context I will also illustrate how various dynamical systems methods, such as, e.g., monotonic functions, center manifold techniques, averaging methods, can yield a global understanding of the solution spaces as well as approximations, complementing, e.g., the slow-roll approximation.