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Despite its amazing quantitative successes and contributions to revolutionary technologies, physics currently faces many unsolved mysteries ranging from the meaning of quantum mechanics to the nature of the dark energy that will determine the future of the Universe. It is clearly prohibitive for the general reader, and even the best informed physicists, to follow the vast number of technical papers published in the thousands of specialized journals. For this reason, we have asked the leading experts across many of the most important areas of physics to summarise their global assessment of some of the most important issues. In lieu of an extremely long abstract summarising the contents, we invite the reader to look at the section headings and their authors, and then to indulge in a feast of stimulating topics spanning the current frontiers of fundamental physics from The Future of Physics by William D Phillips and What characterises topological effects in physics? by Gerard t Hooft through the contributions of the widest imaginable range of world leaders in their respective areas. This paper is presented as a preface to exciting developments by senior and young scientists in the years that lie ahead, and a complement to the less authoritative popular accounts by journalists., Comment: 54 pages, 13 figures

This white paper submitted for the 2020 Canadian Long-Range Planning process (LRP2020) presents the prospects for Canada and the Square Kilometre Array (SKA) from 2020-2030, focussing on the first phase of the project (SKA1) scheduled to begin construction early in the next decade. SKA1 will make transformational advances in our understanding of the Universe across a wide range of fields, and Canadians are poised to play leadership roles in several. Canadian key SKA technologies will ensure a good return on capital investment in addition to strong scientific returns, positioning Canadian astronomy for future opportunities well beyond 2030. We therefore advocate for Canada's continued scientific and technological engagement in the SKA from 2020-2030 through participation in the construction and operations phases of SKA1., Comment: 14 pages, 4 figures, 2020 Canadian Long-Range Plan (LRP2020) white paper

In the era of precision cosmology, even percentage level effects are significant on cosmological observables. The recent tension between the local and global values of $H_0$ is much more significant than this, and any possible solution might rely on us going beyond the standard $\Lambda$CDM cosmological model. For much smaller, yet potentially significant effects, spatial curvature from averaging and cosmological backreaction on observational predictions could play a role. This is especially true with the higher precision of new observational data and improved statistical techniques. In this paper, we discuss the observational viability of a class of physically motivated cosmologies which can be parametrized by a phenomenological two-scale backreaction model with decoupled spatial curvature parameters and two Hubble scales. Using the latest JLA Supernovae data together with some of the latest BAO data, we perform a Bayesian model selection analysis and find that the phenomenological models are not favoured over the standard $\Lambda$CDM cosmological model. Although there is still a preference for non-zero and unequal dynamic and geometric spatial curvatures, there is little evidence for differing Hubble scales within these phenomenological template models., Comment: arXiv admin note: text overlap with arXiv:1606.00634, arXiv:1609.09781 by other authors

We present a link between parametrizations of alternative theories of gravity on large and small scales in cosmology. This relationship is established using theoretical consistency conditions only. We find that in both limits the "slip" and "effective Newton's constant" can be written in terms of a set of four functions of time, two of which are direct generalizations of the $\alpha$ and $\gamma$ parameters from post-Newtonian physics. This generalizes previous work that has constructed frameworks for testing gravity on small scales, and is to the best of our knowledge the first time that a link between parametrizations of gravity on such very different scales has been established. We expect our result to facilitate the imposition of observational constraints, by drastically reducing the number of functional degrees of freedom required to consistently test gravity on multiple scales in cosmology., Comment: 5 pages, 1 figure

The post-Newtonian (PN) perturbative framework has been successful in understanding the slow-motion, weak field limit of Einstein's theory of gravity on solar system scales, and for isolated astrophysical systems. The parameterized post-Newtonian (PPN) formalism extended the PN framework to put very tight constraints on deviations from Einstein's theory on the aforementioned scales and systems. In this work, we extended and applied the post-Newtonian formalism to cosmological scales. We first used it to construct a cosmological model to understand the effect of regularly arranged point sources on the background expansion. Here we found that at higher orders we obtained a small radiation-like correction to the standard Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) equations, for a matter-dominated universe. This radiation-like correction was purely due to the inhomogeneity of our model, and the non-linearity of Einstein's field equations. We also extended the post-Newtonian formalism to include other forms of matter that are cosmologically relevant, such as radiation and a cosmological constant, and studied the non-linear effects they might have on the background expansion. Then we constructed an extension of the parameterized post-Newtonian formalism (PPN) to cosmological scales. We used it to parameterize the background expansion of the universe as well as first-order perturbations in cosmology, using four functions of time. We gave examples of how our parameterization would work for dark energy models and scalar-tensor and vector-tensor theories of gravity. In the final part of this work, we studied how light propagation behaves in an inhomogeneous post-Newtonian cosmology with matter and a cosmological constant. We used it to understand the effect that inhomogeneities would have on observables., Comment: 213 pages, 30 figures, 7 tables, PhD Thesis

On small scales the observable Universe is highly inhomogeneous, with galaxies and clusters forming a complex web of voids and filaments. The optical properties of such configurations can be quite different from the perfectly smooth Friedmann-Lema\^itre-Robertson-Walker (FLRW) solutions that are frequently used in cosmology, and must be well understood if we are to make precise inferences about fundamental physics from cosmological observations. We investigate this problem by calculating redshifts and luminosity distances within a class of cosmological models that are constructed explicitly in order to allow for large density contrasts on small scales. Our study of optics is then achieved by propagating one hundred thousand null geodesics through such space-times, with matter arranged in either compact opaque objects or diffuse transparent haloes. We find that in the absence of opaque objects, the mean of our ray tracing results faithfully reproduces the expectations from FLRW cosmology. When opaque objects with sizes similar to those of galactic bulges are introduced, however, we find that the mean of distance measures can be shifted up from FLRW predictions by as much as $10\%$. This bias is due to the viable photon trajectories being restricted by the presence of the opaque objects, which means that they cannot probe the regions of space-time with the highest curvature. It corresponds to a positive bias of order $10\%$ in the estimation of $\Omega_{\Lambda}$ and highlights the important consequences that astronomical selection effects can have on cosmological observables., Comment: 37 pages, 15 figures, 1 table, v2: matches published version

Einstein's theory of gravity has been extensively tested on solar system scales, and for isolated astrophysical systems, using the perturbative framework known as the parameterized post-Newtonian (PPN) formalism. This framework is designed for use in the weak-field and slow-motion limit of gravity, and can be used to constrain a large class of metric theories of gravity with data collected from the aforementioned systems. Given the potential of future surveys to probe cosmological scales to high precision, it is a topic of much contemporary interest to construct a similar framework to link Einstein's theory of gravity and its alternatives to observations on cosmological scales. Our approach to this problem is to adapt and extend the existing PPN formalism for use in cosmology. We derive a set of equations that use the same parameters to consistently model both weak fields and cosmology. This allows us to parameterize a large class of modified theories of gravity and dark energy models on cosmological scales, using just four functions of time. These four functions can be directly linked to the background expansion of the universe, first-order cosmological perturbations, and the weak-field limit of the theory. They also reduce to the standard PPN parameters on solar system scales. We illustrate how dark energy models and scalar-tensor and vector-tensor theories of gravity fit into this framework, which we refer to as "parameterized post-Newtonian cosmology" (PPNC)., Comment: 30 pages, no figures, v2: matches published version

We construct high-precision models of the Universe that contain radiation, a cosmological constant, and periodically distributed inhomogeneous matter. The density contrasts in these models are allowed to be highly non-linear, and the cosmological expansion is treated as an emergent phenomenon. This is achieved by employing a generalised version of the post-Newtonian formalism, and by joining together inhomogeneous regions of space-time at reflection symmetric junctions. Using these models, we find general expressions that precisely and unambiguously quantify the effect of small-scale inhomogeneity on the large-scale expansion of space (an effect referred to as "back-reaction", in the literature). We then proceed to specialize our models to the case where the matter fields are given by a regular array of point-like particles. This allows us to derive extremely simple expressions for the emergent Friedmann-like equations that govern the large-scale expansion of space. It is found that the presence of radiation tends to reduce the magnitude of back-reaction effects, while the existence of a cosmological constant has only a negligible effect., Comment: 14 pages, 12 figures, v2: matches published version, v3: typo corrected in Eq. (59)

We construct a framework to probe the effect of non-linear structure formation on the large-scale expansion of the universe. We take a bottom-up approach to cosmological modelling by splitting our universe into cells. The matter content within each cell is described by the post-Newtonian formalism. We assume that most of the cell is in the vicinity of weak gravitational fields, so that it can be described using a perturbed Minkowski metric. Our cells are patched together using the Israel junction conditions. We impose reflection symmetry across the boundary of these cells. This allows us to calculate the equation of motion for the boundary of the cell and, hence, the expansion rate of the universe. At Newtonian order, we recover the standard Friedmann-like equations. At post-Newtonian orders, we obtain a correction to the large-scale expansion of the universe. Our framework does not depend on the process of averaging in cosmology. As an example, we use this framework to investigate the cosmological evolution of a large number of regularly arranged point-like masses. At Newtonian order, the Friedmann-like equations take the form of dust and spatial curvature. At post-Newtonian orders, we get corrections to the dust term and we get an additional term that takes the same form as radiation. The radiation-like term is a result of the non-linearity of Einstein's equations, and is due to the inhomogeneity present in our model., Comment: 4 pages, 1 figure, Contribution to the proceedings of MG14