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Gravitational microlensing is one of the methods to detect exoplanets–planets outside our solar system. Here, we focus on the theoretical modeling of systems with three lensing objects and in particular circumbinary systems. Circumbinary systems include two stars and at least one planet and are estimated to represent a sizeable portion of all exoplanets. Extending a method developed for binary lenses to the three lens case, we explore the parameter space of circumbinary systems, producing exact magnification maps and light curves.

Eternal inflation requires upward fluctuations of the energy in a Hubble volume, which appear to violate the energy conditions. In particular, a scalar field in an inflating spacetime should obey the averaged null energy condition, which seems to rule out eternal inflation. Here we show how eternal inflation is possible when energy conditions (even the null energy condition) are obeyed. The critical point is that energy conditions restrict the evolution of any single quantum state, while the process of eternal inflation involves repeatedly selecting a subsector of the previous state, so there is no single state where the conditions are violated., 24 pages, 9 figures

The null energy condition (NEC), an important assumption of the Penrose singularity theorem, is violated by quantum fields. The natural generalization of the NEC in quantum field theory, the renormalized null energy averaged over a finite null segment, is known to be unbounded from below. Here, we propose an alternative, the double smeared null energy condition (DSNEC), stating that the null energy smeared over two null directions has a finite lower bound. We rigorously derive DSNEC from general worldvolume bounds for free quantum fields in Minkowski spacetime. Our method allows for future systematic inclusion of curvature corrections. As a further application of the techniques we develop, we prove additional lower bounds on the expectation values of various operators such as conserved higher spin currents. DSNEC provides a natural starting point for proving singularity theorems in semi-classical gravity., Comment: 33 pages, 1 figure

This review summarizes the current status of the energy conditions in general relativity and quantum field theory. We provide a historical review and a summary of technical results and applications, complemented with a few new derivations and discussions. We pay special attention to the role of the equations of motion and to the relation between classical and quantum theories. Pointwise energy conditions were first introduced as physically reasonable restrictions on matter in the context of general relativity. They aim to express e.g. the positivity of mass or the attractiveness of gravity. Perhaps more importantly, they have been used as assumptions in mathematical relativity to prove singularity theorems and the non-existence of wormholes and similar exotic phenomena. However, the delicate balance between conceptual simplicity, general validity and strong results has faced serious challenges, because all pointwise energy conditions are systematically violated by quantum fields and also by some rather simple classical fields. In response to these challenges, weaker statements were introduced, such as quantum energy inequalities and averaged energy conditions. These have a larger range of validity and may still suffice to prove at least some of the earlier results. One of these conditions, the achronal averaged null energy condition, has recently received increased attention. It is expected to be a universal property of the dynamics of all gravitating physical matter, even in the context of semiclassical or quantum gravity., Comment: 61 pages, 6 figures, Topical review accepted by Classical and Quantum Gravity

Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a nonnegative effective energy density (EED), thereby guaranteeing the timelike convergence property. However, there are models that do not satisfy the SEC and therefore lie outside the scope of Hawking's hypotheses, an important example being the massive Klein-Gordon field. Here we derive lower bounds on local averages of the EED for solutions to the Klein-Gordon equation, allowing nonzero mass and nonminimal coupling to the scalar curvature. The averages are taken along timelike geodesics or over spacetime volumes, and our bounds are valid for a range of coupling constants including both minimal and conformal coupling. Using methods developed by Fewster and Galloway, these lower bounds are applied to prove a Hawking-type singularity theorem for solutions to the Einstein-Klein-Gordon theory, asserting that solutions with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete., 19 pages

The original singularity theorems of Penrose and Hawking were proved for matter obeying the Null Energy Condition or Strong Energy Condition respectively. Various authors have proved versions of these results under weakened hypotheses, by considering the Riccati inequality obtained from Raychaudhuri's equation. Here, we give a different derivation that avoids the Raychaudhuri equation but instead makes use of index form methods. We show how our results improve over existing methods and how they can be applied to hypotheses inspired by Quantum Energy Inequalities. In this last case, we make quantitative estimates of the initial conditions required for our singularity theorems to apply., Comment: 27 pages

Many theories beyond the Standard Model include extra dimensions, though these have yet to be directly observed. In this work we consider the possibility of a compactification mechanism which both allows extra dimensions and is compatible with current observations. This compactification is predicted to leave a signature on the CMB by altering the amplitude of the low l multipoles, dependent on the amount of inflation. Recently discovered CMB anomalies at low multipoles may be evidence for this. In our model we assume the spacetime is the product of a four-dimensional spacetime and flat extra dimensions. Before the compactification, both the four-dimensional space- time and the extra dimensions can either be expanding or contracting independently. Taking into account physical constraints, we explore the observational consequences and the plausibility of these different models., 21 pages, 5 Figures (references added)

Quantum inequalities bound the extent to which weighted time averages of the renormalized energy density of a quantum field can be negative. They have mostly been proved in flat spacetime, but we need curved-spacetime inequalities to disprove the existence of exotic phenomena, such as closed timelike curves. In this work we derive such an inequality for a minimally-coupled scalar field on a geodesic in a spacetime with small curvature, working to first order in the Ricci tensor and its derivatives. Since only the Ricci tensor enters, there are no first-order corrections to the flat-space quantum inequalities on paths which do not encounter any matter or energy., Comment: 21 pages

Quantum inequalities are bounds on negative time-averages of the energy density of a quantum field. They can be used to rule out exotic spacetimes in general relativity. We study quantum inequalities for a scalar field with a background potential (i.e., a mass that varies with spacetime position) in Minkowski space. We treat the potential as a perturbation and explicitly calculate the first-order correction to a quantum inequality with an arbitrary sampling function, using general results of Fewster and Smith. For an arbitrary potential, we give bounds on the correction in terms of the maximum values of the potential and its first three derivatives. The techniques we develop here will also be applicable to quantum inequalities in general spacetimes with small curvature, which are necessary to rule out exotic phenomena., 20 pages

The Averaged Null Energy Condition (ANEC) states that the integral along a complete null geodesic of the projection of the stress-energy tensor onto the tangent vector to the geodesic cannot be negative. Exotic spacetimes, such as those allow wormholes or the construction of time machines are possible in general relativity only if ANEC is violated along achronal geodesics. Starting from a conjecture that flat-space quantum inequalities apply with small corrections in spacetimes with small curvature, we prove that ANEC is obeyed by a minimally-coupled, free quantum scalar field on any achronal null geodesic surrounded by a tubular neighborhood whose curvature is produced by a classical source.

We generalize the concept of Fermi normal coordinates adapted to a geodesic to the case where the tangent space to the manifold at the base point is decomposed into a direct product of an arbitrary number of subspaces, so that we follow several geodesics in turn to find the point with given coordinates. We compute the connection and the metric as integrals of the Riemann tensor. In the case of one subspace (Riemann normal coordinates) or two subspaces, we recover some results previously found by Nesterov, using somewhat different techniques., 9 pages, 4 figures

Hawking's singularity theorem concerns matter obeying the strong energy condition (SEC), which means that all observers experience a non-negative effective energy density (EED). The SEC ensures the timelike convergence property. However, for both classical and quantum fields, violations of the SEC can be observed even in the simplest of cases, like the Klein-Gordon field. Therefore there is a need to develop theorems with weaker restrictions, namely energy conditions averaged over an entire geodesic and weighted local averages of energy densities such as quantum energy inequalities (QEIs). We present lower bounds of the EED for both classical and quantum scalar fields allowing nonzero mass and nonminimal coupling to the scalar curvature. In the quantum case these bounds take the form of a set of state-dependent QEIs valid for the class of Hadamard states. We also discuss how these lower bounds are applied to prove Hawking-type singularity theorems asserting that, along with sufficient initial contraction, the spacetime is future timelike geodesically incomplete., 15th Marcel Grossmann conference proceedings, 6 pages