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"A positive cosmological constant allows arbitrarily many different quantum states, but apparently only if there can be big bangs and/or big crunches. Without any big bang or big crunch, the entropy may be limited by the Gibbons-Hawking entropy of pure deSitter, and the matter entropy might even more limited by a value roughly the three-fourths power of the Gibbons-Hawking entropy. A classical analogue of an upper limit on the entropy is the finite canonical measure for nonsingular cosmologies. Inflation by itself does not explain the arrow of time and seems to require that not only a small region have suitable initial conditions. A special quantum state, such as the Symmetric-Bounce one, with a suitable measure, such as volume averaging and Agnesi weighting, appears possible to explain the arrow of time and other observations, such as high order that would not be expected for Boltzmann brain observations, distant stars that would not be expected if we were fluctuations of empty de Sitter spacetime, and positive cosmological constant that appears not to dominate many other measures."

Closed systems never evolve to lower entropy states -- except when they do, which is if one waits a time that is exponential in the entropy change. Thus macroscopic decreases in entropy are 'never' observed. Yet in cosmology there are eternal systems in which downward entropy fluctuations of any magnitude eventually happen. What is the nature of such fluctuations? I will argue that these can be understood in a simple and general way that sheds light on our understanding of various interesting cosmological processes such as bubble nucleation, black/white hole formation, eternal stochastic inflation, Boltzmann brain formation, and other processes.

Our universe may have formed via bubble nucleation in an eternally-inflating background. Furthermore, the background may have a compact dimension---the modulus of which tunnels out of a metastable minimum during bubble nucleation---which subsequently grows to become one of our three large spatial dimensions. We discuss some potential observational signatures of this scenario.

We propose a novel cosmological scenario, in which standard inflation is replaced by an expanding phase with a drastic violation of the Null Energy Condition (NEC): \dot H >> H^2. The model is based on the recently introduced Galileon theories, which allow NEC violating solutions without instabilities. The unperturbed solution describes a Universe that is asymptotically Minkowski in the past, expands with increasing energy density until it exits the regime of validity of the effective field theory and reheats. This solution is a dynamical attractor and the Universe is driven to it, even if it is initially contracting. Adiabatic perturbations turn out to be cosmologically irrelevant. The model, however, suggests a new way to produce a scale invariant spectrum of isocurvature perturbations, which can be later converted to adiabatic: the Galileon is forced by symmetry to couple to the other fields as a dilaton; the effective metric it yields on the NEC violating solution is that of de Sitter space, so that all light scalars will automatically acquire a nearly scale-invariant spectrum of perturbations.

I will review some recent work on infrared issues for scalar fields in exact and quasi de Sitter space. Renewed interest in this topic has been driven by the observational potential for a more accurate determination of statistics of the primordial curvature perturbations, especially non-Gaussianity. Interestingly, the resulting questions are not only relevant for mapping inflationary models to observation but also link directly to more fundamental questions about the initial state, eternal inflation, and the long time dynamics of interacting quantum fields in curved space. Infrared questions provide a precisely calculable way to put pressure on inflation as a rigorous and consistent framework, ready to confront future observations.

We explore simple but novel solutions of general relativity which, classically, approximate cosmologies cycling through an infinite set of ``bounces." These solutions require curvature K=+1, and are supported by a negative cosmological term and matter with -1 < w < -1/3. They can be studied within the regime of validity of general relativity. We argue that quantum mechanically, particle production leads eventually to a departure from the regime of validity of semiclassical general relativity, likely yielding a singular crunch.

I provide a mathematical model of holographic cosmology whose coarse grained description is that of a homogeneous isotropic, flat universe, which makes a transitions from an FRW to an eternal de Sitter regime.&nbsp;Based on this model, I suggest some heuristic ideas which explain the low initial entropy of the universe and may provide a description of an inflationary era with small fluctuations.

I will argue that anthropic reasoning is unnecessary or misleading when the universe/multiverse is small enough that another observer with exactly your memories is unlikely to exist. Instead, one can evaluate theories or make predictions in the standard Bayesian way, based on the conditional probability of something unknown given all that you do know. Things are not so clear when the universe is large enough that all competing theories predict that an observer with your exact memories exists with probability close to one. I will discuss issues that arise in such large or infinite universes, such as "Boltzmann brains", and will argue that pending better understanding of these issues one should be hesitant to draw conclusions different from those that would apply to a small iverse.

We present a holographic framework for inflationary universes, in particular those that are either asymptotically de Sitter or asymptotically power-law. This framework reveals how cosmological observables, including the primordial power spectrum and non-Gaussianities, are encoded in the correlation functions of a three-dimensional non-gravitational quantum field theory. Introducing a simple yet general class of holographic models, we obtain distinctive observational predictions that are compatible with current observational data and may be either confirmed or excluded by Planck.

Problematic growths of curvature and anisotropy are found in nonsingular bouncing cosmologies that include both an ekpyrotic phase and a bouncing phase. Classically, initial curvature and anisotropy that are suppressed during the ekpyrotic phase will grow back exponentially during the nonsingular bouncing phase. Besides, curvature perturbations and anisotropy are generated by quantum fluctuations during the ekpyrotic phase. In the bouncing phase, an adiabatic curvature perturbation grows to dominate and gives rise to a blue spectrum that spoils the scale-invariance. Meanwhile, a scalar shear perturbation grows nonlinear and creates an overwhelming anisotropy that disrupts the nonsingular bounce altogether.

Einstein's theory of General Relativity and its couplings to matter in 3+1 dimensions can be slightly enlarged with the requirement of a local scale (conformal) symmetry and the corresponding gauge degrees of freedom. This form of the theory is a prediction from 2T-gravity in 4+2 dimensions. It has no dimensionful constants, not even the gravitational constant, and requires all scalar fields to be conformally coupled to gravity and to the rest of matter. The theory can be gauge fixed to the usual gravity theory in the Einstein frame, thus generating the gravitational constant. Other physically equivalent forms of gauge fixing lead to the complete set of exact analytic solutions of the usual Friedmann equations, including radiation, curvature, anisotropy and a special potential for a scalar field coupled minimally to gravity. These analytic cosmological solutions, which are geodesically complete at singularities, reveal many surprising properties that are not noticeable with approximate cosmological solutions. Some aspects of the exact solutions will be reviewed in this lecture. In particular, it is predicted that the universe is cyclic and furthermore is has a period of antigravity between every big crunch and the following big bang.