Daniel Cartin

Lattice refining loop quantum cosmology, anisotropic models and stability

A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.

Generating function techniques for loop quantum cosmology

Loop quantum cosmology leads to a difference equation for the wavefunction of a universe, which, in general, has solutions changing rapidly even when the volume changes only slightly. For a semiclassical regime, such small-scale oscillations must be suppressed by choosing the parameters of the solution appropriately. For anisotropic models, this is not possible to do numerically by trial and error; instead, it is shown here, for the Bianchi I LRS model, how this can be done analytically, by using generating function techniques. Those techniques can also be applied to more complicated models, and the results gained allow conclusions about initial value problems for other systems.

Wave functions for the Schwarzschild black hole interior

Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for preclassical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter {gamma} to the quantum ambiguity {delta} appearing in the model. This bound is violated when we use a natural value for {delta} based on loop quantum gravity together with a recent proposal for {gamma}. We also present numerical solutions of the constraint.

Numerical techniques in loop quantum cosmology

In this article, we review the use of numerical techniques to obtain solutions for the quantum Hamiltonian constraint in loop quantum cosmology (LQC). First, we summa- rize the basic features of LQC, and describe features of the constraint equations to solve { generically, these are difference (rather than differential) equations. Important issues such as differing quantization methods, stability of the solutions, the semi-classical limit, and the relevance of lattice refinement in the difference equations are discussed. Finally, the cosmo- logical models already considered in the literature are listed, along with typical features in these models and open issues.

Upper limits on the probability of an interstellar civilization arising in the local Solar neighbourhood

At this point in time, there is very little empirical evidence on the likelihood of a space-faring species originating in the biosphere of a habitable world. However, there is a tension between the expectation that such a probability is relatively high (given our own origins on Earth), and the lack of any basis for believing the Solar System has ever been visited by an extraterrestrial colonization effort. This paper seeks to place upper limits on the probability of an interstellar civilization arising on a habitable planet in its stellar system, using a percolation model to simulate the progress of such a hypothetical civilizations colonization efforts in the local Solar neighborhood. To be as realistic as possible, the actual physical positions and characteristics of all stars within 40 parsecs of the Solar System are used as possible colony sites in the percolation process. If an interstellar civilization is very likely to have such colonization programs, and they can travel over large distances, then the upper bound on the likelihood of such a species arising per habitable world is on the order of

Exploration of the local solar neighbourhood I: Fixed number of probes

10^{-3}

Conserved quantities in isotropic loop quantum cosmology

; on the other hand, if civilizations are not prone to colonize their neighbors, or do not travel very far, then the upper limiting probability is much larger, even of order one.

Separable wave functions in Bianchi I loop quantum cosmology

Previous work in studying interstellar exploration by one or several probes has focused primarily either on engineering models for a spacecraft targeting a single star system, or large-scale simulations to ascertain the time required for a civilization to completely explore the Milky Way galaxy. In this paper, a simulated annealing algorithm is used to numerically model the exploration of the local interstellar neighborhood (i.e. on the order of ten parsecs of the Solar System) by a fixed number of probes launched from the Solar System; these simulations use the observed masses, positions and spectral classes of targeted stars. Each probe visits a pre-determined list of target systems, maintains a constant cruise speed, and only changes direction from gravitational deflection at each target. From these simulations, it is examined how varying design choices -- differing the maximum cruise speed, number of probes launched, number of target stars to be explored, and probability of avoiding catastrophic system failure per parsec -- change the completion time of the exploration program and the expected number of stars successfully visited. In addition, it is shown that improving this success probability per parsec has diminishing returns beyond a certain point. Future improvements to the model and possible implications are discussed.

Loop quantum cosmology in the Schwarzschild interior

We develop an action principle for those models arising from isotropic loop quantum cosmology, and show that there is a natural conserved quantity Q for the discrete difference equation arising from the Hamiltonian constraint. This quantity Q relates the semi-classical limit of the wave function at large values of the spatial volume, but opposite triad orientations. Moreover, there is a similar quantity for generic difference equations of one parameter arising from a self-adjoint operator.