Janna Levin is a theoretical cosmologist and holds a PhD in theoretical physics from the Massachusetts Institute of Technology, granted in 1993, and a Bachelor of Science in Astronomy and Physics from Barnard College, granted in 1988. Much of her work deals with looking for evidence to support the proposal that our universe might be finite in size due to its having a nontrivial topology. Other work includes black holes and Chaos theory. Since January 2004, she has been an assistant professor in astronomy and physics at Barnard College.
Levin is the author of the popular science book How the Universe Got Its Spots. In 2006, she published A Madman Dreams of Turing Machines, a novel of ideas recounting the lives and deaths of Kurt Gödel and Alan Turing. In an interview with Sylvie Myerson in the Brooklyn Rail, Levin said of her book:
"There was a lot that made me want to write it as a novel, one being this whole idea that sometimes truth cannot come out as a theorem even in mathematics, let alone in a retelling of two people’s lives. Sometimes you have to step outside of the perfect linear logic of biographical facts."
The book won several awards, including the prestigious PEN/Bingham Fellowship Prize for Writers and the MEA Mary Shelley Award for Outstanding Fictional Work. It was also a runner-up for the Hemingway Foundation/PEN Award.
Inflation from Extra Dimensions is a paper about a theoretical higher dimensional universe; the following is an abstract and excerpt of that publication.
Abstract
A gravity-driven inflation is shown to arise from a simple higher dimensional
universe. In vacuum, the shear of n > 1 contracting dimensions is
able to inflate the remaining three spatial dimensions. Said another way, the
expansion of the 3-volume is accelerated by the contraction of the n-volume.
Upon dimensional reduction, the theory is equivalent to a four dimensional
cosmology with a dynamical Planck mass. A connection can therefore be
made to recent examples of inflation powered by a dilaton kinetic energy.
Unfortunately, the graceful exit problem encountered in dilaton cosmologies
will haunt this cosmology as well.
Imagine a spacetime which is created empty. Even in the absence of matter as a source of fuel, the evolution of the spacetime can still involve interesting dynamics. The shearing and twisting of dimensions can feed each other, driving some dimensions to expand and others to contract.
Suppose a universe is created initially with some spatial dimensions contracting and the others expanding. The shear of the contracting dimensions has been shown to expand the other dimensions at a humble space. The expansion of the spatial sections decelerates with time. It is the intent of this paper to present an additional possibility. The contraction of some dimensions can actually inflate the expanding space; that is, the expansion of the spatial sections can accelerate with time.
Consider a universe with n extra spatial dimensions, in addition to the 3 space and 1 time dimensions we occupy. A general vacuum solution describing n such contracting dimensions and 3 expanding spatial dimensions was found nearly fourteen years ago. For this solution, the 3 dimensional space begins singular and decelerates as the universe evolves.
If the number of extra dimensions exceeds 1, then there exists another interesting branc of solutions. For this other branch, the n > 1 contracting dimensions drive the 3 spatial dimensions not only to expand, but what is more, to inflate.
There is no cosmological constant nor potential driving the inflation. Thus higher dimensional theories exhibit an example of gravity-driven, kinetic inflation. The possibility of kinetic inflation, in the absence of all conventional potential sources, was first discussed in the context of dynamical Planck mass theories. It is simple to confirm that upon dimensional reduction the higher dimensional theory is in fact equivalent to a 4-dimensional theory with a variable Planck mass.
The superstring dilaton also demonstrates this property of dynamical Planck fields. A kinetic inflation from the dilaton of superstring theories was discussed independently in Ref.
Recently, there have been interesting studies of an inflating superstring cosmology in higher dimensions. In those papers it is the dilaton which drives the inflation. The extra dimensions exist only as a consequence of the nature of string theory. By contrast, in this paper it is the extra dimensions themselves which drive the inflation. As will be discussed, the extra dimensions act very much like the superstring dilaton.
While it is possible to drive the three dimensional world to inflate, a successful completion of kinetic inflation has yet to be found. For the particular higher dimensional model discussed in this paper, there will be a graceful exit problem, as there is for the superstring cosmology. For a discussion of the general obstacle facing the simplest dynamical Planck mass models see Ref.
In the presence of matter sources, Kaluza-Klein models are already known to show inflationary behavior. It was discovered firstly in a radiation dominated universe.
The focus of Ref. was on the effective radiation entropy increase. Even if the net N-dimensional entropy is conserved, the effective 3-dimensional entropy can increase. In essence, the contracting dimensions squeeze entropy into the expanding dimensions. Unfortunately, unless there is some tuning of parameters, the increase in entropy is too small to solve the cosmological problems. Since these original papers, many source terms have been exploited to fuel inflation in a universe with extra dimensions. Anti-symmetric tensor fields predicted from supergravity, stringy fluids, as well as curvatures have all been considered as sources.
The difference here is that there are no matter sources at all. The solutions presented in this paper are strictly vacuum solutions. There exists only the locally flat spacetime itself.
By stripping the universe down to essentials, a connection can be made with the dilaton cosmologies mentioned above.
Since the writing of this paper, I have discovered reference which considers dust in a higher dimensional universe. The inflationary solution can be recovered as limiting case of.
In addition to the familiar 3 space and 1 time dimensions, consider the addition of n extra spacelike dimensions. Let the 3-volume expands while the n-volume contracts. It is an initial condition choice to begin a universe with all spatial dimensions expanding, all contracting, or a few of each. The naturalness of these initial choices will not be defended in this paper. A natural mechanism to compactify additional dimensions has long been sought in theories which involve higher dimensions. If such a mechanism exists, then the anisotropy of the spacetime is not a special initial condition but instead is generated dynamically. While the mechanics of a realistic model will likely be more complicated than those of the simple model studied here, it seems fair to conjecture that the basic features will be generic.
Take the action to be the Einstein theory of gravity generalized to higher dimensions; where the total dimension of the spacetime is N = n+4. The N-dimensional Planck mass is MN. The assumption is made that the metric divides simply into an N-dimensional homogeneous but anisotropic metric. Three spacelike dimensions are taken to expand isotropically while the other n spacelike dimensions are taken to contract isotropically.
The indices i, j run from 1...3 and the indices m, n run from 4...n+3. The scale factor of the 3-space is a(t) and that of the n-space is b(t).
The vacuum Einstein equations lead to the following set of equations. It is well known that upon dimensional reduction, the radius of the extra dimensions acts just as a dynamical Planck mass. Thus the higher dimensional theory can be related to a generalized Jordan-Brans-Dicke (JBD) theory. Dynamical Planck fields are often generated in particle theories, either through simple quantum corrections or through the full machinery of theories such as superstrings. In a higher dimensional universe, the dynamical Planck mass has a geometric interpretation. It is related to the radius of the compact internal dimensions. To my mind, a geometric source for the dynamical Planck mass is more aesthetic than a particle theory source. It reinforces the suggestion that the entire scenario is purely and truly gravity driven.
As is often done, the action can be reduced to a four-dimensional theory by integrating over the extra dimensions.
The kinetic coupling parameter cp is a negative constant for n > 1. Notice, there is no potential nor cosmological constant in the dimensionally reduced action. The N-dimensional accelerating solution is thus identical to an example of the kinetic driven acceleration from a dynamical Planck mass in (3 + 1) dimensions. In references and it was shown that the peculiar kinetic energy of a dynamical Planck mass could drive the scale factor of the universe to accelerate. Large families of such theories are possible and can be identified by a bound on the kinetic coupling parameter cp.
In the example of this paper, the effective cp is a negative constant, the branch of solutions corresponds to the required branch, and the decrease in the Planck mass is effected by the decrease in the scale of the extra dimensions.
The dilaton of superstring theories in 4-D can also provide an accelerating branch of solutions. The superstring dilaton is akin to a JBD theory with sd = −1. The dimensionally reduced theory of this paper, giving a JBD theory with sd = −1 + 1/n, is thus very similar to the superstring dilaton. Unfortunately, it was found in the string case that inflation could not be completed with success. To be specific, our universe today does not connect smoothly onto the branch of solutions which corresponds to inflation. A mechanism to induce a branch change and provide a graceful exit seems thus far to have eluded the string cosmology. The graceful exit problem which challenges superstring inflation will challenge higher-dimensional inflation as well.
Regardless of its ultimate fate as an inflationary model, the higher-dimensional theory exhibits another interesting feature. The universe begins cold and empty. With just a kick as an initial condition, the cosmology evolves into a future singularity from a completely regular state. This mild and quiet universe, perhaps a simple thing to create, evolves into an era of quantum gravity. Thus a possible explanation is offered as to why the early universe was so fiercely energetic. The cold beginning remains a fascinating alternative to the standard hot big bang lore.
