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Tilt Classifications in Perfect Fluid Cosmology

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Tilt Classifications in Perfect Fluid Cosmology Utah State University DigitalCommons@USU Physics Capstone Project Physics Student Research 5-11-2017 Tilt Classifications in erP fect Fluid Cosmology Bryant Ward Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/phys_capstoneproject Part of the Physics Commons Recommended Citation Ward, Bryant, "Tilt Classifications in erP fect Fluid Cosmology" (2017). Physics Capstone Project. Paper 54. https://digitalcommons.usu.edu/phys_capstoneproject/54 This Report is brought to you for free and open access by the Physics Student Research at DigitalCommons@USU. It has been accepted for inclusion in Physics Capstone Project by an authorized administrator of DigitalCommons@USU. For more information, please contact [email protected]. Tilt Classifications in Perfect Fluid Cosmology Bryant Ward∗ and Charles Torre Utah State University (Dated: May 11, 2017) We classify all known perfect fluid cosmological solutions of the Einstein equations according to whether they are tilted or non-tilted. A non-tilted universe will have observers who see a homo- geneous, isotropic universe with matter at rest with respect to them. A tilted universe will have observers who see matter moving relative to them. These classifications are useful when considering fluid models of the universe in that the Hubble parameter and expansion are observer dependent and can be different in a tilted versus a non-tilted universe. This gives more insight when fitting these models with observations of our real universe. We make these tilt classifications by establish- ing whether the 4-velocity of each model's fluid is aligned with the normal of the hyper-surfaces of homogeneity spanned by the Killing vectors for the space-time, which we obtain for each solution. These computations are performed using the Differential Geometry software package being devel- oped at Utah State University. We incorporate the Killing vector fields and the tilt classification into a library of solutions to Einstein's Field Equations as part of the package, providing users with access to the solutions and their physical and geometric properties. I. INTRODUCTION A. Perfect Fluid Cosmology There are over a thousand known solutions to Ein- One of the most commonly used models of the universe stein's field equations, representing a wide range of phys- are perfect fluid solutions to Einstein's field equations ical and mathematical situations. In an effort to make accessing and using these solutions quick and easy, our 1 research group at Utah State University has been compil- Rµν − Rgµν + Λgµν = κ0Tµν (1) ing all known solutions into a library for Maple as part 2 of the Differential Geometry package also developed at where R is the Ricci tensor, R is the Ricci scalar, Λ is Utah State University [2]. Examples of these computa- µν the cosmological constant, T is the energy-momentum tions, and how to access this library using Maple 18 can µν tensor, g is the metric tensor for the space-time in ques- be found in the appendix. µν tion, and κ = 8πG . The metric tensor signature used By not only including the solutions themselves, but 0 c4 for all the space-times looked at here is (-,+,+,+). their intrinsic physical and geometric properties as well, Physically, these field equations imply that all the in- this library will provide a powerful tool for research in formation (i.e. energy, momentum, curvature) about a General Relativity and Differential Geometry. In a few given system is contained within the metric tensor g , simple keystrokes, one can have access to virtually every µν the solution to equation (1). This means that given a known solution, as well as all of their physical and ge- metric tensor, all the physical and geometric properties ometric properties, saving time making or finding those of that space-time can be extracted from it. calculations. It also makes it possible to search for and find a solution of interest that has whatever physical or Perfect fluid cosmologies treat the universe as being geometric properties one might be interested in. filled with matter that behaves approximately like a per- fect fluid, making the heat flux and viscosity terms neg- This in turn, gives a fast and easy solution to what ligible in the energy momentum tensor that appears in has aptly been named the Equivalence Problem. It is cen- equation (1). This makes solving the field equations far tered around whether, given two metric tensors, there ex- simpler and as a first order approximation, appears to fit ists a coordinate transformation that makes them equiv- well with observations, so most known cosmological so- alent. Einstein's theory is based on his postulate that lutions are of this form. The (2; 0) version of the energy the laws of physics are invariant under coordinate trans- momentum tensor then is given by formations from one inertial reference frame to another. This means that any given solution's physical and geo- p T µν = ρ + U µU ν + pηµν (2) metric properties should also be invariant under coordi- c2 nate transformation that go from one inertial frame to another. Thus these properties are what uniquely de- where ρ is the energy density of the fluid, p is the pres- fine a solution, and this library will provide the means to sure, and U µ is the 4-vector velocity of the fluid. identify a known solution based solely off of them. The pressure and energy density for most of these mod- els are related by a constant γ in the gamma law equation of state ∗ Also at Physics Department, Utah State University. p = ρ(γ − 1) (3) 2 Many of these solutions as special cases, can be reduced If N and U are linearly dependent, then the solution down to simpler forms with either stiff matter or dust is aligned. If they are linearly independent then the so- equations of state p = ρ or p = 0 respectively[4]. lution is tilted. A quick and easy way of checking this To find the four-velocity of the fluid, if p and ρ are is by taking the wedge product of the two. An aligned known for the space-time, plugging equation (2) into (1), solution will necessarily have a vanishing wedge product. gives four equations for each of the four indices. The only unknowns are then the four components of the four- Aligned : U ^ N = 0 (7) velocity, thus making it straightforward to solve for each component. A tilted solution will always satisfy. B. Tilt Classifications in Cosmology T ilted : U ^ N 6= 0 (8) Tilt tends to complicate the equations represented in An intrinsic property of each solution is whether or not (1) due to diagonalization methods and theorems not be- a homogeneous observer is co-moving with the matter in ing applicable to tilted solutions. This being the case, the universe. If so, the solution is \aligned", otherwise it there are far fewer tilted solutions known than aligned is \tilted". We assume a universe foliated into space-like cases. However, tilted cases are of interest for multiple hypersurfaces determined by the model's group orbits. cosmological considerations. For example, some tilted so- Then there can be found two time-like vector fields that lutions permit periods of rapid inflation in cosmological define two congruences. The first is the geometric con- history that their aligned counterparts do not. They are gruence naturally defined as the vector field N, normal able to represent more dynamic and sophisticated mod- to the group orbits and thus orthogonal to the surfaces els that could potentially better fit descriptions of our of transitivity[3]. This vector field must be geodesic, as real universe. Tilt can also affect the predicted behavior well as vorticity and acceleration free. of the expanding universe and its ultimate fate. Within Each solution has unique families of world lines or- these models, it is possible for an observer experiencing thogonal to the hypersurfaces of homogeneity spanned tilt to see a collapsing universe, while an aligned observer by its Killing vector fields, or in other words, a set of could see the universe infinitely expanding. The in-depth observers who see a homogeneous universe. These span- details and derivations of these cosmological considera- ning Killing vector fields, X, are found from the Killing tions are beyond the scope of this paper but are explored equation given in local coordinates by by Coley, Hervik and Lim [3]. Another reason tilt classifications are of interest goes rµXν + rν Xµ = 0 back to the Equivalence Problem discussed earlier. Be- cause tilt is intrinsic, it is one of the defining properties or more consisely, as the vanishing Lie derivative of the that can be used to help identify a particular solution. So metric tensor if a solution is transformed into a new coordinate system, its tilt classification is invariant. LX gµν = 0 (4) The set of vector fields that satisfy (4) form a Lie alge- II. EXAMPLES bra and are generators of the isometry group Gn, acting on each space-time. They are tangent to the homoge- neous hypersurfaces defined by the level sets of the func- The collection of solutions used for this project are tion λ on the given spacetime. This implies that any taken from chapter 14 of the extensive solution encyclo- function f(λ) is invariant under the isometry group and pedia published by Stephani [4]. This includes 65 perfect thus satisfy fluid cosmological solutions, and of those 65 we found that a total of 8 were tilted models. The following sec- LX f(λ) = 0 (5) tions give examples of the classification process and the Killing vector fields for titled and aligned solutions for This then gives a prescription for finding the geometric each class of metric that was analyzed.
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