Cosmological Perturbation in f(T) Gravity Revisited
Cosmological Perturbation in f(T) Gravity Revisited Ong Yen Chin Outline of the Talk
Part I: Mathematical Structures of teleparallel and f(T) theories. Part II: Linear perturbation theory on flat FLRW background. Part III: What can we learn about f(T) gravity from the
linear perturbative analysis? Talk based on arXiv:1212.5774 [grqc], joint work with Keisuke Izumi. Outline of the Talk
Part I: Mathematical Structures of teleparallel and f(T) theories. Part II: Linear perturbation calculation on flat FLRW background. Part III: What can we learn about f(T) gravity from the linear perturbative analysis? Teleparallel Gravity:
Einsteins Other Theory of Gravity Originated from Einsteins attempt to unify general relativity and electromagnetism. In the tranquility of my sickness, I have laid a wonderful egg in the area of general relativity. Whether the bird that will hatch
from it will be vital and long-lived only the gods know. So far I am blessing my sickness that has endowed me with it. 2/41 - Einstein, 1928. Teleparallel Gravity: Einsteins Other Theory of Gravity
You may be amused to hear that one of our great Department Stores [Selfridges] has pasted up in its window your paper [the six pages pasted up side by side] so that passers by can read it all through. Large crowds gather round to read it! [Eddington to Einstein, 11 February, 1929] Field equations in teleparallel spacetime: Einstein's
Fernparallelismus approach towards unified field theory arXiv:physics/0405142v1 [physics.hist-ph] Motivation What we are not trying to investigate: Cosmological models in the context of f(T) gravity, e.g. structure formation, cosmic acceleration, future fate of universe.
What we are interested in: Theoretical aspects of f(T) gravity, in particular: What linear perturbation theory tells us about the viability of f(T) gravity. The Mathematical Structure of Teleparallel Theories Coordinate frame
Non-Coordinate Frame Semi-Riemannian Manifold, Tangent Space, Is Parallelizability too Strong a Demand?
The Following are Equivalent: A manifold M is parallelizable. The tangent bundle TM is trivial. The frame bundle FM has a global section. [In 4-dimension] Vanishing of the second StiefelWhitney characteristic class. [In the general case the vanishing of the second characteristic classes of Stiefel- Whitney, Chern and Pontryagin are necessary but not sufficient conditions
for a manifold to be parallelizable.] Is Parallelizability too Strong a Demand? Fact: Many manifolds are not parallelizable. E.g. among the n-sphere, only S0 , S1 , S3 , and S7 are parallelizable. [R. Bott, J. Milnor, On the Parallelizability of Spheres, Bull. Amer. Math. Soc. 64 (1958)] [The only normed division algebras are
Raoul Bott (September 24, 1923 December 20, 2005) John Milnor (February 20, 1931 - )
] Is Parallelizability too Strong a Demand? Fact: All orientable 3-manifolds are parallelizable [Steenrods theorem], consequently all 4dimensional spacetimes with orientable spatial section are parallelizable.
Fact: A non-compact 4dimensional spacetime admits a spin structure if and only if it is parallelizable [Gerochs theorem]. Norman Steenrod (April 22, 1910 October 14, 1971)
Robert Geroch (1 June 1942 - ) The Mathematical Structure of Teleparallel Theories Notation:
The Mathematical Structure of Teleparallel Theories Weitzenbck connection: Connection Coefficient
Torsion: Roland Weitzenbck (26 May 1885 24 July 1955) A Story: Weitzenbcks Acrostic In 1923 Weitzenbck published a monograph on the theory
of invariants on manifolds that included tensor calculus. In the Preface of this monograph one can read an offensive acrostic. The first letter of the first word in the first 21 sentences spell out: NIEDER MIT DEN FRANZOSEN (Down with the French)
The Mathematical Structure of Teleparallel Theories Riemann Curvature Endomorphism: since That is, the geometry is flat.
The Mathematical Structure of Teleparallel Theories Contortion: The difference between the Weitzenbck connection coefficient and Christoffel symbol of Levi-Civita connection is a tensor:
Physics is Where the Action Is Hilbert-Einstein action: can be re-written as TEGR action: since Covariant derivative of Levi-Civita connection.
Note local Lorentz invariance is broken! Teleparallel Theories and f(T) TEGR f(T)
We will suppress overscript w from now on. Outline of the Talk Part I: Mathematical Structures of teleparallel and f(T) theories.
Part II: Linear perturbation calculation on flat FLRW background. Analysis is done up to second order Important to check ghost condition. Part III: What can we learn about f(T) gravity from the linear perturbative analysis? Background
Hubble parameter: A. Friedmann G. Lematre H.P. Robertson
A.G. Walker Background Gravitational action: Matter [Scalar Field] action: Decomposition to Various Modes
Most general form of decomposition: Coupling between vectors and pseudovector: Gauge Transformation Coordinate transformation:
Gauge Fixing Scalar Perturbation All equality in the analysis is understood to be up to second order only.
Summary: Perturbation of All Modes Up to second order linear perturbation, Scalar: Pseudoscalar: No propagating pseudoscalar degree of freedom. Tensor: Vector and Pseudovector: No propagating vector and pseudovector degree of freedom.
As yet no sign of ghosts. Outline of the Talk Part I: Mathematical Structures of teleparallel and f(T) theories. Part II: Linear perturbation calculation on flat FLRW background. Part III: What can we learn about f(T) gravity from
the linear perturbative analysis? Nonlinearity of f(T) Theory Recall: f(T) gravity has 5 degrees of freedom, i.e. 3 extra ones compared to TEGR. Absence of the extra degrees of freedom at second order linear perturbation indicates that the theory is
highly nonlinear. Remark: Li et al. already pointed out that extra degrees of freedom do not appear in linear perturbative analysis of Minkowski background. Important to know what the hidden ones do at nonlinear level! Nonlinearity of f(T) Theory
The situation is very similar to that in nonlinear massive gravity theory: It has generically, 5 degrees of freedom, but in the second order action on open FLRW background we can catch only two tensor degrees of freedom. Nonlinearity of f(T) Theory The hidden degrees of freedom cause
nonlinear instability in nonlinear massive gravity. Conclusion: One cannot conclusively say anything about stability issues in f(T) gravity, unless we know how the hidden degrees of freedom behave at nonlinear level.
Acknowledgement James Nester, Shinji Mukohyama, Pisin Chen, Je-An Gu, Brett McInnes, Wu-Hsing Huang. End of Talk Extra Slide: Nonlinear Massive Gravity
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