Stability of Einstein Universe
in f(T) Theory
Jung-Tsung Li
National Tsing Hua University,
Taiwan
Apr. 25, 2012
Based on:
Puxun Wu and Hongwei Yu, Phys. Lett. B 703: 223-227 (2011)
Christian G. Bomer, PHYSICAL REVIEW D 76, 084005 (2007)
Outline
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Teleparallel Gravity
Einstein Static Universe in f(T)
Exponential and Logarithmic model
Summary
Teleparallel Gravity
feature
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Use Weitzenbock connection, which has no
curvature but torsion only.
An alternative gravity theory, equivalent to
General Relativity.
Teleparallel Gravity
Introduction
The dynamic object is vierbein field
,which is an orthogonal basis at point
The orthogonality has the relation:
The metric tensor is given by
Teleparallel Gravity
Introduction
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Curvatureless Weitzenbock connection:
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Torsion tensor is:
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Riemann curvature vanish under Weitzenbock
connection
Teleparallel Gravity
Introduction
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We can construct Teleparallel Lagrangian:
The Lagrangian
differ a surface term from
Einstein Lagrangian
where R is constructed by Levi-Civita connection
The action is:
Teleparallel Gravity
Introduction
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Varying the action with respect to the vierbein
gives the field equation:
where
is energy-momentum tensor.
Einstein Static universe
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Consider the FRW metric :
The vierbein is :
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The torsion scalar
Einstein Static universe
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Now we do the linear homogeneous scalar
perturbations on static Einstein universe
Combine with modified Friedmann equation
We can obtain:
f(T) model in closed and open
universe
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We would like to try exponential and
logarithmic f(T) model in both closed and
open universe.
Consider two kinds of Lagrangian
Exponential:
Logarithmic:
Exponential f(T)
Substitute this into the stable condition:
Closed:
Open:
open
closed
Logarithmic f(T)
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Substitute this into the stable condition:
Closed:
Open:
open
closed
Summary
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Teleparallel gravity is an alternative gravity theory
of the universe.
Exponential f(T) gives the broader equation of
state w than GR allows, and it can also be reduced
back to GR form
Logarithmic also gives broader equation of state
w, but it can’t go back to GR, which gives