Evolution of Universe
In Eddington-inspired Born-Infeld
Gravity
Inyong Cho
Seoul National University of Science & Technology
2012 International Workshop on String Theory and
Cosmology
Pusan, Korea, Jun 14-16, 2012
- In preparation with Hyeong-Chan Kim (CNU) & Taeyoon Moon (CQUeST)
Outline
1. Introduction to EiBI Gravity
A. Formalism
B. Field Equations
C. Some Physical Solutions
2. Evolution of Universe filled with Perfect Fluid
A. Isotropic case
B. Anisotropic case
3. Conclusions
Eddington-inspired Born-Infeld Gravity
Formalism
Einstein Gravity
Field:
Eddington Gravity (1924)
Field:
Varying S, Integrating by parts, Eliminating a vanishing trace,
we get
Therefore, Eddington’s action
:- viable and alternative starting point to GR
::- dual to GR
However, incomplete : NOT including MATTER
Later attempts to couple matter with
:- start with Palatini gravitational action
coupled to matter
-- no derivatives in g
:- EOM for g 
 back into
 can eliminate g
:-
: complicated,
but Dynamics is fully equivalent
to the original metric theory
Eddington-inspired Born-Infeld Gravity
One parameter (k) theory
:- coupling matter insisting neither on Affine action, nor on Einstein action
(Vollick 2004)
:-
: independent
:- Matter is in usual way (Not in sqrt)
:- For large
:- For small
-limit  Eddington limit
-limit  Einstein limit
In vacuum  Equivalent to Einstein Gravity
Field Equations
: Energy-Momentum Conservation
 Matter plays in the background metric
Some Physical Solutions (Banados-Ferreira 2010)
Corrections to Poisson Equation
By expanding Field Equations to 2nd order in k :
Metric :
with
and
: Poisson Equation
Black Hole Solutions
In vacuum  Equivalent to Einstein Gravity
Therefore, EiBI in vacuum or with only CC is the same with EH
Schwartzchild-de Sitter BH
SAME with Einstein Solution
Charged BH : non vacuum
: electric field
Perfect Fluid : FRW Universe
Metric :
EM tensor :
For radiation,
Banados & Ferreira (2010)
1) Typy-I :
near
2) Typy-II :
near
Bouncing
Non
Singular
Other Phenomenological Results
:- star (sun) formation simulation by using Poisson Eq.
 puts viable bounds on the value of k
:- dark matter
:- density perturbation
Etc…
Evolution of Universe with Perfect Fluid
Metric & Auxiliary Metric
Energy-Momentum Tensor
EOM 1
From these, we get
EOM 2
From these, we get
Friedmann Equations
Volume Part :
Anisotropic Part :
:- In r >> and r << limits, consider cases of
i) w>0
ii) w=0
iii) w<0
iv) w=-1
:- Consider k>0 case only
1) w>0
2) w=0
3) -1<w<0
Isotropic Case : like Einstein
Anisotropic Case: differs from Einstein for -1<w<1/3
Anisotropy
1) w>0 case
:-
decays to 0 exponentially at both ends
 Initial Singularity in Anisotropy is REMOVED !
 Late-time anisotropy approaches a constant
:-
has a maximum in the intermediate period
2) w<0 case
:-
decays to 0 exponentially
 Late-time anisotropy approaches a constant
:-
diverges initially like in Einstein
4) w=-1
Conclusions
1. EiBI : Palatini-type Gravity + Matter
2. Cosmological solution : avoids initial singularity, or bouncing
3. For perfect fluid
w>0 :
non-singular initial beginning
w=0 :
initial de Sitter
-1<w<0 : similar to Einstein Gravity
w=-1 :
same with Einstein Gravity
4. Late time evolution : similar to Einstein Gravity