nonstandard lagrangian in cosmology - constrained system · 15 years of the seenet-mtp cooperation...
TRANSCRIPT
15 years of the SEENET-MTP cooperation
Dragoljub D. Dimitrijevic1, Goran S. Djordjevic1, Milan Milosevic1,
Marko Dimitrijevic2
1 Faculty of Sciences and Mathematics, University of Nis, Serbia2 Faculty of Electronic Engineering, University of Nis, Serbia
Nonstandard Lagrangian in Cosmology -
Constrained System
Department of Theoretical Physics, IFIN-HH
Department of Theoretical Physics, Faculty of Physics
Bucharest
23 April, 2018
1 Introduction
2 Cosmology
3 Cosmological Perturbations
4 Non-standard Lagrangian
5 Extension
6 Constraints
7 Conclusion
Introduction
• We study real scalar field with non-standard Lagrangian
density, imortant (at least for us) in Cosmological context.
• General space-time metric:
• Metric tensor:
• Components of metric tensor:
• Determinant of metric tensor:
2 , , 0,1,2,3ds g dx dx
g
g
ˆdetg g
Introduction
• Mostly used:
• Minkowski space-time (Special Relativity):
• Friedmann-Robertson-Walker-Lemaitre (FRWL) flat space-
time (General Relativity/Cosmology):
• Scale factor (how much the Universe changes its size): a(t)
• Lapse function: N(t)
2 , , 0,1,2,3ds g dx dx
2 2 2 2 2 2 2ds dt dx dy dz dt dx
2 2 2 2 2( ) ( )ds N t dt a t dx
Cosmology
• Real scalar field with non-standard Lagrangian density.
• General Lagrangian action:
• Lagrangian (Lagrangian density) of the standard form:
• Non-standard Lagrangian:
( , ) ( ) 1 2 ( )tach T X V T X T L
( , ) ( ) ( )X V L
4 ( ( ), )S d x g X Ld
1
2X g T T
Cosmology
• The action:
• In cosmology, scalar fields can be connected with a perfect
fluid which describes (dominant) matter in the Universe.
• Components of the energy-momentum tensor:
4 ( , )S d x g X T Ld
2 ST
gg
( )T P u u Pg
Cosmology
• Pressure, matter density and velocity 4-vector, respectively:
( , ) ( , )P X T X T L
( , ) 2 ( , )X T X X TX
LL
2
Tu
X
( )T P u u Pg
Cosmology
• Total action: term which describes gravity (Ricci scalar,
Einstein-Hilbert action) plus term that describes cosmological
fluid (scalar field minimally coupled to gravity):
• Einstein equations:
4 ( , )S d x g R X T d L
1
2R Rg T
Cosmological Perturbations
• Very important: cosmological perturbations around the
classical background (give us information about Universe at
the very beginning, i.e. CMB and cosmological structure
formation).
• ADM decomposition of the metric:
4 ( , )ij i
ij G i GS d x g N N X T d H H L
2 2 2 ( )( )i i j j
ijds g dx dx N dt N dt dx N dt dx
Cosmological Perturbations
• Perturb everything:
• Action for classical background:
(0) (1) (2) ...S S S S
( )ij ij ijt
( )ij ij ijt
( ) ...T T t T
(2) (2) (2)
QUAD INTS S S
(0)S(1) 0S
Cosmological Perturbations
• Path integral quantization of cosmological perturbations:
• Transition amplitude (tells us how the system evolve quantum-
mehanically): K
• Path integral measure:
• If you are lucky, „standard“ Feynman measure (no constraints)
• If you are less lucky, Faddeev-Popov measure (first class
constraints)
• No luck at all, Senjanovic measure (both first and second class
constraints)
(2) (2)exp[ ]QUAD INTK D iS iS
D
Cosmological Perturbations
• Background FLRW space-time:
• Friedmann equation (H-Hubble parameter):
• Acceleration equation:
• Continuity equation:
2 2 2 2( )FLRWds g dx dx dt a t dx
2 1( , ),
3
aH X T H
a
1
( , ) 3 ( , )6
aX T P X T
a
3 ( , ) ( , )H X T P X T
Non-standard Lagrangian
• Effective field theory of non-standard lagrangian of
DBI-type (tachyon scalar field):
• EoM:
( ) 1tach V T g T T
L
2
2
1(1 ( ) )
1 ( ) ( )
T T dVg T T T
T V T dT
Non-standard Lagrangian
• Minisuperspace model for the background:
• Total Lagrangian:
( , , ) + ( , , , )g tachN a a T T N a L L L
3 2 2( , , , ) ( ) 1tach T T N a Na V T N T L
2 2 2 2 2( ) ( )FLRWds N t dt a t dx
2
( , , ) 6 + 6g
aaN a a kNa
N
L
Extension
• Minisuperspace model for the background :
• Auxiliary field :
• Total Lagrangian:
( , , ) + ( , , , , )g extN a a T T N a L L L
23 2
2
1 1 1( , , , , ) [ ( ) ]
2 2 2ext
TT T N a Na V T
N
L
2 2 2 2 2( ) ( )FLRWds N t dt a t dx
2
( , , ) 6 + 6g
aaN a a kNa
N
L
Extension
• From the EoM for :
2 21( , , ) 1
( )f T T N N T
V T
( , , )( , , , , ) | ( , , , )ext tachf T T NT T N a T T N a
L L
2 3 2 33 21
6 + 6 ( )2 2 2
aa a T NakNa Na V T
N N
L
(0)S dt L
Constraints
• We need to know what are the constraints in order to proceed
with .
• Conugated momenta:
1212
a a
aa Np a p
a N a
L
3
3T T
a T Np T p
T N a
L
0NpN
L
0p
L
(2)S
Constraints
• Primary constraints:
• Canonical Hamiltonian:
• Total Hamiltonian (add all constraints to canonical
Hamiltonian):
1 0 0Np
2 0 0p
0c NH H2 3
2 3 2
0 3
1 16 ( )
24 2 2 2
aT
p aka p a V T
a a
H
1 1 2 2T c H H
Constraints
• Consistency conditions:
• Secondary constraints:
• Consistency conditions on secondary constraints:
1 1, 0T H
1 0 0 H3
2 3 2
2 3 2
1 1[ ( ) ] 02 2 2
T
aN p a V T
a
2 2 , 0T H
1 1, 0T H
2 2 , 0T H
Constraints
• (Reminder) First class constraints: their Poisson brackets with
other constraints vanish:
• (Reminder) Constraints that are not of the first class are called
second class.
• (Reminder) The number of second class constraints is always
even.
• For this extended model:
– two first class constraints and
– two second class constraints.
1 , . 0st any constr
Conclusion
• There are both type of constraints.
• So, Senjanovic method/measure should be used.
• As already mentioned, extended model is classically
equivalent to the initial one.
• As already mentioned, extended model is more suitable for
path integral quantization.
References
• D.A. Steer and F. Vernizzi, Phys. Rev. D 70, 043527 (2004).
• N. Bilic, D.D. Dimitrijevic, G.S. Djordjevic and M. Milosevic, Int. J. Mod. Phys. A32, 1750039 (2017).
• A. Sen, JHEP 04, 048 (2002).
• G.S. Djordjevic, D.D. Dimitrijevic and M. Milosevic, Rom. Rep. Phys. 68, No. 1, 1 (2016).
• S. Anderegg and V. Mukhanov, Phys. Lett. B331, 30-38 (1994).
• T. Prokopec and G. Rigopoulos, Phys. Rev. D82, 023529 (2010).
• J.O. Gong, M.S. Seo and G. Shiu, JHEP 07, 099 (2016).
• This work is supported by the SEENET-MTP
Network under the ICTP grant NT-03.
• The financial support of the Serbian Ministry for
Education and Science, Projects OI 174020 and OI
176021 is also kindly acknowledged.
T H A N K Y O U!
Х В А Л А !