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 !