cosmological implications of the group field theory …kcl...cosmological implications of the group...

Cosmological implications of the group field theory approach

to quantum gravityMarco de Cesare

King’s College Londonin collaboration with D.Oriti, A.Pithis, M.Sakellariadou:

MdC,MS: Phys.Lett. B764 (2017) 49-53, ArXiv:1603.01764 MdC,AP,MS: Phys.Rev. D94 (2016) no.6, 064051, ArXiv:1606.00352

MdC,DO,AP,MS: ArXiv: 1709.00994

COST training school: “Quantum Spacetime and Physics Models” Corfu, Greece, 16-23 September 2017

2

• Emergent cosmology scenario in group field theory (GFT)

• Some brief remarks on GFT

• Effective Friedmann dynamics

• Cosmological implications of GFT models

• No interactions between quanta

• Interactions are allowed

• Anisotropies in the EPRL model

• Conclusions

Plan of the Talk

Group Field Theory (GFT)• GFT is a non-perturbative and background

independent approach to Quantum Gravity (Oriti ’11, ’13)

• Fundamental degrees of freedom are discrete objects carrying pre-geometric data (holonomies, fluxes)

• The dynamics of the GFT is obtained from a path-integral

• The interaction potential contains the information about the gluing of quanta to form 4D geometric objects.

Z =

Ze�S['] S = K + V

Figures: Gielen,Sindoni ‘16

4

• Quantum spacetime as a many-body system

• Spacetime is an emergent concept, determined by the collective dynamics of ‘quanta’ of geometry

• Cosmological background obtained by considering the dynamics of a condensates of such ‘quanta’ (Gielen,Oriti,Sindoni ’13)

• Evolution is defined in a relational sense (matter clock )

Cosmology from GFT

Main advantages• Cosmology can be obtained from the full theory

• The formalism naturally allows for a varying number of ‘quanta’

• It may offer a way to go beyond standard cosmology

�

5

• mean field

• spin representation

• Enforce isotropy (tetrahedra have equal faces)

• Homogeneous and isotropic background can be studied by looking at the expectation value of the volume operator

Cosmological Background

V (�) =X

j

Vj |�j(�)|2 Vj ⇠ j3/2`3Pl

' = '(g1, g2, g3, g4;�)

'j1,j2,j3,j4,◆(�)

�j(�) = 'j,j,j,j,◆?(�)

gi 2 SU(2)

6

• The dynamics of yields effective equations for the evolution of

• Simplification obtained when restricting to one spin j

• Last assumption is justified by results concerning the emergence of a low-spin phase (Gielen’16) (Pithis,Sakellariadou,Tomov’16)

Emergent Friedmann dynamics�j

V

@�V

V= 2

@�⇢

⇢

@2�V

V= 2

"@2�⇢

⇢+

✓@�⇢

⇢

◆2#

@�V

V=

2P

j Vj⇢j@�⇢jPj Vj⇢2j

(Oriti,Sindoni,Wilson-Ewing ’16)

⇢j = |�j |

7

• Single spin effective action (index j dropped)

• Global U(1) conserved charge , interpreted as the momentum canonically conjugated to

• Polar decomposition of the mean field

• Equation of motion of the radial part

Effective Friedmann equation

Seff =

Zd�

�A |@��|2 + V(�)

�

The non-interacting case

�⇡� = a3�

Q

V(�) = B|�(�)|2

� = ⇢ ei✓ Q ⌘ ⇢2@�✓

@2�⇢�

Q2

⇢3� B

A⇢ = 0 E = (@�⇢)

2 +Q2

⇢2� B

A⇢2

(Oriti,Sindoni,Wilson-Ewing ’16)(MdC,Sakellariadou ’16)

8

• The effective Friedmann equation can be recast in the form

• The effective gravitational constant can be computed exactly in the free model

• In the large volume limit one recovers the ordinary Friedmann evolution

• There is a bounce when . This takes place when

Bouncing Universe

H2 =8⇡Geff

3", " =

�2

2

Geff (�) =G�E2

+ 12⇡GQ2�sinh

2⇣2

p3⇡G(�� �)

⌘

⇣E �

pE2

+ 12⇡GQ2cosh

⇣2

p3⇡G(�� �)

⌘⌘2 .

� ! 1

Geff = 0 (at � = �)H2 = 0

(MdC,Sakellariadou ’16)

(a = V 1/3)

Bouncing Universe

-4 -2 2 4ϕ

0.2

0.4

0.6

0.8

1.0

Geff

-4 -2 2 4ϕ

0.2

0.4

0.6

0.8

1.0

1.2

1.4Geff

E<0 E>0

profile of for opposite signs of the conserved charge E. The behaviour is generic for Q 6= 0

Geff

10

• An early era of accelerated expansion is usually assumed in order to solve the classic cosmological puzzles.

• We seek a relational definition of the acceleration

• Classically, for a minimally coupled scalar field, we have:

Accelerated expansion (geometric inflation)

a

a=

1

3

⇣⇡�

V

⌘2"@2�V

V� 5

3

✓@�V

V

◆2#

(a = V 1/3)

Accelerated expansion

0.5 1.0 1.5 2.0 2.5 3.0ϕ

2

4

6

8

0.5 1.0 1.5 2.0 2.5 3.0ϕ

1

2

3

4

5

6

7

E<0 E>0

a

a=

1

3

⇣⇡�

V

⌘2"@2�V

V� 5

3

✓@�V

V

◆2#

N =

1

3

log

✓Vend

Vbounce

◆=

2

3

log

✓⇢end

⇢bounce

◆number of e-folds

too small in the free caseN ⇠ 0.1

(MdC,Sakellariadou ’16)

12

• The role of interactions in GFT is crucial for two main reasons: • Interactions prescribe the gluing of quanta to form 4D geometric

objects

• They have important consequences for cosmology

• Phenomenological approach

The impact of interactions

Seff =

Zd�

�A |@��|2 + V(�)

�

V(�) = B|�(�)|2 + 2

nw|�|n +

2

n0w0|�|n

0w0 > 0, A,B < 0

(MdC,Pithis,Sakellariadou ’16)

�(ϕ)��

0.0 0.5 1.0 1.5 2.0 2.5

50

100

150

200

ϕ

13

Cosmological evolution is periodic. Endless sequence of expansions and contractions, matched with bounces

Cyclic Universe

14

Number of e-folds: switching on the interactions

log(�) log(-�)

10 20 30 40 50 60 70

50

100

150

200

250

300

������ �� �-�����

���(�)

0.2 0.4 0.6 0.8 1.0

1.0

1.5

2.0

2.5

3.0

3.5

������ �� �-�����

log(�) log(-�)

59.5 60.0 60.5 61.0

260

265

270

275

������ �� �-�����V(�) = B|�(�)|2 + 2

nw|�|n +

2

n0w0|�|n

0

(MdC,Pithis,Sakellariadou ’16) No intermediate deceleration stage

N =

2

3

log

✓⇢end

⇢bounce

◆

15

Number of e-folds: switching on the interactions

log(�) log(-�)

10 20 30 40 50 60 70

50

100

150

200

250

300

������ �� �-�����

(MdC,Pithis,Sakellariadou ’16)

can be arbitrarily large!N

Bonus

n0 > n

n � 5

Constraints:

w < 0

V(�) = B|�(�)|2 + 2

nw|�|n +

2

n0w0|�|n

0

N =

2

3

log

✓⇢end

⇢bounce

◆

• GFT for the EPRL model

• Perturb the isotropic background

Microscopic anisotropiesS = K + V5 + V 5

' = '0 + �'

g2, j2

g1, j1

g3, j3

g4, j4

-2 -1 0 1 2

1.0

1.2

1.4

1.6

1.8

2.0

ϕ

A

V

area-to-volume ratio

V =1

5

Zd�

X

ji,mi,◆a

'j1j2j3j4◆1m1m2m3m4

'j4j5j6j7◆2�m4m5m6m7

'j7j3j8j9◆3�m7�m3m8m9

'j9j6j2j10◆4�m9�m6�m2m10

'j10j8j5j1◆5�m10�m8�m5�m1

(MdC,Oriti,Pithis,Sakellariadou ’17)

⇥10Y

i=1

(�1)ji�mi V5(j1, . . . , j10; ◆1, . . . ◆5)

17

Within the GFT framework, homogeneous and isotropic cosmologies are described as coherent states of basic building blocks.

Our results:

• Bouncing cosmologies are a generic feature of the theory

• Interactions between quanta lead to a cyclic Universe

• Suitable choices of the interactions can make the early Universe accelerate for an arbitrarily large number of e-folds

• Anisotropies decay away from the bounce in a region of parameter space

Conclusions

18

• MdC,Sakellariadou’16: "Accelerated expansion of the Universe without an inflaton and resolution of the initial singularity from Group Field Theory condensates", arXiv:1603.01764

• MdC,Pithis,Sakellariadou’16: “Cosmological implications of interacting Group Field Theory models: cyclic Universe and accelerated expansion", arXiv:1606.00352

• MdC,Oriti,Pithis,Sakellariadou’17: “Dynamics of anisotropies close to a cosmological bounce in quantum gravity", arXiv:1709.00994

• Oriti’11: “The microscopic dynamics of quantum space as a group field theory”, arXiv:1110.5606

• Oriti’13: “Group field theory as the 2nd quantization of Loop Quantum Gravity", arXiv:1310.7786

• Gielen,Oriti,Sindoni’13: “Cosmology from Group Field Theory Formalism for Quantum Gravity", arXiv:1303.3576

• Oriti,Sindoni,Wilson-Ewing’16: “Bouncing cosmologies from quantum gravity condensates", arXiv:1602.08271

• Gielen’16: “Emergence of a low spin phase in group field theory condensates", arXiv:1604.06023

References

EXTRAS

200 2 4 6 8 10 12 14

-1000

-500

0

500

ρ

U(ρ)

n=4, n'=5, λ=-1, μ=0.1, Q2=10, m2=10

0 2 4 6 8 10 12 14-60

-40

-20

0

20

40

60

ρ

ρ′

• Polar decomposition:

• Dynamics of the radial part:

Dynamics in the interacting model

� = ⇢ ei✓

@2�⇢ = �@⇢U

U(⇢) = �1

2m2⇢2 +

Q2

2⇢2+

�

n⇢n +

µ

n0 ⇢n0

E =1

2(@�⇢)

2 + U(⇢)

Q ⌘ ⇢2@�✓

and conserved quantities

E Q

� = �w

Aµ = �w0

A> 0with:

21

Effective Friedmann equation (including interaction terms)

H2 =8Q2

9

"ma6

+"Ea9

+"Qa12

+"µ

a9�32n

�

"m =B

2A"E = VjE"Q = �Q2

2V 2j "µ = � w

nAV 1�n/2j

• Quantum gravity corrections • Compare with ekpyrotic models