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