gr group up fie ield ld the heory y co condensate ate co - - PowerPoint PPT Presentation

gr group up fie ield ld the heory y co condensate ate co cosmolo logy gy

king’s college london

mair iri sakella llaria iado dou

• motivation
• elements of group field theory and group field theory condensate cosmology
• quantum dynamics of the mean field for a GFT condensate cosmology model
• cosmological consequences of the modified friedmann equation
• conclusions
• utline

pithis, sakellariadou, tomov, PRD94 94 (2016) 064056 pithis, sakellariadou, PRD95 95 (2017) 064004 de cesare, sakellariadou , PLB764 764 (2017) 49 de cesare, pithis, sakellariadou , PRD94 94 (2016) 064051

motivation

classical cosmology is built upon general relativity (GR) and the cosmological principle but

• GR is a classical effective theory valid at low energies
• the assumption of a continuous spacetime characterised by

homogeneity and isotropy on large scales is valid at low energies to describe the physics near the big bang, a quantum gravity theory with the appropriate space-time geometry is needed

at very high energy scales, quantum gravity corrections can no longer be neglected, spacetime as a continuum medium may no longer be valid, and geometry may altogether lose its familiar meaning

• can we resolve the initial singularity ?
• can we find an accelerated expansion without introducing an

inflaton field with an ad hoc potential?

early universe cosmology is in need of a rigorous underpinning in quantum gravity cosmological data represent the best chance for testing quantum gravity, thus guiding the formulation of the complete theory

quantum gravity approaches:

• top-down approach
• string/brane model
• non-perturbative approach
• - wheeler-de witt
• - loop quantum gravity
• - causal dynamical triangulations
• - causal sets
• - group field theory
• bottom-up approach
• non-commutative spectral geometry
• asymptotic safety

quantum gravity approaches:

• top-down approach
• string/brane model
• non-perturbative approach
• - wheeler-de witt
• - loop quantum gravity
• - causal dynamical triangulations
• - causal sets
• - group field theory
• bottom-up approach
• non-commutative spectral geometry
• asymptotic safety

string gas scenario

quantum gravity approaches:

• top-down approach
• string/brane model
• non-perturbative approach
• - wheeler-de witt
• - loop quantum gravity
• - causal dynamical triangulations
• - causal sets
• - group field theory group field theory condensate cosmology
• bottom-up approach
• non-commutative spectral geometry

construct a theory of quantum geometry and identify states in its hilbert space which represent a macroscopic, spatially homogeneous, isotropic universe

elements of group field theory (GFT) & group field theory condensate cosmology (GFC)

• riti (2007, 2014)

gielen, oriti, sidoni (2013, 2014) gielen, sidoni (2016)

approach: group field theory: spacetime and geometry should be emergent, as an effective description of the collective behaviour of different pre-geometric fundamental degrees of freedom functional renormalisation group analyses of GFT models support the idea of a phase transition separating a symmetric from a broken/condensate phase, as the “mass” parameter changes its sign to negative values in the IR limit of the theory the process responsible for the emergence of a continuum geometric phase is a condensation of bosonic GFT quanta, each representing an atom of space group field theory quantum cosmology: identify this condensate phase to a continuum spacetime and derive for the corresponding GFT condensate states an effective dynamics through mean field techniques and give these states a cosmological interpretation

GFTs are QFTs defined over group manifolds with combinatorially nonlocal interaction terms intending to generalise matrix models for 2d QG in higher dimensions basic idea: all data encoded in the fields is exclusively of combinatorial and algebraic nature which turns GFT into a background independent FT simplest case: this data consists of group elements attached to the edges of a graph group field theory (GFT)

GFTs are QFTs defined over group manifolds with combinatorially nonlocal interaction terms intending to generalise matrix models for 2d QG in higher dimensions basic idea: all data encoded in the fields is exclusively of combinatorial and algebraic nature which turns GFT into a background independent FT

• choice of lie group G, interpreted as local gauge group of gravity: SU(2)

(for models aiming at providing a second quantised reformulation of the kinematics of canonical LQG, G=SU(2) is the gauge group of ashtekar-barbero gravity) group field theory (GFT)

GFTs are QFTs defined over group manifolds with combinatorially nonlocal interaction terms intending to generalise matrix models for 2d QG in higher dimensions basic idea: all data encoded in the fields is exclusively of combinatorial and algebraic nature which turns GFT into a background independent FT

• choice of lie group G, interpreted as local gauge group of gravity: SU(2)

(for models aiming at providing a second quantised reformulation of the kinematics of canonical LQG, G=SU(2) is the gauge group of ashtekar-barbero gravity)

• choice of the combinatorial structure of elementary building blocks :

the elementary building block of 3dim space is a (quantum) tetrahedron

field theory defined

• n 4 copies of G:

quantum tetrahedra

group field theory (GFT)

fundamental quanta of GFT field which are created or annihilated by 2nd quantised field operators

GFTs are QFTs defined over group manifolds with combinatorially nonlocal interaction terms intending to generalise matrix models for 2d QG in higher dimensions basic idea: all data encoded in the fields is exclusively of combinatorial and algebraic nature which turns GFT into a background independent FT

• choice of lie group G, interpreted as local gauge group of gravity: SU(2)

(for models aiming at providing a second quantised reformulation of the kinematics of canonical LQG, G=SU(2) is the gauge group of ashtekar-barbero gravity)

• choice of the combinatorial structure of elementary building blocks :

the elementary building block of 3dim space is a (quantum) tetrahedron

dual picture from LQG and spin foams: central vertex with 4 open outgoing links, with group-theoretic data associated to these links GFT quanta as open spin network vertices

group field theory (GFT)

GFTs are QFTs defined over group manifolds with combinatorially nonlocal interaction terms intending to generalise matrix models for 2d QG in higher dimensions basic idea: all data encoded in the fields is exclusively of combinatorial and algebraic nature which turns GFT into a background independent FT

• choice of lie group G, interpreted as local gauge group of gravity: SU(2)

(for models aiming at providing a second quantised reformulation of the kinematics of canonical LQG, G=SU(2) is the gauge group of ashtekar-barbero gravity)

• choice of the combinatorial structure of elementary building blocks :

the elementary building block of 3dim space is a (quantum) tetrahedron

• specify theory by the choice of a type of field - complex scalar field - and a

corresponding action – a kinetic quadratic term and a sum of interaction polynomials weighted by coupling constants - encoding the dynamics group field theory (GFT)

4d QG complex scalar field  living on d=4 copies of the lie group G= SU(2) : group elements g correspond to parallel transports on the gravitational connection along a link impose invariance of the field under the right diagonal action of the group G

• n G

discrete gauge invariance at the vertex from which d links emanate

where

I

4

equivalent to the closure constraint of a quantum tetrahedron

4d QG complex scalar field  living on d=4 copies of the lie group G= SU(2) :

where

action

kinetic: local interaction term: nonlinear and nonlocal convolution of GFT field with itself

classical e.o.m.

• f the field

=0

fock vacuum no space state (no topological, no quantum geometric information) excitation of a GFT quantum creation of a single open 4-valent

• ver the fock vacuum

LQG spin network vertex

a field corresponds to a an atom of space itself

fock vacuum no space state (no topological, no quantum geometric information) excitation of a GFT quantum creation of a single open 4-valent

• ver the fock vacuum

LQG spin network vertex field operators obey canonical commutation relations:

• btain quantum geometric observable data via second-quantised hermitian operators

vertex number operator vertex volume operator

in terms of matrix elements

• f first quantised LQG

volume operator

construct N-particle states (to describe extended quantum 3-geometries)

fock vacuum no space state (no topological, no quantum geometric information) excitation of a GFT quantum creation of a single open 4-valent

• ver the fock vacuum

LQG spin network vertex quantum theory: dynamics defined by the partition function the appropriate action yields a sum-over-histories for 4dim quantum gravity

group field theory condensate cosmology (GFC) goal: model homogeneous continuum 3-geometries and their cosmological evolution by means of GFT condensate states and their effective dynamics conjecture: a phase transition in a GFT system gives rise to a condensate phase which corresponds to a non-perturbative vacuum of the specific model described by a larger number N=<N>

• f bosonic GFT quanta which have

relaxed into a common ground state

• rthogonal to the fock vacuum

^

suitable to model spatially homogeneous quantum geometries

group field theory condensate cosmology (GFC)

the condensate mean field the non-condensate contributions

field operator,

with nonzero expectation value

simple trial state:

constructed from quantum tetrahedra which encode the same quantum geometric information

with

coherent states (as eigenstates of the field operator) effective dynamics from:

analogue of gross- pitaevskii eq. for real bose condensates

quantum cosmology equation

with

 is a massless scalar field (relational clock)

analysis of the quantum dynamics of the mean field to investigate the geometrogenesis (emergence of low-spin phase) static model in an isotropic restriction

• free case
• interacting case

relational evolution of anisotropic GFT condensates

pithis, sakellariadou, tomov , PRD94 94 (2016) 064056 pithis, sakellariadou, PRD95 95 (2017) 064004

“static” mean field neglect all interactions

symmetry reduction

isotropisation

the mean field is defined on a domain space parametrised by 6 invariant coordinates, assumed to be equal (isotropisation) and expressed in terms of angular coordinate 

“static” mean field neglect all interactions assuming “near-flatness” condition (as a boundary condition): eigensolutions

probability density of the mean free field over 

a concentration of the probability density around small  corresponds to a concentration around small curvature values building blocks are almost flat (smooth continuum 3-sphere) isotropisation

“static” mean field neglect all interactions assuming “near-flatness” condition (as a boundary condition): expectation value

• f the LQG volume
• perator

isotropisation

fourier coefficients

assume interactions between GFT quanta as sub-dominant

normalised spectrum of volume operator wrt eigensolutions σ)

j

• eigensolutions for smaller j have bigger volume;

j=3/2 (=m) has the biggest volume

• the volume is finite for all j

a general solution of the quantum cosmology equation, decomposed in terms of eigensolutions, describes a finite space with largest contributions from low spin modes m labels SU(2) irreducible representations

dominance of the ½-representation assume interactions between GFT quanta as sub-dominant

normalised spectrum of area operator wrt eigensolutions σ)

j

study impact of simplified interactions onto static GFT QG condensate describing effective 3-geometries tensorial interactions

even powered in the modulus of the field

symmetry assumption

• nly for μ<0 and κ>0 we have a nontrivial (nonperturbative) vacuum with

σ so that there is agreement with the condensate state ansatz study impact of simplified interactions onto static GFT QG condensate describing effective 3-geometries the system would settle into one of the 2 minima and describe a condensate consider the interaction term as a perturbation of the free case and compute the effect of perturbation

• nto the spectra of geometric operators

probability density of the interacting mean field over  for the finiteness of the free solutions at the origin is lost due to the interactions the concentration of the probability densities around the origin can still be maintained, giving rise to nearly flat solutions, as long as |κ| does not become too big

normalised spectrum of volume operator normalised spectrum of area operator wrt interacting mean field  for κ=0.22 (triangles) compared to free solutions (dots)

• perturbations increase both the volume and the area;

in the weakly nonlinear regime they remain finite

• the effects are more pronounced for small j (small|μ |)

j

the condensate consists of many discrete building blocks predominantly of the smallest nontrivial size an effectively continuous geometry may emerge from the collective behaviour of a discrete pregeometric GFT substratum solutions around the nontrivial minima in order to understand the condensate phase

normalised discrete spectrum of the volume and area operators wrt interacting mean field

the dominant contribution to V and A comes from fourier coefficients with m=1/2

similar results for simplicial interactions: with or to have a non-trivial (non-perturbative) vacuum in agreement with the condensate state ansatz

differences between the dynamical behaviour of the isotropic and the anisotropic part of the mean field

probability density of the mean field for the isotropic and anisotropic parts for small values of the relational clock probability density of the mean field for the isotropic and anisotropic parts for large values of the relational clock

anisotropies only play an important role at small values of the relational clock (small volumes), whereas at late times the isotropic mode dominates

conclusions:

• a free condensate configuration in an isotropic restriction

settles dynamically into a low-spin configuration of the quantum geometry

• anisotropic GFT condensate configurations tend to isotropise

as the value of the relational clock grows (i.e., as the volume grows)

cosmological consequences through the effective friedmann equation

• turn off GFT interactions (connectivity of graph structures)
• switch on interactions (glueing fundamental building blocks)

de cesare, pithis, sakellariadou , prd 94 (2016)064051 de cesare, sakellariadou , plb 764 (2017) 49

assume interactions between GFT quanta as sub-dominant

equation of motion for 

j

m can be expressed in terms

• f coefficients in the GFT theory

2 j

a minimally coupled massless scalar field introduced to play the role of a relational clock a complex scalar field representing the bose condensate of GFT quanta a representation index

• riti, sidoni, wilson-ewing (2016)

assume interactions between GFT quanta as sub-dominant

equation of motion for 

j

separate σ into its modulus and phase

constant quantity: the conserved U(1) charge constant charge: the GFT energy

j

m can be expressed in terms

• f coefficients in the GFT theory

j 2

• riti, sidoni, wilson-ewing (2016)

the condensate field peaks on a particular representation j:

emergent friedmann equations GFT condensates dynamically reach a low spin phase of many quanta of geometry which are almost entirely characterised by only one spin j the evolution equation of the universe in the classical limit

• f GFT can be written as an effective friedmann equation:

energy density

effective gravitational constant effective gravitational constant from the collective behaviour of spacetime quanta

gielen (2016)

effective gravitational constant effective gravitational constant from the collective behaviour of spacetime quanta quantum geometry effects may lead to stochastic fluctuations of the gravitational constant, which can be thus considered as a macroscopic effective dynamical quantity a time-dependent dark energy term in the modified field equation can be expressed in terms of a time-dependent dynamical gravitational constant the late-time accelerated expansion of the universe may be ascribed to quantum fluctuations in the geometry of spacetime rather than the vacuum energy from the matter sector

de cesare, lizzi ,sakellariadou , PLB760 760 (2016) 498

the condensate field peaks on a particular representation j:

emergent friedmann equations GFT condensates dynamically reach a low spin phase of many quanta of geometry which are almost entirely characterised by only one spin j the evolution equation of the universe in the classical limit

• f GFT can be written as an effective friedmann equation:

energy density

effective gravitational constant effective gravitational constant from the collective behaviour of spacetime quanta bounce (happening at ) when g() vanishes

gielen (2016)

a bounce replacing the classical singularity

asymptotic value for large  is the same in both cases and coincides with newton’s constant

effective gravitational constant bounce (happening at ) when g() vanishes

a bounce replacing the classical singularity

asymptotic value for large  is the same in both cases and coincides with newton’s constant

energy density has a max at the bounce where volume reaches its minimum the singularity is always avoided for E<0 and provided Q is nonzero, it is also avoided for E>0

inflation

the inflationary paradigm

• at high energies, field theory is the correct framework to describe matter
• the universe is homogeneous and isotropic: spin 0 particle
• energy density and pressure:

the potential must be flat

slow-roll

reheating

inflation is a quasi exponential expansion of spacetime

standard cosmology: introduce proper time t and scale factor α with express condition that the universe has positive acceleration in purely relational times classical condition for accelerated expansion:

standard cosmology: introduce proper time t and scale factor α with express condition that the universe has positive acceleration in purely relational times classical condition for accelerated expansion: valid also in the absence of classical spacetime and absence of proper time

near bounce: positive zero

accelerated expansion in the absence of an inflaton field with a tuned potential

r.h.s. l.h.s.

accelerated expansion

l.h.s. r.h.s.

accelerated expansion, followed by a maximal deceleration

standard cosmology: introduce proper time t and scale factor α with express condition that the universe has positive acceleration in purely relational times classical condition for accelerated expansion: valid also in the absence of classical spacetime and absence of proper time

near bounce: positive zero

accelerated expansion in the absence of an inflaton field with a tuned potential can one get sufficiently e-folds?

can one obtain ?

• non-interacting case : for all values of m and Q

2 2

GFT cosmology in the absence of interactions between building blocks cannot replace the standard inflationary scenario

consider effective action for an isotropic GFT condensate

remark:

• spin foam models for 4d QG are mostly based on interaction terms of power 5 (simplicial)
• tensor modes are based on even powers of the modulus field (tensorial)

consider effective action for an isotropic GFT condensate with two conserved quantities:

consider effective action for an isotropic GFT condensate e.o.m. of a classical point particle with potential:

E E E

1 2 3

consider effective action for an isotropic GFT condensate

E E E

1 2 3 1 2

E E

3

E

• rbits are periodic

and describe

• scillations around

stable equilibrium point, given by absolute min of U

for a given GFT energy the solutions are cyclic motions describing

• scillations around a stable minimum

cyclic universe volume that has a positive minimum corresponding to a bounce

• ccurrence of a bounce and an early epoch
• f accelerated expansion found in the free

theory, holds in the interacting case in addition, interactions induce recollapse leading to cyclic cosmologies

can one obtain ?

• interacting case :

quadratic term and 2 interaction terms

• r

additional requirement to have an inflation-like era: no intermediate stage of deceleration between the bounce and end of inflation λ negative

GFT cosmology can lead to an inflation-like era for certain types of interactions between quanta of geometry

comment dynamical effective gravitational constant alternatively with definitions

comment with definitions the exponents of the denominators can be related to the w coefficients in the equation of state p=w of some effective fluids each term scales with the volume as at early times (small volumes) the occurrence of the bounce is determined by the negative sign of  which is the term corresponding to highest w (classical singularity is prevented but interactions are needed to get successful inflation)

Q

remark:

• for real-valued condensate fields, solutions which avoid the singularity problem and

grow exponentially after the bounce can only be found for negative “GFT energy”

• stability properties of an evolving isotropic system:

the condensate models give rise to effectively continuous, homogeneous and isotropic 3-geometries built from many smallest and almost flat building blocks of the quantum geometry

• stability properties of an evolving anisotropic system:
• isotropisation of the system for increasing values of the clock (i.e., the volume)
• the singularity avoidance is not altered by the occurrence of anisotropies

possible further studies:

• check the validity of the assumption for a phase transition into a

condensate phase

• to study the implications of GFT interactions for the geometry of emergent

space-time

• investigate whether the effective dynamics of GFT can reduce at some

limit, to the dynamics of loop quantum cosmology

• investigate the connection between GFT approach (where the pre-

geometric degrees of freedom are algebraic in nature) and non-commutative spectral geometry (NCSG), which may help to quantise NCSG

de cesare, oriti, pithis, sakellariadou (in progress)

conclusions

in the framework of the group field theory approach to quantum gravity, we have investigated the connection between the quantum gravity era and the classical cosmological late-time period in this framework, spacetime geometry emerges from the collective behaviour of quanta of geometry (the fundamental degrees of freedom of the gravitational field),

• nce the macroscopic limit is taken

we have found:

• the initial singularity of the standard cosmology is replaced by a bounce
• interactions between the quanta of geometry lead to a recollapse of the universe
• depending on the type of interactions an early epoch of accelerated expansion

with an arbitrarily large number of e-folds may be achieved