group field theory condensate cosmology mairi sakellariadou - qg... · 2017-03-28 · approach:...

group field theory condensate cosmology

king’s college london

mairi sakellariadou

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

outline

pithis, sakellariadou, tomov, PRD94 (2016) 064056

pithis, sakellariadou, PRD95 (2017) 064004

de cesare, sakellariadou , PLB764 (2017) 49

de cesare, pithis, sakellariadou , PRD94 (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

o string/brane model

o non-perturbative approach

-- wheeler-de witt

-- loop quantum gravity

-- causal dynamical triangulations

-- causal sets

-- group field theory

bottom-up approach

o non-commutative spectral geometry

o asymptotic safety


top-down approach



-- wheeler-de witt



-- causal sets

-- group field theory

bottom-up approach


o asymptotic safety

string gas scenario


top-down approach



-- wheeler-de witt



-- causal sets

-- group field theory group field theory condensate cosmology

bottom-up approach


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)

oriti (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)



(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

on 4 copies of G:

quantum tetrahedra


fundamental quanta of GFT

field which are created or

annihilated by 2nd quantised

field operators



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




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


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

on 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.

of the field

0 =0

fock vacuum no space state

(no topological, no quantum geometric information)

excitation of a GFT quantum creation of a single open 4-valent

over the fock vacuum LQG spin network vertex

a field corresponds to a an atom of space itself





field operators obey canonical commutation relations:

obtain quantum geometric observable data via second-quantised hermitian operators

vertex number operator vertex volume operator in terms of matrix elements

of first quantised LQG

volume operator

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





quantum theory:

dynamics defined by

the partition function

the appropriate action yields a sum-over-histories for 4dim quantum gravity


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>

of bosonic GFT quanta which have

relaxed into a common ground state

orthogonal to the fock vacuum

^

suitable to model spatially homogeneous

quantum geometries


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 (2016) 064056

pithis, sakellariadou, PRD95 (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



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



assuming “near-flatness” condition (as a boundary condition):

expectation value

of the LQG volume

operator

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


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

only 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 onto 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


equation of motion for j

m can be expressed in terms of 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

oriti, sidoni, wilson-ewing (2016)


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 of coefficients in the GFT theory

j 2

oriti, 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 of 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)


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 (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 of GFT can be written as an effective friedmann equation:

energy density


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:




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




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


with two conserved quantities:

consider


e.o.m. of a classical point particle with potential:

E

E

E

1

2

3

consider


E

E

E

1

2

3

1

2 E E 3

E orbits are periodic and describe oscillations around stable equilibrium point, given by absolute min of U

for a given GFT energy

the solutions are cyclic motions describing oscillations around a stable minimum

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

occurrence of a bounce and an early epoch of 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

or

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:

o isotropisation of the system for increasing values of the clock (i.e., the volume) o 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), once 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

group field theory condensate cosmology mairi sakellariadou - qg... · 2017-03-28 · approach:...

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