group field theory condensate cosmology mairi sakellariadou - qg... · 2017-03-28 · approach:...
TRANSCRIPT
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
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
string gas scenario
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 group field theory condensate cosmology
bottom-up approach
o 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)
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)
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
on 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
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
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
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)
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
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>
of bosonic GFT quanta which have
relaxed into a common ground state
orthogonal 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 (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
“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
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
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
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
assume interactions between GFT quanta as sub-dominant
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)
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 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)
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 (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
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 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