15 research outputs found
The asymptotic safety scenario and scalar field inflation
We study quantum gravity corrections to early universe cosmology as resulting
within the asymptotic safety scenario. We analyse if it is possible to obtain
accelerated expansion in the regime of the renormalisation group fixed point in
a theory with Einstein-Hilbert gravity and a scalar field. We show how this
phase impacts cosmological perturbations observed in the cosmic microwave
background.Comment: Contribution to the proceedings of the Thirteenth Marcel Grossmann
Meeting, Stockholm, 201
Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation
We review and extend in several directions recent results on the asymptotic safety approach to quantum gravity. The central issue in this approach is the search of a Fixed Point having suitable properties, and the tool that is used is a type of Wilsonian renormalization group equation. This will be reviewed in chapter 2 after a general overview in the introductory chapter 1. Then we discuss various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff procedure. We compare the beta functions of the gravitational couplings obtained with different schemes, studying first the contribution of matter fields and then the so\u2013called Einstein\u2013Hilbert truncation in chapter 3, where only the cosmological constant and Newton\u2019s constant are retained. In this context we make connection with old results, in particular we reproduce the results of the epsilon expansion and the perturbative one loop divergences. We discuss some possible phenomenological consequences leading to modified dispersion relations and show connections to phenomenological models where Lorentz invariance is either broken or deformed. We then apply the Renormalization Group to higher derivative gravity in chapter 4. In the case of a general action quadratic in curvature we recover, within certain approximations, the known asymptotic freedom of the four\u2013derivative terms, while Newton\u2019s constant and the cosmological constant have a nontrivial fixed point. In the case of actions that are
polynomials in the scalar curvature of degree up to eight we find that the theory has a fixed point with three UV\u2013attractive directions, so that the requirement of having a continuum limit constrains the couplings to lie in a three\u2013dimensional subspace, whose equation is explicitly given. We emphasize throughout the difference between scheme\u2013dependent and scheme\u2013independent results, and provide several examples of the fact that only dimensionless couplings can have \u201cuniversal\u201d behavior
Asymptotically Safe Cosmology
We study quantum modifications to cosmology in a Friedmann-Robertson-Walker
universe with and without scalar fields by taking the renormalisation group
running of gravitational and matter couplings into account. We exploit the
Bianchi identity to relate the renormalisation group scale with scale factor
and derive the improved cosmological evolution equations. We find two types of
cosmological fixed points where the renormalisation group scale either freezes
in, or continues to evolve with scale factor. We discuss the implications of
each of these, and classify the different cosmological fixed points with and
without gravity displaying an asymptotically safe renormalisation group fixed
point. We state conditions of existence for an inflating ultraviolet
cosmological fixed point for Einstein gravity coupled to a scalar field. We
also discuss other fixed point solutions such as "scaling" solutions, or fixed
points with equipartition between kinetic and potential energies.Comment: 8 pages; v2: explanations and references added, accepted for
publication in JCA
Investigating the Ultraviolet Properties of Gravity with a Wilsonian Renormalization Group Equation
We review and extend in several directions recent results on the asymptotic
safety approach to quantum gravity. The central issue in this approach is the
search of a Fixed Point having suitable properties, and the tool that is used
is a type of Wilsonian renormalization group equation. We begin by discussing
various cutoff schemes, i.e. ways of implementing the Wilsonian cutoff
procedure. We compare the beta functions of the gravitational couplings
obtained with different schemes, studying first the contribution of matter
fields and then the so-called Einstein-Hilbert truncation, where only the
cosmological constant and Newton's constant are retained. In this context we
make connection with old results, in particular we reproduce the results of the
epsilon expansion and the perturbative one loop divergences. We then apply the
Renormalization Group to higher derivative gravity. In the case of a general
action quadratic in curvature we recover, within certain approximations, the
known asymptotic freedom of the four-derivative terms, while Newton's constant
and the cosmological constant have a nontrivial fixed point. In the case of
actions that are polynomials in the scalar curvature of degree up to eight we
find that the theory has a fixed point with three UV-attractive directions, so
that the requirement of having a continuum limit constrains the couplings to
lie in a three-dimensional subspace, whose equation is explicitly given. We
emphasize throughout the difference between scheme-dependent and
scheme-independent results, and provide several examples of the fact that only
dimensionless couplings can have "universal" behavior.Comment: 86 pages, 13 figures, equation (71) corrected, references added, some
other minor changes. v.5: further minor corrections to eqs. (20), (76), (91),
(94), (A9), Tables II, III, Appendix
Ultraviolet properties of f(R)-Gravity
We discuss the existence and properties of a nontrivial fixed point in
f(R)-gravity, where f is a polynomial of order up to six. Within this
seven-parameter class of theories, the fixed point has three
ultraviolet-attractive and four ultraviolet-repulsive directions; this brings
further support to the hypothesis that gravity is nonperturbatively
renormalizabile.Comment: 4 page
Emergent Complex Network Geometry
Networks are mathematical structures that are universally used to describe a
large variety of complex systems such as the brain or the Internet.
Characterizing the geometrical properties of these networks has become
increasingly relevant for routing problems, inference and data mining. In real
growing networks, topological, structural and geometrical properties emerge
spontaneously from their dynamical rules. Nevertheless we still miss a model in
which networks develop an emergent complex geometry. Here we show that a single
two parameter network model, the growing geometrical network, can generate
complex network geometries with non-trivial distribution of curvatures,
combining exponential growth and small-world properties with finite spectral
dimensionality. In one limit, the non-equilibrium dynamical rules of these
networks can generate scale-free networks with clustering and communities, in
another limit planar random geometries with non-trivial modularity. Finally we
find that these properties of the geometrical growing networks are present in a
large set of real networks describing biological, social and technological
systems.Comment: (24 pages, 7 figures, 1 table
Complex Quantum Network Manifolds in Dimension d > 2 are Scale-Free
In quantum gravity, several approaches have been proposed until now for the quantum description of discrete geometries. These theoretical frameworks include loop quantum gravity, causal dynamical triangulations, causal sets, quantum graphity, and energetic spin networks. Most of these approaches describe discrete spaces as homogeneous network manifolds. Here we define Complex Quantum Network Manifolds (CQNM) describing the evolution of quantum network states, and constructed from growing simplicial complexes of dimension d. We show that in d = 2 CQNM are homogeneous networks while for d > 2 they are scale-free i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the self-organized evolution of CQNM quantum statistics emerge spontaneously. Here we define the generalized degrees associated with the δ-faces of the d-dimensional CQNMs, and we show that the statistics of these generalized degrees can either follow Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimension of the δ-faces
Renormalisation group improvement of scalar field inflation
We study quantum corrections to Friedmann-Robertson-Walker cosmology with a
scalar field under the assumption that the dynamics are subject to
renormalisation group improvement. We use the Bianchi identity to relate the
renormalisation group scale to the scale factor and obtain the improved
cosmological evolution equations. We study the solutions of these equations in
the renormalisation group fixed point regime, obtaining the time-dependence of
the scalar field strength and the Hubble parameter in specific models with
monomial and trinomial quartic scalar field potentials. We find that power-law
inflation can be achieved in the renormalisation group fixed point regime with
the trinomial potential, but not with the monomial one. We study the transition
to the quasi-classical regime, where the quantum corrections to the couplings
become small, and find classical dynamics as an attractor solution for late
times. We show that the solution found in the renormalisation group fixed point
regime is also a cosmological fixed point in the autonomous phase space. We
derive the power spectrum of cosmological perturbations and find that the
scalar power spectrum is exactly scale-invariant and bounded up to arbitrarily
small times, while the tensor perturbations are tilted as appropriate for the
background power-law inflation. We specify conditions for the renormalisation
group fixed point values of the couplings under which the amplitudes of the
cosmological perturbations remain small.Comment: 17 pages; 2 figure