Weyl-squared Term Research Articles

Renormalization-group techniques for single-field inflation in primordial cosmology and quantum gravity

We study inflation as a ‘cosmic’ renormalization-group (RG) flow. The flow, which encodes the dependence on the background metric, is described by a running coupling α, which parametrizes the slow roll, a de Sitter free, analytic beta function and perturbation spectra that are RG invariant in the superhorizon limit. Using RG invariance as a guiding principle, we classify the main types of flows according to the properties of their spectra, without referring to their origins from specific actions or models. Novel features include spectra with essential singularities in α and violations of the relation r + 8n t = 0 to the leading order. Various classes of potentials studied in the literature can be described by means of the RG approach, even when the action includes a Weyl-squared term, while others are left out. In known cases, the classification helps identify the models that are ruled out by data. The RG approach is also able to generate spectra that cannot be derived from standard Lagrangian formulations.

Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term

We derive the predictions of quantum gravity with fakeons on the amplitudes and spectral indices of the scalar and tensor fluctuations in inflationary cosmology. The action is R +R2 plus the Weyl-squared term. The ghost is eliminated by turning it into a fakeon, that is to say a purely virtual particle. We work to the next-to-leading order of the expansion around the de Sitter background. The consistency of the approach puts a lower bound (mχ > mϕ /4) on the mass mχ of the fakeon with respect to the mass mϕ of the inflaton. The tensor-to-scalar ratio r is predicted within less than an order of magnitude (4/2 < N2r <12 to the leading order in the number of e-foldings N). Moreover, the relation r ≃ –8nT is not affected by the Weyl-squared term. No vector and no other scalar/tensor degree of freedom is present.

Quasi-conformal models and the earlyuniverse

Extensions of the Standard Model and general relativity featuring a UVfixed point can leave observable implications at accessible energies. Although massparameters such as the Planck scale can appear through dimensional transmutation,all fundamental dimension-4 operators can (at least approximately) respect Weylinvariance at finite energy. An example is the Weyl-squared term, whose consistencyand observational consequences are studied. This quasi-conformal scenario emergesfrom the UV complete quadratic gravity and is a possible framework for inflation. Wefind two realizations. In the first one the inflaton is a fundamental scalar with aquasi-conformal non-minimal coupling to the Ricci scalar. In this case the fieldexcursion must not exceed the Planck mass by far. An example discussed in detail ishilltop inflation. In the second realization the inflaton is a pseudo-Goldstoneboson (natural inflation). In this case we show how to obtain an elegant UVcompletion within an asymptotically free QCD-like theory, in which the inflaton is acomposite scalar due to new strong dynamics. We also show how efficient reheatingcan occur. Unlike the natural inflation based on Einstein gravity, thetensor-to-scalar ratio is well below the current bound set by Planck. In bothrealizations mentioned above, the basic inflationary formulæ are computedanalytically and, therefore, these possibilities can be used as simple benchmarkmodels.

Noether–Wald energy in Critical Gravity

Criticality represents a specific point in the parameter space of a higher-derivative gravity theory, where the linearized field equations become degenerate. In 4D Critical Gravity, the Lagrangian contains a Weyl-squared term, which does not modify the asymptotic form of the curvature. The Weyl2 coupling is chosen such that it eliminates the massive scalar mode and it renders the massive spin-2 mode massless. In doing so, the theory turns consistent around the critical point.Here, we employ the Noether–Wald method to derive the conserved quantities for the action of Critical Gravity. It is manifest from this energy definition that, at the critical point, the mass is identically zero for Einstein spacetimes, what is a defining property of the theory. As the entropy is obtained from the Noether–Wald charges at the horizon, it is evident that it also vanishes for any Einstein black hole.

Noether-Wald Charges in Critical Gravity

Critical Gravity theory is defined by a particular combination of quadratic couplings in the curvature added on top of 4D Einstein-Hilbert action with negative cosmological constant. As the Lagrangian is given by a Weyl-squared term, the asymptotic form of the curvature is not modified. The coupling of the W eyl2 term is such the massive scalar mode is eliminated and the massive spin-2 mode become massless, rendering the theory consistent around the critical point.In the present work, we construct the Noether-Wald charges for the action of Critical Gravity. Such construction makes manifest a defining property of this theory: both the energy and entropy for Einstein black holes vanish identically.

Inflationary perturbations in no-scale theories

We study the inflationary perturbations in general (classically) scale-invariant theories. Such scenario is motivated by the hierarchy problem and provides natural inflationary potentials and dark matter candidates. We analyse in detail all sectors (the scalar, vector and tensor perturbations) giving general formulae for the potentially observable power spectra, as well as for the curvature spectral index n_mathrm{s} and the tensor-to-scalar ratio r. We show that the conserved Hamiltonian for all perturbations does not feature negative energies even in the presence of the Weyl-squared term if the appropriate quantisation is performed and argue that this term does not lead to phenomenological problems at least in some relevant setups. The general formulae are then applied to a concrete no-scale model, which includes the Higgs and a scalar, “the planckion”, whose vacuum expectation value generates the Planck mass. Inflation can be triggered by a combination of the planckion and the Starobinsky scalar and we show that no tension with observations is present even in the case of pure planckion inflation, if the coefficient of the Weyl-squared term is large enough. In general, even quadratic inflation is allowed in this case. Moreover, the Weyl-squared term leads to an isocurvature mode, which currently satisfies the observational bounds, but it may be detectable with future experiments.

Tensor instabilities at the end of theΛCDMuniverse

The unphysical spin-2 massive degrees of freedom in higher derivative gravity may be either massive unphysical ghosts or tachyonic ghosts. In the last case there is no Planck-scale threshold protecting vacuum cosmological solutions from instabilities. Within the anomaly-induced action formalism the photon-driven IR running of the coefficient of the Weyl-squared term makes the ghost eventually becoming tachyon, that should produce a gravitational explosion of vacuum. This effect is stable under higher loop corrections and takes place also in known versions of perturbative quantum gravity. However, the contribution of massless fields in the far IR are not the same in flat and de~Sitter spaces. In the asymptotically deSitter case one can observe a kind of IR decoupling, which protects the cosmological solution from the future tachyonic instabilities.

Entanglement temperature with Gauss–Bonnet term

We compute the entanglement temperature using the first law-like of thermodynamics, ΔE=TentΔSEE, up to Gauss–Bonnet term in the Jacobson–Myers entropy functional in any arbitrary spacetime dimension. The computation is done when the entangling region is the geometry of a slab. We also show that such a Gauss–Bonnet term, which becomes a total derivative, when the co-dimension two hypersurface is four dimensional, does not contribute to the finite term in the entanglement entropy. We observe that the Weyl-squared term does not contribute to the entanglement entropy. It is important to note that the calculations are performed when the entangling region is very small and the energy is calculated using the normal Hamiltonian.

Primordial massive gravitational waves from Einstein-Chern-Simons-Weyl gravity

We investigate the evolution of cosmological perturbations during de Sitter inflation in the Einstein-Chern-Simons-Weyl gravity. Primordial massive gravitational waves are composed of one scalar, two vector and four tensor circularly polarized modes. We show that the vector power spectrum decays quickly like a transversely massive vector in the superhorizon limit z → 0. In this limit, the power spectrum coming from massive tensor modes decays quickly, leading to the conventional tensor power spectrum. Also, we find that in the limit of m2 → 0 (keeping the Weyl-squared term only), the vector and tensor power spectra disappear. It implies that their power spectra are not gravitationally produced because they (vector and tensor) are decoupled from the expanding de Sitter background, as a result of conformal invariance.

CAUSALITY CONSTRAINT ON NONCRITICAL EINSTEIN–WEYL GRAVITY

We explore, in the context of AdS/CFT correspondence, the causality constraints on the Noncritical Einstein–Weyl (NEW) gravity model in five dimensions. The scalar and shear channels are considered as small metric perturbations around an AdS black brane background. Our results show that the causality analysis on the propagation of these two channels imposes a new bound on the coupling of the Weyl-squared term in the NEW gravity. This new bound imposes more stringent restrictions than those by the tachyon-free condition, improving the predictive power of the theory.

Tensor ghosts in the inflationary cosmology

Theories with curvature-squared terms in the action are known to contain ghost modes in general. However, if we regard curvature-squared terms as quantum corrections to the original theory, the emergence of ghosts may be simply due to the perturbative truncation of a full non-perturbative theory. If this is the case, there should be a way to live with ghosts. In this paper, we take the Euclidean path integral approach, in which ghost degrees of freedom can be, and are integrated out in the Euclideanized spacetime. We apply this procedure to Einstein gravity with a Weyl curvature-squared correction in the inflationary background. We find that the amplitude of tensor perturbations is modified by a term of O(α2H2) where α2 is a coupling constant in front of the Weyl-squared term and H is the Hubble parameter during inflation.