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In the past few years, a possibility is investigated, where curvature itself behaves as a source of dark energy. So, it is natural to think whether curvature can produce dark matter too. It is found that, at classical level, higher-derivative gravity yields curvature inspired particles namely riccions.³¹ Here, it is probed whether riccion can be a possible source of dark matter. Further, it is found that the late universe accelerates. Here, it is interesting to see that acceleration is obtained from curvature without using any dark energy source of exotic matter.

This work explores an alternative solution to the problem of renormalizability in Einstein gravity. In the proposed approach, Einstein gravity is transformed into the renormalizable theory of four-derivative gravity by applying a local field redefinition containing an infinite number of higher derivatives. It is also shown that the current–current amplitude is invariant with the field redefinition, and thus the unitarity of Einstein gravity is preserved.

Within the background field formalism of quantum gravity, I show that if the quantum fluctuations are limited to diffeomorphic gauge transformations rather than the physical degrees of freedom, as in conventional quantum field theory, all the quantum corrections vanish on shell and the effective action is equivalent to the classical action. In principle, the resulting theory is finite and unitary, and requires no renormalization. I also show that this is the unique parameterization that renders the path integral independent of the on-shell condition for the background field, a form of background independence. Thus, a connection is established between background independence and renormalizability and unitarity.

Perturbatively renormalizable higher-derivative gravity in four space–time dimensions with arbitrary signs of couplings has been considered. Systematic analysis of the action with arbitrary signs of couplings in Lorentzian flat space–time for no-tachyons, fixes the signs. Feynman $+i𝜖$ prescription for these signs further grants necessary convergence in path-integral, suppressing the field modes with large action. This also leads to a sensible wick rotation where quantum computation can be performed. Running couplings for these sign of parameters make the massive tensor ghost innocuous leading to a stable and ghost-free renormalizable theory in four space–time dimensions. The theory has a transition point arising from renormalization group (RG) equations, where the coefficient of $R^2$ diverges without affecting the perturbative quantum field theory (QFT). Redefining this coefficient gives a better handle over the theory around the transition point. The flow equations push the flow of parameters across the transition point. The flow beyond the transition point is analyzed using the one-loop RG equations which shows that the regime beyond the transition point has unphysical properties: there are tachyons, the path-integral loses positive definiteness, Newton’s constant $G$ becomes negative and large, and perturbative parameters become large. These shortcomings indicate a lack of completeness beyond the transition point and need of a nonperturbative treatment of the theory beyond the transition point.

In the present paper, a cosmological scenario is proposed in which dark energy emerges spontaneously from modified gravity. In this scenario, the universe inflates for ~10^-37 s in the beginning and the late universe accelerates after 8.58 Gyrs. During the long intermediate period, the universe decelerates driven by radiation and subsequently by matter. The gravitational dark energy that emerges in this model mimics quintessence and its density falls by 115 orders from 2.18 × 10⁶⁸ GeV⁴ initially to its current value 2.19 × 10^-47 GeV⁴.

I discuss various aspects of background independence in the context of string theory, for which so far we have no manifestly background independent formulation. After reviewing the role of background independence in classical Einstein gravity, I discuss recent results implying that there is a conflict in string theory between manifest background independence and manifest duality invariance when higher-derivative corrections are included. The resolution of this conflict requires the introduction of new gauge degrees of freedom together with an enlarged gauge symmetry. This suggests more generally that a manifestly background independent and duality invariant formulation of string theory requires significantly enhanced gauge symmetries.

We review the history of the ghost problem in quantum field theory from the Pauli–Villars regulator theory to currently popular fourth-order derivative quantum gravity theories. While these theories all appear to have unitarity-violating ghost states with negative norm, we show that in fact these ghost states only appear because the theories are being formulated in the wrong Hilbert space. In these theories, the Hamiltonians are not Hermitian but instead possess an antilinear symmetry. Consequently, one cannot use inner products that are built out of states and their Hermitian conjugates. Rather, one must use inner products built out of states and their conjugates with respect to the antilinear symmetry, and these latter inner products are positive. In this way, one can build quantum theories of gravity in four spacetime dimensions that are unitary.

Second-order-derivative plus fourth-order-derivative gravity is the ultraviolet completion of second-order-derivative quantum Einstein gravity. While it achieves renormalizability through states of negative Dirac norm, the unitarity violation that this would entail can be postponed to Planck energies. As we show in this paper, the theory has a different problem, one that occurs at all energy scales, namely that the Dirac norm of the vacuum of the theory is not finite. To establish this, we present a procedure for determining the norm of the vacuum in any quantum field theory. With the Dirac norm of the vacuum of the second-order-derivative plus fourth-order-derivative theory not being finite, the Feynman rules that are used to establish renormalizability are not valid, as is the assumption that the theory can be used as an effective theory at energies well below the Planck scale. This lack of finiteness is also manifested in the fact that the Lorentzian path integral for the theory is divergent. Because the vacuum Dirac norm is not finite, the Hamiltonian of the theory is not Hermitian. However, it turns out to be $PT$ symmetric. When one continues the theory into the complex plane and uses the $PT$ symmetry inner product, viz. the overlap of the left-eigenstate of the Hamiltonian with its right-eigenstate, one then finds that for the vacuum this norm is both finite and positive, the Feynman rules now are valid, the Lorentzian path integral now is well behaved, and the theory now can serve as a low energy effective theory, one that we show to have a real classical gravity limit. Consequently, the theory can now be offered as a fully consistent, unitary and renormalizable theory of quantum gravity.

Through use of the Pauli–Villars regulator procedure, we construct a second- plus fourth-order derivative theory of gravity that serves as an ultraviolet completion of standard second-order derivative quantum Einstein gravity that is ghost-free, unitary and power-counting renormalizable.

We consider the possibility to enlarge the class of symmetries realized in standard local field theories by introducing infinite order derivative operators in the actions, which become nonlocal. In particular, we focus on the Galilean shift symmetry and its generalization in nonlocal (infinite derivative) field theories. First, we construct a nonlocal Galilean model which may be UV finite, showing how the ultraviolet behavior of loop integrals can be ameliorated. We also discuss the pole structure of the propagator which has infinitely many complex conjugate poles, but satisfies tree level unitarity. Moreover, we will introduce the same kind of nonlocal operators in the context of linearized gravity. In such a scenario, the graviton propagator turns out to be ghost-free and the spacetime metric generated by a point-like source is non-singular.

In this short paper we study how black hole singularities can be tackled in the context of nonlocal ghost-free gravity, in which the action is characterized by the presence of non-polynomial differential operators containing infinite order covariant derivatives. The ghost-freeness condition can be preserved by requiring that such nonlocal operators are made up of exponential of entire functions, thus avoiding the emergence of extra unhealthy poles in the graviton propagator. We will mainly focus on how infinite order derivatives can regularize the singularity at the origin by making explicit computations in the linear regime. In particular, we will show that this kind of non-polynomial operators can smear out point-like distribution and that the Schwarzschild metric can not be a solution of the field equations in the ghost-free infinite derivative gravity.

In this talk we discuss some classical aspects of general polynomial higher-derivative gravity. In particular, we describe the behaviour of the weak-field solutions associated to a point-like mass at small distances and provide necessary and sufficient conditions for the metric to be regular. We also consider the metric for a collapsing thick null shell, and verify that it is regular if the aforementioned conditions are valid.

Higher-derivative gravity, i.e. the system defined by General Relativity’s Lagrangian augmented by curvature-squared terms, is a renormalizable gravity model, along with its matter couplings. This model has two free parameters, α and β, which couple the higher-order terms R² and $R_{\mu \nu }^2$, respectively. In this work we study the bending of light in the framework of higher-derivative gravity utilizing both classical and semiclassical approaches. We show that the Ricci-squared sector is associated to a repulsive interaction and, at the tree-level, yields dispersive propagation of photons yet in first order. Also, a comparison between the predicted results and experimental data allows us to set an upper bound on the coupling constant β.

Fourth-derivative gravity has two free parameters, α and β, which couple the curvature-squared terms R² and $R_{\mu \nu }^2$. Relativistic effects and short-range laboratory experiments can be used to provide upper limits to these constants. In this work we briefly review both types of experimental results in the context of higher-derivative gravity. The strictest limit follows from the second kind of test. Interestingly enough, the bound on β due to semiclassical light deflection at the solar limb is only one order of magnitude larger.

We study at classical level two ranges of weakly nonlocal gravitational theories that are super-renormalizable or finite at quantum level. We explicitly prove that for one out of the two classes of theories all Ricci-flat spacetimes are exact vacuum solutions of the classical equations of motion (EOM). For the second class of theories the EOM are exactly solved by all these FRW spacetimes, which are sourced by a traceless energy tensor. Therefore, the Big-Bang singularity persists when the nonlocal gravity is coupled to radiation. We here prove the above statements in three simple theorems.

We present a canonical Hamiltonian formulation of gravity theories whose Lagrangian is an arbitrary function of the Riemann tensor. Our approach allows a unified treatment of various subcases and an easy identification of the degrees of freedom of the theory.