Improved version, with minor changes. 58 pages, including 7 figures and one table. Accepted for publication in General Relativity and Gravitation (GERG)
We present an alternative cosmology based on conformal gravity, as originally introduced by H. Weyl and recently revisited by P. Mannheim and D. Kazanas. Unlike past similar attempts our approach is a purely kinematical application of the conformal symmetry to the Universe, through a critical reanalysis of fundamental astrophysical observations, such as the cosmological redshift and others. As a result of this novel approach we obtain a closed-form expression for the cosmic scale factor R(t) and a revised interpretation of the space-time coordinates usually employed in cosmology. New fundamental cosmological parameters are introduced and evaluated. This emerging new cosmology does not seem to possess any of the controversial features of the current standard model, such as the presence of dark matter, dark energy or of a cosmological constant, the existence of the horizon problem or of an inflationary phase. Comparing our results with current conformal cosmologies in the literature, we note that our kinematic cosmology is equivalent to conformal gravity with a cosmological constant at late (or early) cosmological times. The cosmic scale factor and the evolution of the Universe are described in terms of several dimensionless quantities, among which a new cosmological variable delta emerges as a natural cosmic time. The mathematical connections between all these quantities are described in details and a relationship is established with the original kinematic cosmology by L. Infeld and A. Schild. The mathematical foundations of our kinematical conformal cosmology will need to be checked against current astrophysical experimental data, before this new model can become a viable alternative to the standard theory.
We continue our presentation of an alternative cosmology based on conformal gravity, following our kinematical approach introduced in a recent paper. In line with the assumptions of our model, which proposes a closed-form expression for the cosmic scale factor R(t), we first revise the Hubble and deceleration parameters and also introduce modified cosmological distances, analyzing in particular the case of the luminosity distance. Our kinematical conformal cosmology is then able to explain the anomalous acceleration of the Pioneer spacecraft, as due to a local region of gravitational blueshift…
First, we review the $C^\ast$-algebraic foundations of loop quantization, in particular, the construction of quantum configuration spaces and the implementation of symmetries. Then, we apply these results to loop quantum gravity, focusing on the space of generalized connections and on measures thereon. Finally, we study the realm of homogeneous isotropic loop quantum cosmology: once viewed as the loop quantization of classical cosmology, once seen as the symmetric sector of loop quantum gravity. It will turn out that both theories differ, i.e., quantization and symmetry reduction do not commut…
We consider here kinematical quantization: a first and often overlooked step in quantization procedures. $\mathbb{R}$, $\mathbb{R}_+$ and the interval are considered, as well as direct (Cartesian) products thereof. Some simple minisuperspace models, and mode by mode consideration of slightly inhomogeneous cosmology, have indefinite signature versions of such kinematical quantizations. The examples in the current paper build in particular toward the case of vacuum $\mathbb{S}^3$ slightly inhomogeneous cosmology's mode configuration space, which is mathematically a finite time interval slab of M…
In this work we analyze kinematical conformal cosmology (KCC), an alternative cosmological model based on conformal Weyl gravity (CG), and test it against current type Ia supernova (SNIa) luminosity data and other astrophysical observations. Expanding upon previous work on the subject, we revise the analysis of SNIa data, confirming that KCC can explain the evidence for an accelerating expansion of the Universe without using dark energy or other exotic components. We obtain an independent evaluation of the Hubble constant, H_{0} = 67.53 km/s Mpc, very close to the current best estimates. The m…
We consider cosmological consequences of a conformal invariant formulation of Einstein's General Relativity where instead of the scale factor of the spatial metrics in the action functional a massless scalar (dilaton) field occurs which scales all masses including the Planck mass. Instead of the expansion of the universe we get the Hoyle-Narlikar type of mass evolution, where the temperature history of the universe is replaced by the mass history. We show that this conformal invariant cosmological model gives a satisfactory description of the new supernova Ia data for the effective magnitude -…
In these lectures my aim is to review enough of conformal differential geometry in four dimensions to give an account of Penrose's conformal cyclic geometry.
We consider two consecutive conformally flat eons in Penrose's Conformal Cyclic Cosmology and study how the perfect fluid matter content of the past eon determines the matter content of the present eon by means of Penrose's reciprocity hypothesis.
In recent years, the use of conformal transformation techniques has become widespread in the literature on gravitational theories alternative to general relativity, on cosmology, and on nonminimally coupled scalar fields. Typically, the transformation to the Einstein frame is generated by a fundamental scalar field already present in the theory. In this context, the problem of which conformal frame is the physical one has to be dealt with and, in the general case, it has been clarified only recently; the formulation of a theory in the ``new'' conformal frame leads to departures from canonical …
This article is an extended version of a talk given in Oxford in June 2021 as part of an online meeting `Ninety minutes of CCC' to mark the 90th birthday of Sir Roger Penrose. I assemble some questions that I have been asked or have asked myself about CCC.
It is stressed that the kinematical problems of the standard cosmological model can be solved by a phase of accelerated contraction of the cosmic scale factor. Such a behaviour is only a particular case of a more general ``pre-big-bang" inflationary scenario, arising naturally in a string cosmology context, and driven asymptotically by the free dilaton field without any self-interaction potential and/or vacuum contribution.
We define the cosmological parameters $H_{c,0}$, $Ω_{m,c}$ and $Ω_{Λ, c}$ within the Conformal Cosmology as obtained by the homogeneous approximation to the conformal-invariant generalization of Einstein's General Relativity theory. We present the definitions of the age of the universe and of the luminosity distance in the context of this approach. A possible explanation of the recent data from distant supernovae Ia without a cosmological constant is presented.
In this paper we find the cosmological solutions of the massive conformal gravity field equations in the presence of matter fields. In particular, we show that the solution of negative curvature is in good agreement with the late universe.
The conformal cosmological model presented by Mannheim predicts a negative value for the effective gravitational constant, G. It also involves a scalar field, S, which is treated classically. In this paper we point out that a classical treatment of S is inappropriate, because the Hamiltonian is non-Hermitean, and the theory must be developed in the way pioneered by Bender and others. When this is done, we arrive at a Hamiltonian with an energy spectrum that is bounded below, and also a G that is positive. The resulting theory closely resembles the conventional cosmology based on Einstein relat…
We consider the problem of matching two spacetimes, the previous and present aeons, in the Conformal Cyclic Cosmology model. The common boundary between them inherits two sets of constraints -- one for each solution of the Einstein field equations extended to the conformal boundaries. The previous aeon is assumed to be an asymptotically de Sitter spacetime, so the standard conformal formulation of the Einstein field equations suffice to derive the constraints on the future null infinity. For the future aeon, which is supposed to evolve from an initial singularity, they are obtained with the us…
Operator-ordering in quantum cosmology is a major as-yet unsettled ambiguity with not only formal but also physical consequences. We determine the Lagrangian origin of the conformal invariance that underlies the conformal operator-ordering choice in quantum cosmology. It is particularly naturally and simply manifest in relationalist product-type actions (such as the Jacobi action for mechanics or Baierlein-Sharp-Wheeler type actions for general relativity), for which all that is required for the kinetic and potential factors to rescale in compensation to each other. These actions themselves ma…
A new class of Bianchi - I cosmological model with magnetic field within the frame work of Lyra geometry has been presented . Exact solutions to the field equations for the model are obtained . The physical and kinematical behaviors of the model have been discussed.
Conformal Cyclic Cosmology (CCC) is a cyclic model of the universe put forward by Sir Roger Penrose. A conformal invariance assumption in the neighbourhood of the crossover region between cycles (which Penrose calls aeons) allows successive space-times to be related by a conformal rescaling. A major open problem is how to choose the conformal factor in a unique way, and is a fundamental hurdle to further study. Proposals have been put forward by Newman, Tod and Nurowski, but they disagree in one way or another with Penrose's original assumptions as well as each other. In this paper we compare …
The framework of quantum symmetry reduction is applied to loop quantum gravity with respect to transitively acting symmetry groups. This allows to test loop quantum gravity in a large class of minisuperspaces and to investigate its features - e.g. the discrete volume spectrum - in certain cosmological regimes. Contrary to previous studies of quantum cosmology (minisuperspace quantizations) the symmetry reduction is carried out not at the classical level but on an auxiliary Hilbert space of the quantum theory before solving the constraints. Therefore, kinematical properties like volume quantiza…
We present a new perspective on early cosmology based on Loop Quantum Gravity. We use projected spinnetworks, coherent states and spinfoam techniques, to implement a quantum reduction of the full Kinematical Hilbert space of LQG, suitable to describe inhomogeneous cosmological models. Some preliminary results on the solutions of the Scalar constraint of the reduced theory are also presented.
By imposing natural geometrical and kinematical conditions on a conformal Killing vector in Bianchi I spacetime, we show that a class of axisymmetric metrics admits a conformal motion. This class contains new exact solutions of Einstein's equations, including anisotropic radiation universes that isotropise at late times.
We propose a holographic dual for (pseudo-)conformal cosmological scenario, with a scalar field that forms a moving domain wall in adS_5. The domain wall separates two vacua with unequal energy densities. Unlike in the existing construction, the 5d solution is regular in the relevant space-time domain.
In a previous paper (P. D. Mannheim, Phys. Rev. D 102, 123535 (2020)) we studied cosmological perturbation theory in the cosmology associated with the fourth-order derivative conformal gravity theory, and provided an exact solution to the theory in the recombination era. In this paper we present an exact solution that holds all the way from recombination until the current era.
Following recent fit of supernovae data to Brans-Dicke theory which favours the model with $ω= - 3/2$ \cite{fabris} we discuss the status of this special case of Brans-Dicke cosmology in both isotropic and anisotropic framework. It emerges that the limit $ω= -3/2$ is consistent only with the vacuum field equations and it makes such a Brans-Dicke theory conformally invariant. Then it is an example of the conformal relativity theory which allows the invariance with respect to conformal transformations of the metric. Besides, Brans-Dicke theory with $ω= -3/2$ gives a border between a standard sca…
It was recently shown that the homogeneous and isotropic cosmology of a massless scalar field coupled to general relativity exhibits a new hidden conformal invariance under Mobius transformation of the proper time, additionally to the invariance under time-reparamterization. The resulting Noether charges form a $sl(2,\mathbb{R})$ Lie algebra, which encapsulates the whole kinematics and dynamics of the geometry. This allows to map FLRW cosmology onto conformal mechanics and formulate quantum cosmology in $\text{CFT}_1$ terms. Here, we show that this conformal structure is embedded in a larger $…
It is shown that any homogeneous and isotropic universe, independently of its spatial topology and matter content, allows for the presence of a conformal stealth, i.e. a nontrivial conformally invariant scalar field with vanishing energy-momentum tensor, which evolves along with the universe without causing even the smallest backreaction. Surprisingly, this gravitationally invisible universal witness is inhomogeneous with zero consequences for the underlying cosmology. Additionally, it is shown that these results are not exclusive of a four-dimensional universe by generalizing them to higher d…
We discuss conformal issues of pure and extended teleparallel gravity. In particular, we present formulations of conformal transformation in teleparallel gravity. Furthermore, we propose conformal scalar and gauge field theories in teleparallel gravity and study conformal torsion gravity. We explicitly demonstrate that a power-law acceleration (including the $Λ$CDM universe) as well as the de Sitter expansion of the universe can be realized in extended teleparallel gravity with a conformal scalar field, and that pure teleparallel gravity with a conformal scalar field can lead to the $Λ$CDM mod…
Approximate de Sitter symmetry of inflating Universe is responsible for the approximate flatness of the power spectrum of scalar perturbations. However, this is not the only option. Another symmetry which can explain nearly scale-invariant power spectrum is conformal invariance. We give a short review of models based on conformal symmetry which lead to the scale-invariant spectrum of the scalar perturbations. We discuss also potentially observable features of these models.
From higher dimensional theories, e.g. string theory, one expects the presence of non-minimally coupled scalar fields. We review the notion of conformal frames in cosmology and emphasize their physical equivalence, which holds at least at a classical level. Furthermore, if there is a field, or fields, which dominates the universe, as it is often the case in cosmology, we can use such notion of frames to treat our system, matter and gravity, as two different sectors. On one hand, the gravity sector which describes the dynamics of the geometry and on the other hand the matter sector which has su…
Conformally invariant GUT-like model including gravity based on Riemann - Cartan space-time $U_4$ is considered. Cosmological scenario that follows from the model is discussed and standard quantum gravitational formalism in the Arnowitt-Deser-Misner form is developed. General formalism is then illustrated on Bianchi-IX minisuperspace cosmological model. Wave functions of the universe in the de Sitter minisuperspace model with Vilenkin and Hartle-Hawking boundary conditions are considered and corresponding probability distributions for the scalar field values are calculated.
In this paper we demonstrate that there are large classes of Friedmann-Robertson-Walker (FRW) cosmologies that admit isotropic conformal structures of Quiescent Cosmology. FRW models have long been known to admit singularities such as Big Bangs and Big Crunches [1, 2] but recently it has been shown that there are other cosmological structures that these solutions contain. These structures are Big Rips, Sudden Singularities and Extremality Events [1, 2]. Within the Quiescent Cosmology framework [3] there also exist structures consistent with a cosmological singularity known as the Isotropic Pas…
We study coherent states for Bianchi type I cosmological models, as examples of semiclassical states for time-reparametrization invariant systems. This simple model allows us to study explicitly the relationship between exact semiclassical states in the kinematical Hilbert space and corresponding ones in the physical Hilbert space, which we construct here using the group averaging technique. We find that it is possible to construct good semiclassical physical states by such a procedure in this model; we also discuss the sense in which the original kinematical states may be a good approximation…
According to Belinsky, Khalatnikov and Lifshitz, gravity near a space-like singularity reduces to a set of decoupled one-dimensional mechanical models at each point in space. We point out that these models fall into a class of conformal mechanical models first introduced by de Alfaro, Fubini and Furlan (DFF). The deformation used by DFF to render the spectrum discrete corresponds to a negative cosmological constant. The wave function of the universe is the zero-energy eigenmode of the Hamiltonian, also known as the spherical vector of the representation of the conformal group SO(1,2). A new cl…
We introduce a new framework for loop quantum gravity: mimicking the spinfoam quantization procedure we propose to study the symmetric sectors of the theory imposing the reduction weakly on the full kinematical Hilbert space of the canonical theory. As a first application of Quantum-Reduced Loop Gravity we study the inhomogeneous Bianchi I model. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could elucidate the relationship between Loop Quantum Cosmology and the full th…
As quotient spaces, Minkowski and de Sitter are fundamental, non-gravitational spacetimes for the construction of physical theories. When general relativity is constructed on a de Sitter spacetime, the usual Riemannian structure is replaced by a more general structure called de Sitter-Cartan geometry. In the contraction limit of an infinite cosmological term, the de Sitter-Cartan spacetime reduces to a singular, flat, conformal invariant four-dimensional cone spacetime, in which our ordinary notions of time interval and space distance are absent. It is shown that such spacetime satisfies all p…
Definitions of non-relativistic conformal transformations are considered both in the Newton-Cartan and in the Kaluza-Klein-type Eisenhart/Bargmann geometrical frameworks. The symmetry groups that come into play are exemplified by the cosmological, and also the Newton-Hooke solutions of Newton's gravitational field equations. It is shown, in particular, that the maximal symmetry group of the standard cosmological model is isomorphic to the 13-dimensional conformal-Newton-Cartan group whose conformal-Bargmann extension is explicitly worked out. Attention is drawn to the appearance of independent…
We study cosmological perturbation theory in the cosmology associated with conformal gravity, establish the validity of the decomposition theorem for it, and then use the theorem to provide an exact solution to the theory in the recombination era. Central to our approach is the use of a fully gauge invariant formulation of the cosmological fluctuations equations. In the recombination era not only is perturbation theory applicable, because of its specific structure in the conformal case the fluctuation equations are found to greatly simplify. Using a master equation for scalar, vector and tenso…
We study teleparallel gravitational theories with are invariant under the conformal transformations. Wide family of the gravitational Lagrangians that are invariant under conformal transformations have investigated. Cosmological solutions inspired by the observational data for a flat Universe in the vacuum has been found. To add matter fields to the cosmological models by preserving the conformal symmetry there are significant limitations. Friedmann-Robertson-Walker (FRW) equations based on the effective terms of energy density and pressure have been rewriting and continuity equation for the e…
M-theory compactified on S^7/Z_k allows for a four-dimensional, asymptotically AdS cosmology. The holographic dual consists of ABJM theory with a non-supersymmetric marginal deformation. At weak 't Hooft coupling the dual theory possesses a UV fixed point, allowing the construction of a conformal field theory dual. The dual theory has an infinite family of instantons, which we are able to sum completely. The resulting theory spontaneously breaks global conformal symmetry but is manifestly unitary and non-singular. Our findings point to a holographic resolution of the singularity, measure and c…
We review the fundamental results of a new cosmological model, based on conformal gravity, and apply them to the analysis of the early data of the Pioneer anomaly. We show that our conformal cosmology can naturally explain the anomalous acceleration of the Pioneer 10 and 11 spacecraft, in terms of a local blueshift region extending around the solar system and therefore affecting the frequencies of the navigational radio signals exchanged between Earth and the spacecraft. On the contrary, conformal gravity corrections alone would not be able to account for dynamical effects of such magnitude to…
The configuration space of general relativity is superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace - the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Taking this conformal nature seriously leads to a new theory of gravity which although very similar to general relativity has some very different features particularly in cosmology and quantisation. It should reproduce the standard tests of general relativity. The cosmology is stud…
One possibility to explain the current accelerated expansion of the universe may be related with the presence of cosmologically evolving scalar whose mass depends on the local matter density (chameleon cosmology). We point out that matter quantum effects in such scalar-tensor theory produce the chameleon scalar field dependent conformal anomaly. Such conformal anomaly adds higher derivative terms to chameleon field equation of motion. As a result, the principal possibility for instabilities appears. These instabilities seem to be irrelevant at small curvature but may become dangerous in the re…
In this work, the cosmological implications of brane world scenario are investigated when the gravitational coupling $G$ and the cosmological term $Λ$ are not constant but rather there are time variation of them. From observational point of view, these time variations are taken in the form $\frac{\dot{G}}{G}\sim H$ and $Λ\sim H^{2}$. The behavior of scale factors and different kinematical parameters are investigated for different possible scenarios where the bulk cosmological constant $Λ_{5}$ can be zero, positive or negative.
We analyze the functional integral for quantum Conformal Gravity and show that with the help of a Hubbard-Stratonovich transformation, the action can be broken into a local quadratic-curvature theory coupled to a scalar field. A one-loop effective action calculation reveals that strong fluctuations of the metric field are capable of spontaneously generating a dimensionally transmuted parameter which in the weak-field sector of the broken phase induces a Starobinsky-type f(R)-model with a gravi-cosmological constant. A resulting non-trivial relation between Starobinsky'sparameter and the cosmol…
Gravitation and the standard model of particle physics are incorporated within a single conformal scalar-tensor theory, where the scalar field is complex. The Higgs field has a dynamical expectation value, as has the Planck mass, but the relative strengths of the fundamental interactions are unchanged. Initial cosmic singularity and the horizon problem are avoided, and spatial flatness is natural. There were no primordial phase transitions; consequently, no topological defects were produced. Quantum excitations of the dilaton phase induced a slightly red-tilted spectrum of gaussian and adiabat…
An intrinsic time of homogeneous models is global. The Friedmann equation by its sense ties time intervals. Exact solutions of the Friedmann equation in Standard cosmology and Conformal cosmology are presented. Theoretical curves interpolated the Hubble diagram on latest supernovae are expressed in analytical form. The class of functions in which the concordance model is described is Weierstrass meromorphic functions. The Standard cosmological model and Conformal one fit the modern Hubble diagram equivalently. However, the physical interpretation of the modern data from concepts of the Conform…
In the Big Rip cosmological scenario, a FRW universe containing dark energy with $w < -1$ in its equation of state (phantom energy) expands in such way that, in a finite time span, the scale factor diverges to infinity and the size of the cosmic horizon goes to zero. Here we revisit this scenario in light of the fact that Hawking radiation is expected to be generated at the apparent horizon of a FRW universe, and show that the energy density and temperature of that radiation both diverge at the Big Rip. We then use this fact to propose a new variant of Penrose's conformal cyclic cosmology mode…
In this paper, using $f(R)$ theory of gravity we explicitly calculate cosmological evolution in the presence of a perfect fluid source in four and five dimensional, space time in which this cosmological evolution in self-creation is presented by Reddy et al 2009 Int. J. Theor. Phys. 48 10. An exact cosmological model is presented using a relation between Einstein gravity field equation components due to a metric with the same component from $f(R)$ theory of gravity. Some physics and kinematical properties of the model are also discussed.
We present a new cosmological model derived from Loop Quantum Gravity. The formulation is based on a projection of the kinematical Hilbert space of the full theory down to a subspace representing the proper arena for an inhomogeneous Bianchi I model. This procedure gives a direct link between the full theory and its cosmological sector. The emerging quantum cosmological model represents a simplified arena on which the complete canonical quantization program can be tested. The achievements of this analysis could also shed light on Loop Quantum Cosmology and its relation with the full theory.
We study late-time cosmic accelerating dynamics from Born-Infeld-$f(R)$ gravity in a simplified conformal approach. We find that a variety of cosmic efects such as Little Rip, $Λ$CDM universe and dark energy cosmology with finite-time future singularities may occur. Unlike the convenient Born-Infeld gravity where in the absence of matter only de Sitter expansion may emerge, apparentlly any FRW cosmology maybe reconstructed from this conformal version of the Born-Infeld-$f(R)$ theory. Despite the fact that the explicit form of $f(R)$ is fixed by the conformal ansatz, the relation between the tw…
We introduce the hypothesis that the matter content of the universe can be a product of the decay of primordial vector bosons. The effect of the intensive cosmological creation of these primordial vector $W, ~Z $ bosons from the vacuum is studied in the framework of General Relativity and the Standard Model where the relative standard of measurement identifying conformal quantities with the measurable ones is accepted. The relative standard leads to the conformal cosmology with the z-history of masses with the constant temperature, instead of the conventional z-history of the temperature with …
In this letter, we stress that the simplest cosmological model consisting in a massless scalar field minimally coupled to homogeneous and isotropic gravity has an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetry naturally provides an efficient way to constrain the quantization of this cosmological system whatever the quantization scheme and allows in particular to fix the quantization ambiguities arising in the canonical quantization program. Applying this method to the loop quantization of the FLRW cosmology leads to a new loop quantum cosmology model which preserves the $\SL(2…
Conformal higher spin (CHS) field theory, which is a solid part of recent advanced checks of AdS/CFT correspondence, finds applications in cosmology. Hidden sector of weakly interacting CHS fields suggests a resolution of the hierarchy problem in the model of initial conditions for inflationary cosmology driven by a conformal field theory. These initial conditions are set by thermal garland type cosmological instantons in the sub-planckian energy range for the model of CHS fields with a large positive coefficient $β$ of the Gauss-Bonnet term in their total conformal anomaly and a large number …
