We analyse some open debates in cosmology in light of the most updated quasar (QSO) sample, covering a wide redshift range up to $z\sim7.5$, combined with type Ia supernovae (SNe) and baryon acoustic oscillations (BAO). Indeed, extending the cosmological analyses with high-redshift data is key to distinguishing between different cosmological models that are degenerate at low redshifts, and allowing better constraints on a possible dark energy (DE) evolution. Also, we discuss combinations of BAO, SNe, and QSO data to understand their compatibility and implications for extensions of the standard cosmological model. Specifically, we consider a flat and non-flat $Λ\mathrm{CDM}$ cosmology, a flat and non-flat DE model with a constant DE equation of state parameter ($w$), and four flat DE models with variable $w$, namely the Chevallier-Polarski-Linder and Jassal-Bagla-Padmanabhan models, and an "exponential" and Barboza-Alcaniz parameterisations. We find that a joint analysis of QSO+SNe with BAO is only possible in the context of a flat Universe. Indeed BAO confirms the flatness condition assuming a curved geometry, whilst SNe+QSO show evidence of a closed space. We also find $Ω_{M,0}=0.3$ in all data sets assuming a flat $Λ\mathrm{CDM}$ model. Yet, all the other models show a statistically significant deviation at 2-3$σ$ with the combined SNe+QSO+BAO data set. In the models where DE density evolves with time, SNe+QSO+BAO data always prefer $Ω_{M,0}>0.3$, $w_{0}<-1$ and $w_{a}>0$. This DE phantom behaviour is mainly driven by SNe+QSO, while BAO are closer to the flat $Λ\mathrm{CDM}$ model.
Using a new recently compiled milliarcsecond compact radio data set of 120 intermediate-luminosity quasars in the redshift range $0.46< z <2.76$, whose statistical linear sizes show negligible dependence on redshifts and intrinsic luminosity and thus represent standard rulers in cosmology, we constrain three viable and most popular $f(T)$ gravity models, where $T$ is the torsion scalar in teleparallel gravity. Our analysis reveals that constraining power of the quasars data (N=120) is comparable to the Union2.1 SN Ia data (N=580) for all three $f(T)$ models. Together with other standard ruler …
We present a novel, purely geometric probe of cosmology based on measurements of differential time delays between images of strongly lensed quasars due to finite source effects. Our approach is solely dependent on cosmology via a ratio of angular diameter distances, the image separation, and the source size. It thereby entirely avoids the challenges of lens modelling that conventionally limit time delay cosmography, and instead entails the lensed reverberation mapping of the quasar Broad Line Region. We demonstrate that differential time delays are measurable with short cadence spectroscopic m…
Our group has developed a technique that makes use of the observed non-linear relation between the ultraviolet and the X-ray luminosity in quasars to provide an independent measurement of their distances, thus turning quasars into standardizable candles. This technique, at present, it is mostly based upon quasar samples with data from public catalogues both in the X-rays and in the optical/ultraviolet and extends the Hubble diagram of supernovae to a redshift range still poorly explored (z>2). From the X-ray perspective, we are now on the eve of a major change, as the upcoming mission eROSITA …
We use 118 H$β$ quasar (QSO) observations in the redshift range $0.0023 \leq z \leq 0.89$ to simultaneously constrain cosmological model parameters and QSO 2-parameter radius-luminosity ($R-L$) relation parameters in six different cosmological models. We find that the $R-L$ relation parameters for these QSOs are independent of the assumed cosmology so these QSOs seem to be standardizable through the $R-L$ relation (although there is a complication that might render this untrue). Cosmological constraints obtained using these QSOs are weak, more favor currently decelerated cosmological expansion…
We consider extensions of General Relativity based on the non-local function $f(R, \Box^{-1} R)$, where $R$ is the Ricci curvature scalar and the non-locality is due to the term $\Box^{-1} R$. We focus on cosmological minisuperspaces and select viable models by the Noether Symmetry Approach. Then we find viable exact solutions pointing out the role of non-locality in cosmology.
The non-linear relation between X-ray and UV luminosity in quasars can be used to estimate their distance. Recently, we have shown that despite the large dispersion of the relation, a Hubble Diagram made of large samples of quasars can provide unique constraints on cosmology at high redshift. Furthermore, the dispersion of the relation is heavily affected by measurement errors: until now we have used serendipitous X-ray observations, but dedicated observations would significantly increase the precision of the distance estimates. We discuss the future role of XMM in this new field, showing (1) …
I show that the de Sitter Equilibrium cosmology generically predicts observable levels of curvature in the Universe today. The predicted value of the curvature depends only on the ratio of the density of non-relativistic matter to energy density in cosmological constant, and the value of the curvature from the initial bubble that starts the inflation. The result is independent of the scale of inflation, the shape of the potential during inflation, and many other details of the cosmology. Future cosmological measurements of the matter density, cosmological constant and curvature will open up a …
We measure the cross-correlation of Atacama Cosmology Telescope CMB lensing convergence maps with quasar maps made from the Sloan Digital Sky Survey DR8 SDSS-XDQSO photometric catalog. The CMB lensing-quasar cross-power spectrum is detected for the first time at a significance of 3.8 sigma, which directly confirms that the quasar distribution traces the mass distribution at high redshifts z>1. Our detection passes a number of null tests and systematic checks. Using this cross-power spectrum, we measure the amplitude of the linear quasar bias assuming a template for its redshift dependence, and…
Although the Hubble constant $H_0$ and spatial curvature $Ω_{K}$ have been measured with very high precision, they still suffer from some tensions. In this paper, we propose an improved method to combine the observations of ultra-compact structure in radio quasars and strong gravitational lensing with quasars acting as background sources to determine $H_0$ and $Ω_{K}$ simultaneously. By applying the distance sum rule to the time-delay measurements of 7 strong lensing systems and 120 intermediate-luminosity quasars calibrated as standard rulers, we obtain stringent constraints on the Hubble con…
We analytically and numerically show that through the cycles with nonsingular bounce the amplitude of curvature perturbation on large scale will be amplified and the power spectrum will be redden. In some sense, this amplification will eventually destroy the homogeneity of background, which will lead to the ultimate end of cycles of global universe. We argue that for the model with increasing cycles, it might be possible that a fissiparous multiverse will emerge after one or several cycles, in which the cycles will continue only at corresponding local regions.
The main sequence offers a method for the systematization of quasar spectral properties. Extreme FeII emitters (or extreme Population A, xA) are believed to be sources accreting matter at very high rates. They are easily identifiable along the quasar main sequence, in large spectroscopic surveys over a broad redshift range. The very high accretion rate makes it possible that massive black holes hosted in xA quasars radiate at a stable, extreme luminosity-to-mass ratio. After reviewing the basic interpretation of the main sequence, we report on the possibility of identifying virial broadening e…
The absolute distance scale measurements in cosmology have always been an important mission. In particular, in recent years, the Hubble constant tension between the measurements from early and late universe has become a new crisis for cosmology, which calls for new, independent absolute cosmological distance measurements. Recently, a result of measuring the parallax distance to 3C 273 through spectroastrometry and reverberation mapping was reported. We comment on this novel method in this News & Views paper.
Several recent studies have considered the implications for astrophysics and cosmology of some possible nonclassical properties of spacetime at the Planck scale. The new effects, such as a Planck-scale-modified energy-momentum (dispersion) relation, are often inferred from the analysis of some quantum versions of Minkowski spacetime, and therefore the relevant estimates depend heavily on the assumption that there could not be significant interplay between Planck-scale and curvature effects. We here scrutinize this assumption, using as guidance a quantum version of de Sitter spacetime with know…
Quasars accreting matter at very high rates (known as extreme Population A [xA] or super-Eddington accreting massive black holes) provide a new class of distance indicators covering cosmic epochs from the present-day Universe up to less than 1 Gyr from the Big Bang. The very high accretion rate makes it possible that massive black holes hosted in xA quasars radiate at a stable, extreme luminosity-to-mass ratio. This in turns translates into stable physical and dynamical conditions of the mildly ionized gas in the quasar low-ionization line emitting region. In this contribution, we analyze the …
Last year (2013) was more or less the 50th anniversary of the discovery of quasars. It is an interesting time to review what we know (and don't know) about them both empirically and theoretically. These compact sources involving line emitting plasma show extraordinary luminosities extending to one thousand times that of our Milky Way in emitting volumes of a few solar system diameters (bolometric luminosity log L$_{bol} \sim $ 44-48 [erg s$^{-1}$]: D=1-3 light months $\sim$ $10^3$ - $10^4$ gravitational radii). The advent of 8-10 meter class telescopes enables us to study them spectroscopicall…
Recently Risaliti \& Lusso [Nature Astron. 3 (2019) 3 272] reported new measurements of the expansion rate of the Universe by constructing the Hubble diagram of 1598 quasars in the redshift range $0.5<z<5.5$. It is claimed a $4σ$ tension with the standard concordance $Λ$CDM concerning both the fractionary matter density $Ω_{m0}$ and the dark energy equation of state parameter $w_{de}$ standard values. In this work we promote an independent analysis of the same data set using a model-independent estimator for cosmic acceleration. Our results corroborate that the source of such tension can be re…
Testing the distance-sum-rule in strong lensing systems provides an interesting method to determine the curvature parameter $Ω_k$ using more local objects. In this paper, we apply this method to a quite recent data set of strong lensing systems in combination with intermediate-luminosity quasars calibrated as standard rulers. In the framework of three types of lens models extensively used in strong lensing studies (SIS model, power-law spherical model, and extended power-law lens model), we show that the assumed lens model has a considerable impact on the cosmic curvature constraint, which is …
We study cosmological dynamics and late-time evolution of an extended induced gravity braneworld scenario. In this scenario, curvature effects are taken into account via the Gauss-Bonnet term in the bulk action and there is also a Chaplygin gas component on the brane. We show that this model mimics an effective phantom behavior in a relatively wider range of redshifts than previously formulated models. It also provides a natural framework for smooth crossing of the phantom-divide line due to presence of the Chaplygin gas component on the brane. We confront the model with observational data fro…
In the non-relativistic theory of gravity recently proposed by Horava, the Hamiltonian constraint is not satisfied locally at each point in space. The absence of the local Hamiltonian constraint allows the system to have an extra dark-matter-like component as an integration constant. We discuss consequences of this fact in the context of cosmological perturbations, paying a particular attention to the large scale evolution of the curvature perturbation. The curvature perturbation is defined in a gauge invariant manner with this ``dark matter'' taken into account. We then clarify the conditions…
Quasars accreting matter at very high rates (known as extreme Population A [xA] quasars, possibly associated with super-Eddington accreting massive black holes) may provide a new class of distance indicators covering cosmic epochs from present day up to less than 1 Gyr from the Big Bang. At a more fundamental level, xA quasars are of special interest in studies of the physics of AGNs and host galaxy evolution. However, their observational properties are largely unknown. xA quasars can be identified in relatively large numbers from major optical surveys over a broad range of redshifts, thanks t…
We consider the cosmology derived from $f(T,B)$ gravity where $T$ is the torsion scalar and $B=\frac{2}{e}\partial_μ(e T^μ)$ a boundary term. In particular we discuss how it is possible to recover, under the same standard, the teleparallel $f(T)$ gravity, the curvature $f(R)$ gravity and the teleparallel-curvature $f(R,T)$ gravity, which are particular cases of $f(T,B)$. We adopt the Noether Symmetry Approach to study the related dynamical systems and to find out cosmological solutions.
It is necessary to make assumptions in order to derive models to be used for cosmological predictions and comparison with observational data. In particular, in standard cosmology the spatial curvature is assumed to be constant and zero (or at least very small). But there is, as yet, no fully independent constraint with an appropriate accuracy that gaurentees a value for the magnitude of the effective normalized spatial curvature $Ω_{k}$ of less than approximately $0.01$. Moreover, a small non-zero measurement of $Ω_{k}$ at such a level perhaps indicates that the assumptions in the standard mod…
It is well known that the standard scalar field mimetic cosmology provides a dark matter-like energy density component. Considering $SU(2)$ gauge symmetry, we study the gauge field extension of the mimetic scenario in spatially flat and curved FLRW spacetimes. Because of the mimetic constraint, the standard Yang-Mills term plays the role of the cosmological constant while the mimetic term provides two different contributions: one is the standard radiation scaling like $ a^{-4}$ while the other contribution in energy density scales as $\propto a^{-2}$. Consequently, in the Friedmann equation we…
We present a new model-independent method to determine the spatial curvature and to mitigate the circularity problem affecting the use of quasars as distance indicators. The cosmic-chronometer measurements are used to construct the curvature-dependent luminosity distance $D^{\rm cal}_{L}(Ω_{K},z)$ using a polynomial fit. Based on the reconstructed $D^{\rm cal}_{L}(Ω_{K},z)$ and the known ultraviolet versus X-ray luminosity correlation of quasars, we simultaneously place limits on the curvature parameter $Ω_{K}$ and the parameters characterizing the luminosity correlation function. This model-i…
Recently a new no-global-recollapse argument was given for some inhomogeneous and anisotropic cosmologies that utilizes surface deformation by the mean curvature flow. In this paper we discuss important properties of the mean curvature flow of spacelike surfaces in Lorentzian manifolds. We show that singularities may form during cosmic evolution and the theorems forbidding the global recollapse lose their validity. The time evolution of the spatial scalar curvature that may kinematically prevent the recollapse is determined in normal coordinates, which shows the impact of inhomogeneities expli…
As is known, in modified cosmological theories of gravity many of the cosmologies which could not be generated by standard Einstein gravity, can be consistently described by $F(R)$ theories. Using known reconstruction techniques, we investigate which $F(R)$ theories can lead to a Hubble parameter describing two types of cosmological bounces, the superbounce model, related to supergravity and non-supersymmetric models of contracting ekpyrosis and also the Loop Quantum Cosmology modified ekpyrotic model. Since our method is an approximate method, we investigate the problem at large and small cur…
We consider cosmic chronometer (CC) data for the Hubble parameter, quasar (QSO) luminosities data of X-rays and ultraviolet rays emission, and the latest measurements of the present value of the Hubble parameter from 2018 Planck mission (PL18), and SH0ES observations (SHOES) to constrain the present value of cosmic curvature density parameter. We consider three kinds of dark energy models: the $Λ$CDM model, the wCDM model, and the CPL parametrization. In all these three models, we find higher values of the matter-energy density parameter, $Ω_{\rm m0}$ compared to the one obtained from the Plan…
We present an holographic model of the Dvali-Gabadadze-Porrati scenario with a Gauss-Bonnet term in the bulk. We concentrate on the solution that generalises the normal Dvali-Gabadadze-Porrati branch. It is well known that this branch cannot describe the late-time acceleration of the universe even with the inclusion of a Gauss-Bonnet term. Here, we show that this branch in the presence of a Gauss-Bonnet curvature effect and an holographic dark energy with the Hubble scale as the infra-red cutoff can describe the late-time acceleration of the universe. It is worthwhile to stress that such an en…
In this work we study non-singular{ bounce cosmology} in the context of the Lagrange multiplier generalized $F(R)$ gravity theory of gravity. We specify our study by using a specific variant form of the well known matter{ bounce cosmology}, with scale factor $a(t) = \left(a_0t^2 + 1 \right)^n$, and we demonstrate that for $n < 1/2$, the primordial curvature perturbations are generated deeply in the contraction era. Particularly, we show explicitly that the perturbation modes exit the horizon at a large negative time during the contraction era, which in turn makes the ``low-curvature'' regime, …
$f(R)$ modifications of Einstein's gravity is an interesting possibility to explain the late time acceleration of the Universe. In this work we explore the cosmological viability of one such $f(R)$ modification proposed in (Kruglov:2013). We show that the model violates fifth-force constraints. The model is also plagued with the issue of curvature singularity in a spherically collapsing object, where the effective scalar field reaches to the point of diverging scalar curvature.
We compare the cosmology of conformal gravity (CG), (Mannheim 2006), to $Λ$CDM. CG cosmology has repulsive matter and radiation on cosmological scales, while retaining attractive gravity at local scales. Mannheim (2003) finds that CG agrees with $Λ$CDM for supernova data at redshifts $z<1$. We use GRBs and quasars as standard candles to contrast these models in the redshift range $0<z<8$. We find CG deviates significantly from $Λ$CDM at high redshift and that $Λ$CDM is favoured by the data with $Δχ^2=48$. Mannheim's model has a bounded dark energy contribution, but we identify a $λ$ fine-tunin…
The loop quantum dynamics of Kantowski-Sachs spacetime and the interior of higher genus black hole spacetimes with a cosmological constant has some peculiar features not shared by various other spacetimes in loop quantum cosmology. As in the other cases, though the quantum geometric effects resolve the physical singularity and result in a non-singular bounce, after the bounce a spacetime with small spacetime curvature does not emerge in either the subsequent backward or the forward evolution. Rather, in the asymptotic limit the spacetime manifold is a product of two constant curvature spaces. …
In this paper, by using the recently compiled set of 120 intermediate-luminosity quasars (ILQSO) observed in a single-frequency VLBI survey, we propose an improved model-independent method to probe cosmic curvature parameter $Ω_k$ and make the first measurement of the cosmic curvature referring to a distant past, with redshifts up to $z\sim 3.0$. Compared with other methods, the proposed one involving the quasar data achieves constraints with higher precision in this redshift range. More importantly, our results indicate that the measured $Ω_k$ is in good agreement with zero cosmic curvature, …
Theoretical arguments and cosmological observations suggest that Einstein's theory of general relativity needs to be modified at high energies. One of the best motivated higher-curvature extensions of general relativity is Einstein-scalar-Gauss-Bonnet gravity, in which a scalar field is coupled to quadratic curvature invariants. This theory is inspired by an effective string-theory model and its predictions dramatically differ from Einstein's theory in high-curvature regions - such as the interior of black holes and the early universe - where it aims at resolving curvature singularities. In th…
We study the evolution of cosmological perturbations in an anti-de-Sitter (AdS) bulk through a cosmological singularity by mapping the dynamics onto the boundary conformal fields theory by means of the AdS/CFT correspondence. We consider a deformed AdS space-time obtained by considering a time-dependent dilaton which induces a curvature singularity in the bulk at a time which we call $t = 0$, and which asymptotically approaches AdS both for large positive and negative times. The boundary field theory becomes free when the bulk curvature goes to infinity. Hence, the evolution of the fluctuation…
On the Reliability of Quasars as Cosmological Distance Indicators
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional (Cinvestav)（墨西哥国立理工学院高级研究中心）；Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学研究所）；Facultad de Ciencias, Universidad Nacional Autónoma de México（墨西哥国立自治大学理学院）；Colegio de Ciencias y Humanidades Plantel Sur, Universidad Nacional Autónoma de México（墨西哥国立自治大学人文科学学院南校区）
We investigate the occurrence of various exotic spacelike singularities in the past and the future evolution of $k = \pm 1$ Friedmann-Robertson-Walker model and loop quantum cosmology using a sufficiently general phenomenological model for the equation of state. We highlight the non-trivial role played by the intrinsic curvature for these singularities and the new physics which emerges at the Planck scale. We show that quantum gravity effects generically resolve all strong curvature singularities including big rip and big freeze singularities. The weak singularities, which include sudden and b…
Mimetic gravity can be described as a formulation capable of mimicking different evolutionary scenarios regarding the universe dynamics. Notwithstanding its initial aim of producing a similar evolution to the one expected from the dark components of the standard cosmology, a recent association with loop quantum cosmology could also provide interesting results. In this work, we reinterpret the physics behind the curvature potential of mimetic gravity description of loop quantum cosmology. Furthermore, we also test the compatibility of our formulation for a Higgs-type field, proving that the mim…
Universe history in $R^2$-gravity is studied from "beginning" up to the present epoch. It is assumed that initially the curvature scalar $R$ was sufficiently large to induce the proper duration of inflation. Gravitational particle production by the oscillating $R(t)$ led to a graceful exit from inflation, but the cosmological evolution in the early universe was drastically different from the standard one till the universe age reached the value of the order of the inverse decay rate of the oscillating curvature $R(t)$. This deviation from the standard cosmology might have a noticeable impact on…
In this paper, we use a newly compiled sample of ultra-compact structure in radio quasars and strong gravitational lensing systems with quasars acting as background sources to constrain six spatially flat and non-flat cosmological models ($Λ$CDM, PEDE and DGP). These two sets of quasar data (the time-delay measurements of six strong lensing systems and 120 intermediate-luminosity quasars calibrated as standard rulers) could break the degeneracy between cosmological parameters ($H_0$, $Ω_m$ and $Ω_k$) and therefore provide more stringent cosmological constraints for the six cosmological models …
Recent technological advances have led to a flood of new data on cosmology rich in information about the formation and evolution of the universe, e.g., the data collected in Sloan Digital Sky Survey (SDSS) for more than 200 million objects. The analyses of such data demand cutting edge statistical technologies. Here, we have used the concept of mixture model within Bayesian semiparametric methodology to fit the regression curve with the bivariate data for the apparent magnitude and redshift for Quasars in SDSS (2007) catalogue. Associated with the mixture modeling is a highly efficient curve-f…
Hořava-Lifshitz gravity is a potentially UV complete theory with important implications for the very early universe. In particular, in the presence of spatial curvature it is possible to obtain a non-singular bouncing cosmology. The bounce is realized as a consequence of higher order spatial curvature terms in the gravitational action. Here, we extend the study of linear cosmological perturbations in Hořava-Lifshitz gravity coupled to matter in the case when spatial curvature is present. As in the case without spatial curvature, we find that there is no extra dynamical degree of freedom for sc…
In addition to shear and vorticity a homogeneous background may also exhibit anisotropic curvature. Here a class of spacetimes is shown to exist where the anisotropy is solely of the latter type, and the shear-free condition is supported by a canonical, massless 2-form field. Such spacetimes possess a preferred direction in the sky and at the same time a CMB which is isotropic at the background level. A distortion of the luminosity distances is derived and used to test the model against the CMB and supernovae (using the Union catalog), and it is concluded that the latter exhibit a higher-than-…
We generalize the time-variable dark energy scalar field $Φ$ model ($Φ$CDM) to nonflat space. We show that even in the space-curvature-dominated epoch the scalar field solution is a time-dependent fixed point or attractor, with scalar field energy density that grows relative to the energy density in spatial curvature. This is the first example of a physically consistent and complete model of dynamical dark energy in a nonflat geometry.
Quasars have been proposed as a new class of standard candles analogous to Supernovae, as their large redshift range and high luminosities make them excellent candidates. Reverberation mapping (RM) method enables to estimate the distance to the source from the time delay measurement of the emission lines with respect to the continuum, since the time delay depends on the absolute luminosity of the source. This radius-luminosity (RL) relation exhibits a low scatter and it was thus proposed for cosmological purposes. However, in recent years the increase in the studied sample, and in particular t…
With the distance sum rule in the Friedmann-Lema\^ıtre-Robertson-Walker metric, model-independent constraints on both the Hubble constant $H_0$ and spatial curvature $Ω_{K}$ can be obtained using strong lensing time-delay data and Type Ia supernova (SN Ia) luminosity distances. This method is limited by the relative low redshifts of SNe Ia, however. Here, we propose using quasars as distance indicators, extending the coverage to encompass the redshift range of strong lensing systems. We provide a novel and improved method of determining $H_0$ and $Ω_{K}$ simultaneously. By applying this techni…
We test the FLRW cosmology by reconstructing in a model-independent way both the Hubble parameter $H(z)$ and the comoving distance $D(z)$ via the most recent Hubble and Supernovae Ia data. In particular we use: data binning with direct error propagation, the principal component analysis, the genetic algorithms and the Padé approximation. Using our reconstructions we evaluate the Clarkson {\it et al} test known as $Ω_K(z)$, whose value is constant in redshift for the standard cosmological model, but deviates elsewise. We find good agreement with the expected values of the standard cosmological …
The cosmic curvature density parameter has been constrained in the present work independent of any background cosmological model. The reconstruction is performed adopting the non-parametric Gaussian Processes (GP). The constraints on $Ω_{k0}$ are obtained via a Markov Chain Monte Carlo (MCMC) analysis. Late-time cosmological probes viz., the Supernova (SN) distance modulus data, the Cosmic Chronometer (CC) and the radial Baryon Acoustic Oscillations ($r$BAO) measurements of the Hubble data have been utilized for this purpose. The results are further combined with the data from redshift space d…
We formulate and solve the problem of Newtonian cosmology under the assumption that the absolute space of Newton is non-Euclidean. In particular, we focus on the negatively-curved hyperbolic space, H3. We point out the inequivalence between the curvature term that arises in the Friedmann equation in Newtonian cosmology in Euclidean space and the role of curvature in the H3 space. We find the generalisation of the inverse-square law and the solutions of the Newtonian cosmology that follow from it. We find the generalisations of the Euclidean Michell 'black hole' in H3 and show that it leads to …
