Cosmological Constant Research Articles

General perturbations in D+1 standard and brane cosmology revisited

In this work, we generalize the theory of perturbations in a ([Formula: see text])-dimensional space–time with cosmological constant, studying scalar, vector and tensor perturbations, as well as its structure in Newtonian and Synchronous gauge. We also show the theory of perturbations in the context of brane cosmology, where branes are embedded in a set of D-spatial dimensions, a temporal dimension, and an additional spatial dimension. In both standard and brane cosmology, an unperturbed space–time is provided with a Friedmann–Lemaitre–Robertson–Walker metric and arbitrary sectional curvature, the matter content has the shape of a perfect fluid. In addition, we consider the arbitrary sectional curvature, obtaining the respective equations in the Newtonian and Synchronous gauge. We highlight that the results presented in this paper can be used to treat brane cosmology with two concentric branes or tackle the [Formula: see text] tension with a braneworld approach. Finally, as an example of the utility of all the technology presented in this paper, we show an application to the energy flux and the repercussions in the framework of the braneworlds.

Thermodynamics of Taub-NUT-AdS spacetimes

We apply the generalised Komar method proposed in [arXiv:2208.05494] to Taub-NUT-AdS spacetime in the theory of Einstein gravity plus a cosmological constant. Based on a generalised closed 2-form, we derive the total mass and NUT charge of the Taub-NUT-AdS spacetime. Together with other thermodynamic quantities calculated through standard method, we conform the first law and Smarr relation. Then, we consider charged AdS NUT spacetimes in Einstein–Maxwell theory with a cosmological constant, and show that the generalised Komar method works, too. We obtain all the thermodynamic quantities, and the first law and Smarr relation are checked to be satisfied automatically.

Scalar fields in Bonnor-Melvin-Lambda universe with potential: a study of dynamics of spin-zero particles-antiparticles

In this study, our primary focus is on exploring the relativistic quantum dynamics of spin-zero scalar particles in a magnetic space-time background. Our investigation revolves around solving the Klein–Gordon (KG) equation within the framework of an electrovacuum space-time, while incorporating an external scalar potential. Specifically, we consider a cylindrical symmetric Bonnor-Melvin magnetic universe featuring a cosmological constant, where the magnetic field aligns parallel to the symmetry axis. Our approach involves deriving the radial equation of the wave equation, initially considering a linear confining potential and subsequently incorporating a Cornell-type scalar potential. We successfully obtain an approximate analytical solution for the eigenvalues of the quantum system under examination. Worth noting is our observation that the energy spectrum and the corresponding radial wave function experience notable modifications due to the presence of various factors including the cosmological constant, the topological parameter characterizing the space-time geometry, and the potential parameters.

Electric shocks: bounding Einstein-Maxwell theory with time delays on boosted RN backgrounds

The requirement that particles propagate causally on non-trivial backgrounds implies interesting constraints on higher-derivative operators. This work is part of a systematic study of the positivity bounds derivable from time delays on shockwave backgrounds. First, we discuss shockwaves in field theory, which are infinitely boosted Coulomb-like field configurations. We show how a positive time delay implies positivity of four-derivative operators in scalar field theory and electromagnetism, consistent with the results derived using dispersion relations, and we comment on how additional higher-derivative operators could be included.We then turn to gravitational shockwave backgrounds. We compute the infinite boost limit of Reissner-Nordström black holes to derive charged shockwave backgrounds. We consider photons traveling on these backgrounds and interacting through four-derivative corrections to Einstein-Maxwell theory. The inclusion of gravity introduces a logarithmic term into the time delay that interferes with the straightforward bounds derivable in pure field theory, a fact consistent with CEMZ and with recent results from dispersion relations. We discuss two ways to extract a physically meaningful quantity from the logarithmic time delay — by introducing an IR cutoff, or by considering the derivative of the time delay — and comment on the bounds implied in each case. Finally, we review a number of additional shockwave backgrounds which might be of use in future applications, including spinning shockwaves, those in higher dimensions or with a cosmological constant, and shockwaves from boosted extended objects.

Convexity restoration from hairy black hole in Einstein-Maxwell-charged scalar system in AdS

In the Einstein-Maxwell-charged scalar system with a negative cosmological constant in arbitrary dimensions higher than three, there exists a horizonless charged soliton solution, which we construct explicitly for an arbitrary mass of the scalar in perturbative series in small charge. We find that the stability of the soliton is determined by the validity of the AdS weak gravity conjecture. The existence of a stable soliton might endanger the convexity of the (free) energy as a function of the charge because the phase transition between the soliton and the extremal Reissner-Nordstrom black hole would be discontinuous. We, however, argue that the existence of the hairy black hole solution circumvents the violation of convexity. The thermodynamic properties of the hairy black hole show that the phase transition becomes continuous irrespective of whether the AdS weak gravity conjecture holds. When it holds, the phase transition occurs between the soliton and the hairy black hole, and when it is violated, the phase transition occurs between the extremal Reissner-Nordstrom black hole and the hairy black hole.

Heisenberg soft hair on Robinson-Trautman spacetimes

We study 4 dimensional (4d) gravitational waves (GWs) with compact wavefronts, generalizing Robinson-Trautman (RT) solutions in Einstein gravity with an arbitrary cosmological constant. We construct the most general solution of the GWs in the presence of a causal, timelike, or null boundary when the usual tensor modes are turned off. Our solution space besides the shape and topology of the wavefront which is a generic compact, smooth, and orientable 2d surface Σ, is specified by a vector over Σ satisfying the conformal Killing equation and two scalars that are arbitrary functions over the causal boundary, the boundary modes (soft hair). We work out the symplectic form over the solution space using covariant phase space formalism and analyze the boundary symmetries and charges. The algebra of surface charges is a Heisenberg algebra. Only the overall size of the compact wavefront and not the details of its shape appears in the boundary symplectic form and is canonical conjugate to the overall mass of the GW. Hence, the information about the shape of the wavefront can’t be probed by the boundary observer. We construct a boundary energy-momentum tensor and a boundary current, whose conservation yields the RT equation for both asymptotically AdS and flat spacetimes. The latter provides a hydrodynamic description for our RT solutions.

Nonassociative cosmological solitonic R-flux deformations in gauge gravity and G. Perelman geometric flow thermodynamics

We elaborate on a model of nonassociative and noncommutative gauge gravity for the de Sitter gauge group SO(4,1) embedding extensions of the affine structure group Af(4,1) and the Poincaré group ISO(3,1). Such nonassociative gauge gravity theories are determined by star product R-flux deformations in string theory and can be considered as new avenues to quantum gravity and geometric and quantum information theories. We analyse physically important and geometric thermodynamic properties of new classes of generic off-diagonal cosmological solitonic solutions encoding nonassociative effective sources. Particularly, we focus on modelling by such solutions of locally anisotropic and inhomogeneous dark matter and dark energy structures generated as nonassociative solitonic hierarchies. Such accelerating cosmological evolution scenarios cannot be described in the framework of the Bekenstein–Hawking thermodynamic formalism. This motivates a change in the gravitational thermodynamic paradigm by considering nonassociative and relativistic generalizations of the concept of W-entropy in the theory of Ricci flows. Finally, we compute the corresponding modified G. Perelman’s thermodynamic variables and analyse the temperature-like evolution of cosmological constants determined by nonassociative cosmological flows.

Signature of f(R) gravity via Lemaître–Tolman–Bondi inhomogeneous perturbations

We analyze inhomogeneous cosmological models in the local Universe, described by the Lemaître–Tolman–Bondi (LTB) metric and developed using linear perturbation theory on a homogeneous and isotropic Universe background. Focusing on the different evolution of spherical symmetric inhomogeneities, we compare the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, in which the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} is included in the LTB formalism, with inhomogeneous cosmological models based on fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} modified gravity theories viewed in the Jordan frame. We solve the system of field equations for both inhomogeneous cosmological models adopting the method of separation of variables: we integrate analytically the radial profiles of local perturbations, while their time evolution requires a numerical approach. The main result of the analysis concerns the different radial profiles of local inhomogeneities due to the presence of a non-minimally coupled scalar field in the Jordan frame of fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} gravity. While radial perturbations follow a power-law in the Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}LTB model, Yukawa-like contributions appear in the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} theory. Interestingly, this latter peculiar behavior of radial profile is not affected by the choice of the fR\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f\\left( R\\right) $$\\end{document} functional form. The numerical solution of time-dependent perturbations exhibits a non-diverging profile. This work suggests that investigations about local inhomogeneities in the late Universe may allow us to discriminate if the present cosmic acceleration is caused by a cosmological constant term or a modified gravity effect.

Holographic batteries

We study a three-dimensional holographic conformal field theory under the influence of a background electric field on a spacetime containing two black hole horizons. The electric background is fixed such that there is potential difference between the two boundary black holes, inducing a conserved current. By constructing the holographic duals to this setup, which are solutions to the Einstein-Maxwell equations with a negative cosmological constant in four dimensions, we calculate, to a fully nonlinear level, the conductivity of the conformal field theory in this background. Interestingly, we find that the conductivity depends nontrivially on the potential difference. The bulk solutions are flowing geometries containing black hole horizons which are non-Killing and have nonzero expansion. We find a novel property that the past boundary of the future horizon lies deep in the bulk and show this property remains present after small perturbations of the temperature difference of the boundary black holes. Published by the American Physical Society 2024

Einstein gravity with generalized cosmological term from five-dimensional AdS-Maxwell-Chern-Simons gravity

Some time ago, the standard geometric framework of Einstein gravity was extended by gauging the Maxwell algebra as well as the so called AdS-Maxwell algebra. In this paper it is shown that the actions for these four-dimensional extended Einstein gravities can be obtained from the five-dimensional Chern-Simons gravities actions by using the Randall-Sundrum compactification procedure. It is found that the Inönü-Wigner contraction procedure, in the Weimar-Woods sense, can be used both to obtain the Maxwell-Chern-Simons action from the AdS-Maxwell-Chern-Simons action and to obtain the Maxwell extension of Einstein gravity in 4D from the four-dimensional extended AdS-Maxwell-Einstein-Hilbert action. It is also shown that the extended four-dimensional gravities belongs to the Horndeski family of scalar-tensor theories.

Probing the Bardeen–Kiselev black hole with the cosmological constant caused by Einstein equations coupled with nonlinear electrodynamics using quasinormal modes and greybody bounds

In this work, we investigate a static and spherically symmetric Bardeen–Kiselev black hole (BH) with the cosmological constant, which is a solution of the Einstein-non-linear Maxwell field equations. We compute the quasinormal frequencies for the Bardeen–Kiselev BH with the cosmological constant due to electromagnetic and gravitational perturbations. By varying the BH parameters, we discuss the behavior of both real and imaginary parts of the BH quasinormal frequencies and compare these frequencies with the Reissner–Nordström–de Sitter BH surrounded by quintessence (RN-dSQ). Interestingly, it is shown that the responses of the Bardeen–Kiselev BH with the cosmological constant and the RN-dSQ under electromagnetic perturbations are different when the charge parameter q, the state parameter w and the normalization factor c are varied; however, for the gravitational perturbations, the responses of the Bardeen–Kiselev BH with the cosmological constant and the RN-dSQ are different only when the charge parameter q is varied. Therefore, compared with the gravitational perturbations, the electromagnetic perturbations can be used to understand nonlinear and linear electromagnetic fields in curved spacetime separately. Another interesting observation is that, due to the presence of Kiselev quintessence, the electromagnetic perturbations around the Bardeen–Kiselev BH with the cosmological constant damps faster and oscillates slowly; for the gravitational perturbations, the quasinormal mode decays slowly and oscillates slowly. We also study the reflection and transmission coefficients along with the absorption cross section in the Bardeen–Kiselev BH with the cosmological constant; it is shown that the transmission coefficients will increase due to the presence of Kiselev quintessence.

Cosmological solutions in the Brans–Dicke theory via invariants of symmetry groups

We proceed to obtain an exact analytical solution of the Brans–Dicke (BD) equations for the spatially flat ([Formula: see text]) Friedmann–Lamaître–Robertson–Walker (FLRW) cosmological model in both cases of the absence and presence of the cosmological constant. The solution method that we use to solve the field equations of the BD equations is called the “invariants of symmetry groups method” (ISG method). This method is based on the extended Prelle–Singer (PS) method and it employs the Lie point symmetries, [Formula: see text]-symmetries, and Darboux polynomials (DPs). Indeed, the ISG method tries to provide two independent first-order invariants associated to the one-parameter Lie groups of transformations keeping the ordinary differential equations (ODEs) invariant, as solutions. It should be noted for integrable ODEs that the ISG method guarantees the extraction of these two invariants. In this work, for the BD equations in FLRW cosmological model, we find the Lie point symmetries, [Formula: see text]-symmetries, and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating factors). By making use of the extended PS method we find two independent first-order invariants in such a way that appropriate cosmological solutions from solving these invariants as a system of algebraic equations are simultaneously obtained. These solutions are wealthy in that they include many known special solutions, such as the O’Hanlon–Tupper vacuum solutions, Nariai’s solutions, Brans–Dicke dust solutions, inflationary solutions, etc.

Cosmological transition epoch from gamma-ray burst correlations

The redshift zt and the jerk parameter jt of the transition epoch are constrained by using two model-independent approaches involving the direct expansion of the Hubble rate and the expansion of the deceleration parameter around z=zt. To extend our analysis to high-redshifts, we employ the Amati, Combo, Yonetoku and Dainotti gamma-ray burst correlations. The circularity problem is prevented by calibrating these correlations through the Bézier interpolation of the updated observational Hubble data. Each gamma-ray burst data set is jointly fit with type Ia supernovae and baryonic acoustic oscillations through a Monte Carlo analysis, based on the Metropolis-Hastings algorithm, to obtain zt, jt and the correlation parameters. The overall results are compatible with the concordance model with some exceptions. We also focus on the behaviors of the dark energy, verifying its compatibility with a cosmological constant, and the matter density Ωm and compare them with the expectations of the concordance paradigm.

Prediction of the non-SUSY AdS conjecture on the lightest neutrino mass revisited

We study the constraint of the nonsupersymmetric (non-SUSY) anti–de Sitter (AdS) conjecture on the three-dimensional vacua obtained from the compactification of the Standard Model coupled to Einstein gravity on a circle where the three-dimensional components of the four-dimensional metric are general functions of both noncompact and compact coordinates. We find from studying the wave function profile of the three-dimensional metric in the compactified dimension that the radius of the compactified dimension must be quantized. Consequently, the three-dimensional vacua are constrained by not only the non-SUSY AdS conjecture but also the quantization rule of the circle radius, leading to both upper and lower bounds for the mass of the lightest neutrino as 2≤mν/Λ4<3, where Λ4≃5.06×10−84 GeV2 is the observed cosmological constant. This means that the lightest neutrino should have a mass around 10−32 eV or it would be approximately massless. With this prediction, we reconstruct the light neutrino mass matrix that is fixed by the neutrino oscillation data and in terms of three new mixing angles and six new phases for both the normal ordering and inverted ordering. In the situation where the light neutrino mass matrix is Hermitian, we calculate its numerical value in the 3σ range. Published by the American Physical Society 2024

The density of virialized clusters as a probe of dark energy

ABSTRACT We use the spherical collapse model to demonstrate that the observable average density of virialized clusters depends on the properties of dark energy along with the properties of gravity on cluster scales and can therefore be used as a probe of these properties. As an application of this approach, we derive the predicted virialized densities and radii of cluster mass structures for a wide range of values of the cosmological constant (including negative values) as a function of the turnaround redshift. For the value of ΩΛ,0 = −0.7 (with Ωm,0 = 0.3), we find an amplification of the density of virialized clusters which can be as large as 80 per cent compared to Planck18/lambda Cold Dark Matter (ΛCDM) for a turnaround redshift zmax ≳ 2. Such an amplification may lead to more efficient early galaxy formation in this class of models, in accordance with the recent findings of JWST, which may be partially pertinent to the Λ sign switching models (ΛsCDM), which have been suggested as potential solutions to the observed Hubble and S8 discrepancies.

Algebraic classification of 2+1 geometries: a new approach

We present a convenient method of algebraic classification of 2+1 spacetimes into the types I, II, D, III, N and O, without using any field equations. It is based on the 2+1 analogue of the Newman–Penrose curvature scalars ΨA of distinct boost weights, which are specific projections of the Cotton tensor onto a suitable null triad. The algebraic types are then simply determined by the gradual vanishing of such Cotton scalars, starting with those of the highest boost weight. This classification is directly related to the specific multiplicity of the Cotton-aligned null directions and to the corresponding Bel–Debever criteria. Using a bivector (that is 2-form) decomposition, we demonstrate that our method is fully equivalent to the usual Petrov-type classification of 2+1 spacetimes based on the eigenvalue problem and determining the respective canonical Jordan form of the Cotton–York tensor. We also derive a simple synoptic algorithm of algebraic classification based on the key polynomial curvature invariants. To show the practical usefulness of our approach, we perform the classification of several explicit examples, namely the general class of Robinson–Trautman spacetimes with an aligned electromagnetic field and a cosmological constant, and other metrics of various algebraic types.

Fill-ins with scalar curvature lower bounds and applications to positive mass theorems

Given a constant C and a smooth closed (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-dimensional Riemannian manifold (Σ,g)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma , g)$$\\end{document} equipped with a positive function H, a natural question to ask is whether this manifold can be realised as the boundary of a smooth n-dimensional Riemannian manifold with scalar curvature bounded below by C and boundary mean curvature H. That is, does there exist a fill-in of (Σ,g,H)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Sigma ,g,H)$$\\end{document} with scalar curvature bounded below by C? We use variations of an argument due to Miao and the author (Int Math Res Not 7:2019, 2019) to explicitly construct fill-ins with different scalar curvature lower bounds, where we permit the fill-in to contain another boundary component provided it is a minimal surface. Our main focus is to illustrate the applications of such fill-ins to geometric inequalities in the context of general relativity. By filling in a manifold beyond a boundary, one is able to obtain lower bounds on the mass in terms of the boundary geometry through positive mass theorems and Penrose inequalities. We consider fill-ins with both positive and negative scalar curvature lower bounds, which from the perspective of general relativity corresponds to the sign of the cosmological constant, as well as a fill-in suitable for the inclusion of electric charge.

Anisotropic Generalization of the ΛCDM Universe Model with Application to the Hubble Tension

Anisotropic Generalization of the ΛCDM Universe Model with Application to the Hubble Tension

ASKING IF VACUUM ENERGY allows COMPUTATIONAL “BITS” PRESENT AT START OF COSMOLOGICAL EVOLUTION, and its possible ties into torsion physics

Based on Torsion as given by de Sabbata and Sirvaram, Erice 1990 . The 1990 article claims that Torsion cancels Cosmological vacuum energy. We consider if relic black holes at the start of inflation may allow for the observed cosmological constant. Using the 1990 text if thermal energy is used at start of inflation creates conceptual issues, whereas an energy term based on Corda’s treatment of black holes may provide a solution. i.e. a left over cosmological constant 10^-121 times vacuum energy . Considerations as to how black hole physics may contribute to torsion and the cosmological constant are considered in several numerical cases Also we present entanglement entropy in the early universe with a shrinking scale factor, due to Muller and Lousto , and show that there are consequences due to initial entanged for a time dependent horizon radius in cosmology, with (flat space conditions) for conformal time . This construction preserves a minimum non zero vacuum energy, and in doing so keep the bits, for computational bits cosmological evolution. The bits are ascribed to initial torsion for reasons which we elaborate on in this manuscript

Image of Kerr–de Sitter black holes illuminated by equatorial thin accretion disks

To explore the influence of the cosmological constant on black hole images, we have developed a comprehensive analytical method for simulating images of Kerr–de Sitter black holes illuminated by equatorial thin accretion disks. Through the application of explicit equations, we simulate images of Kerr–de Sitter black holes illuminated by both prograde and retrograde accretion disks, examining the impact of the cosmological constant on their characteristic curves, relative sizes, and observed intensities. Our findings reveal that, in comparison to Kerr black holes, the cosmological constant not only diminishes the relative size of a black hole but also amplifies its luminosity. Moreover, an observer’s relative position in the universe (r0/rC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0/r_C$$\\end{document}) can influence both the relative size and luminosity of a black hole, where r0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_0$$\\end{document} is the distance from the observer to the black hole, rC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r_C$$\\end{document} is the cosmological horizon determined by the value of the cosmological constant Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document}.