Particle Horizon Research Articles

Modified hoop conjecture in expanding spacetimes and primordial black hole production in FRW universe

According to a variant of the hoop conjecture, if we localize two particles within the Schwarzschild radius corresponding to their center of mass energy, then a black hole will form. Despite a large body of work on the formation of primordial black holes, so far this conjecture has not been generalized to expanding spacetimes. We derive a formula which gives the distance within which two particles must be localized to give a black hole, and which crucially depends on the expansion rate of the background space. In the limit of a very slow expansion, we recover the flat spacetime case. In the opposite limit of the large expansion rate when the inverse Hubble radius is smaller than the Schwarzschild radius of a “would be” black hole, the new critical distance between two particles that can make a black hole becomes equal to the particle horizon, which is just a requirement that the particles are in a causal contact. This behavior also nicely illustrates why the Big Bang singularity is not a black hole. We then use our formula to calculate the number density, energy density and production rate of black holes produced in collisions of particles. We find that though black holes might be numerous at high temperatures, they never dominate over the background radiation below the Planck temperature.

Covariant generalized holographic dark energy and accelerating universe

We propose the generalized holographic dark energy model where the infrared cutoff is identified with the combination of the FRW universe parameters: the Hubble rate, particle and future horizons, cosmological constant, the universe lifetime (if finite) and their derivatives. It is demonstrated that with the corresponding choice of the cutoff one can map such holographic dark energy to modified gravity or gravity with a general fluid. Explicitly, F(R) gravity and the general perfect fluid are worked out in detail and the corresponding infrared cutoff is found. Using this correspondence, we get realistic inflation or viable dark energy or a unified inflationary-dark energy universe in terms of covariant holographic dark energy.

Cosmological horizons, uncertainty principle, and maximum length quantum mechanics

The cosmological particle horizon is the maximum measurable length in the Universe. The existence of such a maximum observable length scale implies a modification of the quantum uncertainty principle. Thus due to non-locality of quantum mechanics, the global properties of the Universe could produce a signature on the behaviour of local quantum systems. A Generalized Uncertainty Principle (GUP) that is consistent with the existence of such a maximum observable length scale $l_{max}$ is $\Delta x \Delta p \geq \frac{\hbar}{2}\;\frac{1}{1-\alpha \Delta x^2}$ where $\alpha = l_{max}^{-2}\simeq (H_0/c)^2$ ($H_0$ is the Hubble parameter and $c$ is the speed of light). In addition to the existence of a maximum measurable length $l_{max}=\frac{1}{\sqrt \alpha}$, this form of GUP implies also the existence of a minimum measurable momentum $p_{min}=\frac{3 \sqrt{3}}{4}\hbar \sqrt{\alpha}$. Using appropriate representation of the position and momentum quantum operators we show that the spectrum of the one dimensional harmonic oscillator becomes $\bar{\mathcal{E}}_n=2n+1+\lambda_n \bar{\alpha}$ where $\bar{\mathcal{E}}_n\equiv 2E_n/\hbar \omega$ is the dimensionless properly normalized $n^{th}$ energy level, $\bar{\alpha}$ is a dimensionless parameter with $\bar{\alpha}\equiv \alpha \hbar/m \omega$ and $\lambda_n\sim n^2$ for $n\gg 1$ (we show the full form of $\lambda_n$ in the text). For a typical vibrating diatomic molecule and $l_{max}=c/H_0$ we find $\bar{\alpha}\sim 10^{-77}$ and therefore for such a system, this effect is beyond reach of current experiments. However, this effect could be more important in the early universe and could produce signatures in the primordial perturbation spectrum induced by quantum fluctuations of the inflaton field.

The Cosmic Causal Mass

In order to provide a better understanding of rotating universe models, and in particular the G\"{o}del universe, we discuss the relationship between cosmic rotation and perfect inertial dragging. In this connection, the concept of \emph{causal mass} is defined in a cosmological context, and discussed in relation to the cosmic inertial dragging effect. Then, we calculate the mass inside the particle horizon of the flat $\Lambda$CDM-model integrated along the past light cone. The calculation shows that the Schwarzschild radius of this mass is around three times the radius of the particle horizon. This indicates that there is close to perfect inertial dragging in our universe. Hence, the calculation provides an explanation for the observation that the swinging plane of a Foucault pendulum follows the stars.

Inflation via logarithmic entropy-corrected holographic dark energy model

We study the inflation via logarithmic entropy-corrected holographic dark energy LECHDE model with future event horizon, particle horizon and Hubble horizon cut-offs, and compare the results with those of obtained in the study of inflation by holographic dark energy HDE model. In comparison, the spectrum of primordial scalar power spectrum in the LECHDE model becomes redder than the spectrum in HDE model. Moreover, the consistency with the observational data in LECHDE model of inflation, constrains the reheating temperature and Hubble parameter by one parameter of holographic dark energy and two new parameters of logarithmic corrections.

Evolution of density and velocity profiles of dark matter and dark energy in spherical voids

We analyse the evolution of cosmological perturbations which leads to the formation of large isolated voids in the Universe. We assume that initial perturbations are spherical and all components of the Universe (radiation, matter and dark energy) are continuous media with perfect fluid energy--momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations for every component in the comoving to cosmological background reference frame are obtained from equations of energy and momentum conservation and Einstein's ones and are integrated numerically. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is much larger than the particle horizon. Results show how the profiles of density and velocity of matter and dark energy are formed and how they depend on parameters of dark energy and initial conditions. In particular, it is shown that final matter density and velocity amplitudes change within range $\sim$4-7\% when the value of equation-of-state parameter of dark energy $w$ vary in the range from --0.8 to --1.2, and change within $\sim$1\% only when the value of effective sound speed of dark energy vary over all allowable range of its values.

Dynamic cancellation of a cosmological constant and approach to the Minkowski vacuum

The q-theory approach to the cosmological constant problem is reconsidered. The new observation is that the effective classical q-theory gets modified due to the back-reaction of quantum-mechanical particle production by spacetime curvature. Furthermore, a Planck-scale cosmological constant is added to the potential term of the action density, in order to represent the effects from zero-point energies and phase transitions. The resulting dynamical equations of a spatially-flat Friedmann–Robertson–Walker universe are then found to give a steady approach to the Minkowski vacuum, with attractor behavior for a finite domain of initial boundary conditions on the fields. The approach to the Minkowski vacuum is slow and gives rise to an inflation-type increase of the particle horizon.

Evolution of density and velocity profiles of matter in large voids

We analyse the evolution of cosmological perturbations which leads to the formation of large voids in the distribution of galaxies. We assume that perturbations are spherical and all components of the Universe - radiation, matter and dark energy - are continuous media with ideal fluid energy-momentum tensors, which interact only gravitationally. Equations of the evolution of perturbations in the comoving to cosmological background reference frame for every component are obtained from equations of conservation and Einstein's ones and are integrated by modified Euler method. Initial conditions are set at the early stage of evolution in the radiation-dominated epoch, when the scale of perturbation is mush larger than the particle horizon. Results show how the profiles of density and velocity of matter in spherical voids with different overdensity shells are formed.

Causal structures in inflation

This article reviews the properties and limitations associated with the existence of particle, visual, and event horizons in cosmology in general and in inflationary universes in particular, carefully distinguishing them from `Hubble horizons'. It explores to what extent one might be able to probe conditions beyond the visual horizon (which is close in size to the present Hubble radius), thereby showing that visual horizons place major limits on what are observationally testable aspects of a multiverse, if such exists. Indeed these limits largely prevent us from observationally proving a multiverse either does or does not exist. We emphasize that event horizons play no role at all in observational cosmology, even in the multiverse context, despite some claims to the contrary in the literature.

Apparent horizon and gravitational thermodynamics of the Universe: Solutions to the temperature and entropy confusions and extensions to modified gravity

The thermodynamics of the Universe is restudied by requiring its compatibility with the holographic-style gravitational equations which govern the dynamics of both the cosmological apparent horizon and the entire Universe, and possible solutions are proposed to the existent confusions regarding the apparent-horizon temperature and the cosmic entropy evolution. We start from the generic Lambda Cold Dark Matter ($\Lambda$CDM) cosmology of general relativity (GR) to establish a framework for the gravitational thermodynamics. The Cai-Kim Clausius equation for the isochoric process of an instantaneous apparent horizon indicates that, the Universe and its horizon entropies encode the \emph{positive heat out} thermodynamic sign convention, which encourages us to adjust the traditional positive-heat-in Gibbs equation into the positive-heat-out version $dE_m=-T_mdS_m-P_mdV$. It turns out that the standard and the generalized second laws (GSLs) of nondecreasing entropies are always respected by the event-horizon system as long as the expanding Universe is dominated by nonexotic matter $-1\leq w_m\leq 1$, while for the apparent-horizon simple open system the two second laws hold if $-1\leq w_m<-1/3$; also, the artificial local equilibrium assumption is abandoned in the GSL. All constraints regarding entropy evolution are expressed by the equation of state parameter, which show that from a thermodynamic perspective the phantom dark energy is less favored than the cosmological constant and the quintessence. Finally, the whole framework is extended from GR and $\Lambda$CDM to modified gravities with field equations $R_{\mu\nu}-Rg_{\mu\nu}/2=8\pi G_{\text{eff}} T_{\mu\nu}^{\text{(eff)}}$. Furthermore, this paper argues that the Cai-Kim temperature is more suitable than Hayward, and the Bekenstein-Hawking and Wald entropies cannot unconditionally apply to the event and particle horizons.

Cosmological constraints on the variable modified Chaplygin gas model by using MCMC approach

We investigate the cosmological constraints on the variable modified Chaplygin gas (VMCG) model from the latest observational data: Union2 dataset of Type Ia supernovae (SNIa), the observational Hubble data (OHD), the baryon acoustic oscillations (BAO) and the cosmic microwave background (CMB) data. By using the Markov chain Monte Carlo (MCMC) method, we obtain the mean values of parameters in the flat model: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Furthermore, we investigate the thermodynamical properties of VMCG model at apparent horizon, event horizon and particle horizon respectively.

Separate universe problem: 40 years on

The claim that an overdense (positive curvature) region in the early universe cannot extend beyond some maximum scale and remain part of our universe, first made 40 years ago, has recently been questioned by Kopp et al. Their analysis is elucidating and demonstrates that one cannot constrain the form of primordial density perturbations using this argument. However, the notion of a separate-universe scale still applies and it places an important upper limit on the mass of primordial black holes forming at any epoch. We calculate this scale for equations of state of the form $p = k \rho c^2$ with $-1 <k < \infty$, refining earlier calculations on account of the Kopp et al. criticisms. For $-1/3 < k < \infty$, the scale is always of order the cosmological particle horizon size, with a numerical factor depending on $k$. This confirms the earlier claim that a primordial black hole cannot be much larger than the particle horizon at formation. For $-1 < k< -1/3$, as expected for some periods in the history of the universe, the situation changes radically, in that a sufficiently large positive-curvature region produces a baby universe rather than a black hole. There is still a separate-universe scale but the interpretation of these solutions requires care.

A study of different horizons in inhomogeneous LTB cosmological model

This work deals with a detailed study of the dynamics of the apparent, event and particle horizons in the background of the inhomogeneous LTB spacetime. The comparative study among these horizons shows a distinct character for apparent horizon compared to the other horizons. The apparent horizon will be a trapping horizon if its acceleration is positive. The Kodama vector is also defined and its causal character is found to be similar to that in the FRW model.

LRS Bianchi Type II tilted barotropic fluid cosmological model with heat conduction in general relativity

A tilted cosmological model is studied for a barotropic fluid distribution with heat conduction in the framework of a compact Bianchi Type II space-time. To get a deterministic solution in terms of cosmic time t, we have used the supplementary condition R = S n between the metric potentials R and S, where n is a constant. The shear tensor σ ij for the tilted model satisfies the trace-free condition σ ij v j = 0, where v i is the fluid velocity vector. In a special case, the model becomes non-tilted. The heat conduction vector q i also satisfies the trace-free condition q i v i = 0. The spatial volume increases as time increases, and the model represents an inflationary scenario. The model for a barotropic fluid distribution represents decelerating and accelerating phases of the universe, matching with the astronomical observations. Anisotropy is maintained throughout, but in a special case the model isotropizes. The particle horizon and entropy are discussed. We have also discussed the stiff fluid case with physical and geometrical aspects of the models.

C-field cosmology for barotropic fluid distribution with variable bulk viscosity and vacuum energy (Λ) in Friedmann–Robertson–Walker (FRW) space–time

A cosmological model that admits barotropic fluid distribution and a negative energy massless scalar creation field as a source in the presence of bulk viscosity and vacuum energy (Λ) in Friedmann–Robertson–Walker space–time is investigated. It has been shown that the model satisfies observational tests and is thus an alternative to the standard Big Bang model. The model is free from real singularity and particle horizon. The creation field increases with time, which matches Hoyle and Narlikar (Proc. Roy. Soc. A, 282, 178 (1964). doi:10.1098/rspa.1964.0225 ). To get the deterministic results, we have assumed that the coefficient of bulk viscosity, ζαρ1/2, where ρ is the matter density, ρ = 3H2, Λ ∼ H2, where H is the Hubble parameter. The model satisfies the energy conditions (weak, dominant, and strong). The deceleration parameter, q &lt; 0, shows that the model represents an accelerating phase of the universe.

Magnetization of fluid phonons and large-scale curvature perturbations

The quasinormal mode of a gravitating and magnetized fluid in a spatially flat, isotropic and homogeneous cosmological background is derived in the presence of the fluid sources of anisotropic stress and of the entropic fluctuations of the plasma. The obtained gauge-invariant description involves a system of two coupled differential equations of which the physical content is analyzed in all the most relevant situations. The Cauchy problem of large-scale curvature perturbations during the radiation dominated stage of expansion can be neatly formulated, and its general solution is shown to depend on five initial data assigned when the relevant physical wavelengths are larger than the particle horizon. The consequences of this approach are explored.

A relativistic view on large scale N-body simulations

We discuss the relation between the output of Newtonian N-body simulations on scales that approach or exceed the particle horizon to the description of general relativity. At leading order, the Zeldovich approximation is correct on large scales, coinciding with the general relativistic result. At second order in the initial metric potential, the trajectories of particles deviate from the second order Newtonian result and hence the validity of second order Lagrangian perturbation theory initial conditions should be reassessed when used in very large simulations. We also advocate using the expression for the synchronous gauge density as a well behaved measure of density fluctuations on such scales.

On the accuracy of N-body simulations at very large scales

We examine the deviation of Cold Dark Matter particle trajectories from the Newtonian result as the size of the region under study becomes comparable to or exceeds the particle horizon. To first order in the gravitational potential, the general relativistic result coincides with the Zel'dovich approximation and hence the Newtonian prediction on all scales. At second order, General Relativity predicts corrections which overtake the corresponding second order Newtonian terms above a certain scale of the order of the Hubble radius. However, since second order corrections are very much suppressed on such scales, we conclude that simulations which exceed the particle horizon but use Newtonian equations to evolve the particles, reproduce the correct trajectories very well. The dominant relativistic corrections to the power spectrum on scales close to the horizon are at most of the order of $\sim 10^{-5}$ at $z=49$ and $\sim 10^{-3}$ at $z=0$. The differences in the positions of real space features are affected at a level below $10^{-6}$ at both redshifts. Our analysis also clarifies the relation of N-body results to relativistic considerations.

Reconstructing f(R) theory from pilgrim dark energy

We analyze pilgrim dark energy model in the background of f(R) gravity by taking IR cut-offs as particle and event horizons as well as conformal age of the universe. We regard the f(R) theory as an effective description for the pilgrim dark energy and reconstruct the function f(R) with the parameter μ<0 (which supports the phantom regime). Finally, we show the distinctive behavior of f(R) for the future cosmic evolution.

Building non-commutative spacetimes at the Planck length for Friedmann flat cosmologies

We propose physically motivated spacetime uncertainty relations (STUR) for flat Friedmann–Lemaître cosmologies. We show that the physical features of these STUR crucially depend on whether a particle horizon is present or not. In particular, when this is the case we deduce the existence of a maximal value for the Hubble rate (or equivalently for the matter density), thus providing an indication that quantum effects may rule out a pointlike big bang singularity. Finally, we construct a concrete realization of the corresponding quantum Friedmann spacetime in terms of operators on a Hilbert space.