Genesis of Dark Energy: Dark Energy as Consequence of Release and Two-Stage Tracking of Cosmological Nuclear Energy

Recent observations on Type-Ia supernovae and low density (Ω _m =0.3) measurement of matter including dark matter suggest that the present-day universe consists mainly of repulsive-gravity type ‘exotic matter’ with negative-pressure often said ‘dark energy’ (Ω _x =0.7). But the nature of dark energy is mysterious and its puzzling questions, such as why, how, where and when about the dark energy, are intriguing. In the present paper the authors attempt to answer these questions while making an effort to reveal the genesis of dark energy and suggest that ‘the cosmological nuclear binding energy liberated during primordial nucleo-synthesis remains trapped for a long time and then is released free which manifests itself as dark energy in the universe’. It is also explained why for dark energy the parameter w=-\frac23w=-\frac{2}{3} . Noting that w=1 for stiff matter and w=\frac13w=\frac{1}{3} for radiation; w=-\frac23w=-\frac{2}{3} is for dark energy because “−1” is due to ‘deficiency of stiff-nuclear-matter’ and that this binding energy is ultimately released as ‘radiation’ contributing “ +\frac13+\frac{1}{3} ”, making w=-1+\frac13=-\frac23w=-1+\frac{1}{3}=-\frac{2}{3} . When dark energy is released free at Z=80, w=-\frac23w=-\frac{2}{3} . But as on present day at Z=0 when the radiation-strength-fraction (δ), has diminished to δ→0, the w=-1+d\frac13=-1w=-1+\delta\frac{1}{3}=-1 . This, almost solves the dark-energy mystery of negative pressure and repulsive-gravity. The proposed theory makes several estimates/predictions which agree reasonably well with the astrophysical constraints and observations. Though there are many candidate-theories, the proposed model of this paper presents an entirely new approach (cosmological nuclear energy) as a possible candidate for dark energy.     

Dissipative Future Universe Without Big Rip

The present study deals with dissipative future universe without big rip in context of Eckart formalism. The generalised Chaplygin gas, characterised by equation of state p=-\fracAr^\frac1ap=-\frac{A}{\rho^{\frac{1}{\alpha}}}, has been considered as a model for dark energy due to its dark-energy-like evolution at late time. It is demonstrated that, if the cosmic dark energy behaves like a fluid with equation of state p=ωρ; ω<−1 as well as Chaplygin gas simultaneously then the big rip problem does not arise and the scale factor is found to be regular for all time.     

Curvature induced late time acceleration

We formulate a modified theory of gravity to an equivalent second order gravity theory for a Lagrangian containing R and \frac1R{\frac{1}{R}} terms by introducing an auxiliary variable in a spatially homogeneous and isotropic background. We present a few analytical solutions of evolution equation for the deceleration parameter q as a function of the scale factor; specially in one solution, the universe evolves continuously from q = 1 (i.e. like a radiation dominated era) to q = -\frac12{q= -\frac{1}{2}} dark energy dominated late time accelerating phase when the universe is sufficiently old. The solution is supported by numerical results.     

CMC Hypersurfaces on Riemannian and Semi-Riemannian Manifolds

In this paper we generalize the explicit formulas for constant mean curvature (CMC) immersion of hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces given in Perdomo (Asian J Math 14(1):73–108, 2010; Rev Colomb Mat 45(1):81–96, 2011) to provide explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-Riemannian manifolds with constant sectional curvature. In particular, we prove that every h ? [-1,-\frac2?{n-1}n)h\in[-1,-\frac{2\sqrt{n-1}}{n}) can be realized as the constant curvature of a complete immersion of S₁^n-1×\mathbbRS_1^{n-1}\times \mathbb{R} in the (n + 1)-dimensional de Sitter space S₁ⁿ⁺¹\hbox{\bf S}_1^{n+1}. We provide 3 types of immersions with CMC in the Minkowski space, 5 types of immersion with CMC in the de Sitter space and 5 types of immersion with CMC in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.     

Critical Hardy–Lieb–Thirring Inequalities for Fourth-Order Operators in Low Dimensions

This paper considers Hardy–Lieb–Thirring inequalities for higher order differential operators. A result for general fourth-order operators on the half-line is developed, and the trace inequality

Statistics of Advective Stretching in Three-dimensional Incompressible Flows

We present a method to quantify kinematic stretching in incompressible, unsteady, isoviscous, three-dimensional flows. We extend the method of Kellogg and Turcotte (J. Geophys. Res. 95:421–432, 1990) to compute the axial stretching/thinning experienced by infinitesimal ellipsoidal strain markers in arbitrary three-dimensional incompressible flows and discuss the differences between our method and the computation of Finite Time Lyapunov Exponent (FTLE). We use the cellular flow model developed in Solomon and Mezic (Nature 425:376–380, 2003) to study the statistics of stretching in a three-dimensional unsteady cellular flow. We find that the probability density function of the logarithm of normalised cumulative stretching (log S) for a globally chaotic flow, with spatially heterogeneous stretching behavior, is not Gaussian and that the coefficient of variation of the Gaussian distribution does not decrease with time as t^-\frac12t^{-\frac{1}{2}} . However, it is observed that stretching becomes exponential log S∼t and the probability density function of log S becomes Gaussian when the time dependence of the flow and its three-dimensionality are increased to make the stretching behaviour of the flow more spatially uniform. We term these behaviors weak and strong chaotic mixing respectively. We find that for strongly chaotic mixing, the coefficient of variation of the Gaussian distribution decreases with time as t^-\frac12t^{-\frac{1}{2}} . This behavior is consistent with a random multiplicative stretching process.     

On the Massive Wave Equation on Slowly Rotating Kerr-AdS Spacetimes

The massive wave equation ${\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0}The massive wave equation \square_gy- a\fracL3y = 0{\square_{g}\psi - \alpha \frac{\Lambda}{3}\psi = 0} is studied on a fixed Kerr-anti de Sitter background (M,g_M,a,L){\left(\mathcal{M},g_{M,a,\Lambda}\right)}. We first prove that in the Schwarzschild case (a = 0), ψ remains uniformly bounded on the black hole exterior provided that a < \frac94{\alpha < \frac{9}{4}}, i.e. the Breitenlohner-Freedman bound holds. Our proof is based on vectorfield multipliers and commutators: The total flux of the usual energy current arising from the timelike Killing vector field T (which fails to be non-negative pointwise) is shown to be non-negative with the help of a Hardy inequality after integration over a spacelike slice. In addition to T, we construct a vectorfield whose energy identity captures the redshift producing good estimates close to the horizon. The argument is finally generalized to slowly rotating Kerr-AdS backgrounds. This is achieved by replacing the Killing vectorfield T = ∂ _t with K=?_t + l?_f{K=\partial_t + \lambda \partial_\phi} for an appropriate λ ~ a, which is also Killing and–in contrast to the asymptotically flat case–everywhere causal on the black hole exterior. The separability properties of the wave equation on Kerr-AdS are not used. As a consequence, the theorem also applies to spacetimes sufficiently close to the Kerr-AdS spacetime, as long as they admit a causal Killing field K which is null on the horizon.     

A Study of Holographic Dark Energy Models in Chern-Simon Modified Gravity

This paper is devoted to study some holographic dark energy models in the context of Chern-Simon modified gravity by considering FRW universe. We analyze the equation of state parameter using Granda and Oliveros infrared cut-off proposal which describes the accelerated expansion of the universe under the restrictions on the parameter α. It is shown that for the accelerated expansion phase \( -1<\omega _{\Lambda }<-\frac {1}{3}\), the parameter α varies according as \(1<\alpha <\frac {3}{2}\). Furthermore, for 0<α<1, the holographic energy and pressure density illustrates phantom-like theory of the evolution when ω_Λ<?1. Also, we discuss the correspondence between the quintessence, K-essence, tachyon and dilaton field models and holographic dark energy models on similar fashion. To discuss the accelerated expansion of the universe, we explore the potential and the dynamics of quintessence, K-essence, tachyon and dilaton field models.     

Invariance of the White Noise for KdV

We prove the invariance of the mean 0 white noise for the periodic KdV. First, we show that the Besov-type space [^(b)]^s_p,￥{\widehat{b}^s_{p,\infty}} , sp < −1, contains the support of the white noise. Then, we prove local well-posedness in [^(b)]^s_{p, ￥}{\widehat{b}^s_{p, \infty}} for p = 2 + , s = -\frac12+{s = -\frac{1}{2}+} such that sp < −1. In establishing the local well-posedness, we use a variant of the Bourgain spaces with a weight. This provides an analytical proof of the invariance of the white noise under the flow of KdV obtained in Quastel-Valko [21].     

VCSEL based Faraday rotation spectroscopy with a modulated and static magnetic field for trace molecular oxygen detection

Faraday Rotation Spectroscopy (FRS) is a useful technique for quantification of paramagnetic trace gases with significantly higher sensitivity when compared to direct absorption techniques. Our prototype system based on the openPHOTONS sensor core measures the concentration of molecular oxygen (O₂) in the A band using a 763-nm vertical cavity surface emitting laser. We provide detailed analysis of two measurement methods based on FRS using the same sensor configuration: one with a modulated magnetic field, and one with a static magnetic field in combination with wavelength modulation. Our spectra signal-to-noise ratios agree well with our simulations via modeling of the FRS signal. For alternating magnetic field, we achieve an equivalent minimum detectable absorption (MDA) of 8.86×10^-7/Hz^\frac128.86\times 10^{-7}/\mathrm{Hz}^{\frac{1}{2}} resulting in a minimum detection limit of 30 ppmv⋅m/ Hz^\frac12\mathrm{Hz}^{\frac{1}{2}} of O₂, limited by detector noise and laser noise. For the same system configuration in the static field case, parasitic etalon fringes limited the MDA to 4.8×10^-6/Hz^\frac124.8\times 10^{-6}/\mathrm{Hz}^{\frac{1}{2}}. In both cases, we describe methods to improve signal-to-noise ratio based on our data and models.     

On parity-violating three-nucleon interactions and the predictive power of few-nucleon EFT at very low energies

We address the typical strengths of hadronic parity-violating three-nucleon interactions in “pion-less” Effective Field Theory (EFT) in the nucleon-deuteron (iso-doublet) system. By analysing the superficial degree of divergence of loop diagrams, we conclude that no such interactions are needed at leading order, O(eQ^-1)\ensuremath {O}(\epsilon Q^{-1}) . The only two distinct parity-violating three-nucleon structures with one derivative mix ²S_\frac12\ensuremath ^2S_{\frac{1}{2}} and ²P_\frac12\ensuremath ^2P_{\frac{1}{2}} waves with iso-spin transitions D \Delta I = 0 or 1. Due to their structure, they cannot absorb any divergence ostensibly appearing at next-to-leading order, O(eQ⁰)\ensuremath {O}(\epsilon Q^0) . This observation is based on the approximate realisation of Wigner’s combined SU(4) spin-isospin symmetry in the two-nucleon system, even when effective-range corrections are included. Parity-violating three-nucleon interactions thus only appear beyond next-to-leading order. This guarantees renormalisability of the theory to that order without introducing new, unknown coupling constants and allows the direct extraction of parity-violating two-nucleon interactions from three-nucleon experiments.     

Thermodynamic Formalism and Variations of the Hausdorff Dimension of Quadratic Julia Sets

Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial z? z²+cz\mapsto z^2+c. The function d restricted to [0,+X) is real analytic in [0,\frac14)è(\frac14,+￥)[0,\frac{1}{4})\cup (\frac{1}{4},+\infty) ([Ru²]), is left-continuous at ¼ ([Bo,Zi]) but not continuous ([Do,Se,Zi]). We prove that c? d￠(c)c\mapsto d'(c) tends to + X from the left at ¼ as (\frac14-c)^{d(\frac14)-\frac32}(\frac{1}{4}-c)^{d(\frac{1}{4})-\frac{3}{2}}. In particular the graph of d has a vertical tangent on the left at ¼, a result which supports the numerical experiments.     

General Dynamical Equations for Dingle’s Space-Times Filled with a Charged Non-perfect Fluid

In this paper we have assumed charged non-perfect fluid as the material content of the space-time. The expression for the “mass function-M(r,y,z,t)” is obtained for the general situation and the contributions from the Ricci tensor in the form of material energy density ρ, pressure anisotropy [\fracp₂+p₃2-p₁][\frac{p_{2}+p_{3}}{2}-p_{1}] , electromagnetic field energy ℰ and the conformal Weyl tensor, viz. energy density of the free gravitational field ε (=\frac-3Y₂4p)(=\frac{-3\Psi_{2}}{4\pi}) are made explicit. This work is an extension of the work obtained earlier by Rao and Hasmani (Math. Today XIIA:71, 1993; New Directions in Relativity and Cosmology, Hadronic Press, Nonantum, 1997) for deriving general dynamical equations for Dingle’s space-times described by this most general orthogonal metric,

Critical Rotational Speeds in the Gross-Pitaevskii Theory on a Disc with Dirichlet Boundary Conditions

We study the two-dimensional Gross-Pitaevskii theory of a rotating Bose gas in a disc-shaped trap with Dirichlet boundary conditions, generalizing and extending previous results that were obtained under Neumann boundary conditions. The focus is on the energy asymptotics, vorticity and qualitative properties of the minimizers in the parameter range |log ε|≪Ω≲ε ⁻²|log ε|⁻¹ where Ω is the rotational velocity and the coupling parameter is written as ε ⁻² with ε≪1. Three critical speeds can be identified. At \varOmega = \varOmega_c1 ~ |loge|\varOmega=\varOmega_{\mathrm{c_{1}}}\sim |\log\varepsilon| vortices start to appear and for |loge| << \varOmega < \varOmega_c2 ~ e^-1|\log\varepsilon|\ll\varOmega< \varOmega_{\mathrm{c_{2}}}\sim \varepsilon^{-1} the vorticity is uniformly distributed over the disc. For \varOmega 3 \varOmega _c2\varOmega\geq\varOmega _{\mathrm{c_{2}}} the centrifugal forces create a hole around the center with strongly depleted density. For Ω≪ε ⁻²|log ε|⁻¹ vorticity is still uniformly distributed in an annulus containing the bulk of the density, but at \varOmega = \varOmega_c3 ~ e^-2|loge|^-1\varOmega=\varOmega_{\mathrm {c_{3}}}\sim\varepsilon ^{-2}|\log\varepsilon |^{-1} there is a transition to a giant vortex state where the vorticity disappears from the bulk. The energy is then well approximated by a trial function that is an eigenfunction of angular momentum but one of our results is that the true minimizers break rotational symmetry in the whole parameter range, including the giant vortex phase.     

Phase diagram for the Grover algorithm with static imperfections

We study effects of static inter-qubit interactions on the stability of the Grover quantum search algorithm. Our numerical and analytical results show existence of regular and chaotic phases depending on the imperfection strength e\varepsilon . The critical border e_c\varepsilon_c between two phases drops polynomially with the number of qubits n _q as e_c ~ n_q^-3/2\varepsilon_c \sim n_q^{-3/2} . In the regular phase (e < e_c)(\varepsilon < \varepsilon_c) the algorithm remains robust against imperfections showing the efficiency gain e_c / e\varepsilon_c / \varepsilon for e >~2^-n_q/2\varepsilon \gtrsim 2^{-n_q/2} . In the chaotic phase $(\varepsilon > \varepsilon_c)$(\varepsilon > \varepsilon_c) the algorithm is completely destroyed.     

Higher Dimensional Cosmological Model with Quark and Strange Quark Matter

We study higher dimensional homogeneous cosmological model in the presence of quark and strange quark matter. The dynamical behavior of the model for the strange quark matter equation of state of the form p = \frac13 (r- 4 B_c)p= \frac{1}{3} (\rho- 4 B_{c}) are studied.     

Unifying dark energy and dark matter with the modified Ricci model

In this paper, two modified Ricci models are considered as the candidates of unified dark matter–dark energy. In model one, the energy density is given by r_MR=3M_pl(aH²+b[(H)\dot])\rho_{\mathrm{MR}}=3M_{\mathrm{pl}}(\alpha H^{2}+\beta\dot{H}), whereas, in model two, by r_MR=3M_pl(\fraca6 R+g[(H)\ddot]H^-1)\rho_{\mathrm{MR}}=3M_{\mathrm{pl}}(\frac{\alpha}{6} R+\gamma\ddot{H}H^{-1}). We find that they can explain both dark matter and dark energy successfully. A constant equation of state of dark energy is obtained in model one, which means that it gives the same background evolution as the wCDM model, while model two can give an evolutionary equation of state of dark energy with the phantom divide line crossing in the near past.     

Virial Theorem for Angular Displacement and Torque

The usual Virial theorem is expressed through the coordinate and the force, 2áT? = áX\fracdVdX? = -áXF?2\langle T\rangle =\langle X\frac{dV}{dX}\rangle =-\langle XF\rangle , F=-\fracdVdXF=-\frac{dV}{dX}, XF is the work done by the force F, T is the kinetic energy. In this paper we extend the usual discussion on the Virial theorem about coordinate-force variables to the case of angular displacement-torque variables. By virtue of introducing the entangled state representation and the bosonic operator realization of the Hamiltonian of quantum pendulum system we derive the Virial theorem for angular variable and torque.     

Relationship between capture coefficient and capture cross-section: Average velocity of a maxwellian distribution of carriers in a medium with an anisotropic effective mass tensor

The capture cross section of a trapping or recombination center for a charge carrier has been defined as the quotient of the capture coefficient and the average thermal velocity of the carrier distribution. For a Maxwellian distribution in a semiconductor band with an ellipsoidal effective mass tensor, this average velocity can be expressed as