Statistical analysis with cosmic-expansion-rate measurements and two-point diagnostics

Direct measurements of Hubble parameters H(z) are very useful for cosmological model parameters inference. Based on them, Sahni, Shafieloo and Starobinski introduced a two-point diagnostic \(Omh^2(z_i, z_j)\) as an interesting tool for testing the validity of the \(\Lambda \hbox {CDM}\) model. Applying this test they found a tension between observations and predictions of the \(\Lambda \hbox {CDM}\) model. We use the most comprehensive compilation H(z) data from baryon acoustic oscillations (BAO) and differential ages (DA) of passively evolving galaxies to study cosmological models using the Hubble parameters itself and to distinguish whether \(\Lambda \hbox {CDM}\) model is consistent with the observational data with statistical analysis of the corresponding \(Omh^2(z_i, z_j)\) two-point diagnostics. Our results show that presently available H(z) data significantly improve the constraints on cosmological parameters. The corresponding statistical \(Omh^2(z_i, z_j)\) two-point diagnostics seems to prefer the quintessence with \(w>-1\) over the \(\Lambda \hbox {CDM}\) model. Better and more accurate prior knowledge of the Hubble constant, will considerably improve the performance of the statistical \(Omh^2(z_i, z_j)\) method.     

Observational Data Fitting to Constrain Variable Modified Chaplygin Gas in the Background of Horava-Lifshitz Gravity

FRW universe in Horava-Lifshitz (HL) gravity model filled with a combination of dark matter and dark energy in the form of variable modified Chaplygin gas (VMCG) is considered. The permitted values of the VMCG parameters are determined by the recent astrophysical and cosmological observational data. Here we present the Hubble parameter in terms of the observable parameters Ω _{d m0}, Ω _{v m c g0}, H ₀, redshift z and other parameters like α, A, γ and n. From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the χ ² test. The best-fit values of the parameters are obtained by 66 %, 90 % and 99 % confidence levels. Next due to joint analysis with BAO and CMB observations, we have also obtained the bounds of the parameters (A, γ) by fixing some other parameters α and n. The best fit value of distance modulus μ(z) is obtained for the VMCG model in HL gravity, and it is concluded that our model is perfectly consistent with the union2 sample data.     

<Emphasis Type="Italic">O</Emphasis>(1) eV sterile neutrino in <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>) gravity

We refer [1] to the role of an additional O(1) eV sterile neutrino in modified gravity models. We find parameter constraints in particular f(R) gravity model using following up-to-dated cosmological data: measurements of the cosmic microwave background (CMB) anisotropy, the CMB lensing potential, the baryon acoustic oscillations (BAO), the cluster mass function and the Hubble constant. It was obtained for the sterile neutrino mass 0.47 eV < m _ν,sterile < 1 eV (2σ) assuming that the sterile neutrinos are thermalized and the active neutrinos are massless, not significantly larger than in the standard cosmology model within the same data set: 0.45 eV < m _ν,sterile < 0.92 eV (2σ). But, if the mass of sterile neutrino is fixed and equals ≈ 1.5 eV according to various anomalies in neutrino oscillation experiments, f(R) gravity is much more consistent with observation data than the CDM model.     

Testing the Distance-Duality Relation from Hubble,Galaxy Clusters and Type Ia Supernovae Data with Model Independent Methods

In this paper, we perform cosmological-model-independent tests for the distance-duality (DD) relation η(z)=D _L(1+z)^?2/D _A by combining the angular diameter distance D _A(or comoving distances D _c ) with the luminosity distance D _L. The D _A are provided by two galaxy clusters samples compiled by De Filippis et al. (the elliptical β model), Bonamente et al. (the spherical β model), the D _c are obtained from Hubble parameter data and D _L are given from the Union2.1 supernovae (SNe) Ia compilation. We employ two methods, i.e., method A: binning the SNe Ia data within the range Δz=|z?z _SNe|<0.005, and method B: reconstructing the D _L(z) by smoothing the noise of Union2.1 data set over redshift with the Gaussian smoothing function, to obtain D _L associated with the redshits of the observed D _A or D _c. Four parameterizations for η(z), i.e., η(z)=1+η ₀ z, η(z)=1+η ₀ z/(1+z), η(z)=1+η ₀ z/(1+z)² and η(z)=1?η ₀ ln(1+z), are adopted for the DD relation. We find that DD relation is consistent with the present observational data, and the results we obtained are not sensitive to the method and parameterization.     

Diagnosing ΛHDE model with statefinder hierarchy and fractional growth parameter

Recently, a new dark energy model called ΛHDE was proposed. In this model, dark energy consists of two parts: cosmological constant Λ and holographic dark energy(HDE). Two key parameters of this model are the fractional density of cosmological constant ?_(Λ0), and the dimensionless HDE parameter c. Since these two parameters determine the dynamical properties of DE and the destiny of universe, it is important to study the impacts of different values of ?_(Λ0) and c on the ΛHDE model. In this paper,we apply various DE diagnostic tools to diagnose ΛHDE models with different values of ?_(Λ0) and c; these tools include statefinder hierarchy{S_3~(1), S_4~(1)}, fractional growth parameter ?, and composite null diagnostic(CND), which is a combination of{S_3~(1), S_4~(1)}and ?. We find that:(1) adopting different values of ?_(Λ0) only has quantitative impacts on the evolution of the ΛHDE model, while adopting different c has qualitative impacts;(2) compared with S_3~(1), S_4~(1) can give larger differences among the cosmic evolutions of the ΛHDE model associated with different ?_(Λ0) or different c;(3) compared with the case of using a single diagnostic, adopting a CND pair has much stronger ability to diagnose the ΛHDE model.     

A Note on Holographic Dark Energy with Varying <Emphasis Type="Italic">c</Emphasis><Superscript>2</Superscript> Term

We reconsider the holographic dark energy (HDE) model with a slowly time varying c ²(z) parameter in the energy density, namely \(\rho _{D}=3{M_{p}^{2}} c^{2}(z)/L^{2}\), where L is the IR cutoff and z is the redshift parameter. As the system’s IR cutoff we choose the Hubble radius and the Granda-Oliveros (GO) cutoffs. The latter inspired by the Ricci scalar curvature. We derive the evolution of the cosmological parameters such as the equation of state and the deceleration parameters as the explicit functions of the redshift parameter z. Then, we plot the evolutions of these cosmological parameters in terms of the redshift parameter during the history of the universe. Interestingly enough, we observe that by choosing L = H ^?1 as the IR cutoff for the HDE with time varying c ²(z) term, the present acceleration of the universe expansion can be achieved, even in the absence of interaction between dark energy and dark matter. This is in contrast to the usual HDE model with constant c ² term, which leads to a wrong equation of state, namely that for dust w _D =0, when the IR cutoff is chosen the Hubble radius.     

Redshift drift constraints on holographic dark energy

The Sandage-Loeb(SL) test is a promising method for probing dark energy because it measures the redshift drift in the spectra of Lyman-α forest of distant quasars, covering the "redshift desert" of 2 z 5, which is not covered by existing cosmological observations. Therefore, it could provide an important supplement to current cosmological observations. In this paper, we explore the impact of SL test on the precision of cosmological constraints for two typical holographic dark energy models, i.e., the original holographic dark energy(HDE) model and the Ricci holographic dark energy(RDE) model. To avoid data inconsistency, we use the best-fit models based on current combined observational data as the fiducial models to simulate 30 mock SL test data. The results show that SL test can effectively break the existing strong degeneracy between the present-day matter density ?_(m0) and the Hubble constant H0 in other cosmological observations. For the considered two typical dark energy models, not only can a30-year observation of SL test improve the constraint precision of ?_(m0) and h dramatically, but can also enhance the constraint precision of the model parameters c and α significantly.     

Recent observational constraints on generalized Chaplygin gas in UDME scenario

Recent observational predictions suggest that our Universe is passing through an accelerating phase in the recent past. This acceleration may be realized with the negatively pressured dark energy. Generalized Chaplygin gas may be suitable to describe the evolution of the Universe as a candidate of unified dark matter energy (UDME) model. Its EoS parameters are constrained using (i) dimensionless age parameter (H ₀ t ₀) and (ii) the observed Hubble (H(z)?z) data (OHD) + baryon acoustic oscillation (BAO) data + cosmic microwave background (CMB) shift data + supernovae (Union2.1) data. Dimensionless age parameter puts loose bounds on the EoS parameters. Best-fit values of the EoS parameters H ₀, A _s and α (A _s and α are defined in the energy density for generalized Chaplygin gas (GCG) and in EoS) are then determined from OHD + BAO + CMB + Union2.1 data and contours are drawn to obtain their allowed range of values. The present age of the Universe (t ₀) and the present Hubble parameter (H ₀) have been estimated with 1σ confidence level. Best-fit values of deceleration parameter (q), squared sound speed (\(c_{\mathrm {s}}^{2}\)) and EoS parameter (ω) of this model are then determined. It is seen that GCG satisfactorily accommodates an accelerating phase and structure formation phase.     

Electro-optical properties of strontium-barium niobate crystals and their relation to the domain structure of the crystals

The electro-optical coefficients r _ij and half-wave voltage V_λ/2 of strontium-barium niobate crystals poled in the ferroelectric phase are shown to vary along the polar axis. The r _ij (z) and V_λ/2(z) dependences indicate the presence of a residual domain density D(z) and clearly depend on the sign of the polarizing field, with r _ij being minimum (D being maximum) near the negative electrode. This character of the D(z) distribution and, hence, the r _ij (z) and V_λ/2(z) coordinate dependences can be explained by predominant domain nucleation near the negative electrode, which is revealed when the switching processes are studied using 90° (Rayleigh) light scattering from domain walls.     

Cosmic acceleration in non-flat <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) cosmology

We study f(T) cosmological models inserting a non-vanishing spatial curvature and discuss its consequences on cosmological dynamics. To figure this out, a polynomial f(T) model and a double torsion model are considered. We first analyze those models with cosmic data, employing the recent surveys of Union 2.1, baryonic acoustic oscillation and cosmic microwave background measurements. We then emphasize that the two popular f(T) models enable the crossing of the phantom divide line due to dark torsion. Afterwards, we compute numerical bounds up to 3-\(\sigma \) confidence level, emphasizing the fact that \(\Omega _{k0}\) turns out to be non-compatible with zero at least at 1\(\sigma \). Moreover, we underline that, even increasing the accuracy, one cannot remove the degeneracy between our models and the \(\Lambda \)CDM paradigm. So that, we show that our treatments contain the concordance paradigm and we analyze the equation of state behaviors at different redshift domains. We also take into account gamma ray bursts and we describe the evolution of both the f(T) models with high redshift data. We calibrate the gamma ray burst measurements through small redshift surveys of data and we thus compare the main differences between non-flat and flat f(T) cosmology at different redshift ranges. We finally match the corresponding outcomes with small redshift bounds provided by cosmography. To do so, we analyze the deceleration parameters and their variations, proportional to the jerk term. Even though the two models well fit late-time data, we notice that the polynomial f(T) approach provides an effective de-Sitter phase, whereas the second f(T) framework shows analogous results compared with the \(\Lambda \)CDM predictions.     

Interacting Dark Energy in the Dark <Emphasis Type="Italic">S</Emphasis><Emphasis Type="Italic">U</Emphasis>(2)<Subscript><Emphasis Type="Italic">R</Emphasis></Subscript> Model

We explore the cosmological implications of the interactions among the dark particles in the dark SU(2) _R model. It turns out that the relevant interaction is between dark energy and dark matter, through a decay process. With respect to the standard ΛCDM model, it changes only the background equations. We note that the observational aspects of the model are dominated by degeneracies between the parameters that describe the process. Thus, only the usual Λ CDM parameters such as the Hubble expansion rate and the dark energy density parameter (interpreted as the combination of the densities of the dark energy doublet) could be constrained by observations at this moment.     

Diagnosing holographic type dark energy models with the Statefinder hierarchy,composite null diagnostic and <Emphasis Type="Italic">w</Emphasis>-<Emphasis Type="Italic">w′</Emphasis> pair

The main purpose of this work is to distinguish various holographic type dark energy (DE) models, including the ΛHDE, HDE, NADE, and RDE model, by using various diagnostic tools. The first diagnostic tool is the Statefinder hierarchy, in which the evolution of Statefinder hierarchy parmeter S ⁽¹⁾ ₃(z) and S ⁽¹⁾ ₄(z) are studied. The second is composite null diagnostic (CND), in which the trajectories of {S ⁽¹⁾ ₃, ?} and {S ⁽¹⁾ ₄, ?} are investigated, where ? is the fractional growth parameter. The last is w-w′ analysis, where w is the equation of state for DE and the prime denotes derivative with respect to lna. In the analysis we consider two cases: varying current fractional DE density Ω _de0 and varying DE model parameter C. We find that: (1) both the Statefinder hierarchy and the CND have qualitative impact on ΛHDE, but only have quantitative impact on HDE. (2) S ⁽¹⁾ ₄ can lead to larger differences than S ⁽¹⁾ ₃, while the CND pair has a stronger ability to distinguish different models than the Statefinder hierarchy. (3) For the case of varying C, the {w,w′} pair has qualitative impact on ΛHDE; for the case of varying Ω _de0, the {w, w′} pair only has quantitative impact; these results are different from the cases of HDE, RDE, and NADE, in which the {w,w′} pair only has quantitative impact on these models. In conclusion, compared with HDE, RDE, and NADE, the ΛHDE model can be easily distinguished by using these diagnostic tools.     

Long-range spin correlations in a honeycomb spin model with a magnetic field

We consider the spin-1/2 model on the honeycomb lattice [A. Kitaev, Ann. Phys. 321, 2 (2006)] in the presence of a weak magnetic field h _α ? J. Such a perturbation treated in the lowest nonvanishing order over h _α leads [K.S. Tikhonov, M.V. Feigel’man, and A.Yu. Kitaev, Phys. Rev. Lett. 106, 067203 (2011)] to a powerlaw decay of irreducible spin correlations 《s ^z (t, r)s ^z (0, 0)》 ∝ h _z ² f(t, r), where f(t, r) ∝ [max(t, Jr)]^–4. We have studied the effects of the next order of perturbation in h _z and found an additional term of the order h _z ⁴ in the correlation function 《s ^z (t, r)s ^z (0, 0)》 which scales as h _z ⁴ cosγ/r ³ at Jt? r, where γ is the polar angle in the 2D plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by the weak imaginary vector potential A _x ∝ i h _z ² .     

Stabilizer of a function in the gage group

It is proved that, for the dimension d of the stabilizer of an analytic function z(x, y) in the gage pseudogroup G = {z(x, y) → c(z(a(x), b(y))}, there are precisely four possibilities: (1) d = ∞ and the complexity of z is zero, (2) d = 3 and the complexity of z is equal to one, (3) d = 1 and z is equivalent the function r(x + y) ? x of complexity two, (4) d = 0 in all remaining cases.     

Photon distribution functions of the fluorescence photons from a single nanoparticle in various photon counting methods

Simple expressions have been derived for three photon distribution functions w _N ^M (T), w _N ^Z (T), and w _N ^O (T) corresponding to three different methods for counting fluorescence photons from a single nanoparticle excited by continuous laser radiation. In contrast to the previously derived expressions represented in the form of N multiple integrals, the new expressions contain only single or double integrals of Poisson functions, which makes it possible to easily perform the numerical calculation of the photon distribution. The simplest photon counting method corresponds to the lengthiest function w _N ^M (T); on the contrary, the simplest function w _N ^O (T) corresponds to the most complex photon counting method. The functions w _N ^M (T), w _N ^Z (T), and w _N ^O (T) are noticeably different in short time intervals T; however, the distributions calculated using these functions are almost indistinguishable from each other in long T intervals. This circumstance makes it possible to use the simplest function w _N ^O (T) to consider the photon statistics measured by the simplest method. This possibility is particularly important for investigating the fluorescence photon statistics, where the intensity fluctuates.     

A new procedure for investigating three-dimensional stress fields in a thin plate with a through-the-thickness crack

In the paper, a new procedure is proposed to investigate three-dimensional fracture problems of a thin elastic plate with a long through-the-thickness crack under remote uniform tensile loading. The new procedure includes a new analytical method and high accurate finite element simulations. In the part of theoretical analysis, three-dimensional Maxwell stress functions are employed in order to derive three-dimensional crack tip fields. Based on the theoretical analysis, an equation which can describe the relationship among the three-dimensional J-integral J(z), the stress intensity factor K(z) and the tri-axial stress constraint level T _z (z) is derived first. In the part of finite element simulations, a fine mesh including 153360 elements is constructed to compute the stress field near the crack front, J(z) and T _z (z). Numerical results show that in the plane very close to the free surface, the K field solution is still valid for in-plane stresses. Comparison with the numerical results shows that the analytical results are valid.     

The analysis of the densities of <Emphasis Type="Italic">s</Emphasis>-wave neutron resonances separated with respect to spin

The density ratio of s-wave neutron resonances z=ρ(J₁)/ρ(J₂) was analyzed on the basis of the experimental data for 22 atomic nuclei and the Gilbert-Cameron formula for ρ(J). Here, J₁=I_x—1/2 and J₂=I_x+1/2, where I_x denotes the spin of the target nucleus in the ground state. Our aim was to verify whether the factor η(I_x), as a multiplier, can be applied in the expression describing ρ(J₁), with the assumption that ρ(J₂) values remain unchanged, or whether the factor 1η(I_x) can be applied, as a multiplier with ρ(J₂), while the ρ(J₁) values remain unchanged. The final conclusions, e.g., the confirmation or the negation of the fact that it may be necessary to apply the η(I_x) factor, depend on the values of “real” errors Δz of the z variable, which can be calculated if the optimal values of Δρ(J₁) and Δρ(J₂) are known.     

A modification of Einstein–Schrödinger theory that contains both general relativity and electrodynamics

We modify the Einstein–Schrödinger theory to include a cosmological constant Λ _z which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant Λ _z is assumed to be nearly cancelled by Schrödinger’s cosmological constant Λ _b which multiplies the nonsymmetric fundamental tensor, such that the total Λ =  Λ _z +  Λ _b matches measurement. The resulting theory becomes exactly Einstein–Maxwell theory in the limit as |Λ _z | → ∞. For |Λ _z | ~ 1/(Planck length)² the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are < 10^?16 of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein–Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein–Infeld–Hoffmann (EIH) equations of motion match the equations of motion for Einstein–Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein–Schrödinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner–Nordström solution except for additional terms which are ~10^?66 of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein–Maxwell theory.     

Violation of light beam reversibility principle in optical media with a random quasi-zero refractive index

The relic abundance of light millicharged particles (MCPs) with the electric charge e′ = 5 × 10^–5 e and with the mass slightly below or above the electron mass is calculated. The abundance depends on the mass ratio η = m _X /m _e and for η < 1 can be high enough to allow MCPs to be the cosmological dark matter or to make a noticeable contribution to it. On the other hand, for η ? 1 the cosmological energy density of MCPs can be quite low, Ω _X h ₀ ² ≈ 0.02 for scalar MCPs, and Ω _X h ₀ ² ≈ 0.001 for spin 1/2 fermions. But even the lowest value of Ω _X h ₀ ² is in tension with several existing limits on the MCP abundances and parameters. However, these limits have been derived under some natural or reasonable assumptions on the properties of MCPs. If these assumptions are relaxed, a patch in the mass–charge plot of MCPs may appear, permitting them to be dark matter particles.     

Model-independent reconstruction of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">T</Emphasis>) teleparallel cosmology

We propose a model-independent formalism to numerically solve the modified Friedmann equations in the framework of f(T) teleparallel cosmology. Our strategy is to expand the Hubble parameter around the redshift \(z=0\) up to a given order and to adopt cosmographic bounds as initial settings to determine the corresponding \(f(z)\equiv f(T(H(z)))\) function. In this perspective, we distinguish two cases: the first expansion is up to the jerk parameter, the second expansion is up to the snap parameter. We show that inside the observed redshift domain \(z\le 1\), only the net strength of f(z) is modified passing from jerk to snap, whereas its functional behavior and shape turn out to be identical. As first step, we set the cosmographic parameters by means of the most recent observations. Afterwards, we calibrate our numerical solutions with the concordance \(\Lambda \)CDM model. In both cases, there is a good agreement with the cosmological standard model around \(z\le 1\), with severe discrepancies outer of this limit. We demonstrate that the effective dark energy term evolves following the test-function: \(f(z)={\mathcal {A}}+{\mathcal {B}}{z}^2e^{{\mathcal {C}}{z}}\). Bounds over the set \(\left\{ {\mathcal {A}}, {\mathcal {B}}, {\mathcal {C}}\right\} \) are also fixed by statistical considerations, comparing discrepancies between f(z) with data. The approach opens the possibility to get a wide class of test-functions able to frame the dynamics of f(T) without postulating any model a priori. We thus re-obtain the f(T) function through a back-scattering procedure once f(z) is known. We figure out the properties of our f(T) function at the level of background cosmology, to check the goodness of our numerical results. Finally, a comparison with previous cosmographic approaches is carried out giving results compatible with theoretical expectations.