We perform a cosmological-model-independent test for the distance-duality (DD) relation η(z) = D L (z)(1 + z) −2 /D A (z), where D L and D A are the luminosity distance and angular diameter distance respectively, with a combination of observational data for D L taken from the latest Union2 SNe Ia and that for D A provided by two galaxy clusters samples compiled by De Filippis et al. and Bonamente et al.. Two parameterizations for η(z), i.e., η(z) = 1 + η 0 z and η(z) = 1 + η 0 z/(1 + z), are used. We find that the DD relation can be accommodated at 1σ confidence level (CL) for the De Filippis et al. sample and at 3σ CL for the Bonamente et al. sample. We also examine the DD relation by postulating two more general parameterizations: η(z) = η 0 + η 1 z and η(z) = η 0 + η 1 z/(1 + z), and find that the DD relation is compatible with the results from the De Filippis et al. and the Bonamente et al. samples at 1σ and 2σ CLs, respectively. Thus, we conclude that the DD relation is compatible with present observations.

We propose a novel enhancement mechanism of the curvature perturbations in the nonminimal derivative coupling inflation model with a coupling parameter related to the inflaton field. By considering a special form of the coupling parameter as a function of the inflaton, a period of ultraslow-roll inflation can be realized due to the gravitationally enhanced friction, and the resulting power spectrum of the curvature perturbations has a sharp peak, which is large enough to produce the primordial black holes. Under this mechanism, we can easily obtain a sharp mass spectrum of primordial black holes around specific masses such as O(10)M , O(10 −5 )M , and O(10 −12 )M , which can explain the LIGO events, the ultrashort-timescale microlensing events in OGLE data, and the most of dark matter, respectively.

We propose a parametrization for the growth index of the linear matter perturbations, γ(z) = γ 0 + z 1+z γ 1 . The growth factor of the perturbations parameterized as Ω γ m is analyzed for both the wCDM model and the DGP model with our proposed form for γ. We find that γ 1 is negative for the wCDM model but is positive for the DGP model. Thus it provides another signature to discriminate them. We demonstrate that Ω γ m with γ taking our proposed form approximates the growth factor very well both at low and high redshfits for both kinds of models. In fact, the error is below 0.03% for the ΛCDM model and 0.18% for the DGP model for all redshifts when Ω m0 = 0.27. Therefore, our parametrization may be robustly used to constrain the growth index of different models with the observational data which include points for redshifts ranging from 0.15 to 3.8, thus providing discriminative signatures for different models.

Emergent theory assumes that the universe originates from an Einstein static (ES) state rather than the big bang singularity, and, thus, provides a possible way to resolve the singularity problem. The stability against all kinds of perturbations that ensures the past eternity of the ES state is crucial for a successful realization of the emergent scenario. Recently, it has been found that in the context of the Jordan-Brans-Dicke (JBD) theory there exists a stable ES solution under homogenous and anisotropic perturbations. In this paper, we extend the analysis to the stability against tensor and inhomogeneous scalar perturbations. We find that, different from general relativity and fðRÞ theory, a stable ES solution is allowed in the JBD theory when different kinds of perturbations are considered, although the stability conditions are tighter for tensor and inhomogeneous scalar perturbations than those for homogenous and anisotropic ones.

We study the scalar induced gravitational wave (GW) background in inflation with gravitationally enhanced friction (GEF). The GEF mechanism, which is realized by assuming a nonminimal derivative coupling between the inflaton field and gravity, is used to amplify the small-scale curvature perturbations to generate a sizable amount of primordial black holes. We find that the GW energy spectra can reach the detectable scopes of the future GW projects, and the power spectrum of curvature perturbations has a power-law form in the vicinity of the peak. The scaling of the GW spectrum in the ultraviolet regions is two times that of the power spectrum slope, and has a lower bound. In the infrared regions, the slope of the GW spectrum can be described roughly by a log-dependent form. These features of the GW spectrum may be used to check the GEF mechanism if the scalar induced GWs are detected in the future.

We study the emergent scenario, which is proposed to avoid the big bang singularity, in the Einstein-Cartan (EC) theory with a positive cosmological constant and a perfect fluid by analyzing the existence and stability of the Einstein static (ES) solutions. We find that there is no stable ES solution for a spatially flat or open universe. However, for a spatially closed universe, the stable ES solution does exist, and in the same existence parameter regions, there also exists an unstable one. With the slow decrease of the equation of state w of the perfect fluid, the stable and unstable critical points move close gradually and coincide once w reaches a critical value, so that the stable critical point becomes an unstable one. As a result, if w approaches a constant at t → −∞, the universe can stay at the stable ES state past eternally, and furthermore it can naturally exit from this state and evolve into an inflationary era if w decreases slowly as time goes forward. Therefore, the emergent scenario that avoids the big bang singularity can be successfully implemented in the EC theory of gravity.

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