Bug Searching in Smart Contract

With the frantic development of smart contracts on the Ethereum platform, its market value has also climbed. In 2016, people were shocked by the loss of nearly $50 million in cryptocurrencies from the DAO reentrancy attack. Due to the tremendous amount of money flowing in smart contracts, its security has attracted much attention of researchers. In this paper, we investigated several common smart contract vulnerabilities and analyzed their possible scenarios and how they may be exploited. Furthermore, we survey the smart contract vulnerability detection tools for the Ethereum platform in recent years. We found that these tools have similar prototypes in software vulnerability detection technology. Moreover, for the features of public distribution systems such as Ethereum, we present the new challenges that these software vulnerability detection technologies face.Comment: 8 pages, 9 figure

Power spectra and spectral indices of kk-inflation: high-order corrections

$k$-inflation represents the most general single-field inflation, in which the perturbations usually obey an equation of motion with a time-dependent sound speed. In this paper, we study the observational predictions of the $k$-inflation by using the high-order uniform asymptotic approximation method. We calculate explicitly the slow-roll expressions of the power spectra, spectral indices, and running of the spectral indices for both the scalar and tensor perturbations. These expressions are all written in terms of the Hubble and sound speed flow parameters. It is shown that the previous results obtained by using the first-order uniform asymptotic approximation have been significantly improved by the high-order corrections of the uniform asymptotic approximations. Furthermore, we also check our results by comparing them with the ones obtained by other approximation methods, including the Green's function method, WKB approximation, and improved WKB approximation, and find the relative errors.Comment: Typos are also corrected. Phys. Rev. D90, 103517 (2014). arXiv admin note: text overlap with arXiv:1405.530

Inflationary cosmology with nonlinear dispersion relations

We present a technique, {\em the uniform asymptotic approximation}, to construct accurate analytical solutions of the linear perturbations of inflation after quantum effects of the early universe are taken into account, for which the dispersion relations generically become nonlinear. We construct explicitly the error bounds associated with the approximations and then study them in detail. With the understanding of the errors and the proper choice of the Liouville transformations of the differential equations of the perturbations, we show that the analytical solutions describe the exact evolution of the linear perturbations extremely well even only in the first-order approximations. As an application of the approximate analytical solutions, we calculate the power spectra and indices of scalar and tensor perturbations in the slow-roll inflation, and find that the amplitudes of the power spectra get modified due to the quantum effects, while the power spectrum indices remain the same as in the linear case.Comment: The analysis of power spectra and indices of scalar and tensor perturbations are enlarged. Two figures added. arXiv admin note: text overlap with arXiv:1308.110

Pre-inflationary universe in loop quantum cosmology

The evolutions of the flat FLRW universe and its linear perturbations are studied systematically in the dressed metric approach of LQC. When it is dominated by the kinetic energy of the inflaton at the quantum bounce, the evolution of the background can be divided into three different phases prior to the preheating, {\em bouncing, transition and slow-roll inflation}. During the bouncing phase, the evolution is independent of not only the initial conditions, but also the inflationary potentials. In particular, the expansion factor can be well described by the same exact solution in all the cases considered. In contrast, in the potential dominated case such a universality is lost. It is because of this universality that the linear perturbations are also independent of the inflationary models and obtained exactly. During the transition phase, the evolutions of the background and its linear perturbations are found explicitly, and then matched to the ones given in the other two phases. Hence, once the initial conditions are imposed, the linear scalar and tensor perturbations will be uniquely determined. Considering two different sets of initial conditions, one imposed during the contracting phase and the other at the bounce, we calculate the Bogoliubov coefficients and find that the two sets yield the same results and all lead to particle creations at the onset of the inflation. Due to the pre-inflationary dynamics, the scalar and tensor power spectra become scale-dependent. Comparing with the Planck 2015 data, we find constraints on the total e-folds that the universe must have expanded since the bounce, in order to be consistent with current observations.Comment: revtex4, 23 figures, and 5 tables. Some typos were corrected. Phys. Rev. D 96, 083520 (2017

High-order Primordial Perturbations with Quantum Gravitational Effects

In this paper, we provide a systematic investigation of high-order primordial perturbations with nonlinear dispersion relations due to quantum gravitational effects in the framework of {\em uniform asymptotic approximations}. Because of these effects, the equation of motion of the mode function in general has multiple-turning points. After obtaining analytically approximated solutions to any order in different regions, associated with different types of turning points, we match them to the third one. To this order the errors are less than $0.15\%$. General expressions of the power spectra of the primordial tensor and scalar perturbations are derived explicitly. We also investigate effects of back-reactions of the quantum gravitational corrections, and make sure that inflation lasts long enough in order to solve the underlying problems, such as flatness, horizon and monopole. Then, we study various features of the spectra that are observationally relevant. In particular, under a moderate assumption about the energy scale of the underlying theory of quantum gravity, we have shown that the quantum gravitational effects may alter significantly the ratio between the tensor and scalar power spectra, thereby providing a natural mechanism to alleviate the tension between observations and certain inflationary models, including the one with a quadratic potential.Comment: revtex4, 9 figures, no tables. Typos are corrected, and sections are rearranged. Version to appear in PR

Primordial non-Gaussianity and power asymmetry with quantum gravitational effects in loop quantum cosmology

Loop quantum cosmology (LQC) provides a resolution of the classical big bang singularity in the deep Planck era. The evolution, prior to the usual slow-roll inflation, naturally generates excited states at the onset of the slow-roll inflation. It is expected that these quantum gravitational effects could leave its fingerprints on the primordial perturbation spectrum and non-Gaussianity, and lead to some observational evidences in the cosmic microwave background (CMB). While the impact of the quantum effects on the primordial perturbation spectrum has been already studied and constrained by current data, in this paper we continue studying such effects on the non-Gaussianity of the primordial curvature perturbations. In this paper, we present detailed and analytical calculations of the non-Gaussianity and show explicitly that the corrections due to quantum effects are in the same magnitude of the slow-roll parameters in the observable scales and thus are well within current observational constraints. Despite this, we show that the non-Gaussianity in the squeezed limit can be enhanced at superhorizon scales and further, these effects may yield a large statistical anisotropy on the power spectrum through the Erickcek-Kamionkowski-Carroll mechanism.Comment: revtex4, one figure and no table

Energy loss of heavy and light quarks in holographic magnetized background

We systematically study holographic effects on the magnetic field dependence of the drag force, diffusion coefficient, jet quenching parameter of heavy quarks and the shooting string energy loss of light quarks in the RHIC and LHC energy regions by using the AdS/CFT correspondence in this paper. This study is motivated by the phenomena of strong magnetic field and jet quenching, which have been found in relativistic heavy ion collisions. The probe's direction of motion is perpendicular and parallel to the direction of magnetic field $B$. The effects of magnetic field on energy loss when moving perpendicular to the magnetic field direction are larger than moving parallel to the magnetic field direction, which implies that the magnetic field tends to suppress more quarks and jets when moving in the transverse direction than in the parallel direction. It is found that the diffusion coefficient decreases with the magnetic field in the transverse direction, but increases with the magnetic field in the parallel direction, which indicates that the quark may diffuse farther when moving parallel to the magnetic field direction. We also find that the magnetic field will enhance the energy loss of the light quarks when moving in the transverse direction than in the parallel direction.Comment: 26 pages, 8 figure

Scalar and tensor perturbations in loop quantum cosmology: High-order corrections

Loop quantum cosmology (LQC) provides promising resolutions to the trans-Planckian issue and initial singularity arising in the inflationary models of general relativity. In general, due to different quantization approaches, LQC involves two types of quantum corrections, the holonomy and inverse-volume, to both of the cosmological background evolution and perturbations. In this paper, using {\em the third-order uniform asymptotic approximations}, we derive explicitly the observational quantities of the slow-roll inflation in the framework of LQC with these quantum corrections. We calculate the power spectra, spectral indices, and running of the spectral indices for both scalar and tensor perturbations, whereby the tensor-to-scalar ratio is obtained. We expand all the observables at the time when the inflationary mode crosses the Hubble horizon. As the upper error bounds for the uniform asymptotic approximation at the third-order are $\lesssim 0.15\%$, these results represent the most accurate results obtained so far in the literature. It is also shown that with the inverse-volume corrections, both scalar and tensor spectra exhibit a deviation from the usual shape at large scales. Then, using the Planck, BAO and SN data we obtain new constraints on quantum gravitational effects from LQC corrections, and find that such effects could be within the detection of the forthcoming experiments.Comment: revtex4, one figure. JCAP10 (2015) 05

Detecting quantum gravitational effects of loop quantum cosmology in the early universe?

We derive the primordial power spectra and spectral indexes of the density fluctuations and gravitational waves in the framework of loop quantum cosmology (LQC) with holonomy and inverse-volume corrections, by using the uniform asymptotic approximation method to its third-order, at which the upper error bounds are $\lesssim 0.15\%$, and accurate enough for the current and forthcoming cosmological observations. Then, using the Planck, BAO and SN data we obtain the tightest constraints on quantum gravitational effects from LQC corrections, and find that such effects could be well within the detection of the current and forthcoming cosmological observations.Comment: revtex4, 2 figutes and 1 table. ApJL 807 (2015) L1

Preinflationary perturbations from the closed algebra approach in loop quantum cosmology

In this paper, the scalar and tensor perturbations in the closed algebra approach of loop quantum cosmology are studied. Instead of the distant past in the contracting phase, we choose the moment at which the initial conditions are imposed to be the silent point, which circumvents the problem due to the signature change in the super-inflationary phase and results in a well-defined Cauchy problem. For the ultraviolet and infrared modes, different approaches are applied in order to obtain analytical solutions with high accuracy. While previous numerical simulations reveal an exponentially divergent power spectrum in the ultraviolet regime, when the initial conditions are imposed in the remote contracting phase, we find a special set of initial conditions at the silent point, which can reproduce results that are consistent with current observations.Comment: Corrected some typos. Phys. Rev. D99, 103536 (2019