Unequal Time Correlators Research Articles

Contribution of domain wall networks to the CMB power spectrum

We use three domain wall simulations from the radiation era to the late-time dark energy domination era based on the PRS algorithm to calculate the energy–momentum tensor components of domain wall networks in an expanding universe. Unequal time correlators in the radiation, matter and cosmological constant epochs are calculated using the scaling regime of each of the simulations. The CMB power spectrum of a network of domain walls is determined. The first ever quantitative constraint for the domain wall surface tension is obtained using a Markov chain Monte Carlo method; an energy scale of domain walls of 0.93 MeV, which is close but below the Zel'dovich bound, is determined.

CMB power spectrum of Nambu-Goto cosmic strings

We improve predictions of the cosmic microwave background (CMB) power spectrum induced by cosmic strings by using source terms obtained from Nambu-Goto network simulations in an expanding universe. We use three high-resolution cosmic string simulations that cover the entire period from recombination until late-time $\Lambda$ domination to calculate unequal time correlators (UETCs) for scalar, vector and tensor components of the cosmic-string energy-momentum tensor. We calculate the CMB angular power spectrum from strings in two ways. First, to aid comparison with previous work, we fit our simulated UETCs to those obtained from different parameter combinations from the unconnected segment model and then calculate the CMB power spectra using these parameters to represent the string network. Second and more accurately, we decompose the UETCs into their corresponding eigenvalues and eigenvectors and input them directly into an Einstein-Boltzmann solver to calculate the power spectrum for each of the three simulation time periods. We combine the three simulations together, using each of them in its relevant redshift range and we obtain overall power spectra in temperature and polarisation channels. Finally, we use the power spectra obtained with the latest Planck and BICEP2 likelihoods to obtain constraints on the cosmic string tension.

The bispectrum of matter perturbations from cosmic strings

We present the first calculation of the bispectrum of the matter perturbations induced by cosmic strings. The calculation is performed in two different ways: the first uses the unequal time correlators (UETCs) of the string network - computed using a Gaussian model previously employed for cosmic string power spectra. The second approach uses the wake model, where string density perturbations are concentrated in sheet-like structures whose surface density grows with time. The qualitative and quantitative agreement of the two gives confidence to the results. An essential ingredient in the UETC approach is the inclusion of compensation factors in the integration with the Green's function of the matter and radiation fluids, and we show that these compensation factors must be included in the wake model also. We also present a comparison of the UETCs computed in the Gaussian model, and those computed in the unconnected segment model (USM) used by the standard cosmic string perturbation package CMBACT. We compare numerical estimates for the bispectrum of cosmic strings to those produced by perturbations from an inflationary era, and discover that, despite the intrinsically non-Gaussian nature of string-induced perturbations, the matter bispectrum is unlikely to produce competitive constraints on a population of cosmic strings.

Empirical microeconomics action functionals

A statistical generalization of microeconomics has been made in Baaquie (2013), where the market price of every traded commodity, at each instant of time, is considered to be an independent random variable. The dynamics of commodity market prices is modeled by an action functional–and the focus of this paper is to empirically determine the action functionals for different commodities. The correlation functions of the model are defined using a Feynman path integral. The model is calibrated using the unequal time correlation of the market commodity prices as well as their cubic and quartic moments using a perturbation expansion. The consistency of the perturbation expansion is verified by a numerical evaluation of the path integral. Nine commodities drawn from the energy, metal and grain sectors are studied and their market behavior is described by the model to an accuracy of over 90% using only six parameters. The paper empirically establishes the existence of the action functional for commodity prices that was postulated to exist in Baaquie (2013).

On exact statistics and classification of ergodic systems of integer dimension.

We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This definition generalizes the construction of Hamiltonian systems by a Hamiltonian and a symplectic form. Some low dimensional examples are given, as well as a discretized field theory with a large number of degrees of freedom and a local nearest neighbor interaction. We also evaluate unequal-time correlations of these systems without direct numerical simulation, by Padé approximants of a short-time expansion. We briefly speculate on the possibility of constructing chaotic dynamical systems with non-integer dimension and exactly known statistics. In this case there is no probability density, suggesting an alternative construction in terms of a Hopf characteristic function and a two-form.

Cosmic microwave background temperature and polarization anisotropies from the large-Nlimit of global defects

We determine the full ${C}_{\ensuremath{\ell}}$ spectra and correlation functions of the temperature and polarization anisotropies in the cosmic microwave background (CMB), generated by a source modeled by the large-$N$ limit of spontaneously broken global $O(N)$-theories. We point out a problem in the standard approach of treating the radiation-matter transition by interpolating the eigenvectors of the unequal-time correlators of the source energy-momentum tensor. This affects the CMB predictions from all types of cosmic defects. We propose a method to overcome this difficulty. We find that in the large-$N$ global model that we study, differences in the final CMB power spectra amplitudes reach $\ensuremath{\sim}10%--20%$ in all channels (TT, EE, BB and TE) when compared to implementations of the eigenvector interpolation technique. We also discuss how to optimally search for the contribution in the CMB from active sources such as cosmic defects in experiments like Planck, COrE and PRISM.

Statistical microeconomics

A statistical generalization is made of microeconomics in the spirit of going from classical to statistical mechanics. The price and quantity of every commodity11The term commodities is used for goods and services. traded in the market, at each instant of time, is considered to be an independent random variable: all prices and quantities are considered to be stochastic processes, with the observed market prices being a random sample of the stochastic prices. The dynamics of market prices is determined by an action functional and, for concreteness, a specific model is proposed. The model can be calibrated from the unequal time correlation of the market commodity prices. A perturbation expansion for the correlation functions is defined in powers of the inverse of the total budget of the aggregate consumer and the propagator for the market prices is evaluated.

Evolution of singularities in unequal time correlator in thermalization of quark gluon plasma

We studied thermalization of strongly coupled gauge theory using a gravitational collapse model. In particular, we studied unequal time correlator for the glue ball operator. We found singularities of the correlator for zero momentum sector is consistent with a geometric optics picture in the gravitational theory. The singularities of the correlator indicates strong temporal correlation, thus the time after which the singularities disappear provides a measure of the temporal decoherence in the thermalization process.

Fast analytic computation of cosmic string power spectra

We present analytic expressions for the cosmic string unequal time correlator (UETC) in the context of the Unconnected Segment Model. This eliminates the need to simulate the many thousands of network realizations needed to estimate the UETC numerically. With our approach we can compute the UETC very accurately, over all scales of interest, in the order of $\ensuremath{\sim}20--30$ seconds on a single CPU. Our formalism facilitates an efficient approach to computing cosmic microwave background anisotropies for cosmic strings. Discretizing the UETC and performing an eigendecomposition to act as sources in the CAMB cosmic microwave background code, the power spectrum can be calculated by summing over a finite number of eigenmodes. A much smaller number of eigenmodes are required compared to the conventional approach of averaging power spectra over a finite number of realizations of the string network. With the additional efficiency and performance improvements offered by the OpenMP CAMB code, the time required to compute string power spectra is significantly reduced compared to the standard serial CMBACT code. The latter takes $\ensuremath{\sim}30$ hours on a modern single threaded CPU for 2000 network realizations. Similar percent level accuracy can be achieved with our approach on a moderately threaded CPU (eight threads) in only $\ensuremath{\sim}15$ minutes. If accuracy is only required at the 10% level and the CPU is more highly threaded, cosmic string power spectra are now possible in $\ensuremath{\sim}2--3$ minutes. This makes exploration of the string parameter space now possible for Markov chain Monte Carlo analysis.

Thermalization from gauge/gravity duality: evolution of singularities in unequal time correlators

We consider a gauge/gravity dual model of thermalization which consists of a collapsing thin matter shell in asymptotically Anti-de Sitter space. A central aspect of our model is to consider a shell moving at finite velocity as determined by its equation of motion, rather than a quasi-static approximation as considered previously in the literature. By applying a divergence matching method, we obtain the evolution of singularities in the retarded unequal time correlator $G^R(t,t')$, which probes different stages of the thermalization. We find that the number of singularities decreases from a finite number to zero as the gauge theory thermalizes. This may be interpreted as a sign of decoherence. Moreover, in a second part of the paper, we show explicitly that the thermal correlator is characterized by the existence of singularities in the complex time plane. By studying a quasi-static state, we show the singularities at real times originate from contributions of normal modes. We also investigate the possibility of obtaining complex singularities from contributions of quasi-normal modes.

Path integral for equities: Dynamic correlation and empirical analysis

This paper develops a model to describe the unequal time correlation between rate of returns of different stocks. A non-trivial fourth order derivative Lagrangian is defined to provide an unequal time propagator, which can be fitted to the market data. A calibration algorithm is designed to find the empirical parameters for this model and different de-noising methods are used to capture the signals concealed in the rate of return. The detailed results of this Gaussian model show that the different stocks can have strong correlation and the empirical unequal time correlator can be described by the model’s propagator. This preliminary study provides a novel model for the correlator of different instruments at different times.

Study of gravitational radiation from cosmic domain walls

In this paper, following the previous study, we evaluate the spectrum of gravitational wave background generated by domain walls which are produced if some discrete symmetry is spontaneously broken in the early universe. We apply two methods to calculate the gravitational wave spectrum: One is to calculate the gravitational wave spectrum directly from numerical simulations, and another is to calculate it indirectly by estimating the unequal time anisotropic stress power spectrum of the scalar field. Both analysises indicate that the slope of the spectrum changes at two characteristic frequencies corresponding to the Hubble radius at the decay of domain walls and the width of domain walls, and that the spectrum between these two characteristic frequencies becomes flat or slightly red tilted. The second method enables us to evaluate the GW spectrum for the frequencies which cannot be resolved in the finite box lattice simulations, but relies on the assumptions for the unequal time correlations of the source.

Cosmic string power spectrum, bispectrum, and trispectrum

We use analytic calculations of the post-recombination gravitational effects of cosmic strings to estimate the resulting CMB power spectrum, bispectrum and trispectrum. We place a particular emphasis on multipole regimes relevant for forthcoming CMB experiments, notably the Planck satellite. These calculations use a flat sky approximation, generalising previous work by integrating string contributions from last scattering to the present day, finding the dominant contributions to the correlators for multipoles l > 50. We find a well-behaved shape for the string bispectrum (without divergences) which is easily distinguishable from the inflationary bispectra which possess significant acoustic peaks. We estimate that the nonlinearity parameter characterising the bispectrum is approximately f_NL \sim -20 (given present string constraints from the CMB power spectrum. We also apply these unequal time correlator methods to calculate the trispectrum for parrallelogram configurations, again valid over a large range of angular scales relevant for WMAP and Planck, as well as on very small angular scales. We find that, unlike the bispectrum which is suppressed by symmetry considerations, the trispectrum for cosmic strings is large. Our current estimate for the trispectrum parameter is tau_NL \sim 10^5, which may provide one of the strongest constraints on the string model as estimators for the trispectrum are developed.

The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition

We analytically derive the spectrum of gravitational waves due to magneto-hydrodynamical turbulence generated by bubble collisions in a first-order phase transition. In contrast to previous studies, we take into account the fact that turbulence and magnetic fields act as sources of gravitational waves for many Hubble times after the phase transition is completed. This modifies the gravitational wave spectrum at large scales. We also model the initial stirring phase preceding the Kolmogorov cascade, while earlier works assume that the Kolmogorov spectrum sets in instantaneously. The continuity in time of the source is relevant for a correct determination of the peak position of the gravitational wave spectrum. We discuss how the results depend on assumptions about the unequal-time correlation of the source and motivate a realistic choice for it.Our treatment gives a similar peak frequency as previous analyses but the amplitude of the signal is reduced due to the use of a more realistic power spectrum for the magneto-hydrodynamical turbulence. For a strongly first-orderelectroweak phase transition, the signal is observable with the space interferometer LISA.

CMB power spectrum contribution from cosmic strings using field-evolution simulations of the Abelian Higgs model

We present the first field-theoretic calculations of the contribution made by cosmic strings to the temperature power spectrum of the cosmic microwave background (CMB). Unlike previous work, in which strings were modeled as idealized one-dimensional objects, we evolve the simplest example of an underlying field theory containing local U(1) strings, the Abelian Higgs model. Limitations imposed by finite computational volumes are overcome using the scaling property of string networks and a further extrapolation related to the lessening of the string width in comoving coordinates. The strings and their decay products, which are automatically included in the field theory approach, source metric perturbations via their energy-momentum tensor, the unequal-time correlation functions of which are used as input into the CMB calculation phase. These calculations involve the use of a modified version of CMBEASY, with results provided over the full range of relevant scales. We find that the string tension required to normalize to the WMAP 3-year data at multipole [script-l]=10 is G=[2.040.06(stat.)0.12(sys.)]10-6, where we have quoted statistical and systematic errors separately, and G is Newton's constant. This is a factor 23 higher than values in current circulation.

A matrix ansatz for the diffusion of an impurity in the asymmetric exclusion process

We study the fluctuations of the position of an impurity in the asymmetric exclusion process on a ring with an arbitrary number of particles and holes. The steady state of this model is exactly known and four different phases appear in the limit of a large system. We calculate the diffusion constant of the impurity by using a matrix product method and also obtain a representation for unequal time correlation functions. We show that our results found by the matrix ansatz agree with those obtained previously by the Bethe ansatz.

Schwinger-Dyson approach to nonequilibrium classical field theory

In this paper we discuss a Schwinger-Dyson (SD) approach for determining the time evolution of the unequal time correlation functions of a nonequilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0.$ We focus on $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ space-time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation (BVA), is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for $1/N$ expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\ensuremath{\varphi}$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.

Ordering dynamics of heisenberg spins with torque: crossover, spin waves, and defects

We study the effect of a torque induced by the local molecular field on the phase ordering dynamics of the Heisenberg model when the total magnetization is conserved. The torque drives the zero-temperature ordering dynamics to a new fixed point, characterized by exponents z=2 and lambda approximately 5. This "torque-driven" fixed point is approached at times such that t>>g(2), where g is the strength of the torque. All physical quantities, like the domain size L(t) and the equal and unequal time correlation functions, obey a crossover scaling form over the entire range of g. An attempt to understand this crossover behavior from the approximate Gaussian closure scheme fails completely, implying that the dynamics at late times cannot be understood from the dynamics of defects alone. We provide convincing arguments that the spin configurations can be decomposed in terms of defects and spin waves which interact with each other even at late times. In the absence of the torque term, the spin waves decay faster, but even so we find that the Gaussian closure scheme is inconsistent. In the latter case the inconsistency may be remedied by including corrections to a simple Gaussian distribution. For completeness we include a discussion of the ordering dynamics at T(c), where the torque is shown to be relevant, with exponents z=4-varepsilon/2 and lambda=d (where varepsilon=6-d). We show to all orders in perturbation theory that lambda=d as a consequence of the conservation law.

Cosmological parameter dependence in local string theories of structure formation

We perform a new and accurate study of the dependence on cosmological parameters of structure formation with local cosmic strings. The crucial new ingredients are the inclusion of the effects of gravitational back reaction on the evolution of the network, and the accurate evolution of the network through the radiation to matter transition. Our work re-iterates the fact that expanding universe numerical simulations only probe a transient regime, and we incorporate our results into the unequal time correlators recently measured. We then compute the CMB and CDM fluctuations' power spectra for various values of the Hubble constant ${H}_{0}$ and baryon fraction ${\ensuremath{\Omega}}_{b}.$ We find that, whereas the dependence on ${\ensuremath{\Omega}}_{b}$ is negligible, there is still a strong dependence on ${H}_{0}.$

Dynamical correlations of two-dimensional vortex-like defects

High-order dynamical correlations of defects (quantum vortices, disclinations, etc.) in thin films are examined by starting from the Langevin equation for the defect motion. It is demonstrated that the dynamical correlation functions F 2n of the vorticity or disclinicity behave as F 2n ∼y 2/r 4n , where r is the characteristic scale and y is the renormalized fugacity. Therefore below the Berezinskii-Kosterlitz-Thouless transition temperature the F 2n are characterized by anomalous scaling exponents. The behavior differs strongly from the normal law F 2n ∼F 2 n obeyed by equal-time correlation functions; the unequal-time correlation functions appear to be much larger. The phenomenon resembles intermittency in turbulence.