9406.json

{
    "9406/astro-ph9406014_arXiv.txt": {
        "abstract": " ",
        "introduction": "The existence of anisotropies in the CMB radiation as recently detected by {\\em COBE} \\cite{COBE92} and subsequently confirmed both by balloon--borne scans at shorter wavelength \\cite{Ganga+Co93} and by ground--based intermediate angular scale observations \\cite{Hancock+Co94} has triggered a large body of literature dealing with the non trivial task of finding the correct statistics able to disentangle the relevant information out of the primeval radiation maps. Both small and large angle scale probes of the microwave sky have been and are the real data our theoretical models must reproduce before we might call them {\\it viable}. Discrimination between the two main theories for the origin of primordial perturbations, namely, whether these are due to topological defects \\cite{DefectLect93} produced during a GUT phase transition or to early inflationary quantum fluctuations \\cite{TurLect93}, has by now become a difficult matter.  In this paper we will work in the frame of inflation and, in particular, we will be mainly concerned with single--field inflationary potentials. Some of the most popular models are characterized by their simplicity and universality (such as quadratic and quartic chaotic potentials), by their being exact solutions of the equations of motion for the inflaton (like power--law  and intermediate inflation) or by their particle physics motivation (as natural inflation with an axion--like potential). In contrast with these simpler models where one has just one relevant parameter more general potentials, with more freedom, were also considered in the literature. One example of this is the polynomial potential \\cite{Hodges+Co90} which for an adequate choice of the parameters was found to lead to broken scale invariant spectra on a wide range of scales with interesting consequences for large scale structure.  One should also worry about initial conditions \\cite{GoldwirthPiran92}. While for single--field models the only effect of kinetic terms consists in slightly changing the initial value of $\\phi$ (leaving invariant the phase space of initial field values leading to sufficient inflation), for models with more than one scalar field initial conditions can be important, e.g., for double inflation \\cite{DoubleInflation} (with two stages of inflation each one dominated by a different inflaton field) leading to primordial non--Gaussian perturbations on cosmological interesting scales. Within the latter models, however, the question of how probable it is that a certain initial configuration will be realized in our neighbouring universe should be addressed. More recently other examples of interesting multiple--field models with broken scale invariance have also been considered (see e.g. Ref.\\cite{Salopek92}). Here all scalar fields contribute to the energy density and non--Gaussian features are produced when the scalar fields pass over the interfaces of continuity of the potential. Extension of the single--field stochastic approach developed in Ref.\\cite{3point} for calculating the CMB angular bispectrum generated through Sachs--Wolfe effect from primordial curvature perturbations in the inflaton in order to include many scalar fields is therefore needed \\cite{progress}.  The plan of the paper is as follows. In Section 2 a brief overview of some general results is given while in Section 3 we concentrate on trying to extract numerical values for the non--Gaussian signal (the dimensionless skewness in this case) predicted in the frame of three different inflationary models. Section 4 contains some general conclusions. ",
        "conclusions": "In the present paper we have presented explicit calculations (in the frame of some well motivated inflationary models) of the dimensionless skewness ${\\cal S}$ predicted for the large angular scale temperature anisotropies in the CMB radiation as well as the evaluation of the primordial spectral index of the density perturbations originating these anisotropies. These computations were performed to full second order in perturbation theory. In all three models (even in the case of the polynomial inflaton potential where more parameters were at our disposal) very low values for the non--Gaussian signal were obtained.  In fact, the explicit values for ${\\cal S}$ were found generically much smaller than the dimensionless {\\em rms} skewness calculated from an underlying Gaussian density field and are therefore hidden by this theoretical noise, making experimental detection impossible.  One may try to resort to many--field models in the hope of shifting the non--zero ${\\cal S}$ window to larger values. In this respect a potential of the form $V(\\phi_i) \\sim \\exp (\\sum_i\\lambda_i\\phi_i)$ is likely to do well the job \\cite{Salopek92}. In this case the resolution of a set of coupled Langevin--type equations for the coarse--grained fields (suitably smoothed over a scale larger than the Hubble radius)  should be faced. In contrast to previous many--field analyses where one of the fields was assumed to dominate at a certain stage, in our case (for the aim of computing the three--point function) we need to make a second--order perturbative expansion in $\\delta\\phi_i$ around the classical solutions $\\phi_i^{class}$ but {\\rm keeping} $V(\\phi_i)$ fully dependent on all the fields. Non--Gaussian fluctuations can indeed be generated within this model and thus the prospect of getting a non--negligible value for the dimensionless skewness should be tested. This is the subject of our current research.  \\vspace{5mm}  \\noindent{\\bf Acknowledgements:} The author is grateful to S. Matarrese for valuable discussions. He also thanks A. Conti for his kind help in the numerical calculations. This work was supported by the Italian MURST."
    },
    "9406/hep-ph9406247_arXiv.txt": {
        "abstract": "I show the existence of a new type of vortex solution which is non-static but stationary and carries angular momentum. This {\\it spinning vortex} can be embedded in models with trivial vacuum topology like a model with $SU(2)_{global}\\times U(1)_{local} \\rightarrow U(1)_{global}$ symmetry breaking. The stability properties of the embedded spinning vortex are also studied in detail and it is shown that stability improves drastically as angular momentum increases. The implications of this result for vortices embedded in the electroweak model are under study. ",
        "introduction": " ",
        "conclusions": ""
    },
    "9406/astro-ph9406042_arXiv.txt": {
        "abstract": "Femtolensing is a gravitational lensing effect in which the magnification is a function not only of the positions and sizes of the source and lens, but also of the wavelength of light. Femtolensing is the only known effect of $(10^{-13}-10^{-16} M_{\\sun})$ dark-matter objects and may possibly be detectable in cosmological gamma-ray burst spectra. We present a new and efficient algorithm for femtolensing calculations in general potentials. The physical-optics results presented here differ at low frequencies from the semi-classical approximation, in which the flux is attributed to a finite number of mutually coherent images. At higher frequencies, our results agree well with the semi-classical predictions. Applying our method to a point-mass lens with external shear, we find complex events that have structure at both large and small spectral resolution. In this way, we show that femtolensing may be observable for lenses up to $10^{-11}$ solar masses, much larger than previously believed. Additionally, we discuss the possibility of a search for femtolensing of white dwarfs in the LMC at optical wavelengths. ",
        "introduction": "The possibility of interference effects in gravitational lensing has been considered by several authors (\\cite{Man:81}; \\cite{ss:85}; \\cite{dw:86}; \\cite{pf:91}; \\cite{g:92}; \\cite{spg:93}). Of particular interest are diffractive variations in flux with frequency $\\nu$ when the source, lens, and observer occupy fixed positions. If the difference in time delay between a pair of images satisfies $\\nu\\Delta t\\gg\\gg 1$, the fringe spacing is $\\Delta\\nu\\approx \\Delta t^{-1}\\ll\\nu$. We may call $\\nu\\Delta t\\gg 1$ the ``semi-classical'' regime, because diffractive phenomena are produced by mutually coherent images whose positions, magnifications, and time delays can be determined using geometric optics. On the other hand, if $\\nu\\Delta t\\lesssim 1$, regions of the lens plane other than the geometric-optics images contribute importantly to the flux. The semi-classical approach then breaks down, and one must use the methods of physical optics.  If observed, interference effects would reveal dark-matter objects in a mass range to which few other tests are sensitive. The characteristic time delay produced by a lens of mass $M$ is \\begin{equation} \\label{char_time} \\Delta t(M)=2GM/c^3= R_{\\rm Sch}/c, \\end{equation} where $R_{\\rm Sch}$ is the Schwarzschild radius. Hence the condition $\\nu\\Delta t\\sim 1$ is equivalent to $\\lambda\\sim R_{\\rm Sch}$. Since the Schwarzschild radius of the Sun is $\\approx 3 \\mbox{km}$, broad-band fringes ($\\Delta\\nu\\sim\\nu$) require decidedly sub-stellar but nevertheless macroscopic lensing objects. In particular, Gould (1992) has shown that lens masses $M\\sim 10^{-16}-10^{-13} M_{\\sun}\\sim 10^{17}-10^{20}\\mbox{g}$ could produce observable fringes in gamma-ray burst spectra at energies $E\\sim 1\\mbox{MeV}$ ($\\lambda\\sim 10^{-10}\\mbox{cm}$). Because the angular separation of the images produced by such a lens is $\\sim 10^{-15}$ arc sec, Gould has coined the name ``femtolensing'' for this phenomenon. On the other hand, the probability that a randomly placed and cosmologically distant point-like source should be lensed is $\\sim\\Omega_{\\rm lens}$, where the latter is the mean mass density in lensing objects expressed as a fraction of the critical density $3H_0^2/8\\pi G$ (\\cite{pg:73}). Given that $\\sim 10^3$ gamma-ray bursts have been detected to date (CGRO Science Report 157 1994), a single well-established case of femtolensing would indicate that objects $\\sim 10^{-16}M_{\\sun}$ contribute significantly to the mass density of the universe.  Even if copious lenses exist in the appropriate mass range, visible fringes can be seen only if the (incoherent) source is smaller than the Fresnel length $(\\lambda D)^{1/2}$, where $D$ is the distance. For source redshifts of order unity, this translates to $R_{\\rm source}\\le 10^{14}\\lambda_{\\mbox{cm}}^{1/2}\\mbox{cm}$. An additional constraint requires that in order to have significant magnification, the source size must be smaller than the Einstein ring radius. For cosmological distances, $R_{\\rm source}\\le 5 \\times 10^{8} M/(10^{-16} M_{\\sun})$~cm.  The rapid time variability of gamma-ray bursts (GRB) of 0.2 msec (\\cite{bh:92}) as well as the cosmological distances of $\\sim 0.5$~Gpc for bright BATSE bursts found from Log~$N$-Log~$P$ studies (\\cite{fe1:93}) suggest, however, that {\\it GRB are sufficiently compact}. There has been some confusion as to whether the appropriate linear source size should be taken from $\\gamma c\\Delta t$ or $\\gamma c t$ where $\\gamma$ is the bulk Lorentz factor, $c$ is the speed of light, $\\Delta t$ is the smallest time variation detected, and $t$ is the total event duration. For many proposed cosmological scenarios, the appropriate measure is $\\gamma c\\Delta t$, because the last-scattering-surface remains at approximately the same radius even though the relativistic ejecta may reach quite large distances in the course of the burst. For $\\gamma$'s of 100--300 (e.g. \\cite{fe:93}), the (linear) size of a GRB last-scattering-surface is of order $5\\times 10^{8}$~cm. Other models predict emission from patches on a relativistically expanding shell which can become extremely large ($\\gamma c t$). In the latter case, it would be difficult to observe femtolensing because the interference patterns would differ from patch to patch. An observation of femtolensing could distinguish between the two scenarios.  Stanek, Paczyn\\'nski, \\& Goodman (1993, henceforth SPG) have discussed the possibility that line features in burst spectra may have been produced by femtolensing. Such lines have been seen or inferred in GINGA data (\\cite{Mu:88}, \\cite{fe:88}) and KONUS data (\\cite{Maz:81})  and have been attributed to cyclotron absorption. Like cyclotron lines, interference fringes would be evenly spaced in photon energy. No convincing evidence for lines has yet been seen in the largest homogeneous data set available, the BATSE experiment on the Compton Gamma-Ray Observatory, although its capability of line detection is lower (\\cite{T:93}).  The femtolensing calculations cited above have considered only the simplest possible case, which is an isolated point-mass lens. The simplicity and symmetry of such a lens allow the physical optics problem to be solved in terms of known functions (\\cite{dw:86}). In the present paper, we present an efficient physical-optics method for computing frequency-space fringes produced by general lenses. We assume that the lensing mass distribution is confined to a layer thin compared to the observer-source distance (single-screen approximation). We also neglect time dependence of the lensing geometry, which is permissible if the time-delay difference between any pair of images changes by less than $\\nu^{-1}$ during an observation. The latter assumption is probably justified for femtolensing of gamma-ray bursts (\\cite{g:92}).  For definiteness, and because it is the simplest lens not yet treated in physical optics, we apply our methods to a single point mass with external shear. The computational approach taken, however, would apply equally well to an arbitrary surface density of lensing mass. Computational savings are achieved mainly by taking advantage of the achromaticity of gravitational lensing: that is, the time delays and excess optical path lengths are independent of frequency.  The plan of our paper is as follows. The physical-optics problem is posed in \\S II. We show how the scalar diffraction amplitude at the observer can be determined as a function of frequency by first calculating its Fourier transform, which is a function of time delay. An efficient numerical procedure for finding the latter function is developed in terms of contour integrals on the lens plane. The role of the geometric-optics images and the correspondence with the semi-classical approximation is explained. In \\S III we describe certain details of our numerical implementation of the method and show that in the case of an isolated-point-mass lens, our results are consistent with those already obtained by \\cite{dw:86} and SPG for the isolated-point-mass lens. In \\S IV we present results for the more complex case of a point mass with external shear, which can produce up to four images. Finally, \\S V briefly summarizes the main points of our approach and prospects for observing femtolensing. ",
        "conclusions": "We find that even in the simplified case of a point mass with external shear, very complex interference patterns can be formed. In reality, however, the patterns should be even more complex. In addition to external shear, one should account for local shear from neighbors for femtolensing matter in galactic halos. Complex, many image geometries will result, and as found in the case of a point mass with external shear, images with smaller-than-characteristic difference in time delay (see Eq. \\ref{char_time}) will cause surprisingly long period interference patterns (e.g. Figure \\ref{energy2}) which allow for detection of larger masses. For an external shear of 0.1, we expect these complex interference patterns to occur 3\\% of the time that a source is found inside the Einstein radius. In general, for a halo  mass distribution similar to a singular isothermal sphere, the shear would have a scale length of 5 kpc, so that between 10 and 50 kpc, the shear would be 0.5-0.1, so these complex patterns would occur much more often. Furthermore, there would be a magnification bias (see Appendix C) towards these complex events, which could likely increase the observed fraction by an order of magnitude.  Cosmological gamma-ray bursts are considered the best candidates to show femto-lensing because of their extremely small angular size. If the line of sight to a gamma-ray burst passed through a galaxy, and if the dark halo mass are composed of $\\sim 10^{-11}-10^{-16} M_{\\sun}$ (we increase the range of masses by a factor of 100 as a result of the long period events discussed above) then one could expect to see, in addition to macro or microlensing, femtolensing effects as well. The femtolensing could easily mimic other emission or absorption line processes. We suggest, then, that if a spectral absorption or emission feature is definitively observed in a gamma-ray burst due to femtolensing, that it is likely that the burst would be macrolensed as well. Additionally, macrolensing would produce multiple images (bursts) which would likely be femtolensed and therefore increase the number of expected femtolensed events.  Although the source size requirements exclude the possibility of ever detecting femtolensing in main sequence stars in nearby galaxies, in the future, it may be possible to detect femtolensing of white dwarfs. (We are grateful to B. Paczy\\'nski for pointing this out.) In order for the fringes to have separation of 0.2 eV (a typical optical bandwidth), the lensing masses would be in the range $M \\sim 10^{-9} - 10^{-11} M_{\\sun}$ for short to characteristic time delays, respectively. In the LMC, the maximum magnification possible for such events as determined by the ratio of the Einstein ring to source size would be $\\sim 2-20$ where the larger masses cause the higher magnification. For a white-dwarf source in the LMC observed at 1 eV, the requirement that the source be smaller than the Fresnel length, $(\\lambda D)^{1/2} = 3\\times 10^{9}$ is met ($D=D_{LMC}/2 =25 \\mbox{kpc}$). These events would be quite short ($\\le 1$~minute), due to relative velocities and a small Einstein ring, relative to current microlensing events (e.g. \\cite{p:86}). However, it is not yet possible to monitor white dwarves in the LMC, so one would have to adopt an observing strategy for transients---possibly in the ultraviolet---similar to that of supernovae searches. In contrast to current microlensing studies, detection techniques would be based on anti-correlation of the flux variation in different wavelength bands."
    },
    "9406/astro-ph9406050_arXiv.txt": {
        "abstract": "We present a simple, yet accurate approximation for calculating the cosmic microwave background anisotropy power spectrum in adiabatic models. It consists of solving for the evolution of a two-fluid model until the epoch of recombination and then integrating over the sources to obtain the CMB anisotropy power spectrum. The approximation is useful both for a physical understanding of CMB anisotropies, as well as for a quantitative analysis of cosmological models. Comparison with exact calculations shows that the accuracy is typically better than 20 percent over a large range of angles and cosmological models, including those with curvature and cosmological constant. Using this approximation we investigate the dependence of the CMB anisotropies on the cosmological parameters. We identify six dimensionless parameters that uniquely determine the anisotropy power spectrum within our approximation. CMB experiments on different angular scales could in principle provide information on all these parameters. In particular, mapping of the Doppler peaks would allow an independent determination of baryon mass density, matter mass density and Hubble constant. ",
        "introduction": "Observations of fluctuations in cosmic microwave background (CMB) can provide important constraints on cosmological models. Large angular scale ($>10^0$) observations probe the initial conditions, in particular the amplitude and the slope of primordial power spectrum (\\cite{Smoot92}; \\cite{Gorski94}; \\cite{Wright94}). These scales could also provide information on the geometry and the matter content of the universe (\\cite{Kofman85}; \\cite{Kamio94}; \\cite{Sugiyama94}). However, theoretical interpretation of measurements on these scales is complicated by cosmic variance and this intrinsically limits the accuracy with which these parameters can be estimated using large angular scale measurements alone. While small scale measurements suffer less from cosmic variance, their interpretation is complicated by the microphysics during recombination and/or reionization. Theoretical models often give wildly different predictions for the anisotropy power spectra when the parameter values are only slightly changed, while some combinations of parameters seem to provide nearly identical spectra (e.g. \\cite{Bond94}). The purpose of this Letter is to clarify which combinations of physical parameters affect the CMB fluctuations and what are the physical processes that lead to these fluctuations. Our two-fluid model for adiabatic fluctuations, presented in \\S  2, generalizes previous theoretical approximations that investigated CMB fluctuations in the limiting cases of large and small angles (\\cite{SW66}; \\cite{Jor93}). The model is accurate enough that it can be used for a quantitative analysis of various models, yet it is also simple enough that it can clearly separate between different physical processes that affect the CMB fluctuations. In \\S  3 we use this model to identify the physical parameters that can be determined using CMB measurements over a large angular range, extending previous studies that were limited to a smaller range of angles and/or parameters (\\cite{Bond94}; \\cite{Kamio94b}; \\cite{GS94}; \\cite{Sugiyama94}; \\cite{Muciaccia93}). In  \\S 4 we present the conclusions. ",
        "conclusions": "The approximation for calculating anisotropy power spectrum presented here is a generalization of the Sachs-Wolfe approximation, which itself is only valid on scales larger than the Hubble sphere radius at recombination. By modelling the cosmological perturbations as a two-component fluid plasma we extended this approach to all scales. The approximation is useful both for developing the physical understanding of processes that affect CMB fluctuations, as well as for a quantitative prediction of multipole moments for various cosmological models. The main approximations used in our model are a two-fluid approximation, neglection of anisotropic shear, a simplified treatment of Thomson scattering effects, neglection of curvature effects and neglection of vector and tensor contributions. None of these assumptions is essential for the method and one can generalize the approach presented here to obtain exact results (Seljak 1994). This will lead to a computationally more demanding system of equations, but the main physical effects that lead to the creation of Doppler peaks will still be determined by the equations presented in this Letter.  By rewriting the equations in their dimensionless form we identified all the dimensionless parameters that affect the anisotropy power spectra. Measurements of CMB fluctuations can only determine these parameters. For example, neutrinos enter into our equations indirectly through the Friedmann equation \\ref{friedmann} and through the energy-momentum constraint equations \\ref{constr}. The presence of a massive neutrino only weakly changes these equations and the resultant multipole moments are almost indistinguishable from the ones with the massless neutrino. Therefore, the question of whether neutrino has a mass has little hope to be answered using the CMB measurements.  The most interesting aspect of the CMB power spectra is the peculiar pattern of the Doppler peaks, which allows a simultaneous determination of $\\Omega_bh^2$ and $\\Omega_mh^2$. This would provide an independent test of nuclesynthesis prediction of $\\Omega_bh^2$ (e.g. \\cite{Walker91}) and would also constrain the parameter space on $\\Omega_m$ and $h$. In addition, the position of the first Doppler peak determines $\\Omega_m$ in curvature dominated model. In cosmological constant dominated model the position of the first Doppler peak does not allow one to determine $\\Omega_m$ accurately, but positions of secondary Doppler peaks could be used to constrain $\\Omega_m$ (although for accurate determination exact calculations should be used in this case). Another way to break the degeneracy between $\\Omega_m$, $\\Omega_b$ and $h$ is to determine the Silk damping scale $x_s$, which depends only on these three parameters and cannot be expressed as a combination of $\\Omega_bh^2$ and $\\Omega_mh^2$. This would require a separation of Silk damping from the damping due to the finite thickness of LSS in reionized models and from the $n<1$ suppression of small scales relative to large scales (including the possible tensor contribution). This is possible, because the three effects suppress the small scale power differently. Thus, a combination of CMB measurements over a large range of angles could be used to separately determine the baryon mass density, matter mass density and the Hubble constant value.  I would like to thank Ed Bertschinger for useful discussions. This work was supported by grants NSF AST90-01762 and NASA NAGW-2807."
    },
    "9406/hep-th9406208_arXiv.txt": {
        "abstract": "\\normalsize We present a ``topological'' formulation of arbitrarily shaped vortex strings in four dimensional field theory. By using a large Higgs mass expansion, we then evaluate the effective action of the closed Abrikosov-Nielsen-Olesen vortex string. It is shown that the effective action contains the Nambu-Goto term and an extrinsic curvature squared term with negative sign. We next evaluate the topological $\\FtF$ term and find that it becomes the sum of an ordinary self-intersection number and Polyakov's self-intersection number of the world sheet swept by the vortex string. These self-intersection numbers are related to the self-linking number and the total twist number, respectively. Furthermore, the $\\FtF$ term turns out to be the difference between the sum of the writhing numbers and the linking numbers of the vortex strings at the initial time and the one at the final time. When the vortex string is coupled to fermions, the chiral fermion number of the vortex string becomes the writhing number (modulo $\\bZ$) through the chiral anomaly. Our formulation is also applied to ``global'' vortex strings in a model with a broken global $U(1)$ symmetry. ",
        "introduction": "\\label{sec:Int}  The study of string-like objects has been actively pursued from both theoretical and experimental interests in various fields including condensed matter physics and biology. In particle physics and cosmology, the topological vortex string arising in field theory is one of the most interesting string-like objects. In particular, the Abrikosov-Nielsen-Olesen (ANO) vortex string is the simplest one that splendidly shows typical properties of the vortex string \\cite{NO}. Toward a better understanding of the physics on the vortex string, it is important to examine its geometric and topological properties in four space-time dimensions. We are especially interested in four dimensional extrinsic properties such as the entanglement of the vortex strings. For studying them, we need a systematic formulation of the vortex string. The method used so far in evaluating the effective action of the vortex string in arbitrary shape is based on F\\\"orster's parameterization of coordinates \\cite{Fo} and the collective coordinates method \\cite{GS}. In this method, however, the cut-off dependence of the theory is not so clear and topological structures such as the self-intersection of the world sheet swept by the vortex string cannot be so easily investigated. Therefore, it is desirable to construct a more systematic and efficient formulation satisfying the following: (i) changes of the shapes of the vortex strings can be described, (ii) using perturbative expansions by appropriate parameters such as coupling constants, masses or cut-offs, one can perform systematic approximations, (iii) topological features of the vortex string can be easily examined.  In this paper, we present a ``topological'' formulation which satisfies the above three requirements and apply it to the arbitrarily shaped vortex strings in field theories with broken local or global $U(1)$ symmetries. This is a relativistic generalization of the ``topological'' formulation used in the study of quantized vortices in superfluid helium \\cite{HYAT}. One of the characteristic features in our formulation is the appearance of an antisymmetric tensor field and a so-called topological BF term \\cite{BBRT}. We also adopt a manifestly Lorentz invariant Gaussian-type regularization for the $\\de$-functions in vorticity tensor currents. Using our formulation, we first evaluate the effective action of the ANO vortex string in the Abelian Higgs model, showing that it contains not only the Nambu-Goto term but also an extrinsic curvature squared term with negative sign. Second, we examine the topological $\\FtF$ term, which for example appears as the chiral anomaly and the $\\theta$ term. The evaluation of this term tells us that there are interesting relations between several geometric or topological quantities: Polyakov's self-intersection number, ordinary self-intersection number, total twist number, self-linking number, writhing number and linking number. The expectation value of the $\\FtF$ term turns out to be the sum of Polyakov's self-intersection number and the ordinary self-intersection number of the world sheet swept by the vortex string at the leading order of our approximation. In addition, the $\\FtF$ term can be written as the difference of the sum of the writhing number and the linking number at the final time and the one at the initial time. Furthermore, we discuss the chiral fermion number of the ANO vortex strings in arbitrary shape by using the chiral anomaly and find it to be the sum of the writhing numbers of each vortex string (modulo $\\bZ$). To make the validity of our formulation clearer, we also study the dynamics of vortex strings in a model with a broken global $U(1)$ symmetry. (In cosmology, the former ANO vortex strings are called ``local strings'' and the latter ones are called ``global strings''.) In both models, it is also shown that the large Higgs mass expansions are good approximations. As a whole, it is demonstrated that our ``topological'' formulation is useful to study the ``effective'' vortex strings. (The ``effective'' vortex string means the vortex string remaining after integration over massive fields in field theory.)  Our formulation can be applied to several phenomenologies, although we do not completely discuss them in this paper. First, when we consider grand unified models with extra broken $U(1)$ symmetries, then there can exist vortex strings which are topologically stable. Their string tension is of order a GUT-scale squared or possibly a fundamental string scale squared (because the gauge coupling is smaller than 1). Furthermore, when the models have anomalous $U(1)$ symmetries, it is interesting to investigate whether any fermion number can be violated through the effect of the vortex string. Second, in the Weinberg-Salam theory, there appears the so-called $Z$ string which is equivalent to the ANO vortex string if one neglects other degrees of freedom \\cite{Na,Va}. It is a kind of sphaleron \\cite{KO}, which perhaps seems to be related to the weak-scale baryogenesis through the chiral anomaly \\cite{tH,KRS}. Therefore, the investigation of the vortex string in arbitrary shape is important from the point of view of the fermion number violation. Third, our theory can be directly applied to the cosmic string model \\cite{Vi} and superconductor systems. Finally, it should be noticed that the study of the effective vortex string would give us a new angle in understanding extrinsic properties of fundamental strings in four dimensional space-time.  The paper is organized as follows: in sect. 2 and 3, we study the Abelian Higgs model with the vortex string. First, in sect. 2, we present our ``topological'' formulation and evaluate the effective action of the vortex string. Next, in sect. 3, the $\\FtF$ term is examined and geometric or topological relations are shown. We also discuss the chiral fermion number of the vortex string in arbitrary shape. In sect. 4, we examine a model with the broken global $U(1)$ symmetry. In sect. 5, we give conclusions and compare our results with previous ones. In appendix A, we explain the relation between Polyakov's self-intersection number and the total twist number. In appendix B, we derive the relation between the intersection number and the linking number. ",
        "conclusions": "In the preceding sections, we have developed the ``topological'' formulation which allows the systematic analysis of the effective vortex string in arbitrary shape and have applied to the Abelian Higgs model and the model with a broken global $U(1)$ symmetry. Using our formulation, in particular, we have evaluated the effective action of the vortex string and the expectation value of the topological $\\FtF$ term. As a result, many geometric and topological quantities concerning the vortex string have been derived. {}From the effective action of the ANO vortex string including the Nambu-Goto term and the extrinsic curvature squared term with negative sign, one can realize the motion of the vortex string, which indicates that the vortex string prefers curving as far as it is smooth enough. Furthermore, we have found that the ANO vortex string has a non-zero chiral fermion number related to the writhing number (modulo $\\bZ$) and have suggested that interesting phenomena such as the anomalous fermion production might occur through intersection processes of the ANO vortex string. It should be emphasized that the chiral fermion number of the ANO vortex string depends on its ``shape''. In addition, we have shown remarkable relations between $I$, $PS_{i}$, $SI_{n}$, $T_{w}$, $SL_{k}$ and $W_{r}$ in (\\ref{eqn:I}), (\\ref{eqn:PSi-Tw}), (\\ref{eqn:SIn-SLk}), (\\ref{eqn:SLk-Tw}), (\\ref{eqn:DWr}), (\\ref{eqn:Wr+Lk}) and (\\ref{eqn:I-Wr+Lk}). They must be useful themselves in studying geometric or topological properties of the vortex string and the role of the $\\th$ term.  In our ``topological'' formulation, there have appeared non-zero extrinsic quantities of the ANO vortex string such as the extrinsic curvature squared term, Polyakov's self-intersection number and the writhing number. On the other hand, it was argued that there appears no extrinsic curvature squared term in the effective action of the ANO vortex string in the formulation based on F\\\"orster's parameterization of coordinates \\cite{Gr}, which we call the F\\\"orster-Gregory (FG) formulation. Furthermore, if one evaluates the topological $\\FtF$ term by using the FG formulation, this term turns out to be zero at least at the leading order. This is due to the fact that the electric field vanishes in the static ANO vortex solution, which is applied at the leading order in the FG formulation. Note that $\\FtF\\approx\\bB\\cdot\\bE$, where $\\bB$ is a magnetic field and $\\bE$ an electric field. In our formulation, as we have said, we have found the $\\FtF$ term to take a non-zero value at the leading order. It is not easy to compared the FG formulation with ours, because in the FG formulation the equation of motion is used instead of the path integral representation which we have adopted. However, these discrepancies between the FG formulation and ours likely come from the differences between the parameterizations of coordinates and the regularizations of the vortex core in each formulation.  We would like to stress the following points. (i) In our formulation, the Lorentz invariance and the conservation of the vorticity are manifestly satisfied all the time. Indeed, we have used the Lorentz invariant Guassian-type regularization for the $\\de$-function in the vorticity tensor current. Furthermore, the static ANO vortex solution, which is not Lorentz invariant at first sight, is not used at all. (ii) Our perturbative calculation (e.g. by the large Higgs mass expansion) is systematic and efficient. In addition, the path integral representation is convenient to evaluate physical quantities on the vortex string in a systematic manner. (iii) Our formulation can be applied to the model with a broken global $U(1)$ symmetry as shown in sect. 4, while it is difficult to adopt the FG formulation for that model. (iv) Using our formulation, the dynamics of quantized vortices in superfluid can be examined \\cite{HYAT}. In this case, the effective action of the vortex string turns out to be of the same form as the action of a vortex string in an incompressible perfect fluid, so that we can explain the phenomena in experiments on the quantized vortex by applying our formulation. On the grounds mentioned above, our ``topological'' formulation is reliable.  Our formulation can be directly applied to superconductor systems, the cosmic string model and grand unified models with extra $U(1)$ symmetries. In particular, it is of interest to investigate the possibility of the fermion number violation by the vortex string in more detail in various cases including the Weinberg-Salam theory. It will be also suggestive to examine strings and gravity from the point of view of the effective string.  \\vskip1cm \\centerline{\\large\\bf Acknowledgements} We would like to thank our colleagues in Kyoto University for encouragement.  \\appendix \\newpage \\noindent \\begin{center} {\\Large {\\bf Appendices}} \\end{center}"
    },
    "9406/hep-ph9406319_arXiv.txt": {
        "abstract": "We explore constraints on the spectral index $n$ of density fluctuations and the neutrino energy density fraction $\\Omega_{HDM}$, employing data from a variety of large scale observations.  The best fits occur for $n\\approx 1$ and $\\Omega_{HDM} \\approx 0.15 - 0.30$, over a range of Hubble constants $40-60$ km s$^{-1}$ Mpc$^{-1}$.  We present a new class of inflationary models based on realistic supersymmetric grand unified theories which do not have the usual `fine tuning' problems.  The amplitude of primordial density fluctuations, in particular, is found to be proportional to $(M_X /M_P)^2$, where $M_X (M_P)$ denote the GUT (Planck) scale, which is reminiscent of cosmic strings!  The spectral index $n = 0.98$, in excellent agreement with the observations provided the dark matter is a mixture of `cold' and `hot' components. ",
        "introduction": " ",
        "conclusions": ""
    },
    "9406/hep-ph9406270_arXiv.txt": {
        "abstract": "The standard model satisfies Sakharov's conditions for baryogenesis but the CP violation in the KM matrix appears too small to account for the observed asymmetry. In this letter we explore a mechanism through which CP violation can be greatly amplified. First CP is {\\it spontaneously} broken through dynamical effects on bubble walls, and the two CP conjugate phases grow through phase ordering. Direct competition between macroscopic regions of both phases then amplifies the microscopic CP violation to a point where one of the two phases predominates. This letter is devoted to a demonstration that spontaneous CP violation may indeed occur on propagating bubble walls via the formation of a condensate of the longitudinal $Z$ boson. \\\\ ",
        "introduction": " ",
        "conclusions": ""
    },
    "9406/hep-ph9406308_arXiv.txt": {
        "abstract": "By coupling axions strongly to a hidden sector, the energy density in coherent axions may be converted to radiative degrees of freedom, alleviating the ``axion energy crisis''. The strong coupling is achieved by mixing the axion and some other Goldstone boson through their kinetic energy terms, in a manner reminiscent of paraphoton models. Even with the strong coupling it proves difficult to relax the axion energy density through particle absorption, due to the derivative nature of Goldstone boson couplings and the effect of back reactions on the evolution of the axion number density. However, the distribution of other particle species in the hidden sector will be driven from equilibrium by the axion field oscillations. Restoration of thermal equilibrium results in energy being transferred from the axions to massless particles, where it can redshift harmlessly without causing any cosmological problems. ",
        "introduction": "Goldstone and pseudo--goldstone bosons play a large role in the speculations of particle physicists and cosmologists.  Some oft--discussed examples are axions,$^1$ majorons,$^2$ familons,$^3$ and the non--abelian goldstone fields of texture models.$^4$  The salient feature of the dynamics of goldstone bosons is that they couple derivatively.  If the associated global continuous symmetry is broken at a scale $f$, then the coupling always involves powers of $f^{-1}\\partial_\\mu b$, where $b$ is the goldstone boson.  The consequence of this is that if $f$ is very large compared to the relevant masses and momenta, the goldstone boson decouples.  This fact is exploited in invisible axion models,$^5$ where for $f_a \\stackrel{_>}{_\\sim} 10^{10}\\GeV$ the axion's couplings are small enough for it to have evaded detection and avoided conflict with astrophysical observations.$^6$  This fact also lies behind the well--known ``axion energy problem'',$^7$ which is that if $f_a \\stackrel{_>}{_\\sim} 10^{12}\\GeV$ the energy in coherent oscillations of the axion field after the QCD phase transition in the early universe is unable to dissipate due to the axion's weak coupling and eventually overcloses the universe.  In this letter we show that different goldstone fields can mix with each other through kinetic terms in the Lagrangian in a way reminiscent of photon--paraphoton mixing,$^8$ and that a goldstone boson can thus acquire effective couplings at low energy to certain fields which are much larger than the typical $p/f$.  After demonstrating this point we show how it makes possible scenarios in which the energy in coherent oscillations of the cosmological axion field can be dissipated and the upper limit on $f_a$ removed. ",
        "conclusions": "We have established that it is possible to dissipate the primordial energy density in axions through their interactions with other particles. A key feature of the process we envision is to mix the axion with some other goldstone boson $b$, whose decay constant $f_b$ is much smaller than $f_a$. The mixing may be large, and this allow axions to couple strongly to `$b$ matter'. By adjusting the matter content of the $b$-sector we can arrange for the axion oscillations to drive oscillations in the energy of $b$ matter. This pushes the $b$ matter out of thermal equilibrium, and ultimately allows the axion energy density to dissipate in the $b$-sector.  Although, we have constructed a model whereby the axions dissipate, the addition of new particles may have other troublesome consequences, chief of which is that $b$ matter may itself come to dominate the universe. The easiest way to avoid this is to adjust the masses and couplings of the $b$-sector so the real part of the $\\Phi$ field is the lightest $b$-particle other than the $b$ goldstone boson itself. Then as the universe cools, first all $b$ matter will end up as $\\rho_b$'s which will then decay with a rate of order $f_b$ into $b$ goldstone bosons.  We argued earlier that it is difficult to dissipate axion energy through the process of single axion absorption, yet, in the present scenario axion energy is indeed absorbed. It is natural to ask if there is a ``Feynman diagram'' explanation of our process. The answer is yes. Imagine a process whereby a single axion is absorbed from the coherent state, as in Fig. (3a) and a second process whereby two axions are absorbed, as in Fig. (3b). The ratio of the rates for these two processes is \\begin{equation} R = \\Gamma_2/\\Gamma_1 \\sim n_a {{\\epsilon^2 m_a^2} \\over {f_b^2}} ~, \\end{equation} where $n_a$ is the occupation number for the state. As stated earlier the effective single axion absorption rate is reduced by a factor of $m_a /T$ when back reactions are included. However, this is also true of the two axion absorption rate, and so the ratio in Eq. (21) is correct even in the presence of backreactions. Now, for the coherent state, $n_a$ is approximated by $n_a \\sim f_a^2/H^2$, so the ratio becomes $R \\sim \\epsilon^2 M_{pl}^2/\\Lambda_{QCD}^2$, which is quite large. This leads to the situtation that two axion absorption formally exceeds one axion absorption. The three axion rate would be even larger... In this situation what one must do is sum over diagrams with any number of axions attached to the particles participating in the scattering process. This is equivalent to solving for the propagator of these particles in the presence of the coherent oscillating axion field, as we have done in this paper.  Along these lines, even in a normal axion model with no mixing each additional axion brings a factor of $M_{pl}/f_a$ to the amplitude, so here also one should not calculate individual absorption rates. Rather, one should calculate the propagators for the other particles, including the time dependent axion field, to arrive at the axion dissipation rate. We are presently considering the question of what dissipation may result from the non-derivative couplings to mesons, and plan to present results in a separate paper. Although derivative couplings may seem important too, one must remember that such couplings are less effective at damping the non-relativistic coherent axion oscillations due to the factor of $m_a$ in the coupling.  Besides damping the coherent oscillations in the early universe, the kind of mixing of axions with other goldstone bosons we have been discussing could lead to other interesting effects.  In particular, it is possible now to contemplate that even invisible axions with $f_a \\sim M_{GUT}$ or $M_{Pl}$ can have sizable couplings to some kinds of matter, perhaps even to particles that carry standard model gauge charges.  This would mean that the ``invisible axion'' may not be quite as invisible as was thought.  Also worth further investigation is whether similar mixing in familiar majoron, familon or texture models could lead to interesting phenomena."
    },
    "9406/nucl-th9406003_arXiv.txt": {
        "abstract": "We develop the Redfield equation for delta-correlated gaussian noise and apply it to the case of two neutrino flavor or spin precession in the presence of a noisy matter density or magnetic field, respectively. The criteria under which physical fluctuations can be well approximated by the delta-correlated gaussian noise for the above cases are examined. Current limits on the possible neutrino magnetic moment and solar magnetic field suggest that a reasonably noisy solar magnetic field would not appreciably affect the solar electron neutrino flux. However, if the solar electron density has fluctuations of a few percent of the local density and a small enough correlation length, the MSW effect is suppressed for a range of parameters. ",
        "introduction": "\\indent  Neutrino oscillations in the presence of matter and magnetic fields have been an area of intense study for approximately the last ten years. In the Mikheyev - Smirnov - Wolfenstein (MSW) effect, electron neutrinos on their journey from the core, are resonantly transformed into muon or tau neutrinos \\cite{msw,us}. If neutrinos are Majorana fermions with transition magnetic moments, they can undergo a magnetic resonant transformation into muon or tau antineutrinos \\cite{lam}. Neutrinos from a supernovae explosion can be transformed from one flavor to another as they pass through the outer part of the star \\cite{pant}. In many stellar situations, the matter density and/or magnetic fields may fluctuate about a mean value. Some well known examples where fluctuations are likely to exist include the magnetic field and the matter density in the solar convective zone and also the turbulence of the post-supernovae matter which has been blown off by the explosion.  A general approach to the neutrino oscillations in inhomogeneous matter was developed in Ref. \\cite{sawyer}. A Study of matter fluctuations which are not random, but harmonic \\cite{koonin,wick}, or occur as a jump-like change in the solar density \\cite{denvar}, are available in the literature. Matter currents and density changes effect neutrino flavor oscillations in a similar way and have also been examined  in Ref. \\cite{wick}. Although matter current effects become important only if the velocity is somewhat close to the speed of light, noisy mixing of matter would also mimic a fluctuating matter density. A priori, fluctuations in such fields may be well approximated by random noise added to an average value. In this paper we show how such noise will affect neutrino oscillations for the situation in which the correlation length of the randomly fluctuating part of either the matter density or the magnetic field is small compared with the neutrino oscillation length.  The case of neutrino spin precession in a noisy magnetic field was considered by Nicolaidis for neutrinos in vacuum \\cite{nic}. The noise was taken to be well approximated by a delta-correlated gaussian distribution with the result that the normal oscillations become damped with a relaxation time of $t_{rel} = (2\\mu^2 <B_r^2> \\tau_c)^{-1}$, where $B_r$ is the randomly fluctuating part of the magnetic field and $\\tau_c$ is its correlation time. Enqvist and Semikoz considered neutrino oscillations in randomly fluctuating magnetic field, approximately, by averaging the coefficients in the third order differential equation for the z-component of the neutrino spin, in a constant matter density \\cite{enq}. Their result suggested that the effect of the randomly fluctuating part of the magnetic field was similar to the effect of a larger (constant) matter density, hence reducing the neutrino precession. Our result is quite different; namely that both a noisy magnetic field and a noisy density (for a constant averaged density) act to depolarize the neutrinos. For a noisy magnetic field, if the probability of transition is greater than one half without the random fluctuations, the inclusion of the random fluctuations will reduce the transition probability, and if the transition probability without the random fluctuations is less than one half, the inclusion of the fluctuations will increase the transition probability. If the randomly fluctuating part of the magnetic field is strong enough or is allowed to act for a long enough time, a complete depolarization of particles-antiparticles will occur. For a noisy density, we find that the MSW transition probability is suppressed. For the case of strongly adiabatic MSW transitions and large fluctuations, the averaged transition probability saturates at one half.  In Section II, we develop the equations governing the time evolution of the averaged probabilities of being found in the N${th}$ level of an N-level system, subject to random and non-random potentials. These equations are derived for the case where the randomly fluctuating part of the field is taken to be a delta-correlated gaussian. In Section III, we show several analytically solvable examples for the case of a two level problem, and numerically examine cases the analytical solutions of which are not instructive. In Section IV, we investigate the conditions under which a real, physical fluctuating field will be well approximated by the equations developed for the delta-correlated gaussian case. We then apply these conditions and examine the cases of spin-flavor and flavor precession of neutrinos in the sun. Section V presents a discussion of results and our conclusions. ",
        "conclusions": "\\indent  We have derived a Redfield differential equation for the time dependence of averaged values of functions of the probability of finding an N level system in the N$^{th}$ level after being subjected to a randomly fluctuating field, of the delta-correlated gaussian type. This formalism applies as an approximation to the case where the fluctuating field can be described by  a finite correlation time. This approximation is valid if the product of the correlation time and the energy scale of the Hamiltonian is small. Upon applying this to neutrino flavor or spin-flavor precession, we have shown that the probability will relax eventually to a value of one half, if the neutrino spends a sufficiently long time in a medium with randomly fluctuating matter density or a randomly fluctuating magnetic field. In the case of a fluctuating magnetic field, the relaxation is independent of the matter density, assuming the correlation time of the field fluctuations is small compared to the neutrino oscillation length in matter. In the case of fluctuations added to a constant matter density, the probability again will eventually relax to a value of one half, but the relaxation time is greatly increased if one is far from the resonant condition.  We have also examined the case of neutrino spin-flavor precession in the sun, for a purely random magnetic field and no flavor mixing. It appears that the current limits on the neutrino magnetic moment and the guesses concerning the maximum values of the solar magnetic field, combine to give only a small effect on the average electron neutrino flux. When there is flavor mixing and a purely random field, the combined average probabilities of the neutrinos is again a simple exponentially decreasing function and therefore the results of a purely random field and no flavor mixing apply to this case as well. For the case of a randomly fluctuating electron density, the MSW effect can be strongly suppressed for rms fluctuations of 4 \\% of the local electron density. However, this requires correlation lengths of about 40km, and seems to give a significant effect only for neutrinos which have their MSW resonant transition deep in the sun.  In spite of these problems, we believe that a numerical study, which should give similar results for correlation times which do not badly violate Eq. (35), may bring to light many interesting effects. Implications of the density fluctuations discussed here on stellar collapse and supernova dynamics will be published elsewhere \\cite{fl}. \\vskip .2in \\centerline{\\bf ACKNOWLEDGMENTS} \\vskip .2in We thank to G. Fuller, W. Haxton, and Y. Qian for very useful discussions. This research was supported in part by the U.S. National Science Foundation Grant No. PHY-9314131 and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. F. N. L.'s research was supported in part by a grant from Mr. E. J. Loreti. We would also like to thank the Institute for Nuclear Theory at the University of Washington for its hospitality and the Department of Energy for partial support during the completion of this work. \\vskip .2in \\centerline{\\bf Appendix } \\vskip .2in \\indent  For $\\hat H_0(t)$ given in Eq. (5), one selects the element of largest value during the time the system is in the presence of the random field. Let this largest value be $E_{max}$. We consider the case where $ \\tau E_{max} \\ll 1$. We take $f_{ij}$ in Eqs. (10) and (11) to be given by, \\begin{equation} f_{ij} = \\theta( \\tau - |t_i - t_j| ), \\end{equation} where again the average of odd products vanish. We rewrite the average of Eq. (9) as, \\begin{equation} <\\hat {\\rho}_I(t)> = \\hat {\\rho}_0 + <\\hat {\\rho}_I^{(2)}(t)> + <\\hat {\\rho}_I^{(4)}(t)> + <\\hat {\\rho}_I^{(6)}(t)> + \\cdots \\end{equation} where, \\begin{eqnarray} <\\hat {\\rho}_I^{(2N)}(t)> &=&  (-1)^N \\alpha^{2N} \\int_0^t dt_{2N} \\int_0^{t_{2N}} dt_{2N-1}\\cdots \\nonumber \\\\ &\\phantom{=}&+ \\int_0^{t_1} dt_1 F_{2N} [\\hat M(t_{2N}), [\\hat M(t_{2N-1}), \\cdots \\hat M(t_1)]\\cdots ]] \\end{eqnarray} and, \\begin{equation} F_{2N} = \\sum_{n_1\\cdots n_{2N}}^{P(1,2,\\cdots ,2N)} f_{n_1n_2}f_{n_3n_4} f_{n_5n_6}\\cdots f_{n_{2N-1}n_{2N}}, \\end{equation} where, P(1,2,$\\cdots$ ,2N) means all permutations. We explicitly show the second and third terms: \\begin{eqnarray} &\\phantom{=}&<\\hat {\\rho}_I^{(2)}(t)>\\ \\sim -\\alpha^2 \\int_0^t dt_1 [\\hat M(t_1), \\int_{t_1-\\tau}^{t_1}[\\hat M(t_1) + {d\\hat M(t_1) \\over dt} (t_2 - t_1), \\hat {\\rho}_0]] \\phantom{oiueoe} \\nonumber \\\\ &\\sim & -\\alpha^2 \\tau \\int_0^t dt_1 [\\hat M(t_1),[\\hat M(T_1), \\hat {\\rho}_0]] + \\alpha^2 {\\tau^2 \\over 2} \\int_0^t dt_1 [\\hat M(t_1), [{d\\hat M(t_1) \\over dt} , \\hat {\\rho}_0]], \\end{eqnarray} where $ d\\hat M(t_1)/dt = i \\hat U_0^{\\dagger} [\\hat H_0, \\hat M^{\\prime}] \\hat U_0 $ and is therefore proportional to $E_{max}$. In the third term one has a sum of three products of two f's, ($ f_{12}f_{34} + f_{13}f_{24} + f_{14}f_{32}$) only the first of which has the times in the order of the times appearing in the nested integrals. The first of these three terms gives, \\begin{eqnarray} <\\hat {\\rho}_I^{(4)}(t)>_{12,34} &\\sim& \\alpha^4 \\tau^2 \\int_0^t dt_1 [\\hat M(t_1), \\bigl\\{ [\\hat M(t_1), \\int_0^{t_1} dt_2 [\\hat M(t_2), [\\hat M(t_2),\\hat {\\rho}_0]]]] \\phantom{oioo} \\nonumber \\\\ &-& {\\tau \\over 2}\\bigl( [{d\\hat M(t_1) \\over dt},\\int_0^{t_1}dt_2 [\\hat M(t_2), [\\hat M(t_2), \\hat {\\rho}_0]]]] \\nonumber \\\\ &+& [\\hat M(t_1),\\int_0^{t_1}dt_2 [\\hat M(t_2),[{d\\hat M(t_1) \\over dt}, \\hat {\\rho}_0]]]] \\\\ &\\phantom{+}& + [\\hat M(t_1),[\\hat M(t_1),[\\hat M(t_1),\\hat {\\rho}_0]]]] \\bigr) \\bigr\\} + O(\\tau^4). \\nonumber \\end{eqnarray} The last term, however, is not proportional to $E_{max}$. This will cause a problem since it could contribute to the next order term of the average . It turns out that it cancels the largest term coming from the remaining two f products. That these remaining f-products are of largest order $\\tau^3$ can be seen by noting that when the argument of the theta function in an f connects two non-sequential times, the intermediate time(s) must also be within $\\tau$ of the larger time in the theta function. For example, the remaining two contributions to $<\\hat {\\rho}_I^{(4)}(t)>$ are, \\begin{eqnarray} <\\hat {\\rho}_I^{(4)}(t)>_{13,24} &+& <\\hat {\\rho}_I^{(4)}(t)>_{14,23} = \\phantom{oiueoi}\\nonumber \\\\ &\\phantom{=}& \\alpha^4 \\int_0^t dt_1 [\\hat M(t_1),[\\hat M(t_1),[\\hat M(t_1),[\\hat M(t_1), \\hat {\\rho}_0]]]] \\nonumber \\\\ &\\times& \\int_{t_1-\\tau}^{t_1} dt_2 \\int_{t_1-\\tau}^{t_2} dt_3 \\Bigl( \\int_{t_2-\\tau}^{t_3} dt_4  + \\int_{t_1-\\tau}^{t_3} dt_4 \\Bigr) \\\\ &=& \\alpha^4 {\\tau^3 \\over 2} \\int_0^t dt_1 [\\hat M(t_1),[\\hat M(t_1),[\\hat M(t_1),[\\hat M(t_1),\\hat {\\rho}_0]]]]. \\nonumber \\end{eqnarray} This feature appears to continue throughout each term in the entire expression. Therefore, if $\\tau E_{max} \\ll 1$, and neglecting terms of order $\\tau E_{max}$ and smaller, \\begin{eqnarray} <\\hat {\\rho}_I^{(2N)}(t)> &=& (-1)^N \\alpha^{2N} \\tau^N \\int_0^t dt_{2N} [\\hat M(t_{2N}),[\\hat M(t_{2N}), \\phantom{oiueoiueoieoie}\\nonumber \\\\ &\\phantom{=}& \\int_0^{t_{2N}} dt_{2N-2} [\\hat M(t_{2N-2},[\\hat M(t_{2N-2}), \\int_0^{t_{2N-2}} \\cdots \\\\ &\\phantom{+}& \\int_0^{t_4} dt_2 [\\hat M(t_2),[\\hat M(t_2), \\hat {\\rho}_0]]\\cdots ]]]]. \\nonumber \\end{eqnarray} which leads to Eqs. (15) \\& (16).  In the above derivation we have assumed $\\alpha $ to be a constant. If $\\alpha$ depends on time, the condition, $\\tau E_{max} \\ll 1$ , should be replaced by the condition, \\begin{equation} \\tau [ (d/dt \\log \\alpha(t))^2 + E_{max}^2]^{1\\over 2} \\ll 1. \\end{equation} \\vfill \\eject"
    }
}