String Density Research Articles

BPS dynamics of the triple ( p, q) string junction

We study the dynamics of the triple junction of ( p, q) strings in Type IIB string theory. We probe the tension and mass density of ( p, q) strings by studying harmonic fluctuations of the triple junction. We show that they agree perfectly with the BPS formula provided a suitable geometric interpretation of the junction is given. We provide a precise statement of the BPS limit and force-balance property. At weak coupling and sufficiently dense limit, we argue that a ( p, q) string embedded in the string network is a ‘wiggly string’, whose low-energy dynamics can be described via a renormalization group evolved, smooth effective non-relativistic string. We also suggest the possibility that, upon Type IIB strings being promoted to the M-theory membrane, there can exist ‘evanescent’ bound-states at the triple junction in the continuum.

Thermodynamics of Cosmic String Densities in U(1) Scalar Field Theory

We present a full characterization of the phase transition in U(1) scalar field theory and of the associated vortex string thermodynamics in 3D. We show that phase transitions in the string densities exist and measure their critical exponents, both for the long string and the short loops. Evidence for a natural separation between these two string populations is presented. In particular, our results strongly indicate that an infinite string population will only exist above the critical temperature. Canonical initial conditions for cosmic string evolution are shown to correspond to the infinite temperature limit of the theory.

Cosmic string formation from correlated fields

We simulate the formation of cosmic strings at the zeros of a complex Gaussian field with a power spectrum $P(k)\ensuremath{\propto}{k}^{n}$, specifically addressing the issue of the fraction of length in infinite strings. We make two improvements over previous simulations: we include a nonzero random background field in our box to simulate the effect of long-wavelength modes, and we examine the effects of smoothing the field on small scales. The inclusion of the background field significantly reduces the fraction of length in infinite strings for $n<\ensuremath{-}2$. Our results are consistent with the possibility that infinite strings disappear at some ${n=n}_{c}$ in the range $\ensuremath{-}3<~{n}_{c}<\ensuremath{-}2.2$, although we cannot rule out ${n}_{c}=\ensuremath{-}3$, in which case infinite strings would disappear only at the point where the mean string density goes to zero. We present an analytic argument which suggests the latter case. Smoothing on small scales eliminates closed loops on the order of the lattice cell size and leads to a ``lattice-free'' estimate of the infinite string fraction. As expected, this fraction depends on the type of window function used for smoothing.

The string density of states from the convolution theorem

We study the microcanonical density of states and the thermal properties of a bosonic string gas starting from a calculation of the Helmholtz free energy in the S-representation. By adding more and more strings to the single string system, we induce that, for infinite volume, there is no negative specific heat region but a transition at a finite value of the energy per string from the low energy regime to a region of infinite specific heat at the Hagedorn temperature. Forcing the description of this phase in terms of strings gives a picture in which there is a very fat string in a sea of low energetic ones. We argue that the necessary changing of this description should not change the fact that perturbatively T H is a maximum temperature of the system.

Linking dynamical and thermal models of ultrarelativistic nuclear scattering.

To analyze ultrarelativistic nuclear interactions, usually either dynamical models like the string model are employed, or a thermal treatment based on hadrons or quarks is applied. String models encounter problems due to high string densities; thermal approaches are too simplistic, considering only average distributions and ignoring fluctuations. We propose a completely new approach, providing a link between the two treatments, and avoiding their main shortcomings: based on the string model, connected regions of high energy density are identified for single events, such regions being referred to as quark matter droplets. Each individual droplet hadronizes instantaneously according to the available $n$-body phase space, making use of special Monte Carlo techniques.

String field theory in curved spacetime and the resolution of spacelike singularities

We attempt to understand the fate of spacelike gravitational singularities in string theory via the quantum stress tensor for string matter in a fixed background. We first approximate the singularity with a homogeneous anisotropic background and review the minisuperspace equations describing the evolution of the scale factors and the dilaton. We then review and discuss the behaviour of large strings in such models. In a simple model which expands isotropically for a finite period of time we compute the number density of strings produced by quantum pair production and find that this number, and thus the stress tensor, becomes infinite when the Hubble volume of the expansion exceeds the string scale, in a manner reminiscent of the Hagedorn transition. Based on this calculation we argue that either the region near the singularity undergoes a phase transition when the density reaches the order of a string mass per string volume, or that the backreaction of the produced string matter dramatically modifies the geometry.

Statistical properties of strings

We investigate numerically the configurational statistics of strings. The algorithm models an ensemble of global U(1) cosmic strings, or equivalently vortices in superfluid 4He. We use a new method which avoids the specification of boundary conditions on the lattice. We therefore do not have the artificial distinction between short and long string loops or a “second phase” in the string network statistics associated with strings winding around a toroidal lattice. Our lattice is also tetrahedral, which avoids ambiguities associated with the cubic lattices of previous work. We find that the percentage of infinite string is somewhat lower than on cubic lattices, 63% instead of 80%. We also investigate the Hagedorn transition, at which infinite string percolate, controlling the string density by rendering one of the equilibrium states more probable. We measure the percolation threshold, the critical exponent associated with the divergence of a suitably defined susceptibility of the string loops, and that associated with the divergence of the correlation length.

Hot string soup: Thermodynamics of strings near the Hagedorn transition.

Above the Hagedorn energy density closed fundamental strings form a long string phase. The dynamics of weakly interacting long strings is described by a simple Boltzmann equation which can be solved explicitly for equilibrium distributions. The average total number of long strings grows logarithmically with total energy in the microcanonical ensemble. This is consistent with calculations of the free single string density of states provided the thermodynamic limit is carefully defined. If the theory contains open strings the long string phase is suppressed.

Back reaction of strings in self-consistent string cosmology

We compute the string energy-momentum tensor and {\bf derive} the string equation of state from exact string dynamics in cosmological spacetimes. $1+1,~2+1$ and $D$-dimensional universes are treated for any expansion factor $R$. Strings obey the perfect fluid relation $ p = (\gamma -1) \rho $ with three different behaviours: (i) {\it Unstable} for $ R \to \infty $ with growing energy density $ \rho \sim R^{2-D} $, {\bf negative} pressure, and $ \gamma =(D-2)/(D-1) $; (ii){\it Dual} for $ R \to 0 $, with $ \rho \sim R^{-D} $, {\bf positive} pressure and $\gamma = D/(D-1) $ (as radiation); (iii) {\it Stable} for $ R \to \infty $ with $ \rho \sim R^{1-D} $, {\bf vanishing} pressure and $\gamma = 1 $ (as cold matter). We find the back reaction effect of these strings on the spacetime and we take into account the quantum string decay through string splitting. This is achieved by considering {\bf self-consistently} the strings as matter sources for the Einstein equations, as well as for the complete effective string equations. String splitting exponentially suppress the density of unstable strings for large $R$. The self-consistent solution to the Einstein equations for string dominated universes exhibits the realistic matter dominated behaviour $ R \sim (X^0)^{2/(D-1)}\; $ for large times and the radiation dominated behaviour $ R \sim (X^0)^{2/D}\; $ for early times. De Sitter universe does not emerge as solution of the effective string equations. The effective string action (whatever be the dilaton, its potential and the central charge term) is not the appropriate framework in which to address the question of string driven inflation.

Remarks on thermal strings outside black hole

We discuss the semiclassical approximation to the level density of (super) strings propagating in non-compact coset manifolds G H . We show that the WKB ansatz agrees with heuristic red-shift arguments with respect to the “exact” sigma-model metric, up to some deviations from minimal coupling, parametrized by the dilaton. This approximation is used to study thermal ensembles of free strings in black holes, with the “brick wall” horizon regularization of 't Hooft. In two dimensions the entropy diverges logarithmically with the horizon thickness, and a local Hagedorn transition occurs in higher dimensional models. We also observe that suppersymmetry improves the regularity of strings art the horizon

String cosmology in a Bianchi type I universe

We study the dynamics of a cloud of massive strings in the context of relativity. The general solution to the gravitational field is obtained in a parametrical form. Solutions are evaluated, for larget, for two equations of states for string density. Its is shown that, as strings disappear leaving only particles, the mean anisotropy vanishes, too

Phantom studies simulating the impact of trabecular structure on marrow relaxation time, T2'.

Phantom studies were conducted to investigate the impact of trabecular structure on the T2' signal measured by MRI. For a separation of density from structural effects, several phantoms were built. They consisted of parallel polyethylene strings arranged in a variety of different patterns to simulate a) a constant uniform trabecular distribution with increasing trabecular thickness and b) different structures with identical overall trabecular density. An asymmetric spin echo sequence was used to determine the apparent relaxation time T2'. Changes in T2' are induced by susceptibility differences between the polyethylene strings simulating trabeculae and Gd-DTPA doped saline simulating bone marrow. The results showed an increasing T2' decay rate with a) decreasing spacing while the string density was constant and b) with increasing string density while the string arrangement was constant. The results demonstrate that the T2' signal is affected not only by density but also by spatial distribution. However, the results also indicate that a separation of the two effects is not possible from a T2' measurement alone, but that e.g., a matching CT slice that would provide purely density information is additionally needed. Theoretical simulations confirmed these results.

Evolution of cosmic string configurations.

We extend and develop our previous work on the evolution of a network of cosmic strings. The new treatment is based on an analysis of the probability distribution of the end-to-end distance, or extension, of a randomly chosen segment of left-moving string of given length. The description involves three distinct length scales: \ensuremath{\xi}, related to the overall string density, \ensuremath{\xi}\ifmmode\bar\else\textasciimacron\fi{}, the persistance length along the string, and \ensuremath{\zeta}, describing the small-scale structure, which is an important feature of the numerical simulations that have been done of this problem. An evolution equation is derived describing how the distribution develops in time due to the combined effects of the universal expansion, of intercommuting and loop formation, and of gravitational radiation. With plausible assumptions about the unknown parameters in the model, we confirm the conclusions of our previous study that if gravitational radiation and small-scale structure effects are neglected the two dominant length scales both scale in proportion to the horizon size. When the extra effects are included, we find that while \ensuremath{\xi} and \ensuremath{\xi}\ifmmode\bar\else\textasciimacron\fi{} grow, \ensuremath{\zeta} initially does not. Eventually, however, it does appear to scale, at a much lower level, due to the effects of gravitational back reaction.

Compact QED in the Landau gauge: A lattice-gauge-fixing case study.

We derive different representations of compact QED fixed to Landau gauge by the lattice Faddeev-Popov procedure. Our analysis finds that (A)Nielsen-Olesen vortices arising from the compactness of the gauge-fixing action are {\it quenched\/}, that is, the Faddeev-Popov determinant cancels them out and they do not influence correlation functions such as the photon propagator; (B)Dirac strings are responsible for the nonzero mass pole of the photon propagator. Since in $D=3+1$ the photon mass undergoes a rapid drop to zero at $\beta_c$, the deconfinement point, this result predicts that Dirac strings must be sufficiently dilute at $\beta > \beta_c$. Indeed, numerical simulations reveal that the string density undergoes a rapid drop to near zero at $\beta\sim \beta_c$.

Coarsening dynamics in uniaxial nematic liquid crystals.

We study the coarsening dynamics of defects in the uniaxial nematic liquid crystal 4-cyano-4'-n-pentylbiphenyl, subjected to a rapid pressure jump from the isotropic to the nematic phase. The defect tangle formed at the phase transition is dominated by type-1/2 strings (disclination lines), but also contains type-1 strings and monopoles. We also give a general discussion of the existence of scaling solutions to the nematodynamic equations for loop collapse and two-dimensional vortex-antivortex annihilation. We have measured the density of type-1/2 strings and loops and the monopole density as a function of time. We have also measured the rate at which the intercommutation of two strings takes place. Surprisingly, although the type-1/2 string density \ensuremath{\rho} scales as expected as a function of time, \ensuremath{\rho}\ensuremath{\sim}${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, loop densities and monopole densities do not seem to scale over the time interval studied. A possible explanation is given.

String cosmology in spherically symmetric space-time

Cosmological solutions for the string model are done in a spherically symmetric Kantowski–Sachs metric ansatz. Solutions are evaluated for three equations of states for string density. Also, an analytic relation between the metric coefficients is assumed. A first-order autonomous system is developed from the field equations and a line singularity is found.

Examples and counter-examples for a string density formula in the case of a discontinuity

In a recent paper, Pranger [l] proved a formula for the density function of a vibrating string with fixed ends in terms of its eigenvalues and their spatial rates of change (l.la) where s(x) = 1 [w;(x)] ~ ‘; n = 1, 2, . (l.lb) Here, w,,(L) are the eigenfrequencies of u” + wf,p(x)u = 0, OdsdL (1.2a) subject to u(O)=O=u(L). (1.2b) This formula demonstrates explicitly that the density function is deter- mined for a single set of boundary conditions by both the mode-number dependence of the eigenvalues and their functional dependence on length, in contrast to an earlier formula of Barcilon [2] which required spectral sequences for two suitable boundary configurations. As observed by Pranger [ 11, if p = 1 then o,(L)=nx/L and S(X) = (.u’/lr*) C( l/n*) = +u2/6, and one recovers p from (1.1). There is one continuous iso-spectral partner of the constant density case (as far as mode-number dependence is concerned) [3,4], viz. p(x) = (1+.X))” with o,(L) = (1 + L)(m/L), i.e., with different L-dependence. 363

Light-transmission study of coarsening in a nematic liquid crystal.

We have used light transmission to follow for four decades in time the annealing of defects generated by pressure jumps in a uniaxial liquid crystal. Two distinct scaling regimes for the light intensity as a function of time were observed: an early-time regime in which light propagates diffusively to the detector, and a late-time regime in which unscattered light dominates the signal reaching the detector. The measured values for the scaling exponent \ensuremath{\nu} for the string density ${\mathrm{\ensuremath{\rho}}}_{\mathit{s}}$\ensuremath{\propto}${\mathit{t}}^{\mathrm{\ensuremath{-}}\ensuremath{\nu}}$ are within 10% of the expected value \ensuremath{\nu}=1 over the time interval 10 msect100 sec.

What is the Role of Torsion in the Universe?

The effective action from the heterotic string compactification is studied on the manifolds with absolute parallelism. The cosmic string solutions resulting from the effective action are discussed in analogy with the dislocations in three-dimensional crystalline solid. The cosmic string density is concluded to be given by the torsion tensor of space-time, which gives rise to a compactification at least in one-dimensional direction in space-time. It is shown also the contorsion tensor defines the deficit angle. We have found a stringy solution in a sixdimensional model on M 4 × T 2 which is compatible with that of Greene et al. and a cylindrically symmetric exact solution is obtained, which are different from the exact cosmic string solutions of the Einstein theory ever proposed

ARE THE COSMIC STRINGS SEEN AS THE DISLOCATIONS?

The effective action from the heterotic string compactification is studied on the manifolds with absolute parallelism. The cosmic string solutions resulting from the effective action are discussed in analogy with the dislocations in 3-dimensional crystalline solid. The cosmic string density is concluded to be given by the torsion tensor of space-time, which gives rise to a compactification at least in 1-dimensional direction in space-time. It is also shown that the contorsion tensor defines the deficit angle. We have found a stringy solution in a 6-dimensional model on M4 × T2 which coincides with that of Greene et al.,5 but the string energy density depends only on torsion, therefore independent of the space-time metric.