1,244 research outputs found
Semi-Classical Quantization of Circular Strings in De Sitter and Anti De Sitter Spacetimes
We compute the {\it exact} equation of state of circular strings in the (2+1)
dimensional de Sitter (dS) and anti de Sitter (AdS) spacetimes, and analyze its
properties for the different (oscillating, contracting and expanding) strings.
The string equation of state has the perfect fluid form with
the pressure and energy expressed closely and completely in terms of elliptic
functions, the instantaneous coefficient depending on the elliptic
modulus. We semi-classically quantize the oscillating circular strings. The
string mass is being the Casimir operator,
of the -dS [-AdS] group, and is
the Hubble constant. We find \alpha'm^2_{\mbox{dS}}\approx 5.9n,\;(n\in N_0),
and a {\it finite} number of states N_{\mbox{dS}}\approx 0.17/(H^2\alpha') in
de Sitter spacetime; m^2_{\mbox{AdS}}\approx 4H^2n^2 (large ) and
N_{\mbox{AdS}}=\infty in anti de Sitter spacetime. The level spacing grows
with in AdS spacetime, while is approximately constant (although larger
than in Minkowski spacetime) in dS spacetime. The massive states in dS
spacetime decay through tunnel effect and the semi-classical decay probability
is computed. The semi-classical quantization of {\it exact} (circular) strings
and the canonical quantization of generic string perturbations around the
string center of mass strongly agree.Comment: Latex, 26 pages + 2 tables and 5 figures that can be obtained from
the authors on request. DEMIRM-Obs de Paris-9404
Renormalization Group Flow and Fragmentation in the Self-Gravitating Thermal Gas
The self-gravitating thermal gas (non-relativistic particles of mass m at
temperature T) is exactly equivalent to a field theory with a single scalar
field phi(x) and exponential self-interaction. We build up perturbation theory
around a space dependent stationary point phi_0(r) in a finite size domain
delta \leq r \leq R ,(delta << R), which is relevant for astrophysical applica-
tions (interstellar medium,galaxy distributions).We compute the correlations of
the gravitational potential (phi) and of the density and find that they scale;
the latter scales as 1/r^2. A rich structure emerges in the two-point correl-
tors from the phi fluctuations around phi_0(r). The n-point correlators are
explicitly computed to the one-loop level.The relevant effective coupling turns
out to be lambda=4 pi G m^2 / (T R). The renormalization group equations (RGE)
for the n-point correlator are derived and the RG flow for the effective
coupling lambda(tau) [tau = ln(R/delta), explicitly obtained.A novel dependence
on tau emerges here.lambda(tau) vanishes each time tau approaches discrete
values tau=tau_n = 2 pi n/sqrt7-0, n=0,1,2, ...Such RG infrared stable behavior
[lambda(tau) decreasing with increasing tau] is here connected with low density
self-similar fractal structures fitting one into another.For scales smaller
than the points tau_n, ultraviolet unstable behaviour appears which we connect
to Jeans' unstable behaviour, growing density and fragmentation. Remarkably, we
get a hierarchy of scales and Jeans lengths following the geometric progression
R_n=R_0 e^{2 pi n /sqrt7} = R_0 [10.749087...]^n . A hierarchy of this type is
expected for non-spherical geometries,with a rate different from e^{2 n/sqrt7}.Comment: LaTex, 31 pages, 11 .ps figure
Mass Spectrum of Strings in Anti de Sitter Spacetime
We perform string quantization in anti de Sitter (AdS) spacetime. The string
motion is stable, oscillatory in time with real frequencies and the string size and energy are bounded. The
string fluctuations around the center of mass are well behaved. We find the
mass formula which is also well behaved in all regimes. There is an {\it
infinite} number of states with arbitrarily high mass in AdS (in de Sitter (dS)
there is a {\it finite} number of states only). The critical dimension at which
the graviton appears is as in de Sitter space. A cosmological constant
(whatever its sign) introduces a {\it fine structure} effect
(splitting of levels) in the mass spectrum at all states beyond the graviton.
The high mass spectrum changes drastically with respect to flat Minkowski
spacetime. For {\it
independent} of and the level spacing {\it grows} with the
eigenvalue of the number operator, The density of states grows
like \mbox{Exp}[(m/\sqrt{\mid\Lambda\mid}\;)^{1/2}] (instead of
\rho(m)\sim\mbox{Exp}[m\sqrt{\alpha'}] as in Minkowski space), thus {\it
discarding} the existence of a critical string temperature.
For the sake of completeness, we also study the quantum strings in the black
string background, where strings behave, in many respects, as in the ordinary
black hole backgrounds. The mass spectrum is equal to the mass spectrum in flat
Minkowski space.Comment: 31 pages, Latex, DEMIRM-Paris-9404
Explicit formula in a imaginary quadratic number field
In \cite{Haran_1990}, Haran, using Riesz potentials, presents a version of
the classical explicit formula for the Riemann zeta function that treats all
places equally. In this article, we extend Haran's results to the case of an
imaginary quadratic extension of . To this end, we define Riesz
kernels for both the totally ramified and complex cases. Although Haran
implicitly addressed the unramified extension, we also include this case for
the sake of completeness. Following Haran's approach with the Riemann zeta
function, we demonstrate that these Riesz kernels naturally arise in connection
with the contributions of the totally ramified and complex places to the
Dedekind zeta function of the imaginary quadratic extension
String dynamics in cosmological and black hole backgrounds: The null string expansion
We study the classical dynamics of a bosonic string in the --dimensional
flat Friedmann--Robertson--Walker and Schwarzschild backgrounds. We make a
perturbative development in the string coordinates around a {\it null} string
configuration; the background geometry is taken into account exactly. In the
cosmological case we uncouple and solve the first order fluctuations; the
string time evolution with the conformal gauge world-sheet --coordinate
is given by , where
are given by Eqs.\ (3.15), and is the exponent of the conformal factor
in the Friedmann--Robertson--Walker metric, i.e. . The string
proper size, at first order in the fluctuations, grows like the conformal
factor and the string energy--momentum tensor corresponds to that of
a null fluid. For a string in the black hole background, we study the planar
case, but keep the dimensionality of the spacetime generic. In the null
string expansion, the radial, azimuthal, and time coordinates are
and The first terms of the series represent a
{\it generic} approach to the Schwarzschild singularity at . First and
higher order string perturbations contribute with higher powers of . The
integrated string energy-momentum tensor corresponds to that of a null fluid in
dimensions. As the string approaches the singularity its proper
size grows indefinitely like . We end the paper
giving three particular exact string solutions inside the black hole.Comment: 17 pages, REVTEX, no figure
String Driven Cosmology and its Predictions
We present a minimal model for the Universe evolution fully extracted from
effective String Theory. This model is by its construction close to the
standard cosmological evolution, and it is driven selfconsistently by the
evolution of the string equation of state itself. The inflationary String
Driven stage is able to reach enough inflation, describing a Big Bang like
evolution for the metric. By linking this model to a minimal but well
established observational information, (the transition times of the different
cosmological epochs), we prove that it gives realistic predictions on early and
current energy density and its results are compatible with General Relativity.
Interestingly enough, the predicted current energy density is found Omega = 1
and a lower limit Omega \geq 4/9 is also found. The energy density at the exit
of the inflationary stage also gives | Omega |_{inf}=1. This result shows an
agreement with General Relativity (spatially flat metric gives critical energy
density) within an inequivalent Non-Einstenian context (string low energy
effective equations). The order of magnitude of the energy density-dilaton
coupled term at the beginning of the radiation dominated stage agrees with the
GUT scale. The predicted graviton spectrum is computed and analyzed without any
free parameters. Peaks and asymptotic behaviours of the spectrum are a direct
consequence of the dilaton involved and not only of the scale factor evolution.
Drastic changes are found at high frequencies: the dilaton produces an
increasing spectrum (in no string cosmologies the spectrum is decreasing).
Without solving the known problems about higher order corrections and graceful
exit of inflation, we find this model closer to the observational Universe than
the current available string cosmology scenarii.Comment: LaTex, 22 pages, Lectures delivered at the Chalonge School, Nato ASI:
Phase Transitions in the Early Universe: Theory and Observations. To appear
in the Proceedings, Editors H. J. de Vega, I. Khalatnikov, N. Sanchez.
(Kluwer Pub
Sinh-Gordon, Cosh-Gordon and Liouville Equations for Strings and Multi-Strings in Constant Curvature Spacetimes
We find that the fundamental quadratic form of classical string propagation
in dimensional constant curvature spacetimes solves the Sinh-Gordon
equation, the Cosh-Gordon equation or the Liouville equation. We show that in
both de Sitter and anti de Sitter spacetimes (as well as in the black
hole anti de Sitter spacetime), {\it all} three equations must be included to
cover the generic string dynamics. The generic properties of the string
dynamics are directly extracted from the properties of these three equations
and their associated potentials (irrespective of any solution). These results
complete and generalize earlier discussions on this topic (until now, only the
Sinh-Gordon sector in de Sitter spacetime was known). We also construct new
classes of multi-string solutions, in terms of elliptic functions, to all three
equations in both de Sitter and anti de Sitter spacetimes. Our results can be
straightforwardly generalized to constant curvature spacetimes of arbitrary
dimension, by replacing the Sinh-Gordon equation, the Cosh-Gordon equation and
the Liouville equation by higher dimensional generalizations.Comment: Latex, 19 pages + 1 figure (not included
The quantum inflaton, primordial perturbations and CMB fluctuations
We compute the primordial scalar, vector and tensor metric perturbations
arising from quantum field inflation. Quantum field inflation takes into
account the nonperturbative quantum dynamics of the inflaton consistently
coupled to the dynamics of the (classical) cosmological metric. For chaotic
inflation, the quantum treatment avoids the unnatural requirements of an
initial state with all the energy in the zero mode. For new inflation it allows
a consistent treatment of the explosive particle production due to spinodal
instabilities. Quantum field inflation (under conditions that are the quantum
analog of slow roll) leads, upon evolution, to the formation of a condensate
starting a regime of effective classical inflation. We compute the primordial
perturbations taking the dominant quantum effects into account. The results for
the scalar, vector and tensor primordial perturbations are expressed in terms
of the classical inflation results. For a N-component field in a O(N) symmetric
model, adiabatic fluctuations dominate while isocurvature or entropy
fluctuations are negligible. The results agree with the current WMAP
observations and predict corrections to the power spectrum in classical
inflation.Such corrections are estimated to be of the order of m^2/[N H^2]
where m is the inflaton mass and H the Hubble constant at horizon crossing.
This turns to be about 4% for the cosmologically relevant scales. This quantum
field treatment of inflation provides the foundations to the classical
inflation and permits to compute quantum corrections to it.Comment: 23 pages, no figures. Improved version to appear in Phys. Rev.
Multi-String Solutions by Soliton Methods in De Sitter Spacetime
{\bf Exact} solutions of the string equations of motion and constraints are
{\bf systematically} constructed in de Sitter spacetime using the dressing
method of soliton theory. The string dynamics in de Sitter spacetime is
integrable due to the associated linear system. We start from an exact string
solution and the associated solution of the linear system , and we construct a new solution differing from
by a rational matrix in with at least four
poles . The periodi-
city condition for closed strings restrict to discrete values
expressed in terms of Pythagorean numbers. Here we explicitly construct solu-
tions depending on -spacetime coordinates, two arbitrary complex numbers
(the 'polarization vector') and two integers which determine the string
windings in the space. The solutions are depicted in the hyperboloid coor-
dinates and in comoving coordinates with the cosmic time . Despite of
the fact that we have a single world sheet, our solutions describe {\bf multi-
ple}(here five) different and independent strings; the world sheet time
turns to be a multivalued function of .(This has no analogue in flat space-
time).One string is stable (its proper size tends to a constant for , and its comoving size contracts); the other strings are unstable (their
proper sizes blow up for , while their comoving sizes tend to cons-
tants). These solutions (even the stable strings) do not oscillate in time. The
interpretation of these solutions and their dynamics in terms of the sinh-
Gordon model is particularly enlighting.Comment: 25 pages, latex. LPTHE 93-44. Figures available from the auhors under
reques
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