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Page 1

DYNAMICAL SYSTEMS AND COSMOLOGY

Page 2

ASTROPHYSICS AND
SPACE SCIENCE LIBRARY

VOLUME 291

EDITORIAL BOARD

Chainnan

W.B. BURTON, National Radio Astronomy Observatory, Charlottesville, Virginia, U.S.A.
([email protected]); University of Leiden, The Netherlands ([email protected])

Executive Committee

J. M. E. KUIJPERS, Faculty of Science, Nijmegen, The Netherlands
E. P. J. VAN DEN HEUVEL, Astronomical Institute, University of Amsterdam,

The Netherlands
H. VAN DER LAAN, Astronomical Institute, University of Utrecht,

The Netherlands

MEMBERS

1. APPENZELLER, Landesstemwarte Heidelberg-Konigstuhl, Gennany
J. N. BAHCALL, The Institute for Advanced Study, Princeton, U.S.A.

F. BERTOLA, Universita di Padova, Italy
J. P. CASSINELLI, University of Wisconsin, Madison, U.S.A.

C. J. CESARSKY, Centre d'Etudes de Saclay, Gif-sur-Yvette Cedex, France
O. ENGVOLD, Institute of Theoretical Astrophysics, University of Oslo, Norway

R. McCRAY, University of Colorado, JILA, Boulder, U.S.A.
P. G. MURDIN, Institute of Astronomy, Cambridge, U.K.

F. PACINI, Istituto Astronomia Arcetri, Firenze, Italy
V. RADHAKRISHNAN, Raman Research Institute, Bangalore, India

K. SATO, School of Science, The University of Tokyo, Japan
F. H. SHU, University of California, Berkeley, U.S.A.

B. V. SOMOV, Astronomical Institute, Moscow State University, Russia
R. A. SUNYAEV, Space Research Institute, Moscow, Russia

Y. TANAKA, Institute of Space & Astronautical Science, Kanagawa, Japan
S. TREMAINE, CITA, Princeton University, U.S.A.

N. O. WEISS, University of Cambridge, U.K.

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93

with two eigenvalues with positive real parts (or three eigenvalues with positive real
parts). Associated with this dynamical nesting are cosmological models with very
interesting physical properties. This will also be true in the case of n scalar fields.
There will be a unique stable n-scalar field assisted inflationary solution. There will
then be n of the (n - 1)-scalar field models which are saddles with one eigenvalue
with positive real part. There will be ~n(n - 1) of the (n - 2)-scalar field models
which are saddles with two eigenvalues with positive real parts. And so on. Finally,
there will be n of the (1 )-scalar field models which are saddles with n - 1 (or
n - 2) eigenvalues with positive real parts. As one "goes up" the nested structure
the equilibrium points respectively become "stronger attractors" (Le., the stable
manifold of the equilibrium points increases in dimension).

There is also a three-field curvature scaling solution corresponding to the equi-
librium point given by

2 2
{ 'T' cJ>. - -}

'l'i = - V6ki ' • - V3ki

whose associated eigenvalues are given by

-1 ± V1 + 4K-2(2 - K2), -1 ± V3i, -1 ± V3i.

This equilibrium point is a sink whenever K2 > 2, in which case it represents an
FRW model with negative curvature (2K- 2 - 1) (else it is a saddle and represents a
positive curvature model). Finally, there are saddle equilibrium points correspond-
ing to the Milne model and the one- and two-field curvature scaling solutions, and
a set of equilibrium points with n=~=l Wi2 = 1, cJ>i = O} corresponding to massless
scalar field models, a subset of which are sources.

A complete qualitative analysis can be done for n-scalar field models. All results
can be proven by induction (see, for example, [269]). The n-scalar field assisted
inflationary solution is given by [229]

and

R(t) <X tP;
n 1

P =2'"'->1 - ~ 2 '
i=l k i

ki<Pi = kj<pj; V1 ::; i i= j ::; n.

We note that in the two-scalar field model, although inflation can occur for poten-
tials that are steeper than in the single-field case, it cannot occur for arbitrarily
steep potentials. For example, if kl = k2 == k, then inflation occurs if k2 < 4.
However, for n-fields, if k i = k for all i, then inflation occurs if k2 < 2n (e.g., k2 < 8
for four scalar field models).

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D. Discussion

We have studied multi-scalar field FRW cosmological models with exponential
potentials and barotropic matter. We have used dynamical systems techniques,
and by establishing a monotonic function in the complete dynamical phase space
(which includes both matter and curvature), we have been able to deduce global
results.

In Section VII A a comprehensive qualitative analysis was presented in the case
of two scalar fields with no matter. We concluded that the two-field assisted in-
flationary solution A is the global attractor when 2:;=1 k;2 > ! and the two-field
curvature scaling solution as is the global attractor when 2:;=1 k;2 < !. A subset
of the massless scalar field solutions M SF are always the early-time attractors. Con-
sequently, we found that in all cases both scalar fields are non-negligible in generic
late-time behaviour, contrary to popular belief. We note that both the assisted
inflationary solution and the massless scalar field early-time attractors correspond
to zero-curvature models. However, the curvature is not dynamically negligible
asymptotically in the two-field curvature scaling solution. The zero-curvature as-
sisted inflationary FRW scaling solutions [229] are of particular physical importance
since, through the combined effect of multiple uncoupled scalar fields, each having
an exponential potential, power-law inflation is possible even when each individual
scalar field need not be a source for inflation. For the parameter values in which
the assisted inflationary solution is the global late-time attractor the two single field
power-law solutions were shown to be saddles, and we showed that there are allow-
able parameter values for which either or both are inflationary, perhaps leading to
new interesting physical scenarios.

In Section VII B we studied the two-scalar field model with barotropic matter. A
monotonic function was established in the resulting phase space, thereby proving
that the matter must be negligible at late times. We found that A and as are the
only global sinks and that consequently assisted inflation and the two-field curvature
scaling solution are the global late-time attractors in their appropriate respective
parameter ranges. This confirmed the result that both scalar fields must be dynam-
ically non-negligible in generic late-time behaviour, and establishes the stability of
the two-field assisted inflationary model when matter is included. The monotonic
function also implies that the early-time attractors lie in the zero-curvature invari-
ant set, and we showed that they consist of a subset of the massless scalar field
models. For 'Y > ~, all of the equilibrium points with n -:F 0 were shown to be sad-
dles (see Table XIV). The two-field matter scaling solution MS was shown to be a
"stronger" attractor than the single-field matter scaling solutions MS1 and MS2 .
We note that when the curvature is zero, the two-field matter scaling solution is the
late-time attractor, consistent with the stability analysis in [45]. These one- and
two-field matter scaling solutions give rise to new transient dynamical behaviour
and may be of physical import. For example, there are solutions which spend a pe-
riod of time with the scalar field mimicking the barotropic fluid in which there is a
non-negligible scalar field energy density (corresponding to a matter scaling saddle

Page 202

Volume 259: The DynllD2ic Sun, edited by Arnold Hanslmeier, Mauro
Messerotti, Astrid Veronig
Hardbound, ISBN 0-7923-6915-7, May 2001

Volume 258: Electrohydrodynamics in Dusty and Dirty Plasmas- Gravito-
Electrodynamics and EHD, by Hiroshi Kikuchi
Hardbound, ISBN 0-7923-6822-3, June 2001

Volume 257: Stellar Pulsation -Nonlinear Studies, edited by Mine Takeuti,
Dimitar D_ Sasselov
Hardbound, ISBN 0-7923-6818-5, March 2001

Volume 256: Organizations and Strategies in Astronomy, edited by Andre
Heck
Hardbound, ISBN 0-7923-6671-9, November 2000

Volume 255: The Evolution of the Milky Way- Stars versus Clusters, edited
by Francesca Matteucci, Franco Giovannelli
Hardbound, ISBN 0-7923-6679-4, January 2001

Volume 254: Stellar Astrophysics, edited by K.S. Cheng, Hoi Fung Chau,
Kwing Lam Chan, Kam Ching Leung
Hardbound, ISBN 0-7923-6659-X, November 2000

Volume 253: The Chemical Evolution of the Galaxy, by Francesca Matteucci
Hardbound, ISBN 0-7923-6552-6, May 2001

Volume 252: Optical Detectors for Astronomy H -State-of-the-art at the Turn
of the Millennium, edited by Paola Amico, James W. Beletic
Hardbound, ISBN 0-7923-6536-4, December 2000

Volume 251: Cosmic Plasma Physics, by Boris V. Somov
Hardbound, ISBN 0-7923-6512-7, September 2000

Volume 250: Information Handling in Astronomy, edited by Andre Heck
Hardbound, ISBN 0-7923-6494-5, October 2000

Volume 249: The Neutral UpperAtmosphere, by S.N. Ghosh
Hardbound, ISBN 0-7923-6434-1, July 2002

Volume 247: Large ScaJe Structure Formation, edited by Reza Mansouri,
Robert Brandenberger
Hardbound, ISBN 0-7923-6411-2, August 2000

Page 203

Volume 246: The Legacy of J.c. Kapteyn -Studies on Kapteyn and the
Development of Modern. Astronomy, edited by Piet C. van der Kruit, Klaas
van Berkel
Hardbound, ISBN 0-7923-6393-0, August 2000

Volume 245: Waves in Dusty Space Plasmas, by Frank Verheest
Hardbound, ISBN 0-7923-6232-2, April 2000

Missing volume numbers have not yet been published.
For further information about this book series we refer you to the following web site:
http://www.wkap.n1JprodlsJASSL

To contact the Publishing Editor for new book proposals:
Dr. Harry (J.J.) Blom: [email protected]

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