Integrable Cosmological Modelsin Diverse Dimensions

Integrable Cosmological Modelsin Diverse Dimensions

Vitaly N. MelnikovCenter for Gravitation and Fundamental Metrology,

VNIIMS,and Institute of Gravitation and Cosmology, PFUR,

Moscow, Russia

Vitaly N. MelnikovCenter for Gravitation and Fundamental Metrology,

VNIIMS,and Institute of Gravitation and Cosmology, PFUR,

Moscow, Russia

Center for Gravitation andFundamental Metrology, (1967) -

present*

Center for Gravitation andFundamental Metrology, (1967) -

present*• Gravitation, Cosmology, Theory of Gravitational

Experiments, Gravimetry and Gradiometry, Space Experiments

• Fundamental Physical Constants: G, h, k, c, m, … and Basic Standards

• Solid State Physics, Liquid Crystalls, Surface Phenomena, Synchrotron Radiation

• Statistical Physics• Elementary Particles Staff: 50 15 during last 15 years-----------------------------------------------------------------------*Russian State Committee for Standards

• Gravitation, Cosmology, Theory of Gravitational Experiments, Gravimetry and Gradiometry, Space Experiments

• Fundamental Physical Constants: G, h, k, c, m, … and Basic Standards

• Solid State Physics, Liquid Crystalls, Surface Phenomena, Synchrotron Radiation

• Statistical Physics• Elementary Particles Staff: 50 15 during last 15 years-----------------------------------------------------------------------*Russian State Committee for Standards

Topics (Gravitation and Cosmology Group CCFM)

4D:

Topics (Gravitation and Cosmology Group CCFM)

4D:1. Exact Solutions with Fields: φ, e-m, their

interactions, Λ in GR: Particle-Like and Cosmological Solutions (1970-79)

2. Quantum Cosmology (φ, Λ, 71-72)3. Quantum Effects in Cosmology (bounce

due to SSB, 1978; Vacuum Polarization).4. FPC, their stability, G measurements,

theory of space experiments. SEE (from 93).

1. Exact Solutions with Fields: φ, e-m, their interactions, Λ in GR: Particle-Like and Cosmological Solutions (1970-79)

2. Quantum Cosmology (φ, Λ, 71-72)3. Quantum Effects in Cosmology (bounce

due to SSB, 1978; Vacuum Polarization).4. FPC, their stability, G measurements,

theory of space experiments. SEE (from 93).

Topics (Gravitation and Cosmology Group CCFM) Contd.

DD:

Topics (Gravitation and Cosmology Group CCFM) Contd.

DD:4. Multidimensional Cosmology

(Fields, Λ, PF, Forms)

(VNM et al., 1988 – present)

5. Multidimensional BH’s, Stability of BH’s

(VNM + Bronnikov)

6. Stochastic Behavior. Billiards for systems:

PF, 94; p-branes, 99; (φ + V(φ)), 2003.

(VNM + Ivashchuk)

7. Observational Windows of Extra Dimensions

4. Multidimensional Cosmology

(Fields, Λ, PF, Forms)

(VNM et al., 1988 – present)

5. Multidimensional BH’s, Stability of BH’s

(VNM + Bronnikov)

6. Stochastic Behavior. Billiards for systems:

PF, 94; p-branes, 99; (φ + V(φ)), 2003.

(VNM + Ivashchuk)

7. Observational Windows of Extra Dimensions

Topics Contd.Topics Contd.

1. NLED in GR. Dilaton interaction with e-m.2. First QC model with Λ [creation from nothing]

(1972); first QC model with minimal and conformal scalar fields (1971)

3. First nonsingular cosmological model with SSB (1978-79) of conformal nonlinear scalar field.

4. First non-singular field particle-like solution with gravitational field (1979);

5. Only G may vary with respect to atomic time (78) (System of Measurements).

1. NLED in GR. Dilaton interaction with e-m.2. First QC model with Λ [creation from nothing]

(1972); first QC model with minimal and conformal scalar fields (1971)

3. First nonsingular cosmological model with SSB (1978-79) of conformal nonlinear scalar field.

4. First non-singular field particle-like solution with gravitational field (1979);

5. Only G may vary with respect to atomic time (78) (System of Measurements).

IntroductionThe second half of the 20th century in the field of gravitation was devoted mainly to theoretical study and experimental verification of general relativity and alternative theories of gravitation

with a strong stress on relations between macro and microworld fenomena or, in other words,

between classical gravitation and quantum physics.

Very intensive investigations in these fields were done in Russia by M.A.Markov, K.P.Staniukovich, Ya.B.Zeldovich, A.D.Sakharov and

their colleagues starting from mid 60’s.

As a motivation there were:

- singularities in cosmology and black hole physics,

- role of gravity at large and very small (planckian) scales,

- attempts to create a quantum theory of gravity as for other physical fields,

- problem of possible variations of fundamental physical constants etc.

A lot of work was done along such topics as :

- particle-like solutions with a gravitational field,

- quantum theory of fields in a classical gravitational background,

- quantum cosmology with fields like a scalar one, Λ, …

- self-consistent treatment of quantum effects in cosmology,

- development of alternative theories of gravitation:

scalar-tensor, gauge, with torsion, bimetric etc.

As all attempts to quantize general relativity in a usual manner failed and it was proved that it is not renormalizable, it became clear that the promising trend is along the lines of unification of all physical interactions which started in the 70’s. About this time the experimental investigation of gravity in strong fields and gravitational waves started giving a powerful speed up in theoretical studies of such objects as pulsars, black holes, QSO’s, AGN’s, Early Universe etc., which continue now.But nowadays, when we think about the most important lines of future developments in physics, we may forsee that

gravity will be essential not only by itself, but as a missing cardinal link of some theory, unifying all existing physical interactions: week, strong and electromagnetic ones.

Even in experimental activities some crucial next generation gravitational experiments verifiing predictions of unified schemes will be important.

Epoch Now: Unified ModelsEpoch Now: Unified ModelsSG, S, SS, M-theory…SG, S, SS, M-theory…

Gravity via UT!!Gravit.

EarlyUniverse

Lab for HEP

BH’sOtherSFO

GWAppl.

BasicFundam. Exp.Terrestrial,

G,G-dot, ISLEP, Clocks, 2nd order…

SpaceExperiments.

- new generation.

E-M WEAK STRONG

GUTE-W

Cosmology as a lab for Super HEP! New revolution > 1998, acceleration, DM, DE. 95% -???

Tests of UT models via G

We may predict as well that a thorough study of

gravity itself and within the unified models will give in the next century and millennium even more applications for our everyday life as electromagnetic theory gave us in the 20th century

after very abstract fundamental investigations of Faraday, Maxwell, Poincare, Einstein and others, which never dreamed about such enormous applications of their works.

Other very important feature, which may be envisaged, is an increasing role of fundamental physics studies, gravitation, cosmology and astrophysics in particular, in space experiments.

Unique microgravity environments and modern technology outbreak give nearly ideal place for gravitational experiments which suffer a lot on Earth from its relatively strong gravitational field and gravitational fields of nearby objects due to the fact that there is no ways of screening gravity.

In the developement of relativistic gravitation and dynamical cosmology after A. Einstein and A. Friedmann, we may notice three distinct stages:

1. Investigation of models with matter sources in the form of a perfect fluid, as was originally done by Einstein and Friedmann.

2. Studies of models with sources as diferent physical fields, starting from electromagnetic and scalar ones, both in classical and quantum cases.

3. Which is really topical now, application of ideas and results of unified models for treating fundamental problems of cosmology and black hole physics, especially in high energy regimes, using ideas of extra dimensions and p-branes as sources.

Multidimensional gravitational models play an essential role in the latter approach.

The necessity of studying multidimensional models of gravitation and cosmology is motivated by several reasons.

First, the main trend of modern physics is the unification of all known fundamental physical interactions: electromagnetic, weak, strong and gravitational ones. During the recent decades there has been a significant progress in unifying weak and electromagnetic interactions, some more modest achievements in GUT, supersymmetric and superstring theories.

Now, theories with membranes, p-branes and more vague M-theory are being created and studied.

Having no definite successful theory of unification now, it is desirable to study the common features of these theories and their applications to solving basic problems of modern gravity and cosmology.

Moreover, if we really believe in unified theories, the early stages of the Universe evolution and black hole physics, as unique superhigh energy regions, are the most proper and natural arena for them.

Second, multidimensional gravitational models, as well as scalar-tensor theories of gravity, are theoretical frameworks for describing possible temporal and range variations of fundamental physical constants .

These ideas have originated from the earlier papers of E. Milne (1935) and P. Dirac (1937) on relations between the phenomena of micro- and macro-worlds, and up till now they are under thorough study both theoretically and experimentally.

Lastly, applying multidimensional gravitational models to basic problems of cosmology and black hole physics, we hope to find answers to such long-standing problems as -singular or nonsingular initial states, -creation of the Universe, creation of matter and its entropy, -acceleration, cosmological constant, origin of inflation and specific scalar fields which may be necessary for its realization, -isotropization and graceful exit problems, -stability and nature of fundamental constants , -possible number of extra dimensions, their stable compactification etc. Bearing in mind that multidimensional gravitational models are certain generalizations of general relativity which is tested reliably for weak fields up to 0.0001 and partially in strong fields (binary pulsars), it is quite natural to inquire about their possible observational or experimental windows. From what we already know, among these windows are:– possible deviations from the Newton and law, or new interactions,– possible variations of the effective gravitational constant with a time rate smaller than the Hubble one,– possible existence of monopole modes in gravitational waves,– different behaviour of SFO, such as multidimensional black holes, wormholes, astrophysical sources, - standard cosmological tests etc.

- Possible non-conservation of energy in SFO and accelerators (e.g. LHC), if BW-models ideas about gravity in the bulk turn out to be true…

-------------------------------------------------------------------------------------------

Since modern cosmology has already become a unique laboratory for testing standard unified models of physical interactions at energies that are far beyond the level of the existing and future man-made accelerators and other installations on Earth, there exists a possibility of using cosmological and astrophysical data for discriminating between future unified schemes.

As no accepted unified model exists, in our approach we adopted simple, but general from the point of view of number of dimensions, models based on multidimensional Einstein equations with or without sources of different nature:

– cosmological constant,

– perfect and viscous fluids,

– scalar and electromagnetic fields,

– their possible interactions,

– dilaton and moduli fields,

– fields of antisymmetric forms (related to p-branes) etc.

Our program’s main objective was and is to obtain

-exact self-consistent solutions (integrable models) for these models and then

-- to analyze them in cosmological, spherically and axially symmetric cases.

In our view this is a natural and most reliable way to study highly nonlinear systems. It is done mainly within Riemannian geometry.

Some simple models in integrable Weyl geometry and with torsion were studied as well.

The Model of MGCThe Model of MGC1. n Einstein spaces of constant curvature with

(m+1)-component perfect fluid, Λ (fields, form-fields,…)

2. Metric (e.g. cosmological):

3. Manifold :

where ( , ) is the Einstein space

of dimension :

1. n Einstein spaces of constant curvature with (m+1)-component perfect fluid, Λ (fields, form-fields,…)

2. Metric (e.g. cosmological):

3. Manifold :

where ( , ) is the Einstein space

of dimension :

n

i

ii gtxdtdttg1

)()(2exp)(2exp

iM)(ig

nMMRM ...1

iN

; , 2,...,1)()( nniggR inm

iinm iiii

4. EMT:

-----------------------------------------------------------------------

5.EOS :

and/or EMT for fields (scalar, e-m, forms,…), Λ,…

4. EMT:

-----------------------------------------------------------------------

5.EOS :

and/or EMT for fields (scalar, e-m, forms,…), Λ,…

m

MN

NM TT

0

)(

nn

mkn

mk

MN tptptdiagT )(,...,)(),( )()(

1)()( 1

1

m,...,0

0)( NMNT

mniconstu

tN

utp

i

i

ii

,...,0,...,1

)(1)(

)(

)()(

)(

, ,

6. Non-zero components of Ricci tensor:

7. Conservation law constraint for

Using EOS

6. Non-zero components of Ricci tensor:

7. Conservation law constraint for

Using EOS

ni

XNXXXgR

XXXNR

n

i

ii

iiiinmnm

n

i

iiii

iiii

,...,1

22exp

)(

1

1

200

,

m,...,0

n

ii

ii pXN

1

)()()( 0

constA

tXutXNAt ii

ii

)(

)()()( )()(2exp)(

8. Let (harmonic time gauge),

then Einstein multidimensional Eqs.

are equivalent to Lagrange Eqs. with Lagrangian

minisuperspace metric

and potential

Exponential potential !!! Combined form for PF, Λ and λi

8. Let (harmonic time gauge),

then Einstein multidimensional Eqs.

are equivalent to Lagrange Eqs. with Lagrangian

minisuperspace metric

and potential

Exponential potential !!! Combined form for PF, Λ and λi

n

i

iiXN

10

MN

MN

MN TRR 2

2

1

XVXXGxL jiij 00 exp)(exp

2

1

jiijiij NNNG

mi

i

mi

i

n

i

ii

i

XuA

XuAXXNXV

0

)(

0

)()(2

10

exp

exp)(22exp2

1)(

mAAnmm ,...,0)(2 , ,

• Curvature terms:

where

-----------------------------------------------------------------

Cosmological constant term: and

So,

And 0-component describes Λ

------------------------------------------------------------------

10. Diagonalization of metric

It has signature so by linear

Transformation we diagonalize it

• Curvature terms:

where

-----------------------------------------------------------------

Cosmological constant term: and

So,

And 0-component describes Λ

------------------------------------------------------------------

10. Diagonalization of metric

It has signature so by linear

Transformation we diagonalize it

ii

im NA 2

1 ji

jim

j Nu 2)(

nji ,...,1,

0AnjNu jj ,...,12)0( ,

nipi ,...,1)0()0( ,

jiij dxdxGG

,...,,ia

ia XeZ

1

1

00n

i

iibaab dzdzdzdzdzdzG 1,...,0, nba

• We come to: and

D is dimension of M.

11. Reduction to σ-model and Toda-like systems,

further to Liouville, Abel, Emden-Fowler Eqs. etc.

12. Behaviour of extra spaces:- constant,- dynamically compactified,- torus,- large, but with barriers, walls,…

• We come to: and

D is dimension of M.

11. Reduction to σ-model and Toda-like systems,

further to Liouville, Abel, Emden-Fowler Eqs. etc.

12. Behaviour of extra spaces:- constant,- dynamically compactified,- torus,- large, but with barriers, walls,…

1,...,1,1 diagabab

n

ii

i

ijij N

NG

1

1,2

1D

D

Realized program in MGC (from 1988)I. Cosmology, exact solutions in DD for:- Λ, (Λ+φ):nonsingular, dynamically compactified, inflationary;- PF, (PF+φ) : nonsingular, inflationary;- Viscous fluid: nonsingular, generation of mass, entropy;

quintessence and coincidence in 2-component model;- Stochastic behaviour near the singularity, billiards in

Lobachevsky space, D=11 critical, φ destroys billiards (94);- Ricci-flat solutions above for any n, also with curvature in

one factor-space;- with curvatures in 2 factor-spaces only for N=10,11;- fields: scalar, dilatons, forms of arbitrary rank (JMP, 98) -

inflationary, Λ generation; billiads (99);- quantum variants (WDW-equation) for above cases;- dilatonic fields with potentials, billiads;

Realized program in MGC (from 1988)I. Cosmology, exact solutions in DD for:- Λ, (Λ+φ):nonsingular, dynamically compactified, inflationary;- PF, (PF+φ) : nonsingular, inflationary;- Viscous fluid: nonsingular, generation of mass, entropy;

quintessence and coincidence in 2-component model;- Stochastic behaviour near the singularity, billiards in

Lobachevsky space, D=11 critical, φ destroys billiards (94);- Ricci-flat solutions above for any n, also with curvature in

one factor-space;- with curvatures in 2 factor-spaces only for N=10,11;- fields: scalar, dilatons, forms of arbitrary rank (JMP, 98) -

inflationary, Λ generation; billiads (99);- quantum variants (WDW-equation) for above cases;- dilatonic fields with potentials, billiads;

II. Solutions depending on r in DD:- Generalized Schwarzchild, Tangerlini (BH’s), also with φ(no

BH’s),- Generalized R-N (BH’s), plus φ (no BH’s);- Multi-temporal;- Dilaton-like interaction of φ and e.-m. fields (BH’s only for

special case);- Stability studies (stable only BH’s case above);- Same with dilaton-forms interaction, stability only in some

cases, e.g. for one form in particular;

--------------------------------------------------------------------------- Some simple axially-symmetric; - with torsion;- in integrable Weyl cosmology

II. Solutions depending on r in DD:- Generalized Schwarzchild, Tangerlini (BH’s), also with φ(no

BH’s),- Generalized R-N (BH’s), plus φ (no BH’s);- Multi-temporal;- Dilaton-like interaction of φ and e.-m. fields (BH’s only for

special case);- Stability studies (stable only BH’s case above);- Same with dilaton-forms interaction, stability only in some

cases, e.g. for one form in particular;

--------------------------------------------------------------------------- Some simple axially-symmetric; - with torsion;- in integrable Weyl cosmology

Main Publications in 4D, DD Main Publications in 4D, DD- VNM, Staniukovich. Hydrodynamics Fields and Constants in

Gravitation Theory, Moscow, Energoatomizdat, 1983 (in Russian). English version of VNM part in CBPF-MO-02/02 (4D).

- VNM. Multidimensional Classical and Quantum Cosmology and Gravitation: Exact Solutions and Variations of Constants.

CBPF-NF-051/93, also in “Cosmology and Gravitation”, Ed. M.Novello, Frontieres, Singapore, 1994.

- VNM. Multidimensional Cosmology and Gravitation, CBPF-MO-002/95, also in “Cosmology and Gravitation II”, Ed. M.Novello, Frontieres, Singapore, 1996.

- VNM and Ivashchuk, in “Lecture Notes in Physics”, 2000, v.157, p.214.

- VNM and Ivashchuk. Exact Solutions in Multidimensional Gravity with Antisymmetric Forms. Class. Quant. Grav., 2001, Topical Review, v.18, pp. R1-R66.

- VNM, Staniukovich. Hydrodynamics Fields and Constants in Gravitation Theory, Moscow, Energoatomizdat, 1983 (in Russian). English version of VNM part in CBPF-MO-02/02 (4D).

- VNM. Multidimensional Classical and Quantum Cosmology and Gravitation: Exact Solutions and Variations of Constants.

CBPF-NF-051/93, also in “Cosmology and Gravitation”, Ed. M.Novello, Frontieres, Singapore, 1994.

- VNM. Multidimensional Cosmology and Gravitation, CBPF-MO-002/95, also in “Cosmology and Gravitation II”, Ed. M.Novello, Frontieres, Singapore, 1996.

- VNM and Ivashchuk, in “Lecture Notes in Physics”, 2000, v.157, p.214.

- VNM and Ivashchuk. Exact Solutions in Multidimensional Gravity with Antisymmetric Forms. Class. Quant. Grav., 2001, Topical Review, v.18, pp. R1-R66.

- VNM. Multidimensional Gravitation and Cosmology. II.

CBPF-MO-003/ 02, Rio de Janeiro. 2002.

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See also in gr-qc, hep-th for VNM.

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Recent papers with scalar fields in 4D:- VNM, Gavrilov, Dehnen. General Solutions for Flat Friedmann

Universe Filled by PF and Scalar Field with Exponential Potential.

Grav. & Cosm., 2003, v.9, p. 189 (acceleration, coincidence).

- VNM, Gavrilov, Abdyrakhmanov. Friedmann Universe with Dust and Scalar Field with Multiple Exponential Potential. GRG, 2004, v.36, N 7, p. 1579 (acceleration, recollapse, coincidence).

- VNM. Multidimensional Gravitation and Cosmology. II.

CBPF-MO-003/ 02, Rio de Janeiro. 2002.

---------------------------------------------------------------------------------------

See also in gr-qc, hep-th for VNM.

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Recent papers with scalar fields in 4D:- VNM, Gavrilov, Dehnen. General Solutions for Flat Friedmann

Universe Filled by PF and Scalar Field with Exponential Potential.

Grav. & Cosm., 2003, v.9, p. 189 (acceleration, coincidence).

- VNM, Gavrilov, Abdyrakhmanov. Friedmann Universe with Dust and Scalar Field with Multiple Exponential Potential. GRG, 2004, v.36, N 7, p. 1579 (acceleration, recollapse, coincidence).

In DD :In DD :• VNM, Ivashchuk, Selivanov. Cosmological Solutions in

Multidimensional Model with SF, Multiple Exponential Potential. JHEP, 0309 (2003) 059 (classical and quantum, acceleration,)

• VNM, Ivashchuk, S.-W. Kim. S-brane Solutions with Acceleration in Models with Forms and Multiple Exponential Potential. Crav. & Cosm., 2004, v. 10, N1-2, p. 141.

• VNM, Ivashchuk, Selivanov. Composite S-brane Solutions on Product of Ricci-Flat Spaces. GRG, 2004, v. 36, N 7(accelerat.)

• VNM, Alimi,Gavrilov. Multicomponent Perfect Fluid with Variable Parameters in n Ricci-Flat Spaces. JKPS, v.44, p. S148 (acceleration, isotropisation in 4D, compactification in DD)

• VNM, Alimi, Ivashchuk. Non-singular solutions in multidimensional model with scalar fields and exponential potential. Grav. & Cosm., 2005, N1-2.

• VNM, Ivashchuk, Selivanov. Cosmological Solutions in Multidimensional Model with SF, Multiple Exponential Potential. JHEP, 0309 (2003) 059 (classical and quantum, acceleration,)

• VNM, Ivashchuk, S.-W. Kim. S-brane Solutions with Acceleration in Models with Forms and Multiple Exponential Potential. Crav. & Cosm., 2004, v. 10, N1-2, p. 141.

• VNM, Ivashchuk, Selivanov. Composite S-brane Solutions on Product of Ricci-Flat Spaces. GRG, 2004, v. 36, N 7(accelerat.)

• VNM, Alimi,Gavrilov. Multicomponent Perfect Fluid with Variable Parameters in n Ricci-Flat Spaces. JKPS, v.44, p. S148 (acceleration, isotropisation in 4D, compactification in DD)

• VNM, Alimi, Ivashchuk. Non-singular solutions in multidimensional model with scalar fields and exponential potential. Grav. & Cosm., 2005, N1-2.

G-dot in (4+N)-dimensional cosmology with multicomponent anisotropic fluid (VNM+ Ivashchuk,

2003, JKPS)We consider here a (4+N)-dimensional cosmology with an isotropic 3-space and an arbitrary Ricci-°at internal space. The Einstein equations provide a relation between and other cosmological parameters.

1 The model

Let us consider (4 + N)-dimensional theory described by the action

where is the fundamental gravitational constant. Then the gravitational field equations are

where is a (4+N)-dimensional energy-momentum tensor, , and M, P = 0, …, N + 3.

For the (4 + N)-dimensional manifold we assume the structure

where is a 3-dimensional space of constant curvature, , , for k = +1, 0, -1, respectively, and is a N-dimensional compact Ricci-flat Riemann manifold.

The metric is taken in the form

where i, j, k = 1, 2, 3; m, n, p = 4, …, N + 3; , , and are, respectively, the metrics and scale factors for and . For we adopt the expression of the multi-component (anisotropic) fluid form

Under these assumptions the Einstein equations take the form

The 4-dimensional density is

where we have normalized the factor b(t) by putting

On the other hand, to get the 4-dimensional gravity equations one should put

. Consequently, the e®ective 4-dimensional gravitational " constant" G(t) is defined by

whence its time variation is expressed as

2 Cosmological parametersSome inferences concerning the observational cosmological parameters can be extracted just from the equations without solving them [3]. Indeed, let us de¯ne the Hubble parameter H, the density parameters and the "deceleration" parameter q referring to a ¯xed instant in the usual way

Besides, instead of G let us introduce the dimensionless parameter

Then, excluding b from (1.6) and (1.8), we get

with

where

When is small we get from (1.15)

Note that (1.18) for N = 6, m = 1, (so that ) coincides with the corresponding relation of Wu and Wang [1] obtained for large times in case k = -1 (see also [2]).If k = 0, then in addition to (1.18), one can obtain a separate relation between and , namely,

(this follows from the Einstein equation , which is certainly a linear combination of (1.6)-(1.8).

The present observational upper bound on is

if we take in accord with [4, 5]

and

Let us consider two component case: m = 2. Let the first component (called "matter") be a dust, i.e.

3 Two-component example: dust + (N -1)-brane

and the second one (called "quintessence") be a (N - 1)-brane, i.e.

We remind that as it was mentioned in [6] the multidimensional cosmological model on product manifold with fields of forms (for review see [8]) may be decribed in terms of multicomponent "perfect" fluid [7] with the following equations of state for component: if p-brane worldvolume contains and in opposite case. Thus, the field of form matter leads us either to , or to stiff matter equations of state in internal spaces.In this case we get from (1.18) for small a

and for k = 0 and small g we obtain from (1.19)

Now we illustrate the formulas by the following example when N = 6 ( may be a Calabi-Yau manifold) and

We get from (1.24)

in agreement with (1.20).

In this case the second fluid component corresponds to magnetic (Euclidean) NS5-brane (in D = 10 type I, Het or II A string models). Here we consider for simplicity the case of constant dilaton field.

SCALAR-TENSOR COSMOLOGY AND VARIATIONS OF G(VNM+Bronnikov, Novello, G&C, 2002)

The purpose of this note is to estimate the order of magnitude of variations of the gravitationalconstant G due to cosmological expansion in the framework of scalar-tensor theories (STT) ofgravity.

Consider the general (Bermann-Wagoner-Nordtvedt) class of STT where gravity is chara-cterized by the metric and the scalar field ; the action is

Here is the scalar curvature, , and U are certain functions of , varying from theory to theory, is the matter Lagrangian.

This formulation of the theory corresponds to the Jordan conformal frame, in which matter particles move along geodesics and hence the weak equivalence principle is valid, and non-gravitational fundamental constants do not change. In other words, this is the frame well describing the existing laboratory, geophysical and cosmological observations.

Among the three functions of entering into (1) only two are independent since there is

a freedom of transformations . We use this arbitrariness, choosing , as is done, e.g., in Ref. [1]. Another standard parametrization is to put and

( the Brans-Dicke parametrization of the general theory (1)). In our para-metrization , the Brans-Dicke parameter , the Brans-Dicke parameter is ; here and henceforth, the subscript denotes a derivative with respect to . The Brans-Dicke STT is the particular case , so that in (1)

The field equations that follow from (1) read

where is the D'Alembert operator, and the last term in (4) is matter energy-momentum tensorodf matter.

Consider now isotropic cosmological models with the standard FRW metric

where is the scale factor of the Universe, and k = −1, 0, 1 for closed, spatially flat and hyperbolic models, respectively. Accordingly, we assume and the energy-momentum tensor of matter in the perfect fluid form ( is the density and is the presuure).

The field equations in this case can be written as follows:

To connect these equations with observations, let us fix the time t at the present epoch ((i.e., consider the instantaneous values of all quantities) and introduce the standard observables:

where is the critical density, or, in our model, the r.h.s. of Eq. (7) in case k = 0: .This is slightly different from the usual definition where G is the Newtoniangravitational constant. The point is that the locally measured Newtonian constant in STT differs from ; provided the derivatives and are sufficiently small, one has [1]

Since, according to the solar-system experiments, , for our order-of-magnitude reasoning we can safely put , and, in particular, our definition of now coincides with the standard one.

The time variation of G, to a good approxiamtion, is

where, for convenience, we have introduced the coefficient expressing in terms of the Hubble parameter H.

Eqs. (6)(8) contain too many arbitrary parameters for making a good estimate of . Let us now introduce some restrictions according to the current state of observational cosmo-logy:

(i) k = 0 (a spatially flat cosmological model, so that the total density of matter equals );

(ii) p = 0 (the pressure of ordinary matter is negligible compared to the energy density);

(iii) (the ordinary matter, including its dark component, contributes to only 0.3 of the critical density; unusual matter, which is here represented by the scalar field, comprises the remaining 70 per cent).

Then Eqs. (7) and (8) can be rewritten in the form

Subtracting (8) from (7), we exclude the cosmological constant U , which can be quite largebut whose precise value is hard to estimate. We obtain

The first term in Eq. (13) can be represented in the form

and can be replaced with . The term can be neglected for our estimation purposes. To see this, let us use as an example the Brans-Dicke theory, in which . We then have

here the first term is the same as the first term in Eq. (13), times the small parameter . Assuming that is of the same order of magnitude as (or slightly greater), we see that,generically, . Note that our consideration is not restricted to the Brans-Dicke theory andconcerns the model (1) with an arbitrary function and an arbitrary potential . Neglecting , we see that (13), divided by , leads to an algebraic equation with respect to :

where .

According to modern observations, the Universe is expanding with an acceleration, so thatthe parameter q is, roughly, -0.5 ± 0.2, hence we can take .

In case we simply obtain . Assuming

and , we come to the estimate

Where is, by modern views, close to 0.7.

For nonzero values of , solving the quadratic equation (14) and assuming , we arrive at the estimate , so that, taking and again , we have instead of (15)

We conclude that, in the framework of the general STT, modern cosmological observations,taking into account the solar-system data, restrict the possible variation of G to values within . This estimate may be considerably tightened if the matter density parameter and the (negative) deceleration parameter will be determined more precisely.

Thanks !Thanks !

Merci !

Danke !

Спасибо !

Merci !

Danke !

Спасибо !

Special Thanks to Local Organizers

Special Thanks to Local Organizers

Multidimensional models In the 80’s the supergravitational theories were “replaced” by superstring models. Now it is heated by expectations connected with the overall M-theory. In all these theories, 4-dimensional gravitational models with extra fields were obtained from some multidimensional model by dimensional reduction based on the decomposition of the manifold M = M4×Mint,where M4 is a 4-dimensional manifold and Mint is some internal manifold (mostly considered to be compact).

The earlier papers on multidimensional gravity and cosmology dealt with multidimensional Einstein equations and with a block-diagonal cosmological or spherically symmetric metric defined on the manifold M = R ×M0 × . . . ×Mn of the form

where (Mr, gr) are Einstein spaces, r = 0, . . . , n. In some of them a cosmological constant and simple scalar fields were also used. Such models are usually reduced to pseudo-Euclidean Toda-like systems with the Lagrangian

and the zero energy constraint E = 0.

It should be noted that pseudo-Euclidean Toda-like systems are not yet well-studied. There exists a special class of equations of state that gives rise to Euclidean Toda models [9].

At present there exists a special interest to the so-called M- and F-theories etc. These theories are “supermembrane” analogues of the superstring models in D = 11, 12 etc. The low-energy limit of these theories leads to models governed by the Lagrangian

where g is the metric, Fa = dAa are forms of rank na, and φa are scalar fields.

It was shown that, after dimensional reduction the manifold may be

M0 ×M1 × . . . ×Mn

and when the composite p-brane ansatz is considered, the problem is reduced to the gravitating self-interacting σ-model with certain constraints.