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low energy effective theories derived from the ns - ns sector of string theories contain the gravity , @xmath0 , the dilation , @xmath1 , and the antisymmetric tensor field , @xmath2 . the existence of the last degree of freedom leads to intriguing implications in string cosmology @xcite . in four spacetime dimensions , the massless antisymmetric tensor field is dual to the a pseudo - scalar ( axion ) field @xcite , and the axion dilaton system is known to develop an unobserved anisotropy in our universe , which can be diluted away at late times only in a contracting universe @xcite . such a disastrous cosmological situation can be resolved also in string theory which contains another dynamical object called d - brane @xcite . the gauge invariance on the d - brane is maintained through the coupling of the gauge field strength to the antisymmetric tensor field @xcite . then , the effective action derived on the d - brane describes the antisymmetric tensor field as a massive and self - interacting 2-form field . cosmological evolution of such a tensor field has been investigated in ref . @xcite assuming that the dilaton is fixed to a reasonable value . although the time - dependent magnetic @xmath3 field existing in the early universe develops an anisotropy in the universe , it was realized that the matter - like behavior of the @xmath3 field ( b - matter ) ensures a dilution of the anisotropy at late times and thus the isotropy is recovered in reasonable cosmological scenarios @xcite . in such sense the effect of antisymmetric tensor field on the d - brane is distinguished from that of field strength of the u(1 ) gauge field @xcite .    in this paper , we investigate the cosmological evolution of the b - matter  dilaton system in our universe which is assumed to be imbedded in the d - brane . the usual string cosmology with the dilaton suffers from the notorious runaway problem , which is also troublesome in the d - brane universe . in our study , the dilaton obtains two exponential potential terms due to the curvature of extra dimensions @xmath4 , and the d - brane tension ( the mass term of the b - matter ) @xmath5 . it is interesting to observe that the dilaton can be stabilized for negative @xmath4 @xcite which , however , leads to a contracting universe due to the effective negative cosmological constant in our universe . when @xmath4 is positive , the b - matter dominance will be overturned by the @xmath4 dominance as the dilaton runs away to the negative infinity . as a consequence of this , the initial anisotropy driven by the b - matter can also be diluted away at late times .    for a realistic low energy effective theory , string theory must be endowed with a certain mechanism generating an appropriate vacuum expectation value for the dilaton . in such a situation , the dilaton is expected to be stabilized at some stage of the cosmological evolution affecting the dynamics of the antisymmetric tensor field . taking an example of the dilaton stabilization , we will also examine the cosmological evolution of the b - matter and the dilaton in which the essential features of ref . @xcite are reproduced . this paper is organized as follows . in section  [ sec:2 ] , we describe the low energy effective action of the d - brane universe and the corresponding field equations . before our main discussion , section  [ sec:3 ] is devoted to presenting homogeneous solutions in flat spacetime illustrating some interesting features of the b - matter  dilaton system in a simple way . in section  [ sec:4 ] , we find semi - analytic and numerical cosmological solutions to observe an intriguing interplay of the curvature @xmath4 and the d - brane tension @xmath5 . in section  [ sec:5 ] , we consider the evolution of the anisotropic universe with the dilaton stabilization which leads to a satisfactory cosmology of the d - brane universe . we conclude in section  [ sec : conc ] . the main idea of the d - brane world is that we reside on a d@xmath6-brane imbedded in 10 ( or 11 ) dimensional spacetime with extra - dimensions compactified . the bosonic ns - ns sector of the d - brane world consists of the u(1 ) gauge field @xmath7 living on the d@xmath6-brane and the bulk degrees including the graviton @xmath0 , the dilaton @xmath1 , and the antisymmetric tensor field of rank - two @xmath2 . in the presence of the brane , the gauge invariance of @xmath2 is restored through its coupling to a u(1 ) gauge field @xmath7 and the gauge invariant field strength is @xcite @xmath8    even though we are assumed to live on the d - brane , we adopt the conventional compactification in the sense that the extra dimensions are compact and stabilized , and thus static . in the presence of the d - brane , this would in general require an additional setup like additional branes and fluxes along the extra dimensions  @xcite , and the working of non - perturbative effects  @xcite . the extra dimensions are warped due to branes and fluxes . the warping of extra dimensions gives a warp factor in the definition of the four - dimensional planck scale  @xcite , and also induces the potential for the dilaton  @xcite as described below . if the fluxes in the extra dimensions significantly affect the compactification , the existence of the fluxes on the d - brane seems also natural and it is intriguing to tackle their effect in the early universe . however , simultaneous consideration of the fluxes along both the extra dimensions and the d - brane lets the computation almost intractable . therefore , the conventional compactification is an appropriate setup at the present stage , and then we take the following four - dimensional effective action of the bosonic sector in the string frame  @xcite @xmath9,\end{aligned}\ ] ] where the tilde denotes the string frame quantity , @xmath10}$ ] and @xmath11 with @xmath12 . the parameter @xmath5 is defined by @xmath13 where @xmath14 is the effective brane tension . note that we omitted the chern - simons like term assuming the trivial r - r background . if we assume that the six extra - dimensions are compactified with a common radius @xmath15 , one finds @xmath16 where @xmath17 and @xmath18 is the four - dimensional planck mass . the qualitative features of our results do not depend on specific values of @xmath6 . thus , we take @xmath19 for simplicity . the @xmath4 term comes from the scalar curvature or the condensate @xmath20 of extra dimensions integrated over the whole extra dimensions . the action , and thus also field equations , can be written in a more familiar form in the einstein metric , which is defined by @xmath21 we will work in this metric from now on . in terms of einstein metric , the action becomes @xmath22.\end{aligned}\ ] ] the field equations derived from the action ( [ action - e ] ) are @xmath23 @xmath24 @xmath25 where the dilaton potential is @xmath26 , \label{dilaton - potential}\end{aligned}\ ] ] and the energy - momentum tensor is given by @xmath27    in the subsequent sections , we examine the equations of motion ( [ b - eq1])([e - eq1 ] ) and find homogeneous solutions . cosmological implication of the d - brane universe is of our main interest , including stabilization of the dilaton and condensation of the antisymmetric tensor field . for the sake of the simplicity , let us first look into the dynamics of our system in flat spacetime . with @xmath28 , the einstein equations ( [ e - eq1 ] ) become simple and gauge invariance of the field variables @xmath29 and @xmath30 which directly appear in the expression of energy - momentum ( [ tb ] ) allows a homogeneous configuration @xmath31    for homogeneous configuration ( [ hom ] ) , @xmath32-components ( _ electric _ components ) of the equation of the antisymmetric tensor field ( [ b - eq1 ] ) reduce to an algebraic equation @xmath33 @xmath34-components ( _ magnetic _ components ) give @xmath35 where the overdot denotes differentiation with respect to the rescaled dimensionless time @xmath36 these equations ( [ 0i])([ij ] ) tell us that the _ magnetic _ components @xmath37 are _ dynamical _ but the _ electric _ components are determined by the _ constraint _ equation ( [ 0i ] ) . dynamics of the dilaton is governed by ( [ p - eq1 ] ) which becomes @xmath38 where the dimensionless parameter @xmath39 is defined by @xmath40    the constraint equation ( [ 0i ] ) forces its numerator to vanish except a trivial solution @xmath41 , however vanishing numerator allows only a _ vanishing _ electric field solution @xmath42 . then the equations for the magnetic components ( [ ij ] ) and the dilaton ( [ de ] ) reduce to @xmath43 in the following , we will find solutions of ( [ m1 ] ) and ( [ m2 ] ) to see how the dilaton and the magnetic component behave . when the magnetic components of the antisymmetric tensor field @xmath45 vanish , we have only the dilaton equation from ( [ m2 ] ) @xmath46 where the dilaton potential @xmath47 is given by the sum of the curvature term and the brane tension term @xmath48 as shown in figure  [ fig1 ] .    . for @xmath49 , @xmath47 monotonically increases ( dashed line ) . for @xmath50 , @xmath47 has a minimum at @xmath51 ( solid line ) . ]    for @xmath49 , the dilaton potential @xmath47 monotonically increases , so the dilaton finally rolls to the minimum @xmath52 as @xmath53 . for @xmath50 , the dilaton potential @xmath47 has the minimum value @xmath54 at @xmath55 . if @xmath56 , the dilaton is stuck at @xmath57 . when @xmath58 , the dilaton oscillates around the minimum . therefore , the dilaton is stabilized with mass @xmath59 , and due to exponential potential terms , its mass can be much different from the vacuum expectation value of the dilaton @xmath60 . as @xmath39 increases , the dilaton mass rapidly grows . when @xmath61 , the dilaton again rolls to negative infinity . it means that the dilaton is easily destabilized by its fluctuations , if the dilaton mass becomes too small . though the negative vacuum energy @xmath62 does not affect the dynamics of the dilaton and the antisymmetric tensor field in flat spacetime , its coupling to gravity leads to singular cosmological evolution as we shall see in section  [ sec:4 ] . therefore , for the proper description of cosmology , some additional term is needed to adjust the value of @xmath63 to vanish as we shall discuss in section  [ sec:5 ] . the runaway or stabilized behavior of the dilaton remains essentially unchanged even with @xmath64 although the dependence on the initial conditions alters as we will see in the next subsection .      without loss of generality , let us assume that the direction of the magnetic component of antisymmetric tensor field is fixed as @xmath65 . although the equations of motion ( [ m1])([m2 ] ) are still complicated even under this assumption , the presence of the magnetic components do not alter the story of the dilaton much . first , let us consider a simple case of zero brane tension ( @xmath66 ) , which is analytically tractable . then in ( [ m1])([m2 ] ) , only the @xmath39-dependent term survives on the right - hand side . ( for this case , we will use the original @xmath4 and time @xmath67 since @xmath66 . ) the solutions to these equations are @xmath68 + \ln c_1 \,,\ ] ] and @xmath69 where @xmath70 , @xmath71 , and @xmath72 are integration constants , and the time has been rescaled to @xmath73 by absorbing another constant @xmath74 . the solutions are still remaining valid under @xmath75 , i.e. , under the change of the sign of @xmath72 . therefore , we can fix the time @xmath76 to flow to the positive direction and @xmath77 . the solutions approach their asymptotic configurations at large @xmath76 , @xmath78 we observe that the dilaton rolls linearly in time to negative infinity , and the antisymmetric tensor field condensates to a constant @xmath79 which depends on @xmath4 . there are two topologically distinct condensation processes depending on the signature of @xmath70 which is to be set by the initial conditions ; the change in @xmath4 contributes to @xmath80 oppositely . with the brane tension term turned on , we numerically study due to the complexity of field equations . the solutions are classified into two classes as in the pure dilaton case in section  [ subsec:3.a ] , depending on the signature of @xmath39 .    for @xmath49 , the dilaton approaches negative infinity irrespective of initial conditions , @xmath81 and @xmath82 . the condensed value of the magnetic component deviates from the initial value in the beginning , and then stabilizes to a constant as shown in figure  [ fig2 ] . we observed from the numerical results that there exist two branches of the magnetic condensation depending on the initial conditions as discussed above , but we show only one branch in the figure . like in the @xmath66 case , once the magnetic component is condensed , the condensation survives as a constant value . it is different from the case without the dilaton , in which the magnetic component permanently oscillates in flat spacetime  @xcite .     and @xmath3 in the einstein frame for @xmath83 ( solid lines ) , @xmath84 ( dashed lines ) , and @xmath85 ( dotted lines ) . the initial conditions are @xmath86 and @xmath87 . the configurations are very similar regardless of @xmath88.,height=226 ]     and @xmath3 in the einstein frame for @xmath89 , @xmath87 , and @xmath90 the dilaton stabilization is observed as well as the periodically waving oscillation of @xmath3 . ]    when @xmath39 is negative , there are two classes of solutions , depending on the initial value of the dilaton @xmath81 ( or equivalently the energy density ) for a fixed initial magnetic condensation @xmath82 . for large @xmath81 or @xmath82 , the configurations are almost the same as those for nonnegative @xmath39 ( see figure  [ fig2 ] ) . for small enough @xmath81 and @xmath82 , the dilaton shows the oscillating behavior as expected from the dilaton potential in figure  [ fig1 ] . according to the dilaton oscillation , the magnetic component is also oscillating around a condensed value ( see figure  [ fig3 ] ) . if we involve the radiation of perturbative modes of the dilaton and the antisymmetric tensor field , which is missing in this classical configuration , the classical solution will damp to a stabilized value of the dilaton and the final condensed value of the magnetic component . in this section we study cosmological solutions of the d - brane universe in the presence of both the dilaton and the antisymmetric tensor field . as we have done in the previous paper @xcite , we assume spatially homogeneous configurations for the antisymmetric tensor field and the dilaton , and look for the time evolution of these fields and the expansion of the universe .    the non - vanishing homogeneous antisymmetric tensor field , in general , implies the anisotropic universe . to consider the simplest form of anisotropic cosmology , we take only a single magnetic component of @xmath91 to be nonzero , namely @xmath92 and @xmath93 . then the metric consistent with this choice of field configuration is of bianchi type i @xmath94 then , in the einstein frame , the field equation for @xmath3 ( [ b - eq1 ] ) reads @xmath95 the dilaton - field equation ( [ p - eq1 ] ) is @xmath96 and einstein equations ( [ e - eq1 ] ) are @xmath97 where @xmath98 it is more convenient to employ the dimensionless time variable @xmath99 of ( [ ti ] ) and to introduce the variables @xmath100 and @xmath101 defined by @xmath102 then the above equations ( [ eq = beq])([eq = g33 ] ) are rewritten as @xmath103b & = & 0,\end{aligned}\ ] ] @xmath104 @xmath105 where , by dividing @xmath106 from each @xmath107 in eqs . ( [ br1])([br2 ] ) , we have @xmath108 ^ 2,\quad \rho_\lambda=\lambda e^{2\phi } , & \nonumber\\&\displaystyle \rho_b=\frac12e^{3\phi}(1+b^2)^{1/2},\quad \tilde\rho_b=\frac12e^{3\phi}(1+b^2)^{-1/2 } , & \end{aligned}\ ] ] where @xmath39 is defined in ( [ co ] ) . in the subsequent subsections , we look for the solutions to the above field equations for various cases beginning with some simple solutions . the dilaton in the system of equations ( [ eq - b])([eq - a3 ] ) has the exponential potential up to the correction due to the antisymmetric tensor field . it is well - known that the scalar field with the exponential potential possesses the scaling solution in which the energy density of the scalar field mimics the background fluid energy density @xcite . this scaling solution is also an attractor , so that the late time behavior of the solutions are universal irrespective of initial conditions . this is an attractive property of the exponential potential . for the potential of the form @xmath109 , there is an attractor solution @xmath110,\quad \alpha=\left(\frac{2}{\beta}\right)^2\ln t,\ ] ] for @xmath111 . the scale factor obeys the power - law time - dependence , implying that the rolling of @xmath1 constitutes the matter having an equation of state @xmath112 where @xmath113 varies from @xmath114 to @xmath115 for the aforementioned range of @xmath116 . we found this type of particular solutions of the eqs . ( [ eq - b])([eq - a3 ] ) , which can be found when we have a single exponential term in the potential , that is , for the case of @xmath117 and for the case of @xmath66 . for both cases we start from an ansatz of the form @xmath118 where we suppressed constant terms in @xmath119 and @xmath120 which correspond to simple rescaling of coordinates . for the case of @xmath117 , we obtain two distinguished solutions @xmath121 and @xmath122 the first solution is nothing but the solution ( [ sol - a ] ) with @xmath123 . the second solution has the non - vanishing antisymmetric tensor field . for the case of @xmath66 , we introduce a new rescaled dimensionless time variable @xmath124 instead of @xmath125 of ( [ ti ] ) and then get the continuous set of solutions from eqs . ( [ eq = beq])([eq = g33 ] ) @xmath126 @xmath127 and @xmath128 are same as those in eq . ( [ sol - a ] ) with @xmath129 , while we have the non - vanishing @xmath3 field condensate . these solutions are the solutions to the eqs . ( [ eq - b])([eq - a3 ] ) for the specific initial conditions . however , the importance of these solutions , as noted in the first paragraph , arises from the fact that they are attractors , which means that after enough time the solutions with different initial conditions approach these solutions . we will confirm this through numerical analysis in the next subsection . the solution ( [ sol-1 ] ) applies for the brane tension dominated case where the dilaton potential is approximated by @xmath130 from eq . ( [ pu ] ) . the evolution of the dilaton under this potential produces matter with the equation of state @xmath131 . once the antisymmetric tensor field is turned on , the anisotropy appears as in the solution ( [ sol-2 ] ) . the measure of anisotropy is @xmath132 this result is contrasted with that in ref . @xcite where the dilaton is assumed to be stabilized . the rolling of the dilaton makes the difference . it affects the dynamics of @xmath3 field in such a way that @xmath101 remains constant at @xmath133 instead of oscillating about the potential minimum @xmath134 and the anisotropy is maintained . let us turn to the third solution ( [ sol-3 ] ) . it is relevant when the dilaton potential arising from the curvature of extra dimensions , @xmath135 , dominates over other contributions . in our scheme this happens as the dilaton rolls down the potential . when @xmath101 vanishes , the transition point at which @xmath136 starts to dominate over @xmath137 is at @xmath138 . thus , this solution describes the late time behavior of all the solutions with various initial conditions when @xmath4 is positive . it is very intriguing since we achieve the isotropic universe in the end . the rolling of dilaton makes the brane tension term which causes the anisotropy when we have non - vanishing @xmath3 field less important than the extra dimension curvature term which recovers the isotropy . the rolling of the dilaton under the potential @xmath136 now forms the matter having the equation of state @xmath139 , thus giving marginal inflation . we have the second order differential equations for four variables @xmath140 , @xmath141 , @xmath142 , @xmath143 . thus we need eight initial values @xmath81 , @xmath144 , @xmath82 , @xmath145 , @xmath146 , @xmath147 , @xmath148 , @xmath149 to specify the solution . among these , @xmath150 can always be set to zero by coordinate rescaling . @xmath151 must obey the constraint equation ( [ eq - constraint ] ) , but this does not fix the ratio @xmath152 . we choose the isotropic universe as a natural initial condition which leads to @xmath153 . since the dilaton potential is composed of exponential terms , the shift of the dilaton field by a constant can be traded for the redefinition of mass scale . we use this property to take the initial value of the dilaton to be zero without loss of generality . in our numerical analysis , we take the dimensionless time variable as @xmath154 where @xmath155 and use the variable @xmath156 with its initial value @xmath157 . this means that the proper time scale for the cosmological evolution is not @xmath158 , but @xmath159 . for convenience s sake , we take the initial time as @xmath160 . the other mass scale @xmath4 is also affected by this shift and we can treat it by replacing the parameter @xmath39 with @xmath161 . now we need three initial values @xmath144 , @xmath82 , and @xmath162 , to fix the functional form of the solution . the initial values @xmath163 and @xmath164 are related to @xmath82 and @xmath145 by @xmath165 and @xmath166 . if we assume @xmath168 , we can study the role of the dilaton interacting with gravity more clearly as was also true in the subsection  [ subsec:3.a ] . we keep the brane term with the tension @xmath5 . the spacetime is now isotropic . the parameter @xmath4 in the string - frame action , which can be interpreted as the curvature of the compact internal manifold , plays a very important role , and the solutions are topologically different depending on the signature of @xmath4 .    when @xmath117 , we have an attractor solution ( [ sol-1 ] ) . since we consider the dilaton only , the only relevant initial condition is @xmath144 . the initial condition for the attractor solution is @xmath169 in the setup described in the previous subsection . for a few other values of @xmath144 , we plot the numerical solutions in figure  [ fig4 ] . regardless of the initial value @xmath144 , all the solutions approach universally the attractor solution ( up to rescaling for the scale factor ) after some time .     and @xmath117 with initial value @xmath169 ( solid curve , the attractor ) , @xmath170 ( dotted curve ) , @xmath171 ( dashed curve ) and @xmath172 ( dash - dotted curve ) , respectively . for the evolution of the scale factor , we plot @xmath173 for it becomes constant @xmath174 when the scale factor behaves as @xmath175.,title="fig:",height=245 ]   and @xmath117 with initial value @xmath169 ( solid curve , the attractor ) , @xmath170 ( dotted curve ) , @xmath171 ( dashed curve ) and @xmath172 ( dash - dotted curve ) , respectively for the evolution of the scale factor , we plot @xmath173 for it becomes constant @xmath174 when the scale factor behaves as @xmath175.,title="fig:",height=245 ]    when @xmath176 , the evolution is divided into two stages . when @xmath1 is large so that @xmath177 is much larger than @xmath178 , the solution approaches the attractor ( [ sol-1 ] ) . as @xmath1 rolls down to around @xmath138 , @xmath179 becomes larger than @xmath180 . after that point the solution approaches the attractor ( [ sol-3 ] ) with @xmath134 . in figure [ fig5 ] , numerical solutions show this two stage evolution explicitly .     and @xmath176 . when we set the parameter @xmath181 , the curves stand for the solutions with the initial value @xmath169 ( solid curve ) and @xmath170 ( dotted curve ) , @xmath171 ( dashed curve ) and @xmath172 ( dash - dotted curve ) , respectively.,title="fig:",height=245 ]   and @xmath176 . when we set the parameter @xmath181 , the curves stand for the solutions with the initial value @xmath169 ( solid curve ) and @xmath170 ( dotted curve ) , @xmath171 ( dashed curve ) and @xmath172 ( dash - dotted curve ) , respectively.,title="fig:",height=245 ]    since @xmath4 measures the curvature of the extra dimensions , its value can be negative . when @xmath182 , the dilaton potential has the global minimum at @xmath183 with negative cosmological constant @xmath184 as discussed in the subsection  [ subsec:3.a ] . the disastrous consequence of this negative energy minimum is that the scale factor collapses in the end . this can not be avoided because the dilaton rolls down toward the minimum and stays where the potential is negative while the kinetic energy is diluted by the expansion , thus the total energy becomes negative and it derives the universe to collapse . it is different from the case of flat spacetime in the subsection  [ subsec:3.a ] , where the dilaton experiences either the permanent oscillation around @xmath185 for negative initial energy density or the attractor solution for nonnegative initial energy density . now , we turn on the antisymmetric tensor field along the @xmath187 direction , @xmath188 . the spacetime becomes anisotropic , @xmath189 , in general . the solutions are classified again by the signature of @xmath4 .     and @xmath117 . the initial conditions are as follow . solid line : @xmath190 , @xmath191 , @xmath192 ; dotted line : @xmath193 , @xmath194 , @xmath192 ; dashed line : @xmath193 , @xmath195 , @xmath192 ; dash - dotted line : @xmath169 , @xmath194 , @xmath192 . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath190 , @xmath191 , @xmath192.,title="fig:",height=245 ]   and @xmath117 . the initial conditions are as follow . solid line : @xmath190 , @xmath191 , @xmath192 ; dotted line : @xmath193 , @xmath194 , @xmath192 ; dashed line : @xmath193 , @xmath195 , @xmath192 ; dash - dotted line : @xmath169 , @xmath194 , @xmath192 . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath190 , @xmath191 , @xmath192.,title="fig:",height=245 ] +   and @xmath117 . the initial conditions are as follow . solid line : @xmath190 , @xmath191 , @xmath192 ; dotted line : @xmath193 , @xmath194 , @xmath192 ; dashed line : @xmath193 , @xmath195 , @xmath192 ; dash - dotted line : @xmath169 , @xmath194 , @xmath192 . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath190 , @xmath191 , @xmath192.,title="fig:",height=245 ]   and @xmath117 . the initial conditions are as follow . solid line : @xmath190 , @xmath191 , @xmath192 ; dotted line : @xmath193 , @xmath194 , @xmath192 ; dashed line : @xmath193 , @xmath195 , @xmath192 ; dash - dotted line : @xmath169 , @xmath194 , @xmath192 . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath190 , @xmath191 , @xmath192.,title="fig:",height=245 ]    for @xmath117 , we have an attractor ( [ sol-2 ] ) under the initial conditions of @xmath190 , @xmath196 , and @xmath197 . in figure  [ fig6 ] , we plotted numerical solutions for a few different initial conditions in addition to the attractor . it is confirmed again that all the solutions approach the attractor as the time goes on . the final value of @xmath101 is either @xmath198 or @xmath199 depending on the initial conditions . in the lower - right panel in figure  [ fig6 ] we provide the evolution of each component of energy density . the kinetic energy of the dilaton @xmath200 catches up the potential energy @xmath180 and the ratio of them becomes constant . this is a characteristic feature of the scaling solution @xcite . the kinetic energy of @xmath3 field is kept much smaller than both of them , but the anisotropy is still maintained due to the difference between @xmath180 and @xmath201 .     and @xmath176 . when @xmath181 , the solutions of four initial conditions are given : @xmath190 , @xmath191 , @xmath192 for the solid curves , @xmath193 , @xmath194 , @xmath192 for the dotted curves , @xmath193 , @xmath195 , @xmath192 for the dashed curves , and @xmath169 , @xmath194 , @xmath192 for the dash - dotted curves . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath169 , @xmath194 , @xmath192.,title="fig:",height=245 ]   and @xmath176 . when @xmath181 , the solutions of four initial conditions are given : @xmath190 , @xmath191 , @xmath192 for the solid curves , @xmath193 , @xmath194 , @xmath192 for the dotted curves , @xmath193 , @xmath195 , @xmath192 for the dashed curves , and @xmath169 , @xmath194 , @xmath192 for the dash - dotted curves . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath169 , @xmath194 , @xmath192.,title="fig:",height=245 ] +   and @xmath176 . when @xmath181 , the solutions of four initial conditions are given : @xmath190 , @xmath191 , @xmath192 for the solid curves , @xmath193 , @xmath194 , @xmath192 for the dotted curves , @xmath193 , @xmath195 , @xmath192 for the dashed curves , and @xmath169 , @xmath194 , @xmath192 for the dash - dotted curves . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath169 , @xmath194 , @xmath192.,title="fig:",height=245 ]   and @xmath176 . when @xmath181 , the solutions of four initial conditions are given : @xmath190 , @xmath191 , @xmath192 for the solid curves , @xmath193 , @xmath194 , @xmath192 for the dotted curves , @xmath193 , @xmath195 , @xmath192 for the dashed curves , and @xmath169 , @xmath194 , @xmath192 for the dash - dotted curves . the lower - right panel shows the evolution of each component of energy density for initial conditions @xmath169 , @xmath194 , @xmath192.,title="fig:",height=245 ]    for @xmath176 , the evolution is again divided into two stages as in the @xmath167 case . in the first stage where @xmath180 is dominant , the solution approaches an attractor ( [ sol-2 ] ) . in the second stage where @xmath179 is dominant , it approaches another attractor ( [ sol-3 ] ) . thus the universe recovers the isotropy . the final value of @xmath101 is a certain constant which is determined by initial conditions and can differ from @xmath202 . numerical solutions for a few initial conditions are shown in figure  [ fig7 ] . the kinetic energy of the dilaton @xmath200 now catches up the potential energy @xmath180 in the first stage and @xmath179 in the second stage . the ratios , @xmath203 and @xmath204 , approach constants in each stage . the kinetic energy of @xmath3 field is kept much smaller as in @xmath117 case .    for @xmath182 , the solution becomes singular as in the @xmath167 case . here we skip the description of such singular solutions which are not suitable for the evolution of our universe . the vacuum expectation value of the dilaton determines both the gauge and gravitational coupling constants of the low energy effective theory . therefore , the dilaton must be stabilized at some stage of the evolution for the action ( [ action - e ] ) to have something to do with the reality . in this section , we study the cosmological evolution when the dilaton is stabilized . as for the correct mechanism of dilaton stabilization , the consensus has not been made yet . our goal here is to illustrate an example of the dilaton stabilization and look into the effect of it on the dynamics of @xmath3 field and the cosmological evolution , since the overall features of which are insensitive to the detailed mechanism of stabilization . our starting point is the dilaton potential ( [ dilaton - potential ] ) . this potential possesses the minimum for @xmath182 , but the value of the potential at the minimum is negative and need to be set to zero by fine - tuning of the constant shift . however , the constant shift of the potential has no motivation in the context of string theory . instead , we introduce a term @xmath205 in the potential , which can arise from the effect of various form field fluxes in extra dimensions  @xcite . thus , our potential for the dilaton for @xmath167 looks like @xmath206 this potential has a global minimum for any value of @xmath4 and @xmath207 . to have sensible cosmology , the potential at the minimum must be zero . for @xmath182 , this can be done through a fine - tuning of the parameters in the potential @xmath208 where @xmath209 . then the minimum is located at @xmath210 the shape of this fine - tuned potential for @xmath211 is shown in figure  [ fig8 ] . now the equations ( [ eq - phi])([eq - a3 ] ) are modified accordingly @xmath212 where @xmath213 , while the equation ( [ eq - b ] ) for @xmath101 is not changed .    with the stabilizing potential for the dilaton , the cosmological evolution is completely changed . the dilaton and the antisymmetric tensor field rapidly come to the oscillation about the potential minimum @xmath214 and @xmath134 . oscillating @xmath1 and @xmath3 fields behave like ordinary matter satisfying the equation of state @xmath215 @xcite . the universe becomes isotropic and matter dominated . the numerical solutions for the stabilizing dilaton potential ( [ eq = v1 ] ) with @xmath211 are plotted in figure  [ fig8 ] . one can see the oscillation of @xmath1 and @xmath101 , and that both @xmath216 and @xmath217 approach to @xmath218 , indicating the matter domination . damping of @xmath1 and @xmath101 oscillations is due to the expansion of the universe . ( 500,430)(0,0 ) ( 0,240 ) for @xmath211 . the minimum of the potential is at @xmath219 with @xmath220 . ( b ) , ( c ) and ( d ) show the numerical solutions of @xmath221 , @xmath222 and @xmath223 , respectively , for @xmath211 and initial conditions @xmath224 , @xmath193 and @xmath194 . @xmath221 , @xmath222 , and @xmath225 approach @xmath226 , @xmath170 , and @xmath227 , respectively.,title="fig:",height=245 ] ( 248,240 ) for @xmath211 . the minimum of the potential is at @xmath219 with @xmath220 . ( b ) , ( c ) and ( d ) show the numerical solutions of @xmath221 , @xmath222 and @xmath223 , respectively , for @xmath211 and initial conditions @xmath224 , @xmath193 and @xmath194 . @xmath221 , @xmath222 , and @xmath225 approach @xmath226 , @xmath170 , and @xmath227 , respectively.,title="fig:",height=245 ] ( 2 , 0 ) for @xmath211 . the minimum of the potential is at @xmath219 with @xmath220 . ( b ) , ( c ) and ( d ) show the numerical solutions of @xmath221 , @xmath222 and @xmath223 , respectively , for @xmath211 and initial conditions @xmath224 , @xmath193 and @xmath194 . @xmath221 , @xmath222 , and @xmath225 approach @xmath226 , @xmath170 , and @xmath227 , respectively.,title="fig:",height=245 ] ( 240 , 0 ) for @xmath211 . the minimum of the potential is at @xmath219 with @xmath220 . ( b ) , ( c ) and ( d ) show the numerical solutions of @xmath221 , @xmath222 and @xmath223 , respectively , for @xmath211 and initial conditions @xmath224 , @xmath193 and @xmath194 . @xmath221 , @xmath222 , and @xmath225 approach @xmath226 , @xmath170 , and @xmath227 , respectively.,title="fig:",height=245 ] ( 120,435)(a ) ( 360,435)(b ) ( 120,195)(c ) ( 360,195)(d ) we investigated cosmology of a four - dimensional low energy effective theory arising from the ns - ns sector of string theory with a d - brane which contains the dynamical degrees of freedom such as the gravity , the dilaton , and the antisymmetric tensor field of second rank , coupling to the gauge field strength living on the brane . the dynamics of the system crucially depends on the curvature @xmath4 and the brane tension @xmath5 through which the dilaton obtains a potential of the form ; @xmath228 . here , the latter becomes the effective mass of the antisymmetric tensor field ( b - matter ) .    in terms of the homogeneous solution in flat spacetime , we first showed how the dilaton @xmath140 and the non - vanishing magnetic component of the tensor field @xmath141 , in one direction evolve in time . for positive @xmath4 , one finds that the dilaton runs away to negative infinity , and @xmath141 reaches a constant value at late time . when @xmath4 is negative , one can see that the dilaton is stabilized at a finite value and then the dilaton ( and also @xmath141 ) shows an oscillatory behavior around that value , or runs away again depending on the choice of the initial conditions . such a dilaton stabilization , however , produces a negative effective cosmological constant , and thus leads to a collapsing universe in string cosmology . when @xmath141 is turned on , the universe undergoes an anisotropic expansion described by the bianchi type - i cosmology . we found the attractor solutions showing the overall features of general solutions and confirmed it through numerical analysis . the dilaton @xmath140 decreases and settles to a logarithmic decrease in time to negative infinity . when the brane tension term dominates , the anisotropy is sustained . if there is a positive curvature term , it dominates finally over the brane tension term as the dilaton rolls down to negative infinity . then the expansion of the universe turns to be isotropic and linear in time . accordingly , @xmath141 decreases inversely proportional to time . there have been various proposals to stabilize the dilaton . in order to study the dynamics of the b - matter and the stabilized dilaton system , we adopted an option of generating a dilaton mass term of the form @xmath229 . then the potential for the dilaton has a global minimum and the cosmological constant of our universe can be fine - tuned to a desired value with negative @xmath4 . while the dilaton evolved to a stabilized value , @xmath3 shows an oscillatory matter - like behavior , and the universe expands as in the usual matter - dominated era recovering the isotropy . the obtained result is consistent with that of ref . this work is the result of research activities ( astrophysical research center for the structure and evolution of the cosmos ( arcsec ) ) and was supported by grant no . r01 - 2006 - 000 - 10965 - 0 from the basic research program of the korea science @xmath230 engineering foundation ( i.c . @xmath230 y.k . ) , by the science research center program of the korea science and engineering foundation through the center for quantum spacetime(cquest ) of sogang university with grant number r112005021 ( h.b.k ) . d.  s.  goldwirth and m.  j.  perry , `` string dominated cosmology , '' phys . d * 49 * , 5019 ( 1994 ) [ arxiv : hep - th/9308023 ] ; 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we analyze homogeneous anisotropic cosmology driven by the dilaton and the self - interacting `` massive '' antisymmetric tensor field which are indispensable bosonic degrees with the graviton in the ns - ns sector of string theories with d - branes . 
 we found the attractor solutions for this system , which show the overall features of general solutions , and confirmed it through numerical analysis . 
 the dilaton possesses the potential due to the presence of the d - brane and the curvature of extra dimensions . in the presence of the non - vanishing antisymmetric tensor field , 
 the homogeneous universe expands anisotropically while the d - brane term dominates . 
 the isotropy is recovered as the dilaton rolls down and the curvature term dominates . with the stabilizing potential for the dilaton 
 , the isotropy can also be recovered .