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in a previous work@xcite , we introduced a new formalism where , instead of implementing a dynamical foliation by taking a spatial dependence of the fifth coordinate including its time dependence , we considered another extra dimension , the sixth dimension , making possible the implementation of two dynamical foliations in a sequential manner . the first one was considered by choosing the fifth coordinate depending of the cosmic time , and the second one by choosing the sixth coordinate as dependent of the 3d spatial coordinates ( in our case considered as isotropic ) . of course , all of these choices preserve the continuity of the metric . in addition , the 6d metric must be ricci - flat . this requirement is a natural extension of the vacuum condition used in the stm theory@xcite , in which 5d ricci - flat metrics are used@xcite and the cylinder condition has been eliminated in favor of retaining the metric s dependence on the extra coordinate . in simple words , we used the campbell - magaard theorem@xcite and its extensions for embedding a 5d ricci - flat space - time in a 6d ricci - flat space - time . the conditions of 6d ricci - flatness and the continuity of the metric gives us the foliation of the sixth coordinate . these conditions specify the sixth dimension as a function of the 3d spatial coordinates , in order to establish the foliation . this function , for a particular 6d metric , can be seen in 4d as a gravitational potential related to a localized compact object that has the characteristics of a white hole . from a more general point of view , this is a mechanism for inducing localized matter onto a time - varying 4d hypersurface by establishing a spatial foliation of a sixth coordinate from a 6d ricci - flat metric . the importance of this approach lies in that it can describe matter at both , cosmological and astrophysical scales , in an expanding universe . the use of 6d physics is currently popular in particle physics@xcite , but with the sixth dimension as time - like , rather than space - like . + in this letter we aim to study some predictions of this model on astrophysical scales . the power spectrum of matter is one of the most important statistics to describe the large - scale and astrophysical - scale structures of the universe . the studies developed in the last years have shown that on astrophysical scales the power spectrum of galaxies and clusters of galaxies can be satisfactorily expressed by a power law with an index between @xmath0 and @xmath1@xcite . on larger scales the spectrum turns over reaching a maximum on scales of @xmath2 . we shall suppose that the evolution of structure in the universe is only due to gravity . in order to describe a 6d vacuum , we consider the recently introduced 6d riemann - flat metric@xcite @xmath3 -d\psi^2 - d\sigma^2\ ] ] which defines a 6d vacuum state @xmath4 ( @xmath5 ) . we consider the 3d spatial space in spherical coordinates : @xmath6 ; here @xmath7 . the metric ( [ ini ] ) resembles the 5d ponce de leon one@xcite , but with one additional space - like dimension . furthermore , the coordinate @xmath8 is dimensionless and the extra ( space - like ) coordinates @xmath9 and @xmath10 are considered as noncompact . we define a physical vacuum state on the metric ( [ ini ] ) through the action for a scalar field @xmath11 , which is nonminimally coupled to gravity @xmath12,\ ] ] where @xmath13 is the ricci scalar and @xmath14 gives the coupling of @xmath11 with gravity . implementing the coordinate transformation @xmath15 and @xmath16 on the frame @xmath17 ( considering @xmath18 as a constant ) , followed by the foliation @xmath19 on the metric ( [ ini ] ) , we obtain the effective 5d metric @xmath20 - d\sigma^2,\ ] ] which is not ricci - flat because @xmath21 . however , it becomes riemann - flat in the limit @xmath22 i.e. @xmath23 , so that in this limit a 5d vacuum given by @xmath24 , ( @xmath25 ) . hence , we can take the foliation @xmath26 in the metric ( [ 5d ] ) on the sixth coordinate , and we obtain the effective 4d metric @xmath27,\ ] ] where @xmath28 is the cosmic time , and @xmath29 is the hubble parameter for the scale factor @xmath30 , with @xmath31 . the einstein equations for the effective 4d metric ( [ 4d ] ) are @xmath32 ( @xmath33 ) , where @xmath34 is represented by a perfect fluid : @xmath35 , @xmath36 and @xmath37 being the pressure and the energy density on the effective 4d metric ( [ 4d ] ) . in a previous work@xcite @xmath38 was found for a puntual mass @xmath39 ( @xmath40 is the planckian mass ) located at @xmath41 , in the absence of expansion ( @xmath42 ) @xmath43 here , @xmath44 is the value of @xmath45 such that @xmath46 and @xmath47 the gravitational constant . hence , the function @xmath38 describes the geometrical deformation of the metric induced from a 5d flat metric , by a mass @xmath48 . this function is @xmath49 ( or @xmath50 ) for @xmath51 ( @xmath52 ) , respectively . furthermore , @xmath53 , and thereby the effective 4d metric ( [ 4d ] ) is ( in their 3d ordinary spatial components ) asymptotically flat . in this analysis we are considering the usual 4-velocities @xmath54 . the equation of state for a given @xmath45 is@xcite @xmath55 e^{-2 h t } } { \left[h^2 - \frac{2 g m_n}{r^3 } e^{-2 h t}\right ] } , \ ] ] being @xmath56 . from the equation ( [ state ] ) we can see that at the end of inflation , when the number of e - folds is sufficiently large , the second term in ( [ state ] ) becomes negligible on cosmological scales [ on the infrared ( ir ) sector ] , and the equation of state on this sector describes an asymptotic vacuum dominated ( inflationary ) expansion : @xmath57    on the other hand , for @xmath58 , we obtain ( on arbitrary scales ) @xmath59}{\left [ h^2 - \frac{2gm_n}{r^3}\right]},\ ] ] which for a non - expanding universe ( @xmath60 ) gives us , for @xmath51 , the equation of state for primordial galaxies @xmath61 which means that primordial galaxy formation should be possible on scales smaller than @xmath44 . the effective 4d action for the universe is @xmath62,\ ] ] where @xmath63 is the effective 4d ricci scalar for the effective 4d metric ( [ 4d ] ) , @xmath64 gives the coupling of the scalar field @xmath65 with gravity on the background induced by the foliation of the first extra dimension @xmath9 at @xmath66 and @xmath67 gives the coupling of @xmath11 with gravity , on the background induced by the foliation of the second extra dimension @xmath10 at @xmath68 . the equation of motion for the field @xmath11 on the metric ( [ 4d ] ) is @xmath69 - \frac{1}{(1 + 2\phi_n(r))^2 } \frac{\partial\phi_n}{\partial r } \frac{\partial \varphi}{\partial r } \right . \nonumber \\ & & \left.+ \frac{1}{r^2 { \rm sin}(\theta ) } \frac{\partial}{\partial\theta } \left[{\rm sin}(\theta ) \frac{\partial\varphi}{\partial\theta}\right ] \right . \nonumber \\ & & + \left . \frac{1}{r^2 { \rm sin}^2(\theta ) } \frac{\partial^2\varphi}{\partial\phi^2 } \right ] + \left [ \xi_1 \left.^{(4)}{\cal r}\right|_{\phi_n=0 } + \xi^{n , l}_2(r ) \left.^{(4 ) } { \cal r}\right|_{h=0 } e^{-2 h t}\right ] \varphi = 0.\label{5}\end{aligned}\ ] ] notice that under this approach , the expansion is affected by a geometrical deformation induced by the gravitational attraction of a white hole of mass @xmath70 . this deformation is described by the function @xmath38 , which tends to zero on cosmological scales [ see figure ( [ f1 ] ) ] .      in this section we study the weak field approximation for the equation ( [ 5 ] ) . in that limit approximation the function @xmath38 in ( [ sphi ] ) can be written as @xmath71 @xmath48 being the mass of the compact object located at @xmath41 . note that the function ( [ 6 ] ) , as well as the exact one ( [ sphi ] ) , goes to zero at @xmath72 . for @xmath73 there is a stable equilibrium for test particles at @xmath74 and it exhibits a gravitational repulsion ( antigravity ) for @xmath75 . hence , this object has the properties of a white hole @xcite . in the figure ( [ f1 ] ) we have plotted @xmath38 given by the eqs . ( [ sphi ] ) [ the exact expression plotted with a continuous line ] and ( [ 6 ] ) [ the weak field approximated expression plotted with a pointed line ] , respectively , for @xmath52 . notice that the difference between both is more important on smaller scales . furthermore , it is evident that the exact expression of @xmath38 is more sensitive to the interaction . in order to obtain solutions of the equation ( [ 5 ] ) we propose @xmath76 . with this choice and using the fact that [ see the eq . ( 11 ) in @xcite ) ] : @xmath77 , we obtain @xmath78 & + & \left [ 3 g m_n ( 1 + 2\phi_n ) + r\phi_n\right ] \frac{\partial \varphi_r}{\partial r } \nonumber \\ & = & \varphi_r \left [ \frac{12 g m_n}{r } \xi^{n , l}_2(r ) -\alpha_t r^2 + \alpha_r\right ] ( 1 + 2\phi_n ) , \label{1b } \\ { \rm sin}(\theta ) \frac{\partial}{\partial\theta } \left [ { \rm sin}(\theta ) \frac{\partial \varphi_{\theta,\phi } } { \partial\theta}\right ] & + & \frac{1}{{\rm sin}^2(\theta ) } \frac { \partial^2\varphi_{\theta,\phi}}{\partial\phi^2 } = \alpha_r \varphi_{\theta,\phi } , \label{1c}\end{aligned}\ ] ] where @xmath79 and @xmath80 are separation constants . the solution of the equation ( [ 1c ] ) is given by the spherical harmonics @xmath81 [ @xmath82 @xmath83 where @xmath84 is a separation constant and @xmath85 are the legendre polynomials : @xmath86 ( 1-x^2)^{m/2 } ( d^{l+m}/ dx^{l+m } ) ( x^2 - 1)^l$ ] .    in order to study astrophysical implications for the solutions of eqs . ( [ 1a ] ) , ( [ 1b ] ) and ( [ 1c ] ) , we shall concentrate on the dispersive case ( @xmath87 ) for @xmath65 . in this case @xmath88 is the wavenumber related to the coordinate @xmath45 . on the other hand , for solving the equation ( [ 1b ] ) , we propose @xmath89 such that @xmath90 $ ] . with this choice ( [ 1b ] ) can be replaced by the equations @xmath91=0 , \label{3a } \\ & & \frac{d\varphi_{k_r}}{dr } \left[6 g m_n(1 + 2\phi_n ) + 2r\phi_n\right ] + \varphi_{k_r } \left [ r^2 \left(f^2_n - \frac{df_n}{dr}\right ) - 2 r f_n + 2\phi_n \left(k^2_r r^2 - l(l+1)\right ) \right.\nonumber \\ & & - \left . 3 g m_n ( 1 + 2\phi_n ) f_n - r \phi_n f_n - \frac{12 g m_n \xi^{n , l}_2(r)}{r } \left(1 + 2\phi_n \right)\right]=0 . \label{3b}\end{aligned}\ ] ] the solution for the equation ( [ 3a ] ) is @xmath92,\ ] ] where @xmath93 are the bessel functions . on the other hand , from the equation ( [ 3b ] ) we obtain the coupling @xmath67 @xmath94 - \left [ r^2\left(f^2_n - \frac{df_n}{dr}\right ) \right . \nonumber \\ & - & \left.\left . 2 r f_n + 2\phi_n \left[k^2_r r^2 - l(l+1)\right ] -3 g m_n \left ( 1 + 2\phi_n \right ) f_n - r \phi_n f_n\right]\right\ } ,    \label{coup}\end{aligned}\ ] ] with @xmath95 \left[3 g m_n \left(1 + 2\phi_n(r)\right ) + r \phi_n(r)\right]$ ] . the solution of the equation ( [ 1a ] ) for the dispersive case is @xmath96,\ ] ] @xmath97 being a constant , @xmath98 and @xmath99 is the hankel function of second kind . the complete solution for the field @xmath65 can be written as @xmath100 } { \large\int } d^3 k_r \sum^{l}_{m =- l } \sum^{n-1}_{l=0 } \left [ a_{k_r l m } y_{l , m}(\theta,\phi ) \varphi_{k_r}(r ) \xi_{k_r}(t ) \right . \nonumber \\ & + & \left . a^{\dagger}_{k_r l m } y^*_{l , m}(\theta,\phi ) \varphi^*_{k_r}(r ) \xi^*_{k_r}(t)\right ] , \label{var}\end{aligned}\ ] ] where @xmath101 and @xmath102 are given respectively by the expressions ( [ 3c ] ) and ( [ 26 ] ) . in this section we shall study the structure formation during inflation on astrophysical scales . in the framework of cosmological scales , we understand that these are small scales , on which one feels the presence of the compact object [ and thus we shall consider @xmath103 . we must understand the present astrophysical scales ( @xmath104 100 mpc ) . we shall refer to this part of the spectrum as the small - scale ( ss ) spectrum . it is known from observation that galaxies and clusters of galaxies are correlated . this should be responsible for differences in the clustering properties of the populations in the nearby universe . one population is characteristic for rich superclusters , and the other for poorer ones . the former population has a power spectrum with a sharp peak and a correlation function with zero crossing near @xmath105 . the later population has a flatter power spectrum and a zero crossing of the correlation function near @xmath106 . the effective small - scale squared fluctuations in presence of @xmath107 are given by[see @xcite ] @xmath108 } \int _ { \epsilon k_{h}}^{k_{h}}\frac{dk_{r}}{k_{r}}k_{r}^{3}\left.\left[\xi _ { k_r}\xi % % @ _ { k_r}^{*}\right]\right|_{ir}^{(\phi_n(r)\simeq 0)}\ ] ] where @xmath109 is a dimensionless constant parameter , and @xmath110 is the wavenumber related with the hubble radius at the time @xmath111 ( when the horizon re - enters ) , and @xmath112 is the planckian wavenumber . the asymptotic expression for the modes on the ir sector is@xcite @xmath113 inserting ( [ modosir ] ) into ( [ sma3 ] ) we obtain @xmath114 } k_{h}^{3 - 2\nu}(1-\epsilon)^{3 - 2\nu}.\ ] ]    hence , the expression ( [ sma4 ] ) becomes @xmath115 } \left<\varphi^2\right>_{ir}^{(\phi_n(r)\simeq 0)},\ ] ] where@xcite @xmath116 finally , we can write the power spectrum of @xmath117 , making @xmath118 and @xmath119 @xmath120 } k^{3 - 2\nu}_r.\ ] ] in the figure ( [ f2 ] ) we have plotted the coupling parameter @xmath67 in eq . ( [ coup ] ) on astrophysical scales [ @xmath121 , for @xmath122 , @xmath123 and @xmath124 . notice that we have used @xmath101 given in eq . ( [ 3c ] ) and @xmath38 given in eq . ( [ 6 ] ) . it is evident that the coupling parameter becomes more important as @xmath45 increases ( i.e. , for bigger scales ) . in the figure ( [ f3 ] ) we shows the power spectrum @xmath125 as a function of the wavenumber @xmath88 on astrophysical scales . there are three indices that are important on the whole spectrum . two of these dominate on astrophysical scales . the first one for @xmath126 , the second one on the range @xmath127 . the third index @xmath128 dominates on cosmological scales ( @xmath129 ) . we have used @xmath130 ( which corresponds to @xmath131 ) in all the graphics . we have studied the power spectrum on astrophysical scales of the @xmath132-expectation value . the model here studied predicts a spectrum that agrees qualitatively with experimental data@xcite . an interesting result is that we detect two different sectors with different power indices , which dominate on different astrophysical scales . the third index is @xmath133 , which is relevant on cosmological scales . another interesting result is the periodicity of the coupling @xmath67 . notice that its amplitude increases with @xmath45 . this coupling should be responsible for the correlation of the galaxies on astrophysical scales .    for simplicity , in this letter we are considered a de sitter expansion , where the hubble parameter @xmath18 is a constant . however , the formalism could be extended to whatever @xmath134 . the possibility of having a dynamical foliation using only 5d ( in models without gravitational sources ) , was explored@xcite . in our model the fifth dimension is responsible for the 4d de sitter expansion , which is physically driven by the inflaton field @xmath11 . from the physical point of view , the sixth dimension is responsible for the spatial curvature induced by the mass of the white hole ( located at @xmath41 ) . in more general terms , the fifth dimension is physically related to the vacuum energy density which is the source of the effective 4d global inflationary expansion whereas the sixth one induces local gravitational sources . 99 j. e. madriz aguilar and m. bellini , phys * b648 * , 19 ( 2007 ) . j. overduin , p. s. wesson , phys . rept . * 283 * , 303 ( 1997 ) . p. s. wesson , gen . rel * 16 * , 193 ( 1984 ) ; p. wesson , gen . . grav . * 22 * , 707 ( 1990 ) ; p. s. wesson , phys . * b276 * , 299 ( 1992 ) ; p. s. wesson and j. ponce de leon , j. math . phys . * 33 * , 3883 ( 1992 ) ; h. liu and p. s. wesson , j. math . phys . * 33 * , 3888 ( 1992 ) ; p. wesson , h. liu and p. lim , phys . lett . * b298 * , 69 ( 1993 ) . n. i. katzourakis , _ bundle - theoretical globalization of campbell - magaard embedding theorem in the context of md gravity_. e - print : math - ph/0407067 . c. romero , r. tavakol , gen . * 28 * , 365 ( 2007 ) ; s. s. seahra , p. s. wesson , class . 20 * , 1321 ( 2003 ) . i. bars , yueh - cheng kuo , phys . lett . * 99 * : 041801 ( 2007 ) . j. einasto , _ et . al _ , astroph . j. * 519 * , 441 ( 1999 ) . j. ponce de leon , gen . rel . grav . * 20 * , 539 ( 1988 ) . r. kallosh , a. linde , phys . rev . * d52 * , 7137 ( 1995 ) . review of particle physics : phys . b592 * , 207 ( 2004 ) . madriz and m. bellini , eur . j. * c42 * , 349 - 354 ( 2005 ) .
using a previously introduced model in which the expansion of the universe is driven by a single scalar field subject to gravitational attraction induced by a white hole during the expansion ( from a 6d vacuum state ) , we study the origin of squared inflaton fluctuations spectrum on astrophysical scales .    epsf.sty