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by now , the inflationary universe @xcite is generally recognized to be a likely scenario that leads to the big bang .
so far , its predictions of the flatness and the scale - invariant power spectrum of the density perturbation that seeds structure formations are in good agreement with observations @xcite .
future data will test inflationary predictions to a very high accuracy .
in fact , there is widespread hope that the enormous growth in scale during inflation may provide a kind of planck scale microscope , allowing us one day to probe stringy and/or brane world effects from precision cosmological measurements @xcite .    in conventional inflationary models ,
the physics lies in the inflaton potential .
although there are constraints from observations , such as enough number of e - foldings , the amplitude of the density perturbation etc . , there are very few theoretical constraints on the underlying dynamics of the inflaton field . as a result
, this scalar field potential can take a large variety of shapes . in hybrid inflation
@xcite or other variations , additional fields and/or parameters are introduced , allowing even more freedom .
however , the problem in a conventional 4-dimensional theory is the difficulty of finding an appropriate inflaton potential . in general , a potential that yields the correct magnitude of density perturbation and satisfies the slow - roll condition is not well - motivated and requires some fine - tuning .
generically , such fine - tuning is not preserved by quantum corrections .
recently , motivated by the idea of brane world @xcite , the brane inflationary scenario@xcite was proposed , where the inflaton is identified with an inter - brane separation : @xmath0 .
inflation ends when the branes collide , heating the universe that starts the big bang . in this scenario ,
the inflaton potential has a geometric interpretation .
in particular , higher dimensional gravity ( or more generally , bulk modes or closed string states ) dictates the form of the inflaton potential .
for example , the inflaton potential may simply be the newton s potential between branes .
this visualization of the brane dynamics allows one to implement inflation physics pictorially .
suitable inflaton potentials naturally appear in the brane world scenario , and their forms are robust under quantum corrections . in this paper
, we demonstrate further this unusual feature of brane inflation . if the fundamental string scale is substantially above a few tev , brane inflation will be a valuable testing ground for the brane world scenario .
the motion of the branes is dictated by interbrane ( which can be brane - brane , brane - orientifold or brane - antibrane ) forces . moreover , a velocity - dependent term in the potential is present generically .
so when branes move , this term may become important .
this velocity - dependent term is calculable in string theory ; the precise values of the parameters that appear in this term depend on the details of the model .
in fact , one might expect such a velocity - dependent term from the post - newtonian approximation in higher dimensional gravity . in this paper
, we also examine the impact of such a velocity - dependent force on the slow - roll and the power spectrum of the density perturbation .
we find that the effect on the tilt of the spectral index of the density perturbations may be small .
however , such a term is likely to have an observable effect on the ratio of the tensor to scalar perturbations .
moreover , the scale dependence of the spectral index is also modified .
therefore , a global analysis of data from various measurements may allow us to distinguish these effects from that of the conventional inflationary scenario .
our starting part is the effective action @xmath1 for the canonically normalized inflaton @xmath2 of the form : @xmath3 where @xmath4 is the effective potential .
the precise form of @xmath5 will be specified later on .
note that the field @xmath2 in the brane world is related to the separation @xmath6 between the branes by @xmath7 , where @xmath8 is the string scale . to lowest order @xmath9 .
there are also higher order corrections to @xmath10 . in the usual inflationary scenario ,
the deviation of @xmath11 from unity is due to quantum effects .
typically , it takes the form of @xmath12 , where @xmath13 is a constant and @xmath14 is the coupling constant . here
, we will consider a different type of contribution to @xmath11 which is motivated by brane world physics ( or higher dimensional gravity ) .    in an expanding universe , @xmath15 , then , @xmath16    for inflation to take place , there is some choice of @xmath2 so that the potential @xmath17 satisfies the slow - roll conditions : latexmath:[\ ] ] the solution of @xmath171 does not change greatly with the velocity - dependent term , @xmath172 furthermore , @xmath158 and so the velocity - dependent term is also negligible .
therefore , @xmath173 where @xmath136 .
the spectrum is tilted to the red . if the lowest non - vanishing contribution to the potential comes from the dipole of a brane - orientifold plane system , @xmath174 which gives a slightly larger red tilt .
in this paper , we have examined the density perturbation spectrum in the brane inflationary scenario .
we have also studied the effect of a velocity - dependent potential .
the form of the inflation potential as well as the velocity - dependent potential is strongly motivated from higher dimensional gravity ( in particular , string theory ) . in the usual four - dimensional inflationary scenario ,
the deviation of @xmath11 from unity is due to quantum effects and is usually small . in brane inflation , however , the velocity - dependent potential can give rise to a significant effect in @xmath11 .
generically , the effect on the spectral index is comparable with that from the slow - roll parameter @xmath43 ( from eq.([index ] ) ) .
therefore , in models where @xmath43 is not small , this may be a measurable effect . even though this effect on @xmath46 can be small in a generic model
, the scale - dependence of @xmath46 has a stronger dependence on the velocity dependent term , as it scales @xmath175 by a factor of @xmath176 ( see eq .
( [ ktilt ] ) ) .
more importantly , the velocity - dependent potential can also modify the ratio of tensor to scalar fluctuations ( see eq.([t / s ] ) ) . with a global analysis of data from various measurements ,
one should be able to distinguish these effects from those that can be obtained from a conventional inflationary scenario .
we expect that a few percent deviation of @xmath39 from unity may be measurable .
it seems that the power spectrum index in a generic inflationary scenario in the brane world has a red tilt of a few percent .
the velocity dependent potential can further tilt the density perturbation spectrum to the red or to the blue , depending on the specific form of the potential .
the blue tilt due to the velocity - dependent term does not require extra fields , in contrast to other inflationary models ( _ e.g. _ , hybrid inflation ) .
the velocity - dependent potential in the brane flationary scenario is similar to that appears in k - inflation @xcite .
however , unlike k - inflation , in our scenario , inflation is still driven by the static potential .
the additional velocity dependent term provides modifications of the usual slow - roll inflation .
furthermore , the kinetic term and the static potential in the brane inflation scenario are related by string theory or by post - newtonian approximation ( see eq.([postnewton ] ) ) .
potential of the type @xmath177 has been studied in the context of quintessence @xcite .
it was argued in @xcite that this type of potential appear naturally in supersymmetric theory .
as we have described , this type of potential is also motivated when we study cosmology involving branes .
furthermore , the velocity dependent term has a close resemblance to that in k - essence @xcite
. it would be interesting to explore whether the specific form of k - essence motivated by brane dynamics may offer an explanation for the present accelerating universe .
the research of g.s . was supported in part by the doe grant fg02 - 95er40893 , the university of pennsylvania school of arts and sciences dean s fund , and the nsf grant phy99 - 07949 .
the research of s .- h.h.t . was partially supported by the national science foundation .
g.s . would like to thank the m - theory workshop at itp , santa barbara , and the theory division at cern for support and hospitality during the final stage of this work .
in this appendix , we derive the equation for the quantum fluctuation @xmath25 in the presence of a non - trivial @xmath11 , and show that at late time , @xmath25 and @xmath26 obey the same differential equation with respect to time .
we used this fact to derive eq.([gpformula ] ) analogous to the original derivation of @xcite .    to study quantum fluctuations , we split @xmath2 into the classical piece @xmath178 ( which satisfies ( [ eom ] ) ) and the quantum piece @xmath25 : @xmath179 the classical piece is homogeneous , whereas the fluctuations satisfy : @xmath180 where we have dropped the subscript @xmath13 for the classical part of @xmath2 .
we note that for scales outside the horizon ( hence the gradient term is negligible ) , @xmath25 and @xmath26 obeys the same time - dependent differential equation . by expanding @xmath25 in fourier modes : @xmath181
the equation of motion for @xmath182 is @xmath183 it is convenient to rewrite it in terms of @xmath184 . in the conventional case , @xmath9 , @xmath185 where prime denotes derivative with respect to conformal time @xmath186 . for de sitter space
, the solution is @xmath187 which reduces to @xmath188 for @xmath189 and @xmath190 for @xmath191 .    in the presence of a velocity - dependent potential , @xmath192 for scales inside the horizon , when the fluctuations are generated
, the dominant term is still @xmath193 .
therefore , @xmath194 is still given by the bunch - davies vacuum , @xmath195 hence the quantum fluctuations @xmath182 is still given by the de sitter temperature , @xmath196                        v.f .
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the inflaton potential in four - dimensional theory is rather arbitrary , and fine - tuning is required generically . by contrast , inflation in the brane world scenario has the interesting feature that the inflaton potential is motivated from higher dimensional gravity , or more generally , from bulk modes or string theory . 
 we emphasize this feature with examples . 
 we also consider the impact on the spectrum of density perturbation from a velocity - dependent potential between branes in the brane inflationary scenario . 
 it is likely that such a potential can have an observable effect on the ratio of tensor to scalar perturbations .    
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