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one of the most puzzling problems nowadays in physics is the issue of the late - time acceleration of the universe @xcite . a possible approach to tackle this problem is within the frame - work of self - accelerating universes @xcite ; i.e. could a modification of gravity at late - time and on large scale be the cause of the current inflationary phase of the universe ?
in other words , could this modification of gravity on large scales provide an _ effective negative pressure _ that would fuel the late - time acceleration of the universe ?    a possible approach to tackle these questions is the dvali , gabadadze and porrati ( dgp ) scenario @xcite , which corresponds to a five - dimensional ( 5d ) model . in this model ,
our universe is a brane ; i.e. a 4d hyper - surface , embedded in a flat space - time .
the dgp model has two types of solutions : the self - accelerating branch and the normal one .
the self - accelerating brane is asymptotically de sitter .
this feature takes place without invoking any unknown dark energy component . on the other hand
, the normal branch requires a dark energy component to accommodate the current observations @xcite . despite the nice features of the self - accelerating dgp branch
, it suffers from serious theoretical problems like the ghost issue @xcite . in this paper
we propose a mechanism to self - accelerate the normal branch which is known to be free from the ghost issue @xcite .
this mechanism will be based on a modified hilbert - einstein action on the brane @xcite and the simplest gravitational option is an @xmath0 term @xcite .
we consider a brane , described by a 4d hyper - surface ( @xmath1 , metric g ) , embedded in a 5d bulk space - time ( @xmath2 , metric @xmath3 ) , whose action is given by @xmath4\;\right\}%\nonumber \\
+ \int_{h } d^4x\ , \sqrt{-g}\ ; \left\{\frac{1}{\kappa_5 ^ 2 } k\;+\alpha f(r ) + \mathcal{l}_m \right\}\ , , \label{action}\end{aligned}\ ] ] where @xmath5 is the 5d gravitational constant , @xmath6 $ ] is the scalar curvature in the bulk and @xmath7 the extrinsic curvature of the brane in the higher dimensional bulk . on the other hand
, @xmath8 is the scalar curvature of the induced metric on the brane , @xmath9 , and @xmath10 is a constant that measures the strength of the generalised induced gravity term @xmath0 and has mass square units .
notice that therefore the function @xmath0 has mass square units .
finally , @xmath11 corresponds to the matter lagrangian of the brane .
we will assume that the brane splits the bulk in two symmetric pieces .
the previous action , includes as a particular case the dgp model @xcite for @xmath12 and @xmath13 where @xmath14 is proportional to the 4d gravitational constant .
it can be shown that the total energy density of the brane is conserved ( we refer the reader to @xcite for more details ) .
in particular , the energy density of matter on the brane is conserved .    in what follows ,
we consider a homogeneous and isotropic brane .
the matter sector on the brane can be described by a perfect fluid with energy density @xmath15 and pressure @xmath16 , where @xmath15 is conserved as we have pointed above . on the other hand ,
an effective energy density and an effective pressure associated to the energy momentum tensor coming from the @xmath0 term on action can be defined as follows @xcite @xmath17 , \label{rhof}\\ p^{(f)}&= & 2\alpha\left\{\left(2\dot h+3h^2+\frac{k}{a^2}\right)f'-\frac12(rf'-f ) \left[\ddot r f''+(\dot r)^2f'''+2h\dot r f''\right ] \right\},\label{pf}\end{aligned}\ ] ] notice that the definition of @xmath18 and @xmath19 is different from the standard 4d definition in @xmath0 models @xcite . on the other hand ,
the energy density is conserved on the brane .
the modified friedmann equation on the brane can be written as @xmath20 while , the spatial component of einstein equation can be expressed as @xmath21 where the energy density @xmath22 and the pressure @xmath23 are defined as @xmath24    for simplicity , on equations ( [ friedmann ] ) and ( [ rhop ] ) we have used the spatially flat chart of the brane .
a de sitter universe is the simplest cosmological solution that exhibits acceleration and therefore it is worthwhile to prove the existence of this solution in our model and study its stability .
this would be a first step towards describing in a realistic way the late - time acceleration of the universe in an @xmath0 brane - world model .
this approach will also enable us to look for self - accelerating solutions on the modified normal dgp branch .
so , in this section , we first obtain the fixed points of the model corresponding to a de sitter space - time and then we study their stability under homogeneous perturbations .      in our model , the hubble parameter for de sitter solutions can be expressed as term we will disregard the matter content in our analysis of de sitter branes . ]
@xmath25}%\nonumber\\\end{aligned}\ ] ]    where @xmath26 , the subscript 0 stands for quantities evaluated at the de sitter space - time , @xmath27 and @xmath28 .
we recover the dgp model for @xmath12 .
in fact , in that case , the de sitter self - accelerating dgp branch is obtained for @xmath29 and the normal dgp branch or the non - self - accelerating solution for @xmath30 .
when the brane action contains curvature corrections to the hilbert - einstein action given by the brane scalar curvature , the branch with @xmath30 is no longer flat and accelerates ( cf .
[ ploth ] ) .
therefore , an @xmath0 term on the brane action induce in a natural way self - acceleration on the normal branch .
most importantly , it is known that such a branch is free from the ghost problem ( see @xcite and references therein ) .
the reason behind the self - acceleration of the generalised normal brane is the presence of the effective energy density @xmath31 on the modified friedmann equation on the brane .
this can be easily shown by comparing the friedmann equation ( [ desitterh1 ] ) with that of modified gravity on brane world - models @xcite     for the two branches that generalise the dgp solution versus the rescaled energy density @xmath32 defined as @xmath33 .
the blue star corresponds to the normal dgp branch which is flat .
the red star corresponds to the self - accelerating dgp branch . on the other hand
, the blue curve corresponds to the generalised ( by the inclusion of the @xmath0 term ) self - accelerating branch , while the red curve corresponds to the generalised ( by the inclusion of the @xmath0 term ) normal branch .
the figure on the right corresponds to a zoom of the normal branch as it appears on the figure of the left.,title="fig:",width=264 ] for the two branches that generalise the dgp solution versus the rescaled energy density @xmath32 defined as @xmath33 .
the blue star corresponds to the normal dgp branch which is flat .
the red star corresponds to the self - accelerating dgp branch . on the other hand
, the blue curve corresponds to the generalised ( by the inclusion of the @xmath0 term ) self - accelerating branch , while the red curve corresponds to the generalised ( by the inclusion of the @xmath0 term ) normal branch .
the figure on the right corresponds to a zoom of the normal branch as it appears on the figure of the left.,title="fig:",width=264 ]      we next analyse the stability of de sitter solutions under homogeneous perturbations up to first order on @xmath34 .
we will follow the method used in @xcite .
the perturbed friedmann equation ( [ friedmann ] ) implies an evolution equation for @xmath35 : @xmath36 where @xmath37 is defined as @xmath38 where @xmath39,\nonumber \\
m_{\rm{pert}}^2&=&\frac{f_0}{3f_{rr}}\left[1-\sqrt{1+\frac23\alpha^2\kappa_5 ^ 4f_0(f_0-\kappa_5 ^ 2uf_0)}\right]^{-1}\nonumber.\end{aligned}\ ] ] and @xmath40 .
all this quantities are evaluated at the de sitter background solution .
any de sitter solution is stable as long as @xmath37 is positive .
the terms defined on eq .
( [ othermass ] ) have the following physical meaning : ( i ) @xmath41 is the analogous quantity to @xmath37 in a 4d f(r ) model @xcite , ( ii ) @xmath42 is a purely background effect due to the shift on the hubble parameter respect to the standard 4d case and ( iii ) @xmath43 is a purely perturbative extra - dimensional effect .
if we assume that we are close to the 4d regime ; i.e. the hubble rate of the brane is close to its analogous quantity in a 4d @xmath0 model , then @xmath44 and @xmath45 .
consequently , @xmath42 tends to make the perturbation heavier .
however , the perturbative effect encoded on @xmath43 would make the perturbation lighter .
it can be shown that the extra - dimension has a _ benigner _ effect in the 4d f(r ) model ; i.e. @xmath46 , as long as ; i.e. the effective gravitational constant of the brane is positive . on the other hand , we have also assumed that we are slightly perturbing the hilbert - einstein action of the brane , i.e. @xmath47 .
therefore , @xmath48 is positive because @xmath49 . ]
we have presented a mechanism to self - accelerate the normal dgp branch which unlike the original self - accelerating dgp branch is known to be free of the ghost problem . the mechanism is based in including curvature modifications on the brane action . for simplicity , we choose those terms to correspond to an @xmath0 contribution , which in addition is known to be the only higher order gravity theories that avoid the so called ostrogradski instability in 4d models .    * acknowledgements *    m.b.l .
is supported by the portuguese agency fundao para a cincia e tecnologia through the fellowship sfrh / bpd/26542/2006 .
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despite the nice features of the dvali , gabadadze and porrati ( dgp ) model to explain the late - time acceleration of the universe , it suffers from some theoretical problems like the ghost issue . 
 we present a way to self - accelerate the normal dgp branch , which is known to be free of the ghost problem , by means of an f(r ) term on the brane action . 
 we obtain the de sitter self - accelerating solutions of the model and study their stability under homogeneous perturbations .