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the inflationary mechanism is still the most popular alternative to solve the main problems of the standard cosmology : the flatness , homogeneity , horizon and isotropy problems . but this mechanism is not the only one .
the ekpyrotic mechanism manages to solve these problems by means of a previous phase of contraction before the current phase of expansion ( see @xcite for a review ) . in this slow contraction phase ,
the universe is dominated by a stiff fluid with a determined equation of state ( eos ) : @xmath0 ( @xmath1 and @xmath2 are pressure and energy density , respectively ) . with this fluid ,
the universe became flat , homogeneous and isotropic before the expansion phase , and the seeds of the structure formation were generated during this contraction phase .
these seeds , generated by quantum fluctuations , are nearly scale - invariant and are in agreement with the latest data @xcite
.    among the researches of the ekpyrotic universe , there exists the belief that this mechanism solves the problems of the standard cosmology as well as avoids the problems of the inflationary paradigm @xcite . according to these researches , the inflationary mechanism has two important problems : the initial conditions and multiverse - unpredictability . on the other hand
, the ekpyrotic cosmology does not have both problems andwith the aid of other ingredients , for example , the galileon cosmology and ghost condensate @xcitemanages to avoid another issue in standard cosmology : the problem of the initial singularity . as we can see , the ekpyrotic cosmology offers us an interesting option to the inflation .
moreover , the ekpyrotic cosmology may be extended to a cyclic cosmology . in this cyclic scenario @xcite ,
one has a endless sequence where the universe expands and contracts .
there is no beginning for the timetherefore , the problem of initial conditions is solved in this extended version .
the ekpyrotic mechanism is able to remove debris generated in a previous cycle , such as black holes ( bhs ) .
thus the existence of such a mechanism is crucial to build cyclic cosmologies .
any modern scientific cosmology has the isotropy as a feature . from the beginning of the modernity
, the cosmology assumes that there is no favourite place or direction in the universe ( contrary to both the plato s cosmology in his _ timaeus _ and the aristotle s view in _ on the heavens _ ) .
the physics , from this time , describes the universe as isotropic at large scales .
hence , the current scientific cosmology , an einsteinian cosmology , is assumed to be isotropic .    in this work , by using a determined approach , we intend to show that bhs supported by an isotropic fluid exclude an eos given by the ekpyrotic phase . from the mechanism developed in @xcite , @xcite , @xcite , @xcite , @xcite , @xcite and generalized in @xcite , we study , using the schwarzschild metric as ansatz , the family of deformed bhs with spherical symmetry supported by an isotropic fluid obtained in @xcite .
we point out that these solutions are forbidden when supported by a stiff fluid ( @xmath3 )
. this result may be interpreted as an indirect indication of the ekpyrotic phase as a period that removes inhomogeneities , anisotropies and debris of a previous cosmic phase in cyclic cosmologies that use the ekpyrotic mechanism .
the structure of this paper is presented in the following : in section 2 we presented the ekpyrotic mechanism and its main features ; in section 3 we show the solutions , supported by a isotropic fluid , constructed by means of deformations in the schwarzschild solution ; in section 4 , the final remarks .
we adopt the metric signature @xmath4 and @xmath5 , where @xmath6 is the newtonian constant , and @xmath7 is the speed of light in vacuum .
such as the standard big bang cosmology , the ekpyrotic cosmology assumes a friedmann - lematre - robertson - walker ( flrw ) metric as a solution of the gravitational field equations to describe the spacetime fabric . in the first version @xcite , a brane world model was used .
but , some years later , a four - dimensional version , an effective model , was introduced in @xcite . assuming the flrw metric @xmath8,\label{frlw}\ ] ] where @xmath9 is the scale factor , which depends on the cosmic time @xmath10
, @xmath11 determines the spatial curvature ( @xmath12 flat universe ; @xmath13 open universe ; @xmath14 closed universe ) , and the cosmic matter - energy content is described by a perfect fluid @xmath15 , the generalized friedmann equation reads @xmath16 with @xmath17 , where dot represents a derivative w.r.t .
time @xmath10 ; @xmath18 and @xmath19 mean the energy densities of non - relativistic matter ( including dark matter ) and radiation , respectively .
the term which depends on the @xmath20 indicates the energy density of anisotropies .
lastly , @xmath21 denotes the energy density of the scalar field that generates the ekpyrotic phase  interpreted as a perfect fluid in the friedmann equation .
such a field has the action @xmath22 where @xmath23 is the metric determinant , and an eos written as @xmath24 typically , the potential is given by @xmath25 , with @xmath26 and @xmath7 playing the role of constants .
a very steep potential of this type produces an eos for the ekpyrotic phase with a dominant fluid given by @xmath27 .
such a matter content is dominant in eq .
( [ friedmann ] ) during a contraction phase ( @xmath28 with @xmath29 ) .
the term that describes the anisotropies , @xmath30 is not dominant over the last term in eq .
( [ friedmann ] ) .
then , the slow contraction phase , where the stiff fluid is dominant over other matter contents , produces a homogeneous and isotropic universe , free of the belinsky - khalatnikov - lifshitz ( bkl ) instabilities , before the current expansion phase .
the authors in @xcite have shown that the ekpyrotic mechanism works even when the initial conditions of this slow contraction phase are highly inhomogeneous and anisotropic .    however , the standard ekpyrotic mechanism with one scalar field is considered tachyonically unstable ( see , for example , @xcite ) . to solve this problem , another scalar field
is introduced to generate a stable mechanism with nearly scale - invariant perturbations and small non - gaussianity @xcite .    in the next section
, we will show how to build bh solutions supported by an isotropic fluid . to obtain these bhs , the approach which deforms bhs will be presented .
contrary to the work @xcite , in this work , we do not study the initial conditions of the ekpyrotic phase and its evolution . in this article , we assume an isotropic fluid ( as a linear constraint in the deformation approach ) and show that an ekpyrotic phase does not support these deformed bhs .
that is , an isotropic universe with eos @xmath3 , or an ekpyrotic phase where the universe is already isotropic ( close to the bounce ) , does not support the deformed geometries .
the novel part of this work is the use of the deformation approach to show these results .
it is possible to build isotropic bh solutions by means of deformations . from the standard singular solutions , such as schwarzschild ,
reissner - nordstrm , schwarzschild-(a)-ds and others @xcite , and regular solutions @xcite , at linear level , deformed bhs are obtained by imposing a linear constraint in the energy - momentum tensor ( the approach was generalized in @xcite , where other constraints were used ) .
these deformed solutions are close to the standards bhs .
a linear constraint in the energy - momentum tensor to generate isotropic solutions means : @xmath31 where @xmath32 @xmath33 and @xmath34 are the radial and tangential pressures , respectively , and assume the same value due to constraint ( [ constraint_2 ] ) throughout this work .
the constraint ( [ constraint_2 ] ) used in the field equations ( [ field_equations ] ) , with the aid of the general metric with spherical symmetry @xmath35 yields to @xmath36-\frac{1}{2}\left[\frac{a'(r)b'(r)}{a(r)}-\left(\frac{a'(r)}{a(r)}\right)^{2}\right]\ ] ] @xmath37-\frac{1}{2r}b'(r)-\frac{a''(r)b(r)}{2a(r)}=0.\label{constraint_3}\ ] ] it is worth noting that the deformed solution has @xmath38 in this sense , the birkhoff theorem is not valid .    to obtain a set of solutions close to the known solutions , we choose a function @xmath39 , and the approach yields to a function @xmath40 close to the standard case where @xmath41 in eq .
( [ metric ] ) .
the approach modifies only the function @xmath40 because this modification leads to the solutions with a different spacetime structure ( event horizons , as we will see below ) . in this work ,
we follow @xcite and deal with deformations from the schwarzschild metric , where @xmath42 the ansatz is completed by @xmath43 the subscript @xmath44 stands for linear , and @xmath45 may be seen as the first - order expansion coefficient . then
, we want to obtain deformed solutions until the linear level ( in @xcite there is a discussion on the possibility of generating deformed solutions beyond the linear level ) . according to @xcite ,
the constant @xmath46 fixes basically three kind of solutions : for @xmath47 one has either regular or singular black holes , @xmath48 may determine wormholes , and for @xmath49 , one has extremal black holes . for @xmath50
the standard cases are restored .    in the linear regime ,
the function @xmath51 is determined from @xmath52,\label{blin2}\ ] ] which is solution of eq .
( [ constraint_3 ] ) using the ansatz ( [ a_0])-([b ] ) , where @xmath53 the zeros of @xmath54 are important because may determine the singularities of the spacetime . in this work , @xmath54 has one real root , @xmath55 . using the ansatz ( [ a_0 ] ) in the eqs .
( [ blin2])-([h(r ) ] ) , one has @xmath56,\label{blin3}\ ] ] and the metric ( [ metric ] ) is completely determined . the zeros of @xmath57 fix the horizons ( when a metric is written in the form ( [ metric ] ) , the localization of horizons is given by the zeros of @xmath58 ) . according to @xcite , bhs are possible only with @xmath47 ( our value of @xmath46 corresponds to the @xmath59 in the cited paper ) .    for @xmath60
, the function @xmath40 has two zeros , @xmath61 .
the largest zero corresponds to the event horizon @xmath62 .
the metric is singular at @xmath63 .
then , the spacetime structure reads @xmath64 this case corresponds to a singular bh in a noncompact universe .
when @xmath65 , the situation is quite different : there are three zeros in @xmath40 ( the third zero is not zero of @xmath66 ) .
there is a maximum value of @xmath67 @xmath68 . according to @xcite , these values of @xmath46
lead to a singular bh in a compact universe .
the spacetime structure is given by @xmath69 with @xmath70 then , these are the two types of solutions when the ansatz is given by eqs .
( [ a_0])-([b ] ) assuming isotropy .    with @xmath66 and @xmath40 fixed , the values of the components of the energy - momentum tensor are : @xmath71 with these components in the energy - momentum tensor , an eos is possible : @xmath72 which is almost constant and always negative outside the event horizon ( see fig .
1 ) . for large @xmath67
, @xmath73 is about @xmath74 . in the case @xmath60 ,
one has the simple limit @xmath75 at the event horizon , one has @xmath76 . in this sense , these deformed metrics may not be supported ( such as the schwarzschild-(a)-ds metric is by a fluid with eos @xmath77 ) by a stiff fluid ( @xmath3 ) , dominant in the ekpyrotic phase . then , at linear level , bhs supported by an isotropic stiff fluid are ruled out .
but not only a stiff fluid : as we can see , these metrics are not supported by either a fluid with eos @xmath78 ( radiation ) or a phantom - like fluid ( @xmath79 ) . outside the event horizon
, the eos assumes @xmath80 , which are values corresponding to a quintessence fluid .     obtained from isotropic deformed solutions . as we can see , this eos is pretty much constant outside the event horizon and rules out a stiff fluid @xmath81 , dominant in the ekpyrotic phase . the dashed line indicates the event horizon . in this graphic
we use @xmath82 . ]
the ekpyrotic cosmology is an option for the inflationary mechanism to solve the problems of the standard cosmology ( including the initial singularity ) and to generate nearly scale - invariant quantum fluctuations .
this alternative cosmology is characterized by a slow phase of contraction , where the equation of state ( eos ) of the matter - energy contend in this phase is dominated by a stiff fluid @xmath83 .
we show that  by means of a procedure of generating deformed isotropic black holes ( bhs ) at linear level in @xmath40 , adopting a linear constraint in the energy - momentum tensor  the metrics with spherical symmetry obtained from the schwarzschild metric are not supported by a stiff fluid .
these metrics are not supported by a stiff fluid in the same sense that the schwarzschild-(a)-ds metrics may be interpreted as solutions with spherical symmetry supported by a fluid with eos @xmath77 .
we believe that this result may be seen as an _ indirect _ indication ( it is not a strong proof ) of the ability of the ekpyrotic phase to leave the universe free of inhomogeneities and debris ( and bhs may be interpreted as debris ) in a previous phase before the current phase of cosmic expansion
. this result may be important for cosmological cyclic models that assume an ekpyrotic phase .
the ekpyrotic phase ( a slow contraction cosmic phase before the current expansion phase ) manages to solve the main problems of the standard cosmology by means of a scalar field interpreted as a cosmic fluid in the friedmann equation . 
 moreover , this phase generates a nearly scale - invariant spectrum of perturbations in agreement with the latest data . 
 then , the ekpyrotic mechanism is a serious possibility to the inflationary model . in this work  by using the approach of deforming metrics at linear level  we point out that it is impossible to generate a black hole with spherical symmetry supported by an isotropic fluid in this cosmological scenario .