Unified Model of k-Inflation, Dark Matter & Dark Energy
aa r X i v : . [ a s t r o - ph . C O ] N ov Uni(cid:28)ed Model of k -In(cid:29)ation, Dark Matter & Dark EnergyNilok Bose ∗ and A. S. Majumdar † S. N. Bose National Centre for Basi S ien es, Blo k JD, Se tor III, Salt Lake, Cal utta 700098, India(Dated: November 13, 2018)We present a k-essen e model where a single s alar (cid:28)eld is responsible for the early expansion ofthe universe through the pro ess of k-in(cid:29)ation and at appropriate subsequent stages a ts both asdark matter and dark energy. The Lagrangian ontains a potential for the s alar (cid:28)eld as well as anon- anoni al kineti term, and is of the form F ( X ) V ( φ ) whi h has been widely used as a k-essen eLagrangian. After the period of in(cid:29)ation is over the model an be approximated as purely kineti k-essen e, generating dark matter and dark energy at late times. We show how observational resultsare used to put onstraints on the parameters of this model.PACS numbers: 98.80.-k, 98.80.Cq, 95.36.+xI. INTRODUCTIONTill date the nature of both dark matter and dark en-ergy is largely unknown and they onstitute one of thebiggest puzzles of modern osmology. The dynami s ofthe pro ess driving the urrent a eleration of the uni-verse is still un lear but there exist a wide variety ofapproa hes that ould theoreti ally a ount for this a - eleration. The ombination of observations of high red-shift supernovae, CMBR and large s ale stru ture have ategorized the urrent energy density of the universe to onsist of approximately 73% dark energy, whi h drivesthe late time a eleration of the universe, and approxi-mately 23% dark matter whi h lusters and is responsiblefor the formation of large-s ale stru ture in the universe(see [1℄ and referen es therein). These observations, in- luding those of the nearly s ale-independent density per-turbations, are also in onformity with the widely heldview that the early universe underwent a brief period ofa elerated expansion, dubbed as in(cid:29)ation.Sin e a elerated expansion is a ommon feature forboth the very early and the very late universe, it is plau-sible that some ommon me hanism ould be responsiblefor both. Several models have been onstru ted to ex-plain in(cid:29)ation and dark energy using a single s alar (cid:28)eld(see, for example, quintessential in(cid:29)ation [2℄). It is alsopossible for the two dark omponents of the universe tobe the manifestations of a single entity, and a onsider-able number of models an be found in the literature thattry to unify dark matter and dark energy (for instan e[3℄, [4℄). Apart from the above s hemes there are modelsthat try to unify in(cid:29)ation and dark matter (for instan e[5℄) and also those that attempt to unify all three, viz.in(cid:29)ation, dark matter and dark energy (for instan e [6℄).In many of these uni(cid:28) ation models the dynami s ofone or more s alar (cid:28)elds plays the entral role. In fa t,the idea of k -essen e driven by s alar (cid:28)eld with a non- anoni al kineti term motivated from the Born-Infeld ∗ Ele troni address: nilokbose.res.in † Ele troni address: ar hanbose.res.in a tion of string theory [7℄, was (cid:28)rst introdu ed as a pos-sible model for in(cid:29)ation [8℄. Later, it was noted thatk -essen e ould also yield interesting models for the darkenergy [9℄, [10℄. An interesting attempt was made tounify dark matter and dark energy using kineti k -essen ein [3℄. Though this model had its share of problems (itis worth noting that a purely kineti k -essen e leads to astati universe when the late time energy density of theuniverse is expressed simply as a sum of a osmologi al onstant and a dark matter term [11℄), extensions of theformalism to extra t out dark matter and dark energy omponents within a uni(cid:28)ed framework have been usedalso in subsequent works [12℄.Re ently, we [11℄ have proposed a k -essen e model thatreprodu es the essential features of in(cid:29)ation, dark matterand dark energy within a uni(cid:28)ed framework. We foundthat a ouple of parameters of this model had to be tunedin order to onform with various observational featurespertaining to both the early and the late time eras ofthe universe. The Lagrangian hosen in this model wasof the form where the kineti and potential terms werede oupled in the standard way. However, it may be re- alled that in most k -essen e models [9℄, [10℄ in ludingthe original k -in(cid:29)ation idea [8℄, the distinguishing fea-ture was the use of non- anoni al kineti terms in theLagrangian of the form F ( X ) V ( φ ) . In the present pa-per we return to su h a Lagrangian with the motivationof reprodu ing the features of in(cid:29)ation in the early uni-verse, and also generating dark matter and dark energyat late times. We (cid:28)nd that after the early expansion isover, our present model an be approximated as kineti k -essen e, i.e., the dynami s be omes dominated by onlythe kineti omponent of the s alar (cid:28)eld. We show thatthe late time energy density reprodu es a osmologi al onstant and a matter like term whi h we all dark mat-ter. We then onsider observational results from the boththe early and late eras, whi h are used to put onstraintson the parameters of this model.II. THE MODELWe begin with a Lagrangian for a s alar (cid:28)eld φ of theform L = F ( X ) V ( φ ) (1)where X is de(cid:28)ned as X = 12 ∂ µ φ∂ µ φ Throughout this paper we will work with a (cid:29)atRobertson-Walker metri having signature (+ , − , − , − ) .Taking the s alar (cid:28)eld to be homogeneous in spa e, whi his the usual ase, we get X = 12 ˙ φ .The fun tional forms of F and V are taken to be F ( X ) = KX − m P l L √ X + m P l M (2) V ( φ ) = 1 + e − φ/φ c (3)where the parameters K , L and M are dimensionless, andare taken to be positive. The parameter φ c is also takento be positive and learly has the dimension of φ . Wework in natural units and onsider V to be dimensionless.As is the usual ase, the s alar (cid:28)eld φ has the dimensionof mass. From the de(cid:28)nition of X it turns out that Xand hen e F has dimension M .The energy density in this ase is given by ρ = V ( φ )(2 XF X − F ) (4)where F X ≡ dF/dX . So substituting the forms of F andV in (4) we get ρ = (1 + e − φ/φ c )( KX − m P l M ) (5)The pressure, whi h is simply the Lagrangian, turns outto be p = (1 + e − φ/φ c )( KX − m P l L √ X + m P l M ) (6)The equation of state parameter is given by w = F XF X − F (7)whi h in our model evaluates to w = KX − L √ X + MKX − M (8)The sound speed, or the speed at whi h perturbationstravel, is de(cid:28)ned to be [13℄ c s ≡ ∂p/∂X∂ρ/∂X = F X XF XX + F X (9)where F XX = d F/dX . Note that this de(cid:28)nition isdi(cid:27)erent from the usual de(cid:28)nition of the adiabati sound speed (namely, c s = dpdρ ). However, it has been shown re- ently [14℄ that perturbations in su h models travel witha speed de(cid:28)ned as above, where the authors also de(cid:28)nethis to be the (cid:16)phase speed(cid:17).Now, the equation of motion for the k -essen e s alar(cid:28)eld is given by (2 XF XX + F X ) ˙ X + 6 HF X X + ˙ VV (2 XF X − F ) = 0 (10)whi h has been written in terms of X. If V is a onstantor varies very slowly with time so that the third term in(10) is negligible then the situation orresponds to kineti k -essen e and the (cid:28)eld equation an be written as (2 XF XX + F X ) ˙ X + 6 HF X X = 0 (11)This an be integrated exa tly [3℄ to give the solution √ XF X = ka (12)where k is a onstant of integration. This solution waspreviously derived in a slightly di(cid:27)erent form in Ref. [10℄.The above result holds irrespe tive of the spatial urva-ture of the universe.The energy onservation equation states that ˙ ρ = − H ( ρ + p ) = − HF X XV (13)This shows that the (cid:28)xed points of the equation orre-spond to the extrema of F [8℄, whi h from equations (1)and (4) yields ρ = − p . Moreover ρ de reases with timewhen ρ > − p and in reases when ρ < − p showing thatany point orresponding to ρ = − p is an attra tor and,as is well known, will lead to exponential in(cid:29)ation.In our model the extrema of F orrespond to X = 0 ,or X = m P l L K . The point X = 0 is of no signi(cid:28) an esin e that orresponds to energy density and pressurewhi h are onstant in time. We take X = m P l L K (14)whi h leads form de(cid:28)nition of X, to ˙ φ = m P l L √ K (15)where we have taken the positive sign for ˙ φ . For theabove value of X the energy density and pressure turnout to be ρ = V ( φ ) (cid:18) L K − M (cid:19) m P l = − p (16)A tually, X orresponds to an instantaneous attra -tive (cid:28)xed point and X evolves slowly away from thatpoint, whi h is the analog of (cid:16)slow-roll(cid:17) potential drivenin(cid:29)ation in whi h the potential dominates the kineti term and evolves slowly. Hen e, in dire t analogy, theabove al ulated values of ρ and p an be alled the (cid:16)slow-roll(cid:17) values. In our model we assume that the exponentialterm inside V is mu h larger than during the ourseof in(cid:29)ation, for whi h we must have φ /φ c < , and also | φ /φ c | ≫ . From Eq.(13)we an write φ = ˙ φ t + C φ ,where C φ is an integration onstant. This onstant anhave a negative value, hen e making φ < . Thus, we hoose φ c > , su h that the onditions φ /φ c < , and | φ /φ c | ≫ are satis(cid:28)ed during in(cid:29)ation. Sin e ˙ φ > ,it follows that φ be omes less and less negative with time.V an be quite a urately approximated as e − φ/φ c . Thisenables us to (cid:28)nd the number of e-folds of expansion N,under this (cid:16)slow-roll(cid:17) approximation as N = t e Z t i H dt = φ e Z φ i H dφ ˙ φ (17)whi h turns out to be N ≃ r π m − P l (cid:18) L K − M (cid:19) / √ KL φ c (cid:16) √ V i − √ V e (cid:17) (18)where the subs ripts `i' and `e' refer to the intial and (cid:28)nalvalues respe tively.The slow-roll ondition for k -in(cid:29)ation is given by [ δX/X ] ≪ . Now, during the post slow-roll stage we an write X = X + δX . Also, from Eq.(13), one has F X ( KX − m P l M ) = − X ˙ VHV (19)Retaining terms up to the (cid:28)rst order in δX we get δXX ≃ X (cid:18) L K − M (cid:19) √ π Lφ c X m P l (cid:18) L K − M (cid:19) / √ V − Km P l (20)In(cid:29)ation ends when δXX ∼ . Using this fa t in Eq.(20)we an (cid:28)nd the expression for the (cid:28)nal value of the po-tential, V e to be √ V e ≃ m P l √ π Lφ c (cid:18) L K − M (cid:19) / + m P l √ π L Kφ c (cid:18) L K − M (cid:19) − / (21)The kinemati s of the in(cid:29)ationary era in our modelmay be viewed in the following way. We start with somerepresentative point in the ( ρ, p ) plane orresponding tosome initial value of φ su h that the slow-roll ondi-tion is satis(cid:28)ed. In fa t, during the (cid:28)rst evolutionarystage the representative point takes only a few e-folds torea h the nearest in(cid:29)ationary attra tor that orresponds to ρ = − p . After this initial stage the representativepoint follows the post slow-roll motion, X = X + δX with δX/X ≪ , thereby staying near but not exa tlyon the ρ = − p line. The value of δX is positive (as we willshow later in the Se tion on observational onstraints).Hen e X slowly moves away from the value X . As theevolution ontinues, the slow-roll ondition is satis(cid:28)ed toa less and lesser extent till a time is rea hed when theslow-roll ondition is a tually violated ( δX/X ∼ ), andone naturally exits the in(cid:29)ationary stage.Now, after in(cid:29)ation ends we have X > X , meaningthat the time evolution of φ is faster than during in(cid:29)a-tion, and hen e its value in reases very qui kly and orre-spondingly de reases the value of the exponential part in V , so that one gets V ≃ . In order for su h a behaviourto ensue, we must have φ/φ c > after in(cid:29)ation is over.Sin e we have already hosen φ c to be positive, then φ has to be ome positive after in(cid:29)ation where previously itwas negative, and this is exa tly its behaviour as pointedout earlier, i.e., ˙ φ is positive. Note that even if the ra-tio φ/φ c is not too big ompared to 1, the exponentialpart of the potential will be negligible. Thus, after thein(cid:29)ationary expansion is over the exponential part in Vqui kly de ays away (we will present an estimate of thetime taken for this pro ess in the se tion on observational onstraints on the model). When the exponential termbe omes quite negligible we have V ≈ , ˙ V ≈ So the (cid:28)eld equation e(cid:27)e tively be omes of the form ofEq.(11) and the dynami s an be approximated quite wellby the purely kineti form of k -essen e. On using Eq.(12)to (cid:28)nd X as a fun tion of a we get X = 1 K (cid:18) m P l L ka (cid:19) (22)Therefore the orresponding expression for the k -essen eenergy density turns out to be ρ = m P l (cid:18) L K − M (cid:19) + m P l kLKa + k Ka (23)The subsequent evolution of the universe is des ribedas follows. After the end of in(cid:29)ation the universe is in akineti dominated period when the third term in Eq.(23)dominates, whi h orresponds to p = ρ ∼ a − . But thisterm be omes small qui kly in omparison to radiationwhi h goes as ∼ a − and a period of radiation dominationin the universe ensues. The se ond term in Eq.(23) gainsprominen e in the epo h of matter domination and weidentify it with dark matter. But as the universe evolvestowards the present era the (cid:28)rst term begins to dominateand a ts like a osmologi al onstant giving rise to thelate time a eleration of the universe. The equation ofstate parameter after in(cid:29)ation is over is given by w = k Ka − m P l (cid:18) L K − M (cid:19) m P l (cid:18) L K − M (cid:19) + m P l kLKa + k Ka (24)with the following values of w orresponding to the var-ious epo hs: w ≈ after the end of in(cid:29)ation andbefore radiation domination w ≈ during matter domination w → − as a → ∞ Using Eq.(9) the sound speed is found to be c s = 1 m P l La k + 1 (25)From the above equation it is lear that the sound speedde reases as the universe expands.III. OBSERVATIONAL CONSTRAINTSSo far we have seen that the model onsidered by usprodu es the primary features of k -in(cid:29)ation in the earlyuniverse and reprodu es dark matter as well as a os-mologi al onstant in the later period of evolution. Wewill now use various observational features to onstrainthe parameters of our model. A notable feature [8, 9℄ inour model is that the potential and the kineti part are oupled. So parameters that are relevant during the latetime era annot be determined independently of the pa-rameters relevant during the in(cid:29)ationary era. It is thuspra ti al to (cid:28)rst arry out the analysis in the late timeera and then use the al ulated values of the relevant pa-rameters in the in(cid:29)ationary era. We have provided theexpression for the k -essen e energy density after in(cid:29)ationis over in Eq.(23). Using the urrent observed value ofthe osmologi al onstant, we get m P l (cid:18) L K − M (cid:19) ≃ − ( GeV ) (26)Also, observations put the urrent dark matter densityto be about / rd of the urrent dark energy density.This enables us to write kLKa ≈ m P l (cid:18) L K − M (cid:19) (27)where the subs ript ` ' signi(cid:28)es the present epo h. Weknow from observations that the fra tion of the urrentenergy density ontained in radiation is (Ω R ) ≃ × − orresponding to the present radiation density ( ρ R ) ≃ . × − ( GeV ) . Denoting the third term in Eq.(23)as ρ k , and assuming that ρ R rosses over ρ k at a redshiftof z ∼ (prior to the nu leosynthesis at a redshift of ), we get z = ( ρ R ) Ka k ⇒ ka = K / z ( ρ R ) / (28)Now from Eq.(27) and Eq.(28) we get m P l LK / ≃ × − ( GeV ) − (29)From Eqs.(28) and (29) it an be seen that the the ross-over between dark matter and ρ k o urs at a redshift of ∼ and that between radiation and dark matter at aredshift of ∼ , i.e., at the epo h of matter-radiationequality. We also (cid:28)nd the present value of ρ k to be ( ρ k ) = k Ka = ( ρ R ) z ≈ . × − ( GeV ) (30)The sound speed at the epo h of matter radiation equal-ity turns out to be (cid:0) c s (cid:1) eq = 1 m P l La eq k + 1 = 1 m P l La z eq k + 1 ≃ . × − (31)Now, we an rexpress w from Eq.(24) in terms of theredshift z. Sin e ρ k is negligible in omparison to theother omponents, we have w ≈ − (cid:18) L K − M (cid:19)(cid:18) L K − M (cid:19) + m − P l kLKa ( z + 1) (32)Therafter, it is possible to (cid:28)nd dw/dz . Its value at the urrent epo h, i.e., at redshift z = 0 using Eqs.(26) and(27) turns out to be (cid:18) dwdz (cid:19) ≈ . × − (33)One an also estimate the urrent value of the equationof state parameter in our model, whi h using (32) andputting z = 0 turns out to be w ≈ − . (34)We an further (cid:28)nd out the value of the redshift at whi hthe universe started its transition from the matter dom-inated de elerating era to its presently a elerating era.Knowing that for a eleration to begin we must have w = − / , from Eq.(32) we (cid:28)nd that z acc ≈ . (35)Su h a value for the redshift is quite ompatible withpresent observations [15℄. But, from Eqs.(26) and (29)we (cid:28)nd that m P l M = 4 × − − − ( GeV ) (36)showing that a tuning of the parameter M is needed.This is expe ted sin e it is simply a rephrasal of the o-in iden e problem asso iated with the present window ofa eleration of the universe.We now revisit the in(cid:29)ationary era for analyzingthe observational onstraints pertaining to it. FromRef.[13℄ the spe trum of s alar density perturbations ink -in(cid:29)ation is given by P = 169 m − P l c s ρ p/ρ = − m − P l c s r πG ρ / ˙ ρ = 32 √ √ √ πm − P l c s √ Kφ c L (cid:18) L K − M (cid:19) / V / i (37)where in the se ond step we have used the energy onser-vation law and also used the Friedmann equation. Usingthe COBE normalization √ P ∼ × − , and assumingthat 60 e-folds of expansion takes pla e, we an rewriteEq.(37) to get an expression for V i to be p V i = (27) / c / s m − / P l π / (cid:18) P LKφ c (cid:19) / (cid:18) L K − M (cid:19) − / (38)Using Eqs.(38) and (21) in Eq.(18) we an write c / s φ / c = 4(27) / π / m / P l (cid:18)
KP L (cid:19) / " √ π L (cid:18) L K − M (cid:19) + 1 √ π L K + N L / K r π (39)Now from Eq.(9) we see that in slow-roll approximationwhen F X = 0 we get c s = 0 . But, in the post slow-rollstage, X = X + δX , and F X does not vanish. To (cid:28)rstorder in δX we an write F X ≈ ( F XX ) δX . Using thisin Eq.(9) we get c s ≃ δX X (40)Stability requires δX > and we show now that this isindeed the ase. From Eqs.(38) and (20) we al ulate δX/X when V = V i , to get δXX = K L (cid:16) L K − M (cid:17) √ π K L h √ π L (cid:0) L K − M (cid:1) + √ π L K + NL / K q π i − K (41)It is to be noted that in order to evaluate the above equa-tion the a tual value of K or L is not required, instead the ratio L/ √ K from Eq.(29) serves the purpose. Sub-stituting the various values we (cid:28)nd that δXX ≃ . × − (42)whi h is positive as laimed. The sound speed is thereforefound to be c s ≃ δXX ≃ . × − (43)Having found the sound speed and using the values of P and N , we now use Eq.(29) in Eq.(39) to get φ c √ K ≃ . × ( GeV ) − (44)We now have all the parameter values to evaluate thevalue of V at the beginning and at the end of k -in(cid:29)ationwhi h we write below V i ≃ . × (45) V e ≃ . × (46)The orresponding energy densities are ρ i = V i (cid:18) L K − M (cid:19) m P l ≃ . × ( GeV ) (47) ρ e = V e (cid:18) L K − M (cid:19) m P l ≃ . × ( GeV ) (48)The tensor-to-s alar ratio is given by [13℄ r = 24 c s (cid:18) pρ (cid:19) = − c s m P l √ π ˙ ρρ / = r π c s φ c (cid:18) L K − M (cid:19) − / L √ K √ V i (49)where in the se ond step we have used the energy on-servation and Friedmann's equation. On substituting theparameter values we get r = 9 . × − (50)The s alar spe tral index an be obtained from the rela-tion [13℄ n s − − (cid:18) pρ (cid:19) − H ddt ln (cid:18) pρ (cid:19) − H ddt ln c s = 2 ˙ ρρH − ¨ ρρH + ˙ HH − H ˙ c s c s (51)To evaluate n s the values of the following quantites arerequired ˙ ρρ = ¨ ρ ˙ ρ = − ˙ φ φ c = − m P l φ c L √ K = − . × GeVH = r πG ρ i = 2 . × GeV ˙ H = − πG ρ i + p i ) = 4 πG ρ i H = − . × ( GeV ) ˙ c s c s = 2 . × − GeV
All the above values have been al ulated using theslow-roll approximation pertaining to the beginning ofk -in(cid:29)ation. Therefore, using these values in Eq.(51) weget n s = 0 . (52)This value is quite lose to what is predi ted by modelsof potential driven in(cid:29)ation. Eq.(51) di(cid:27)ers from the ap-propriate expression in the ase of usual in(cid:29)ation by theterm proportional to the derivative of the sound speed.Sin e in standard in(cid:29)ation c s = 1 , this term vanishes andone obtains n s to be very lose to , i.e., a s ale invariantspe trum. But in k -in(cid:29)ation models, c s = 1 , and a tiltedspe trum with n s < is generally predi ted. However, inour model this term in Eq.(51) makes a vanishingly small ontribution, and hen e we get a spe tral index that isagain quite lose to . Only the value of the tensor-to-s alar ratio in our model makes it distinguishable fromstandard in(cid:29)ation where typi ally a value of about 0.12to 0.15 is obtained.Now, the duration of in(cid:29)ation in our model is found tobe t e − t i = φ e Z φ i dφ ˙ φ = √ KL ( φ e − φ i ) m − P l ≈ . × − s (53)After the end of in(cid:29)ation, the stage of kineti domi-nated evolution sets in very qui kly. In order to have anidea as to how mu h time it takes for the exponential partof the potential to be ome negligible, we assume that forargument's sake, X ≃ X . This assumption is only madeto perform a simple al ulation and get an upper boundon the time required for the exponential part to de ay(the a tual time taken is mu h smaller sin e X > X and φ evolves more rapidly ompared to its linear evolu-tion during in(cid:29)ation). The time taken after in(cid:29)ation forthe exponential part to attain the value e − φ/φ c ≃ . ,is about . × − s . Thus, the time required for thek -essen e (cid:28)eld to e(cid:27)e tively behave as kineti k -esen e isof the order of − s . This again justi(cid:28)es our analysis of the previous se tion pertaining to the post in(cid:29)ationaryperiod being dominated by the dynami s of purely ki-neti k -essen e. It should be noted that the estimate forthe time required for the universe to enter into a kineti dominated era after in(cid:29)ation is a tually an upper bound.In reality the time required is mu h shorter sin e X > X and the s alar (cid:28)eld evolves more rapidly with time thanduring the in(cid:29)ationary era (the potential de reases veryqui kly to assume an almost onstant value).Reheating in this model ould be aused by grav-itational parti le produ tion. The pro ess of gravi-tational reheating in the presen e of kineti domina-tion by a s alar (cid:28)eld is not yet understood very well[16℄. However, standard al ulations [17℄ give the den-sity of parti les produ ed at the end of in(cid:29)ation to be ρ R ≃ . × g ( GeV ) where g is the number of(cid:28)elds whi h produ e parti les at this stage, likely tobe between 10 and 100. This energy density if imme-diately thermalized would give rise to a temperature of T e ≃ . × (cid:18) gg ∗ (cid:19) / GeV , where g ∗ is the total num-ber of spe ies in the thermal bath and maybe somewhathigher than g. Assuming that immediately after the endof in(cid:29)ation there is omplete kineti domination so thatthe s alar (cid:28)eld density falls as a − , it is estimated thatin our model the universe has to expand by a fa tor ofabout for radiation domination to set in. After thatexpansion the temperature whi h goes as T ∝ /a omesout as T ≃ . × − (cid:18) gg ∗ (cid:19) GeV . So we see thatthe temperature is not high enough for a su essful nu- leosynthesis for whi h a temperature around 1 MeV isneeded. Now, if we hange our parameters somewhatsu h that the value of the redshift for the ross-over be-tween ρ R and ρ k is , then we (cid:28)nd that the reheat tem-perature turns out to be T ≃ . × − (cid:18) gg ∗ (cid:19) / GeV whi h is roughly about the order of . MeV. There havebeen some re ent studies whi h indi ate that very lowreheating temperatures ould also be a viable option forsu essful nu leosynthesis (see, for instan e [18℄). Theseideas have to be analyzed in detail in the ontext of k -essen e s enarios in order to he k how far gravitationalreheating ould be su essful in our model.IV. CONCLUSIONSTo summarize we have onsidered a k -essen e modelthat produ es in(cid:29)ationary expansion in the early universeby the pro ess of k -in(cid:29)ation and later on generates bothdark matter and dark energy at appropriate subsequentstages. For our Lagrangian we have onsidered that formwhi h has been widely used for k -essen e models [9℄. In ontrast to an earlier model studied by us [11℄, the po-tential and the kineti parts of the s alar (cid:28)eld are notde oupled, leading to oupling between the in(cid:29)ationaryera and the late time parameters. A signi(cid:28) ant featureof this fa t an be found in the expression for the energydensity. It thus follows that the generated osmologi al onstant whi h dominates the dynami s at late times, de-rives its value from in(cid:29)ationary parameters. It needs tobe mentioned here that our model is unable to addressthe oin iden e problem. The addressal of this problemwithin the ontext of k -essen e is made possible by theexisten e of (cid:28)xed points in the radiation and matter era.In order to have these (cid:28)xed points, it is ne essary that thepotential has the form V ( φ ) = 1 /φ . It was shown thatsu h models that solve the oin iden e problem su(cid:27)erfrom superluminal propagation of the (cid:28)eld perturbations[19℄ (whi h, however, may not a(cid:27)e t ausality [20℄). Butthe hoi e of the potential in our model does not allowthe existen e of (cid:28)xed point in the radiation and matterera. Consequently, this model does not su(cid:27)er from theproblem of superluminal propagation.Our model is able to reprodu e the basi features ofk -in(cid:29)ation. Although in general k -in(cid:29)ation predi ts that n s < , our model gives rise to a value whi h is nearlythe same with what is obtained in standard potential di- ven in(cid:29)ation, predi ting an almost s ale invariant densityperturbation spe trum. But, the value of the al ulatedtensor-to-s alar ratio is quite di(cid:27)erent from what is ob-tained in standard in(cid:29)ationary models. After the in(cid:29)a-tion is over the potential qui kly be omes onstant andwe are able to approximate the model as purely kineti k -essen e. The late time energy density and the soundspeed in terms of the s ale fa tor aa