k-essence classical Hamiltonian approach for an accelerated expansion of the Universe with ω≈−1
aa r X i v : . [ phy s i c s . g e n - ph ] J un EPJ manuscript No. (will be inserted by the editor) k-essence classical Hamiltonian approach for an acceleratedexpansion of the Universe with ω ≈ − Somnath Mukherjee Department of Physics, Dharsa Mihirlal Khan Institution[H.S], P.O:-New G.I.P Colony, Dist:-Howrah-711112, India, e-mail: [email protected]
Received: date / Revised version: date
Abstract.
We obtain lagrangian for k -essence scalar field φ ( r, t ) with scalar curvature k of Friedmann-Lemaitre-Robertson-Walker (FLRW) metric . Obtained lagrangian has two generalised co-ordinates φ andlogarithm of scale factor ( q = ln a ). Classical Hamiltonian ( H ) is obtained in terms of two correspondingconjugate momentum p q and p φ . Solving Hamilton’s equation of motion , we obtain classical solutionfor scale factor a ( t ), energy density ρ , equation of state parameter ω and deceleration parameter q . Atlate time as t → ∞ , we have an exponential growth of scale factor with time, energy density ρ becomesconstant, which we can identify as dark energy density, equation of state parameter becomes ω ≈ − q ≈ −
1. All this results indicates an accelerated expansion of universedriven by negative pressure known as dark energy.
From the observation of Type 1a Supernovae (SNe 1a)by The Supernova Cosmology Project [1,2,3,4,5,6] andthe High-Z-Supernova search team [7,8,9] it was first es-tablished that the universe is undergoing accelerated ex-pansion. Recent observations with WMAP satellite [10,11] and Planck satellite [12] also ensures an acceleratedexpansion of the universe, driven by negative energy nowknown as dark energy which accounts for 70 percent con-stituents of the universe.Several cosmological model has been developed to under-stand the role of dark energy in the universe, out of whichwe have chosen k -essence model of scalar field φ ( r, t ) withnon-canonical kinetic term X as our field of study.The Lagrangian for the k − essence field is taken as [15,16,17,18,19,20,21,22,23,24,25,26,27,28] L = − V ( φ ) F ( X ) (1)where X = 12 ∂ µ φ∂ ν φ = 12 ˙ φ −
12 ( ∇ φ ) (2)Energy density ρ for k-essence field is given by ρ = V ( φ )[ F ( X ) − XF X ] (3)with F X = ∂F∂X .and the pressure P is given by P = L = − V ( φ ) F ( X ) (4) Send offprint requests to : Somnath Mukherjee
Correspondence to : Dharsa Mihirlal Khan Institution[H.S],P.O:-New G.I.P Colony, Dist:-Howrah-711112, India.
The conservation equation is given by˙ ρ + 3 H ( ρ + P ) = 0 (5)where H is a Hubble parameter, defined in terms of scalefactor a as H = ˙ aa . Considering homogeneity and isotropyof the universe φ ( r, t ) ≈ φ ( t ), so that X = ˙ φ , we getfrom (3),(4) and (5)( F X + 2 XF XX ) ¨ φ + 3 HF X ˙ φ + (2 XF X − F ) V φ V = 0 (6)Considering constant potential ( V ( φ )=constant, so that ∂V∂φ = 0) the conservation equation becomes( F X + 2 XF XX ) ˙ X + 6 HXF X = 0 (7)Solving this gives the scaling relation XF X = Ca − (8)This is the scaling relation [22,23] of k -essence cosmol-ogy. It plays a very important role in determining the la-grangian for the development of cosmological scenario ofobservational importance. k Considering Friedmann-Lemaitre-Robertson-Walker (FLRW)metric of the form ds = − dt + a ( t )[ dr − kr + r dθ + r sin θdφ ] (9) Please give a shorter version with: \authorrunning and \titlerunning prior to \maketitle where a ( t ) is the cosmological scale factor and k is thecurvature constant that describes closed, flat and openuniverse for value k = +1 , − ka + H = 8 πG ρ (10)where H = ˙ aa is the Hubble’s constant and ρ is the darkenergy density. Since we are considering constant poten-tial, hence we can assume V ( φ ) = V as some constant.From (3) and (10) we get XF X = 12 [ F ( X ) − πGV H + 3 k πGV a ] (11)Now considering equation (8) and (11) we get F ( X ) = − √ C √ Xa − − πGV H + 3 k πGV a − (12)thus the k-essence lagrangian (1) becomes L = − √ C √ XV a − − πG H + 3 k πG a − (13)From the consideration of Homogeneity and isotropy ofobserved universe we will consider scalar field as φ ( r, t ) ≈ φ ( t ).Now considering q ( t ) = ln a ( t ), c = πG and c = √ C equation (13) becomes L = − c ˙ q − c ke − q − c V ˙ φe − q (14)This is the k -essence lagrangian with curvature constant k of FRW metric and has two genaralised co-ordinate q ( t )and φ ( t ). For flat universe with k = 0, this reduces tothe lagrangian Ref. [19]. In the subsequent section we willuse this lagrangian to obtain cosmological parameter ofobservational importance. In this section we will derive classical Hamiltonian corre-sponding to classical lagrangian (14).Conjugate momentum corresponding to q is p q = ∂ L ∂ ˙ q = − c ˙ q (15)and conjugate momentum corresponding to φ is p φ = ∂ L ∂ ˙ φ = − c V e − q (16)thus the Hamiltonian corresponding to the k -essence La-grangian H = p q ˙ q + p φ ˙ φ − L (17) is obtained as H = − p q c − c kp φ ( c V ) (18)This is k -essence Hamiltonian. This is different from theearlier framed hamiltonian in the context that this hasnon zero curvature constant. Our objective is to developcosmological parameters out of this Hamiltonian whichwill be consistent with the observed data. Since φ is a cyclic co-ordinate thus p φ is conserved. Con-sidering second term of Hamiltonian (18) in terms of q H = − p q c + c ke − q (19)Hamiltonian equation of motion for the generalised co-ordinate q is determined as follows:-˙ q = ∂ H ∂p q = − p q c (20)˙ p q = − ∂ H ∂q = 2 c ke − q (21)Considering equation (20) and (21) yields¨ q = − ke − q (22)Since q = ln a , this becomes a d adt − ( dadt ) + k = 0 (23)The solution yields a = Be c t − kDe − c t (24)where B , D and c are constant. This is the scale factorwith curvature constant k . For late time cosmology, as t → ∞ , this becomes a ≈ Be c t (25)This shows the exponential increase of scale factor overcosmic time. ρ Considering (10) and (24) we get ρ = c kB − e − c t z + c c (1 + kDB − e − c t ) z (26)where z = (1 − kDB − e − c t ) .As we are considering late time cosmology, hence as t → ∞ energy density becomes ρ ≈ c c ≈ constant (27)This constant energy density can be identified as darkenergy density. This is independent of curvature constant k . Thus we see that energy density remains constant atlate time independent of the nature of universe. We getthis result out of k -essence classical Hamiltonian. lease give a shorter version with: \authorrunning and \titlerunning prior to \maketitle ω From Friedmann equation for the solution of the spatialpart of Einstein’s equation , the relationship betwen pres-sure ( P ) and energy density ( ρ ) is given by:- − πGP = 2 ¨ aa + 8 πG ρ (28)using (24) this yields P = − c πG − ρ − c c − ρ ω = Pρ = − c c ρ −
13 (30)since we are considering only late time cosmology, thusconsidering the form of ρ at t → ∞ , we get ω ≈ − k = +1 , , − q Deceleration parameter in terms of scale factor a ( t ) isgiven by q = − ¨ aa ˙ a (32)Considering the form of scale factor (24) this yields q = − (cid:18) − kDB − e − c t kDB − e − c t (cid:19) (33)Considering late time cosmology, as t → ∞ , this becomes q ≈ − q with val-ues ranging from − ≤ q ≤ − . k = +1 , , −
1) closed, flat and open universe.This satisfies late time acceleration of the universe drivenby negative pressure know as dark energy.
Incorporating scaling relation XF X = Ca − , k -essencelagrangian of non-canonical form L = − V ( φ ) F ( X ) is de-veloped into canonical form with curvature constant k ofFLRW metric. It has two generalised co-ordinate q ( t ) and φ ( t ). From this lagrangian we develop Hamiltonian, cor-responding to two conjugate momentum p q and p φ . Since φ ( t ) is a cyclic co-ordinate thus p φ is conserved. SolvingHamiltonian equation of motion for q ( t ), we obtain cosmo-logical relevant classical solutions. In this paper we studiedthe behaviour of certain cosmological parameter at latetime epoch. Scale factor a ( t ) evolves exponentially at latetime cosmology. Energy density ρ remains constant at latetime cosmology, which can be identified as dark energy.At t → ∞ , equation of state parameter becomes ω ≈ − q ≈ −
1, that sat-isfies an accelerated expansion of the universe.Thus we get all the necessary relevant cosmological param-eters from Hamilton’s equation which predicts an acceler-ated expansion of the universe driven by negative pressureknown as dark energy.
References
1. S.Perlmutter, et al, Nature. , 51 (1998) .2. S.Perlmutter, et al, Astrophys.J. , 565 (1999) .3. A.G.Riess, et al, Astrophys.J. , 1009 (1998) .4. A.G.Riess, et al, Astrophys.J. , 98 (2007) .5. R.R.Caldwell, R.Dave, P.J.Steinhardt, Phys.Rev.Lett. ,1582 (1998) .6. W.M.Wood-Vesey, et al, Astrophys.J. , 694 (2007) .7. B.P.Schimdt, et al, Astrophys.J. , 46 (1998) .8. N.W.Halverson, et al, Astrophys.J. , 38 (2002) .9. E.Komatsu, et al, Astrophys.J.Suppl. , 18 (2011) .10. C.L.Bennett, et al, Astrophys.J.Suppl. , 1 (2003) .11. D.N.Spergel, et al, Astrophys.J.Suppl. , 175 (2003) .12. P.A.R.Ade, et al, Astron.Astrophys. A1 ,571 (2014) .13. N.Aghanim, et al, Astron.Astrophys. Special issue (2020).14. M.Tegmark, et al, Phys.Rev.D. , 103501 (2004) .15. C.A.Picon, T.Damour, V.Mukhanov, Phys.Lett.B. ,209 (1999) .16. J.Garriga, V.Mukhanov, Phys.Lett.B. , 219 (1999) .17. C.A.Picon, V.Mukhanov, P.J.Steinhardt, Phys.Rev.Lett. , 4438 (2000) .18. T.Chiba, T.Okabe, M.Yamaguchi, Phys.Rev.D. , 023511(2000) .19. T.Chiba, Phys.Rev.D. , 063514 (2002) .20. C.A.Picon, V.Mukhanov, P.J.Steinhardt, Phys.Rev.D. ,103510 (2001) .21. L.P.Chimento, A.Feinstein, Mod.Phys.Lett.A. , 761(2004) .22. L.P.Chimento, Phys.Rev.D. , 123517 (2004) .23. R.J.Scherrer, Phys.Rev.Lett. , 011301 (2004) .24. D.Gangopadhyay, S.Mukherjee, Phys.Lett.B ,121(2008) .25. D.Gangopadhyay, S.Mukherjee, Grav.Cosmol. , 349(2011) .26. L.P.Chimento, M.Forte, Phys.Rev.D , 063502 (2006).27. K.Enqvist, Gen.Rel.Grav. , 451 (2008) .28. C.H.Chuang, Class.Quant.Grav.25