Λ -CDM Universe: A Phenomenological Approach With Many Possibilities
aa r X i v : . [ g r- q c ] M a r October 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd
International Journal of Modern Physics Dc (cid:13)
World Scientific Publishing Company
Λ-CDM Universe: A Phenomenological Approach With ManyPossibilities
UTPAL MUKHOPADHYAY
Satyabharati Vidyapith, North 24 Parganas, Kolkata 700 126, West Bengal, India.
SAIBAL RAY
Department of Physics, Barasat Government College, North 24 Parganas, Kolkata 700 124,West Bengal, India ∗ [email protected] S. B. DUTTA CHOUDHURY
Department of Physics, Jadavpur University, Kolkata 700 032, West Bengal, India
Received Day Month YearRevised Day Month YearA time-dependent phenomenological model of Λ, viz., ˙Λ ∼ H is selected to investigatethe Λ-CDM cosmology. Time-dependent form of the equation of state parameter ω isderived and it has been possible to obtain the sought for flip of sign of the decelerationparameter q . Present age of the Universe, calculated for some specific values of theparameters agrees very well with the observational data. Keywords : dark energy, variable Λ, Λ-CDM cosmology
1. Introduction
Modern cosmological research rests heavily on observational data. Any theoreticalmodel should be corroborated with observation for understanding the viability ofthat model. Present cosmological picture, emerging out of this theory-observationcombination, reveals that, the total energy-density of the Universe is dominated bytwo dark components, viz., dark matter and dark energy. Observational evidencefrom various independent sources including SN Ia1 , , , , , , , ∗ Corresponding address 1 ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd U. Mukhopadhyay, Saibal Ray and S. B. Dutta Choudhury
Now, dark matter played a significant role in the early Universe during structureformation because it clumps in sub-megaparsec scales. But, the exact compositionof dark matter is still unknown. Since small density perturbation ( δρ/ρ ∼ − at z ≃ , , ,
12, so most of the dark matter must be cold and non-baryonic.On the other hand, clustering of cold dark matter on small scale9 supports thehierarchical structure formation. Moreover, after introduction of the idea of accel-erating Universe, the previous Standard Cold Dark Matter (SCDM) models havefallen out of grace13 ,
14 and is replaced by Λ-CDM or LCDM model for includingdark energy as a part of the total energy density of the Universe. Λ-CDM model isfound to be in nice agreement with various sets of observations15. An advantage ofΛ-CDM model is that it assumes a nearly scale-invariant primordial perturbationsand a Universe with no spatial curvature. These were predicted by inflationaryscenario16 , , , , ,
22 show that the present accelerating phase was preceded by adecelerating one and observational evidence23 also supports this idea. The presentwork is done with this background in mind.Phenomenological approach is one of the several ways of searching such darkenergy. In a recent work24, the equivalence of three phenomenological variable Λmodels have been shown. The behaviour of the same three forms of Λ have beenstudied when both G and Λ vary25. But, in both those works the equation ofstate parameter ω was considered as a constant because, due to inability of currentobservational data in separating a time-varying ω from a constant one26 ,
27, inmost of the cases a constant value of ω is used. However, ω , in general, is a functionof time28 , ,
30. It has already been commented by Ray et al.24 that for a moreaccurate result, an investigation regarding time evolution of ω may be taken upfor searching better physical features. In fact, the Statefinder diagnostic, used fordistinguishing various dark energy models, can be applied if the equation of stateof scalar potential has a direct relationship with the Hubble parameter and itsderivative31 , , q and ω in mind, an investigation about the Λ-CDMUniverse is done by selecting a specific time-dependent form of Λ, viz., ˙Λ ∼ H .This particular time-varying Λ model was studied by Reuter and Wetterich33 forfinding a mechanism which would explain the present small value of Λ as a result ofthe cosmic evolution. In the present work, the same Λ model is used for investigatinga time evolving equation of state parameter ω along with a possible signature flipof the deceleration parameter q . This change of sign is very important for Λ-CDMcosmology.ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd Λ -CDM Universe: A Phenomenological Approach With Many Possibilities
2. Field Equations
The Einstein field equations are given by R ij − Rg ij = − πG (cid:20) T ij − Λ8 πG g ij (cid:21) (1)where the cosmological term Λ is time-dependent, i.e. Λ = Λ( t ) and c , the velocityof light in vacuum, is assumed to be unity.Let us consider the Robertson-Walker metric ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) (2)where k , the curvature constant, assumes the values −
1, 0 and +1 for open, flatand closed models of the Universe respectively and a = a ( t ) is the scale factor.For the spherically symmetric metric (2), field equations (1) yield Friedmann andRaychaudhuri equations respectively given by3 H + 3 ka = 8 πGρ + Λ , (3)3 H + 3 ˙ H = − πG ( ρ + 3 p ) + Λ (4)where G , ρ and p are the gravitational constant, matter energy density and pres-sure respectively and the Hubble parameter H is related to the scale factor by H = ˙ a/a . In the present work, G is assumed to be constant. The generalized en-ergy conservation law for variable G and Λ is derived by Shapiro et al.34 usingRenormalization Group Theory and also by Vereshchagin and Yegorian35 using aformula of Gurzadyan and Xue36. Vereshchagin and Yegorian37 have presented aphase portrait analysis of the cosmological models relying on the Gurzadyan-Xuetype dark energy formula as mentioned above. A novel interpretation of the physicalnature of dark energy and description of an internally consistent solution for thebehavior of dark energy as a function of redshift are provided by Djorgovski andGurzadyan38 based on the vacuum fluctuations model by Gurzadyan and Xue36.The conservation equation for variable Λ and constant G is a byproduct of thegeneralized conservation law and is given by˙ ρ + 3( p + ρ ) H = − ˙Λ8 πG . (5)Let us consider a relationship between the pressure and density of the physicalsystem in the form of the following barotropic equation of state p = ωρ (6)where ω is the barotropic index which has been considered here as time-dependent.Using equation (6) we get from (5)8 πG ˙ ρ + ˙Λ = − πG (1 + ω ) ρH. (7)ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd U. Mukhopadhyay, Saibal Ray and S. B. Dutta Choudhury
Differentiating (3) with respect to t we get for a flat Universe ( k = 0) − πGρ = ˙ H ω . (8)As already mentioned in the introductory part, equivalence of three phenomeno-logical Λ-models (viz., Λ ∼ ( ˙ a/a ) , Λ ∼ ¨ a/a and Λ ∼ ρ ) have been studied in detail.So, similar type of variable-Λ model may be investigated for a deeper understandingof both the accelerating and decelerating phases of the Universe. Let us, therefore,use the ansatz , ˙Λ ∝ H , so that ˙Λ = AH (9)where A is a proportional constant.Using equations (6), (8) and (9) we get from (4)2(1 + ω ) H d Hdt + 6 H dHdt = A. (10)If we put dH/dt = P , then equation (10) reduces to dPdH + 3(1 + ω ) H = A (1 + ω ) H P . (11)To arrive at fruitful conclusions, let us now solve equation (11) under somespecific assumptions.
3. Solutions3.1. A = 0 A = 0 implies via equation (9), Λ = constant . In this case equation (11) reduces to dPdH + 3(1 + ω ) H = 0 . (12)Solving equation (12) for a ( t ), ρ ( t ) and H ( t ) we get a ( t ) = C t / ω ) , (13) H ( t ) = 23(1 + ω ) 1 t , (14) ρ ( t ) = 16 πG (1 + ω ) t (15)where C is a constant.It may be mentioned here that the above expressions for a ( t ), H ( t ) and ρ ( t )can be recovered from the corresponding expressions of Ray et al.24 for α = 0, i.e.Λ = 0 where α is a parameter for the model Λ ∼ H considered there. This meansthat the results (13), (14) and (15) can be obtained either for constant Λ (as in thepresent case) or for null Λ as in the case of Ray et al.24. The essence of this is that,ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd Λ -CDM Universe: A Phenomenological Approach With Many Possibilities a null Λ or constant Λ will provide equivalent result. It may also be mentioned herethat by abandoning Λ, Einstein obtained the expanding Universe while the sameexpanding Universe was obtained by de Sitter for constant Λ.Again, using equation (14) we can find the expression for the deceleration pa-rameter q as q = − HH ! = (cid:18) ω (cid:19) . (16)From equation (16) we find that for an accelerating Universe, ω < − /
3. But,from equation (13)-(15) it is clear that ω cannot be equal to −
1. Moreover, thepresent value of q lies near − .
539 which can be obtained from equation (16) byputting a value of ω which is equal to − /
3. The sought for signature flipping of q can be obtained from equation (16) if one considers ω as time-dependent.If H and t be the present values of H and t , then from equation (14) we canwrite, t = 23(1 + ω ) H . (17)Putting ω = − / H = 72 kms − Mpc − wefind that the present age of the Universe comes out as 13 .
58 Gyr. which fits verywell within the ranges provided by various sources (for a list of data provided byvarious sources one may consult Ray et al. 24. In this context it may be mentionedthat for stiff-fluid ( ω = 1) Ray et al.40 obtained the present age of the Universe as13 .
79 Gyr. under the ansatz Λ ∼ H . ω = − P/H
By the use of the above substitution Eq. (11) becomes dPdH − P = − AH . (18)Solving equation (18) we get a ( t ) = C e − t/ ( secBt ) / B , (19) H ( t ) = 16 ( tanBt − , (20)Λ( t ) = B (cid:20) B tan Bt + 2 B log ( secBt ) − B tanBt + 2 t (cid:21) , (21) ρ ( t ) = 148 πG ( tanBt − , (22) ω ( t ) = − (cid:20) Bsec Bt ( tanBt − (cid:21) (23)ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd U. Mukhopadhyay, Saibal Ray and S. B. Dutta Choudhury where C is a constant and B = A/ H , tanBt >
1. Then Eq. (23) implies that for a positive B , ω must be less than − B <
0, then ω can be greater than − ω in non-dust case underthe ansatz Λ ∼ ¨ a/a . Simple trigonometric solution for the scale factor was alsoobtained by Banerjee and Das42 in scalar field model. But, they obtained theirsolution by making a special assumption on the deceleration parameter while thepresent solution is a result of a supposition on the equation of state parameter ω .Again, using equation (20) we get q = − (cid:20) Bsec Bt ( tanBt − (cid:21) . (24)From equation (24) we find that, a signature flipping of q is possible if B <
0. So,the merit of this case lies the fact that the same change of sign of q can be obtainedhere by using a time-dependent form of ω and not making any special assumptionon q directly as was done by Banerjee and Das42. This once again shows that theequation of state parameter is a key ingredient of cosmic evolution. ω = − P/ H With the above assumption, Eq. (11) becomes dPdH − PH = − A H. (25)Solving equation (25) we get our solution set as a ( t ) = C t /A , (26) H ( t ) = 6 At , (27) ρ ( t ) = 27 πGA t , (28)Λ( t ) = − A t , (29) ω ( t ) = A − C is an integration constant.Thus, we find that in this case the scale factor admits a power law solution, H varies inversely as t and ρ as well as Λ follow the well known inverse square lawwith t . This type of solution was obtained by Ray et al.24 for Λ ∼ ( ˙ a/a ) , Λ ∼ ¨ a/a and Λ ∼ ρ models. For physical validity A >
0. But, in this case Λ < Λ -CDM Universe: A Phenomenological Approach With Many Possibilities A and hence represents an attractive force. However, Λ can be a repulsive force aswell if A is a complex number. Now, a complex A means a complex scale factor. So,this particular case can be thought of as a phenomenological version of spintessencemodel of Banerjee and Das43 where a complex scalar field of the form φ = e iωt is used to search for the cosmic acceleration. But, in that case ω is a constant.Also, for A = 6 if we take the present value of the Hubble parameter as H = 72kms − Mpc − then, from equation (27) the present age of the Universe comes outas 13 .
58 Gyr. which agrees very well with the estimated value24. Now, for A = 6we have ω = − / q = − (cid:20) A (cid:21) . (31)Equation (31) shows that for a positive A , the Universe expands with a constantacceleration. For A = 6 the amount of acceleration is −
2. So, in this case of thephenomenological model ˙Λ ∼ H , the deceleration parameter q does not show anychange in sign during cosmic evolution.
4. Discussions
The main objectives of the present work were to search for a signature flip of q and to find time-dependent expression for the equation of state parameter. In thatrespect, this work in general has fulfilled its goal. By selecting a time dependentform of the cosmological parameter Λ, through some analytical study, it has beenpossible to show that a change in sign of the deceleration parameter can be achievedunder some special assumptions (Sec. 3.2). It has also been possible to derive timedependent expressions for the equation of state parameter ω . It is found that ω canbe less than − A plays a vital role for studying the cosmic evolution of various phases of theUniverse. For instance, a null A presents us a case of constant Λ (Sec. 3.1) whereaspositive and negative A show the possibility of ω < − ω > − q is also possible for A < A is provided in Sec. 3.3 which hassimilarity with the work of Banerjee and Das43. For a specific value of A the ageof the Universe and the value of q are calculated also (Sec. 3.3). It is interestingto note that similar type of case study for the cosmic evolution has been done byKhachatryan44 using a parameter b for null, positive and negative values of it. So,whether there exists any internal physical relationship between the present workand that of Khachatryan44 may be a subject matter of future investigation.Determination of the present value of the Hubble parameter through analysis ofCMB data from WMAP and HST Key Project suggests that value of the equationof state parameter for dark energy models should be less than − . U. Mukhopadhyay, Saibal Ray and S. B. Dutta Choudhury models, the Statefinder diagnostic satisfies the condition d ¨ adt /aH = 1. Since d ¨ adt /a can be expressed in terms of H , ˙ H and ¨ H , so it is easy to verify that first (Sec. 3.1)and third (Sec. 3.3) cases satisfy the above condition prescribed by the Statefinderdiagnostic for ω = 0 and A = 9 respectively.However, the Λ-CDM Universe with ω = −
1, where the sine hyperbolic form ofthe scale factor can reflect both matter dominated past and accelerated expansionin future45, can not be achieved through this model. Equation (11) shows that for ω = − H grows linearly with time which does not fit with the present cosmolog-ical scenario. However, through the present model, it has been possible to providesome interesting situations which were obtained earlier by different researchers andare already discussed in respective Sections. Some awkward cases, such as constantenergy density (which can be obtained by putting 1 + ω = 2 P in equation (11)) canbe found in the work of Ray46 in relation to electromagnetic mass in n + 2 dimen-sional space-time. Some other works47 , , , , ,
52 also admit constant matterdistribution in their solutions. Finally, it should be mentioned that the presentwork is done making Λ variable and keeping G constant. So, it may be an inter-esting study when the present model is combined with a variable G and obey thegeneralized energy conservation law derived by Shapiro et al.34 and Vereshchaginand Yegorian35. That can be a subject matter of our future investigation. Acknowledgments
One of the authors (SR) is thankful to the authority of Inter-University Centre forAstronomy and Astrophysics, Pune, India, for providing Associateship programmeunder which a part of this work was carried out.
References
1. A. G. Riess et al.,
Astron. J. , 1009 (1998).2. S. J. Perlmutter et al.,
Astrophys. J. , 565 (1999).3. R. A. Knop et al.,
Astrophys. J. , 102 (2003).4. D. N. Spergel
Astrophys. J. Suppl. , 175 (2003).5. A. G. Riess et al.,
Astrophys. J. , 665 (2004).6. M. Tegmark et al.,
Phys. Rev. D , 103501 (2004a).7. P. Astier et al., Astron. Astropohys. , 31 (2005).8. D. N. Spergel et al.,
Astrophys. J. Suppl. , 377 (2007).9. V. Sahni,
Lec. Notes Phys. , 141 (2004).10. O. Elgaroy et al.,
Phys. Rev. Lett. , 021802 (2002).11. H. Minakata and H. Sugiyama, Phys. Lett. B , 305 (2003).12. J. Ellis,
Phil. Trans. Roy. Soc. Lond.
A361 , 2607 (2003).13. G. Efstathiou, W. Sutherland and S. J. Madox,
Nat. , 705 (1990).14. A. C. Pope et al.,
Astrophys. J. , 655 (2004).15. M. Tegmark et al.,
Astrophys. J. , 702 (2004b).16. V. F. Mukhanov and G. B. Chibisov,
JETP Lett. , 532 (1981).17. A. H. Guth and S.-Y. Pi, Phys. Rev. Lett. , 1110 (1982).18. S. W. Hawking, Phys. Lett. B , 295 (1982). ctober 24, 2018 6:31 WSPC/INSTRUCTION FILE um8-ijmpd Λ -CDM Universe: A Phenomenological Approach With Many Possibilities
19. A. A. Starobinsky,
Phys. Lett. B , 175 (1982).20. J. Bardeen, P. J. Steinhardt and M. S. Turner,
Phys. Rev. D , 679 (1983).21. T. Padmanabhan and T. Roychowdhury, Mon. Not. R. Astron. Soc. , 823 (2003).22. L. Amendola,
Mon. Not. R. Astron. Soc. , 221 (2003).23. A. G. Riess,
Astrophys. J. , 49 (2001).24. S. Ray, U. Mukhopadhyay and X. -H. Meng,
Grav. Cosmol. , 142 (2007a).25. S. Ray, U. Mukhopadhyay and S. B. Dutta Choudhury, Int. J. Mod. Phys. D , 1791(2007c).26. J. Kujat et al. Astrophys. J. , 1 (2002).27. M. Bartelmann et al.
New Astron. Rev. , 199 (2005).28. S. V. Chevron and V. M. Zhuravlev, Zh. Eksp. Teor. Fiz. , 259 (2000).29. V. M. Zhuravlev,
Zh. Eksp. Teor. Fiz. , 1042 (2001).30. P. J. E. Peebles and B. ratra,
Rev. Mod. Phys. , 559 (2003).31. A. A. Starobinsky, JETP Lett. , 757 (1998).32. T. D. Saini, S. Raychaudhury, V. Sahni and A. A. Starobinsky, Phys. Rev. Lett. ,1162 (2000).33. M. Reuter and C. Wetterich, Phys. Lett. B , 38 (1987).34. I. L. Shapiro, J. Sol`a and H. ˘Stefan˘ci´c,
J. Cosmol. AstroparticlePhys. , 012 (2005).35. G. V. Vereshchagin and G. Yegorian, Class. Quatum Grav. , 5049 (2006).36. V. G. Gurzadyan and S. -S. Xue, Mod. Phys. Lett. A , 561 (2003).37. G. V. Vereshchagin and G. Yegorian, Phys. Lett. B , 150 (2006).38. S. G. Djorgovski and V. G. Gurzadyan, ‘Dark Energy From Vacuum Fluctuations’ Toappear in Proc. UCLA Conference Dark Matter 2006, eds. D. Cline et al.,
Nucl. Phys.Proc. Suppl. , 6 (2007), astro-ph/0610204.39. V. Sahni,
Pramana , 937 (1999).40. S. Ray and U. Mukhopadhyay, Grav. Cosmol. , 46 (2007b).41. U. Mukhopadhyay and S. Ray, astro-ph/0510550.42. N. Banerjee and S. Das Gen. Relativ. Grav. , 1695 (2005).43. N. Banerjee and S. Das Astrophys. Sp. Sc. , 25 (2006).44. H. G. Khachatryan,
Mod. Phys. Lett. A , 333 (2007).45. V. Sahni and A. A. Starobinski, Int. J. Mod. Phys. D , 373 (2000).46. S. Ray, Int. J. Mod. Phys. D , 917 (2006).47. W. B. Bonnor, Z. Phys. , 59 (1960).48. S. J. Wilson,
Pub. Astron. Soc. Japan , 385 (1968).49. J. M. Cohen and M. D. Cohen, Nuovo Cemento , 241 (1969).50. P. S. Florides, J. Phys. A : Math Gen. , 1419 (1983).51. Ø. Grøn, Phys. Rev. D , 2129 (1985).52. B. V. Ivanov, Phys. Rev. D65