Modified Holographic energy density driven inflation and some cosmological outcomes
aa r X i v : . [ phy s i c s . g e n - ph ] J un Modified Holographic energy density driven inflation and somecosmological outcomes
Gargee Chakraborty and Surajit Chattopadhyay ∗ Department of Mathematics, Amity University, Major Arterial Road,Action Area II, Rajarhat, New Town, Kolkata 700135, India. Department of Mathematics, Amity University, Major Arterial Road,Action Area II, Rajarhat, New Town, Kolkata 700135, India. (Dated: June 15, 2020)
Abstract : Motivated by the work of Nojiri et al. [1], the present study reports a modelof inflation under the consideration that the inflationary regime is originated by a type ofholographic energy density. The infrared cutoff has been selected based on the modifiedholographic model that is a particular case of Nojiri-Odintsov holographic dark energy [13]that unifies phantom inflation with the acceleration of the universe on late-time. On gettingan analytical solution for Hubble parameter we considered the presence of bulk viscosityand the effective equation of state parameter appeared to be consistent with inflationaryscenario with some constraints. It has also being observed that in the inflationary scenariothe contribution of bulk viscosity is not of much significance and its influence is increasingwith the evolution of the universe. Inflationary observables have been computed for themodel and the slow-roll parameters have been computed. Finally, it has been observed thatthe trajectories in n s − r are compatible with the observational bound found by Planck. Ithas been concluded that the tensor to scalar ratio for this model can explain the primordialfluctuation in the early universe as well. Keyword : holographic density; inflation; slow-roll parameters
AMSC:
I. INTRODUCTION
The holographic principle, put forward by [5], has its root in the theory of quantum gravity. Itstates that the entropy of a system scales with the area of the enveloping horizon, rather than thevolume. Inspired by black hole thermodynamics, the holographic principle gives a connection of theshort cutoff of a quantum field theory to a long distance cutoff due to the limit set by the formation ∗ Electronic address: [email protected]; [email protected] of a black hole [6]. This consideration has been applied in the study of the late time accelerationof the universe to a considerable extent. In a cosmological framework of the late time universe,the vacuum energy constitutes a dark energy (DE) sector having a holographic origin. This DE iscalled holographic dark energy (HDE) [7]. During recent years, a considerable amount of effortshave been devoted to the development of the various cosmological aspects and generalizations ofHDE. Literatures in this arena include [9–12]. A detailed review on HDE has been presented in[7]. Applying holographic principle to HDE in a universe with a characteristic length scale L andreduced Planck mass M p the following expression can be derived (see [7]): ρ de = C M p + C M p L − + C L − + ... (1)The first term being disfavoured due to its incompatibility with holographic principle and thevacuum fluctuation estimated from UV-cutoff quantum field theory being ρ de ∼ Λ . M p L − , ithas been prescribed that the above expression should begin from the second term. The third andsubsequent terms being negligible in comparison with the second term, the expression for HDE hascome out to be [7] ρ de = 3 C M p L − (2)Here, C is an arbitrary parameter. For the remaining part of the paper we will consider M p = 1.In a recent paper, Chattopadhyay et al. [8] reported an inflationary universe in f ( T ) frameworkthrough slow-roll formalism and holographic Ricci dark energy as its driving force.In the present paper we will probe the holographic inflation. The inflation theory proposes aperiod of extremely rapid (exponential) expansion of the universe during its first few moments. Itis developed by Alan Guth, Andrei Linde, Paul Steinhardt and Andy Albrecht, offers solutions tothese problems and several other open questions in cosmology. It states that prior to the moregradual Big Bang expansion, there was a period of extremely rapid (exponential) expansion of theuniverse, during which the time energy density of the universe was dominated by a cosmologistconstant type of vacuum energy that later decayed to produce the matter and radiation that fill theuniverse today. Inflation was both rapid and strong. Inflation is now considered an extension ofthe Big Bang theory since it explains the above puzzle so well while retaining the basic assumptionof a homogeneous expanding universe [12]. Rest of the paper is organized as follows: In Section IIwe have demonstrated the inflation with modified holographic density as its driving force. In thissection, we have derived an analytical solution for the inflationary scale factor. In a subsectionunder section II, we have discussed the role of bulk viscosity in the modified holographic inflation.In another subsection we have discussed the slow roll parameters from the Hubble parameterderived under the inflationary settings through modified holographic density. Lastly in this sectionwe have considered some limiting scenarios. We have concluded in Section III. II. HOLOGRAPHIC INFLATION
In this section, we will construct the basic model of holographic inflation. We consider ahomogeneous and isotropic Friedmann-Robertson-Walker (FRW) geometry with metric dS = − dt + a ( t ) (cid:18) dr − k r + r d Ω (cid:19) (3)where, a ( t ) is scale factor and k = 0 , +1 , − k = 0.As already stated, the primary focus of the present work is to demonstrate the inflation drivenby an effective fluid of holographic origin. We denote the effective fluid density responsible fordriving the inflation as ρ inf .As the inflation driving fluid is of holographic origin, we consider its source as modified holo-graphic dark energy (MHDE), where the IR cutoff is a linear combination of H and ˙ H [2, 3] ρ Λ = 2 α − β (cid:20) ˙ H + 3 α H (cid:21) (4)It may be noted that holographic Nojiri-Odintsov DE proposed in [13] is the most general HDEmodel and the holographic inflation follows naturally from that proposal. The form (4) is a specificexample of the HDE proposed in [13]. In studying the inflationary scenario, we will take ρ Λ as ρ inf where ρ inf is the energy density of the effective field that derives the inflation and having thepossibility of originating from a scalar field, modified theory of gravity or from any other sources[23–26]. Here, L − IR = ˙ H + α H . As we are considering an inflationary scenario, we incorporatea correction to IR cutoff because of quantam effect L = q L IR + UV . This approach has alreadybeen attempted in [1]Friedmann’s first equation in inflationary scenario is H = 13 ρ inf (5)Following Elizalde and Timoshkin [27], the conservation equation can be written as˙ ρ inf + 3 H ( ρ inf + p eff ) = 0 (6)where, p eff = p inf + Π, where Π represents the bulk viscous pressure. As we are consideringMHDE as the driving force for the inflation, we have ρ Λ = ρ inf in Eq. (6). Hence, using Eqns. (4)and (6), we get the solution for reconstructed Hubble parameter H as H = s a − α + α − β )Λ4 UV C + 2Λ UV (cid:0) − α + β + α Λ UV (cid:1) (7)As in Eq.(6) we have H expressed in terms of a , we can express the ρ inf as ρ inf = 3 C − /m − C + e C fm + √ fmt f ! m − α + α − β )Λ4 UV + 2Λ UV (cid:0) − α + β + α Λ UV (cid:1) (8)Using H = ˙ aa we have the following analytical solution for reconstructed scale factor a ( t ) as a = 2 − /m − C + e √ fmt +2 fmC f ! m (9)where m = 3 α − α − β )Λ UV and f = UV ( − α + β + α Λ UV ) . Since Λ UV is neither equal to 0 nor to infinity, Eq.(9) provides the evolution of scale factor. If C is very small, then it does not contribute signifi-cantly to the expression − C + e √ fmt +2 fmC and hence we can write (cid:16) − C + e √ fmt +2 fmC (cid:17) m ≈ e ( √ ft +2 fC ). Hence, from (9) we can write a ( t ) ≈ a e H t , which is the de Sitter solution for scalefactor with a = (cid:16) f (cid:17) /m e fC and H = m √ f . Hence, this consideration is consistent with thebasic inflationary features.
1. Consideration of bulk viscous pressure
Ability of bulk viscosity to drive inflationary expansion is being discussed since eighties. Refer-ences [15, 16] argued in favour of the bulk viscosity to drive inflation. However, [17] argued that itis the non perturbative effect that is responsible for the negative pressure during the inflationaryexpansion and ruled out bulk viscosity as a potential candidate for being the driving fore behindthe inflationary expansion of the early universe. In a later study, ref.[18] argued for a non lineargeneralization of causal linear thermodynamics that could describe viscous inflation without parti-cle production. In a very recent work, Bamba and Odintsov [19] demonstrated a fluid model havingEoS that includes bulk viscosity and argued that a fluid description of inflation has an equivalencewith the description of inflation in terms of scalar field theories. They [19] also demonstrated therealization of graceful exit from inflation for the reconstructed models of fluid.Inspired by the work of [19] we incorporate the bulk viscous pressure as time varying form to viewthe consequences if the inflationary fluid includes bulk viscosity. As a specific case the bulk-viscouspressure is considered as Π = − Hξ , where ξ = ξ + ξ aa + ξ aa = ξ + ξ H + ξ ( ˙ H + H ), where ξ , ξ and ξ are positive constants. Under the consideration of ρ Λ = ρ inf we get reconstructed ξ and Π as ξ = ξ + ξ s a − α + α − β )Λ4 UV C + 2Λ UV (cid:0) − α + β + α Λ UV (cid:1) + ξ a − α + α − β )Λ4 UV C + 2Λ UV (cid:0) − α + β + α Λ UV (cid:1) ! (10)and Π = − s a − α + α − β )Λ4 UV C + UV ( − α + β + α Λ UV ) ξ + ξ s a − α + α − β )Λ4 UV C + UV ( − α + β + α Λ UV ) + ξ a − α + α − β )Λ4 UV C + UV ( − α + β + α Λ UV ) ! (11)The absolute of reconstructed bulk viscous pressure in Eq.(11) is plotted in Fig.1 against redshift z for various combinations of α and β with α − β > < ρ inf + 3 H ( ρ inf + p inf + Π) = 0 (12)In the Eq.(12), putting the value of ρ inf from Eq.(8), Π from Eq.(11), H from Eq.(7), we getthermodynamic pressure as p inf p inf = a − α + 3( α − β )Λ4 UV C ( − α + β +( − α )Λ UV ) Λ UV − UV β − α + α Λ UV +3 s a − α + α − β )Λ4 UV C + UV ( β − α + α Λ UV ) ξ + a − α + α − β )Λ4 UV C ξ + ξ Λ UV ( β − α + α Λ UV ) + ξ s a − α + α − β )Λ4 UV C + UV ( β − α + α Λ UV ) (13)Equation of state parameter (EoS) is w eff = p inf +Π ρ inf . Hence in this equation putting the valueof p inf from Eq.(13), Π from Eq.(11) and the value of ρ inf from Eq.(8), we get w eff as w eff = C − /m (cid:18) − C e C fm + √ fmtf (cid:19) m − α + 3( α − β )Λ4 UV ( β − α +( − α )Λ UV ) Λ UV − UV β − α + α Λ UV C (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) − α + α − β )Λ4 UV + UV ( β − α + α Λ UV ) (14)Eq. (14) gives us that w = − . , − . , − . , − . t = 0 , . , . , . w eff = w inf ≈ −
1. Hence, the introduction of bulk viscosity having itsconsequence consistent with inflation [28]. Furthermore, Fig.1 makes it apparent that in the veryearly phase of the universe Π has contribution close to 0. However, with exit from inflation thecontribution of bulk viscous pressure is increasing with the evolution of the universe.
2. Slow roll parameters
As we have obtained an analytic solution for H , it is now possible to obtain the Hubble slowroll parameter ǫ n , where n is positive integer. The Hubble slow roll parameter is defined as [1] ǫ n +1 = dln | ǫ n | dN (15)where with ǫ ≡ H ini /H and N ≡ ln ( a/a ini ) the e-folding number, and where a ini and H ini is thescale factor at the beginning of inflation and the corresponding Hubble parameter (inflation endswhen ǫ = 1). Therefore, we can find the values of the inflationary observables, namely the scalarspectral index of the curvature perturbations n s , its running α s , the tensor spectral index n T andthe tensor-to-scalar ratio r [22]. The slow-roll parameters in terms of Hubble parameter are [1] ǫ = − ˙ HH (16) ǫ = ¨ HH ˙ H − HH (17) ǫ = ( ¨ HH − H ) − " H ˙ H ... H − ¨ H ( ˙ H + H ¨ H ) H ˙ H ! − HH ( H ¨ H − H ) (18)Using Eq.(7)in Eqns. (16), (17)and (18), the slow roll parameters comes out to be: ǫ = 9 C (cid:0) − α + β + α Λ UV (cid:1) UV C ( − α + β ) + 2 (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α − α − β )Λ4 UV Λ UV + 3 C α Λ UV (19) ǫ = − α + 3( α − β )Λ UV + 9 C (cid:0) − α + β + α Λ UV (cid:1) Λ UV C ( − α + β ) + 2 (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α − α − β )Λ4 UV Λ UV + 3 C α Λ UV (20) ǫ = " C (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α Λ4 UV (cid:0) − α + β + α Λ UV (cid:1) × " Λ UV − C (cid:18) − /m (cid:16) − C e C fm + √ fmt f (cid:17) m (cid:19) α Λ4 UV ( α − β )+2 (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) (cid:18) α + β Λ4 UV (cid:19) Λ UV + 3 C (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α Λ4 UV α Λ UV − (21)Putting the value of ǫ , ǫ respectively from Eq.(19),Eq.(20) respectively in n s = 1 − ǫ − ǫ , weget spectral index n s as n s = 1 − C ( − α + β + α Λ UV ) Λ UV C ( − α + β )+2 − /m (cid:18) − C e C fm + √ fmtf (cid:19) m α − α − β )Λ4 UV Λ UV +3 C α Λ UV − − α + α − β )Λ UV + C ( − α + β + α Λ UV ) Λ UV C ( − α + β )+2 − /m (cid:18) − C e C fm + √ fmtf (cid:19) m α − α − β )Λ4 UV Λ UV +3 C α Λ UV (22)Putting the value of ǫ , ǫ , ǫ respectively from Eq.(19),Eq.(20), Eq.(21) respectively in α s = − ǫ ǫ − ǫ ǫ , we get α s as α s = C (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) (cid:18) α + α + β Λ4 UV (cid:19) (cid:0) − α + β + α Λ UV (cid:1) × Λ UV (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) (cid:18) α + β Λ4 UV (cid:19) Λ UV +3 C (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α Λ4 UV (cid:0) − α + β + α Λ UV (cid:1)! − (23)Putting the value of ǫ from Eq.(19) in n T = − ǫ , we get tensor spectral index n T as n T = − C (cid:0) − α + β + α Λ UV (cid:1) Λ UV C ( − α + β ) + 2 (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α − α − β )Λ4 UV Λ UV + 3 C α Λ UV (24) FIG. 1: Evolution of reconstructedBulk Viscous pressure (see Eq. (11))against red-shift z . We have taken ξ = 0 . ξ = 0 . ξ = 0 . C = 0 .
24, Λ UV = 26, β ∈ [ − . , +0 . α = − . , . . n s r FIG. 2: Evolution of tensor to scalarratio r against spectral index n s .We have taken Λ UV = 24 , C =0 . , C = 0 .
48. Lines have beendrawn for { α = 0 . , β = 180 } , { α =0 . , β = 190 } and { α = 0 . , β =200 } . The lines are almost coincident. Putting the value of ǫ from Eq.(19)in r = 16 ǫ , we get tensor to scalar ratio r as r = 72 C (cid:0) − α + β + α Λ UV (cid:1) Λ UV C ( − α + β ) + 2 (cid:18) − /m (cid:16) − C + e C fm + √ fmt f (cid:17) m (cid:19) α − α − β )Λ4 UV Λ UV + 3 C α Λ UV (25)The evolution of tensor to scalar ratio r in Eq.(25) is plotted against spectral index n s in Eq.(22)in Fig.2. It is observed that the trajectories in n s − r plane exhibit a decreasing behaviour,which is consistent with the Jawad et al.[4] observation. Here, r < .
168 (95% CL, Planck TT +LowP),which is the observational bound found by Planck. Hence, our calculated tensor to scalarratio for this model is consistent with the observational bound of Planck. Hence, it can explainthe primordial fluctuation in the early universe.Now, we will deduce the particle horizon R P and the event horizon R E . R P is the maximumdistance from which light could have travelled to the observer in the age of the universe and theevent horizon R E is the largest comoving distance from which light emitted now can ever reachthe observer in future.It is the boundary beyond which the light cannot affect the observer. Anevent horizon is an acknowledged feature of the expanding universe [20, 21]. The particle horizon R P and the event horizon R E is given by the respective differential equations:˙ R P = HR P + 1 (26)and ˙ R E = HR E − H = ˙ aa . Using the expression of a from Eq.(9), we get H as H = (cid:16) C − C + e C fm + √ fmt (cid:17) √ f Now putting this value of H in Eqns. (26) and (27)and solving theequations we get R P = (cid:16) − C + e C fm + √ fmt (cid:17) m C − (cid:16) − C e − C fm −√ fmt (cid:17) m F h m , m , m , C e − C fm −√ fmt i √ f (28) R E = (cid:16) − C + e C fm + √ fmt (cid:17) m C + (cid:16) − C e − C fm −√ fmt (cid:17) m F h m , m , m , C e − C fm −√ fmt i √ f (29)If Λ UV → ∞ then f → m → α . Hence in that case R E and R P will tend to infinity.
3. Limiting Λ UV → ∞ Ultraviolet cutoff Λ UV has a major role in understanding the inflation. Here we will study theparameters when there is no ultraviolet cutoff i.e., Λ UV → ∞ . In this case, scale factor a calculatedin Eq.(9) will also tend to infinity. As a = z which implies that z → − H in Eq.(7) will takethe value √ C a − α . As H is real so C ≥
0. In this case ρ inf in Eq.(8) tends to 0 and also p inf tends to 0. Effective EoS w eff will take the value α − UV → ∞ . As in the inflationaryscenario w ≈ −
1, we may infer that α should be very small. III. CONCLUDING REMARKS
Motivated by the works of Nojiri et al. [1] and Oliveros and Acero [6] the present studyattempted to study a model of inflation under the consideration that the inflationary regime isoriginated by a type of holographic energy density as in references [2, 3]. The holographic densityis basically a The infrared cutoff has been selected based on the modified holographic model0considered here. Because of the high energy scales in the inflationary regime the infrared cutoffhas been corrected by ultraviolet cutoff Λ UV . This procedure helped us in getting an analyticalsolution for Hubble parameter, which in turn could give us a solution for scale factor (see Eqs.(7)) and (9). Afterwards we considered the presence of bulk viscosity Π and the effective equationof state parameter appeared to be consistent with inflationary scenario with the necessity that C and α are small.The C , an integration constant and α , a constant parameter of the infrared cutoff are alreadythoroughly explained in the previous sections. As α is required to be small, in Fig. 1 we havechosen values of | α | <
1. The β has been varied in such a manner that it can get hold of thevarious combinations of { α, β } leading to positive and negative differences. In Fig. 1 it is observedthat for α − β < z ) the bulk viscosity is very low in theinflationary scenario and subsequently it s increasing. Hence, for this combination, it is observedthat in the inflationary scenario the contribution of bulk viscosity is not of much significance and itsinfluence is increasing with the evolution of the universe. It is consistent with our assumption that inthe inflationary scenario only a fluid of holographic origin is considerable and other components arenon-existent. The low bulk viscosity is indicative of the absence of other fluids in the inflationaryphase. However, we can further generate the conditions on α and β to derive the role of bulkviscosity in the early inflationary scenario. If we consider Eq. (11), we can have a further insightinto α and β . Hence, in Fig. 1 we generate two more surfaces that indicate decreasing pattern ofbulk viscosity. However, for this case with α − β > H , the inflationary observables have been computed for thepresent model in Eqs. (22) to (24), and the slow-roll parameters have been computed in Eqs. (16)to (21). Additionally, it has been observed that the trajectories in n s − r plane exhibit a decreasingbehaviour, which is consistent with the Jawad et al.[4] observation. Also, it has been found that r < .
168 (95% CL, Planck TT + LowP),which is the observational bound found by Planck. Hence,our calculated tensor to scalar ratio for this model is consistent with the observational bound ofPlanck. Hence, it can explain the primordial fluctuation in the early universe as well.While concluding, we would like to state that we have incorporated bulk viscosity in the infla-tionary scenario driven by holographic energy density. Although it has been observed that in aholographic fluid driven inflation the contribution of bulk viscosity is negligible, it has also beenobserved that the effect of bulk viscosity is increasing with expansion of the universe. This ap-proach can further be extended to the reconstruction of Starobinsky inflation, and also to modified1gravity like f ( T ) and f ( G ) framework. IV. ACKNOWLEDGEMENT
Authors are thankful to the anonymous reviewer for the constructive suggestions. Surajit Chat-topadhyay acknowledges financial support from the Council of Scientific and Industrial Research(Government of India) with Grant No. 03(1420)/18/EMR-II. Both the authors acknowledge thefacility and hospitality provided by the Inter-University Centre for Astronomy and Astrophysics(IUCAA), Pune, India during a scientific visit from December, 2019 till January, 2020. [1] S. Nojiri, S. D. Odintsov and E. N. Saridakis, Phys. Lett. B , 275-277(2008).[3] L. P. Chimento and M. G. Richarte, Phys. Rev. D, Phys. Rev. D 84, 123507 (2007).[4] A. Jawad , N. Videla , F. Gulshan , Eur.Phys. J.C. , 1 (2004).[8] S. Chattopadhyay, A. Pasqua, I. Radinschi and A. Beesham, Int. J. Geom. Methods Mod. Phys. , 114790 (2019).[11] S. Nojiri, S. D. Odintsov, Eur. Phys. J. C
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