Inflationary phase and role of dark energy: Revisited
IInflationary phase and role of dark energy: Revisited
Indranath Bhattacharyya ∗ Department of Mathematics, Barasat Government College, Barasat, Kolkata 700124, West Bengal, India
Saibal Ray † and Prasenjit Paul ‡ Department of Physics, Government College of Engineering and Ceramic Technology, Kolkata 700010, West Bengal, India (Dated: May 22, 2019)The inflationary phase of the Universe is explored by proposing a toy model related to the scalarfield, termed as inflaton . The potential part of the energy density in the said era is assumed to havea constant vacuum energy density part and a variable part containing the inflaton. The prime ideaof the proposed model constructed in the framework of the closed Universe is based on a fact thatthe inflaton is the root cause of the orientation of the space. According to this model the expansionof the Universe in the inflationary epoch is not approximately rather exactly exponential in natureand thus it can solve some of the fundamental puzzles, viz. flatness as well as horizon problems.It is also predicted that the constant energy density part in the potential may be associated to thedark energy, which is eventually different from the vacuum energy, at least in the inflationary phaseof the Universe. However, the model keeps room for the end of inflationary era.
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I. INTRODUCTION
The cosmology based on an elegant idea that our Uni-verse was started to be expanded from a spacetime sin-gularity, known as Big Bang cosmology. The discoveryof the Cosmic Microwave Background [1] abandonedthe idea of the Steady State theory of the Universe [2–4]. Moreover, the data observed in the ‘Supernova Cos-mology Project (SCP)’ [5, 6] and the ‘High-z Super-nova Search Team (HST)’ [7, 8] show that presently theUniverse is not only expanding but also accelerating.Both the groups concluded that their results are sensi-tive to a linear combination of 0 . M − . Λ (SCP)and 1 . M − Ω Λ (HST), where Ω M and Ω Λ being re-spectively the cosmic matter and vacuum energy densi-ties. The negative sign in the linear combination signifiesthat the matter and vacuum energy have opposite effectson the cosmological acceleration. A positive vacuum en-ergy causes to accelerate the expansion, while the mattertends to slow it down. The high density vacuum energyeventually wins over the matter and makes the Universeto be expanded with an acceleration. Such vacuum en-ergy, characterized by the negative pressure p (cid:39) − ρ , isinterpreted as dark energy, constituting 73 percent of thecontent of the present Universe [9]. The energy associ-ated with primordial inflation is around a few TeV hav-ing extremely long duration and hence dark energy be anatural consequence of inflationary paradigm at the elec-troweak energy scale [10].During early age the Universe was solely filled up withthis vacuum energy. The scale factor a ( t ) grew expo- ∗ Electronic address: i [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] nentially so that the ratio of the vacuum energy densitywould rapidly be approaching towards the critical densityat this era. Such phenomenon of exponential expansionis called inflation . It was Guth [11, 12] who first proposedsuch inflationary model to fix up ‘monopole problem’. Itwas soon observed that the inflation could solve otherlong-standing problems, such as ‘flatness problem’ and‘horizon problem’ also. Guth proposed that the scalarfields might get caught in the local minimum of some po-tential, and then rolled towards a true minimum of thepotential. Kazanas [13] suggested that an exponentialexpansion could eliminate the particle horizon. Accord-ing to Sato [14] exponential expansion would be able toeliminate domain walls, another kind of exotic relic. Ein-horn and Sato [15] argued that their model could solve‘magnetic monopole puzzle’.However, it was then realized that Guth’s model ofinflation had fatal problem as the transition from supercooled initial ‘false vacuum’ [16–19] to the lower energy‘true vacuum’ cannot occur everywhere simultaneously[20, 21]. The Guth version of inflation theory was thennecessarily replaced by a new type ‘slow roll inflation’theory [22, 23]. A scalar field, called ‘inflaton’ was intro-duced to explain such phenomenon. Coleman and Wein-berg [24] introduced the symmetry breaking mechanismto the new inflation model. In the inflation theory Linde[25] added a new dimension by introducing the ‘chaoticinflation’ in which initially one or more inflaton fieldsvary in a random manner with positions. Linde [26, 27]also proposed the theory of ‘hybrid inflation’ theories byintroducing more than one scalar fields.In cosmology essentially inflation is nothing but a rapidexponential expansion of the early Universe by a fac-tor of at least 10 in volume, driven by negative pres-sure. The detailed particle physics mechanism behindsuch phenomenon is still unknown to the researchers. Itis also a super cooled expansion, when the temperature a r X i v : . [ phy s i c s . g e n - ph ] M a y drops down from 10 K to 10 K, although exact dropof temperature is model dependent. Such relatively lowtemperature is maintained throughout the entire infla-tionary phase and at the end of inflation the reheatingoccurs to attain the pre-inflationary temperature.In the present article our motivation is to propose anew kind of inflation model based on the scalar poten-tial with decreasing kinetic energy. The energy density isassumed to consist of a large constant part along with asmall slowly varying part, termed as vacuum energy den-sity. The constant part would generate an exponentialexpansion, whereas the variable part of the energy den-sity participates in the dynamics of inflation. In a nut-shell, therefore, we are strongly motivated by the ideathat during inflationary age the expansion is approxi-mately exponential in nature. In the ongoing work, basi-cally we investigate role of dark energy in the inflationarymodel.The study is organized as follows: We provide the ba-sic mathematical formulation including the Einstein fieldequations in Sec. 2. In Sec. 3 we have presented physicalexplanation of the proposed model, whereas some casestudies have been featured in Sec. 4. The last Sec. 5 isdevoted to provide some concluding remarks along withsalient features.
II. MATHEMATICAL FORMULATION
We consider a scalar field φ , known as the inflaton,along with a potential V ( φ ) to take part to vary the vac-uum energy. At this stage of expansion the Universe isassumed to be filled up with such vacuum energy. The ex-pression of the vacuum energy density as well as vacuumpressure in presence of a spatially homogeneous inflaton φ in the Robertson-Walker spacetime becomes [28] ρ = 12 ˙ φ + V ( φ ) , p = 12 ˙ φ − V ( φ ) , (1)which must be derived from the expression of the gener-alized energy-momentum tensor T µνφ = − g µν (cid:20) g ρσ ∂φ∂x ρ ∂φ∂x σ + V ( φ ) (cid:21) + g µρ g νσ ∂φ∂x ρ ∂φ∂x σ . (2)Such energy-momentum tensor is slightly differentfrom that of the constant vacuum energy T µν , which isproportional to g µν in the general coordinate system ac-cording to the Lorentz invariance condition. The pro-portionality of T µν to g µν corresponds to the conditionof negative pressure as p = − ρ , but by the introductionof inflaton such condition is deviated as neither pressurenor density remains constant.Now the important issue is what should be the expres-sion of V ( φ ). Let us, as a check, consider the expressionof potential in the following form: V ( φ ) = 12 m φ + ρ Λ , (3) where ρ Λ is the constant vacuum density.It looks like the potential of the massive scalar fieldalong with this constant term ρ Λ , which should be muchlarger than the term m φ . The idea behind to intro-duce such potential is that it should be analogous to thevacuum energy density if there is no inflaton field at all.Now, let us recall the Friedmann equation in presenceof inflaton, which yields following two equations:¨ φ + 3 H ˙ φ + m φ = 0 , (4)8 πG H (cid:34) ˙ φ m φ ρ Λ (cid:35) − ka H , (5)where H represents the Hubble parameter and m is themass of inflaton field.It is already mentioned that we are strongly motivatedby the idea that during inflationary age the expansionis approximately exponential in nature. The presence ofinflaton makes the Universe to expand very fast, nearlyexponentially. Therefore, one may assume that the φ field is inversely proportional to the scale factor a . Suchproportionality relation may be expressed as aφ = c k = (cid:115) k πc p , (6)where c p stands for the ratio of the mass of the inflatonto the Planck mass. The above relation implies that theexpansion of the Universe yields the decay of inflatonfield in the same rate, although it is restricted in theinflationary era. Now, using the proportionality relationgiven by the above equation the Hubble parameter canbe calculated from Eq. (5) as H = H I (cid:34) φ ρ Λ (cid:35) , (7)where H I = (cid:114) πGρ Λ . (8)If ρ Λ (cid:29) ˙ φ the Hubble parameter can be approximatedas H ≈ H I + ˙ φ (cid:18) H I ρ Λ (cid:19) . (9)Let us now try to solve Eq. (4) with the aid of Eq. (9).Neglecting the second degree term of ˙ φ one can solve theequation to obtain the expression of inflaton as φ = e − HI t (cid:34) Ae − √ H I − m t + Be √ H I − m t (cid:35) . (10)We would like to obtain the analytical expression ofthe scale factor to realize the nature of the expansion inthe inflationary age. By means of Eqs. (6), (7) and (8)such expression can be calculated as (cid:32) a + (cid:114) a − km (cid:33) e − √ a − km a (11)= (cid:32) a I + (cid:114) a I − km (cid:33) e H I t − √ a I − km aI , where a I is the initial value of a , i.e., the value of scalefactor at the beginning of inflationary era. The value of a , smaller than a I , corresponds to the quantum age. If ma I (cid:29) a ≈ a I e H I t , (12)showing the approximate nature of exponential expan-sion.To have a simplified as well as approximate expressionof φ one can exploit the proportionality relation givenby Eq. (6) as well as approximate expression of a in Eq.(12) and find that the φ is approximately proportional to e H I t . This is possible if in Eq. (10) it is assumed A = 0and B = φ I (where φ I is the value of φ at the beginningof inflation phase) or vise-versa. In addition to that thefollowing relation is to be satisfied, i.e. H I = m √ . (13)This is an important relation showing that m is neitherimaginary nor zero because of the expression of H I beingpositive real quantity. Again, according to Eq. (6) the k is proportional to m and therefore the model proposedhere is restricted only for the closed type of Universe. Itis also to be noted that Eq. (6) shows a relation betweenthe spacetime geometry and inflaton field, which readilysolves the flatness problem and yields an approximateexponential expansion. The flatness condition | V (cid:48) ( φ ) V ( φ ) | (cid:28)√ πG [28] gives an estimate of φ . Utilizing this flatnesscondition and also the condition given by Eq. (10) onecan estimate | φ I | m p (cid:28) .
24 (14)where m p represents the Planck mass.In this model we must keep provision for the Universeto get out of the inflationary era and to begin the ra-diation dominated regime. In the present model φ aswell as ˙ φ diminish exponentially with time and hence ρ a .Eventually ρ a vanishes when t is large enough. This isquite uncomfortable situation as it implies the inflationcould take pretty long time to be completed. To get ridof this situation it may be assumed that ρ a need not van-ish rather would reach to a minimum value ρ min at theend of the inflationary phase. After that ρ a decays into standard model particles including the electromagneticradiation, resulting the radiation dominated phase. It isto be noted that all the particles at this stage remainmassless. Some of these may get mass through sponta-neous symmetry breaking. The process of decaying ρ a into the particles might take place through parametricresonance [29], which is a mechanical excitation and os-cillation at certain frequencies.The above issue can also be handled in a mathematicalframework so that the inflation could end up. The totaldensity ρ of the Universe may be expressed as ρ = ρ a + ρ Λ , (15)where ρ a = 12 ( ˙ φ + m φ ) . (16)From the energy equation of the cosmology, given by˙ ρ = − aa ( p + ρ ) , (17)with p = wρ, it can easily be obtained w = − , if the density variesas a − . If the density would comprise of only constantvacuum density ρ Λ , then w = −
1. In the Eq. (14) ρ a isthe varying density, whereas ρ Λ is the constant vacuumdensity. Now one can treat such varying density as astate and the pressure-density relation may be realizedas p a = W ρ a = wρ a (18)where W is an operator which brings different phasesof the universe and w is the corresponding eigen value.In the inflationary phase of the Universe the eigen value w = − . Now the operator W operating on ρ a bringsa change so that the eigen value w becomes , leadingto the end of inflation. It can be calculated that when w = the density ρ a is proportional to a − and thus theradiation age begins. At the radiation phase the varyingdensity energy becomes ρ a = ρ R (cid:18) aa R (cid:19) − , (19)where ρ R and a R represent the radiation density and thescale factor, respectively, at the moment when inflationends and radiation era begins. Such a radiation density ρ R can be estimated as ρ R ≈ (cid:18) k πG (cid:19) a R . (20)Now a question may arise what will be the nature of φ and ˙ φ at the end of inflation. Relative to the time scaleof inflationary age both of φ and ˙ φ vanish asymptoticallyand thus the potential V reaches to a true vacuum. Butin the global scale of time period φ should have a mini-mum value beyond which no inflaton field exists and theinflation era gradually exits to the radiation age. Thisminimum φ is given by φ min = c k a R . (21)This model does not deal with the dynamics of theUniverse in the radiation age. It only addresses the issueof possible deadlock at the end of inflation.The number of e-foldings can be calculated as N = H I (cid:20) T − (cid:18) H I φ I ρ Λ (cid:19) e − H I T (cid:21) , (22)where T is the total time period for which the inflationlasts.Thus it can be predicted for a large period that thenumber of e-foldings are directly proportional to the timeperiod through which the inflation elapses. To have alarge number of e-foldings therefore the time period of theinflationary phase is to be large enough. The minimumamount of inflation required to solve the various cosmo-logical problems is about 70 e-foldings, i.e. an expansionby a factor of 10 . The total number of e-foldings in-flation described by this model will exceed the numberdepending on the period of inflation as well as mass ofthe inflaton field. III. PHYSICAL EXPLANATION
To this end, let us critically discuss how the presentmodel is advantageous compared to the earlier models.It has already been mentioned that Guth’s model of in-flation [11, 12] had problem that the transition from the‘false vacuum’ [16–19] to the ‘true vacuum’ does notoccur everywhere at a time. Linde’s version could ableto get rid of such drawback and well explain ‘flatness’and ‘horizon’ problem. Likewise Linde’s model [22] ourmodel proposes an approximate nature of such exponen-tial burst.In the framework of the proposed model an importantrelation is deduced in Eq. (13). The left hand side of thesaid equation contains the terms representing the dynam-ics of the expansion, whereas the right hand side standsfor the space-time geometry. Followed by Eq. (13) it isquite evident that k is not only non-zero but it is alsoa positive, interpreting the orientation of the Universe isclosed. In Eq. (5) the term k is cancelled out, which indi-cates that the expansion of the Universe is independentof the orientation of the space. In the closed Universethe massive inflaton field makes the Universe flat in thatstage. Thus the presence of inflaton field not only fix the‘flatness problem’, but it may also be interpreted thatthe origin of the inflaton is solely responsible for the ori-entation of space. The density ρ of the Universe at this stage can be di-vided into two parts: the variable part ρ a and the con-stant part ρ Λ . This constant part ρ Λ does not take partin the dynamical change of the Universe. Now one caninterpret such energy ρ Λ as the widely discussed dark en-ergy. The dark energy identified in this manner is differ-ent from the vacuum energy consisting of both ρ a and ρ Λ in the inflation era. With a minimal prediction one mustconclude that the dark energy, arising by the introduc-tion of cosmological constant, is different from vacuumenergy even beyond the inflationary age, if quintessenceeffect [30] has not been taken into account. However,Basilakos, Lima and Sola [31] have made an attempt atelleviating fundamental cosmic puzzles via a dynamic Λwhich may have definite role for gracefully exits from in-flation to a radiation phase followed by dark matter andvacuum regimes, and, finally, evolves to a late-time deSitter phase. IV. COMPARISON WITH OTHERINFLATIONARY MODELS
In this section we compare our toy model with two re-cent inflationary models, viz. intermediate inflation andlogamediate inflation [32–35], in the light of recent obse-vational data [36–38].Using the potential given in Eq. (3), we can obtain theslow roll parameter as (cid:15) ( φ ) = M p (cid:20) V (cid:48) ( φ ) V ( φ ) (cid:21) = M p (cid:20) m φρ Λ + m φ (cid:21) , (23) η ( φ ) = − M p (cid:20) V (cid:48)(cid:48) ( φ ) V ( φ ) (cid:21) = − (cid:34) M p m ρ Λ + m φ (cid:35) . (24)Generally, inflationary models are characterized by thescalar spectral index n s and the tensor-to-scalar ratio r [39, 40]. The scalar spectral index ( n s ) and the tensor-to-scalar ratio ( r ) of primordial spectrum which are re-lated to the slow-roll parameters ( (cid:15), η ), given by n s − ≈ η − (cid:15), (25) r ≈ (cid:15). (26)Assuming ρ Λ to be large and constant as indicatedearlier for the present model, we can write r ≈ −
83 ( n s − − C, (27)where C = M p m ρ Λ . Hence it is possible to vary C in therange C ∈ [0,0.065] so that our toy model fits well in { r, n s } space with the observational data. FIG. 1: The plot of r versus n s for different values of C (redlines). Outer contour correponds to the Planck + WMAP-9+ BAO data [36, 37] whereas inner contour correponds to thePlanck + BICEP2 + Keck Array data [38]. The dashed linecorresponds to power-law inflationary models with exponen-tial potentials whereas the black lines I and II correspond tointermediate and logamediate inflation model respectively. Now, in the case of intermediate inflation, we have a = exp ( At f ), where A > > f > A = 0 . a = exp [ A (ln t ) λ ], where A > λ > A = 2 . × − and λ =2 [34].Fig. 1 shows that our model agrees fairly well withthe observational data along with these highly modifiedinflationary models. V. CONCLUSION
To summarize, the basic philosophy behind the presentpaper is to explain the nature of exponential expansion in general. However, specifically the investigation providesthe following novel features:(i) The model shows that in the inflationary era theHubble constant, an approximate constant and hence avariable, evolves out of the inflaton mass.(ii) The model keeps room for the end of inflation toavoid the possible deadlock.(iii) It can also be predicted that for a large time periodthe number of e-foldings are directly proportional to thetime period through which the inflation elapses. In otherwords, to have a large number of e-folding the time periodof the inflationary phase is to be large enough.(iv) In the treatment we have divided the density ρ of the Universe into two parts: the variable part ρ a andthe constant part ρ Λ . Though the constant part ρ Λ doesnot take part in the dynamical change of the Universe,however it can be interpreted as the widely discussed darkenergy [30] responsible for inflation through it’s repulsivenature.(v) It has been shown that even in such a simple toymodel the values of the primordial tilt and the tensor-to-scalar ratio are in very good agreement with the obser-vational data and comparable with other modified infla-tionary models.It has already been shown that the proposed modelcan explain the flatness as well as the horizon problems.However, there is a great issue of monopole. It seems thatat this stage monopole problem cannot be tackled by themodel as it is fixed through the process of reheating afterthe inflationary phase. This is therefore beyond purviewof the present work. Acknowledgement
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