Quintessential inflation and cosmological seesaw Mechanism: reheating and observational constraints
Llibert Aresté Saló, David Benisty, Eduardo I. Guendelman, Jaime d. Haro
QQuintessential Inflation and Cosmological See-Saw Mechanism:Reheating and Observational Constraints
L. Aresté Saló, ∗ D. Benisty,
2, 3, 4, † E. I. Guendelman,
2, 3, 5, ‡ and J. d. Haro § School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom Physics Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438 Frankfurt am Main, Germany DAMTP, Centre for Mathematical Sciences, University ofCambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom Bahamas Advanced Study Institute and Conferences, 4A OceanHeights, Hill View Circle, Stella Maris, Long Island, The Bahamas Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
Recently a new kind of quintessential inflation coming from the Lorentzian distribution has beenintroduced in [1, 2]. The model leads to a very simple potential, which basically depends on twoparameters, belonging to the class of α -attractors and depicting correctly the early and late timeaccelerations of our universe. The potential emphasizes a cosmological see-saw mechanism (CSSM)that produces a large inflationary vacuum energy in one side of the potential and a very small valueof dark energy on the right hand side of the potential. Here we show that the model agrees withthe recent observations and with the reheating constraints. Therefore the model gives a reasonablescenario beyond the standard Λ CDM that includes the inflationary epoch.
Keywords: Quintessential Inflation, Instant Preheating, Numerical Simulations, Observational Constraints
I. INTRODUCTION
After the discover of the current cosmic acceleration[3–6], several theoretical mechanisms were developed inorder to explain it. One of them is quintessence (see forinstance [7–13]), where a scalar field is the responsiblefor the late time acceleration of our universe. The nextstep was to unify both acceleration phases: the early ac-celeration of the universe, named inflation [14–16], withthe current acceleration. One of the simplest ways to doit is the so-called quintessential inflation , introduced forthe first time by Peebles and Vilenkin in [17], where theinflaton field is the only responsible for both inflationaryphases.Several authors developed and improved the origi-nal Peebles-Vilenkin toy model [18–47] obtaining modelswhose theoretical results match very well with the ob-servational data provided by the Planck’s team [48–50].However, these models were built using very artificial po-tentials: containing discontinuities, matching two differ-ent types of potentials (one for inflation and the other oneleading to quintessence) or having very complicated ex-pressions and containing many parameters. Fortunately,a very simple model, constructed from the well-knownLorentzian distribution, was recently presented in [1, 2].The model belongs to the category of α -attractors [15],meaning that it provides a power spectrum of perturba-tions agreeing with the observation data, and is able todepict correctly the current cosmic acceleration. Thus, ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] the main goal of this work is to study in detail this sim-ple model, which only depends on two parameters, andto show its viability.The paper is organized as follows: In Section II we in-troduce the Lorentzian quintessential model, studying itspower spectrum during inflation and providing the the-oretical value of the parameters involved in the model.Sections III and IV are devoted to the study of all theevolution of the inflaton field, showing that the theoreti-cal results provided by the model agree with the currentobservational data. Then, in Section V we discuss the vi-ability of other similar models and in Section VI we usea combination of cosmological probes from different datasets to constrain further our model and verify its viability.Finally, Section VII summarises the results. The unitsused throughout the paper are (cid:126) = c = 1 and we denotethe reduced Planck’s mass by M pl ≡ √ πG ∼ = 2 . × GeV.
II. THE LORENTZIAN QUINTESSENTIALINFLATION MODEL
Based on the Cauchy distribution (Lorentzian in thephysics language) in [1, 2] the ansatz to be considered isthe following one, (cid:15) ( N ) = ξπ Γ / N + Γ / , (1)where (cid:15) is the main slow-roll parameter and N denotesthe number of e-folds. From this ansatz, we can find thecorresponding potential of the scalar field, which is of thesame kind than the one considered in this work, V ( ϕ ) = λM pl exp (cid:20) − ξπ arctan (sinh ( γϕ/M pl )) (cid:21) , (2) a r X i v : . [ a s t r o - ph . C O ] F e b V () / ( H M p l ) FIG. 1.
The shape of the scalar potential (2) with ξ ∼ and λ ∼ − . The left side shows the inflationary energydensity and the right side shows the late dark energy density. where λ is a dimensionless parameter: γ ≡ (cid:114) π Γ ξ . We can see the shape of the potential on Fig. 1, wherethe inflationary epoch takes place on the left hand sideof the graph, while the dark energy epoch occurs on theright hand side.The main slow roll parameter is given by (cid:15) ≡ M pl (cid:18) V ϕ V (cid:19) = 2 ξ/ (Γ π )cosh (cid:16) γ ϕM pl (cid:17) = 2 γ ξ /π cosh (cid:16) γ ϕM pl (cid:17) (3)and, since inflation ends when (cid:15) END = 1 , one has toassume that ξ Γ π > to guarantee the end of this period.In fact, we have ϕ END = M pl γ ln (cid:32)(cid:114) ξ Γ π − (cid:114) ξ Γ π − (cid:33) = (4) M pl γ ln (cid:34) √ ξπ (cid:32) γ − (cid:115) γ − π ξ (cid:33)(cid:35) < and we can see that, for large values of γ , one has that ϕ END is close to zero. Thus, we will choose γ (cid:29) ⇒ Γ ξ (cid:28) , which is completely compatible with the condi-tion ξ Γ π > .On the other hand, the other important slow roll pa-rameter is given by η ≡ M pl V ϕϕ V = 2 ξγ π tanh (cid:16) γ ϕM pl (cid:17) cosh (cid:16) γ ϕM pl (cid:17) + 4 γ ξ /π cosh (cid:16) γ ϕM pl (cid:17) . (5)Both slow roll parameters have to be evaluated when thepivot scale leaves the Hubble radius, which will happen FIG. 2.
The Marginalized joint confidence contours for ( n s , r ) at σ and σ CL, without the presence of running of the spec-tral indices. We have drawn the curve for the present modelfor γ (cid:29) from N = 65 to N = 75 e-folds. (Figure courtesyof the Planck2018 Collaboration). for large values of cosh ( γϕ/M pl ) , obtaining (cid:15) ∗ = 2 γ ξ /π cosh (cid:16) γ ϕ ∗ M pl (cid:17) , η ∗ ∼ = 2 ξγ π tanh (cid:16) γ ϕ ∗ M pl (cid:17) cosh (cid:16) γ ϕ ∗ M pl (cid:17) (6)with ϕ ∗ < . Then, since the spectral index is given inthe first approximation by n s ∼ = 1 − (cid:15) ∗ + 2 η ∗ , one getsafter some algebra n s ∼ = 1 + 2 η ∗ ∼ = 1 − γ (cid:112) r/ , (7)where r = 16 (cid:15) ∗ is the ratio of tensor to scalar perturba-tions.Now, we calculate the number of e-folds from the leav-ing of the pivot scale to the end of inflation, which isgiven by N = 1 M pl (cid:90) ϕ END ϕ ∗ √ (cid:15) dϕ = (8) π γ ξ [sinh ( γϕ END /M pl ) − sinh ( γϕ ∗ /M pl )] ∼ = ξ √ (cid:15) ∗ , so we have that n s ∼ = 1 − N , r ∼ = 8 N γ , (9)meaning that our model belongs to the class of α -attractors with α = γ ( see for instance [15]).Finally, it is well-known that the power spectrum ofscalar perturbations is given by P ζ = H ∗ π (cid:15) ∗ M pl ∼ × − . (10)Now, since in our case V ( ϕ ∗ ) ∼ = λM pl e ξ , meaning that H ∗ ∼ = λM pl e ξ , and taking into account that (cid:15) ∗ ∼ = (1 − n s ) γ ,one gets the constraint λγ e ξ ∼ × − , (11)where we have chosen as a value of n s its central value . .Summing up, we will choose our parameters satisfyingthe condition (11), with γ (cid:29) and ξ (cid:29) . Then, tofind the values of the parameters one can perform thefollowing heuristic argument:Taking for example γ = 10 , the constraint (11) be-comes λe ξ ∼ × − . On the other hand, at thepresent time we will have γϕ /M pl (cid:29) where ϕ de-notes the current value of the field. Thus, we will have V ( ϕ ) ∼ λM pl e − ξ , which is the dark energy at thepresent time, meaning that . ∼ = Ω ϕ, ∼ = V ( ϕ )3 H M pl ∼ λe − ξ (cid:18) M pl H (cid:19) . (12)Taking the value H = 67 . Km/sec/Mpc = 5 . × − M pl , we get the equations λe ξ ∼ × − and λe − ξ ∼ − , (13)whose solution is given by ξ ∼ and λ ∼ − .Given that we see that the values of ξ and γ could beset equal in order to obtain the desired results from boththe early and late inflation, we will do that from nowon, since this is statistically stronger than having two in-dependent parameters. Hence, from now on we will set ξ = γ . As we will see later, numerical calculations showthat, in order to have Ω ϕ, ∼ = 0 . (observational datashow that, at the present time, the ratio of the energydensity of the scalar field to the critical one is approxi-mately . ), one has to choose ξ = γ ∼ = 121 . . III. DYNAMICAL EVOLUTION OF THESCALAR FIELD
In this section, we want to calculate the value of thescalar field and its derivative. Analytical calculationscan be done disregarding the potential during kinationbecause during this epoch the potential energy of thefield is negligible. Then, since during kination one has a ∝ t / = ⇒ H = t , using the Friedmann equation thedynamics in this regime will be ˙ ϕ M pl t = ⇒ ˙ ϕ = (cid:114) M pl t = ⇒ (14) ϕ ( t ) = ϕ kin + (cid:114) M pl ln (cid:18) tt kin (cid:19) , where we use by definition [51, 52] as the beginning of thekination the moment when the Equation of State param-eter is close to , which coincides when the derivative of the field is maximum, corresponding to ϕ kin ≈ − . M pl and w ϕ ≈ . for γ = ξ = 122 .Recall that for our choice of the parameters ϕ END isvery close to zero and, looking at the shape of the po-tential, this regime has to start very near from ϕ = 0 .In order to check it numerically, we have integrated thedynamical system ¨ ϕ + 3 H ˙ ϕ + V ϕ = 0 , with initial conditions when the pivot scale leaves theHubble radius, that is, with ϕ i = ϕ ∗ and ˙ ϕ i = 0 , where (1 − n s ) γ = 2 γ ξ π ( γϕ ∗ /M pl ) , n s = 0 . . Thus, at the reheating time, i.e., at the beginning of theradiation phase, one has ϕ rh = ϕ kin + (cid:114) M pl ln (cid:18) H kin H rh (cid:19) , (15)where we assume, as usual, that there is not drop ofenergy from the end of inflation to the beginning of kina-tion, i.e., H kin = H END = √ V ( ϕ END ) √ M pl , which is numeri-cally satisfied, both being of the order of × − M pl .And, using that at the reheating time (i.e., when theenergy density of the scalar field and the one of the rel-ativistic plasma coincide) the Hubble rate is given by H rh = ρ rh M pl , one gets ϕ rh = ϕ kin + (cid:114) M pl ln H kin (cid:113) π g rh T rh M pl (16)and ˙ ϕ rh = (cid:114) π g rh T rh , (17)where we have used that the energy density and the tem-perature are related via the formula ρ rh = π g rh T rh ,where the number of degrees of freedom for the StandardModel is g rh = 106 . [53]. Because of the smoothnessof the potential we consider instant preheating [54–57]and, thus, we will choose as the reheating temperature T rh ∼ = 10 GeV, which is its usual value when the mech-anism to reheat the universe is this one.Effectively, considering a massless scalar X -field con-formally coupled with gravity and interacting with the in-flaton field as follows, L int = − gϕ X [54, 55], where g is the dimensionless coupling constant and where the En-hanced Symmetry Point (ESP) has been chosen at ϕ = 0 because, as we have already shown numerically, the be-ginning of the kination starts at ϕ kin ∼ − . M pl . Then,at the beginning of the kination the number density ofproduced X -particles is given by [58] n X,kin = g / ˙ ϕ / kin π (18)and, since these particles acquire a very heavy effectivemass equal to gM pl , in order to reheat the universe theyhave to decay into lighter ones, whose energy densitywill eventually dominate the one of the inflaton field, ob-taining a reheated universe with a reheating temperaturegiven by [56] T rh = (cid:18) g ∗ π (cid:19) / ρ / X,dec (cid:114) ρ X,dec ρ ϕ,dec (19) ∼ g / (cid:18) M pl Γ (cid:19) / GeV , where g ∗ = 106 . are the degrees of freedom for theStandard Model, Γ is the decay rate and the sub-index“dec” denotes the moment when the X -field decays com-pletely.Assuming now that the X -field decays into fermionsvia a Yukawa type of interaction hψ ¯ ψX with decay rate Γ = h gM pl π , where h is a dimensionless coupling con-stant, one gets T rh ∼ g / h − / GeV , (20)which leads for the narrow range of viable parameters g and h [56] to a reheating temperature around GeV.
Remark III.1
It is important to realise that for smoothpotentials, as the one considered in this study, gravita-tional particle production [59–63] does not work becausean abrupt phase transition is needed, which could onlybe obtained with a discontinuity of the potential (see forexample the Peebles-Vilenkin potential [17]).
Then, at the beginning of the radiation era we have ϕ rh ∼ = 20 M pl ˙ ϕ rh ∼ = 1 . × − M pl . (21) IV. NUMERICAL SIMULATION
First of all, we consider the central values obtained in[48] (see the second column in Table of [48]) of thered-shift at the matter-radiation equality z eq = 3365 ,the present value of the ratio of the matter energydensity to the critical one Ω m, = 0 . , and, onceagain, H = 67 . Km/sec/Mpc = 5 . × − M pl .Then, the present value of the matter energy density is ρ m, = 3 H M pl Ω m, = 3 . × − M pl , and at matter-radiation equality we will have ρ eq = 2 ρ m, (1 + z eq ) =2 . × − M pl = 8 . × − eV . So, at the begin-ning of matter-radiation equality the energy density ofthe matter and radiation will be ρ m,eq = ρ r,eq = ρ eq / ∼ =4 . × − eV .In this way, the dynamical equations after the begin-ning of the radiation can be easily obtained using as atime variable N ≡ − ln(1 + z ) = ln (cid:16) aa (cid:17) . Recasting the energy density of radiation and matter respectively asfunctions of N , we get ρ m ( a ) = ρ m,eq (cid:16) a eq a (cid:17) → ρ m ( N ) = ρ m,eq e N eq − N ) (22)and ρ r ( a ) = ρ r,eq (cid:16) a eq a (cid:17) → ρ r ( N ) = ρ r,eq e N eq − N ) , (23)where N eq ∼ = − . denotes the value of the time N atthe beginning of the matter-radiation equality. To obtainthe dynamical system for this scalar field model, we willintroduce the dimensionless variables x = ϕM pl , y = ˙ ϕH M pl . (24)Taking into account the conservation equation ¨ ϕ +3 H ˙ ϕ + V ϕ = 0 , one arrives at the following dynamical system, x (cid:48) = y/ ¯ H, y (cid:48) = − y − ¯ V x / ¯ H, (25)where the prime is the derivative with respect to N , ¯ H = HH and ¯ V = VH M pl . It is not difficult to see that one canwrite ¯ H = 1 √ (cid:114) y V ( x ) + ¯ ρ r ( N ) + ¯ ρ m ( N ) , (26)where we have defined the dimensionless energy densitiesas ¯ ρ r = ρ r H M pl , ¯ ρ m = ρ m H M pl . (27)Finally, we have to integrate the dynamical system (25),with initial conditions x ( N rh ) = x rh = 20 and y ( N rh ) = y rh = 2 . × imposing that ¯ H (0) = 1 , where N rh de-notes the beginning of reheating, which is obtained im-posing that ρ r,eq e N eq − N rh ) = π g rh T rh , (28)that is, N rh = N eq −
14 ln (cid:18) g rh g eq (cid:19) − ln (cid:18) T rh T eq (cid:19) ∼ = − . , (29)where we have used that ρ eq,r = π g eq T eq with g eq = 3 . and, thus, T eq ∼ = 7 . × − GeV.We have numerically checked that, to obtain the con-dition ¯ H (0) = 1 , the parameters γ and ξ have to be equalto . . Once these parameters have been properly se-lected, the obtained results are presented in Figure 3. V. OTHER SIMILAR POSSIBLE MODELS
An analogously built simplified model would be V ( ϕ ) = λM pl e − α arctan (cid:16) β ϕMpl (cid:17) , (30) - - -
20 20 N - - ω eff FIG. 3.
Left: The density parameters Ω m = ρ m H M pl (orange curve), Ω r = ρ r H M pl (blue curve) and Ω ϕ = ρ ϕ H M pl , from kinationto future times. Right: The effective Equation of State parameter w eff , from kination to future times. As one can see in thepicture, after kination the universe enters in a large period of time where radiation dominates. Then, after the matter-radiationequality, the universe becomes matter-dominated and, finally, near the present, it enters in a new accelerated phase where w eff approaches − . for α and β being its positive parameters, where we havesuppressed the sinh function. Its slow-roll parameters (cid:15) and η are (cid:15) = 12 αβ (cid:16) β ϕM pl (cid:17) , (31) η = αβ (cid:16) β ϕM pl (cid:17) (cid:18) βα ϕM pl (cid:19) . Hence, the slow-roll parameter (cid:15) is also related to aLorentzian distribution, in this case in function of ϕ in-stead of N and with an overall square involved. Usingthat (cid:12)(cid:12)(cid:12) β ϕ ∗ M pl (cid:12)(cid:12)(cid:12) (cid:29) max(1 , | α ) , we get that n s ∼ = 1 + 4 αβ (cid:18) M pl ϕ ∗ (cid:19) , r ∼ = 8 (cid:18) αβ (cid:19) (cid:18) M pl ϕ ∗ (cid:19) . (32)Using the same approximations, one can find that N = 1 M pl (cid:90) ϕ END ϕ ∗ √ (cid:15) dϕ ∼ = − β α (cid:18) ϕ ∗ M pl (cid:19) (33)and, therefore, n s ∼ = 1 − N and r ∼ = 8 (cid:18) α βN (cid:19) / , (34)which does not fall within the allowed range of the val-ues of the spectral index for the number of e-folds beingbetween and (see for instance [27, 28, 64]). So, thismodel does not work. In the same way, the model named“arctan inflation" introduced in [65] V ( ϕ ) = λM pl (cid:18) − α arctan (cid:18) β ϕM pl (cid:19)(cid:19) , (35) does not work as well.To finish this Section a final comment is in order. Onecould also use the original model obtained in [1], V ( ϕ ) = λM pl exp (cid:20) − ξπ arctan (sinh ( γϕ/M pl )) (cid:21) · (cid:18) − γ ξ π γϕ/M pl ) (cid:19) , (36)which is negative around ϕ ∼ = 0 . However, it leads tothe exact same results, given that the only change is thebehavior of the potential for ϕ ∼ = 0 .Effectively, when the pivot scale leaves the Hubble ra-dius one has (cid:18) − γ ξ π γϕ ∗ /M pl ) (cid:19) ∼ = (cid:16) − r π (cid:17) ∼ = 1 , (37)because r ∼ = N γ (cid:28) . Thus, the last term of the poten-tial (36) does not affect to the power spectrum of pertur-bations. In the same way, one can easily check that atthe end of inflation the potential is positive. Therefore,defining once again that kination starts when w ϕ ∼ = 1 ,which occurs when ϕ kin = 0 . M pl (corresponding nowto the time when the potential becomes positive again),everything works as expected. VI. COSMOLOGICAL PROBES
In order to constrain our model, we use a few data sets:
Cosmic Chronometers (CC) exploit the evolution ofdifferential ages of passive galaxies at different redshiftsto directly constrain the Hubble parameter [66]. We useuncorrelated 30 CC measurements of H ( z ) discussed in[67–70]. For Standard Candles (SC) we use measure-ments of the Pantheon Type Ia supernova dataset [71]that were collected in [72] and the measurements from
68 70 72 74 H ( km / sec / Mpc ) r d ( M p c ) m m r d ( Mpc ) CC + BAO + SC + CMBCC + BAO + SC + CMB + R19
FIG. 4.
The posterior distribution for the LQI model with σ and σ . The data set include Baryon Acoustic Oscillationsdataset, Cosmic Chronometers, the Hubble Diagram from Type Ia supernova, Quasars and Gamma Ray Bursts and the CMB.R19 denotes the Riess 2019 measurement of the Hubble constant as a Gaussian prior. Parameter LQI LQI + SH0ES H ( km/sec/Mpc ) 70 . ± .
123 71 . ± . ϕ /M pl . ± .
541 22 . ± . ϕ / ( H M pl ) 10 − . ± .
635 5 . ± . m . ± . . ± . Λ . ± . . ± . ξ . ± .
865 122 . ± . r d ( Mpc ) 145 . ± .
363 143 . ± . TABLE I.
The best fit values for the discussed model for a uniform prior of the Hubble parameter and for a Gaussian prior thatdemonstrates the SH0ES measurement (Supernovae and H0 for the Dark Energy Equation of State). The values ϕ , ˙ ϕ denotethe current values of the scalar field and its derivative. Quasars [73] and Gamma Ray Bursts [74]. The parame- ters of the models are to be fitted with by comparing theobserved µ obsi value to the theoretical µ thi value of thedistance moduli, which is given by µ = m − M = 5 log ( D L ) + µ , (38)where m and M are the apparent and absolute magni-tudes and µ = 5 log (cid:0) H − /M pc (cid:1) + 25 is the nuisance pa-rameter that has been marginalized. The distance moduliis given for different redshifts µ i = µ ( z i ) . The luminositydistance is defined by D L ( z ) = cH (1 + z ) (cid:90) z dz ∗ E ( z ∗ ) , (39)where E ( z ) = H ( z ) H . Here, we are assuming that Ω k = 0 (flat space-time).We use uncorrelated data points from different Baryon Acoustic Oscillations (BAO) collected in[75] from [76–87]. Studies of the BAO feature inthe transverse direction provide a measurement of D H ( z ) /r d = c/H ( z ) r d , where r d is the sound horizon atthe drag epoch and it is taken as an independent param-eter and with the comoving angular diameter distance[88, 89] being D M = (cid:90) z c dz (cid:48) H ( z (cid:48) ) . (40)In our database we also use the angular diameter distance D A = D M / (1+ z ) and D V ( z ) /r d , which is a combinationof the BAO peak coordinates above, namely D V ( z ) ≡ [ zD H ( z ) D M ( z )] / . (41)Finally we take the CMB Distant Prior measurements[90]. The distance priors provide effective information ofthe CMB power spectrum in two aspects: the acousticscale l A characterizes the CMB temperature power spec-trum in the transverse direction, leading to the variationof the peak spacing, and the “shift parameter” R influ-ences the CMB temperature spectrum along the line-of-sight direction, affecting the heights of the peaks, whichare defined as follows: l A = (1 + z d ) πD A ( z d ) r d ,R ( z d ) = √ Ω m H c (1 + z d ) D A ( z d ) , (42)with its corresponding covariance matrix (see table I in[90]). The BAO scale is set by the sound horizon atthe drag epoch z d ≈ when photons and baryonsdecouple, given by r d = (cid:90) ∞ z d c s ( z ) H ( z ) dz, (43)where c s ≈ c (3 + 9 ρ b / (4 ρ γ )) − . is the speed of sound inthe baryon-photon fluid with the baryon and photon den-sities being ρ b ( z ) and ρ γ ( z ) respectively [91]. However,in our analysis we used r d as independent parameter.
68 70 72 74 H ( km / sec / Mpc ) CC + BAO + SC + CMBCC + BAO + SC + CMB + R19
FIG. 5.
The posterior distribution for the LQI model with σ and σ , for the Hubble parameter vs. the parameter ξ .The data set include Baryon Acoustic Oscillations dataset,Cosmic Chronometers, the Hubble Diagram from Type Ia su-pernova, Quasars and Gamma Ray Bursts and the CMB. R19denotes the Riess 2019 measurement of the Hubble constantas a Gaussian prior. We take the complete analyses that combine the like-lihoods from all of the datasets. We use a nested sam-pler as it is implemented within the open-source packaged
P olychord [92] with the
GetDist package [93] to presentthe results. The prior we choose is with a uniform dis-tribution, where Ω r ∈ [0; 1 . ] , Ω m ∈ [0 . ; 1 . ] , ϕ ∈ [20; 25] , ˙ ϕ ∈ [0; 10 − ] Ω Λ ∈ [0 . ; 1 . ] , H ∈ [50; 100] Km/sec/Mpc, ξ ∈ [100; 130] , r d ∈ [130; 160] Mpc. The measurementof the Hubble constant yielding H = 74 . ± . (km/s)/Mpc at CL by [6] has been incorporatedinto our analysis as an additional prior (
R19 ).Figure 4 shows the posterior distribution of the datafit with the best fit values at table I. One can seethat the Gaussian prior of the Hubble parameter doesnot change the results by much. For both cases the χ minimized value gives a good fit, since χ /Dof =[255 . / , . / ∼ , where Dof are the degreesof freedom for the χ distribution. The statement fromthe fit shows that the QI models that we discuss here areviable models and can describe early times as well as latetimes. VII. CONCLUDING REMARKS
In this paper we study the phenomenological implica-tions of a Lorentzian Quintessential Model, dependingonly on two parameters, where the reheating of the uni-verse, due to the smoothness of the corresponding poten-tial, is produced via the well-known
Instant Preheating mechanism. We have shown, analytically and numeri-cally, that for reasonable value of these parameters, thissimple model is able to depict correctly our universe uni-fying its early and late time acceleration. In fact, themodel belongs to the class of the so-called α -attractorsand, thus, matches very well with the observational dataof the power spectrum of perturbation during inflationprovided by the Planck’s team. It currently leads to adark energy density around % of the total one.In addition to the reheating constraints, we havetested the model with different measurements, some ofthem from the late universe such as Type Ia supernova,Gamma Ray Bursts and Quasars and the others from theCosmic Microwave Background from the early universe.The model fits very well to the latest measurements andgives a reasonable scenario beyond the standard Λ CDMthat includes the inflationary epoch.
ACKNOWLEDGMENTS
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