We show that a model of quintessence with exponential potential, which allows
to obtain general exact solutions, can generate locations of CMBR peaks which
are fully compatible with present observational dataComment: 7 pages, no figure

Di Domenico, D.

Rubano, C.

Scudellaro, P.

English

arXiv

We show that a model of quintessence with exponential potential,
which allows to obtain general exact solutions, can generate location of
CMBR peaks which are fully compatible with present observational data

D. DI DOMENICO

RUBANO, CLAUDIO

SCUDELLARO, PAOLO

Archivio della ricerca - Università degli studi di Napoli Federico II

arXiv:astro-ph/0209357v1  18 Sep 2002Testing a quintessence model with CMBR peakslocationD.Di Domenico†, C.Rubano†,•, P.Scudellaro†,•† Dept. of Physical Sciences - Univ. Federico II - Naplesvia Cinthia Napoli Italy•I.N.F.N - Sec. of NaplesE-Mail: Rubano@na.infn.itFebruary 2, 2008AbstractWe show that a model of quintessence with exponential potential,which allows to obtain general exact solutions, can generate location ofCMBR peaks which are fully compatible with present observational data.1 IntroductionIn correspondence of some angular scales, the anisotropy spectrum of the CosmicMicrowave Background Radiation (CMBR) exhibits some peaks of intensity.Standard cosmology succeeds in explaining the origin of these peaks and suppliesan analytical procedure for the calculation of their positions. A particularlyinteresting aspect of this argument is that such locations are very sensitiveto the variation of the cosmological parameters. Like other kinds of analysis,this places strong constraints on the cosmological parameters and discriminatesamong the different models [1].In this paper we want to test a particular model of quintessence. In orderto make this, we first have to calculate, assuming a flat anisotropy spectrum,the theoretical values of the peaks’ positions and compare these results with theexperimental data.Peaks are formed during recombination age. The perturbative scales smallerthan the value of Hubble’s radius at the recombination time ηrec re-enter into thehorizon before that time and behave like acoustic waves; in other words, in theprimordial plasma they generate perturbations comparable to those generatedby a sound wave in a fluid [2].Small perturbations are therefore generated in matter and radiation, givingrise to photon clustering.The radiation pressure acts against this process. The result of these twocompetitive processes is the formation of peaks of intensity of radiation (that1is, of peaks of clustering in the distribution of photons) at several scales, char-acterized from the relation [3]kn =nπrs(ηrec), (1)where η is the conformal time, n is a positive integer and rs is the sound horizonat the instant η, that is,rs(η) =∫ η0csdη′, (2)being cs the sound velocity.Today we observe the CMBR projected on the sky, so that we are interestedin the angular positions of the peaks. Once we suppose that the distribution ofradiation propagates, from the last scattering up to now, in such a way that itdoes not modify the position of the peaks, these ones are observed at angularscales corresponding to the three-dimensional scales expressed in Eq. (1). Weobtain such values noticing that to a scale kn corresponds an angular scale givenbyln ≃ rθ(ηrec)kn , (3)where rθ is the comoving angular distance which in the case of a spatially flatuniverse has the simple expressionrθ = η0 − ηrec. (4)Consequently, we getln =η0 − ηreccsηrecnπ = nlA , (5)where it has been assumed that cs is approximately constant between η andηrec and equal to its average value, while lA is given bylA ≡η0 − ηreccsηrecπ . (6)2 Universe with quintessenceIn a model of universe without quintessence or a Λ-term the positions of thepeaks satisfy relation (5). But the recent observations give data which cannotbe deduced from this relation. On the other hand, if we consider a model withquintessence, the presence of this new component modifies Eq. (5). It has beenshown that such modifications can be accounted for introducing a discrete seriesof parameters ϕn, in function of which Eq. (5) is rewritten as [4]ln = lA(n − ϕn) . (7)2In [4] the empirical formulas giving the values of the parameters ϕn for ageneric model of quintessence are deduced. It isϕn = ϕ + ∆ϕ , (8)withϕ = a1 (rrec)a2 + 0.291Ωϕls , (9)wherea1 = 0.286 + 0.262(Ωbh2) , (10)a2 = 0.1786− 6.308(Ωbh2) + 174.9(Ωbh2)2 − 1168(Ωbh2)3 , (11)rrec =ρr(zrec)ρm(zrec)= 0.042(Ωbh2) zrec103, (12)Ωϕls = η−1rec∫ ηrec0Ωϕ(η)dη . (13)In the last one of these relations, it is assumed that Ωϕ(η) does not changetoo much rapidly before recombination, so that Ωϕls is an effective averaged valueof quintessential energy.In [4] Doran and Lilley obtain the formulas that allow to calculate the the-oretical values of the first three peaks.Therefore, by means of Eq. (5), it is possible to test a given model ofquintessence. In fact, once we fix the values of the parameters by which themodel is specified, we can calculate the values ln of the angular positions of thepeaks and compare them with the experimental data. In order to perform thiscalculation, of course, it is first necessary to find the value of lA.From the definition in Eq. (6), and following [4], we can use the evolutionequations for a universe with quintessence and derive the expressions for η andηrec in terms of Ωϕ. We can also approximate Ωϕ by means of the constantaverage Ωϕls for the period around last scattering.In this way, setting 8πG = c = 1 and denoting with Ωr0 and Ωϕ0 today’s radi-ation and quintessence components, with arec the scale factor at last scattering(having we supposed a0 = 1), and with cs the average sound speed before lastscattering, we obtain3H2(t)(1 − Ωϕls)= ρm(t) + ρr(t) = ρm0 a(t)−3 + ρr0a(t)−4 . (14)Today, the radiation density is negligible, and we have3H20 (t) (1 − Ωϕ0 ) = ρm0 (t) , (15)which we insert in Eq. (14) to obtain(dadη)2= H20(1 − Ωϕls)−1[(1 − Ωϕ0 ) a (η) + Ωr0] . (16)3Integrating this equation by separation of variables, it is possible to obtainη(a, H0, Ωϕls, Ωϕ0 , Ωr0).In particular, we are interested in ηrec, that is,ηrec = 2H−10√1 − Ωϕls1 − Ωϕ0{(arec +Ωr01 − Ωϕ0)1/2−(Ωr01 − Ωϕ0)1/2}. (17)If we define w0 as the Ωϕ weighted equation of state of the universew0 =∫ η00Ωϕ(η)w(η)dη∫ η00 Ωϕ(η)dη, (18)we can integrate the cosmological equation for w ≃ cost.= w0 and rewrite Eq.(16) as(dadη)2= H20 [(1 − Ωϕ0 − Ωr0) a (η) + Ωr0 + Ωϕ0 a(1 − 3w0)] . (19)This equation givesη0 = 2H−10 (1 − Ωϕ0 )−1/2F (Ωϕ0 , w0) . (20)Thus, we now have η0 and ηrec, which we can insert in Eq. (6) for the presenttime.The result is [5]lA = πc−1sF (Ωϕ0 , w0)(1 − Ωϕls)−1/2{(arec +Ωr01 − Ωϕ0)1/2−(Ωr01 − Ωϕ0)1/2}−1− 1,(21)withF (Ωϕ0 , w0) =12∫ 10dx(x +Ωϕ01 − Ωϕ0x(1−3w0) +Ωr0(1 − x)1 − Ωϕ0)−1/2. (22)3 Location of the CMB peaks for an exponentialmodelLet us then calculate the locations of the CMB peaks in a particular model ofquintessence. As it will be seen, through the comparison with the experimentaldata and the forecasts of the simplest models of quintessence, the model we useis in good agreement with the observations.4The model of quintessence here investigated has been studied by Rubanoand Scudellaro [6], starting from a scalar field with a potentialV (ϕ) = Be−λϕ , λ =√3/2 . (23)This choice leads to the following exact solutions for, respectively, the scalefactor, the density and equation of state parametersa =τ2(1 + τ2)τ20 (1 + τ20 ), (24)Ωm =1 + τ2(1 + 2τ2)2 , (25)w = −3 + 2τ24 + 4τ2, (26)with τ = ωt, ω2 = σ2B2 = (3/2)B2 (this time scale depends on the unknownvalue of B, but this does not affect the situation presented here, which dependsonly on τ). The value of τ at present epoch is τ0, so that Ωm0 = Ωm(τ0).The present values of all the relevant parameters, thus, depend only on thechoice of τ0.Such a model is compatible with the data of the observations of the type-IAsupernovae if we choose [7]τ0 = [0.82, 1.40] (with the most probable value t0 = 1.268) ,Ωm0 ∈ [0.12, 0.30] (with the most probable value Ωm0 = 0.15) .This last result is smaller than Ωm0 = 0.30, the commonly accepted one.Indeed, in this model, it is true that Ωm = 0.15 is the most probable value, butthe distribution statistics of the data gives an approximately equal probabilityfor the other values of the interval. Moreover, some experimental evidences existthat justify a consideration of values of Ωm0 smaller than the usual one (see [8],for instance).The model we are testing here has been proved to be compatible also withthe data on the peculiar velocities of galaxies [9].By means of the values given above for τ0 and Ωm0, it is possible to goback to what is foreseen by this model on the locations of the peaks. In thecalculations it has been placed cs = 0.52 and Ωr0 = 9.89·10−5.Choosing Ωb = 0.05 and h = 0.65, we obtain, for l1, l2, and l3, the values inTable I.The range of the values comes from the range of τ0, at 1σ level.Most Probable Value Rangel1 232 218 − 236l2 572 527 − 586l3 884 799 − 9095Table IOn the other hand, Table II shows the values of the angular scales at whichCMBR is experimentally observed.The data in the first column concern the BOOMERANG experiment, whilein the second column the data are obtained from a combined analysis of somerecent observations.BOOMERANG Experiment Combined Datal1 221±14 222±14l2 524±35 539±21l3 850±28 851±31Table IIThe comparison between the intervals of the values for the single peaksshows that the exponential model for quintessence reproduces the observed dataenough faithfully. In particular, it turns out to be more satisfactory when thevalue of τ0 is a little smaller than the most probable value, so correspondingto a value of Ωm0 greater than 0.15, as it is reasonable with respect to otherexperimental evidences.Finally, let us compare our results with the standard theoretical predictionsfor models with and without a Λ-term, which are showed in Table III.Universe with w0 = Ωϕ = 0 Universe with w0 = −1 and ΩΛ = 0.6l1 206 220l2 502 528l3 711 793Table IIIWe see that the model without a Λ-term is out of the observational ranges(as expected), while the other is fully compatible with both the data and ourresults. Therefore, we find no possibility of distinction.4 ConclusionsUp to now the model presented in [6] has revealed fully compatible with ob-servational data. Even if simple exponential potentials for quintessence do notappear as the preferred ones in the literature, not giving the most known andused results, they anyway still seem to be able to describe some phenomena.As a matter of fact, we have now also found that, concerning CMBR, thevalues in Table I are sensibly different from those in Table III.This gives the hope that more precise measurements of the CMBR spectrumwill be sufficient to discriminate among quintessence models. (For a discussionon this point see also [10].)6References[1] W. Hu and N. Sugiyama, Astrophys. J. 444 (1995) 489[2] V. Mukhanov, H. Feldmann, and R. Branderberger, Phys. Rep. 215 (1992)203[3] W. Hu, doctoral thesis, astro-ph/9508126[4] M. Doran and M. Lilley, astro-ph/0104486[5] M. Doran et al., Astrophys. J. 559 (2001) 501[6] C. Rubano and P. Scudellaro, Gen. Rel. Grav. 34 (2002) 307[7] M. Pavlov, C. Rubano, M. Sazhin, and P. Scudellaro, Astrophys. J. 566(2002) 619[8] N. Bachall, R. Cen, R. Davè, J. P. Ostriker, and Q. Yu, Astrophys. J. 541(2000) 1[9] M. Sereno and C. Rubano, astro-ph/0203205, to appeir MNRAS[10] C. Rubano and P. Scudellaro, astro-ph/0203225 to appair Gen. Rel. Grav.7

Testing a quintessence model with CMBR peaks location

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