arXiv:astro-ph/0305334v1 19 May 2003 tt,adbhvo fflcutos hs ffcsaeil- are effects 1. These Figure fluctuations. in lustrated of equation-of- reveal behavior density, to and energy CMB state, dark the the expect oscillations about we acoustic information scales, of angular pattern smaller the on horizon and sound scales, last-scattering degree the on of scales, optics Via angular geometric large supernovae. the on 1a effect Sachs-Wolfe type integrated dark the the the of to presence the complementary of energy, probe a the provide world, di- photons baryonic a CMB to our of with absence clues the interaction Despite seek dark-energy rect energy. to dark — the of Universe nature the accelerating the cosmological of our theories anew consider must data last we powerful 7], of this of 6, epoch theories. light 5, the In 4, from present. [3, Universe of the record to the sharp scattering in a this conditions provided minutes, arc already the several has to angular mission down on sky ongoing anisotropy full the CMB from pre- the scales for of a Designed measurement represents cision cosmology. 2] experimental [1, in satellite milestone (WMAP) Probe Microwave Anisotropy Wilkinson the by (CMB) background crowave l.Teemdl r:(1 oeswt constant a with models (Q1) out are: carry we models equation-of-state, mod- precise, mass These quintessence and of be range To anisotropy els. wide CMB a for 14]. the acceleration spectra of 13, fluctuation cosmic analysis 12, energy extensive the 11, an cosmic for 10, the 9, responsible [8, dominates is energy and density of inhomo- pressure, form negative geneous time-evolving, dynamical, a a oesaesltit pctm ls“atrsec- “matter plus spacetime into split variables, tor” are models cal poten- law inverse-power an under tial. evolving by field described scalar trackers a (Q4) era; recombination density the energy during non-negligible a with models, quintessence ters, ihasml-aaerzd time-evolving simply-parametrized, a with nti ril ets the test we article this In cosmological test to results WMAP the use to aim We mi- cosmic the of measurement precision The h ut fprmtr eciigtecosmologi- the describing parameters of suite The θ omcMcoaeBcgon n uenv osrit nQ on Constraints Supernova and Background Microwave Cosmic Q h pctm n mte etr fthe of sector” “matter and spacetime The . 1 eateto hsc srnm,DrmuhClee 6127 College, Dartmouth Astronomy, & Physics of Department ASnmes 98.80.-k dat m CMB numbers: target PACS ACBAR viable, and identify CBI we the constraints to select fit joint these a Using quintessence provide early also We of models tential. of equation-of-s class constant a a equation-of-state; including: o tempera models models, (CMB) cosmological quintessence quintessence background of microwave predictions the cosmic with the of surements epromadtie oprsno h ikno Microwave Wilkinson the of comparison detailed a perform We θ M w n eaaeqitsec parame- quintessence separate and , including , unesnehypothesis quintessence ocrac ein n agtModels Target and Regions Concordance < w oetR Caldwell R. Robert − ;(2 models (Q2) 1; w Q)early (Q3) ; Dtd oebr3 2018) 3, November (Dated: that , 1 n ihe Doran Michael and qaino-tt,w edol ospecify to constant a only with need models, we of equation-of-state, family simplest the For model. infla- by tion. nearly generated of perturbations spectrum density primordial scale-invariant in- a with this cold models spatially-flat, matter In to dark attention our depth. restrict we optical vestigation hub- and scalar density, index, amplitude, matter spectral perturbation perturbation dark scalar cold parameter, density, ble baryon the are I.1 h atr fCBaiorp a eelinfor- reveal can anisotropy CMB (Ω of abundance quintessence pattern the The about mation 1: FIG. e iteb y,btaedsigihdb h aa h red The data. the by distinguished are but di ( eye, which by models little equation-of-state fer constant of examples are tt ( state unesnemdl r pcfidb h aaee set parameter the by specified θ are models quintessence the at CMB the peak by acoustic rejected first is the [5], of 3 WMAP height by and determined location the matches nitnusal,btdsic ihrsett N.Tebl The SNe. ( to curve respect with distinct but indistinguishable, ino unesnea al ie,mr aaeesare parameters more frac- times, early non-negligible at a quintessence of include tion may the of which time-evolution quintessence, realistic more feature which models w σ M h unesneprmtr ayfo oe to model from vary parameters quintessence The -level. = clrfilswt nivrepwrlwpo- law inverse-power an with fields scalar ; ae ipyprmtie,time-evolving simply-parametrized, a tate; = akeeg.W osdrawd ag of range wide a consider We energy. dark f dl o ute analysis. further for odels − w − { 0 ,adbhvo fflcutos( fluctuations of behavior and ), 0 . Ω . )adbu ( blue and 5) ) lhuhi scnitn ihteSedt and data SNe the with consistent is it although 8), idrLbrtr,Hnvr H03755 NH Hanover, Laboratory, Wilder b h 2 nstoyPoe(MP mea- (WMAP) Probe Anisotropy ueadplrzto anisotropy polarization and ture , Ω ,adtetp asupernovae. 1a type the and a, cdm 1 h 2 − ,n h, , 1 . )cre r ohlow- both are curves 2) s A , S τ , uintessence: r } δ nodr these order, In . .Tetrecurves three The ). θ Q Q ,equation-of- ), = { χ w 2 } CMB- For . ack f- 2 required, e.g. θQ = {w, dw/da, ...}, to characterize the impact on the cosmology in general and the CMB in par- ticular. Our analysis method is as follows: (i) compute the CMB and fluctuation power spectra for a given cosmo- logical model; (ii) compute the relative likelihood of the model with respect to the experimental data; (iii) assem- ble the likelihood function in parameter space to deter- mine the range of viable quintessence models. For step (i) we use both a version of CMBfast [15] modified for quintessence, as well as the newly-available CMBEASY [16]. For step (ii) we supplement the WMAP data with the complementary ACBAR [17] and CBI-MOSAIC data FIG. 3: The constraints on constant equation-of-state models [18, 19] (using the same bins as Ref. [6, 7]), in addi- due to CMB (WMAP, ACBAR, CBI) and type 1a supernovae tion to the current type 1a SNe data [20, 21, 22, 23]. (Hi-Z, SCP) are shown. The starting point for our parameter- Certain other constraints, such as the HST Key Project search, the family of CMB-degenerate models, is shown by the measurement of H0 [24] or the limit from Big Bang Nu- thick, black line. 2 cleosynthesis on Ωbh [25] through the deuterium abun- dance measurement are satisfied as cross-checks. It is 2 2 remarkable that such agreement can be found between with Ωbh = 0.023, Ωcdmh = 0.126, ns = 0.97, and such diverse phenomena. Our focus in the following in- characterized by pairs {w, h} having (nearly) indistin- vestigation, however, is primarily on the CMB and SNe. guishable CMB anisotropy patterns. The pairs {w, h} are shown in Figure 2, and all represent quintessence models with χ2 = 1429 for the WMAP temperature- temperature and temperature-polarization data — a one- parameter family of best-fit models. From this starting point, we explored over 6 × 104 models distributed on a grid filling a 6-cylinder around the best-fit line, vary- 2 2 ing {Ωbh , Ωcdmh , h, ns, τr} at intervals in w. For each model we evaluate the likelihood relative to WMAP, as well as the complementary ACBAR and CBI-MOSAIC data. The 2σ boundary, based on a ∆χ2 test for six degrees of freedom is shown in Figure 3. We have also evaluated the constraint in the w − Ωm plane for the combined Hi-Z/SCP type 1a supernova data set, show- FIG. 2: The one-parameter family of best-fit models, which 2 exploit the geometric degeneracy of the CMB anisotropy pat- ing the 2σ region based on a ∆χ test for two degrees tern, is shown as the thick, red curve in the w − h plane. We of freedom. Our basic conclusion from the overlapping have explored models in a six-dimensional cylinder in the pa- constraint regions is that there exist concordant models rameter space surrounding this “best-fit line.” The HST-Key with −1.25 . w . −0.8 and 0.25 . Ωm . 0.4. We Project 1σ measurement of the Hubble constant is shown by have identified four sample models in Table I for further the shaded band. analysis. Q1: We have analyzed the cosmological constraints on the simplest model of quintessence, characterized by a model w h σ8 constant equation-of-state, w. We have used the equiv- Q1.1 -0.82 0.630 0.84 alence between a scalar field ϕ with potential V (ϕ) and Q1.2 -1.00 0.682 0.89 Q1.3 -1.18 0.737 0.96 the equation-of-state w in order to self-consistently eval- Q1.4 -1.25 0.759 0.97 uate the quintessence fluctuations. For the range w< −1 we employ a k essence model, keeping the sound speed (actually, this is dω2/dk2) fixed at c2 = 1. Since this TABLE I: Sample best-fit models with a constant equation-of- s 2 2 model introduces only one additional parameter beyond state (Q1). All models have Ωbh = 0.023, Ωcdmh = 0.126, the basic set of spacetime plus matter sector variables, ns = 0.97, and τr = 0.11. we adopt a simplistic grid-based search for viable mod- els. The acceptance criteria for the Q1 models is based Our search of the parameter space for the remaining on a ∆χ2-test. The results of our survey of Q1 models models is based on a Bayesian approach, using a Monte- are shown in Figures 2, 3. We have exploited the de- Carlo Markov-Chain (MCMC) search algorithm to iden- generacy of the CMB anisotropy pattern among models tify the best cosmological models. The end product is a with the same apparent angular size of the last scat- realization of the posterior probability distribution func- tering horizon [26]. Hence, there is a family of models tion on the parameter space [27]. Our approach is similar 3

(a) Q2: monotonic evolution w(a) (b) Q3: leaping kinetic quintessence (c) Q4: inverse-power law

(d) Q2: monotonic evolution w(a) (e) Q3: leaping kinetic quintessence (f) Q4: inverse-power law

FIG. 4: The results of our MCMC search of the multi-dimensional parameter search, for models Q2-4, are illustrated in the figures above. In all cases, we have marginalized over the suppressed parameters. The solid lines indicate the 1, 2, 3σ contours based on comparison with the CMB (WMAP, ACBAR, CBI) and type 1a supernovae (Hi-Z, SCP). to the procedure described in [7], whereby the MCMC if we were to simply discard any steps which cross the makes a “smart” walk through the parameter space, ac- boundary. That is, the likelihood in the parameter region cepting or rejecting sampled points based on a running close to the boundary would be artificially suppressed. criteria. For each of Q2-4, after some experimentation we If such a (virtual) jump in a parameter direction i oc- found it practical to use four independent chains in or- curs, we use the following strategy. We try making a der to monitor convergence and mixing according to the gaussian-weighted step but only towards the boundary. criteria of Ref. [28]. Each such chain explored ∼ 3 × 104 If this fails after ten attempts, then we start over again models. We found it advantageous to use a dynamical by making a gaussian step in any direction. By trying stepsize factor r ∈ [0.1, 10]. In each step, all parame- ten times, however, it is very likely that we do make the ters are varied according to a Gaussian with width rσi, step successfully, if we are at least one σi away from the where σi is an initial stepsize for the random walk in the boundary. If, however, we are very close to the boundary direction of the i-th parameter. The stepsize factor r is already and our step would have carried us deep into the adjusted such that the chain neither rejects nor accepts unphysical region, then it is sensible to make the step ls too frequently. Certain parameters, such as w0 and ΩQ in away from the current point, for the chain didn’t intend the early quintessence models, have physical boundaries to stay close to the boundary anyhow. To give us confi- and it is therefore important that our MCMC does not dence in the reliability of this algorithm, we have checked lead to a bias in the sampled distribution close to the by synthetic distributions that this strategy is very well boundaries. As the likelihood in the MCMC approach suited for sampling the likelihood, even in cases where a is given by the frequency with which the chain passes physical boundary cuts off the parameter space. through a given region of parameter space, we would be in danger of depleting the region close to the boundary Q2: We have examined quintessence models with an 4 equation-of-state that evolves monotonically with the more general parameterization, allowing for independent scale factor, as w(a)= w0 + (1 − a)w1. For this case, the w0 and steepness of transition can be found in [32].) Since ls parameters are simply θQ = {w0, w1}. This parametriza- ΩQ is not tied as closely to the expansion rate sampled by tion has been shown to be versatile in describing the late- the supernovae, compared to case Q2, the result is the ls time quintessence evolution for a wide class of scalar field weaker constraint ΩQ . 0.1, as shown in Figure 4(b). models [29]. Based on the degeneracy of models found for Although the limit of a cosmological constant can be ap- Q1, we expect to find a two-dimensional family of equiv- proached in this model, the presence of early quintessence alent best-fit models with the same apparent angular size will then require a sharp transition in the equation-of- of the last scattering horizon, occupying a plane in the state in order to reach w → −1. In addition to the fact {w0, w1, h} space. There are three ways in which this that such models lose the early tracking behavior and plane is pared down: Firstly we confine w ≥ −1 at all instead require some degree of fine-tuning, there is the times. Secondly, the transition from w0 to w0 + w1 takes practical consideration that the sharp transition leads to place at low redshift, z . 1, so that high redshift su- some numerical instability in our code. To avoid this pernovae restrict w0, w1 for these models. Thirdly, this problem, we restrict w > −0.97 as can be seen in Fig- parametrization allows for models in which the dark en- ure 4(b). We believe that we have succeeded at imple- ergy is non-negligible at early times, which influences the menting an MCMC algorithm that does not artificially small-scale fluctuation spectrum. The first two consider- distort the probability distribution near the parameter- ations yield w0 < −0.75 at the 2σ level, marginalizing space boundaries, based on our experimentation with over the suppressed five dimensional parameter space, as synthetic distributions with boundaries. Next, because illustrated in Figure 4(a). There, the shapes of the con- early quintessence suppresses the growth of fluctuations tours indicate that current data can only distinguish be- on small scales compared to large scales, we find that tween fast (dw/da & 0.5) and slow evolution of w(a), and comparable fluctuation spectra can be achieved by mak- ls ls offer only a weak bound on w1. However, in terms of ΩQ, ing a trade-off between ns and ΩQ. As shown in Fig- our third consideration gives a tight upper bound on the ure 4(e), slight suppression of small-scale power can be quintessence density during recombination. As shown accomplished either by a tilt towards the red, ns < 1, ls ls in Figure 4(d), ΩQ < 0.03 at the 2σ level. Three tar- or a rise in ΩQ. Since the effect of early quintessence on get models, with significantly different equation-of-state the small-scale fluctuation power spectrum closely mim- evolution dw/da, are given in Table II for future investi- ics a running spectral index, we have not introduced gations. dns/d ln k as an additional parameter, which would be ls highly degenerate with ΩQ [31]. We expect that im- ls proved measurements of the second and third acoustic model w0 w1 h ΩQ σ8 ls −6 peaks will tighten the constraint on Ω and sharpen the Q2.1 -0.93 0.43 0.66 4 × 10 0.77 Q −4 ls Q2.2 -0.99 0.68 0.64 1 × 10 0.78 degeneracy in the ns − ΩQ plane. Target models, with − Q2.3 -0.92 0.62 0.62 1 × 10 4 0.73 significantly different values of early quintessence abun- dance, are given in Table III.

TABLE II: Sample best-fit models with a monotonically- ls evolving equation-of-state (Q2). Although a range of param- model w0 ΩQ h A w¯ls σ8 eters give equivalently good fits to the observational data, we Q3.1 -0.94 0.006 0.69 0.0026 -0.27 0.89 2 2 have selected this sample with {Ωbh , Ωcdmh , ns, τr} = Q3.2 -0.91 0.024 0.70 -0.0028 -0.21 0.81 {0.023, 0.11, 0.98, 0.16} (Q2.1), {0.023, 0.11, 0.98, 0.15} Q3.3 -0.93 0.043 0.71 -0.0070 -0.19 0.85 ls (Q2.2), {0.023, 0.12, 0.98, 0.08} (Q2.3), Entries for ΩQ and σ8 are the resulting values based on the other parameters. TABLE III: Sample best-fit leaping-kinetic quintessence models (Q3). The parameters A is an equivalent descrip- ls Q3: We have examined models of leaping kinetic tion of ΩQ, which along with the averaged w at last scat- w quintessence, a scalar field evolving under an exponen- tering, ¯ls, can be used more easily with equations 2-4 of Ref. [31] to generate the time-evolution of these models. The tial potential with a non-canonical kinetic term that un- 2 2 non-quintessence parameters are {Ωbh , Ωcdmh , ns, τr} = dergoes a sharp transition at late times, leading to the {0.022, 0.112, 0.96, 0.12} (Q3.1), {0.023, 0.116, 1.0, 0.16} current accelerated expansion [30]. At early times the (Q3.2), {0.024, 0.119, 1.04, 0.26} (Q3.3). field closely tracks the cosmological background with w = 0 during matter domination, appearing as early quintessence [31] before undergoing a steep transition Q4: Finally, we have examined tracker models of towards a strongly negative equation-of-state by the quintessence. Inverse-power law (IPL) models are the present day. The steepness of the transition in w for archetype quintessence models with tracking property a leaping kinetic model is directly connected to the and acceleration [8, 33, 34]. The potential is given by equation-of-state w0 today. Such models can therefore be V ∝ ϕ−α, where the constant of proportionality is deter- ls 0 characterized by the parameters θQ = {ΩQ, w }, where mined by ΩQ. In certain supersymmetric QCD realiza- ls ΩQ is the quintessence density during recombination. (A tions of the IPL [35], α is related to the numbers of color 5 and flavors, and can take on a continuous range of values the tightest constraint methods, using the CMB and α > 0. For α → 0, however, inverse-power law mod- SNe. Our study advances beyond past investigations els behave more and more like a cosmological constant. [37, 39, 40, 41, 42, 43, 44, 45] by treating a wide class Using earlier data, α has been constrained to α < 1.4 of quintessence models with the powerful weight of the [36, 37], although h =0.65 has been fixed in those anal- WMAP data. We have considered four classes of models yses. Keeping h free, a more conservative value of α< 2 which cover the most basic quintessence scenarios, in- [38] was inferred. From our analysis, we see that the 2σ cluding a versatile parametrization, as well as the best bound has not changed dramatically. The 2σ-bound with motivated and most realistic scenarios based on our cur- α . 1 − 2 is consistent with values of h within the range rent understanding of particle physics. Absent from our determined by the HST, as seen in Figure 4(c). In Fig- survey are k essence models, and dark energy models with 2 ure 4(f) we plot the likelihood contours in the Ωmh − α a coupling to other matter fields or gravity. In the former 2 plane: our results agree with the best fit at Ωmh =0.149 case, since the sound speed of fluctuations varies with for α = 0 or w = −1, but show a tolerance for a wider time in these models, however, we can make a simple range for 0 ≤ α ≤ 2. That is, the additional degree of distinction with quintessence models with an underlying freedom in α means that the matter density for the IPL scalar field, wherein the propagation speed is equal to the model is not as well-determined from the peak position speed of light. (See Refs. [46, 47] for analysis of these [5] as compared to the Λ model. However, to maintain models with respect to CMB anisotropy.) For the latter 2 the peak at ℓ = 220, we observe that Ωmh decreases case we refer to Ref. [48] for specific coupled models. We slightly as α increases. We might have inferred the re- also note that the mass fluctuation power spectrum is an sults for the IPL based on the constant equation-of-state important cosmological constraint which we have omit- models: pairs of {α, h} can equivalently determine a ted at this stage, primarily because it constrains energy family of models with degenerate CMB anisotropy pat- density rather than pressure (although there are excep- terns, since the differences in the late-ISW for this model tions), a chief feature distinguishing dark energy from compared to Q1 make only a small contribution to the dark matter. (The constraint curves obtained in Ref. overall χ2. Furthermore, since IPL quintessence can be [49], e.g. Figure 3, are consistent with, but do not de- modeled by the appropriate choice of the Q2 parameters, cisively pare down the parameter regions of our current then improved sensitivity to dw/da is required to tighten results.) Furthermore, the analysis of mass power spec- the constraints here. While α is tightly constrained, our trum observations will have to take into account possible take-away is that IPL models with 0.25 . Ωm . 0.4 influences of a time-dependent w and early quintessence, remain viable. Target models are given in Table IV. which we put off for later investigation. Overall, we have simulated in detail more than 400,000 individual cosmo- model α h σ8 logical models. The stored spectra and parameter-space Q4.1 0.1 0.68 0.90 likelihood functions will be used to evaluate additional Q4.2 0.2 0.68 0.85 constraints that can offer clues to the behavior of the Q4.3 0.8 0.68 0.82 dark energy. The set of thirteen target models, listed in Tables I-IV, which includes a revised concordance Λ model [50], should be of use by researchers to further test TABLE IV: Sample best-fit IPL quintessence mod- the quintessence hypothesis. els (Q4). The non-quintessence parameters are 2 2 {Ωbh , Ωcdmh , ns, τr} = {0.023, 0.122, 0.97, 0.13} (Q4.1), {0.023, 0.116, 0.97, 0.14} (Q4.2), {0.024, 0.102, 1.0, 0.23} Acknowledgments This work was supported by NSF (Q4.3). grant PHY-0099543 at Dartmouth. We thank Pier Ste- fano Corasaniti for useful conversations, and Dartmouth This work provides a capsule summary of the vi- colleagues Barrett Rogers and Brian Chaboyer for use of able quintessence dark energy models, based on two of computing resources.

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