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Exploring parameter constraints on quintessential dark energy: The Albrecht-Skordis model Michael Barnard, Augusta Abrahamse, Andreas Albrecht, Brandon Bozek, and Mark Yashar Department of Physics, One Shields Avenue, University of California, Davis, California 95616, USA (Received 11 January 2008; published 5 May 2008) We consider the effect offuture dark energy experiments on ‘‘Albrecht-Skordis’’ (AS) models of scalar field dark energy using the Monte Carlo Markov chain method. We deal with the issues of parametrization of these models, and have included spatial curvature as a parameter, finding it to be important. We use the Dark Energy Task Force (DETF) simulated data to represent future experiments and report our results in the form of likelihood contours in the chosen parameter space. Simulated data is produced for cases where the background cosmology has a cosmological constant, as well as cases where the dark energy is provided by the AS model. The latter helps us demonstrate the power of DETF Stage 4 data in the context of this specific model. Though the AS model can produce equations of state functions very different from what is possible with the w 0 w a parametrization used by the DETF, our results are consistent with those reported by the DETF. DOI: 10.1103/PhysRevD.77.103502 PACS numbers: 95.36.+x, 98.80.Es I. INTRODUCTION Current astronomical observations are best fit by cos- mologies containing a significant density of energy with negative pressure. This has been dubbed dark energy, and its exact nature remains a mystery. Though the properties of the dark energy are poorly constrained by current data, a number of proposed experiments will have the reach to probe them. Most assessments of proposed experiments use abstract parameters such as the ‘‘ w 0 w a ’’ parame- trization of the dark energy equation of state used by the Dark Energy Task Force (DETF) [1], or generalized forms of scalar field potentials [24]. In order to fully evaluate future experiments it is useful to understand the impact and discriminating power they will have on specific proposed models of dark energy. In this paper, we will focus on one scalar field (or ‘‘quintessence’’) model of dark energy, the so-called ‘‘Albrecht-Skordis’’ (AS) model [5,6]. The AS model is interesting for a number of reasons. Fields of the form needed for the AS model can arise as radius moduli of curled extra dimensions [7] and, while these moduli may be treated as scalar fields in gravitational calculations, they are not subject to the quantum correc- tions that can create serious problems for most quintes- sence models [8,9]. The phenomenology of these models is also interesting because, unlike the majority of dark energy models, the AS quintessence field can contribute a signifi- cant fraction of the total cosmic energy density throughout the history of the universe. Central to this work are the projected data sets (or ‘‘data models’’) created by the DETF [1]. We use the supernova, weak lensing, baryon oscillation, and Planck data sets (though not the cluster data sets, for technical reasons similar to those outlined in [10]), to show how these data sets would impact the range of possible parameters for quintessence models. To this end, we use Monte Carlo- Markov chains (MCMC) on these data sets to analyze dark energy and cosmological parameters, rather than the fisher matrix methods used by the DETF. In addition to using ‘‘standard’’ simulated data based on a universe with a cosmological constant, we create projected data sets based on an AS model universe, and run the MCMC around these to highlight the discriminating power of the projected data sets. This paper is one in a series of papers that consider different scalar field models using similar methods [11]. (Technical information about our data models and MCMC methods will be presented in an appendix of another paper in the series [11]). We show that the stage-by-stage improvement in pa- rameter constraints for the Albrecht-Skordis model is simi- lar to the relative improvement in the w 0 w a constraints seen by the DETF. We do the same for data sets generated around a specific AS model which shows small deviations from wðaÞ¼1 in the present epoch. This AS model is chosen to illustrate a case where if the real universe is described by an AS model, pure cosmological constant dark energy can be ruled out by a large margin using good Stage 4 data. We also note some features in the AS parameter contours when fitting to cosmological constant- based data that reflect the fact that the AS models we consider can only ever duplicate a cosmological constant in an approximate manner. II.PARAMETER SPACE OF THE ALBRECHT- SKORDIS MODEL Scalar field models of dark energy, or quintessence, are considered in the framework of an Friedmann-Robertson- Walker (FRW) cosmology, with an equation of motion of 0 þ 2 _ a a _ 0 þ a 2 @V @0 ¼ 0; (1) where a is the scale factor, 0 is the homogeneous scalar field, and V its potential. The background is assumed to be PHYSICAL REVIEW D 77, 103502 (2008) 1550-7998= 2008=77(10)=103502(8) 103502-1 Ó 2008 The American Physical Society

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Page 1: Exploring parameter constraints on quintessential dark energy: The Albrecht-Skordis model

Exploring parameter constraints on quintessential dark energy: The Albrecht-Skordis model

Michael Barnard, Augusta Abrahamse, Andreas Albrecht, Brandon Bozek, and Mark Yashar

Department of Physics, One Shields Avenue, University of California, Davis, California 95616, USA(Received 11 January 2008; published 5 May 2008)

We consider the effect of future dark energy experiments on ‘‘Albrecht-Skordis’’ (AS) models of scalar

field dark energy using the Monte Carlo Markov chain method. We deal with the issues of parametrization

of these models, and have included spatial curvature as a parameter, finding it to be important. We use the

Dark Energy Task Force (DETF) simulated data to represent future experiments and report our results in

the form of likelihood contours in the chosen parameter space. Simulated data is produced for cases where

the background cosmology has a cosmological constant, as well as cases where the dark energy is

provided by the AS model. The latter helps us demonstrate the power of DETF Stage 4 data in the context

of this specific model. Though the AS model can produce equations of state functions very different from

what is possible with the w0 � wa parametrization used by the DETF, our results are consistent with those

reported by the DETF.

DOI: 10.1103/PhysRevD.77.103502 PACS numbers: 95.36.+x, 98.80.Es

I. INTRODUCTION

Current astronomical observations are best fit by cos-mologies containing a significant density of energy withnegative pressure. This has been dubbed dark energy, andits exact nature remains a mystery. Though the propertiesof the dark energy are poorly constrained by current data, anumber of proposed experiments will have the reach toprobe them. Most assessments of proposed experimentsuse abstract parameters such as the ‘‘ w0 � wa’’ parame-trization of the dark energy equation of state used by theDark Energy Task Force (DETF) [1], or generalized formsof scalar field potentials [2–4]. In order to fully evaluatefuture experiments it is useful to understand the impact anddiscriminating power they will have on specific proposedmodels of dark energy. In this paper, we will focus on onescalar field (or ‘‘quintessence’’) model of dark energy, theso-called ‘‘Albrecht-Skordis’’ (AS) model [5,6].

The AS model is interesting for a number of reasons.Fields of the form needed for the AS model can arise asradius moduli of curled extra dimensions [7] and, whilethese moduli may be treated as scalar fields in gravitationalcalculations, they are not subject to the quantum correc-tions that can create serious problems for most quintes-sence models [8,9]. The phenomenology of these models isalso interesting because, unlike the majority of dark energymodels, the AS quintessence field can contribute a signifi-cant fraction of the total cosmic energy density throughoutthe history of the universe.

Central to this work are the projected data sets (or ‘‘datamodels’’) created by the DETF [1]. We use the supernova,weak lensing, baryon oscillation, and Planck data sets(though not the cluster data sets, for technical reasonssimilar to those outlined in [10]), to show how these datasets would impact the range of possible parameters forquintessence models. To this end, we use Monte Carlo-Markov chains (MCMC) on these data sets to analyze dark

energy and cosmological parameters, rather than the fishermatrix methods used by the DETF. In addition to using‘‘standard’’ simulated data based on a universe with acosmological constant, we create projected data sets basedon an AS model universe, and run the MCMC around theseto highlight the discriminating power of the projected datasets. This paper is one in a series of papers that considerdifferent scalar field models using similar methods [11].(Technical information about our data models and MCMCmethods will be presented in an appendix of another paperin the series [11]).We show that the stage-by-stage improvement in pa-

rameter constraints for the Albrecht-Skordis model is simi-lar to the relative improvement in the w0 � wa constraintsseen by the DETF. We do the same for data sets generatedaround a specific AS model which shows small deviationsfrom wðaÞ ¼ �1 in the present epoch. This AS model ischosen to illustrate a case where if the real universe isdescribed by an AS model, pure cosmological constantdark energy can be ruled out by a large margin usinggood Stage 4 data. We also note some features in the ASparameter contours when fitting to cosmological constant-based data that reflect the fact that the AS models weconsider can only ever duplicate a cosmological constantin an approximate manner.

II. PARAMETER SPACE OF THE ALBRECHT-SKORDIS MODEL

Scalar field models of dark energy, or quintessence, areconsidered in the framework of an Friedmann-Robertson-Walker (FRW) cosmology, with an equation of motion of

€�þ 2_a

a_�þ a2

@V

@�¼ 0; (1)

where a is the scale factor, � is the homogeneous scalarfield, and V its potential. The background is assumed to be

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the homogeneous Friedmann equation

H2 ¼ 8�G

3ð�r þ �m þ ��Þ � k

a2: (2)

The equation of state parameter of the scalar field is

w ¼ p�

��

; (3)

where p� and �� are the pressure and density of the scalar

field. Nonrelativistic matter has w ¼ 0, radiation has w ¼1=3, and a cosmological constant has w ¼ �1; a homoge-neous scalar field can have any behavior in this range,depending on the actual time evolution �ðtÞ. TheAlbrecht-Skordis model [5] postulates a scalar field withan ‘‘exponential-with-prefactor’’ potential,

Vð�Þ ¼ V0½ð�� BÞ2 þ A�e���: (4)

One of the attractions of this model is that realistic cos-mologies can be achieved for cases where the parametersin this potential are all roughly of order unity in Planckunits. Specifically, with such parameter choices the poten-tial can have a local minimumwith a height consistent withthe dark energy density observed today. It is assumed thatthe field starts high up on the potential. The field will thenrapidly approach an attractor solution where the fieldmimics the equation of state of the dominant energy. Itcontinues this tracking behavior until the field reaches the(approximately) quadratic shaped local minimum, atwhich point it begins a damped oscillation around thatminimum, giving its equation of state a wavelike variationas the field becomes the dominant energy. Figure 1 showswðaÞ throughout the history of the universe for a typical ASmodel.

The potential of the AS model has in principle 4 degreesof freedom important to cosmology, most of which arecomplicated combinations of the various parameters.Figure 2 illustrates some of the parameter dependenciesof the potential. Perhaps the most important degree offreedom, because it determines the scalar field fraction ofthe total energy density during the tracking behavior, is the

logarithmic slope, ðlnVÞ0 or V0V ; this is almost entirely

controlled by �. The height of the local minimum setsthe dark energy density today. The curvature of the poten-tial at its minimum is related to the height of the localmaximum. The height of the local maximum determineswhether the field stops in the minimum or rolls off toinfinity, and the curvature at the minimum sets the fre-quency of the late time oscillations. We have found that theoscillation frequency is not measurable by any of thesimulated data sets we considered. Also, the parameterspace for models that roll off to infinity but still giverealistic cosmic acceleration is only an exponentially thinand extremely finely tuned region [12]. We choose toignore this exotic behavior and focus completely on pa-

rameters for which the field (classically) stays in the localminimum.It will be helpful to reexpress Eq. (4) in the following

parameters:

−12 −10 −8 −6 −4 −2 0−1

−0.5

0

0.5

1

log10(a)

w(a

)

FIG. 1. The equation of state w for the scalar field of a typicalAS model. Note that here the a scale is logarithmic in order toshow behavior on all time scales. When the background energydensity drops to a level approaching the initial energy of the fieldduring radiation domination, the energy density goes through atransient before mimicking the equation of state of radiation(w ¼ 1=3). At matter domination, the field again goes through atransient, but the field approaches the local minimum in thepotential and becomes the dominant form of energy (with w !�1) before it can stabilize with a matterlike equation of state(w ¼ 0).

34 34.50

0.5

1

x 10−120

V(φ

)

λ=8

λ=8.01

34 34.50

0.5

1

x 10−120

φ

V0=1

V0=1.5

34 34.50

0.5

1

x 10−120

φ

V(φ

)

A=0.01

A=0.005

34 34.50

0.5

1

x 10−120

B=34

B=34.1

FIG. 2. The effects of the original parameters (from Eq. (4)) onthe potential minimum. All measures are in Planck units. Notethat very small changes in � or B cause substantial changes inthe height of the minimum.

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Vð�Þ ¼ V0½�ð�� �Þ2 þ ��e���: (5)

Because the number of parameters is greater than thenumber of parameters measured by our simulated datathe AS model has a number of degenerate directions.Along these degenerate directions, the parameters canchange without significantly affecting the observables.The most apparent of these directions are the ways thatthe height of the local minimum of the potential can bechanged. This height is exponentially sensitive to the prod-uct of the exponential factor �, and to �, which primarilycontrols the location of the minimum. This degeneracy canbe stabilized by fixing the product of � and �, so that � ¼272� . There is also a dependence of the minimum on � and�2

� . To work around these difficulties, we fix �, and vary �

as a function of � so that the overall constant V0 is the onlyparameter that adjusts the height of the minimum. Thisnecessarily excludes the portion of the parameter spacewhere there is no minimum (with essentially no loss ofgenerality, as discussed above). Also, these parameterconstraints do not allow the oscillation frequency nearthe local minimum to vary independently of the otherparameters. This feature is required to successfully runthe MCMC calculations since none of the data sets aresensitive to the oscillation frequency. Figure 3 shows how� affects the new parametrization.

Aside from degeneracy in the specifics of the potential,there are broad patches of parameter space that are, formost data sets, not detectably different from a cosmologi-cal constant. This happens when the exponential slope issteep, and the field slides very early down to the localminimum and oscillates in a shallow and rapid manner,

resulting in an equation of state very nearly�1 at the timesof interest.The starting value of the scalar field is mostly unimpor-

tant in the AS model, but it should be noted that there arevalues of � and the initial � that can cause problems. Forlarge values of �, if the scalar field starts at a point higheron its potential than the radiation density during nucleo-synthesis, then the scalar field will form a significantfraction of the energy density at that epoch. Such regionsof parameter space would be ruled out by observations ofprimordial element abundances, though our algorithm doesnot contain such considerations.Other related problems with the algorithm specific to the

ASmodel include the last scattering surface. The algorithmdoes not calculate the size of perturbations at the time oflast scattering, but takes them as an input, so the effect ofthe scalar field on that size is not taken into account by thealgorithm even when the scalar field energy density issignificant. With respect to the growth of linear densityperturbations, the algorithm does not use the scalar field inits calculation until a redshift of ten. The effect of theseissues is that some constraining power has been ignored.Albrecht and Skordis investigated this issue [6] to somedegree, but while we do not believe this affects the con-clusions of this paper, this matter remains interesting andworthy of further exploration.In our parametrization, the amplitude of the oscillations

in the equation of state at late times is primarily determinedby the value of �. For small values of � the oscillationamplitudes are very small, but become large for values of�approaching 100. This somewhat unintuitive behavior canbe explained by noting that, when the dominant form ofenergy is scaling as a�n, ��=�c ¼ n

�2 (where �c is the

critical density) [13]. Thus, the smaller � is (and the larger� is), the further the field energy is below the matterdensity during matter domination, and the more Hubblefriction will slow the field. The closer the scalar fieldenergy density is to the matter density during matter domi-nation, the more kinetic energy remains in the field when itbecomes dominant, and thus is free to oscillate. Figure 4shows how the equation of state depends on � in thisparametrization.

III. CONSTRAINING THE AS MODEL AROUNDCOSMOLOGICAL CONSTANT MODEL DATA

SETS

Having chosen to explore the scalar field parameterspace in terms of the parameters �, V0, and the initialvalue of the scalar field (with the other parameters con-strained), we then performed an MCMC analysis on simu-lated data sets generated around a cosmology with acosmological constant. The technical details of this workare presented in the appendix of [11]. These data sets areconsistent with those used by the DETF, using simulatedStage 2, Stage 3, and Stage 4 supernova, weak lensing,

34.4 34.6 34.8 350

2

4

6x 10

−120

φ

V(φ

)

β=34.5

β=34.6

FIG. 3. Under the new parametrization, V0 maintains itsprevious effects, while the new parameter � becomes theprimary parameter for the AS model. � sets the exponentialslope as well as the minimum curvature.

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baryon oscillation, and cosmic microwave background(CMB) observations. We did not use the cluster data be-cause of the difficulty of adapting the DETF constructionto a quintessence cosmology, nor have we included pos-sible improvements from cross correlations [14,15]. In theDETF language Stage 2 encompasses projects that arefully funded and underway, Stage 3 models mediumterm, medium cost proposals, and Stage 4 are the largerprojects, such as a large ground-based survey or a newspace telescope.

Though there are three parameters controlling the stateof the scalar field, the initial value of the scalar field isgenerally washed out by the tracking behavior, and thevariance in height of the minimum is mostly a reflection ofthe uncertainty in the Hubble parameter. This leaves uswith one important parameter, �, that mostly controls theunique properties of the AS model. We include a V0 axis inthe contour plots for convenience. One should take care,though, to remember that V0 is not a parameter that can besaid to have a significant effect on the equation of state ofthe dark energy.

Figure 5 shows the likelihood contours for DETF Stage2 and Stage 3 (optimistic) data. From the point of view ofthese data sets the behavior of the AS model can be thoughtof as trending toward a cosmological constant as � goestoward zero. While these plots show that the constraints onV0, which corresponds to the dark energy density now,improve greatly with stage number, the critical considera-tion here is �. The constraints on � do not appear toimprove in as dramatic a manner. This is because there isa substantial range of � where these data sets cannotdistinguish the AS model from a cosmological constant.As seen in Fig. 4, at � ¼ 50 already one must look back to

a ¼ :2 (z ¼ 4) to see a large deviation from w ¼ �1. Tosome extent this feature will affect any attempt to evaluatethe AS model using data from a universe with a cosmo-logical constant. In that low� region, the small oscillationsof the field persist at late times, and the Stage 4 data setsmay be strong enough to pick up on some effects of theseoscillations. Figure 6 shows likelihood contours for DETFStage 4 space and ground (optimistic) data. There is a kinkin the Stage 4 space contour plot which corresponds to arange of�where the residual oscillations at late times peakin amplitude. As the oscillations represent an energy den-sity above the amount corresponding to the minimum ofthe potential, V0 must be lower to accommodate regionswith larger late time amplitudes.Figure 7 shows contours in !k � � space (!k, the

‘‘curvature density’’ is a measure of the curvature of the

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

FIG. 5. Likelihood contours for the AS model in � and V0 forStage 2 (top) and Stage 3 photometric, optimistic (bottom), fordata generated from a cosmological constant model. Thoughconstants in V0 improve greatly, the AS model has littleobservable difference from a cosmological constant for valuesof � below 50.

0 0.5 1−1

−0.5

0

0.5

1

a

w(a

)

β=50

β=80

β=100

FIG. 4. This figure illustrates how the primary parameter �affects the wðaÞ behavior of the AS model. The AS model-baseddata was generated using � ¼ 80.

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universe). There are well-known degeneracies associatedwith determining curvature and the dark energy equation ofstate simultaneously [16]. Here these degeneracies showup by allowing ‘‘best fits’’ to data from a flat universe usingnonzero curvature for certain values of �, as can be seen incontours for the Stage 2 data. The Stage 4 data set shown inthe lower panel clamps down on this behaviorconsiderably.

IV. MCMC OF THE AS MODEL AROUND ASMODEL-BASED DATA SETS

As a way of further exploring the power of the advanceddata sets, a sample AS model was chosen as a new fiducial

model around which the data sets were generated. Thismodel was chosen to be marginally consistent with thecosmological constant-based Stage 2 data, but differentenough from a cosmological constant that a cosmologicalconstant would be strongly ruled out with Stage 4 data (thepoint being to illustrate the power of Stage 4). The fiducialmodel chosen here has a � ¼ 80, as depicted in Fig. 4.Because the Stage 2 data allowed larger values of � whencurvature was present, the sample AS model was chosenwith nonzero spatial curvature to make it more consistentwith a flat cosmological constant model for this value of �.The value used for curvature,!k ¼ :005, is actually a littlecloser to zero than the value that would be most consistent

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

FIG. 6. Likelihood contours for the AS model in � and V0 forStage 4 space (top) and Stage 4 ground large survey telescope(LST) (bottom), both optimistic cases, for data generated from acosmological constant model. Though constants in V0 improvegreatly, the AS model has little observable difference from acosmological constant for values of � below around 50. Thedifferent regions represent 68.27%, 95.44%, and 99.73% con-fidence regions. This convention is used for all contour plots inthis paper.

β

ωk

20 40 60 80 100 120−0.01

−0.005

0

0.005

0.01

0.015

β

ωk

20 40 60 80 100 120−0.01

−0.005

0

0.005

0.01

0.015

FIG. 7. This figure illustrates the confusion of the AS modelwith curvature. The Stage 2 set (top) allow significant departuresfrom flatness, even when using data generated by a cosmologicalconstant model with no curvature. The Stage 4 (ground, opti-mistic) data set (bottom) are effective at constraining this degen-eracy.

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with a flat, cosmological constant cosmology for � ¼ 80,but is still within the second contour at Stage 2. Theparameters for this fiducial model, along with the parame-ters for the �CDM model, appear in Table I. The results ofthe calculation, as seen in Figs. 8 and 9, show that, for auniverse described by this specific model, the Stage 4experiments will rule out a cosmological constant by sev-eral sigma. In these AS fiducial chains, the curvature

behaves as to be expected from its correlation with �;allowing the � parameter to vary downward was well asupward produces a corresponding increase in the varianceof the curvature. As there is no new information here, andfor the sake of saving space, we have not included contourplots with !k for the AS model.

V. DISCUSSION AND CONCLUSIONS

We have analyzed the impact of the DETF simulateddata sets in the context of the Albrecht-Skordis scalar fieldmodel of dark energy. We find that the effect of the DETFdata sets on the parameter space of the AS model is verysimilar to the DETF findings. There is substantial improve-ment in the constraining power of each successive stage ofexperiments. We presented likelihood contour plots in ASparameter space for a few key combinations of DETFsimulated data. We used these plots to demonstrate thebroad agreement with the DETF results and also to point

TABLE I. A table of the parameters for the �CDM fiducialmodel (left column) and the AS fiducial model (right column).

!DE 0.3796 .3796

!m 0.146 0.146

!k 0.0 0.005

!B 0.024 0.024

ns 1.0 1.0

� — 80

V0 — 0.92

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

FIG. 8. Likelihood contours for the AS model in � and V0 forStage 2 (top) and Stage 3 photometric, optimistic (bottom), fordata generated from the � ¼ 80 AS model.

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

β

V0

20 40 60 80 100 1200.6

0.8

1

1.2

FIG. 9. Likelihood contours for the AS model in � and V0 forStage 4 space (top), and Stage 4 ground LST, both optimistic(bottom) cases, for data generated from the selected AS model.

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out more subtle effects such as degeneracies between thedark energy equation of state and curvature and somepeculiarities of the way the AS model can approximate acosmological constant. In the course of this work we haveproduced similar contour plots for many more instances ofthe DETF simulated data (taking individual techniquesseparately, for example, and using both the DETF optimis-tic and pessimistic data models). We find our overall con-clusions and detailed points apply quite generally acrossthe full range of DETF simulated data. We also find that asone moves to better experiments, the improved discrimi-nating power will definitely rule out interesting portions ofthe AS model space and could completely rule out thecosmological constant if the AS model is correct.

The DETF reported a figure of merit in terms of theinverse of the area inside of likelihood contours in theirmodel space, w0 and wa. This figure of merit showed animprovement of a factor of 3 going from Stage 2 to Stage 3,and a factor of 10 going from Stage 2 to Stage 4, assuminggood Stage 3 and Stage 4 projects. In the parameter spacewe used for the AS model we have only one parameter thatstrongly affects the equation of state. The constraints onthis parameter, �, show improvement by roughly thesquare roots of these factors, appropriate for the reduceddimensionality in the AS parameter space. We also seesome subtle differences between the constraining power ofground and space Stage 4 data models in that the othervariable plotted, V0, is somewhat more strongly con-strained by the Stage 4 ground data set than for the Stage4 space. Other scalar field models [11,17] display theopposite behavior. We are currently investigating this ef-fect further, which may lead to interesting insights into thecomplementarity of ground and space-based Stage 4experiments.

It is interesting to consider the relationship between thiswork and recent work by one of us and G. Bernstein [10].There the same DETF data sets were studied in the contextof an abstract dark energy model where the equation ofstate was modeled by many more than the two parametersused by the DETF. One conclusion of [10] was that high

quality data can make good measurements of significantlymore than two equation of state parameters. We have seenin this paper that the main equation of state parameter � ofthe AS model is constrained by DETF data sets to a similardegree as the DETFwo � wa parameters, even though theydescribe very different functions wðaÞ. As discussed in[18], we believe that this ability to constrain a wide varietyof functions wðaÞ is another manifestation of the richconstraining power demonstrated in [10].A good way to think about this may be to consider the

expansion of the AS family of wðaÞ function as well as thew0 � wa family of functions in terms of the independentlymeasured orthogonal functions wiðaÞ from [10]. The factthat we have seen here (and also in [11,17]) that a variety ofdifferent wðaÞ functions can be constrained as well as theDETF w0 � wa functions appears to reflect the fact thatfundamentally many more functions wðaÞ are measuredthan are contained in any one of these families alone.An upshot of this and the companion work in [11,17] is

that modeling the impact of future experiments using theDETF parameters appears to give a good indication of theimpact on quintessence dark energy models with a similarnumber of parameters. An advantage of the methods usedhere is that one can see explicitly how future data canconstrain real dark energy models in a significant way,and can even eliminate some models entirely.

ACKNOWLEDGMENTS

We wish to acknowledge Lloyd Knox, Jason Dick, andMichael Schneider for conversations and consultation thatcontributed to this paper. We would also like to thankDavid Ring for finding an error in our code and TonyTyson and his group (especially Perry Gee and Hu Zhan)for use of their computational resources. We thank GaryBernstein for providing us with Fischer matrices suitablefor adapting the DETF weak lensing data models to ourmethods. This work was supported in part by DOE GrantNo. DE-FG03-91ER40674 and NSF Grant No. AST-0632901.

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