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Confronting quintessence models with recent high-redshift supernovae data
G. Barro CalvoDepartamento de Astrofısica, Universidad Complutense de Madrid, 28040 Madrid, Spain
A. L. MarotoDepartamento de Fısica Teorica, Universidad Complutense de Madrid, 28040 Madrid, Spain
(Received 24 April 2006; revised manuscript received 22 September 2006; published 26 October 2006)
We confront the predictions of different quintessence models with recent measurements of theluminosity distance from two sets of supernovae type Ia. In particular, we consider the 157 SNe Ia inthe Gold dataset with z < 1:7, and the more recent data containing 71 supernovae obtained by theSupernova Legacy Survey (SNLS) with z < 1. We numerically solve the evolution equations for theRatra-Peebles inverse power-law model, the double exponential and the hyperbolic cosine quintessencemodels. We obtain confidence regions from the two datasets in the M and M w planes for thedifferent models and compare their predictions with dark energy models with constant equation of state.
DOI: 10.1103/PhysRevD.74.083519 PACS numbers: 98.80.k, 98.80.Es
I. INTRODUCTION
The growing observational evidence from high-redshiftsupernovae [1,2] and other cosmological data [3] suggeststhat the dominant component of the universe today is somesort of dark energy fluid with negative pressure [4,5].Indeed a component with constant equation of state pX wXX and wX <0:78 [3] fits the existing data withreasonable goodness (XCDM model). In particular, it in-cludes the most economical explanation for the presentstate of accelerated expansion of the universe, i.e., thepresence of a pure cosmological constant with wX 1.However, there are other models in which the equation ofstate depends on redshift and which are also able to fit thedata with comparable quality. This is the case of the socalled quintessence models [6,7], in which it is the pres-ence of an evolving scalar field with appropriate potentialterm what plays the role of dark energy.
Quintessence models exhibit an interesting propertyusually called tracking behavior [8]. This means that thereis a wide range of initial conditions for which the evolutionof the scalar fields converges to a common evolutionarytrack. Furthermore in such a tracking regime the equationof state of quintessence w remains almost constant.Moreover there are particular models, usually called scal-ing models [9–11], in which w mimics the equation ofstate of the dominant component throughout the cosmo-logical evolution. These properties allow quintessencemodels to alleviate the fine tuning problem of XCDMmodels, although in any case the scale of the quintessencepotential has always to be tuned in order to reproduce thedata.
Apart from quintessence, other models have been pro-posed in the literature. Thus for instance, scalar fields withnoncanonical kinetic terms (k-essence) [12]; the general-ized Chaplygin gas model [13], which in principle allowsfor an unification of dark energy and dark matter; infraredmodifications of General Relativity, as for instance in extra
dimensional theories [14] or the modification of theFriedmann equation in the so called Cardassian models[15,16].
The present SNe Ia data are not very constraining whentrying to determine the nature of dark energy and differentdataset favor different regions of the parameter space. Thusfor instance, it is known that the Gold dataset [1] prefersmodels with wX <1 [17,18], the so called phantom darkenergy models, whereas for the more recent SupernovaLegacy Survey (SNLS) [2], the best fit is closer to a purecosmological constant [19,20]. Therefore it is worth ex-ploring what kind of constraints are imposed in each case,and this is the aim of the present work. However theinformation we will obtain will be very limited becauseeven the highest redshift points in those sets explore onlyrelatively recent epochs, and therefore the present super-novae data alone are not able to discriminate between thedifferent models (although larger future supernovae cata-logues are expected to improve in an important way thepresent constraints). Accordingly we should rely on com-plementary information in order to distinguish quintes-sence models from a cosmological constant or otheralternatives [21,22]. Thus for instance, in the particularcase of quintessence, dark energy is generated by a scalarfield, and there exists the possibility of having density orvelocity perturbations which could affect CMB anisotro-pies [23]. In constrast, the pure cosmological constant casedoes not support such perturbations.
In this work we will concentrate on different quintes-sence models and derive constraints on their parametersfrom the two mentioned Gold and SNLS datasets. Weconsider three models: an inverse power-law Ratra-Peebles (RP) model [6], V M4=, the doubleexponential (DE) potential V M4e e[24,25] which exhibits scaling behavior and the hyperboliccosine (HC) V M4cosh 1 [26], which pos-sesses an oscillatory behavior at recent epochs. Unlikeprevious works [27,28], we do not parametrize the equa-
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tion of state wz in order to obtain explicit expressionsfor the luminosity distance-redshift relation dLz. Instead,we solve numerically the evolution equation for the scalarfield and the universe scale factor. This allows us to nu-merically obtain dLz for each model. The value of thefitted cosmological parameters is known to have a strongdependence on the particular parametrization chosen forthe equation of state (see [5,29] and references therein). Inparticular, at least three parameters are needed in order totake into account properly all the information encodedin the SNIa data. Our approach does not rely on parti-cular parametrizations, but instead we consider the fullredshift dependence by numerically solving the evolutionequations.
The plan of the paper goes as follows, in Sec. II wereview the models properties and obtain the appropriateequations of motion. In Sec. III, we compute the luminos-ity distance expressions and perform the correspondingcosmological fits. Section IV contains the main results ofthe paper with the confidence regions for different spacesof parameters and finally we include a brief section withconclusions.
II. QUINTESSENCE MODELS AND EVOLUTIONEQUATIONS
A. Inverse power-law potential
In the first model that we will consider, the potentialreads:
V M4
(1)
This model has a tracker attractor, but it is not scaling,although the equation of state is determined by that of thedominant component:
w wR;M 2
2(2)
where, for >2, in the radiation or matter eras we havew < wR;M respectively, i.e. the energy density decreasesmore slowly than the dominant component and it eventu-ally becomes dominant at late times. The correspondingequations of motion reduce to the Friedmann equation forthe scale factor, together with the scalar field evolutionequation:
H2 8G
3
_2
2 V
XiM;R
i;0a3wi1
3H _ V0 0 (3)
where i;0=0 i with i M, R and 0 is the criticaldensity. Here a dot denotes derivative with respect tocosmological time.
In order to solve these equations numerically, we trans-form them into dimensionless equations, defining: H0t
and ~ 8G1=2. Now a prime will denote derivativewith respect to :
~00
3 ~0
a0
a ~V0 ~ 0
a0
a
2
~02
6 ~V ~ Ra4 Ma3
(4)
where
~V ~ A~ A
M48Gp
2
3H20
(5)
The initial conditions for ~ and a will be given by: ~0 5 101, ~00 0 and a0 103. We will explore theparameters range 0–5:5 and A 0:3–8 104. Noticethat for those values of the parameters, the initial scalarfield energy density is a small fraction of M and R.Notice also that thanks to the attractor behavior, the latetime evolution is not very sensitive to the particular initialconditions chosen.
B. Double exponential potential
In this case:
V M4e e (6)
where 8Gp
. The energy density in this model fol-lows a scaling behavior at early time, whereas it becomesdominant at late times with appropriate equation of state:w ’ 1. The corresponding dimensionless potential nowreads:
~V ~ Ae ~ e ~ A M48G
3H20
(7)
and the corresponding equations of motion are given in (4).The initial conditions are ~0 0:5, ~00 0 anda0 103. The range of parameters considered are 20, 0:1–1:5 and A 0:4–120, where for simplicitywe have fixed the constant . The and constantsessentially fix the value of w today, and we have checkedthat fixing still allows us to cover a wide range of valuesinw, just varying . Again for those particular values, theinitial energy density in the scalar field is a small fractionof the radiation and matter densities.
C. Hyperbolic cosine potential
In this case the dimensionless potential reads:
~V ~ Acosh ~ 1 A M48G
3H20
(8)
In the limit ~ 1, i.e. at early times, the potentialbehaves as a single exponential and it is possible to choosethe parameters so that the model possesses a scaling be-havior. In the opposite limit, ~ 1 (late times), the
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potential can be approximated by ~V ~ / ~2, and thescalar field oscillates around ~ 0. When oscillationsstart, the equation of state no longer mimics the dominantcomponent, but it also start oscillating with average valuegiven by:
hwi 1
1(9)
Accordingly, the value of determines the present equa-tion of state. For 1 the oscillations behaves as non-relativistic matter, whereas if we require acceleratedexpansion today, i.e. w <1=3 then we should have< 1=2.
The numerical integration of Eqs. (4) now has an addi-tional difficulty. In the interesting case (< 1=2), thepotential derivative ~V0 appearing in the first equation di-verges at the origin. Since both ~ and ~0 are continuous atthe origin, in order to avoid numerical instabilities we havesmoothed the divergence modifiying the potential asV ~ Acosh ~ 1 . We have checked thatfor sufficiently small values, the results do not depend onthe parameter.
The initial conditions considered are ~0 1:6,~00 0, a0 103. The free parameters are 0:02–0:4, A 0:3–14 and we have fixed 5, whichensures again that initially quintessence is not the domi-nant component of the energy density and that it follows ascaling behavior.
III. COSMOLOGICAL DATA FIT
After numerically solving (4), we obtain a discrete timeevolution for the scale factor at and the scalar field tfor a given set of parameters A;. The time dependencewill be used to compute the theoretical values of theluminosity distance to each SNe Ia with a given redshiftz. Thus we use the well known expression, derived fromthe FRW metric under flat prior, expressed in terms of thedimensionless time variable .
dL a0
a1
Z 1
0
da
(10)
The observations of supernovae measure essentially thedistance modulus , which is the diference between theapparent magnitude m and the absolute magnitude M, andrelates to the luminosity distance as:
th mM 5 logdL 5 logcH1
0
Mpc
25
5 logdL ~M (11)
The M value can be assumed constant once the necessarycorrections are applied on mz.
Assuming 0 to be the value for which a0 1, thescale factor depends on redshift as a z 11. As aconsequence, for each supernovae in a given set, we derivea1 from its redshift and obtain 1 from the numericaloutput data of (4). Hence the integral (10) can be numeri-cally evaluated to obtainth for each A;, once the valueof H0 is fixed.
Once we obtain th, the comparison to its observationalvalue will enable us to carry out a 2 statistical analysis.For this purpose, we have considered two sets of super-novae. On one hand, the Gold set, compiled by Riess et al.[1], containing 143 points from previously published data,plus 14 points with z > 1 discovered with the HST, allreduced under the same criteria in order to improve theerrors arising from systematics. On the other, the SNLS set,comprising 71 distant supernovae (0:15< z< 1) discov-ered during the first year of the Supernova Legacy Project(SNLS) [2], alongside with 44 SNe Ia from other sourcesthat feeds the nearby zone (0:0015< z< 0:125), andwhich are also included in the Gold set.
The process followed to obtain 2 is slightly differentfor each set. In the Gold set we have used the observationaldistance modulus exp given by Riess et al. together withits associated error to compute 2 as a function of thefree parameters of the quintessence model.
2A;; ~M XNi1
obs thA;; ~M2
2i
(12)
The dependence on H0 has been accounted for in (12)through the nuissance parameter ~M, which is independentof the data points and the data set. We have marginalized2 over all values of ~M by expanding and minimizing (12)with respect to ~M (see [16,17,19]).
On the other hand, for the SNLS data, a more detailedrelation between and the observational measurementshas to be considered in the statistical analysis, whichimplies recursively fitting two new noncosmological pa-rameters in the calculation of 2.
The expression for the observational distance modulusused by the SNLS team includes these two parameters tomeasure the impact of the rest frame color parameter (c),and the light curve stretch (s), on the distance modulus.
SNLSobs mz M a1s 1 a2c (13)
Introducing this equation into the expression for 2
2;A; a1; a2;M ~M
XNi1
SNLSobs a1; a2;M thA;;H0; ~M2
2i
(14)
we obtain a six parameter dependence (five parameters inpractice, since we can replace M ~M by a single additiveconstant) We minimize following the process suggested in[2], i.e. marginalizing with respect to a1, a2 andM ~M foreach pair of A; values. Notice that marginalizing over
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M or M ~M is equivalent to marginalize with respect toH0. Again and are obtained from the data in [2].
IV. RESULTS AND MODEL COMPARISON
The best resulting cosmologies obtained after fittingeach model to the Gold and SNLS sets are summarizedin Table I and II respectively. In each sample, we find thesame 2 for the best fit with the three quintessence modelsconsidered in the paper, i.e. 2
min 177:0 for the 157 SNeIa of the Gold set, and 2
min 111:0 for the 115 SNe Ia ofthe SNLS set. In addition, the minimun 2 for all modelscorresponds to 0. For this value the RP and HCpotentials turn into a cosmological constant term and wehave used this fact to check our results, finding goodagreement for the M value when compared to the Riesset al., and Astier et al. XCDM model. On the other hand,albeit the DE potential does not strictly behave as a term,due to the nonvanishing exponential, the ! value tendsasymptotically to1, leading to the same result as DE andHC. This implies that a quintessence potential, and a pureCDM model, can not be distinguished only by fittingM;!, since a quintessence model can be tuned toaccurately resemble a cosmological constant, at least inthe redshift interval explored by the actual SNe Ia sets. Inany case, notice that evolves in time for any nonzerovalue of for the three potentials.
Combining the data obtained by solving (4) for eachpotential, we may relate A;, and its correspondingfitting 2, to a single !;M pair which allows us to
plot the confidence regions for the two-parameter combi-nations ;M and !;M.
For the HC model, we use directly h!i 1p1p , instead
of the value derived from Eq. (4), due to the difficultiesarising from the calculation of h!i
num in the cases whenthe oscillations fails to complete a whole period. Moreover,a comparison between h!i and h!i
num shows a smallaverage discrepancy, ! 103, which justifies ourapproximation, and serves as a check of the numericalsolution.
Figures 1–3 show the confidence regions for the modelsin the M and ! M planes, together with theXCDM contour plots. The XCDM regions have been cal-culated using the same code on the Friedmann equationwith a dark energy term X;!X instead of a coupledscalar field. In the M plots the contours grow asymp-totically for decreasing M [30]. Nevertheless, as !
follows the same growing trend, it is possible to restrictthe maximum value of !.
At the 95% confidence level and for M > 0:1, we find!gold <0:56, !SNLS
<0:58, for the RP model, and
!gold <0:55, !SNLS
<0:56, for the DE model. TheHC potential also shows similar results for M > 0:15,limiting !gold
<0:58, and !SNLS <0:64. Notice that
we do not extract any result for values below M 0:1 forRP and DE or M 0:15 for HC, since the computationaleffort increases in those regions. In fact in the HC modelthat region is also highly sensitive to the value of thesmoothing factor, .
By comparing the plots for the two datasets, we see thatthe width of the contours is essentially the same. However,the SNLS 68% C.L. contour never closes for M > 0:06,whereas the Gold set constraints the cosmological parame-ters, for all models, within 0:25<M < 0:36 and 1<! <0:75, at this significance level. For the RP poten-tial, these results can be compared to analysis done withprevious supernovae data in [21,30].
Concerning the XCDM contours, one remarkable fea-ture is that as ! 1, the SNLS confidence regions forquintessence and XCDM tends to overlap. This is an ex-pected fact, since the best fitting value for XCDM accord-ing to Astier et al., is ! 1:02, quite close to thequintessence best fit, which is precisely the degeneratecase 0. On the other hand, the best fitting value forthe Gold set, assuming no priors, is ! 2:3. Thisimplies an important difference in 2 with respect to thequintessence models, which prevents the confidence re-gions from matching. In spite of this, it is noticeable thesimilarity between contours given the fact that the 68%C.L. region of XCDM is out of the plotted interval. Finally,for all models, the quintessence 99% contours place anouter boundary containing all the XCDM regions, imply-ing that any of the XCDM cosmologies, might be obtainedfrom a quintessence potential.
TABLE II. Best fitting cosmological parameters, and upperbounds at 95% C.L. for the SNLS set. The constraints on thecosmological paramaters hold for M > 0:06 for the RP and DEpotentials, and M > 0:12 for the HC.
SNLSM ! 2
RP <0:36 < 0:58 >0 111.0HC <0:36 < 0:61 <0:3 111.0DE <0:36 < 0:54 <1:27 111.0
TABLE I. Best fitting cosmological parameters, with theirassociated 68% errors for the Gold set. The degeneracy in 0 leads to the same value of M for the three potentials in a set.Moreover, due to the form of the quintessence Lagrangian valuesof ! below 1 cannot be obtained.
GoldM ! 2
RP 0:300:060:11 10:24
01:20 177.0
HC 0:300:060:10 10:24
01:35 177.0
DE 0:300:060:07 10:22
01:08 177.0
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ΩM
α
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 DEΩ
M=0.30±0.06
0.07
α =0±1.08
Gold Set
ΩM
ωφ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1
−0.9
−0.8
−0.7
−0.6
−0.5 ΩM =0.30 ±0.060.07
ωφ=−1±0.22
Gold Set
DE
α
ΩM
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
DE
ΩM<0.36 at 95% C.L.
α<1.27 at 95% C.L.SNLS Set
ΩM
ωφ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1
−0.95
−0.9
−0.85
−0.8
−0.75
−0.7
−0.65
−0.6
−0.55
−0.5
ΩM<0.36 at 95% C.L.ωφ<−0.54 at 95% C.L.
SNLS Set
DE
FIG. 2 (color online). Confidence regions at the 68%, 95% and 99% C.L. (corresponding to 2 2:3, 6 and 11.8 for a two-parameter fit) for ;M (left panels) and !;M (right panels), in the Double Exponential quintessence potential, obtained usingthe Gold (top panels) and the SNLS (bottom panels) samples. The dotted lines, on the right panels, show the probability contours for aXCDM model: at the 99% and 95% for the Gold set, and at the 99%, 95%, and 68% for the SNLS set.
ΩM
α
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
ΩM
=0.30 ± 0.060.11
α =0 ±1.2
Gold Set
RP
ΩM
ωφ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1
−0.95
−0.9
−0.85
−0.8
−0.75
−0.7
−0.65
−0.6
−0.55
−0.5 RPΩM=0.30 ±0.06
0.1ωφ=−1±0.24
Gold Set
ΩM
α
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
ΩM<0.36 at 95% C.L.
SNLS Set
RP
ΩM
ωφ
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1
−0.95
−0.9
−0.85
−0.8
−0.75
−0.7
−0.65
−0.6
−0.55
−0.5
RPΩM<0.36 at 95% C.L.ωφ <−0.58 at 95% C.L.
SNLS Set
FIG. 1 (color online). Confidence regions at the 68%, 95% and 99% C.L. (corresponding to 2 2:3, 6 and 11.8 for a two-parameter fit) for ;M (left panels) and !;M (right panels), in the Ratra-Peebles quintessence potential, obtained using theGold (top panels) and the SNLS (bottom panels) samples. The dotted lines, on the right panels, show the probability contours for aXCDM model: at the 99% and 95% for the Gold set, and at the 99%, 95%, and 68% for the SNLS set.
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V. CONCLUSIONS
We have compared the predictions of three quintessencemodels with the measurements of the luminosity distancevs redshift from two SNe Ia catalogues, (Gold and SNLSsurveys). We have obtained the corresponding confidenceregions in the ;M and !;M planes and comparedthem with the predictions of dark energy models withconstant equation of state (XCDM). From the plots wesee that the different confidence regions for quintessenceand XCDM models cover essentially the same areas in theM;w plots. This makes it difficult to discriminatebetween the different models just from SNe Ia data alone.In all the three cases, the best fit cosmology corresponds tothe 0 case, in which the quintessence potential re-duces effectively to a cosmological constant. AlthoughSNLS data favor a pure cosmological constant, they yield
larger confidence regions than the Gold dataset. Bounds onthe cosmological parameters M, ! and the potentialparamaters have also been obtained. In principle it couldbe interesting to use the differences in constraints betweenthe two sets as an estimate of possible residual systematicsin the SNe Ia data. However, in practice, since both data-sets have been obtained and reduced in different ways andpossibly the actual data is still dominated by statisticalerrors, it would be difficult to perform such estimations.In any case this is beyond the scope of the present work.
ACKNOWLEDGMENTS
We would like to thank P. Astier for useful comments.This work has been partially supported by DGICYT(Spain) under project numbers FPA 2004-02602 and FPA2005-02327
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ΩM
α
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.05
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ΩM =0.30 ± 0.060.10
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ΩM
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SNLS Set
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FIG. 3 (color online). Confidence regions at the 68%, 95% and 99% C.L. (corresponding to 2 2:3, 6 and 11.8 for a two-parameter fit) for ;M (left panels) !;M (right panels), in the Hyperbolic Cosine quintessence potential, obtained using theGold (top panels) and the SNLS (bottom panels) samples. The dotted lines, on the right panels, show the probability contours for aXCDM model: at the 99% and 95% for the Gold set, and at the 99%, 95%, and 68% for the SNLS set.
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