PHYSICAL REVIEW D, VOLUME 63, 023503Properties of cosmologies with dynamical pseudo Nambu-Goldstone bosons
S. C. Cindy Ng* and David L. Wiltshire
Department of Physics and Mathematical Physics, University of Adelaide, Adelaide, S.A. 5005, Australia~Received 12 April 2000; published 18 December 2000!
We study observational constraints on cosmological models with a quintessence field in the form of adynamical pseudo Nambu-Goldstone boson. After reviewing the properties of the solutions, from a dynamicalsystems phase space analysis, we consider the constraints on parameter values imposed by luminosity distancesfrom the 60 type Ia supernovae published by Perlmutteret al., and also from gravitational lensing statistics ofdistant quasars. In the case of the type Ia supernovae we explicitly allow for the possibility of evolution of thepeak luminosities of the supernovae sources, using simple empirical models which have been recently dis-cussed in the literature. We find weak evidence to suggest that the models with supernovae evolution fit thedata better in the context of the quintessence models in question. If source evolution is a reality then thegreatest challenge facing these models is the tension between the current value of the expansion age,H0t0, andthe fraction of the critical energy density,Vf0, corresponding to the scalar field. Nonetheless there are rangesof the free parameters which fit all available cosmological data.
DOI: 10.1103/PhysRevD.63.023503 PACS number~s!: 98.80.Cq, 95.35.1d, 98.80.Esthfeeinel
Scalar fields have played a central role in models ofvery early universe for the past 20 years. In the pastyears attention has turned to models in which a scalar fiplays a dynamical role at late times, rather than simply befrozen in as a static relic vacuum energy. Such modwhich have been dubbed quintessence models@1#, couldin principle provide a dynamical solution to the cosmologicconstant problemnamely the question of why the magtude of the vacuum energy at the present epoch is so msmaller than one might naively expect from particle physmodels such as various supergravity theories. A dynamsolution of the cosmological constant problem woulamount to a demonstration that a particular dynamical elution of the scalar quintessence field is a natural conquence of the cosmological field equations without fintuning of parameters, given some reasonable physassumptions about the initial conditions.
The most notable recent observational evidence whichdriven the theoretical interest is the measurement of theparent magnitude-redshift relationship using type Ia supevae ~SNe Ia! @2#. These results have been interpreted, incontext of a cosmological model containing pressurelessand a cosmological constantL, as evidence that the universis undergoing accelerated expansion at the present epoch~see@3,4# and references therein!. The validity of this conclusionis currently open to some doubt, however. In particularrecent analysis by Riesset al. @5# indicates that the sample otype Ia supernovae shows a possible evolution in rise timfrom moderate (z;0.3) to large (z;1) redshifts. Althoughthe statistical significance of this result has bediminishedfrom the 5.8s level @5# to the 1.5s level@6#upon a more rigorous treatment of the uncertaintiesthe data@6#, it remains true that while a systematic evolutio
*Electronic address: email@example.comElectronic address: firstname.lastname@example.org/2000/63~2!/023503~15!/$15.00 63 0235ewldgs,
in the rise times of the supernovae is not conclusively ruin, neither is it conclusively ruled out.
Given that an evolution in the shape of the light curvesthe supernovae measured in their rest frame remains apossibility, it would not be surprising if the pealuminositywhich is the effective standard candle usedwere also to evolve. Riesset al. @5# conclude that the type Iasupernovae data could conceivably be explained entiwithin the context of an open Friedmann-Robertson-Waluniverse together with a reasonable astrophysical evolumodel, e.g., a consequence of a time variation of the abdances of relevant heavy elements in the environment ofwhite dwarf supernovae progenitors. Detailed astrophysmodelingsee, e.g.,@7#should hopefully eventually resolve the issue, although at this stage the difference betwour theoretical understanding and the observations remquite substantial@8#.
In many recent papers it has been commonly assumedthe dynamical scalar field,f, should obey an effective equation of statePf.wrf with 21,w,0, at the present epoch, in order to obtain a cosmological acceleration, i.e.negative deceleration parameterq0. Indeed, the conditionthat 21,w,0 is often taken as a defining characteristicquintessence@1#. The broad picture in this cosmologicascenario is that the universe is currently in the early stagean epoch of inflationary expansion. The motivation for thisthat one could then hope to have a model cosmologywhich observations such as the type Ia supernovae appamagnitude-redshift relation could be explained by a cosmlogical acceleration in a similar fashion to models withcosmological constant, but with the possibility of explaininwhy the magnitude of the vacuum energy density andenergy density in ordinary pressureless matter,rm , are com-parable at the present epochthe so-called cosmic coidence problem@9#.
One attractive feature of homogeneous isotropic cosmlogical models with dynamical scalar fields is that manythem possess cosmological scaling solutions@10#, namelysolutions which at late times have energy density com2000 The American Physical Society03-1
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503nents which depend on the cosmic scale factor accordinr}a2m1 andrf}a
2m2 simultaneously, and which act as atractors in the phase space. Ifm2,m1, which is the case, forexample, for simple power-law potentials with inverse poers @11,12# V(f)}f2a, or for certain power-law potentialwith positive powers@10#, then the scalar field dominateslate times, producing a quintessence-dominated cosmowith accelerated expansion at late times. Ifm15m2, which isthe case for exponential potentials@1320#, then the scalingsolutions are self-tuning@16#i.e., the dynamics of thescalar field follows that of the other dominant energy coponent, with a dependencerf}a
24 in the radiation-dominated era and a dependencerf}a
23 in the matter-dominated era.
Even if the ultimate late-time properties of the solutioare not precisely self-tuning in the above sense, modsuch as those with inverse power law potentials caneffectively act as tracking solutions, since for a widrange of initial conditions, the solutions rapidly convergea common, cosmic evolutionary track@12#. Thus there are anumber of ways in which one might hope to solve the comic coincidence problem, though in practice a degreetuning of the parameters has been necessary in all mostudied to date.
In this paper, we consider a form of quintessence,ultra-light pseudo Nambu-Goldstone boson~PNGB! @21#which is still relaxing to its vacuum state. From the viewpoint of quantum field theory PNGB models are the simplway to have naturally ultra-low mass, spin-0 particles ahence perhaps the most natural candidate for a presentlyisting minimally coupled scalar field. The effective potentof a PNGB fieldf can be taken to be of the form@22#
V~f!5M4@cos~f/ f !11#, ~1!
where the constant term is to ensure that the vacuum envanishes at the minimum of the potential. This potentiacharacterized by two mass scales, a purely spontaneousmetry breaking scalef and an explicit symmetry breakinscaleM.
The effective PNGB mass ismf;M2/ f . To obtain solu-
tions withVf;1, the energy scales are essentially fixed@21#to valuesM;1023 eV, interestingly close to the neutrinmass scale for the Mikheyev-Smirnov-Wolfenstein~MSW!solution to the solar neutrino problem, andf ;mPl.1019 GeV, the Planck scale. Since these two energy schave values which are reasonable from the viewpoint of pticle physics, one might hope to explain the coincidence tthe vacuum energy is dynamically important at the presepoch.
The cosmology of PNGB models has already been exsively studied in the literature@17,18,20,2224#. In particu-lar, a number of constraints have been placed on the paeters M and f by various sets of observational da@17,18,23,24#. Most recently, Frieman and Waga@24# haveset bounds based on the SNe Ia data of Riesset al. @4# ~here-after R98! on the one hand, and gravitational lensing surveon the other. Comparing these bounds is of interest, sinceSn Ia data have been interpreted as favoring a cosmolog02350to
constant, whereas gravitational lensing data has been usplace upper bounds onL @2527#. They therefore providecomplementary tests of the parameter spaces of modelsa non-trivial vacuum energy.
In this paper it is our intention to critically study thesbounds. First, we will consider how the bounds are affecby the initial value of the scalar field at the beginning of tmatter-dominated epoch. Secondly, we wish to investighow such bounds might be affected in the case of the PNmodel if the observed apparent faintness of type Ia supevae is at least partly due to an intrinsic evolution of tsources over cosmological time scales, which in view ofresults of@5# would appear to be a very real possibility. Threason for focusing on the PNGB models in such an invtigation is suggested by the fact that whereas many quinsence models have been singled out in the literature, persomewhat artificially, simply because they have the propeof yielding an accelerated expansion, many different pobilities arise in the PNGB case. Indeed, at very late times,apparent magnitude-redshift relation for PNGB models umately coincides with that of the Einsteinde Sitter modeven though the density of ordinary matter can be low inPNGB cosmologies. The requirement that the faintnesstype Ia supernovae is entirely due to their cosmological dtances places rather strong restrictions on the values ofparametersM andf @24#, because it requires us to exist at aepoch of the PNGB cosmologies which is still quite far rmoved from our ultimate destiny. If these restrictions arelaxed because of evolutionary effects, then it is quite plsible that other regions of the parameter space of the PNmodels become viable alternatives. Since PNGB cosmgies could therefore still solve the missing energy prolem, even if the evidence for a cosmological acceleratiproves to be ephemeral, we believe it is important to invtigate this possibility quantitatively.
We will begin the paper with a qualitative analysis of thsolutions, to provide some general insights which will heto guide our quantitative discussion. Although these propties are no doubt already known, to the best of our knoedge an analysis of the phase space of the solutions has nbeen presented in the literature. Having completedanalysis in Sec. II we will go on to discuss a numberissues relating to numerical integration in Sec. III, and relthe properties of the solutions found numerically to the exanalysis of Sec. II. In Sec. IV we present the main analysisthe constraints imposed on the (M , f ) parameter space, allowing for the possibility of evolution of peak luminosities ithe type Ia supernova sources. Bounds from gravitatiolensing statistics are updated in Sec. V, and the implicatiof our results are discussed at greater length in Sec. VI.
II. PHASE-SPACE ANALYSIS
We will begin by performing an analysis of the differential equations governing the cosmological evolution inmanner similar to previous studies in inflationary and quiessential models@10,14,15,19,20#.
The classical action for gravity coupled to a scalar fieldfhas the form3-2
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503S5E d4xA2gF S R2k2 2 12 gmn]mf]nf2V~f! D1LG ,~2!
where k is the Planck constant,R is the Ricci scalar,g[detgmn , andL is the Lagrangian density of non-relativistmatter and radiation. For simplicity, we assumef is mini-mally coupled to the curvature, and we work in unitswhich \5c51.
Consider a spatially flat Friedmann-Robertson-Wal~FRW! universe containing a fluid with barotropic equatioof state Pg5(g21)rg , where g is a constant, 0
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503tend I to 6`, with trajectories crossing from one cell toanother at the cell boundaries.
The trajectories which occupy the lower half of Fig. 1 aobtained from those in the upper half by the symmetryI2p2I , J2J of the differential equations~11!~14!.Physically this simply corresponds to the scalar field rollifrom the maximum to the right of the minimum as opposto the one to the left.
An analysis of small perturbations about the critical poiC1 and C2 yield eigenvalues
~1! l52A6mg,2 12 m(A66A614/F 2) at C11 ,~2! l50,6 im/F at C2.Thus C11 attracts a two-dimensional bunch of trajectori
but is a saddle point with respect to trajectories lying inrg50 surface, as is evident from Fig. 1. The 2-dimensiobunch of trajectories which approach C11 are found to cor-respond to an inflationary solution witha}exp(A2/3mt) ast` and fconst52npF, nP Z. The possible role ofscalar fields with PNGB potentials in driving an inflationaexpansion of the early universe has been discussed in@28#.
The point C2 is a degenerate case, in particular with rgard to perturbations orthogonal to therg50 surface~i.e.into the surfacerg.0 region!, for which the eigenvalue iszero. It is a center with respect to the trajectories lying inrg50 surface, and when perturbations of higher orderconsidered it becomes a stable spiral point in therg50 sur-face as can be seen in Fig. 1. Since there is a degenehowever, an alternative choice of phase space variabledesirable. We will defer a discussion of the late time behior of the solution near C2 to Sec. II B.
The points C16 correspond to models with a scalar fiesitting at the maximum of the potential, whereas C2 corre-sponds to the scalar field sitting at the bottom of the potenwell. The separatrices in Fig. 1 which join C16 to C2 corre-spond to the field rolling from the maximum to the minmum. It would appear from Fig. 1 that trajectories whispiral into C2 become arbitrarily close to the separatrixlate times.
The separatrices which join the points C16 to points at
FIG. 1. The projection of the trajectories within therg50 sub-space on theI J plane for values ofI P@0,2p). Within this sub-space, C11 is a saddle point and C2 is a stable spiral.02350s
infinity correspond to solutions for which the scalar fiereaches the top of the potential hill ast6` ~e.g., the right-most trajectory in Fig. 1!. Finally, there are also straight linseparatrices parallel to theH-axis at each of the points C16 ,extending fromH56H1 to infinity, which represent solu-tions with a static scalar field sitting on top of the potenthill.
To examine the critical points at infinity it is convenieto transform to spherical polar coordinatesr , u, andf bydefining
H5r cosu, ~16!
I 5r sinu sinf, ~17!
J5r sinu cosf, ~18!
and to bring the sphere at infinity to a finite distance fromorigin by the transformationr 5r/(12r), 0
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503~2! B6` : two points at
H56`, I /H50, andJ/H50.Since the projection onto spherical polar coordina
~16!~18! is degenerate at the north and south polesu50,p, these points are excluded from the chart (u,f) butcan be included using an alternative hemispherical projec~see Fig. 3!.
~3! C1,2` : two points at
(u,f)P$(p/2,p/2),(p/2,3p/2)% orI 56`, H/I 50, andJ/I 50.An analysis of small perturbations shows that the poi
A11,21` are repellors in all directions of the phase space~cf.
Table I!, while A12,22` are attractors. This therefore repr
sents the most typical early behavior of solutions. ForH.0, A11,21
` correspond to the limitt0. We find thatH;1/(3t) or a}t1/3, while kf;6A2/3lnt, for these solu-tions. The points A12,22
` with H,0 represent the timereversed solutions.
The point B1` (B2
` ) repels ~attracts! a 2-dimensionalbunch of trajectories traveling to~from! finite values ofH,I ,J, but is a saddle point with respect to directions onsphere at infinity. The points are found to correspond tt0 with H;2/(3gt) or a}t2/3g while kf}tn, n.0.
The points C1,2` are the projection of the points C16 and
C2 into the sphere at infinity. The degenerate eigenvalsimply reflect the degeneracy of the projection.
B1` acts as a repellor for trajectories withf.0, f
.const ast0. As shown in Fig. 3, trajectories are drivetowards B1
` before they reach C1` . This is consistent with the
property that whenH is large (3H>mf), the field evolutionis over-damped by the expansion, and the field is effectivfrozen to its initial value (f0).
FIG. 3. The projection of trajectories within theI .0 andI ,0hemispheres at infinity on theH J plane. The unphysical region ishaded.
TABLE I. The critical points on the sphere at infinity and theeigenvalues.
Critical points Eigenvalues~with degeneracies!
` 63 ~2!, 63(22g)
C1,2` 0 ~3!02350s
B. Energy density variables
In view of the eigenvalue degeneracy encountered abowe can alternatively choose to represent the system byvariablesH, x, andy, where
5A2m cos~ I /2!
These are the same variables used by Copeland, LiddleWands@19# for the model with an exponential potential. Aabove, we will considerH.0 only. The field equations thentake the form
We note that in these variables
which is why we have adopted the terminology energy desity variables. The physical region of the phase space wbe constrained to lie within the cylinderx21y2
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503Within the horn portion of each phase space cell the mtion of most trajectories is roughly circular inH5constslices, in a clockwise sense in even cells,C2n , and anti-clockwise in odd cellsC2n11. However, trajectories can crosfrom one cell to another along they56H1 /H boundaries oftheir horns, which correspond to the surfacesI 52np interms of the field variables. For even cells,C2n , trajectoriesjoin the cell C2n along they5H1 /H surface and the celC2n11 along the y52H1 /H surface. Below theH5H1plane solutions cannot cross from one cell to another,remain confined within the cylinderx21y2
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503we see thatgf.sin2(mt/F) at late times, so that it remaintruly variable, varying from 0 to 2 over each cycle. It folows from Eqs.~25!~28! that the scalar energy density prameter obeys the equation
so thatVf0 as t`, which accords with the late timeproperties of the solutions observed above.
III. NUMERICAL INTEGRATION
We will now consider the extent to which models bason the PNGB potential are constrained by the latest obsetional evidence. This work extends the studies previouundertaken by various authors@17,18,23,24,30#. In the mostrecent analysis, Frieman and Waga have compared thestraints imposed by the high redshift supernovae luminodistance on the one hand and gravitational lensing boundthe other@24#. The two measures provide tests which apotentially in opposition. Here we will perform a similaanalysis for the PNGB models, but also taking into accothe possibility of luminosity evolution which has not beeconsidered in previous studies@18,24#. We are therefore considering the model of the previous section withg51.
To proceed it is necessary to integrate the equationsmerically. To do this we introduce the dimensionless vaables
F . ~40!
whereVm0 is the fractional energy density of matter at tpresent epoch,t0. The dynamical system then becomes
F , ~43!
where the Hubble parameter is defined implicitly accordto
H0[Fv31 16 u21 m23H02~cosw11!G
and prime denotes a derivative with respect to the dimsionless time parametert[H0t. Only the variablev differsfrom those used by Frieman and Waga@18#. Our reason for02350a-ly
making the choice~39! is that it allows us to integrate thFriedmann equation directly rather than a second order etion ~11! which follows from the other equations by virtue othe Bianchi identity. This may possibly lead to better nmerical stability since it is not necessary to implementFriedmann constraint separately.
We begin the integration at initial values ofu, v, andwchosen to correspond to initial conditions expected inearly matter-dominated era. The integration proceeds tuntil the rhs of Eq.~44! is equal to 1, thereby determining thvalue of the present epoch,t0, to be the time at whichH5H0. We are then also able to determineVm0, since accord-ing to Eq.~39!
Vm05v3~ t0!. ~45!
The choice of appropriate initial conditions has been pviously discussed@23,17,18#. In particular, since the Hubbleparameter is large at early times, it effectively acts asdamping term in Eq.~3!, driving the scalar field to a statwith f50 initially, i.e. u50. We takev51101 initially,which in view of relation ~39! and the fact thatVm0;0.11 corresponds to the early matter dominated1100
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503readily understood by considering the corresponding plotthe deceleration parameter,q0, which is defined in terms ou(t0), v(t0), andw(t0) by
FIG. 5. Contours ofVf0 in the M , f parameter space for twochoices of initial values:~a! wi51.5; ~b! wi50.5.
FIG. 6. Contours ofH0t0 in the M , f parameter space for twochoices of initial values:~a! wi51.5; ~b! wi50.5.02350of
We display contour plots ofq0 in the M , f parameter spacein Fig. 7. Essentially, asM increases for roughly fixedf forsufficiently largeM, the value ofq0 oscillates over negativeand positive values, from a minimum ofq05
12 (123Vf0) to
a maximumq0512 (113Vf0) about a mean ofq050.5. This
corresponds to the scalar field,f, having undergone moreand more oscillations by the time of the present epoch. Tminimum value ofq0 is attained whenf50 instantaneously,while the maximum value ofq0 is attained whenf is instan-taneously passing through the minimum of its potential.
For smaller values ofM to the left of the plots the scalafield has only relatively recently become dynamical, wherefor larger values ofM, the scalar can already have undergoseveral oscillations by the time of the present epoch, partlarly if f is small. This variation can be understood in termof the asymptotic period of oscillation of solutions whicapproach C1, which by Eqs.~29!,~30! is
ta52pF/m52p f /M2. ~47!
The periodta is shorter for largerM, or for smallerf. Sincethe finalf values plotted in thewi50.5 case are a factor of 2smaller than thewi51.5 case, this also explains why poinwith the same value ofM have undergone more oscillationup to the present epoch for the smaller value ofwi .
The value ofH0t0 oscillates asM increases for roughlyfixed f, according to whether the universe has been acceating or decelerating in the most recent past, with more ravariation for parameter values with shorter asymptotic peods,ta .
FIG. 7. Contours ofq0 in the M , f parameter space for twochoices of initial values:~a! wi51.5; ~b! wi50.5. Valuesq0,0,corresponding to a universe whose expansion is accelerating apresent epoch, are shaded.3-8
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503The wiggles in theVf0 contours are a residual effect othe oscillation of the scalar field asVf settles down to aconstant value according to Eq.~37!. The variation in thevalue of Vf0 with the value ofwi can be understood fromthe fact that for smaller values ofwi the scalar field begins itsevolution in the matter-dominated era closer to the critipoints, C11, corresponding to the maximum of the potentiV(f). For fixed M and f the period of quasi-inflationaryexpansion is therefore longer, and the present value ofVf0larger.
IV. CONSTRAINTS FROM HIGH-REDSHIFT TYPE IASUPERNOVAE
Empirical calibration of the light curveluminosity relationship of type Ia supernovae provides absolute magnituthat can be used as distance indicators. Since the luminodistance of a light source,dL , is defined by
with t5H0t, it is convenient to define an additional variab
v dt, ~49!
which is proportional to the comoving coordinate distanc
H0a~ t0!v~ t0!. ~50!
We then adjoin a differential equation
r 852v ~51!
to the differential equations~38!~40! when performing thenumerical integration. In terms ofr and v, the luminositydistance is then determined according to
r 2r ~ t0!, ~52!
and can be used in the appropriate distance modulus to cpare with the supernovae data.
Frieman and Waga@24# have recently considered constraints on PNGB models from supernovae data~withoutsource evolution! using the data set of Riesset al. @4#. Wewill perform a similar analysis, but we will make use of thlargest available data set, namely the 60 supernovaelished by Perlmutteret al. @3# ~hereafter P98!. Of these, 18low redshift SNe Ia were discovered and measured inCalan-Tololo survey @31#, and the Supernova CosmologProject discovered 42 new SNe Ia at redshifts between 0and 0.83. The peak magnitudes of the supernovae arerected using the stretch factor light curve fitting metho@3,32#. This method is based on fitting a time-stretched vsion of a single standard template to the observed light cuThe stretch factor is then used to estimate the absolute mnitude.02350l,
A. Models without evolution
We use the stretch-luminosity corrected effectiveB-bandpeak magnitude in Table 1 of P98 as the absolute magniand denote it asmB
eff . The distant modulus of a supernovadefined asmB
eff1M0, where M0 is the fiducial absolutemagnitude which has not been given in the literaturesfitting method to obtainM0, using only the 18 Calan/Tololosupernovae, can be found in@33,34#. Perlmutteret al. @3#calculate the confidence regions by simply integrating oM0. In this paper, we will perform an analytic marginaliztion overM0 and obtain the marginal likelihood.
Similarly to thex2 statistic used by Riesset al. @4#, wedefine the quadratic form
x2~M , f !5(i 51
eff 1M02m i !2s i
wherem i(zi ;H0 ,M , f ) is the predicted distance modulus fomodel parameters (M , f ), and
21S 5 log ezi sz,i D
The predicted distance modulus is
m i55 logdL~zi !125, ~55!
if the luminosity distance,dL , as defined by Eq.~48! or Eq.~52!, is given in units of Megaparsecs.
In order to perform analytic marginalization overH0 aswell as overM0 we separate out theH0 and M0 depen-dence fromm i into a quantity,n, which we define by
m i2M05gi1n ~56!
wheregi(zi ;M , f ) depends implicitly only onM and f. Wethen follow the statistical procedures adopted by DreLoredo and Wasserman@35,36# and marginalize overn usinga flat prior that is bounded over some rangeDn.
The marginal likelihood is
L~M , f !5 1DnE dn e2x2/25 sA2pDn e2Q/2 ~57!
Q~M , f !5(i 51
eff 2gi2 n !2
and n(M , f ) is the best-fit value ofn given M and f, withconditional uncertaintys. We identify 1/s2 as the coefficientof n2 in Eq. ~53!,
s i2 , ~59!
and n ass2 times the coefficient of2n, viz.3-9
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503n~M , f !5s2(i 51
60 mB,ieff 2gi
The quadratic form~58! is what one would obtain bycalculating the maximum likelihood forM and f. Sincesis independent ofM and f, it follows from Eq. ~57! that themarginal likelihood is proportional to the maximum likelhood in this case.
Figure 8 shows the 68.3% and 95.4% joint crediblegions forM and f, and is a direct analogue of Fig. 1 of Re@24#, where a similar analysis was performed on 37 supervae given in R98. The position of the region of parametwhich are included at both the 68.3% and 95.4% confidelevels is broadly similar to that obtained from the R98 d@24#. Although Frieman and Waga@24# did not include theparameter regionM.0.004h eV, we have redone theianalysis on the R98 data and find that the parameter valuthe right of Fig. 8 which are admitted at the 95.4% level bexcluded at the 68.3% for the P98 dataset~labeled region IIin Fig. 8!, are in fact excluded also at the 95.4% confidenlevel if the 37 supernovae of the R98 dataset are used.possible that this discrepancy has its origin in the differtechniques used by Riesset al. @4# to determine the distancmoduli. Possible systematic discrepancies in the strefactor method of P98 versus the multi-color light curveand template fitting methods of R98 have been discussin some detail in Ref.@35#.
The importance of the 2s included parameter region tright of Fig. 8 diminishes, however, if one compares it wFigs. 5 and 6, since it largely corresponds to parameterues withVf0*0.9, which can be discounted by dynamicmeasurements ofVm0, whereVm01Vf051. Furthermore,the few allowed values below theVf050.9 contour in thispart of the parameter space have unacceptably small vafor the age of the Universe,H0t0.
The parameter region 0.002h eV,M,0.003h eV,which from Fig. 8 is admitted at both the 68.3% and 95.4levels ~labeled region I!, by contrast corresponds to accepable values of bothVf0 and H0t0. Comparing with Fig. 7,we see that this region has20.1&q0&20.6, correspondingto a Universe with a scalar field still in an early stage
FIG. 8. Confidence limits onM , f parameter values, withwi51.5 and no evolution of sources, for the 60 supernovae Ia inP98 dataset. Parameter values excluded at the 95.4% levedarkly shaded, while those excluded at the 68.3% level are ligshaded.02350-
rolling down the potentialV(f). The label I is thus indica-tive of the fact that the scalar field is rolling down the ptential for the first time~from left to right!, while in region IIthe scalar field is rolling down the potential for the secotime ~from right to left!. In region II q0 is positivehowever, it corresponds to parameter values for which thwould have been a cosmological acceleration at modestshifts in the past, e.g., atz;0.2, well within the range of thecurrent supernovae dataset.
Since conclusions regarding statistically preferred regiof the parameter space can change if evolution of the souoccurs, we think it is important that this possibility is alsexamined, as we will now do.
B. Models with evolution
In view of the recent results of Refs.@5,6#, supernovae Iamay exhibit some evolutionary behavior, at least as fartheir rise times are concerned. A possible evolution inpeak luminosity is therefore a possibility which must be sriously investigated.
In the absence of a detailed physical model to explprecisely how the source peak luminosities vary with reshift, one approach is to assume some particular empirform for the source evolution, and to examine the conquences. Such an analysis has been recently preformeDrell, Loredo and Wasserman@35# in the case of FriedmannLematre models with constant vacuum energy. We will udertake an equivalent analysis for the case of PNGB quessential cosmologies.
Following Drell, Loredo and Wasserman@35# we will as-sume that the intrinsic luminosities of SNe Ia scale aspower of 11z as a result of evolution. This model introducea continuous magnitude shift of the formb ln(11z) to theSNe Ia sample. Equation~53! then becomes
x2~M , f !5(i 51
eff 2gi2n2b ln~11zi !2
The parameterb will be assumed to have a Gaussian prdistribution with meanb0 and standard deviationb. Physi-cally the parameterb0 represents a redshift-dependent evlution of the peak luminosity of the supernovae sourcwhich might be expected to arise as a result of the chemevolution of the environment of the supernovae progenitas abundances of heavier elements increase with cotime. Ultimately, one should hope to account for this evotion with astrophysical modeling of the supernovae expsions @7#. The parameterb would then account for a locavariability in the supernovae environments between regiof individual galaxies at the same redshift which are richerpoorer in metals, or with progenitor populations of differeages and masses, etc.
We now have two parameters to marginalize over,n andb. As in the case of models with no evolution, we will maginalize overn using a flat prior. We use a Gaussian prior fb with meanb0 and standard deviationb, so that
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503b85b2b0 . ~63!
The marginal likelihood is calculated by multiplying thprior ~62! by the likelihood resulting from Eq.~57!, and in-tegrating overb. The resulting likelihood is
L~M , f !5 1DnE db8E dn p~b8!e2x2/2
Q~M , f !52b2
hi~zi ;M , f !5mB,ieff 2gi2b0 ln~11zi !, ~66!
H~M , f !5(i 51
s i2 . ~67!
The conditional best-fit ofb8 is given by
b~M , f !5s2(i 51
60hi@ ln~11zi !2s
s i2 , ~69!
and theb8 uncertainty,s, is given by
60 ln2~11zi !
s i2 . ~70!
Although s is independent ofM and f, the marginal likeli-hood is no longer proportional to the profile likelihood bcauseQ is now given by Eq.~65! rather than by the chi-square type statistic~58!.
We have performed a detailed numerical analysis onP98 dataset, varyingb0 , b and wi . As a result we find abest-fit value ofb0[b* .0.414, forwi51.5 in the PNGBmodels. This would correspond to supernovae being intrically dimmer by 0.17 magnitudes at a redshift ofz50.5,which is an effect of the typical order of magnitude beiaddressed in current attempts to better model the superexplosions@7#. Furthermore, we find that inclusion of a nozero variance,b2, does not alter the prediction of the best-value ofb0, although it naturally does lead to a broadeniof the areas of parameter space included at the 2s level.There is relatively little broadening of the region of paraeter values included at the 1s level in the b05b* plane,however, which is no doubt a consequence of the steepof the q0 contours in Fig. 7~a! in the area corresponding tregion II.
In Figs. 9 and 10 we display the joint credible regionsM and f, for two slices through the 3-dimensional (M , f ,b0)02350e
parameter space forwi51.5: ~a! the best-fit caseb05b* ;and ~b! b050. Analagously, the best-fit case is shownFig. 11 for wi50.2. We seefrom Fig. 10 that once thelikelihood is normalized relative tob* no regions remain intheb50 parameter plane which are admitted at the 1s levelwhen b50. Furthermore, even when a non-zero standdeviation,b, is included, region II of the (M , f ) parameterplane is favored at the 1s level, in contrast to Fig. 8.
We have also undertaken an analysis of the models wb050 but variableb, similarly to the study of Ref.@35#. Inthat case, we once again find that region II of the (M , f )parameter plane is admitted at the 1s level if b50.25 orb50.5. The dependence of the value of
Q[22 ln~LDn!5Q22 lnS A2pssb
D ~71!on the value ofb is displayed in Fig. 12, forb050 as com-pared with the best-fit caseb05b* . The quantityQ is analo
FIG. 9. Confidence limits onM , f parameter values in the besfit b050.414 slice of the (M , f ,b0) parameter space, withwi51.5, for the 60 supernovae Ia in the P98 dataset. Parameter vexcluded at the 95.4% level are darkly shaded, while those excluat the 68.3% level are lightly shaded. For reference, contoursVf0 and H0t0 are superposed as dashed and dotted lines restively.3-11
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503gous to the chi-square statistic of the maximum likelihomethod. We see that theb050 models favor a non-zerovalue ofb.0.36 by a very small margin as compared to tb50 case. Forb50.36 the points of greatest likelihood limainly in region II, in contrast with theb50 case in Fig. 8.
Varying the initial conditionwi does not appear to affecthe best-fit value ofb0 significantly. Forwi50.2 ~cf. Fig.11!, for example, the best-fit value wasb* 50.435, a differ-ence of 5% from thewi51.5 case. Furthermore, the numecal value of the least value ofQ was only 0.2% greater in thewi50.2 case. We have not attempted to find a best-fit vafor wi .
V. CONSTRAINTS FROM LENSING STATISTICS
Gravitational lensing of distant light sources due to taccumulation of matter along the line of sight provide a
FIG. 10. Confidence limits onM , f parameter values in theb050 slice of the (M , f ,b0) parameter space relative to a best-valueb* .0.414, for the 60 supernovae Ia in the P98 dataset, wwi51.5. Parameter values excluded at the 95.4% level are dashaded, while those excluded at the 68.3% level are lightly shaFor reference, contours ofVf0 andH0t0 are superposed as dashand dotted lines respectively.02350e
other relatively sensitive constraint on the cosmologimodels of interest. For cosmology the situation of mostterest is the lensing of high luminosity quasars by interving galaxies. The abundance of multiply imaged quasarsthe observed separation of the images to the source putsstraints on the luminosity-redshift relation and hencemodel parameters. Basically, if the volume of space togiven redshift is larger then on average one can expect mlensing events. This leads to a statistical test, which has bused to put bounds onL @2527# and to test properties osome decayingL or quintessence models@30,37#.
Gravitational lensing statistics are useful since they pvide a test which potentially provides opposing constraintsthose obtained from supernovae magnitude-redshift testsparticular, in the case of models with a vacuum energy pvided by a cosmological constant, the high redshift supervae have been interpreted as favoring relatively large vaof VLPerlmutteret al. @3# give a value ofVL50.72 at 1s
FIG. 11. Confidence limits onM , f parameter values in theb050 slice of the (M , f ,b0) parameter space relative to a best-valueb* .0.435, for the 60 supernovae Ia in the P98 dataset, wwi50.2. Parameter values excluded at the 95.4% level are dashaded, while those excluded at the 68.3% level are lightly shaFor reference, contours ofVf0 andH0t0 are superposed as dasheand dotted lines respectively.3-12
PROPERTIES OF COSMOLOGIES WITH DYNAMICAL . . . PHYSICAL REVIEW D63 023503levelwhereas the gravitational lensing data lead to upbounds onVL : Kochanek@25# quotesVL,0.66 at the 2slevel. A combined likelihood analysis has been performedvarious authors@24,27,30#.
Gravitational lensing constraints on the PNGB modhave been very recently given by Frieman and Waga@24# forwi51.5. However, Frieman and Waga considered a mrestricted range of parameter space,M,0.005h eV, sincethey did not consider the possibility of source evolutionthe case of the type Ia supernovae and therefore took vaof M.0.005h eV to be ruled out. We wish to extend thrange ofM for the gravitational lensing statistics to considparameter values corresponding to region II in the supevae constraint graphs, Figs. 8 and 10, so as to compareconstraints from different tests.
We have thus simply followed the calculation describby Waga and Miceli@30#, who performed a statistical lensinanalysis of optical sources described earlier by Kocha@25#. They used a total of 862 (z.1) high luminosity qua-sars plus 5 lenses from seven major optical surveys@38#.~Another alternative not considered here is to analyze dfrom radio surveyssee, e.g.,@26,27#.! Undertaking a simi-lar analysis for the increased parameter range, we arrivFig. 13, which shows the 68.3% and 95.4% joint crediregions forM and f, for two values ofwi . We refer thereader to Refs.@25,30# for details of the calculation. The onlregions of parameter space excluded at the 2s level turn outto be areas of parameter space for which the deceleratiopresently negative~cf., Fig. 7!, with the scalar field still com-mencing its first oscillation at the present epoch.
Let us now consider the overall implications of the costraints observed above.
First, since empirical models with source evolutionappear to fit the data somewhat better, it would appearwe do have weak evidence for an underlying evolution ofpeak luminosity of the type Ia supernovae sources, at leathe context of the PNGB quintessence models. It mightinteresting to compare the case of other quintessence moor the case of a cosmological constant. However, the PNmodel is qualitatively different from such models sincefinal state corresponds to one in which the ultimate des
FIG. 12. Variation of the least value ofQ522 ln(LDn) as afunction of the standard deviationb of the prior distribution forb0for valuesb050 andb05b* .02350er
of the universe is to expand at the same rate as a spatiallyFriedmann-Robertson-Walker model, rather than to undean accelerated expansion. This is of course precisely whychose the PNGB models as the basis of our investigatrather than models in which a late-time accelerated expsion had been built in by hand. If we wish to test the hypoesis that the faintness of the type Ia supernovae is at lpartly due to an intrinsic variation of their pealuminositieswhich is a very real possibility in view of thresults of@5#then a quintessence model which possessevariety of possibilities for the present-day variation of tscale factor is probably the best type of model to investiga
If only supernovae luminosity distances~cf., Fig. 9! andgravitational lensing statistics~cf., Fig. 13! are comparedthen we see that there is a remarkable concordance betwthe two testsregion II of Fig. 9 coincides with a regioincluded at even the 1s level in Fig. 13. This is perhaps nosurprising, since in view of Fig. 7 region II correspondsparameter values for which the present day universe haready undergone almost one complete oscillation of the slar field about the final critical point C2 of Fig. 1. It is thusalready well on the way towards its asymptotic behaviwhich closely resembles that of a standard spatiallyFriedmann-Robertson-Walker model.
Due to the oscillatory behavior, parameter values ingion II correspond to models in which there has been acent cosmological acceleration~e.g., atz;0.2), butwith aq(z) which changes sign three times over the larger rangredshifts, 0,z,4, in the quasar lensing sample, and thefore differing significantly from Friedmann-Lematre modelsover this larger redshift range. Extending the SNe Ia sam
FIG. 13. Confidence limits from gravitational lensing statistic~a! wi51.5; ~b! wi50.2. Parameter values excluded at the 95.4level are darkly shaded, while those excluded at the 68.3% levelightly shaded. For reference, contours ofVf0 andH0t0 are super-posed as dashed and dotted lines respectively.3-13
tes-nedcted.ivehef aaili-veith auldsu-
S. C. CINDY NG AND DAVID L. WILTSHIRE PHYSICAL REVIEW D 63 023503to include objects at redshifts 0.1,z,0.4 andz.0.85 insubstantial numbers would greatly improve the ability to dcide between models in regions I and II.
There is some cause for concern, however, if we consthe favored values ofVf0 andH0t0. In the case of the models with empirical evolution of the supernovae sources,always found that the best-fit parameter values occurrethe Vf01 boundary of the (M , f ) parameter space. Thoverwhelming evidence of many astronomical observatiover the past two decades@39# would tend to indicate thaVm0.0.260.1, indicating that a vacuum energy fractionVf0;0.70.8 is desirable, andVf0&0.9 in any case. Al-though parameter values withVf0,0.8 certainly fall withinboth the 2s and 1s portions of region II of Fig. 9, for allvalues ofb, there are potentially serious problems if we wito simultaneously obtain large values ofH0t0. In view ofrecent estimates of the ages of globular clusters@40#, a lowerbound of 12 Gyr for the age of the Universe appears tocurrently indicated. Withh.0.65 this would requireH0t0*0.8. Forwi51.5, parameter values withH0t0.0.8 coin-cide with valuesVf0*0.9 in region II, which is phenomenologically problematic.
The tension between the values ofVf0 andH0t0 is some-what mitigated for lower values ofwi . For wi50.2, for ex-ample, we see from Figs. 12 and 13~b! that theVf050.7 andH0t0 meet in region II, and there is a small region of paraeters there with 0.7&Vf0&0.9 andH0t0*0.8, which is alsoconsistent with the other cosmological tests.
Even if the supernovae sources undergo evolution iclear that parameter values in region I of 8, which arevored in the absence of evolution of peak SNe Ia luminoties, are still included at the 2s level in the models withevolution, in view of Fig. 9.
Perhaps the most significant aspect of our results isetl,
fact that the mere introduction of an additional dispersiob.0.17, in the peak luminosities while leaving their mevalue fixed~cf. Fig. 11!, gives rise to a change in the best-region of parameter space from region I to region II.~See@41# for further details.! One would imagine that an increasedispersion is likely to be a feature of many models of souevolution, even if evolutionary effects are of secondary iportance. Thus even if the empirical models with non-zerobare somewhat artificial, more sophisticated scenarios cowell lead to similar changes in regard to the fitting of comological parameters in the PNGB model.
Much tighter bounds on the parameter space of quinsence models, including the present model, will be obtaiover the next decade as more supernovae data are colleWhat we wish to emphasize, however, is that an effectvacuum energy which is cosmologically significant at tpresent epoch should not simply be thought of in terms ocosmic acceleration. A dynamical vacuum energy withvarying effective equation of state allows for many possibties for the evolution of the universe, and overly restrictiassumptions, such as equating quintessence to models wlate period of continuous cosmological acceleration, shobe avoided. If detailed astrophysical modeling of type Iapernovae explosions ultimately shows that the dimnessdistant supernova events is largely due to evolutionaryfects, it does not spell the end for cosmologies with dynacal scalar fields.
We would like to thank Chris Kochanek for supplying uwith the gravitational lensing data, Elisa di Pietro, Don Paand Ioav Waga for helpful discussions about various aspof the paper, and the Australian Research Council for fincial support.s.s.
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