New Constraints from High Redshift Supernovae and Lensing Statistics upon Scalar Field Cosmologies

6 Pages·0.16 MB·English

by  Ioav Waga

New Constraints from High Redshift Supernovae and Lensing Statistics upon Scalar
Field Cosmologies
Ioav Waga1 and Joshua A. Frieman2,3
1Universidade Federal do Rio de Janeiro, Instituto de F´ısica
Rio de Janeiro, RJ, 21945-970, Brazil
2NASA/Fermilab Astrophysics Center,
Fermi National Accelerator Laboratory
PO Box 500, Batavia IL 60510, USA
3Department of Astronomy and Astrophysics
University of Chicago, Chicago, IL 60637 USA
(February 1, 2008)
0
0 sultsandstronglysupportaspatiallyflatcosmologywith
WeexploretheimplicationsofgravitationallylensedQSOs
0 Ω ∼0.3andadarkenergycomponentwithΩ ∼0.7.
m0 X
2 and high-redshift SNe Ia observations for spatially flat cos- Thesemodelsarealsotheoreticallyappealingsinceadark
n mological models in which a classically evolving scalar field energy component that is homogeneous on small scales
currently dominates the energy density of the Universe. We
a consider two representative scalar field potentials that give (20–30h−1 Mpc)reconcilesthespatialflatnesspredicted
J by inflation with the sub-critical value of Ω [6].
rise toeffectivedecayingΛ(“quintessence”) models: pseudo- m0
0 Nambu-Goldstone bosons (V(φ) = M4(1+cos(φ/f))) and The cosmological constant has been introduced sev-
2 aninversepower-lawpotential(V(φ)=M4+αφ−α). Weshow eral times in modern cosmology to reconcile theory with
1 that alarge region ofparameter spaceis consistent with cur- observations [10] and subsequently discarded when im-
proved data or interpretation showed it was not needed.
v rent data if Ωm0 > 0.15. On the other hand, a higher lower
However, it may be that the “genie” will now remain
4 bound for the matter density parameter suggested by large-
5 scale galaxy flows, Ωm0 > 0.3, considerably reduces the al- forever out of the bottle [9]. Although current cosmo-
3 lowed parameter space, forcing the scalar field behavior to logical observations favor a cosmological constant, there
1 approach that of a cosmological constant. is as yet no explanation why its value is 50 to 120 or-
0 ders ofmagnitude below the naive estimates ofquantum
0
fieldtheory. Oneoftheoriginalmotivationsforintroduc-
0
I. INTRODUCTION ing the idea of a dynamical Λ-term was to alleviate this
/
h problem. There are also observational motivations for
p considering dynamical-Λ as opposed to constant-Λ mod-
Recent observations of type Ia supernovae (SNe Ia)
-
o at high redshift suggest that the expansion of the Uni- els. Forinstance,theCOBE-normalizedamplitudeofthe
r verse is accelerating [1,2]: these calibrated ‘standard’ mass power spectrum is in generallowerin a dynamical-
t
s candles appear fainter than would be expected if the ex- Λ model than in a constant-Λ one, in accordance with
a observations [14]. Further, since distances are smaller
pansion were slowing due to gravity. While concerns
:
v about systematic errors (such as possible evolution of (for fixed z and Ωm0), constraints from the statistics of
i the source population and grey dust) remain, the cur- lensed QSOs are weaker in dynamical-Λ models [7,8,12].
X
rentevidenceindicatesthatthehigh-redshiftsupernovae
r
a appear fainter because, at fixed redshift, they are at
II. SCALAR FIELD MODELS
larger distances. According to the Friedmann equation,
a¨/a=−(4πG/3)(ρ+3p), acceleratedexpansionrequires
a dominant component with either negative energy den- AnumberofmodelswithadynamicalΛhavebeendis-
sity,whichisphysicallyinadmissible,oreffectivenegative cussed in the literature [17,12,16,18–20]. We report here
pressure. Dark energy, dynamical-Λ (dynamical vacuum new constraints from gravitational lensing statistics and
energy), or quintessence are different names that have high-zSNeIaontworepresentativescalarfieldpotentials
beenusedtodenotethiscomponent. Acosmologicalcon- that give rise to effective decaying Λ models: pseudo-
stant, with p =−ρ , is the simplest possibility. Nambu-Goldstonebosons(PNGB),withpotentialofthe
Λ Λ
Recentstudies incorporatingnewCMB data[3,4]con- form V(φ) = M4(1+cos(φ/f)), and inverse power-law
firm previous analyses suggesting a large value for the models,V(φ)=M4+αφ−α. Thesetwomodelsarechosen
total density parameter, Ω >0.4, and favor a nearly to be representative of the range of dynamical behav-
total
flat Universe (Ω = 1). A different set of observa- ior of scalar field ‘quintessence’ models. In the PNGB
total
tions [5] now unambiguously point to low values for the model, the scalarfield atearly times is frozenand there-
matter density parameter, Ω = 0.3±0.1. In combi- fore acts as a cosmological constant; at late times, the
m0
nation, these two results provide independent evidence fieldbecomesdynamical,eventuallyoscillatingaboutthe
for the conventional interpretation of the SNe Ia re- potentialminimum, andthe large-scaleequationofstate
1
approaches that of non-relativistic matter (p = 0). The sity decreases more slowly than the background energy
power-law model, on the other hand, exhibits “tracker” density, and the field eventually begins to dominate the
solutions [17,21]: at high redshift, the scalar field equa- dynamics of the expansion. If the field is on track dur-
tion of state is close to that of non-relativistic matter, ing the MDE, its effective adiabatic index is less than
and at late times it approaches that of the cosmological unity—its effective pressure p =(φ˙2/2)−V(φ) is nega-
φ
constant. tive. This condition by itself does not guarantee acceler-
Let us consider first the motivation for the PNGB ated expansion: the field must have sufficiently negative
model. All “quintessence” models involve a scalar field pressureandasufficiently largeenergydensitysuchthat
with ultra-low effective mass. In quantum field the- the total effective adiabatic index (of the field plus the
ory, such ultra-low-mass scalars are not generically nat- matter) is less than 2/3. Moreoever, for inverse power-
ural: radiativecorrectionsgeneratelargemassrenormal- law potentials, at late times Ω → 1, such that when
φ
izations at each order of perturbation theory. To in- the growing Ω starts to become appreciable, γ devi-
φ φ
corporate ultra-light scalars into particle physics, their ates from the above tracking value, decreasing toward
smallmassesshouldbeatleast‘technically’natural,that thevalueγ →0. Thus,evenifα>4,suchthatinitially
φ
is, protected by symmetries, such that when the small γ = γTR > 2/3 in the MDE, when the field begins
φ φ
masses are set to zero, they cannot be generated in any to dominate the energy density and γ decreases, the
φ
order of perturbation theory, owing to the restrictive Universe will enter a phase of accelerated expansion. If
symmetry. Pseudo-Nambu-Goldstone bosons (PNGBs) Ω and α are sufficiently small, this will happen before
m0
are the simplest way to have naturally ultra–low mass, the present time. For inverse power-law potentials, the
spin–0 particles. These models are characterized by two two conditions Ω ∼ 1 and the preponderance of the
φ0
massscales,aspontaneoussymmetrybreakingscalef (at field potential energy over its kinetic energy (the condi-
whichtheeffectiveLagrangianstillretainsthesymmetry) tion for negative pressure) imply M ∼ 1027αα+−412 eV and
and an explicit breaking scale M (at which the effec- φ ∼ m . Since φ ∼ m , quantum gravitational cor-
0 Pl 0 Pl
tive Lagrangiancontainsthe explicitsymmetry breaking rections to the potential may be important and could
term). In order to act approximately like a cosmologi- invalidate this picture [22].
cal constant at recent epochs with Ωφ ∼1, the potential In the very early Universe, in order to successfully
energy density should be of order the critical density, achieve tracking, the scalar field energy density must be
M4 ∼ 3H02m2Pl/8π, or M ≃ 3×10−3h1/2 eV. As usual smaller than the radiation energy density. If, in addi-
we set V = 0 at the minimum of the potential by the tion, ρ is smaller than the initial value of the tracking
φ
assumption that the fundamental vacuum energy of the energydensity,thefieldwillremainfrozenuntiltheyhave
Universe is zero – for reasons not yet understood. Fur- comparable magnitude; at that point, the field starts to
ther, since observations indicate an accelerated expan- follow the tracking solution. On the other hand, if ρ
φ
sion,atpresentthefieldkineticenergymustberelatively is larger than the initial value of the tracking energy
smallcomparedtoitspotentialenergy. Thisimpliesthat density, the field will enter a phase of kinetic energy
the motion of the field is still (nearly) overdamped, that domination (γ ∼ 2); this causes ρ to decrease rapidly
φ φ
is, p|V′′(φ0)| <∼ 3H0 = 5 × 10−33h eV, i.e., that the (ρφ ∝ a−6), overshooting the tracker solution [21]. Sub-
PNGB is ultra-light. The two conditions above imply sequently, as in the case above, the field is frozen and
that f ∼ mPl ≃ 1019 GeV. Note that M ∼ 10−3 eV later begins to follow the tracking solution when its en-
is close to the neutrino mass scale for the MSW solu- ergy density becomes comparable to the tracking energy
tion to the solar neutrino problem, and f ∼mPl ≃1019 density. In either case, there is always a phase before
GeV, the Planck scale. Since these scales have a plau- tracking during which the field is frozen. Consequently,
sible origin in particle physics models, we may have an an important variable is the value of the field energy
explanationforthe‘coincidence’thatthevacuumenergy density when it freezes. For instance, is it smaller or
isdynamicallyimportantatthepresentepoch[12,11,13]. larger than ρ , the mean energy density at the epoch
eq
Moreover,thesmallmassmφ ∼M2/f istechnicallynat- of radiation-matter equality? Did the field have time to
ural. completely achieve tracking or not? In fact, the exact
Next consider the inverse power-law model: this po- constraintsimposed by cosmologicaltests on the param-
tential gives rise to attractor (tracking) solutions. If eter space of this model depend upon this condition.
ρφ and ρB denote the mean scalar and dominant back- In a previous study [15], we numerically evolved the
ground (radiation or matter) densities, then if ρφ ≪ρB, scalar field equations of motion forward from the epoch
the following ‘tracker’ relationship is satisfied: ρTR ∼ ofmatter-radiationequality,assumingthefieldisinitially
φ
a3(γB−γφTR)ρB, where γφTR = γB α/(2 + α) < γB frozen, φ˙(teq)=0. In this case, depending on the values
ofαandΩ ,itmayhappenthatthefielddoesnothave
[17,21]. Here, a(t) is the cosmic scale factor, and γ = m0
B
timetoreachthetrackingsolutionbeforethepresent. In
(p +ρ )/ρ denotes the adiabatic index of the back-
B B B
general, if Ω is large, we observe that at the present
ground (γ = 4/3 during the radiation-dominated era m0
B
γ is still growingawayfrom its initial value γ =0. On
andγ =1duringthematter-dominatedepoch(MDE)). φ φ
B
the other hand, if Ω is sufficiently low, γ will reach
Ifthescalarfieldisinthetrackersolution,itsenergyden- m0 φ
2
a maximum value (not necessarily the tracking value) at 6
some point in the past and at the present time will be
decreasing to the value γ → 0. Here we follow a dif-
φ
ferent approach. In our numerical computation we now
5
starttheevolutionofthescalarfieldduringtheradiation
9955..44
dominated epoch and assume that it is on track early in
the evolution of the Universe.1 When ρ becomes non-
φ
negligible compared to the matter density, γφ starts to 4 qq00==00 6688..33
decrease toward zero. Recently, constraints from high- eV
G
z SNe Ia on power-law potentials with the field rolling 180
1
with this set of initial conditions were obtained by Po- f/
dariu and Ratra [23]. We complement their analysis by 3 6688..33 9955..44
includingthelensingconstraintsaswell. Inthenextsec-
tionweshowusingthescalarfieldequationsthatpresent 6688..33 ΩΩ==00..1155
data prefer low values of α. We also update and expand
2 9955..44 ΩΩ==00..33
the observational constraints on the PNGB models [15].
ΩΩ==00..55
III. OBSERVATIONAL CONSTRAINTS
0.00225 0.0025 0.00275 0.003 0.00325 0.0035 0.00375 0.004
M [h1/2 eV]
In the following we briefly outline our main assump-
tions for lensing and supernovae analysis. Our approach
for lensing statistics is based on Refs: [29,30] and is de- FIG.1. Contoursofconstantlikelihood(95.4%and68.3%)
scribed in more detail in [8]. To perform the statistical arising from lensing statistics (the region above and to the
analysisweconsiderdatafromtheHSTSnapshotsurvey right of the short dashed curves is excluded) and type Ia su-
(498highlyluminousquasars(HLQ)),theCramptonsur- pernovae(solidcurves)areshownforthePNGBmodel. Also
vey (43HLQ), the Yee survey(37 HLQ), the ESO/Liege shownarecontoursofconstantΩm0 andthelimitforpresent
survey (61 HLQ), the HST GO observations (17 HLQ), acceleration, q0 =0. Theshadedregion showstheparameter
the CFA survey (102 HLQ) , and the NOT survey (104 space allowed at 95% C.L. by the lensing, SNe, and cluster
observations.
HLQ) [24]. We consider a total of 862 (z > 1) highly
luminous opticalquasarsplus 5 lenses. The lens galaxies
For SIS, the total lensing optical depth can be
aremodeledassingularisothermalspheres(SIS),andwe
expressed analytically, τ(z ) = F (d (0,z )(1 +
consider lensing only by early-type galaxies, since they S 30 A S
are expected to dominate the lens population. We as- zS))3(cH0−1)−3, where zS is the source redshift,
sume a conserved comoving number density of lenses, dA(0,zS) is its angular diameter distance, and F =
n = n0(1 + z)3, and a Schechter form for the early 16π3n(cH0−1)3(σ∗/c)4Γ(1+α+4/γ) ≃ 0.026 measures
t,ywpiethganlax=yp0.o6p1u×la1ti0o−n2,hn30M=pcR−0∞3na∗nd(cid:0)LLα∗(cid:1)=α−ex1p.0(cid:0)−[2L8L]∗.(cid:1)WdLL∗e tahgeese[ff2e5c].tivWeneecsosr||roefctthteheleonpstiicnalpdreopdtuhcifnogr tmhuelteiffpelcetsimo-f
assume ∗that the luminosity satisfies the Faber-Jackson magnificationbiasandincludetheselectionfunctiondue
relation [26], L/L = (σ /σ )γ, with γ = 4. Since the to finite angular resolution and dynamic range [29,30,8].
∗ ∗
lensing optical depth dep||end|s| upon the fourth power of We assume a mean optical extinction of ∆m =0.5 mag,
as suggested by Falco et al. [31]: this makes the lensing
thevelocitydispersionofanL galaxy,acorrectestimate
∗ statistics for optically selected quasars consistent with
of this quantity is crucial for strong lensing calculations.
the results for radio sources, for which there is no ex-
The image angular separation is also very sensitive to
tinction. When applied to spatially flat cosmological
σ : largervelocitiesgiverisetolargerimageseparations.
∗ constant models, our approach yields the upper bounds
Isne||rvoeudrilmikaegliehoseopdaraantaiolynsiosfwtheetlaekneseidntqouaacsacorsunatndthaedoopbt- ΩΛ <∼ 0.76 (at 2σ) and ΩΛ <∼ 0.61 (at 1σ), with a best-
fit value of Ω ≃ 0.39. Recent statistical analyses using
the value σ =225 km/s, which gives the best fit to the Λ
∗ both HLQ and radio sources slightly tighten these con-
||
observed image separations [30]. straints on a cosmological constant [31]. A combined
(optical+radio) lensing analysis for dynamical-Λ models
isstillinprogress;qualitatively,weexpectthistotighten
the lensing constraints below by approximately 1σ.
1 In fact this is true only if α is not close to zero. The case For the SNe Ia analysis [15], we consider the latest
α= 0 is equivalent to a cosmological constant, and the field publisheddatafromtheHigh-zSupernovaeSearchTeam
remains frozen always. [1] [32]. We use the 27 low-redshift and 10 high-redshift
SNe Ia (including SN97ck) reported in Riess et al. [1]
3
and consider data with the MLCS [33,1]method applied Wenotethatthebulkofthe2σ-allowedparameterspace,
to the supernovae light curves. Following a procedure where the lensing and SNe contours are nearly vertical,
similarto thatdescribedinRiess et al. [1], wedetermine correspondstothescalarfieldbeingnearlyfrozen,i.e.,in
the cosmologicalparameters througha χ2 minimization, this region the model is degenerate with a cosmological
neglecting the unphysical region Ω <0. constant.
m0
In Fig. 2 we show the 95.4% and 68.3% C. L. limits
fromlensing(thickdashedcontours)andtheSNeIadata
(solidcurves)ontheparametersαandΩ oftheinverse
m0
power-law potential. The horizontal dotted line shows a
0.6 lowerbound on the matter density inferred from the dy-
ww00==--00..66 ww00==--00..55 namics of galaxy clusters, Ωm0 = 0.15. We also show
contours of the present equation of state w = γ − 1
0 0
0.5
(thin dotted curves) and the curve q = 0 (long dashed
0
curve). At 95.4% confidence, the SNe Ia and Ω con-
m0
straints require α < 5 and w < −0.5; the latter bound
0.4 0
m0 agreesroughlywiththeconstraintobtainedbyassuming
Ω
a time-independent equation of state [8], an approxima-
0.3 6688..33 qq00==00 tionsometimesusedfortheinversepower-lawmodel. We
alsoobservethatthelensingconstraintsonthemodelpa-
9955..44
rameters are weak, constraining only low values of Ω
m0
0.2 and α. We remark, however, that they are consistent
cclluusstteerrss with the SNe Ia constraints. We can tighten the con-
6688..33
6688..33 9955..44 straints on the equation of state if we consider a higher
0.1
9955..44 valueforthe Ω lowerbound. Forinstance,ifweadopt
m0
Ω > 0.3, as suggested in [34], we obtain w < −0.67
1 2 α3 4 5 6 m0 0
and α < 1.8. In both models, a larger lower bound on
Ω pushes the scalar field behavior toward that of the
m0
cosmologicalconstant (w =−1).
FIG.2. Contoursofconstantlikelihood(95.4%and68.3%)
arising from lensing statistics (the region below the thick
IV. CONCLUSION
dashed curves is excluded) and type Ia supernovae (solid
curves) are shown for the inverse power-law model. Also
shown is the lower bound Ωm0 = 0.15 from clusters and A consensus is beginning to emerge that we live in a
curves of constant present equation of state w0 = pφ0/ρφ0. nearly flat, low-matter-density Universe with Ωm0 ∼0.3
Theshadedregionshowstheparameterspaceallowedat95% and a dark energy, negative-pressure component with
C.L. bythe lensing, SNe, and cluster observations. Ω ∼ 0.7. The nature of this dark energy component
X
is still not well understood; further developments will
In Fig. 1 we show the 95.4% and 68.3% C. L. limits require deeper understanding of fundamental physics as
fromlensing(shortdashedcontours)andtheSNeIadata wellasimprovedobservationalteststomeasuretheequa-
(solid curves) on the parameters f and M of the PNGB tion of state at recent epochs, w(t), and determine if it
potential. As in [15], these limits apply to models with is distinguishable fromthat of the cosmologicalconstant
the initial condition 4√πφ(ti) = 1.5 and dφ(t )= 0, with [35]. Classical scalar field models provide a simple dy-
t = 10 5t ; for othermPchloices, the boudntdinig contours namical framework for posing these questions. In this
i − 0
paper we analyzed two representative scalar field mod-
would shift by small amounts in the f −M plane. We
els, the PNGB and power-law potentials, which span
also plot some contours of constant Ω (dashed) and
m0
the range of expected dynamical behavior. The inverse
the curve q = 0 (long dashed contour) as a function of
0
power-law model displays tracking solutions [21] which
the parameters f and M. The allowed region(shown by
allow the scalar field to start from a wide set of initial
the shaded area in Fig. 1) is limited by the lensing and
conditions. We showed that current data favors a small
SNe Ia 95.4% C. L. contours and also by the constraint
value of the parameter, α < 5. This may be a problem
Ω >0.15, which we interpret as 2σ lower bound from
m0
for these models: in Refs: [21] it was shown that, start-
observations of galaxy clusters. The data clearly favors
ing from the equipartition condition after inflation, it is
acceleratedexpansion(theregionabovetheq =0curve)
0
necessary to have α > 5 for the field to begin tracking
but curiously there is a small region in the parameter
beforematter-radiationequality. Sincethe observational
space, close to the point where the Ω = 0.15 and the
m0
Sne Ia 2σ curves cross, where the Universe is not in ac- constraintsindicate thattrackingcouldonlybe achieved
(if at all) at more recent times, it is not clear what the-
celeratedexpansionby the presenttime. This smallarea
oretical advantage, in terms of alleviating the ‘cosmic
disappears if we adopt the tighter constraint Ω >0.3.
m0
4
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