Precision cosmological measurements: independent evidence for dark energy

0.12 MB·English

by  Greg Bothun

Precision cosmological measurements: independent evidence for dark energy
Greg Bothun,1,∗ Stephen D. H. Hsu,1,2,† and Brian Murray1,2,‡
1Department of Physics, University of Oregon, Eugene, OR 97403
2Institute of Theoretical Science, University of Oregon, Eugene, OR 97403-5203
Using recent precision measurements of cosmological paramters, we re-examine whether these
observations alone, independent of type Ia supernova surveys, are sufficient to imply the existence
ofdarkenergy. Wefindthatbestmeasurementsoftheageoftheuniverset0,theHubbleparameter
H0 and the matter fraction Ωm strongly favor an equation of state defined by (w < −1/3). This
resultisconsistentwiththeexistenceofarepulsive,acceleration-causingcomponentofenergyifthe
universeis nearly flat.
The current era in cosmology seems to be the first in collaborations [6], it is still possible that dust [7], evolu-
8
which local astrophysical measurements are consistent tion effects [8] or exotic particle physics [9] might alter
0
with the generally accepted large scale cosmology. To the interpretation of the extracted redshift-distance re-
0
2 providesomehistoricalcontext,considertheperiodfrom lation. For example, the axion models in [9] account for
1980 to roughly 1995. Inflation offered us a large scale the dimness of distant supernovae by conversion of pho-
n
model for cosmology, requiring Ω = 1, which could tons into axions in background galactic magnetic fields,
a total
J not find verification in measurements on smaller scales. rather than through accelerated expansion. Exotic par-

4 Attempts to dynamically determine Ωtotal (e.g., [1, 2]) ticle physics models which are less well motivated than
consistentlyreturnedresultsofΩ ∼0.25±0.10. This axions, but perhaps no more counterintuitive than the
total
2 led to the notion [3] that, under the Ω = 1 prior, existenceofdarkenergyitself, mightinprinciple explain
total
v
there must be a bias between the distribution of light thesupernovadatawithoutrequiringacceleration. How-
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(e.g. galaxies) and mass (e.g. the dark matter compo- ever, the demonstration that a dominant component of
0
1 nent). Not only did the Univese have to be dark mat- energy with w ≡ p/ρ < −1/3 is strongly favored by the
2 ter dominated, the distribution of that dark matter had observed values of cosmological parameters provides a
1 to be signficantly different than the the distribution of direct and robust argument for acceleration.
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light. Atthe time, this wasthe onlywaytoreconcilethe We seek evidence for a component whichhas equation
0
small scale measurements with the large scale (inflation) of state w≡p/ρ<−1/3. Recall the Einstein equation
/
h requirement.
p R¨ 4πG
- Inthisnotewereinvestigatewhetherrecentdetermina- =− (ρi+3pi) . (1)
o tions of cosmologicalparameters are sufficient, by them- R 3 Xi
r
st selves, to imply the existence of dark energy – specif- The sign of the acceleration R¨ is determined by the sign
a ically, a component of energy with equation of state of (ρ +3p ), where the sum runs over all contribu-
: w ≡ p/ρ < −1/3. In the mid-90’s several authors [4, 5] i i i
v tionPstotheenergymomentumtensor. Strictlyspeaking,
i analyzed aggregate data based on globular cluster ages, w <−1/3 is the threshold for a component to cause ac-
X
clustering of galaxies, big bang nucleosynthesis, and the
celeration when it is the only form of energy. If other
r Hubbleconstantandconcludedthatsomethinglikeacos-
a forms of energy are non-negligible the overallsign of the
mological constant might be necessary to produce a flat
right hand side of (1) might still be negative (i.e., the
Universe. However, the conclusions were not definitive
universe is decelerating, albeit more slowly than other-
at the time due to the large uncertainty in the observa- wise) even in the presence of energy with w < −1/3.
tionalparameters. Ourpurposeistoupdatetheseearlier
Asymptotically, though, the component with the small-
investigations,accountingforimprovementsinprecision.
est positive or most negative value of w will eventually
We will argue that observations of key parameters such
dominate all others. We recall that a cosmological con-
as the age of the universe t0, the Hubble parameter H0 stant has w = −1, while a dynamical scalar model with
and the matter fraction Ωm have become definitive in non-zero vacuum energy typically has −1<w <0. Val-
support of dark energy. One might question the need ues of w less than −1 violate the null energy condition,
for this analysis in the post-WMAP era, but it is impor-
and are generally associated with instabilities [10].
tanttounderstandwhetherincreasinglyprecisemeasure-
Analysis of the 3 year WMAP data [11] favorsa nega-
mentsareconsistentwiththeconcordancecosmologyob-
tive pressure equation of state for models with constant
tained from best fits of WMAP data. Indeed, given the
w when constraints on the matter energy density are in-
dramatic nature and consequences of dark energy, it is
cluded (i.e., from the Sloan Digital Sky Survey or the
important to understand the observational evidence for
2dF Galaxy Redshift Survey). In this note we conduct a
it as broadly and robustly as possible.
simpler analysis in which the priors are transparent and
DespitetheimpressiveresultsofthetypeIasupernova easy to state.
2
We find that best measurements of the age of the measurements using different techniques, each with dif-
universe t , the Hubble parameter H and the matter ferent statistical and systematic errors, is challenging.
0 0
fraction Ω are sufficient to require the existence, dur- However,ourdiscussionatleastallowsareasonableguess
m
ing some cosmologicallysignificantepoch, of a repulsive, at current global best values and uncertainties for these
acceleration-causing (w < −1/3) component of energy, quantities. Examples of more sophisticated Bayesian
assuming the universe is nearly flat. A relation between analysis are given in [12].
these quantities is obtained using Einstein’s equationfor t : Our approach is made possible by relatively re-
0
a Friedmann-Robertson-Walker universe. The analysis cent measurements of t with unprecedented accuracy.
0
itself is not necessarily new, but it can now be applied In the past, estimates of t have been made by either
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for the very first time with stringent constraints due to using model-dependent estimates for the ages of globu-
recent precision measurements of the relevant cosmolog- lar clusters or through nuclear cosmochronometry. The
ical parameters. former method has traditionally suffered from the un-
The age of the universe is given by known role of convection and its effects on the lifetimes
of low mass/low metallicity stars. Krauss and Chaboyer
R(t0) dR
[13] performed a thoroughMonte Carlo analysis that in-
t = (2)
0 Z0 R˙ cludes these uncertainties, to arrive at a firm lower limit
of 11.2 Gyrs for t . However, t as large as 15 Gyrs is
which yields 0 0
still allowable. Using Thorium cosmochronometry, Sne-
1 dx den and Cowan [14] also find a lower limit of 11 Gyrs
t H = , (3)
0 0 Z (Ω x−1+Ω x−1−3w)1/2 for t0 but acknowledge that lower limit could range up-
0 m de
wards by another 3-4 Gyrs. For the reasons cited, we
wherewehavetakenwconstantintimeandneglectedthe donotusethesemethodsorobservationsinconstruction
radiation component as it is numerically small. We also our argument for the most probable value of t .
0
assume flatness,whichimplies Ωde =1−Ωm, andallows Improvements in the precision of measuring t0 have
us to define the integral as I(Ωm,w). The quantities t0, utilized the white dwarfcooling curve andHubble Space
H0 and Ωm then determine w. Telescopemeasurementsofthe haloglobularclusterM4.
In the more general case, where the dark energy com- MeasurementsbyHansenetal. (2002)[15]reportavalue
ponent has time varying equation of state w(t), the sec- of 12.7±0.7 Gyr. Hansen et al. (2004) [16] update this
ond term in the denominator of the integral in (3) (the age to 12.1±0.9 Gyr. The major source of systematic
dark energy term) is more complicated, having the form errorinthisanalysisinvolvesestimatingthe lagtime be-
tweenthe ageof the Universe andthe formationofglob-
1 dx′
Ω exp (1+3w(x′)) . (4) ular clusters. Numerical simulations of the Milky Way
de (cid:20)Z x′ (cid:21)
x and its globular cluster system by Kravtsov and Gnedin
If (1 + 3w(x′)) > 0 for all x < x′ < 1, the dark en- (2005) [17] indicate that the peak formation of Globular
Clusters occurs atz = 3-5. Using a meanformationred-
ergyterm(4)isalwaysdecreasingwithincreasingx, and
shift of z = 4 implies that Globular Clusters formed at
the denominator in (3) is larger for all x than it would
be in the special case w = −1/3, where (4) is constant. 1.2 Gyr after the onset of the Big Bang. This then leads
Therefore, if the dark energy never exhibits a repulsive to a lower limit of t0 = 12.4 Gyr and a mean value of
equation of state, so w(t) > −1/3 at all times, the inte- t0 =13.3+−1.9.1 Gyr.
gral is bounded above:
H : For decades, measurements of H were plagued
0 0
I(Ωm,w >−1/3) < I(Ωm,−1/3) . (5) by noise and biased samples. Today, however, there is
good reason to believe that we have a relatively precise
Similarly, we deduce
measure for this parameter as well. The Hubble Space
I(Ω ,w>w∗) < I(Ω ,w∗) . (6) Telescope Key Projectfor determining the CepheidZero
m m
Point and subsequent distance determinations to nearby
In other words,in the mostgeneralcase,unless the dark galaxies using the Cepheid Period-luminosity relation-
energy behaved repulsively during some earlier epoch, shiphavereturnedavalueof72±3km/s/Mpc[18]. The
the integral I, and hence the product t H , is bounded major source of systematic uncertainty in that measure-
0 0
above by I(Ω ,−1/3). Using measured values of t , H ment lies in the distance to the Large Magellanic Cloud
m 0 0
andΩ ,itisthereforepossibletodeducethatarepulsive (LMC), to which the zeropoint of the Cepheid Luminos-
m
epochmusthaveoccurred. (Noteanepochwithrepulsive ity scale is anchored. Freedman and Madore [19] quote
energydoesnotnecessarilyimplyoverallacceleration,as atotalsystematicerrorof±7km/s/Mpc,butrecentim-
discussed.) proved distance estimates for the LMC (e.g., Benedict
We now review the best measurements of t , H and et al. (2002) [20] and Sebo et al. [21]) have served to
0 0
Ω . Systematically combining the results of distinct lower this systematic error down to ±4 km/s/Mpc (see
m
3
NgeowandKangur(2006)[22]). Moreover,confidencein from analysis of the power spectrum of galaxy cluster-
the precisionofH ,asanchoredbythe LMC distance,is ing. Assuming a flatUniverse, Sanchez et al. (2006)[28]
0
reinforced by recent measurements that are completely find Ω =0.237±0.02. In addition, Mohayee and Tully
m
independent of the distance to the LMC. In the past, (2005)[29]revisitthepeculiarvelocitiesofgalaxiesinthe
thesekindsofmeasurementswerealsoavailablebutthey LocalSuperclustertoderiveΩ =0.22±0.02. Schindler
m
had sufficiently large random error that precluded them (2002) [30] summarizes all techniques to determine Ω
m
from providing meaninful constraints on the value of H (includingthemoreunreliableapproachessuchastheX-
0
asdeterminedfromtraditionaldistancescaleladdertech- ray cluster luminosity function, weak gravitational lens-
niques. The new observations are: ing,orgalaxycluster evolution). Thatsummaryyields a
1) Using a sample of 38 X-ray clusters in combina- modalvalueofΩ =0.3(whichislikelyarealisticupper
m
tionwiththe Sunyaev-Zeldovicheffect, Bonamenteetal. limit giventhe WMAP model) but alsoshows that most
(2006) [23] derive a value of H = 77.6±5 km/s/Mpc. large scale structure studies yield values of Ω in the
0 m
Whiletheremaybesystematicsassociatedwiththenon- range0.20-0.25(whichisconsistentwiththe workdone
spherical shape of clusters, their sample may is suffi- inthe 1980s). Averagingtogetherthe Sanchezetal. and
ciently large (and much larger than past samples) that Mohayee and Tully studies produces a well constrained
this problem is removed by averaging. valueofΩ =0.23± 0.02. Fordiscussionbelowwetake
m
2) Wang et al. (2006) [24] have examined a sample aconservativelylargerangeforΩ ,assuming0.15−0.25
m
of 109 SN of type Ia and have discoveredimportant new to be a one standard deviation range about the central
corrections for metallicity and absorption (by dust) in value.
determining SN Ia peak luminosity. This recalibration
leads to H0 =72±6 km/s/Mpc. An independent treat- Results: InFig. 1,weplotI(Ωm,w)forΩm =.15,.20
ment of SN Ia has been compiled by Riess et al. (2005) and .25. Ω = .15 corresponds to the curve with the
m
[25] which yields a value of H0 =73±4 km/s/Mpc with largest values of t0H0. Taking t0 = 12.4 Gyr and H0 =
possible systematic error of ±5 km/s/Mpc. 69 km/s/Mpc, which are each one standard deviation
3)Koopmansetal. (2003)[26]performadetailedanal- below the favored (central) values in our assumed error
ysis of a gravitational lens system (from which a direct model, we obtain t H = .9, which corresponds to the
0 0
determination of the distance can be determined using a grey horizontal line in the figure. The implications can
modelmassdistributionofthelens)tofindH =75±6.5 be readdirectly from the figure. If w was alwaysgreater
0
km/s/Mpc. than −1/3, then some or all of our parameters must be
Averaging these 5 different results together formally well below their central values.
leads to 74±2.5 km/s/Mpc (error in the mean). Direct From Fig. 1, we see that taking t , H and Ω to
0 0 m
averaging is crude, but gives a characterization of the eachbe one standarddeviationbelow their centralvalue
uncertainty. Averaging over systematic errors as well, (so, t = 12.4 Gyr, H = 69 km/s/Mpc and Ω = .15),
0 0 m
we assume H0 = 74±5 km/s/Mpc in further analysis. an epoch with w < −.4 or so is required, which is just
Incontrast,onecoulduseonlymethod1and3above(as negative enough to imply acceleration (R¨ > 0). Taking
they completely circumvent the LMC distance problem) t = 12.4 Gyr and Ω = .15, one would have to, e.g.,
0 m
to obtain 76 ± 6 as the relevant range. pushH below67km/s/Mpctohavew >−1/3,andbe-
0
low 50 km/s/Mpc to have w >0 (no negative pressure).
Ω : In contrast, Ω remains the most weakly con- We compute the likelihood of no epoch with w < w∗
m m
strained cosmological observable. There are two reli- (for given w∗) as follows. First, we assume uncorrelated
ablemethodsofmeasurement: dynamicaldeterminations Gaussian errors in all three parameters: t = 13.3±1
0
based on infall to clusters of galaxies and/or the nature Gyr, H =74±5 km/s/Mpc and Ω =.2±.05 (all one
0 m
of large scale structure (e.g., Bothun et al. [27]) or by standarddeviation). That is, we assume that the proba-
fitting the Hubble diagram to distant objects. In the bility distribution for the actualvalue each of parameter
first case, an unbiased and fairly large sample is needed is normal, with maximum at the central value and stan-
for precision; in the second case, accurate distance mea- darddeviationgivenbytheerrorestimate. Wethencom-
surements of intermediate redshift galaxies are required, pute, for a particular value of w∗, the total probability
and such measurements are ultimately based on the su- that the parameters take on values for which inequality
pernova luminosity scale. In principle, Ω is highly con- (6) is satisfied. In practice, this was done using Monte
m
strained by the multi-parameter maximum likelihood fit Carlo.
totheWMAPdata;butthisisanindirectdetermination The results are displayed in Fig. 2 (top curve). Using
ofΩ (aswellast )Inthespiritofthisanalysis,weseek this error model the probability of no epoch with w <
m 0
to use values of Ω that have been directly determined. −1/3 is less than 4 percent. This is an overestimate of
m
Note, though, that Ω is now usually determined by the likelihood, since the model allows values of, e.g., t
m 0
assuming a flat Universe as a prior constraint. For in- whicharemuchtoolow: t =12.4Gyris moreplausibly
0
stance, a recent accurate determination of Ω results interpreted a strict minimum than minus one standard
m
4
deviation from the central value. Modifying the error Pr(w >w∗)
model so that values of t < 12.4 Gyr are not allowed 0.175
0
reduces the likelihood of no epoch with w < −1/3 to 0.15
about 1.3 percent. This is represented by the middle
0.125
curve in Fig. 2. Adding a similar constraint that Ω >
m
.15leadstothelowestcurveinthefigure,andalikelihood 0.1
of no epoch with w <−1/3 of about 0.8 percent. Fig. 3
0.075
is identical to Fig. 2 except that we have increased the
one standard deviation error for H to ± 7 km/s/Mpc; 0.05
0
the existence of dark energy is still strongly favored.
0.025
We conclude that, unless systematic errors are signif-
w∗
icantly larger than currently recognized, best measure- -0.4 -0.3 -0.2 -0.1
mentsoftheageoftheuniverset ,theHubbleparameter
0
H and the matter fraction Ω strongly favor the exis- FIG.3: SameasFig. 2exceptwithlargerHubbleuncertainty:
ten0ce of a repulsive dominantmenergy component, also H0=74±7 km/s/Mpc.
known as dark energy. These observations are indepen-
dent of type Ia supernova surveys: specifically, they are
ACKNOWLEDGEMENTS
not sensitive to uncertainties [7, 8, 9] which affect the
direct measurement of the distance-redshift relation at
large z. TheauthorsthankP.Corasanitiforclarificationofhis
work in [7]. S. H. and B. M. are supported by the De-
t H
0 0 partment of Energy under DE-FG02-96ER40969.
1.2
1.1
∗ Electronic address: [email protected]
w∗ † Electronic address: [email protected]
-1 -0.8 -0.6 -0.4 -0.2 ‡ Electronic address: [email protected]
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