Dark Energy, A Cosmological Constant, and Type Ia Supernovae

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by  Lawrence M. Krauss

Dark Energy, A Cosmological Constant, and Type Ia Supernovae
Lawrence M. Krauss and Katherine Jones-Smith
CERCA, Case Western Reserve University, Cleveland OH 44106-7079
Dragan Huterer
Kavli Institute for Cosmological Physics and Department of Astronomy
and Astrophysics, University of Chicago, Chicago, IL 60637
Wefocusonuncertaintiesinsupernovameasurements,inparticularofindividualmagnitudesand
redshifts,toreviewtowhatextentsupernovaemeasurementsoftheexpansionhistoryoftheuniverse
are likely to allow us to constrain a possibly redshift-dependent equation of state of dark energy,
w(z). focus in particular on the central question of how well one might rule out the possibility of
a cosmological constant w = −1. We argue that it is unlikely that we will be able to significantly
7 reduce the uncertainty in the determination of w beyond its present bounds, without significant
0 improvements in our ability to measure the cosmic distance scale as a function of redshift. Thus,
0 unlessthedarkenergysignificantlydeviatesfromw(z)=−1atsomeredshift,verystringentcontrol
2 of the statistical and systematic errors will be necessary to have a realistic hope of empirically
distinguishing exotic dark energy from a cosmological constant.
n
a
J
I. INTRODUCTION unambigiously determine a deviation from −1, if such a
4
deviation indeed exists.
2
Variousproposalsfordynamicaldarkenergyhavebeen
Eight years ago two teams observing distant Type Ia
1 put forth which might produce such deviations. Most
supernovae (SNe Ia) announced evidence that the ex-
v notably, a scalar field rolling down its effective potential
pansion of the universe is speeding up [1, 2]. The dis-
2 can provide the necessary energy density and accelera-
9 tant supernovaeappear dimmer than they would be in a tion of the universe [12, 13, 14, 15, 16, 17]. Generically
6 matter-only universe. If this is a true distance effect, it
many of these possibilities are already ruled out by the
1 implies that about 70% of the energy density of the uni-
data. Among those that are not, none is particularly
0 verse reside in a smooth component with negative pres-
7 compelling as they also typically do not naturally ad-
sure,leavingonlyabout25%isindarkmatterand5%in
0 dress the problem why the observeddark energy density
/ baryonicmatter. Whilecompellingevidenceforprecisely (∼ 10−120M4 ) is so small, nor why dark energy starts
h thiscombinationwaspointedoutseveralyearsearlier[3], PL
to dominate the expansionof the universe only at recent
-p based on measurements of the clustering of galaxies,age times — redshift z <∼1. Nevertheless, before we can ad-
of the universe, measurements of the baryon and dark
o dresssuchpuzzlesweneedtoknowempiricallyifthedark
r matter densities, and the Hubble constant, SNe Ia were
t energyismeasurablydistinguishablefromacosmological
s the “shot heard around the world” as they provided ex-
constant.
a plicit evidence for the largest contributor to the energy
: Here we will be concerned with two specific observa-
v density of the universe. This constituent became known
tional factors, as well as one overriding theoretical con-
i as dark energy.
X straint. Observationally we need to determine both the
Since that time the constraints have improved signif- magnitude and redshift of individual objects in order to
r
a icantly: combination of cosmic microwave background map the universe’s expansion history. Theoretically we
experiments [4], large-scale structure surveys [5, 6] and, havetoaccountforthefactthatdarkenergyhas,apriori,
particularly,newSNeIaobservations[7,8,9,10,11],now no predetermined time dependence, and thus our analy-
constrain the dark energy equation of state, w ≡ p/ρ to ses must allow for arbitrary time variations.
be within about −1±0.1. Essentially all of the consequences of dark energy fol-
Unfortunately, our understanding of dark energy is as low from its effect on the expansion rate:
murky as ever. The simplest model for dark energy is
providedbythe cosmologicalconstantterminEinstein’s
8πG
equations, and is still an excellent fit to observations as H2 = (ρM +ρDE) (1)
3
it predicts w = 1 identically and at all times. However,
even if observations were to pin down the value of w to H2(z)/H02 = ΩM(1+z)3+
be −1, this gives us essentially no insight into the pos- z
′ ′
Ω exp 3 [1+w(z )]dln(1+z )
sible source of dark energy. While a cosmological con- DE
(cid:20) Z (cid:21)
0
stant may be the best bet, it is simple to imagine other
sources of dark energy, including the energy density as- whereΩ andΩ arethedarkmatteranddarkenergy
M DE
sociatedwithafalsevacuummetastablescalarfield,that density relative to critical, respectively, and we have ig-
would produce a similar value. The only way to get any nored the relativistic components. Type Ia supernovae
new theoretical handle on dark energy is to be able to effectively measure the luminosity distance
2
Many other simple parametrizations of the equation
of state have been suggested (e.g. [21, 22]); similar pro-
z dz′
d (z)=(1+z)r(z)= , (2) posalshavebeenextendedtotheHubbleparameter(e.g.
L Z0 H(z′) [23]). Finally,specificcombinationsoftheexpansionhis-
toryparameters,suchasthe“statefinder”[24],havebeen
wherer(z)isthecomovingdistanceandwehaveassumed
proposedasgooddiscriminatorsbetweenphenomenolog-
aflatuniverse. Sinceobservationsofsupernovaeallowus,
ical dark energy descriptions. It is well worth emphasiz-
at leastin principle, to map the expansionhistoryof the
ingthatalloftheaboveparametrizationsareadhoc,and
universe,onecanusethisdatatoconstrainthenatureof
may lead to biases as the true DE model may not follow
dark energy.
the form imposed by these functions. Their advantage,
Thispaperisorganizedasfollows. InSec.II,wereview
however, is in simplicity and the fact that two or three
a variety of parametrizations of the expansion history of
additional parameters in the dark energy sector will be,
the universe, and therefore ways to measure the prop-
inthe nearfuture, measuredto a goodaccuracy,atleast
erties of dark energy. In Sec. III we study the extent
when the data from the various cosmological probes is
to which an increased statistical error in supernova dis-
combined.
tances affects determination of the equation of state of
darkenergyandtestsofitsconsistencywiththevacuum
energy value of −1. We conclude in Sec. IV.
B. Direct reconstruction
II. MAPPING THE EXPANSION HISTORY Going in the opposite direction, the most general way
to probe the background evolution of dark energy has
beenproposedin[25,26,27]: theequationofthedistance
A. Key questions and parametrizations
vs redshift can be inverted to obtain w(z) as a function
ofthefirstandsecondderivativesofthe(SNeIa-inferred,
At the present time, it has become clear that there
for example) comoving distance
are two major goals that upcoming dark energy probes
should address:
1+z 3H2Ω (1+z)2+2(d2r/dz2)/(dr/dz)3
1. Is dark energy consistent with the vacuum energy 1+w(z)= 0 M
3 H2Ω (1+z)3−(dr/dz)−2
scenario – that is, is w(z)=−1? 0 M
(6)
2. Is the equationofstate w constantin time (or red- Similarly, assuming that dark energy is due to a single
shift)? rolling scalar field, one can reconstruct the potential of
the field exactly
Violationof either ofthese two hypotheses would be a
truly momentous discovery: the former would rule out a
pure cosmological constant, while the latter would pro- 1 3 d2r/dz2
V[φ(z)] = +(1+z)
vide further evidence for nature of of dark energy via its 8πG(cid:20)(dr/dz)2 (dr/dz)3(cid:21)
dynamics.
3Ω H2(1+z)3
With this in mind, the most obvious approach is to − M 0 (7)
16πG
parametrize the equation of state of dark energy as a
constant piece plus a redshift-varying one [18, 19, 20]
where the upper (lower) sign applies if φ˙ > 0 (< 0) The
signis arbitrary,as it can be changedby the field redefi-
′ nition, φ↔−φ.
w(z) = w +w z (3)
0
Direct reconstruction is the most general method for
w(z) = w0+waz/(1+z) (4) inferring the dark energy history, and it is the only ap-
w −w proach that is truly model-independent (despite some
f 0
w(z) = w + (5)
0 1+exp[(z−z )/∆] claims in literature to the contrary). However,direct re-
t
construction comes at a steep price — it calls for taking
where the first equation assumes linear evolution with the second derivative of the noisy data. In order to take
redshift, the second is linear with scale factor, and the the second derivative, one essentially must fit the lumi-
third allows for the transition between two asymptotic nosity distance (or SNa apparent magnitude) data with
constant values of the equation of state, with the transi- a smooth function — a polynomial, Pad´e approximant,
tionatredshiftz withthecharacteristicwidthinredshift spline with tension etc. Unfortunately, the parametric
t
∆. Equation4,inparticular,hasbecomecommonlyused natureofthefittingprocessintroducessystematicbiases.
to planprobes ofdarkenergyas itretains the minimally Aftervaliantattemptstodothisusingavarietyofmeth-
required two parameters and does not diverge at high odsforsmoothingorfittingthedata(e.g.[23,28,29,30]),
z, while still allowing for the low-redshift dynamics (e.g. various authors found that direct reconstruction is sim-
variation in the value of w(z)). plytoochallengingandnotrobustevenwithSNeIadata
3
FIG. 1: Example of the direct reconstruction, simulated for future SNa Ia data and assuming the true equation of state is
w(z) = −1. The left, middle and right panels show results when the luminosity distance data are fit with a third, fourth
and fifth order polynomial in redshift, respectively. Note that, depending on which order polynomial is used to fit the data,
significant bias, or statistical error, or both are introduced. Adoptedfrom Ref. [28].
ofexcellentquality. Figure1showsexampleofthedirect practice computed directly and here approximated with
reconstruction, simulated for future SNa Ia data and as- F). Therefore
suming the true equationof state is w(z)=−1(adopted F ≡C−1 =WTΛW (8)
from Ref. [28]). Note that, depending on which order
polynomial is used to fit the data, significant bias, or wherethematrixΛisdiagonalandrowsofthedecorrela-
statistical error, or both are introduced. For an excel- tion matrix W are the eigenvectors e (z), which define a
i
lentreviewofthedarkenergyreconstructionandrelated basisinwhichourparametersareuncorrelated[35]. The
issues, see [31]. original function can be expressed as
N
C. Principal components w(z)= α e (z) (9)
i i
Xi=1
The nextmostgeneralmethod, introducedin[32] (see where e are the “principal components”. Using the or-
i
also [33, 34]), is to compute the principal components of thonormality condition, the coefficients α can be com-
i
the quantity that we are measuring — the equation of puted as
state or energy density of dark energy. Principal com-
N
ponents are the redshift weights (or window functions)
α = w(z )e (z ). (10)
ofthe function in question, andare uncorrelatedby con- i a i a
struction. In this scheme, one simply lets data decide Xa=1
which weights of the function are measured best, and Diagonal elements of the matrix Λ, λi, are the eigen-
which ones are measured most poorly. The principal values which determine how well the parameters (in the
components form a natural basis that parameterizes the new basis) can be measured; σ(α ) = λ−1/2. We have
i i
measurements of any particular survey. ordered the α’s so that σ(α )≤σ(α )≤...≤σ(α ).
1 2 N
To compute the principal components, let us Principalcomponentshaveonedistinctadvantageover
parametrize w(z) (same arguments follow for ρ(z) or fixed parametrizations. They allow the data to deter-
H(z)) in terms of piecewise constant values w (i = mine which weight of the cosmologicalfunction w(z) (or
i
1,...,N), each defined in the redshift range z = [(i− ρ (z), or H(z)) is best determined. In fact, the PCs
i DE
1)∆z,i∆z] where ∆z =zmax/N. In the limit of large N depend on the cosmological probe, on the specifications
this recovers the shape of an arbitrary dark energy his- of the probe (redshift and sky coverage etc), and (more
tory (in practice, N >∼20 is sufficient). We then proceed weakly) on the true cosmological model. These depen-
to compute the covariancematrix forthe parametersw , dencies are features and not bugs, and they make the
i
plusanyothercosmologicalparameterssuchasΩ ,then principal components a useful tool in survey design. For
M
marginalizeoverthe latter. We then havethe covariance example, one can design a survey that is most sensi-
matrix, C, for the w . tive to the dark energy equation of state at some spe-
i
It is then a simple matter to find a basis in which the cificredshift,orstudyhowmanyindependentparameters
parameters are uncorrelated; this is achieved by simply are measured by any given combination of cosmological
diagonalizing the inverse covariance matrix (which is in probes (e.g. [22]).
4
0.5
50
1
1
49 0.5
3
0
)
e(zi0 (z)-0.5
w
4 -1
-1.5
2 -2
-0.5
0 0.5 1 1.5 -2.5
z 0 0.5 z1 1.5 2
FIG. 2: The four best-determined and two worst-determined
principalcomponentsofw(z)forafutureSNeIasurveysuch
as SNAP[36]. Adoptedfrom Ref. [32].
D. Uncorrelated estimates of DE evolution
A useful extension of the principal component formal-
ismistocomputethebandpowersinredshiftofthedark
energy function, w(z) (or ρ (z) or H(z)). These band
DE
powers can be made 100%uncorrelatedby construction,
and the price to pay is a small leakage in the sensitiv-
ity of each band power outside of its redshift range. For
details on how this is implemented, see [37].
FIG. 3: Top panel: constraints on the four band powers of
ThetoppanelofFigure3showstheconstraintsonthe
w(z),adoptedfromRef.[37],assumingtheRiess04SNadata
fourbandpowersofw(z),adoptedfromRef.[37],assum-
[7] and a prior on ΩM. Bottom panel: the same method
ing the Riess04 SNa data [7] and a prior on Ω . The
M appliedtoconstrainthreebandpowers,usingthenewestdata
bottom panel of the Figure shows the same method to
SNaIadatafrom theHubbleSpaceTelescope(adoptedfrom
constrain three band powers, using the newest data SNa Ref.[10])incombination withthebaryonacousticoscillation
Ia data from the Hubble Space Telescope, adopted from measurements [6].
Ref. [10], in combination with the baryon acoustic oscil-
lation measurements [6].. The slight preference for w(z)
increasing with redshift is seen in both panels; however, inferredusingthistechnique. Wenowturntoconsidering
significant systematics still affect the data, in particular this question in more detail
becauseoftheheterogeneityoftheSNaIadatasetsused.
Therefore,itistooearlytoclaimanyevidenceforthede-
parturesfromΛCDM,anditremainstobeseenhowthe III. LAMBDA OR NOT?
resultschangeoncewehaveasystematicallymorehomo-
geneous and statistically more powerful data set (for the
As stressed earlier, one of the most important out-
requirements on the SNa Ia systematics, see [38]).
standingquestionsincosmologyiswhetherdarkenergyis
It interesting to see in Fig. 3 how good the constraints consistentwithacosmologicalconstant. Thishypothesis
are, given that this is current data and that interesting canbetestedinavarietyofways. Thesimplestapproach
constraintsare obtained on3-4 band powers. A superior istocomputeasimplelikelihoodcomparisonbetweenthe
datasetwithanexcellentcontrolofthesystematics,such data and the vacuum energy model w(z)=−1. A much
as that expected from a dedicated space telescope such moresophisticated(butadmittedlylessrobust)approach
as SuperNova/Acceleration Probe (SNAP; [36]) will sig- would be to perform some type of reconstruction of the
nificantly improve the constraints, and also allow a finer energydensityρ (z)andcheckwhetherornotitiscon-
DE
resolution(i.e.morebandpowerparameters)inredshift. sistentwithaconstantvalue(orsimilarly,toreconstruct
Nevertheless,itisclearfromtheexistingsetthateither the equation of state w(z) and check whether it is con-
significantredshift evolutionin w orsignificantimprove- sistent with −1).
ment in the data will be required before either redshift Here we consider the Λ hypothesis test in terms of
evolution or deviation from −1 could be unambiguously principal components (PCs). We consider models in the
5
w -w plane (see Eq. (4)) and as we outline below, we Although the oft-claimed quoted magnitude uncertainty
0 a
usethe bestmeasuredPCstodetermine ifwecanstatis- perlow-redshiftsupernovais0.15wefoundthatafreefit
ticallydistinguishdistinguishthemodelfromΛCDM.In tothedatatendedtopreferaslightlylargervalue. Thus,
thissection,weassumefutureSNaIadatawith3000SNe for purposes of comparison, we considered both redshift
distributed uniformly in 0.1 < z < 1.7; this corresponds independent uncertainties of 0.15 or 0.21 magnitudes, as
roughly (but not exactly) to what is expected from the well as redshift dependent uncertainties 0.15+0.079z or
SNAP space telescope [36]. 0.21+0.079z mag.
Forafixeddarkenergymodeldescribedbysomevalues Figure4,showstheregionsinthew −w thatareruled
0 a
of the cosmological parameters w and w , let the prin- out at 68% C.L. (region outside of the innermost closed
0 a
cipal components take values α , with associated errors curve)95%C.L.(regionoutsideofthemiddlecurve)and
i
σ(α )(note, wemarginalizethe resultsoverthevaluesof 99.7%C.L. (region outside of the outermost curve, coin-
i
Ω ,thus enlargingthe errorsinthe PCs;inthis waywe ciding with the blue shaded region). In other words, the
M
cantalkaboutmodelsinthew -w planewithoutfurther blue region corresponds to dark energy scenarios where
0 a
recourse to Ω ). Further, from Eq. (10) it follows that the ΛCDM model can be ruled out at >3σ confidence.
M
the principal component coefficients for the w(z) = −1 In the four panels of Figure 4 we show cases when the
model are error per SNa is 0.15 or 0.21 magnitudes, and when it is
increasingwith redshiftas 0.15+0.079z or0.21+0.079z
mag. Clearly, increasing magnitude uncertainties with
N
redshift can significantly affect the size of the indistin-
α¯ =(−1) e (z ), (11)
i i a guishable parts of model space. Note too that the shape
Xa=1
of the region that cannot be distinguished from ΛCDM
is of characteristic shape, is elongated roughly along the
where e (z) is the shape of ith PC in redshift. We can
i
now perform a simple χ2 test to determine whether w is 7(w0 +1)+wa ≈ 0 direction — this is a direct conse-
quence of the fact that the physically relevant quantity
constant:
is w(z)=w +w z/(1+z).
0 a
M (α −α¯)2 One other way to get a handle on whether w 6= −1 is
χ2 = i i , (12) to explore the most precise constraint on w that might
σ2(α )
Xi=1 i be obtainable at any single redshift. For, if w 6= −1 at
any redshift, this will establish unambiguously that the
wherewehavechosentokeeponlythefirstM PCs,since
dark energy is not a cosmologicalconstant.
the best-determined modes will contribute the most to
To explore the sensitivity of SNe for this purpose we
thesum(notethatthetestisvalidregardlessofthevalue
parametrize dark energy equation of state as in Eq. (4),
of M; in practice,our M is typically 3-5). Given that we
and study the accuracy in the equation of state w(z) at
haveM degreesoffreedom,eachmodel(w ,w )willlead
0 a the best determined, or pivot, point. The pivot value is
toχ2 whichmayormaybe inconsistentwiththe ΛCDM
given by [29, 39]
model at some fixed confidence level.
In order to determine the ability of future SN surveys
to make this distinction, we focus here on the sensitiv- Cov(w ,w )
0 a
ityoftheresultstothemeasuredmagnitudeuncertainty. wpivot =w0− wa (13)
Cov(w ,w )
a a
Wefocusonthisfactorfortworeason. First,webelieveit
willbethesinglemostimportantdeterminantoftheabil-
whereCov(x,y)standsforelementsofthecovariancema-
ity of future surveys to possibly rule out a cosmological
trix element, while the pivot redshift at which w(z) is
constant, and second, because our examination of past
best determined is given by
surveys suggests that one should consider the possibil-
ityofredshfit-dependentmagnitudemeasurementerrors.
This latter possibility is not unexpected. It is systemat- Cov(w0,wa)
z =− . (14)
ically harder to determine the luminosity of ever fainter pivot Cov(w ,w )+Cov(w ,w )
0 a a a
and more distant objects.
Indeed, it is this possibility that originally motivated Other parameter we are the matter density relative to
[40] the current study. Measuring supernovae at ever critical,Ω andtheoffsetintheSNaIaHubblediagram,
M
higher redshift provide a useful lever-arm to distinguish M. Throughout we assume a flat universe.
between different equations of state, unless the magni- Using a Fisher matrix formalism, we estimate errors
tude uncertainty increases more quickly with redshift for a future SNAP-type survey [36] with 2800 SNe dis-
than the redshift-distance relation diverges for differing tributed in redshift out to z = 1.7 as given by [36],
valuesofthedarkenergyequationofstate. Forpurposes and combined with 300 local supernovae uniformly dis-
of this example analysis, we used 217 SN1a, from the tributed in the z =0.03−0.08range. We study how the
initial large SN surveys [1, 2], supplemented by several errors of the parameter of most interest, w , change
pivot
morerecentmeasurements,fittingthequotedmagnitude asthe individualSNa errorsvary. We againassume that
uncertainty as a function of redshift to a linear relation. the statistical error per SNa scales with redshift as
6
2 σ (z)=0.15 2 σ (z)=0.15+0.079z
m m
1 1
0 0
68% CL 68% CL
a -1 a -1
w w
95.4% CL
-2 -2
95.4% CL
99.7% CL
-3 -3
99.7% CL
-4 -4
ΛCDM Excluded at 99.7% CL ΛCDM Excluded at 99.7% CL
-5 -5
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
w w
0 0
2 σ (z)=0.21 2 σ (z)=0.21+0.079z
m m
1 1
0 0
68% CL 68% CL
a -1 a -1
w w
-2 -2
95.4% CL
95.4% CL
-3 -3
99.7% CL
-4 -4
ΛCDM Excluded at 99.7% CL ΛCDM Excluded at 99.7% CL 99.7% CL
-5 -5
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
w w
0 0
FIG.4: Regionsinthew0−wa thatareruledoutat68%C.L.(regionoutsideoftheinnermostclosedcurve)95%C.L.(region
outsideofthemiddlecurve)and99.7%C.L.(regionoutsideoftheoutermostcurve,coincidingwiththeblueshadedregion). In
otherwords,theblueregioncorrespondstodarkenergyscenarioswheretheΛCDMmodelcanberuledoutat>3σ confidence.
Thistest isperformed by“measuring” theprincipalcomponentsfrom each (w0,wa)modeland comparingthemtothosefrom
the ΛCDM model. We show cases when theerror per SNais 0.15 or 0.21 magnitudes (top and bottom left panels), and when
it is increasing with redshift as 0.15+0.079z or 0.21+0.079z mag (top and bottom right panels).
Forexample,a20%increaseintheSNaerrorbetweenthe
redshift of 0 and 1 (so that A=0.2σ (0)) will lead to a
m
σm(z)=σm(0)+Az, (15) 10% increase in error associated with the pivot value of
the equation of state.
where σ (0)=0.10,0.15or 0.20,andwe now let Avary
m
in order to ascertain sensitivity to this parameter
As can be seen in Fig. 5, the constraints on w are
pivot IV. CONCLUSIONS
dependentonbothσ (0)andA;wecanfittheconstraint
m
via an approximate relation
SNeIaarecurrentlythestrongestcosmologicalprobes
of dark energy, and are likely to remain the most solid
σ(w )≈0.17(2σ (0)+A). (16) source of information in the near future. This implies
pivot m
that control of the statistical errors is crucial. In par-
This equation shows that the slope of the statistical ticular, if we are ever to address the central question of
errorvs.redshiftrelation,A,contributestoσ(w )one cosmology,’Isthedarkenergyduetoacosmologicalcon-
pivot
halfasmuchastheinterceptofthesamerelation,σ (0). stant?’, we need to be able to unambiguously determine
m
7
also demonstrates, allowing for possible variations in w
0.12 with redshift implies to reduce the 95% confidence limit
assuming σ (z) = σ (0) + A z uncertainty in w(z) for a fit near w = −1 and w = 0
m m 0 a
0.1 significantly below 10 % in w0 will be challenging.
This latter point is even more important when con-
0.08
)
sideringwhether one mightutilize the measuredvalue of
ot
v w(z)atthepivotpointtoattempttodiscernsometimeat
pi 0.06
w which w 6= −1. Redshift-dependent uncertainties in su-
(
σ pernovamagnitudescaneasilyalmostdoubletheinferred
0.04
σ (0)=0.20 uncertainty in w at this point. Moreover, only if the
m
σ (0)=0.15 planned large scale SNe surveys can maintain a uniform
0.02 m
σ (0)=0.10 magnitudeuncertaintypersupernova,σ ,lessthan0.10
m m
canwehopetoderivea95%confidencelimituncertainty
0
0 0.02 0.04 0.06 0.08 0.1 in wpivot of less than about 10% (see Fig. 5) which itself
A may not be sufficient to distinguish some non-standard
dark energy models from a cosmologicalconstant.
FIG. 5: Constraints on the pivot value of the equation of
state, wpivot, as a function of the intercept and slope of the It is not clear whether resolving the nature of dark
magnitude error of SNe, σm(0) and A. We have assumed a energy is a 10-year or a 100-year problem. The answer
fiducial surveyof 2800 SNe. partly depends on how much information measurements
of dark energy can provide. Here we have addressed the
former problem by studying the requirements of mag-
that w 6= −1 at some, or all redshifts. The extent to nitude errors in a SNa Ia survey. Our results may be
whichSNe Iameasurementswillallowsuchadetermina- viewed as discouraging, or they may be viewed as in-
tion has been our prime concern in this analysis spiring for those observers who enjoy a demanding chal-
Figures 4 and 5 represent our primary results in this lenge. It is alreadybecoming clear that, inorderto have
regard. Figure4makesclearthatthemodel-independent hope of detecting a deviationfrom the cosmologicalcon-
constraints on w(z), for a two-parameter class of devia- stant scenario, we need nature to be kind by producing
tions aroundw(z)=−1 (the two parametersare w and a non-negligible deviation in w(z) at some redshift, and
0
w ; see Eq. (4)), are quite sensitive to the measurement we need to controlSNa magnitude errors,and systemat-
a
uncertainty in supernova magnitudes. Moreover, both ics in particular, to high precision. If either one of these
Figure4andFig.5demonstratethatitisparticularlyim- requirements is not met, may have to rely on theorists
portanttoattempttomaintaincontrolovermeasurement to understandthe nature ofdark energy,whichis option
uncertaintiesforhigherredshiftsupernovae. As Figure4 over which we may have much less control.
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95.4% CL constraints
2
1
0
-1
a
w
-2
σ
(z)=0.15
-3
m
σ
(z)=0.21
m
σ
-4 (z)=0.21+0.079z
m
-5
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
w
0