Non-parametric Dark Energy Degeneracies

0.96 MB·English

by  Renée Hlozek

Non-parametric Dark Energy Degeneracies
Ren´ee Hlozek1,3, Marina Cortˆes1,2, Chris Clarkson1 and Bruce Bassett1,3
1Cosmology & Gravity Group, Department Mathematics and Applied
Mathematics, University of Cape Town, Rondebosch 7701, South Africa.
2Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom
3 South African Astronomical Observatory, Observatory, Cape Town, South Africa
Westudythedegeneraciesbetweendarkenergydynamics,darkmatterandcurvatureusinganon-
parametricandnon-perturbativeapproach. Thisallowsustoexaminetheknock-onbiasinducedin
thereconstructed dark energy equation ofstate, w(z),when thereis abias inthecosmic curvature
or dark matter content, without relying on any specific parameterisation of w. Even assuming
perfect Hubble, distance and volumemeasurements, we show that for z >1, the bias in w(z) is up
8
to two orders of magnitude larger than the corresponding errors in Ωk or Ωm. This highlights the
0
importance of obtaining unbiased estimators of all cosmic parameters in the hunt for dark energy
0
dynamics.
2

n
a I. INTRODUCTION mic volume with redshift, dV/dz. Secondly we discuss
J the w(z) that would be incorrectly reconstructed from

4 perfect Hubble, distance and volume data if the wrong
Since1998[18,19]evidencehasbeenmountinginsup-
2 value of Ω were used. We assume perfect data for all
portof anacceleratedexpansionof the Universe. Nearly m

observations,whichallowsus toprobefundamental,“in-
] 10yearson,thepuzzleoftheoriginofthisacceleration–
h dubbeddarkenergy–remainsoneofthemostintriguing principle” degeneracies that are not due to finite errors
p andincompleteredshift-coverage. Thisimpliesthatgiven
enigmas in modern day science. Much activity has come
- a specific bias in a cosmological parameter, the degen-
o fromboththetheoreticalandobservationalsectorsofthe
eracies will be true no matter what progress is made in
r physics community in anattempt to pin down its origin.
t improving future cosmic surveys. Furthermore, the key
s The current drive in dark energy studies is focused on
a pointinourreconstructionofw(z)isthatitisperformed
trying to establishits dynamicalbehaviouras a function
[ inafullynon-parametricmanner,andsodoesnotrelyon
ofredshift,w(z). Whilethesimplestexplanationremains
1 aΛCDMuniversewithw = 1forallredshift,dynamics the validity of any particular parameterisation of w(z).
v inw(z)wouldprovideawind−owintonewphysics. There- To illustrate the power of this non-parametricapproach,
7 wecompareourmethodwithastandardequationofstate
fore,uncoveringthedynamicsofdarkenergyasdescribed
4 parameterisation [25, 26], which cannot fully resolve the
by the ratio of its pressure to density, w(z)=p /ρ ,
8 DE DE
above degeneracies.
hasbecome the focusofmulti-billiondollarproposedex-
3
. periments using a wide variety of methods, with several
1
plannedsurveysatredshiftsaboveunity,ashigh-redshift
0
measurementsareusefultoconstraindarkenergyparam- A. Degeneracies in Dark Energy Studies
8
0 etersandtestfordeviationfromtheconcordanceΛCDM
: model(seee.g. [1]). Unfortunatelythesearchfordynam- The success of the inflationary scenario for the early
v
ical behaviour in w is a mani-fold problem. The nature Universe and its standard prediction of flatness to high
i
X ofdarkenergyiselusive: cosmicobservationsdependnot precision (Ω < 10−10) is perhaps the main reason why
k
r onlyondarkenergybutalsoonothercosmicparameters curvature has traditionally been left out in analyses of
a such as the cosmic curvature, Ωk, and the total matter dark energy. However, possible scenarios in which infla-
content, Ωm, leading to degeneracies between these and tion is consistent with non-zero spatial curvature have
w(z)parameters,anissuewhichhasrecentlybeenunder recently been investigated [20]. It is also interesting to
intense scrutiny by the community [2, 3, 13, 16]. Kunz note that the backreaction of cosmological fluctuations
[3] argues that observations are only sensitive to the full may cause effective non-zero curvature that may yield
energy-momentum tensor and thus cannot see beyond practicallimitsonourabilitytomeasurew(z)accurately
a combination of the “dark component” – dark matter at z > 1 (see e.g. [5]). Since measurements of the Cos-
plus darkenergy. The degeneracybetween the geometry mic Microwave Background (CMB) have so far proved
of the universe and the equation of state of dark energy consistent with flatness (e.g. [17]) statistical quantities
has also been discussed in light of the well-known result that measure the necessity of introducing extra param-
that a cosmological constant in the presence of spatial eters (such as Bayesian Evidence or information criteria
curvature can mimic a dynamical dark energy [13]. [21,22, 28])donotfavourthe inclusionofcurvatureasa
In this work we review current constraints on cosmic parameterincurrentanalyses[23]. Howeverthis is more
curvature and extend the approach in [13] in two ways. asymptomoftheinabilityofcurrentdatatoconstrainan
First we include the reconstruction of w(z) that would extraparameterthanaconclusivecaseforaflatuniverse.
follow from measurements of the rate of change of cos- Further,Bayesianevidenceorinformationcriteriadonot
2
takeintoaccountthepowerofthebiasesthatmaybein-
troducedbyfalselyneglectingaparameter. Wewillshow
–0.8
below that the biases introduced in neglecting curvature
are very significant at z >1.
In general constraints on curvature are very fragile to
assumptionsaboutthedarkenergysincetheyareprimar-
–1.0
ilyderivedfromdistancemeasurements(d ord )which
L A
are completely degenerate with curvature [12]. One way
to illustrate the degeneracy between curvature and dy-
namics is as follows. Let us assume that we know all –1.2
cosmicparametersperfectlyotherthanthecurvatureΩ
k
andthe darkenergyequationofstate,w(z). Atanyred-
shift, z∗, a perfect measurement of dL(z∗) (or dA(z∗))
allowsus to measure a single quantity. If we knoww(z∗) –1.4
then that quantity can be Ω . However, if w(z) is truly
k –0.08 –0.06 –0.04 –0.02 0.00 0.02 0.04
a free function, then its value at z∗ is completely free
and we are left trying to find two numbers from a single
observation, which is impossible.
FIG. 1: The curvature-dark energy degeneracy Con-
Only when we start to correlate the values of w(z) at
tours showing the 2D marginalized contours for w and Ωk
different redshifts canwe begin to use distance measure-
based on combined data from WMAP3, 2dFGRS, SDSS and
ments alone to constrain the curvature. The standard supernova surveys. While the slope of the degeneracy differs
waytodo this is to assumethatw(z) canbe compressed for this combination of data, the sign of the degeneracy is
ontoafinite-dimensionalsubspacedescribedbynparam- consistent with the w0 term in Eqs. (18),(19). Taken from
eters, e.g. [17].
w(z)=Σnw zj (1)
j j
Inthiscaseperfectdistancemeasurementsatn+1differ- restriction the resulting constraints on Ω would be in-
k
entredshiftswillallowacompletesolutionoftheproblem dependent of the precise choice of the n parameters, i.e.
and will yield the w and Ω . The most extreme version independent of the parameterisation. Unfortunately a
j k
ofthisistoassumeΛCDM,w(z)= 1. Withinthiscon- little thought makes it clear that this cannot be true. A
−
text it is of course possible to derive very stringent con- parameterisation of w(z) which does not allow mimicry
straints on the curvature. For example, combining the of curvature will provide good, decorrelated constraints
WMAP 3 year data and the SDSS DR5 Luminous Red onthecurvature(whichdoesnotmeanthecorresponding
Galaxy (LRG) sample leads to Ω = 0.003 0.010 as- best-fitwillbeagoodfittothedata)whileamodelwhich
k
− ±
suming w = 1[9]. The additionofextra data is crucial allows perfect mimicry of the dynamics of the curvature
sincetheWM−APdataaloneprovidesonlytheconstraint (i.e. (1+z)2)willshowhighlycorrelatedconstraints(al-
Ωk = 0.3040+0.4067ΩΛ [17]. though for negative Ωk mimicry of distance data is only
−
It is a highly non-trivial statement that flat ΛCDM possible up to a critical redshift as we show later).
models provide such a good fit to all the data, but we This dilemma is visible in various recent studies at-
must be aware that such constraints on the curvature tempting to constrain cosmic curvature in the presence
areartificiallystrongin the sense thatadding moredark ofmultipledarkenergyw(z)parameters[14,15,16]. For
energy parameters will lead to an almost perfect degen- some popular parameterisation constraints on Ω are of
k
eracywiththecurvature. ThisisvisibleinFig. 17ofthe order Ω < 0.05 at 2σ. For other parameterisation the
k
| |
WMAP3 [17] (here Fig. 1) which shows the correlation constraintsevaporateandevenΩ 0.2cannotberuled
k
∼
between a constant w and Ω . out (see Fig. 2, which is taken from [15]).
k
Hence we can currently say very little about the true Alternative cosmic measurements sensitive to curva-
value of the Ω and the belief that the spatial curvature ture include the Integrated Sachs Wolfe (ISW) effect,
k
issmallisessentiallybasedonOccam’sRazor. Although whichissensitivetothegrowthofthemetricfluctuations
one could fit distance measurements with any value of Φ,whichisinturnsensitivetobothdarkenergyandcur-
Ω , the requiredw(z) functions wouldbe disfavouredby vature. Recent work to investigate the ISW effect as a
k
Bayesianmodelselectionwhichpenalizemodelswithex- function of redshift uses the combination of CMB data
tra parameters that do not significantly improve the fit. with informationonlarge scale structure [6]. Combining
Weshowindetaillatertherequiredw(z)functionstodo WMAPwithsuchsuitabletracersoflargescalestructure
precisely this. shows that Φ has been decreasing with cosmic time [31],
At present a well-defined program for measuring the which rules out a large positive curvature which would
spatial curvature of the cosmos does not exist. To illus- have predicted the opposite trend.
tratethis,considerfixingthedarkenergytobedescribed Anothermeasurementsensitivetothegrowthfunction
by only n parameters. One would hope that given this is differential number counts dN/dz, e.g. of clusters.
3
B. Future surveys
We will show in equation (6) that simultaneous mea-
surements of the Hubble rate H(z), distance D d ,d
A L
′ ∝
and D (z) allow for a perfect measurement of Ω . BAO
k
allowforthesimultaneousmeasurementofbothdistance
and Hubble rate at the central redshift [24]. For a flat
′
universeD (z) 1/H(z),butinacurveduniversethisis
∝ ′
not true: the curvedgeodesics mean that D (z) contains
extra information encoded in Ω .
k
′
Measuring D (z) is in principle possible with future
BAO, weak lensing and supernova surveys. In par-
ticular, cross-correlation tomography of deep lensing
surveysappearstobe averypowerfulprobeofcurvature
when combined with BAO surveys [7], assuming that
self-calibration is possible. In principle it should be
possible to measure the cosmic curvature to an accuracy
of about σ(Ω ) 0.01 for an all-sky weak lensing
k
≃
and BAO survey out to z = 10. In principle such a
survey would be able to measure distances to about
10−4f−1/2 in redshift bins of width ∆z = 0.1 out to
sky
z = 2.5 [7]. This relies critically on the combination
of weak lensing and BAO data since constraints from
FIG. 2: The curvature-dark energy degeneracy like-
either observations alone are significantly degraded.
lihood for Ωk for different parameterisations of dark en-
This can also be seen in Fig. (3) which shows the error
ergy. Assuming a constant model for w, allows Ωk to be
ellipses for the parameters in the CPL parameterisation
tightly constrained at 2σ to be near 0. However introduc-
ing dynamics reduces these constraints significantly. Here w(z) = w0 + wa 1+zz assuming flatness (left) and
X(z) = ρX(z)/ρX(0) is the dark energy density, which [15] leaving Ω free (rig(cid:16)ht).(cid:17)Note that although individual
assumeisafreefunctionbelowsomecut-offredshiftzcut. The k
error ellipses are significantly degraded, the combined
value of X at redshifts zi=zcut(i/n),i=1,2..,n are treated
data sets have an almost unchanged error ellipse.
as n independent model parameters that are estimated from
the data. A specific functional form for X is assumed above
the cut-off redshift. The likelihoods are given for two such Our work is organizedas follows: we illustrate the de-
formsofX(z);namelyapowerlaw,X ∝(1+z)αforz >zcut, pendenceofthebackgroundobservablesonthecosmolog-
and an exponential function X ∝eαz. In these figures there icalparametersΩ ,Ω insectionIIAanddiscussobtain-
m k
are n = 3 independent redshifts below a cut-off redshift of ingthe darkenergyequationofstatew fromobservables
zcut =1.4. Taken from [15]. insectionIIB. Theprocessofreconstructingwviaanon-
parametric approach is described in section III. Finally
we link this non-parametric approach to other standard
approachestodarkenergydegeneraciesinsectionIVand
conclude in section V.
This is a potentially sensitive test which, given a con-
stant comoving number of objects, reduces to a test of
therateofchangeofcosmicvolumewithredshift,dV/dz.
II. DARK ENERGY FROM OBSERVATIONS
We discuss in detail below how perfect measurements of
dV/dz allow reconstruction of w(z), and we discuss the
There are three key observables of the backgroundge-
resulting errors on dark energy when systematic biases
ometry which play a pivotal role in determining w(z),
in cosmic parameters are present.
namely measurements of distances, of the expansionhis-
Measurements of the power spectrum from CMB data tory (i.e. the Hubble parameter) and of the change in
and from measurement of Baryon Acoustic Oscillations thefractionalvolumeoftheUniverse(e.g. fromnumber-
(BAO)provideestimatesofthemattercontentoftheuni- counts).
verse. While constraints on Ω are sharpened by com- Theprinciplemethodtodateistorelatemeasurements
m
bining data frommany observations,the best-fit value is of the distances of objects to the cosmology of the Uni-
oftenderivedontheassumptionofflatness[9,27]. Unlike verse. This is done via either standard ‘rulers’ of known
thecaseforcosmiccurvaturethedegeneracybetweenob- length - giving the angular diameter distance d (z) - or
A
servables and the matter content is perfect and we show via standard ‘candles’ of known brightness which results
that incorrectly assuming a particular value for Ω can in the luminosity distance d (z). These are related via
m L
also mimic deviations from ΛCDM. thereciprocityrelationd (z)=(1+z)2d . InanFLRW
L A
4
FIG. 3: Left - 1σ error contours assuming flatness for the dark energy parameters w0 and wa for the CPL parameterisations.
Right-asontheleftbutwithcurvatureleftfreeandmarginalisedover. Notehowpuredistancemeasurementssufferstrongly
evenwiththeverylimitedw(z)parameterisationbutthatwhenallthesurveysarecombinedthefinalerrorellipseisessentially
unaffected. This is to be expected from Equation (6) which shows how Ωk can be determined from simultaneous Hubble rate
and distance measurements. Figure from Knox et al. [8].
model, these are given by dL(z) = c(1 + z)D(z)/H0, Given any two of the above observables we may de-
where we define duce the third. Perfectobservations of these observables
shouldallowus,inprinciple,tobeabletoreconstructtwo
D(z)= 1 sin Ω zdz′ H0 . (2) free functions when in fact we only need to reconstruct
√−Ωk (cid:18)p− kZ0 H(z′)(cid:19) one, namely w(z), as well as two cosmological parame-
ters, Ω and Ω . (Note that if we know H(z) perfectly,
Here, Ω is the usual curvature parameter, and H(z) is m k
k
given by the Friedmann equation, we know H0 = H(0), and so this is no longer a free pa-
rameter in the same sense.) How do we find these?
H(z)2 =H02 Ωm(1+z)3+Ωk(1+z)2+ΩDEf(z) (3) We may determine the curvature directly, and inde-
(cid:2) (cid:3) pendently of the other parameters or dark energy model
where
via the relation [13]
z 1+w(z′)
′
f(z)=exp 3 dz (4)
(cid:20) Z0 1+z′ (cid:21) [H(z)D′(z)]2 H2
Ω = − 0, (6)
and ΩDE = 1 Ωm Ωk. Thus, given a cosmological k [H0D(z)]2
− −
model,wemaycalculateanydistancemeasurewechoose.
The Hubble parameter is in itself an observable which
which may be derived directly from Eq. (2). Such inde-
will play an important role in future dark energy exper-
pendent measurements of the curvature of the universe
iments. Knowledge of H(z) allows us to directly probe
can in turn be used to test the Copernican Principle in
the dynamical behavior of the universe, and it will be
a model-independent way. [32]
directly determined from BAO surveys which simultane-
ously provide the angular diameter distance, d at the
A
same redshift by exploiting the radial and angular views
of the acoustic oscillation scale [24], a fact that will pro-
A. Expansions of the background observables
vide key new data in coming years [4, 9, 10].
The third key background test we will discuss here is
theobservationoffractionalvolumechangeasafunction To illustrate the dependency of the background ob-
of redshift, servables we consider here we expand them in terms of
the cosmological parameters ǫ , Ω and the parameter
d2V c3D(z)2 m k
V′(z)≡ dzdΩ = H2H(z), (5) x=z/(1+z). Here ǫm :=Ωm∗−Ωm, where Ωm∗ is the
0 true value of the matter energy density and Ω is the
m
assumed value, as seen in Eq. (17).
which can in principle be determined via number-counts
′
or the BAO. The expansions for H(x),d (x),V (x) yield
L
5
z
x=
1+z
1
H(x)=H0 1+ 3(1+w0(1 Ωm∗))x (1+3w0)Ωkx 3w0ǫm (7)
(cid:20) 2 − − − (cid:21)
(cid:8) (cid:9)
cx
dL(x)= 1+ (5+3w0(Ωm∗ 1))+(1+3w0)Ωk+3w0ǫm x (8)
H0(cid:20) − (cid:21)
(cid:8) (cid:9)
c3x2
′
V (x)= H3 (cid:20)1+ (−1+3w0(Ωm∗−1))+(1+3w0)Ωk+3w0ǫm x(cid:21) (9)
0 (cid:8) (cid:9)
It can be seen from equations (7, 8, 9) that the lead- rate [2, 11], from Eq. (3), and is given by:
ing termcorrespondsto that ofthe standardflatΛCDM
model. From these equations we can directly compute
1Ω H2(1+z)2+2(1+z)HH′ 3H2
the error on the particular observable as a function of w(z)= k 0 − . (10)
redshift based on the difference between the ‘true’ cos- −3 H02(1+z)2[Ωm(1+z)+Ωk] H2
−
mology and the ‘assumed’ cosmologicalmodel.
This tellsus w(z) provided wealreadyknowΩ andΩ .
m k
B. Obtaining the Dark Energy equation of state However,thisrevealsadegeneracybetweenΩm andw(z)
from Observations whichcannotbeovercomebybackgroundtestsalone[3].
In essence, geometric background tests can measure the
combination Ω +Ω f(z)/(1+z)3, but not the two
Assuming we have ‘perfect’ and uncorrelated data m DE
separately. Another way to view this is by differentiat-
from observations we would like to reconstruct w(z)
ing Eq. (10), and eliminating Ω to give a differential
without assuming a specific parameterisation. Depend- m
′ ′′
equation for w(z) in terms of H,H and H ; the con-
ing on the particular observable of interest, there are
stant arising in the general solution to this differential
different ways to reconstruct w.
equation is Ω .
m
Dark energy from Hubble Similarly, we can reconstruct w(z) from the other two
It is straightforward to find w(z) from the Hubble tests on their own.
Dark energy from distance measurements
From distance measurements, we may invert Eq. (2) to find
2 (1+z) D2Ω +1 D′′ D′ Ω (1+z)2D′2+2Ω D(1+z)D′ 3 3D2Ω
k k k k
w(z)= (cid:0) (cid:1) − h − − i. (11)
3 [Ω +Ω (1+z)](1+z)2D′2 D2Ω 1 D′
k m k
n − − o
Reconstructing w(z) from volume measurements as an analytical formula is rather tricky (as it involves the root
of a quartic power). It is simpler instead to reconstruct w(z) by solving the differential equation for f(z) and then
differentiating to get w(z).
Dark energy from volume measurements
Starting with Eq. (2), we solve for the derivative of the Hubble parameter and equate this with the expression for
′
H in terms of w(z) from Eq. (10) and use
′
(1+z)f
w(z)= 1 (12)
3f −
6
to yield a first order differential equation for f, namely
A(z)+B(z)+C(z)
′
f (z)= , (13)
H02V′ΩDE
−
where
1/2
A(z)= 4 V′H0 c3 f(z)ΩDE+X11+V′H03Ωk(f(z)ΩDE+X11) ,
− (cid:16) (cid:16) p (cid:17)(cid:17)
with
X =(1+z)2(aΩ +bΩ (1+z)),
ab k m
B(z)=2H02V′′(f(z)ΩDE+X11)
and
C(z)=H2V′′ X32 .
0
1+z
Wesolvethisforf(z)andthenuse(12)againtoyieldw(z). Thesolutionforf(z)isuniquesincewedemandf(0)=1.
III. RECONSTRUCTING w(z) function required to yield the same H(z) or dL(z) as in
the actual curved ΛCDM model: e.g.,
If we knew Ω and Ω perfectly then our three ex-
m k d [flat,w(z)]=d [curved,w(z)= 1]. (14)
L L
pressions for w(z) would yield the same function w(z), −
assumingwelivedinanexactFLRWuniverse. Butwhat ForexamplefortheHubbleratethereconstructedw(z)
if–asiscommonlyassumed–weimposeΩk =0whenin can be found analytically to be
fact the truecurvatureis actuallynon-zero? Itis usually
implicitlyassumedthattheerroronw(z)willbeoforder 1Ωk(1+z)2+3ΩDE
w(z)= , (15)
Ωk, but, as was shown in [13] this is not the case. Will −3 Ωk(1+z)2+ΩDE
′
measuring V (z) possibly circumvent this? And further-
without any dependence on a specific parameterisation.
more, are there similar issues from an imperfect knowl-
In the figure we show what happens for ΛCDM: cur-
edge of Ω ?
m
vature manifests itself as evolving dark energy. In the
case of the Hubble rate measurements this is fairly obvi-
ous - we are essentially solving the equation Ω f(z)=
DE
A. Zero curvature assumption ΩΛ + Ωk(1 + z)2 where f(z) is given by Eq. (4). For
Ω > 0, w(z) must converge to 1/3 to compensate for
k
−
We can easily see the implications of incorrectly as- the curvature. For Ω < 0, the opposite occurs and a
k
suming flatness by constructing the functions dL(z) and redshiftisreachedwhenw inanattempttocom-
→−∞
H(z) under the assumption of the ΛCDM in a curved pensate albeit unsuccessfully for the positive curvature.
Universe (i.e. assuming w = 1,Ωk = 0) and inserting Alreadywecanseewhythe assumptionthatthe errorin
− 6
the results into Eqs (10) and (11). w is of order the error in Ω breaks down so drastically.
k
If we then set Ω = 0 in Eqs. (10) and (11) we ar- Interestingly,thecurvedgeodesicsimplythattheerror
k
rive at the two correspondingw(z) functions (if they ex- in w reconstructed from d (z) and H(z) have opposing
L
ist)requiredtoreproducethecurvedformsforH(z)and signs at z &0.9, as can be seen by comparing the panels
d (z) in a flat Universe with dynamic dark energy. This for the Hubble rate and the distance indicator in Fig. 4.
L
wouldapplyequallytod (z)forthatmatter-theresults Above the critical redshift the effect of curvature on the
A
areexactlythe sameforanydistanceindicator. Figure4 geodesicsbecomesmoreimportantthanthepuredynam-
presentsthismethodusingforsimplicitytheconcordance ics,andtheluminositydistanceflipsw(z)intheopposite
valueofw= 1butwehavecheckedthatthequalitative direction to that reconstructed from H(z).
−
results do not depend on the ‘true’ underlying dark en- Inthe case ofvolume measurementsthe reconstructed
ergy model[33]. We assume Ω =0.3 in allexpressions; w(z)hasasimilarformtothewweobtainedfromthedis-
m
numbers quoted are weakly dependent on this. The re- tancemeasurementsD(z). ThiscanbeseenfromEq.(5),
sulting (spurious) w(z) can then be thought of as the where the distance information enters the equation as a
7
FIG. 4: Reconstructing the dark energy equation of state assuming zero curvature when the true curvature is 2%
in a closed ΛCDM universe. The w(z) reconstructed from H(z) is phantom (w < −1) and rapidly acquires an error of order
50% and more at redshift z & 2, and diverges at finite redshift. The reconstructed w(z) from dL(z) for Ωk < 0 is phantom
until z ≈0.86, where it crosses the true value of −1 and then crosses 0 at high redshift, where the bending of geodesics takes
overfromdynamicalbehavior,producingerrorsinopposite directiontotheDEreconstructedfrom H(z). Inordertomakeup
for themissing curvature,the reconstructed dark energy is behaving like a scalar field with a tracking behavior. These effects
arise even if thecurvatureis extremely small (<0.1%). Reprinted from [13].
8
square power. For example in the closed Universe case the concordance value of Ω = 0.3 incorrectly. For ex-
m
the reconstructed w(z) drops to more phantom values ample in this case the w(z) reconstructed from Hubble
( 2.5 compared to 1.3 for the distance measurements) measurements Eq. (10) reduces to
− −
in order to make up for the missing curvature.
Again the effect of curvature on the geodesics domi- 1 2(1+z)HH′ 3H2
nates the effect ofdynamics for largez,andthe distance w(z)= − . (16)
−3H2(1+z)2[Ω (1+z)] H2
contributioninthevolumemeasurementsflipstherecon- 0 m −
structedw(z) atz =1.6. The criticalredshiftof this flip Similarexpressionsarefoundforboththedistanceand
is determined by the redshift at which the curvature of volume measurements. The w(z) curves obtained from
the geodesics affecting distance measurements becomes incorrectly assuming Ω = 0.3 are shown in Fig. 6. If
m
more important than the expansion rate. This playoff we assume flatness for this example we find that chang-
becomes more finely balanced for volume measurements ing the value of Ω can only affect the dark energy den-
m
due to the fact that H(z) appears both in D(z) (as a sity, and thus change the value of H(z). As Ω is only
m
square power) and on its own. Hence w(z) has to work present in all three observables through H(z) or inte-
harder in reproducing curvature to counterbalance the grals of 1/H(z), the reconstructed w(z) is the same for
opposing trends of expansion history and geometry, and all three measurements. Interestingly, the reconstructed
sothebalanceisachievedathigherredshift. Thespecific w(z) curves do not go through w= 1 at z =0, but are
redshiftatwhichthishappensisdependentonΩminthat spreadbetween-0.85and-1.15for0−.2<Ωm <0.4. This
lower values imply higher value of the critical redshift. is also shown in Fig. 7.
w(z) from Volume measurements with changing Ω w(z) from Volume, distances and the Hubble parameter with changing Ω
k m
2.0 0
0.4
0.35
0.31
1.0
−0.2 −0.5
Ω = 0.301
m
0 −0.05 −0.01
w(z) Ω = −0.001 w(z)−1.0
k
−1.0 Ω = 0.299
0.25 m
0.01 0.29
Ωk = 0.001 −1.5
−2.0
0.05
0.2
0.2
−3.0 −2.0
0 2 4 6 8 10 0 2 4 6 8 10
z z
FIG.5: Reconstructed dark energy from volume mea- FIG. 6: Reconstructed dark energy from an incor-
surementswhileincorrectlyassumingflatness-Similar rectlyestimatedmatterdensity-Thereconstructedw(z)
to the case for distance measurements in a closed Universe, forchangingΩm from allthreemeasurements(H,D,dV/dz).
the reconstructed w(z) must initially be phantom in order Since we assume flatness while changing Ωm, all three ob-
to compensate for curvature, and crosses the true value of servables yield the same reconstructed w(z), since Ωm only
w =−1 at a redshift of z 1.6, which is greater than the red- enters the functions through H(z) or integrals of 1/H. For
shift of 0.86 for the distance measurements alone [13]. After Ωm > 0.3 the dark energy tries to compensate for the extra
this point, the w(z) increases to overcome the curvature of matter contribution and so asymptotes to w = 0 as z → ∞.
thegeodesics. For Ωm <0.3 the w(z) is of the same form to what is recon-
structed from neglecting curvature in a closed Universe (see
Fig. 4), and the phantom w tends to −∞ as it attempts to
We have shown that incorrectly assuming flatness can compensate for the‘missing’ matter density.
result in a reconstructed w(z) that mimics dynamics,
yielding errors on w that are much larger than the or-
Given any scenario of an assumed cosmology that dif-
der of errorson Ω . One might then ask if similar errors
k
fers from the ‘true’ Universe, we can derive the value
will result when incorrectly assuming a particular value
of today, w(z = 0) from both the Hubble and distance
for the matter density in the Universe, Ω .
m
measurements as
B. Uncertainties in the Matter content Ωm 3 4Ωk∗ 3Ωm+Ωk
w(0) = − − (17)
6Ωm+6Ωk∗ 3Ωm∗ 3 3Ωk
− − −
We consider the similar case of reconstructing w(z) in ǫm 2Ωk
1,
a flat Universe but here the errors occur when assuming ∼ ( 1+Ωm∗) − 3( 1+Ωm∗) −
− −
9
where ǫm =Ωm∗ Ωm as defined above where the as- From luminosity distance measurements
−
teriskindicatesassumedbutincorrectvaluesofthecorre-
sponding quantities. We vary this equation in one ‘true’
density (Ω or Ω ) at a time, while keeping the other
m k Ωk+3ΩDE
constant at the assumed value of either Ωk = Ωk∗ or w0 =
−3(1 Ω )
Ωm =Ωm∗ to produce the curves in Fig. 7. This param- − m
eter w(z = 0) allows us to easily quantify the affect of 2Ωk(Ωk ΩDE)
w = − (19)
assuminganincorrectcosmologicalmodelontheinferred a −3 (1 Ω )2
m
−
low-redshift value of w.
We plot in Figure 8 the non-parametric reconstructed
−0.8
w(z) along with the reconstructed wCPL(z) from the
−0.9 coefficients given by Eqs. (18, 19) for the observables
H(z) and d (z).
−1 L
Ω
k
−1.1
0)−1.2
=
z Reconstructed w(z) from measurements of the H(z) using series expansion
w(−1.3 Ω −0.8
m
−1.4 −0.85
Ω = 0.05
−1.5 −0.9 k
−1.6 −0.95 0.01
−1
−1.7
−0.2 −0.15 −0.1 −0.05 ∆ 0Ωi 0.05 0.1 0.15 0.2 w(z)−1.05 −0.01
−1.1
FIG. 7: Low redshift variation in w(z) from H(z) and −1.15 Ωk = −0.05
D(z)- incorrectly assuming concordance values of Ωm = 0.3 −1.2
and Ωk = 0 results in a variation in the low-redshift value
−1.25
of w(z) reconstructed from observables. Therelationship be-
tweentheerrorinthecosmological parameterandtherecon- −1.3
0 0.5 1 1.5 2 2.5 3
structed value for w (while keeping the other cosmological z
parameterfixedatthepriorvalue)isshownforbothΩm (the
green curve)and Ωk (theblue curve). Reconstructed w(z) from measurements of dL(z) using series expansion
−0.85
−0.9
Ω = 0.05
k
−0.95
0.01
IV. PARAMETRIC DEGENERACIES
−1
z) −0.01
w(
We nowwantto connectthe non-parametricapproach −1.05
we have followed above with standard approaches to Ω =
k
degeneraciesandsoweexpandEqs. (10)and(11)forthe −1.1 −0.05
Hubble rate and distance measurements to first order in
−1.15
x = z/(1+z). This allows us to link to the parameters
w0,wa used in the Chevallier-Polarski-Linder (CPL)
0 0.5 1 1.5 2 2.5 3
[25, 26] parameterisation wCPL(z) = w0 + wa 1+zz , z
(cid:16) (cid:17)
which is used in the Dark Energy Task Force report [1].
The values of w0,wa obtained using this expansion are FIG. 8: Degeneracies in standard parameterisations -
given below. w(z) = w0+wa1+zz using the coefficients in Eqs. (19) and
(18)(solidlines)comparedwiththefullynon-parametricw(z)
From Hubble rate measurements inferred from Hubble and distance measurements. Using a
limited parameterisation of w(z) like this incorrectly makes
it appear that dark energy and curvatureare not completely
Ωk+3ΩDE degenerate,leadingtoartificiallystrongconstraintsoncurva-
w0 = −3(1 Ω ) tureand w0,wa.
m
−
4 ΩkΩDE
w = (18)
a 3(1 Ω )2
m
−
10
V. CONCLUSIONS AND OUTLOOK from uncertainty in the cosmic parameters are much
larger than the uncertainty in Ω or Ω , especially at
k m
We have explored the degeneracies between the dark large redshifts. We have shown that curvature affects
energyequationofstatew(z) andcosmic parametersus- measurementsofH(z)andD(z)incomplementaryways,
ing a non-parametric approach. This means we are able with the error at high redshift having opposite signs for
towritedowntheprecisew(z)thatwillbereconstructed an error in Ωk. In the case of an Ωm error, Hubble,
from perfect data if slightly wrong or biased values for distance and volume measurements all lead to the same
thecosmicparametersΩ ,Ω areassumed. Thisiscom- erroneously reconstructed w(z), a manifestation of the
k m
plementarytotraditionalmethodswhichtypicallyusean dark matter-dark energy degeneracy highlighted in [3].
aggressive compression of the w(z) function onto a cou- In this review we have assumed perfect data for Hub-
ple of parameters (usually w0,wa) and then study the ble rate, distance and volume at all redshifts. It would
degeneracy between these and other cosmic parameters. be interesting to extend our non-parametricapproachto
Our approachis superior in one way however: degenera- the case of imperfect data which has incomplete redshift
cies between w(z) and some cosmic parameters such as coverage and errors on the observables. This is left to
Ω canappeartobe quiteweakinthe parameterisedap- future work but will allow contact with the approaches
k
proach. However, in the case of distance measurements in [29, 30].
thisiscompletelyartificialandduetostrongassumptions Acknowledgments – we thank Luca Amendola, Chris
about the allowed form of w(z) since the degeneracy is Blake, Thomas Buchert, Daniel Eisenstein, George El-
perfect if w(z) is allowed to be totally free. lis, Martin Kunz, Roy Maartens, Bob Nichol and David
We extend the work of [13] to show the reconstructed Parkinsonfor useful comments and insights. MC thanks
w(z) from measurements of volume for both wronglyas- Andrew Liddle and FTC for support. BB and CC ac-
sumed Ω and Ω . As with Hubble and distance mea- knowledge support from the NRF and RH acknowledges
k m
surements we show that the errors in w(z) that result funding from KAT.
[1] A.Albrechtetal.,ReportoftheDarkEnergyTaskForce, [15] Y. Wang & P. Mukherjee, astro-ph/0703780 (2007)
astro-ph/0609591 (2006) [16] Z.-Y. Huang, B. Wang, & R.-K. Su, Int. Journ. Mod.
[2] E. V.Linder, Astropart.Phys. 24, 391 (2005) Phys. A, 22, 1819 (2007)
[3] M. Kunz,astro-ph/0702615, (2007) [17] D. N. Spergel et al. Ap.J. S. , 170, 377 (2007)
[4] K. Glazebrook and the WFMOS Feasibility Study Dark [18] S. Perlmutter, et al. Ap.J. 483, 565 (1997)
Energy Team, White paper submitted to the Dark [19] A. G. Riess et al.A. J. , 116, 1009 (1998)
Energy Task Force, astro-ph/0507457; B. A. Bassett, [20] B. Freivogel, M. Kleban, M. Rodr´ıguez Mart´ınez, & L.
R.C. Nichol and D.J. Eisenstein [for theWFMOS Col- Susskind,Journal of High Energy Physics, 3, 39 (2006)
laboration], astro-ph/0510272; [21] M. Kunz, R. Trotta, & D. R. Parkinson, Phys. Rev. D
[5] See e.g. T. Buchert, M. Carfora, Phys. Rev. Lett. 90, 74, 023503 (2006)
031101 (2003); N.Li, D.J. Schwarz, gr-qc/0702043; [22] R. Trotta, M. N.R. A.S. 378, 72 (2007)
S.Rasanen, Class. Quant.Grav. 23 (2006) 1823; [23] A. R.Liddle astro-ph/0701113 (2007)
A.A.Coley, N.Pelavas Phys. Rev.D75, 043506 (2007) [24] H. -J. Seo, & D. J. Eisenstein, Ap. J. , 598, 720 (2003);
[6] Seeeg. T.Giannantonio et al.,Phys.Rev.D 74, 063520 C. Blake, & K.Glazebrook, Ap.J. 594, 665 (2003)
(2006); S. Hoet al. arXiv:0801.0642 801 (2008) [25] M.Chevallier&D.PolarskiInt.J.Mod.Phys.D10,213
[7] G. Bernstein, A. J. 637 (2006) 598 (2001)
[8] L. Knox,Y-S.Song, H.Zhan, astro-ph/0605536 (2006) [26] E. LinderP. R.L. 90, 091301 (2003)
[9] M. Tegmark et al., Phys.Rev.D 74, 123507 (2006) [27] W. J., Percival, et al. Ap.J. 657, 51 (2007)
[10] K.Glazebrook et al.,astro-ph/0701876 2007 [28] B. A. Bassett, P. S. Corasaniti & M. Kunz,Ap. J. L. ,
[11] D. Huterer & M. S. Turner, Phys. Rev. D 64, 123527 617, L1 (2004)
2001 [29] C. Espana-Bonet & P. Ruiz-Lapuente hep-ph/0503210
[12] S.Weinberg, ApJL 161, L233 (1970) (2005)
[13] C. Clarkson, M. Cortˆes, & B. Bassett, J. C. A. P. 8, 11 [30] A.Shafieloo,U.Alam,V.Sahni,V.&A.A.Starobinsky,
(2007) M. N.R. A.S. 366, 1081 (2006)
[14] Y.Gong, Q.Wu & A.Wang, arXiv:0708.1817 (2007); [31] A. Aguirre, C. P. Burgess, A. Friedland, & Nolte, D.
L. Mersini-Houghton, Y.Wang, P. Mukherjee & Classical and Quant Gravity 18, 223 (2001)
E. Kafexhiu, arXiv:0705.0332 (2007); E. Wright, A. J. [32] C. Clarkson, B. A. Bassett, & T. Hui-Ching Lu,
664,633(2007);K.Ichikawa,M.Kawasaki,T.Sekiguchi, arXiv:0712.3457, 712 (2007)
& T. Takahashi, J. C. A.P. 12, 5 (2006); C.-B. Zhao, [33] In fact, the results presented here are qualitatively the
J.-Q. Xia, H.Li, et al.,Phys. Lett. B, 648, 8 (2007); sameforanyassumedΩk whichisdifferentfromthetrue
K.Ichikawa, & T. Takahashi, , J. C. A. P., 2, 1 (2007) value.