Non-parametric Dark Energy Degeneracies
aa r X i v : . [ a s t r o - ph ] J a n Non-parametric Dark Energy Degeneracies
Ren´ee Hlozek , , Marina Cortˆes , , Chris Clarkson and Bruce Bassett , Cosmology & Gravity Group, Department Mathematics and AppliedMathematics, University of Cape Town, Rondebosch 7701, South Africa. Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom South African Astronomical Observatory, Observatory, Cape Town, South Africa
We study the degeneracies between dark energy dynamics, dark matter and curvature using a non-parametric and non-perturbative approach. This allows us to examine the knock-on bias induced inthe reconstructed dark energy equation of state, w ( z ), when there is a bias in the cosmic curvatureor dark matter content, without relying on any specific parameterisation of w . Even assumingperfect Hubble, distance and volume measurements, we show that for z >
1, the bias in w ( z ) is upto two orders of magnitude larger than the corresponding errors in Ω k or Ω m . This highlights theimportance of obtaining unbiased estimators of all cosmic parameters in the hunt for dark energydynamics. I. INTRODUCTION
Since 1998 [18, 19] evidence has been mounting in sup-port of an accelerated expansion of the Universe. Nearly10 years on, the puzzle of the origin of this acceleration –dubbed dark energy – remains one of the most intriguingenigmas in modern day science. Much activity has comefrom both the theoretical and observational sectors of thephysics community in an attempt to pin down its origin.The current drive in dark energy studies is focused ontrying to establish its dynamical behaviour as a functionof redshift, w ( z ). While the simplest explanation remainsa ΛCDM universe with w = − w ( z ) would provide a window into new physics. There-fore, uncovering the dynamics of dark energy as describedby the ratio of its pressure to density, w ( z ) = p DE /ρ DE ,has become the focus of multi-billion dollar proposed ex-periments using a wide variety of methods, with severalplanned surveys at redshifts above unity, as high-redshiftmeasurements are useful to constrain dark energy param-eters and test for deviation from the concordance ΛCDMmodel (see e.g. [1]). Unfortunately the search for dynam-ical behaviour in w is a mani-fold problem. The natureof dark energy is elusive: cosmic observations depend notonly on dark energy but also on other cosmic parameterssuch as the cosmic curvature, Ω k , and the total mattercontent, Ω m , leading to degeneracies between these and w ( z ) parameters, an issue which has recently been underintense scrutiny by the community [2, 3, 13, 16]. Kunz[3] argues that observations are only sensitive to the fullenergy-momentum tensor and thus cannot see beyonda combination of the “dark component” – dark matter plus dark energy. The degeneracy between the geometryof the universe and the equation of state of dark energyhas also been discussed in light of the well-known resultthat a cosmological constant in the presence of spatialcurvature can mimic a dynamical dark energy [13].In this work we review current constraints on cosmiccurvature and extend the approach in [13] in two ways.First we include the reconstruction of w ( z ) that wouldfollow from measurements of the rate of change of cos- mic volume with redshift, dV /dz . Secondly we discussthe w ( z ) that would be incorrectly reconstructed fromperfect Hubble, distance and volume data if the wrongvalue of Ω m were used. We assume perfect data for allobservations, which allows us to probe fundamental, “in-principle” degeneracies that are not due to finite errorsand incomplete redshift-coverage. This implies that givena specific bias in a cosmological parameter, the degen-eracies will be true no matter what progress is made inimproving future cosmic surveys. Furthermore, the keypoint in our reconstruction of w ( z ) is that it is performedin a fully non-parametric manner, and so does not rely onthe validity of any particular parameterisation of w ( z ).To illustrate the power of this non-parametric approach,we compare our method with a standard equation of stateparameterisation [25, 26], which cannot fully resolve theabove degeneracies. A. Degeneracies in Dark Energy Studies
The success of the inflationary scenario for the earlyUniverse and its standard prediction of flatness to highprecision (Ω k < − ) is perhaps the main reason whycurvature has traditionally been left out in analyses ofdark energy. However, possible scenarios in which infla-tion is consistent with non-zero spatial curvature haverecently been investigated [20]. It is also interesting tonote that the backreaction of cosmological fluctuationsmay cause effective non-zero curvature that may yieldpractical limits on our ability to measure w ( z ) accuratelyat z > z > d L or d A ) whichare completely degenerate with curvature [12]. One wayto illustrate the degeneracy between curvature and dy-namics is as follows. Let us assume that we know allcosmic parameters perfectly other than the curvature Ω k and the dark energy equation of state, w ( z ). At any red-shift, z ∗ , a perfect measurement of d L ( z ∗ ) (or d A ( z ∗ ))allows us to measure a single quantity. If we know w ( z ∗ )then that quantity can be Ω k . However, if w ( z ) is trulya free function, then its value at z ∗ is completely freeand we are left trying to find two numbers from a singleobservation, which is impossible.Only when we start to correlate the values of w ( z ) atdifferent redshifts can we begin to use distance measure-ments alone to constrain the curvature. The standardway to do this is to assume that w ( z ) can be compressedonto a finite-dimensional subspace described by n param-eters, e.g. w ( z ) = Σ nj w j z j (1)In this case perfect distance measurements at n +1 differ-ent redshifts will allow a complete solution of the problemand will yield the w j and Ω k . The most extreme versionof this is to assume ΛCDM, w ( z ) = −
1. Within this con-text it is of course possible to derive very stringent con-straints on the curvature. For example, combining theWMAP 3 year data and the SDSS DR5 Luminous RedGalaxy (LRG) sample leads to Ω k = − . ± .
010 as-suming w = − k = − . . Λ [17].It is a highly non-trivial statement that flat ΛCDMmodels provide such a good fit to all the data, but wemust be aware that such constraints on the curvatureare artificially strong in the sense that adding more darkenergy parameters will lead to an almost perfect degen-eracy with the curvature. This is visible in Fig. 17 of theWMAP3 [17] (here Fig. 1) which shows the correlationbetween a constant w and Ω k .Hence we can currently say very little about the truevalue of the Ω k and the belief that the spatial curvatureis small is essentially based on Occam’s Razor. Althoughone could fit distance measurements with any value ofΩ k , the required w ( z ) functions would be disfavoured byBayesian model selection which penalize models with ex-tra parameters that do not significantly improve the fit.We show in detail later the required w ( z ) functions to doprecisely this.At present a well-defined program for measuring thespatial curvature of the cosmos does not exist. To illus-trate this, consider fixing the dark energy to be describedby only n parameters. One would hope that given this –0.08 –0.06 –0.04 –0.02 0.00 0.02 0.04–1.4–1.2–1.0–0.8 FIG. 1:
The curvature-dark energy degeneracy
Con-tours showing the 2D marginalized contours for w and Ω k based on combined data from WMAP3, 2dFGRS, SDSS andsupernova surveys. While the slope of the degeneracy differsfor this combination of data, the sign of the degeneracy isconsistent with the w term in Eqs. (18),(19). Taken from[17]. restriction the resulting constraints on Ω k would be in-dependent of the precise choice of the n parameters, i.e.independent of the parameterisation. Unfortunately alittle thought makes it clear that this cannot be true. Aparameterisation of w ( z ) which does not allow mimicryof curvature will provide good, decorrelated constraintson the curvature (which does not mean the correspondingbest-fit will be a good fit to the data) while a model whichallows perfect mimicry of the dynamics of the curvature(i.e. (1 + z ) ) will show highly correlated constraints (al-though for negative Ω k mimicry of distance data is onlypossible up to a critical redshift as we show later).This dilemma is visible in various recent studies at-tempting to constrain cosmic curvature in the presenceof multiple dark energy w ( z ) parameters [14–16]. Forsome popular parameterisation constraints on Ω k are oforder | Ω k | < .
05 at 2 σ . For other parameterisation theconstraints evaporate and even Ω k ∼ . dN/dz , e.g. of clusters. FIG. 2:
The curvature-dark energy degeneracy like-lihood for Ω k for different parameterisations of dark en-ergy. Assuming a constant model for w , allows Ω k to betightly constrained at 2 σ to be near 0. However introduc-ing dynamics reduces these constraints significantly. Here X ( z ) = ρ X ( z ) /ρ X (0) is the dark energy density, which [15]assume is a free function below some cut-off redshift z cut . Thevalue of X at redshifts z i = z cut ( i/n ) , i = 1 , .., n are treatedas n independent model parameters that are estimated fromthe data. A specific functional form for X is assumed abovethe cut-off redshift. The likelihoods are given for two suchforms of X ( z ); namely a power law, X ∝ (1+ z ) α for z > z cut ,and an exponential function X ∝ e αz . In these figures thereare n = 3 independent redshifts below a cut-off redshift of z cut = 1 .
4. Taken from [15].
This is a potentially sensitive test which, given a con-stant comoving number of objects, reduces to a test ofthe rate of change of cosmic volume with redshift, dV /dz .We discuss in detail below how perfect measurements of dV /dz allow reconstruction of w ( z ), and we discuss theresulting errors on dark energy when systematic biasesin cosmic parameters are present.Measurements of the power spectrum from CMB dataand from measurement of Baryon Acoustic Oscillations(BAO) provide estimates of the matter content of the uni-verse. While constraints on Ω m are sharpened by com-bining data from many observations, the best-fit value isoften derived on the assumption of flatness [9, 27]. Unlikethe case for cosmic curvature the degeneracy between ob-servables and the matter content is perfect and we showthat incorrectly assuming a particular value for Ω m canalso mimic deviations from ΛCDM. B. Future surveys
We will show in equation (6) that simultaneous mea-surements of the Hubble rate H ( z ), distance D ∝ d A , d L and D ′ ( z ) allow for a perfect measurement of Ω k . BAOallow for the simultaneous measurement of both distanceand Hubble rate at the central redshift [24]. For a flatuniverse D ′ ( z ) ∝ /H ( z ), but in a curved universe this isnot true: the curved geodesics mean that D ′ ( z ) containsextra information encoded in Ω k .Measuring D ′ ( z ) is in principle possible with futureBAO, weak lensing and supernova surveys. In par-ticular, cross-correlation tomography of deep lensingsurveys appears to be a very powerful probe of curvaturewhen combined with BAO surveys [7], assuming thatself-calibration is possible. In principle it should bepossible to measure the cosmic curvature to an accuracyof about σ (Ω k ) ≃ .
01 for an all-sky weak lensingand BAO survey out to z = 10. In principle such asurvey would be able to measure distances to about10 − f − / sky in redshift bins of width ∆ z = 0 . z = 2 . w ( z ) = w + w a (cid:16) z z (cid:17) assuming flatness (left) andleaving Ω k free (right). Note that although individualerror ellipses are significantly degraded, the combineddata sets have an almost unchanged error ellipse.Our work is organized as follows: we illustrate the de-pendence of the background observables on the cosmolog-ical parameters Ω m , Ω k in section II A and discuss obtain-ing the dark energy equation of state w from observablesin section II B. The process of reconstructing w via a non-parametric approach is described in section III. Finallywe link this non-parametric approach to other standardapproaches to dark energy degeneracies in section IV andconclude in section V. II. DARK ENERGY FROM OBSERVATIONS
There are three key observables of the background ge-ometry which play a pivotal role in determining w ( z ),namely measurements of distances, of the expansion his-tory (i.e. the Hubble parameter) and of the change inthe fractional volume of the Universe (e.g. from number-counts).The principle method to date is to relate measurementsof the distances of objects to the cosmology of the Uni-verse. This is done via either standard ‘rulers’ of knownlength - giving the angular diameter distance d A ( z ) - orvia standard ‘candles’ of known brightness which resultsin the luminosity distance d L ( z ). These are related viathe reciprocity relation d L ( z ) = (1 + z ) d A . In an FLRW FIG. 3:
Left - 1 σ error contours assuming flatness for the dark energy parameters w and w a for the CPL parameterisations. Right - as on the left but with curvature left free and marginalised over. Note how pure distance measurements suffer stronglyeven with the very limited w ( z ) parameterisation but that when all the surveys are combined the final error ellipse is essentiallyunaffected. This is to be expected from Equation (6) which shows how Ω k can be determined from simultaneous Hubble rateand distance measurements. Figure from Knox et al. [8]. model, these are given by d L ( z ) = c (1 + z ) D ( z ) /H ,where we define D ( z ) = 1 √− Ω k sin (cid:18)p − Ω k Z z d z ′ H H ( z ′ ) (cid:19) . (2)Here, Ω k is the usual curvature parameter, and H ( z ) isgiven by the Friedmann equation, H ( z ) = H (cid:2) Ω m (1 + z ) + Ω k (1 + z ) + Ω D E f ( z ) (cid:3) (3)where f ( z ) = exp (cid:20) Z z w ( z ′ )1 + z ′ d z ′ (cid:21) (4)and Ω DE = 1 − Ω m − Ω k . Thus, given a cosmologicalmodel, we may calculate any distance measure we choose.The Hubble parameter is in itself an observable whichwill play an important role in future dark energy exper-iments. Knowledge of H ( z ) allows us to directly probethe dynamical behavior of the universe, and it will bedirectly determined from BAO surveys which simultane-ously provide the angular diameter distance, d A at thesame redshift by exploiting the radial and angular viewsof the acoustic oscillation scale [24], a fact that will pro-vide key new data in coming years [4, 9, 10].The third key background test we will discuss here isthe observation of fractional volume change as a functionof redshift, V ′ ( z ) ≡ d V d z dΩ = c D ( z ) H H ( z ) , (5)which can in principle be determined via number-countsor the BAO. Given any two of the above observables we may de-duce the third. Perfect observations of these observablesshould allow us, in principle, to be able to reconstruct twofree functions when in fact we only need to reconstructone, namely w ( z ), as well as two cosmological parame-ters, Ω m and Ω k . (Note that if we know H ( z ) perfectly,we know H = H (0), and so this is no longer a free pa-rameter in the same sense.) How do we find these?We may determine the curvature directly, and inde-pendently of the other parameters or dark energy modelvia the relation [13]Ω k = [ H ( z ) D ′ ( z )] − H [ H D ( z )] , (6)which may be derived directly from Eq. (2). Such inde-pendent measurements of the curvature of the universecan in turn be used to test the Copernican Principle ina model-independent way. [32] A. Expansions of the background observables
To illustrate the dependency of the background ob-servables we consider here we expand them in terms ofthe cosmological parameters ǫ m , Ω k and the parameter x = z/ (1 + z ). Here ǫ m := Ω m ∗ − Ω m , where Ω m ∗ is thetrue value of the matter energy density and Ω m is theassumed value, as seen in Eq. (17).The expansions for H ( x ) , d L ( x ) , V ′ ( x ) yield x = z zH ( x ) = H (cid:20) (cid:8) w (1 − Ω m ∗ )) x − (1 + 3 w )Ω k x − w ǫ m (cid:9)(cid:21) (7) d L ( x ) = cxH (cid:20) (cid:8) (5 + 3 w (Ω m ∗ − w )Ω k + 3 w ǫ m (cid:9) x (cid:21) (8) V ′ ( x ) = c x H (cid:20) (cid:8) ( − w (Ω m ∗ − w )Ω k + 3 w ǫ m (cid:9) x (cid:21) (9)It can be seen from equations (7, 8, 9) that the lead-ing term corresponds to that of the standard flat ΛCDMmodel. From these equations we can directly computethe error on the particular observable as a function ofredshift based on the difference between the ‘true’ cos-mology and the ‘assumed’ cosmological model. B. Obtaining the Dark Energy equation of statefrom Observations
Assuming we have ‘perfect’ and uncorrelated datafrom observations we would like to reconstruct w ( z )without assuming a specific parameterisation. Depend-ing on the particular observable of interest, there aredifferent ways to reconstruct w . Dark energy from Hubble
It is straightforward to find w ( z ) from the Hubble rate [2, 11], from Eq. (3), and is given by: w ( z ) = −
13 Ω k H (1 + z ) + 2(1 + z ) HH ′ − H H (1 + z ) [Ω m (1 + z ) + Ω k ] − H . (10)This tells us w ( z ) provided we already know Ω m and Ω k .However, this reveals a degeneracy between Ω m and w ( z )which cannot be overcome by background tests alone [3].In essence, geometric background tests can measure thecombination Ω m + Ω DE f ( z ) / (1 + z ) , but not the twoseparately. Another way to view this is by differentiat-ing Eq. (10), and eliminating Ω m to give a differentialequation for w ( z ) in terms of H, H ′ and H ′′ ; the con-stant arising in the general solution to this differentialequation is Ω m .Similarly, we can reconstruct w ( z ) from the other twotests on their own. Dark energy from distance measurements
From distance measurements, we may invert Eq. (2) to find w ( z ) = 2 (1 + z ) (cid:0) D Ω k + 1 (cid:1) D ′′ − D ′ h Ω k (1 + z ) D ′ + 2 Ω k D (1 + z ) D ′ − − D Ω k i n [Ω k + Ω m (1 + z )] (1 + z ) D ′ − D Ω k − o D ′ . (11)Reconstructing w ( z ) from volume measurements as an analytical formula is rather tricky (as it involves the rootof a quartic power). It is simpler instead to reconstruct w ( z ) by solving the differential equation for f ( z ) and thendifferentiating 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 w ( z ) = (1 + z ) f ′ f − f , namely f ′ ( z ) = A ( z ) + B ( z ) + C ( z ) − H V ′ Ω DE , (13)where A ( z ) = − (cid:16) V ′ H (cid:16) c p f ( z )Ω DE + X + V ′ H Ω k ( f ( z )Ω DE + X ) (cid:17)(cid:17) / , with X ab = (1 + z ) ( a Ω k + b Ω m (1 + z )) ,B ( z ) = 2 H V ′′ ( f ( z )Ω DE + X )and C ( z ) = H V ′′ X z . We solve this for f ( z ) and then use (12) again to yield w ( z ). The solution for f ( z ) is unique since we demand f (0) = 1. III. RECONSTRUCTING w ( z ) If we knew Ω m and Ω k perfectly then our three ex-pressions for w ( z ) would yield the same function w ( z ),assuming we lived in an exact FLRW universe. But whatif – as is commonly assumed – we impose Ω k = 0 when infact the true curvature is actually non-zero? It is usuallyimplicitly assumed that the error on w ( z ) will be of orderΩ k , but, as was shown in [13] this is not the case. Willmeasuring V ′ ( z ) possibly circumvent this? And further-more, are there similar issues from an imperfect knowl-edge of Ω m ? A. Zero curvature assumption
We can easily see the implications of incorrectly as-suming flatness by constructing the functions d L ( z ) and H ( z ) under the assumption of the ΛCDM in a curvedUniverse (i.e. assuming w = − , Ω k = 0) and insertingthe results into Eqs (10) and (11).If we then set Ω k = 0 in Eqs. (10) and (11) we ar-rive at the two corresponding w ( z ) functions (if they ex-ist) required to reproduce the curved forms for H ( z ) and d L ( z ) in a flat Universe with dynamic dark energy. Thiswould apply equally to d A ( z ) for that matter - the resultsare exactly the same for any distance indicator. Figure 4presents this method using for simplicity the concordancevalue of w = − m = 0 . w ( z ) can then be thought of as the function required to yield the same H ( z ) or d L ( z ) as inthe actual curved ΛCDM model: e.g., d L [flat , w ( z )] = d L [curved , w ( z ) = − . (14)For example for the Hubble rate the reconstructed w ( z )can be found analytically to be w ( z ) = −
13 Ω k (1 + z ) + 3Ω DE Ω k (1 + z ) + Ω DE , (15)without any dependence on a specific parameterisation.In the figure we show what happens for ΛCDM: cur-vature manifests itself as evolving dark energy. In thecase of the Hubble rate measurements this is fairly obvi-ous - we are essentially solving the equation Ω DE f ( z ) =Ω Λ + Ω k (1 + z ) where f ( z ) is given by Eq. (4). ForΩ k > w ( z ) must converge to − / k <
0, the opposite occurs and aredshift is reached when w → −∞ in an attempt to com-pensate albeit unsuccessfully for the positive curvature.Already we can see why the assumption that the error in w is of order the error in Ω k breaks down so drastically.Interestingly, the curved geodesics imply that the errorin w reconstructed from d L ( z ) and H ( z ) have opposingsigns at z & .
9, as can be seen by comparing the panelsfor the Hubble rate and the distance indicator in Fig. 4.Above the critical redshift the effect of curvature on thegeodesics becomes more important than the pure dynam-ics, and the luminosity distance flips w ( z ) in the oppositedirection to that reconstructed from H ( z ).In the case of volume measurements the reconstructed w ( z ) has a similar form to the w we obtained from the dis-tance measurements D ( z ). This can be seen from Eq. (5),where the distance information enters the equation as a 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 order50% and more at redshift z &
2, and diverges at finite redshift. The reconstructed w ( z ) from d L ( z ) for Ω k < z ≈ .
86, where it crosses the true value of − H ( z ). In order to make upfor the missing curvature, the reconstructed dark energy is behaving like a scalar field with a tracking behavior. These effectsarise even if the curvature is extremely small ( < . square power. For example in the closed Universe casethe reconstructed w ( z ) drops to more phantom values( − . − . z , and the distancecontribution in the volume measurements flips the recon-structed w ( z ) at z = 1 .
6. The critical redshift of this flipis determined by the redshift at which the curvature ofthe geodesics affecting distance measurements becomesmore important than the expansion rate. This playoffbecomes more finely balanced for volume measurementsdue to the fact that H ( z ) appears both in D ( z ) (as asquare power) and on its own. Hence w ( z ) has to workharder in reproducing curvature to counterbalance theopposing trends of expansion history and geometry, andso the balance is achieved at higher redshift. The specificredshift at which this happens is dependent on Ω m in thatlower values imply higher value of the critical redshift. z w ( z ) w(z) from Volume measurements with changing Ω k Ω k = −0.001−0.2 −0.05 −0.010.050.01 Ω k = 0.0010.2 FIG. 5:
Reconstructed dark energy from volume mea-surements while incorrectly assuming flatness -
Similarto the case for distance measurements in a closed Universe,the reconstructed w ( z ) must initially be phantom in orderto compensate for curvature, and crosses the true value of w = − z .
6, which is greater than the red-shift of 0.86 for the distance measurements alone [13]. Afterthis point, the w ( z ) increases to overcome the curvature ofthe geodesics. We have shown that incorrectly assuming flatness canresult in a reconstructed w ( z ) that mimics dynamics,yielding errors on w that are much larger than the or-der of errors on Ω k . One might then ask if similar errorswill result when incorrectly assuming a particular valuefor the matter density in the Universe, Ω m . B. Uncertainties in the Matter content Ω m We consider the similar case of reconstructing w ( z ) ina flat Universe but here the errors occur when assuming the concordance value of Ω m = 0 . w ( z ) reconstructed from Hubblemeasurements Eq. (10) reduces to w ( z ) = −
13 2(1 + z ) HH ′ − H H (1 + z ) [Ω m (1 + z )] − H . (16)Similar expressions are found for both the distance andvolume measurements. The w ( z ) curves obtained fromincorrectly assuming Ω m = 0 . m can only affect the dark energy den-sity, and thus change the value of H ( z ). As Ω m is onlypresent in all three observables through H ( z ) or inte-grals of 1 /H ( z ), the reconstructed w ( z ) is the same forall three measurements. Interestingly, the reconstructed w ( z ) curves do not go through w = − z = 0, but arespread between -0.85 and -1.15 for 0 . < Ω m < .
4. Thisis also shown in Fig. 7. z w ( z ) w(z) from Volume, distances and the Hubble parameter with changing Ω m Ω m = 0.3010.35 0.31 0.4 Ω m = 0.2990.290.250.2 FIG. 6:
Reconstructed dark energy from an incor-rectly estimated matter density -
The reconstructed w ( z )for changing Ω m from all three measurements ( H, D, dV /dz ).Since we assume flatness while changing Ω m , all three ob-servables yield the same reconstructed w ( z ), since Ω m onlyenters the functions through H ( z ) or integrals of 1 /H . ForΩ m > . w = 0 as z → ∞ .For Ω m < . w ( z ) is of the same form to what is recon-structed from neglecting curvature in a closed Universe (seeFig. 4), and the phantom w tends to −∞ as it attempts tocompensate for the ‘missing’ matter density. Given any scenario of an assumed cosmology that dif-fers from the ‘true’ Universe, we can derive the valueof today, w ( z = 0) from both the Hubble and distancemeasurements as w (0) = 3 − k ∗ − m + Ω k m + 6Ω k ∗ − m ∗ − − k (17) ∼ ǫ m ( − m ∗ ) − k − m ∗ ) − , where ǫ m = Ω m ∗ − Ω m as defined above where the as-terisk indicates assumed but incorrect values of the corre-sponding quantities. We vary this equation in one ‘true’density (Ω m or Ω k ) at a time, while keeping the otherconstant at the assumed value of either Ω k = Ω k ∗ orΩ m = Ω m ∗ to produce the curves in Fig. 7. This param-eter w ( z = 0) allows us to easily quantify the affect ofassuming an incorrect cosmological model on the inferredlow-redshift value of w . −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2−1.7−1.6−1.5−1.4−1.3−1.2−1.1−1−0.9−0.8 ∆ Ω i w ( z = ) Ω m Ω k FIG. 7:
Low redshift variation in w ( z ) from H ( z ) and D ( z ) - incorrectly assuming concordance values of Ω m = 0 . k = 0 results in a variation in the low-redshift valueof w ( z ) reconstructed from observables. The relationship be-tween the error in the cosmological parameter and the recon-structed value for w (while keeping the other cosmologicalparameter fixed at the prior value) is shown for both Ω m (the green curve) and Ω k (the blue curve). IV. PARAMETRIC DEGENERACIES
We now want to connect the non-parametric approachwe have followed above with standard approaches todegeneracies and so we expand Eqs. (10) and (11) for theHubble rate and distance measurements to first order in x = z/ (1 + z ). This allows us to link to the parameters w , w a used in the Chevallier-Polarski-Linder (CPL)[25, 26] parameterisation w CPL ( z ) = w + w a (cid:16) z z (cid:17) ,which is used in the Dark Energy Task Force report [1].The values of w , w a obtained using this expansion aregiven below. From Hubble rate measurements w = − Ω k + 3Ω DE − Ω m ) w a = 43 Ω k Ω DE (1 − Ω m ) (18) From luminosity distance measurements w = − Ω k + 3Ω DE − Ω m ) w a = −
23 Ω k (Ω k − Ω DE )(1 − Ω m ) (19)We plot in Figure 8 the non-parametric reconstructed w ( z ) along with the reconstructed w C P L ( z ) from thecoefficients given by Eqs. (18, 19) for the observables H ( z ) and d L ( z ). z w ( z ) Reconstructed w(z) from measurements of the H(z) using series expansion Ω k = 0.050.01 Ω k = −0.05−0.010 0.5 1 1.5 2 2.5 3−1.15−1.1−1.05−1−0.95−0.9−0.85 z w ( z ) Reconstructed w(z) from measurements of d L (z) using series expansion Ω k = 0.050.01 Ω k =−0.05−0.01 FIG. 8:
Degeneracies in standard parameterisations - w ( z ) = w + w a z z using the coefficients in Eqs. (19) and(18) (solid lines) compared with the fully non-parametric w ( z )inferred from Hubble and distance measurements. Using alimited parameterisation of w ( z ) like this incorrectly makesit appear that dark energy and curvature are not completelydegenerate, leading to artificially strong constraints on curva-ture and w , w a . V. CONCLUSIONS AND OUTLOOK
We have explored the degeneracies between the darkenergy equation of state w ( z ) and cosmic parameters us-ing a non-parametric approach. This means we are ableto write down the precise w ( z ) that will be reconstructedfrom perfect data if slightly wrong or biased values forthe cosmic parameters Ω k , Ω m are assumed. This is com-plementary to traditional methods which typically use anaggressive compression of the w ( z ) function onto a cou-ple of parameters (usually w , w a ) and then study thedegeneracy between these and other cosmic parameters.Our approach is superior in one way however: degenera-cies between w ( z ) and some cosmic parameters such asΩ k can appear to be quite weak in the parameterised ap-proach. However, in the case of distance measurementsthis is completely artificial and due to strong assumptionsabout the allowed form of w ( z ) since the degeneracy isperfect if w ( z ) is allowed to be totally free.We extend the work of [13] to show the reconstructed w ( z ) from measurements of volume for both wrongly as-sumed Ω k and Ω m . As with Hubble and distance mea-surements we show that the errors in w ( z ) that result from uncertainty in the cosmic parameters are much larger than the uncertainty in Ω k or Ω m , especially atlarge redshifts. We have shown that curvature affectsmeasurements of H ( z ) and D ( z ) in complementary ways,with the error at high redshift having opposite signs foran error in Ω k . In the case of an Ω m error, Hubble,distance and volume measurements all lead to the sameerroneously reconstructed w ( z ), a manifestation of thedark matter-dark energy degeneracy highlighted in [3].In this review we have assumed perfect data for Hub-ble rate, distance and volume at all redshifts. It wouldbe interesting to extend our non-parametric approach tothe case of imperfect data which has incomplete redshiftcoverage and errors on the observables. This is left tofuture work but will allow contact with the approachesin [29, 30]. Acknowledgments – we thank Luca Amendola, ChrisBlake, Thomas Buchert, Daniel Eisenstein, George El-lis, Martin Kunz, Roy Maartens, Bob Nichol and DavidParkinson for useful comments and insights. MC thanksAndrew Liddle and FTC for support. BB and CC ac-knowledge support from the NRF and RH acknowledgesfunding from KAT. [1] A. Albrecht et al. , Report of the Dark Energy Task Force,astro-ph/0609591 (2006)[2] E. V. Linder, Astropart.Phys. , 391 (2005)[3] M. Kunz, astro-ph/0702615, (2007)[4] K. Glazebrook and the WFMOS Feasibility Study DarkEnergy Team, White paper submitted to the DarkEnergy Task Force, astro-ph/0507457; B. A. Bassett,R. C. Nichol and D. J. Eisenstein [for the WFMOS Col-laboration], astro-ph/0510272;[5] See e.g. T. Buchert, M. Carfora, Phys. Rev. Lett. ,031101 (2003); N. Li, D. J. Schwarz, gr-qc/0702043;S. Rasanen, Class. Quant. Grav. (2006) 1823;A.A. Coley, N. Pelavas Phys. Rev. D , 043506 (2007)[6] See eg. T. Giannantonio et al. , Phys. Rev. D , 063520(2006); S. Ho e t al. arXiv:0801.0642 (2008)[7] G. Bernstein, A. J. (2006) 598[8] L. Knox, Y-S. Song, H. Zhan, astro-ph/0605536 (2006)[9] M. Tegmark et al. , Phys. Rev. D , 123507 (2006)[10] K. Glazebrook et al. , astro-ph/0701876 2007[11] D. Huterer & M. S. Turner, Phys. Rev. D , 1235272001[12] S. Weinberg, ApJL , L233 (1970)[13] C. Clarkson, M. Cortˆes, & B. Bassett, J. C. A. P. , 11(2007)[14] Y. Gong, Q. Wu & A. Wang, arXiv:0708.1817 (2007);L. Mersini-Houghton, Y. Wang, P. Mukherjee &E. Kafexhiu, arXiv:0705.0332 (2007); E. Wright, A. J. , 633 (2007); K. Ichikawa, M. Kawasaki, T. Sekiguchi,& T. Takahashi, J. C. A. P. , 5 (2006); C.-B. Zhao,J.-Q. Xia, H. Li, et al. , Phys. Lett. B, , 8 (2007);K. Ichikawa, & T. Takahashi, , J. C. A. P. , , 1 (2007) [15] Y. Wang & P. Mukherjee, astro-ph/0703780 (2007)[16] Z.-Y. Huang, B. Wang, & R.-K. Su, Int. Journ. Mod.Phys. A, , 1819 (2007)[17] D. N. Spergel et al. Ap. J. S. , , 377 (2007)[18] S. Perlmutter, et al.
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