Dynamical dark energy or simply cosmic curvature?

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ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Dynamical dark energy or simply cosmiccurvature?

Chris Clarkson1, Marina Cortes1,2 and Bruce Bassett1,3

1 Cosmology and Gravity Group, Department of Mathematics and AppliedMathematics, University of Cape Town, Rondebosch 7701, South Africa2 Astronomy Centre, University of Sussex, Brighton BN1 9QH, UK3 South African Astronomical Observatory, Observatory, Cape Town,South AfricaE-mail: chris.clarkson@uct.ac.za, mvc23@sussex.ac.uk and bruce@saao.ac.za

Received 23 April 2007Accepted 17 July 2007Published 9 August 2007

Online at stacks.iop.org/JCAP/2007/i=08/a=011doi:10.1088/1475-7516/2007/08/011

Abstract. We show that the assumption of a flat universe induces criticallylarge errors in reconstructing the dark energy equation of state at z � 0.9even if the true cosmic curvature is very small, O(1%) or less. The spuriouslyreconstructed w(z) shows a range of unusual behaviour, including crossing of thephantom divide and mimicking of standard tracking quintessence models. For1% curvature and ΛCDM, the error in w grows rapidly above z ∼ 0.9 reaching(50%, 100%) by redshifts of (2.5, 2.9) respectively, due to the long cosmologicallever arm. Interestingly, the w(z) reconstructed from distance data and Hubblerate measurements have opposite trends due to the asymmetric influence of thecurved geodesics. These results show that including curvature as a free parameteris imperative in any future analyses attempting to pin down the dynamics of darkenergy, especially at moderate or high redshifts.

Keywords: dark energy theory, cosmological constant experiments, classicaltests of cosmology

ArXiv ePrint: astro-ph/0702670

c©2007 IOP Publishing Ltd and SISSA 1475-7516/07/08011+8$30.00

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Contents

1. Introduction 2

2. Reconstructing the DE equation of state from observations 3

3. Zero curvature assumption 5

4. Conclusions 6

Acknowledgments 7

References 7

1. Introduction

The quest to distinguish between a cosmological constant, dynamical dark energy andmodified gravity has become the dominant obsession in cosmology. Formally elevatedto the status of one of the most important problems in fundamental physics [22, 23]uncovering the true nature of dark energy, as encapsulated in the ratio of its pressureto density, w(z) = pDE/ρDE, has become the focus of multi-billion dollar proposedexperiments using a wide variety of methods, many at redshifts above unity (see e.g. [1]).

Unfortunately these experiments will only measure a meagre number of w(z)parameters to any precision—perhaps two or three [3]4 —since the standard methods allinvolve integrals over w(z) and typically suffer from subtle systematic effects. As a resultof this information limit, studies of dark energy have traditionally fallen into two groups.The first group (see e.g. [10]) have taken their parameters to include (Ωm, ΩDE, w) with wconstant and often set to −1. The second group are interested in dynamical dark energyand have typically assumed Ωk = 0 for simplicity while concentrating on constrainingw(z) parameters (see e.g. [11])5.

In retrospect, the origins of the common practice of ignoring curvature in studiesof dynamical dark energy are clear. Firstly, the curved geodesics add an unwelcomecomplexity to the analysis that has typically been ignored in favour of studies ofdifferent parametrizations of w(z). Secondly, standard analysis of the cosmic microwavebackground (CMB) and baryon acoustic oscillations (BAO) in the context of adiabaticΛCDM also put stringent limits on the curvature parameter, e.g. Ωk = −0.003 ± 0.010from WMAP + SDSS [12, 13]. As a result it was taken for granted that the impact onthe reconstructed w(z) would then be proportional to Ωk and hence small compared toexperimental errors.

Further support for the view that Ωk should not be included in studies of dark energycame from information criteria [8] which showed that for adiabatic ΛCDM includingcurvature is not warranted from a model selection viewpoint. Finally, even two-parametermodels of dark energy suffer from severe limitations [14]. To include Ωk as an additionalparameter would only further dilute constraints on w(z).

4 Note that non-standard methods such as varying alpha offer the possibility of extracting significantly moreinformation [2].5 There is however a small group of studies which have allowed both for dynamical w(z) and Ωk �= 0; e.g. [4]–[6].

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However, we will show that ignoring Ωk induces errors in the reconstructed darkenergy equation of state, w(z), that grow very rapidly with redshift and dominate thew(z) error budget at redshifts (z � 0.9) even if Ωk is very small. The aim of this paperis to argue that future studies of dark energy, and in particular, of observational data,should include Ωk as a parameter to be fitted alongside the w(z) parameters.

Looking back, this conclusion should not be unexpected. Firstly the case for flatnessat the sub-per cent level is not yet compelling: a general ΛCDM analysis [9], allowing forgeneral correlated adiabatic and isocurvature perturbations, found that WMAP, togetherwith large-scale structure and HST Hubble constant constraints, yields Ωk = −0.06±0.02.We will show that significantly smaller values of Ωk lead to large effects at redshifts z � 0.9well within reach of the next generation of surveys.

Secondly, Wright (e.g. [15]) has petitioned hard against the circular logic that onecan prove the joint statement (Ωk = 0, w = −1) by simply proving the two conditionalstatements (Ωk = 0|w = −1) and (w = −1|Ωk = 0). This has been verified in [4, 6]where values of |Ωk| ∼ 0.05 or larger are found to be acceptable at 1− σ if one allows fortwo-parameter varying w(z). Clearly with more dark energy parameters—or correlatedisocurvature perturbations—even larger Ωk would be consistent with the current datasets.Given that the constraints on Ωk evaporate precisely when w deviates most stronglyfrom a cosmological constant, it is clearly inconsistent to assume Ωk = 0 when derivingconstraints on dynamical dark energy. The uncertainty around the current value of Ωk

begs the question, how does the error on w(z) scale with Ωk?We show below and in figure 1 that the growth of the error in w(z), arising from

erroneously assuming the cosmos to be flat, is very rapid, both in redshift and Ωk. Thereason for this rapid growth of the error in w can be easily understood. Consider aΛCDM model with small but non-zero Ωk. The curvature contribution to H(z)2 scalesas Ωk(1 + z)2 while ΩΛ is constant. In addition the curved geodesics drastically alterobserved distances when they are a sizable fraction of the curvature radius. If one triesto reproduce the observations in such a universe with a flat model with varying w(z), itis clear that w(z) must deviate strongly from w = −1 as z increases as the dark energytries to mimic the effects of the rapidly growing curvature and the curved geodesics. Acrude estimate for the redshift for strong deviations from the true w would follow fromsetting Ωk(1 + z)2 ∼ ΩDE. For values of Ωk = 0.05 and ΩDE = 0.7 this yields z ≈ 2.7,within reach of BAO surveys such as WFMOS [17] and VIRUS [24].

We will show in detail below that the actual redshift where problems set in issignificantly lower in the case of luminosity distance measurements due to the addedcurved geodesic effects. In fact, if the curvature is negative, a redshift is reached wherethe dark energy cannot mimic the curvature at all, unless ρDE can change sign, and thereconstructed dark energy has w → −∞. Related work [7] has investigated some of theseissues. However because a restricted parametrization was assumed for w(z) the resultsfound were significantly less severe than we find.

2. Reconstructing the DE equation of state from observations

Here we wish to compare the reconstructed dark energy w(z) from perfect distance (eitherluminosity, dL(z), or area, dA(z)) and Hubble rate (H(z)) measurements as these are thebasis for all modern cosmology experiments including those using Type Ia supernovae,

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Figure 1. Reconstructing the dark energy equation of state assuming zerocurvature 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 errorof order 50% and more at redshift z � 2, and diverges at finite redshift. Thereconstructed w(z) from dL(z) for Ωk < 0 is phantom until z ≈ 0.86, where itcrosses the true value of −1 and then crosses 0 at high redshift, where the bendingof geodesics takes over from dynamical behaviour, producing errors in oppositedirection to the DE reconstructed from H(z). In order to make up for the missingcurvature, the reconstructed DE is behaving like a scalar field with a trackingbehaviour. These effects arise even if the curvature is extremely small <0.1%.

BAO, weak lensing etc. Since we are interested in the effects of ignoring curvature, letus assume we have perfect knowledge of both H(z) and dL(z) across a range of redshifts:

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how would we reconstruct w(z)? The luminosity distance may be written as

dL(z) =c(1 + z)

H0

√−Ωk

sin

(√−Ωk

∫ z

0

dz′H0

H(z′)

)(1)

which is formally valid for all curvatures, where H(z) is given by the Friedmann equation,

H(z)2 = H20

{Ωm(1 + z)3 + Ωk(1 + z)2 + ΩDE exp

[3

∫ z

0

1 + w(z′)

1 + z′dz′

]},

and ΩDE = 1−Ωm −Ωk. Knowledge of H(z) allows us to directly probe the dynamics ofthe universe, while dL(z) is strongly affected by the cosmic curvature which distorts nullgeodesics away from straight lines.

We can invert these functions to give two different expressions for the equation ofstate of the dark energy, w(z): one in terms of H(z) and its first derivative,

w(z) = −1

3

ΩkH20 (1 + z)2 + 2(1 + z)HH ′ − 3H2

H20 (1 + z)2[Ωm(1 + z) + Ωk] − H2

, (2)

where ′ = d/dz; and the other in terms of the dimensionless luminosity distanceDL = (H0/c)dL, and its derivatives

w(z) = 23[(1 + z){[ΩkD

2L + (1 + z)2]D′′

L − 12(ΩkD

′2L + 1)[(1 + z)D′

L − DL]}]× {[(1 + z)D′

L − DL]{(1 + z)[Ωm(1 + z) + Ωk]D′2L − 2[Ωm(1 + z) + Ωk]

× DLD′L + ΩmD2

L − (1 + z)}}−1. (3)

Note that ΩDE has dropped out of both expressions.If, in addition, we also knew Ωm and Ωk perfectly then these two expressions would

yield the same function w(z). But what if—as is commonly assumed—we impose Ωk = 0when in fact the true curvature is actually non-zero? It is usually implicitly assumed thatthe error on w(z) will be of order Ωk, and this is indeed true for z � 0.9 (see figure 1).However, this intuition is strongly violated for z � 0.9, even given perfect knowledge ofdL(z) or H(z).

3. Zero curvature assumption

We may uncover the implications of incorrectly assuming flatness by computing thew = −1, Ωk �= 0 functional forms for dL(z) and H(z) and inserting the results intoequations (2) and (3). If we then set Ωk = 0 in equations (2) and (3) we arrive at the twocorresponding w(z) functions (if they exist) required to reproduce the curved forms forH(z) and dL(z) (or dA(z)—for any distance indicator the results are exactly the same).Figure 1 presents for simplicity the concordance value of w = −1 but we have checkedthat the qualitative results do not depend on the ‘true’ underlying dark energy model6.We assume Ωm = 0.3 in all expressions7; numbers quoted are weakly dependent on this.The resulting w(z) can then be thought of as the function which produces the same H(z)or dL(z) as the actual curved ΛCDM model: e.g.,

dL[flat, w(z)] = dL[curved, w(z) = −1]. (4)

6 In fact, the results presented here are qualitatively the same for any assumed Ωk which is different from the truevalue.7 There is a degeneracy between Ωm and dark energy in general [16].

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In the figure we show what happens for ΛCDM: curvature manifests itself as evolvingdark energy. In the case of the Hubble rate measurements this is fairly obvious: we areessentially solving the equation ΩDEf(z) = ΩΛ + Ωk(1 + z)2 where f(z) is the integral inthe last term of equation (2). For Ωk > 0, w(z) must converge to −1/3 to compensate forthe curvature. For Ωk < 0, the opposite occurs and a redshift is reached when w → −∞in an attempt to compensate (unsuccessfully) for the negative curvature. Already we cansee why the assumption that the error in w is of order the error in Ωk breaks down sodrastically.

Interestingly, the curved geodesics imply that the error in w reconstructed from dL(z)and H(z) have opposing signs at z � 0.9, as can be seen by comparing the panels infigure 1. Above the critical redshift the effect of curvature on the geodesics becomes moreimportant than the pure dynamics, and the luminosity distance flips w(z) in the oppositedirection to that reconstructed from H(z). Given that dark energy reconstructed fromH(z) pulls in the other direction implies that knowledge of H(z) is a powerful tool inmeasuring curvature. In fact, we may directly combine measurements of H(z) and DL(z)to measure curvature independently of other cosmological parameters (except H0) via

Ωk =[H(z)D′(z)]2 − c2

[H0D(z)]2, (5)

where dL(z) = (1 + z)D(z). This is an advantage of BAO surveys which provide asimultaneous measurement of both dA and H(z) at the same redshift and, with Lyman-break galaxies as targets, can easily reach z = 3 [17].

4. Conclusions

We have argued that cosmic curvature must be included as a free parameter in all futurestudies of dark energy. This is particularly crucial at redshifts z � 0.9 where the resultingerror in w(z) can exceed Ωk by two orders of magnitude or more. Even with perfectmeasurements of dL(z) or H(z) the error induced on the measured w(z) by assumingflatness when in fact Ωk = 0.01 reaches 100% by z ≈ 2.9; sub-per cent errors on w(z ∼ 5)would require |Ωk| � 10−5—see figure 2. This implies that we may never do better than1% errors in w(z) since we expect Ωk ∼ 10−5, even if inflation predicts zero curvature, atthe level of perturbations. It is interesting to note that the back-reaction of cosmologicalfluctuations may cause effective non-zero curvature that may yield practical limits on ourability to measure w(z) accurately at z > 1 (see e.g. [18]).

Finally it is interesting to ask what the results imply for other tests of dark energysuch as the integrated Sachs–Wolfe (ISW) effect. Since the ISW effect is sensitive to bothcurvature and dark energy, and since it is possible to probe the ISW effect at z > 1 [19]the effect of curvature may also be important there. Similar sentiments apply to clustersurveys which are sensitive to both the growth function and the volume of space and candetect clusters at high redshift through the Sunyaev–Zel’dovich effect. Since the volumeof space is very sensitive to Ωk (see e.g. [20]) it is likely that the reconstructed w(z) fromthis method will also be subject to large errors if one incorrectly assumes flatness.

As a result, measuring Ωk accurately (to σΩk< 10−3) will play an important role

in the quest to hunt down the true origin of dark energy in the coming decade. Baryon

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Figure 2. The accuracy with which we must know Ωk in order to measure w(z)to within a given error.

acoustic oscillations, in conjunction with other probes of distance such as weak lensingare likely to play a key role in illuminating these shadows of curvature [21].

Acknowledgments

We thank Chris Blake, Thomas Buchert, Daniel Eisenstein, George Ellis, Renee Hlozek,Martin Kunz, Roy Maartens, Bob Nichol and David Parkinson for useful comments andinsights. MC thanks Andrew Liddle and FTC for support. BB and CC acknowledgesupport from the NRF.

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