Redshift drift as a model independent probe of dark energy
aa r X i v : . [ g r- q c ] F e b Redshift drift as a model independent probe of dark energy
Asta Heinesen ∗ Univ Lyon, Ens de Lyon, Univ Lyon1, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F–69007, Lyon, France
It is well known that positive values of redshift drift is a signature of dark energy within theconventionally studied Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe models. Here weshow – without making assumptions on the metric tensor of the Universe – that redshift drift isa promising direct probe of violation of the strong energy condition within the theory of generalrelativity.
INTRODUCTION
In general relativistic theory energy conditions arephysically motivated constraints, which can be appliedto the energy momentum tensor of space-time. Energyconditions are important tools for constraining the pos-sible solutions of the Einstein field equations and for de-riving general theorems about the nature of gravitatingsystems. The Penrose and Hawking singularity theorems[1, 2] use energy conditions to arrive at physical scenarioswhere the developments of singularities are unavoidable.In particular, the strong energy condition is a central as-sumption in the focusing theorem, which states the con-ditions under which a matter congruence develops singu-larities in finite proper time.Within the Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) framework of cosmology the strong energy con-dition is considered abandoned by observations [3, 4].The most direct evidence for violation of the strong en-ergy condition, comes from the observed acceleration ofspace when interpreting data from supernovae of typeIa within the FLRW class of models [5–7]. However, nomodel independent falsification of the strong energy con-dition has been made to date in cosmology.Measurements of the drift of redshift in proper time ofthe observer [8–10] are promising probes of the expan-sion history of the Universe. In the conventionally stud-ied FLRW universe models, a positive value of redshiftdrift is a signature of dark energy [11]. The direct detec-tion of the time-evolution of redshift – expected withinone to a few decades of observation time with facilitiessuch as CODEX and the Square Kilometer Array (SKA)[12, 13] – allows for model independent determination ofkinematic properties of the Universe. The potential fordoing model independent analysis with redshift drift datamust be accompanied by a theoretical understanding ofredshift drift in generic universe models.So far most theoretical studies of redshift drift havebeen done within the FLRW class of models – though see[14–23]. Local inhomogeneities and anisotropies in gen-eral contribute with accumulated effects along null raysand systematic effects from the position of the observer.Thus, local structure alter measurements of redshift drift[23] resulting in a violation of the signal expected fromFLRW modelling. Such effects from local inhomogene- ity are not a priori expected to be subdominant, andthe size of the effects must ultimately be determined bydata. This in turn raises the question if redshift drift asa probe of dark energy – or more generally the strongenergy condition – is valid in universe models that arenot subject to the FLRW idealisation. In this paper wepropose a model independent test of the strong energycondition by redshift drift measurements.Notation and conventions: Units are used in which c = 1.Greek letters µ, ν, . . . label spacetime indices in a generalbasis. Einstein notation is used such that repeated in-dices are summed over. The signature of the spacetimemetric g µν is ( − +++) and the connection ∇ µ is the Levi-Civita connection. Round brackets ( ) containing indicesdenote symmetrisation in the involved indices and squarebrackets [ ] denote anti-symmetrisation. Bold notation V for the basis-free representation of vectors V µ is used oc-casionally. REDSHIFT DRIFT IN A GENERAL SPACE-TIME
In this section we review the expression for redshiftdrift for a generic space-time congruence of physical ob-servers and emitters – for details, see [23]. We considera general space-time congruence of observers and emit-ters (henceforth referred to as the ‘observer congruence’)with worldlines generated by the 4–velocity field u andparametrised by the proper time function τ . The redshiftdrift of light rays generated by the 4-momentum field k and passing from an emitter placed at spacetime point E to an observer situated at spacetime point O can bewritten dzdτ (cid:12)(cid:12)(cid:12) O = (1 + z ) H O − H E + S E→O , (1)with S E→O = E E Z λ O λ E dλ I , I ≡ − k ν ∇ ν (cid:18) e µ ∇ µ EE (cid:19) , (2)where the function λ satisfies k µ ∇ µ λ = 1, and is an affineparameter along each null line. The redshift z and photon For another interesting representation of redshift drift in generalspace-time models, see [22]. energy function E associated with the light rays are givenby z ≡ E E E O − , E ≡ − k µ u µ , (3)and the change of the photon energy E along a given nullray is given by H ≡ − k µ ∇ µ EE = 13 θ − e µ a µ + e µ e ν σ µν . (4)The spatial unit vector e describes the spatial propa-gation direction of the null ray relative to an observercomoving with u , and is defined by the decomposition k µ = E ( u µ − e µ ) . (5)The function H is an observationally natural generalisa-tion of the Hubble parameter of FLRW space-time: H O plays the role of the proportionality constant betweenredshift and distance in the generalised Hubble law validfor arbitrary space-times in the O ( z ) vicinity of the ob-server [25]. The variables θ , σ µν and a µ describe theexpansion, shear and 4–acceleration of the observer con-gruence. Together with the vorticity tensor ω µν , theydescribe the kinematics of the observer congruence ∇ ν u µ = 13 θh µν + σ µν + ω µν − u ν a µ ,θ ≡ ∇ µ u µ , σ µν ≡ h β h ν h αµ i ∇ β u α ,ω µν ≡ h βν h αµ ∇ [ β u α ] , a µ ≡ ˙ u µ , (6)where h νµ ≡ u µ u ν + g νµ is the spatial projection tensordefined in the frame of the 4–velocity field u and wheretriangular brackets hi denote traceless symmetrisation inthe involved indices of a tensor in three dimensions . Theoperator ˙ ≡ u µ ∇ µ denotes the derivative in proper timealong flow lines of u . From geometrical identities, theevolution of the kinematic variables θ , σ µν and ω µν alongthe observer flow lines can be expressed as˙ θ = − θ − σ µν σ µν + ω µν ω µν − u µ u ν R µν + D µ a µ + a µ a µ , (7)˙ σ µν = − θσ µν − σ α h µ σ αν i + ω α h µ ω αν i + 2 a α σ α ( ν u µ ) + D h µ a ν i + a h µ a ν i − u ρ u σ C ρµσν − h α h µ h βν i R αβ , (8)˙ ω µν = − θω µν + 2 σ α [ µ ω ν ] α − D [ µ a ν ] − a α ω α [ µ u ν ] , (9) For two indices we have that the traceless parts of symmet-ric spatial tensors T µν = T ( µν ) = h αµ h βν T ( αβ ) is given by T h µν i = T µν − h µν T . Analogously for a tensor with threeindices satisfying T µνρ = T ( µνρ ) = h αµ h βν h γρ T ( αβγ ) , we have T h µνρ i = T µνρ − ( T µ h νρ + T ν h ρµ + T ρ h µν ). For four indiceswe have T h µνρκ i = T µνρκ − ( T h µν i h ρκ + T h µρ i h νκ + T h µκ i h νρ + T h νρ i h µκ + T h νκ i h µρ + T h ρκ i h µν ) − T h ( µν h ρκ ) . We have usedthe short hand notations T µν ≡ h ρκ T µνρκ , T µ ≡ h νρ T µνρ , and T ≡ h µν T µν . where R µν is the Ricci tensor of the space-time and C µνρσ is the Weyl tensor. The operator D µ is the covariantspatial derivative defined on the 3-dimensional spaceorthogonal to u . The integrand in (2) is convenientlyexpressed in terms of the (differentiated) kinematic vari-ables of the observer congruence in the following seriesexpansion [23] I = I o + e µ I e µ + d µ I d µ + e µ e ν I ee µν + e µ d ν I ed µν + e µ e ν e ρ I eee µνρ (10)with coefficients I o ≡ −
13 (4 ω µν ω µν + D µ a µ + a µ a µ ) − d µ d µ , I e µ ≡ D µ θ + 13 θa µ + 25 D ν σ νµ + 25 a ν σ µν − a ν ω µν , I d µ ≡ − a µ , I ee µν ≡ − (cid:16) ω αµ σ αν + 4 ω α h µ ω αν i + D h ν a µ i + a h µ a ν i (cid:17) , I ed µν ≡ σ µν − ω µν ) , I eee µνρ ≡ D h ρ σ µν i + a h µ σ νρ i , (11)where d µ ≡ h µν e α ∇ α e ν denotes the spatially projected‘4–acceleration’ of e . The magnitude of d can be seenas a measure of the failure of e to define an axis of localrotational symmetry, and is thus a quantification of thelocal departure from isotropy [24]. In deriving (11), ithas been assumed that the null congruence is irrotational,such that ∇ [ α k ν ] = 0. Note that all coefficients in (11)with more than one spacetime index are traceless. MODEL INDEPENDENT TEST OF THESTRONG ENERGY CONDITION
For the purpose of examining the strong energy condi-tion, it is convenient to rewrite the expression for redshiftdrift (1), such that the Ricci curvature of the space-timeappears explicitly in the formula. For this purpose thefollowing identity(1 + z ) H O − H E = E E Z λ O λ E dλ A , A ≡ k ν ∇ ν (cid:18) H E (cid:19) (12)will be useful. Combining (1), (2) and (12) gives dzdτ (cid:12)(cid:12)(cid:12) O = E E Z λ O λ E dλ Π , Π ≡ I + A . (13)In the FLRW limit, the integrand Π reduces to the wellknown length scale acceleration ‘¨ a/a ’, where a is the uni-form FLRW scalefactor. The function −A / H enters in The acting of D µ on a tensor field T γ ,γ ,..,γ m ν ,ν ,..,ν n is defined as: D µ T γ ,γ ,..,γ m ν ,ν ,..,ν n ≡ h α ν h α ν ..h α n ν n h γ β h γ β ..h γ m β m h σµ ∇ σ T β ,β ,..,β m α ,α ,..,α n . the ‘Hubble law’ for generic space-times [25] as an effec-tive deceleration parameter, replacing the FLRW decel-eration parameter in the series expansion of luminositydistance in redshift – see [25] for details. In a similarspirit as for H and I , the function A can be written as atruncated multipole expansion A = A o + e µ A e µ + e µ e ν A ee µν + e µ e ν e ρ A eee µνρ + e µ e ν e ρ e κ A eeee µνρκ (14)with coefficients A o ≡ − u µ u ν R µν + 23 D µ a µ − σ µν σ µν + 13 ω µν ω µν , A e µ ≡ − θa µ + a ν σ µν + a ν ω µν − D µ θ − D ν σ νµ − h νµ ˙ a ν , A ee µν ≡ a h µ a ν i − σ αµ ω αν − σ α h µ σ αν i + ω α h µ ω αν i +2 D h µ a ν i − u ρ u σ C ρµσν − h α h µ h βν i R αβ , A eee µνρ ≡ − D h ρ σ µν i − a h µ σ νρ i , A eeee µνρκ ≡ σ h µν σ ρκ i , (15)where R µν is the Ricci curvature of the space-time, andwhere we have used the deviation equations (7) and (8).In the derivation of (14) we have used the definition (4)to write A = k µ ∇ µ ( H ) /E + H together with the identity k ν ∇ ν e µ E = ( e µ − u µ ) H − e ν (cid:18) θh µν + σ µν + ω µν (cid:19) + a µ . (16)By combining the multipole coefficients in (11) and (15)of the same order, we finally have that the integrand in(13) can be expressed asΠ = Π o + e µ Π e µ + d µ Π d µ + e µ e ν Π ee µν + e µ d ν Π ed µν + e µ e ν e ρ Π eee µνρ + e µ e ν e ρ e κ Π eeee µνρκ (17)with coefficientsΠ o ≡ − u µ u ν R µν + 13 D µ a µ − a µ a µ − d µ d µ − σ µν σ µν − ω µν ω µν , Π e µ ≡ − θa µ + 75 a ν σ νµ − a ν ω µν − h νµ ˙ a ν , Π d µ ≡ − a µ , Π ee µν ≡ a h µ a ν i − σ α h µ σ αν i − ω α h µ ω αν i − σ αµ ω αν + D h µ a ν i − u ρ u σ C ρµσν − h α h µ h βν i R αβ , Π ed µν ≡ σ µν − ω µν ) , Π eee µνρ ≡ − a h µ σ νρ i , Π eeee µνρκ ≡ σ h µν σ ρκ i . (18)The multipole coefficients (18) are given in terms of kine-matic and dynamic variables associated with the observercongruence and the Ricci curvature tensor R µν . While the coefficients in (11) and (15) contain spatial gradi-ents of θ and σ µν , these contributions cancel in (18),and the only spatial gradients that remain are of the4–acceleration a µ . The anisotropic function Π reduces tothe FLRW scale factor acceleration ‘¨ a/a ’ in the idealisedisotropic and homogeneous limit. In the FLRW limit, theonly non-zero contribution is the first term − u µ u ν R µν of the monopole contribution Π o – all of the remainingterms in (18) arise from the contributions of inhomo-geneity and anisotropy along the null rays. These modi-fications of the FLRW law for redshift drift need not besmall or cancel under the integral sign (13), and the mea-surements of redshift drift cannot a priori be expected toobey FLRW predictions in realistic universe models withstructure.The dominant monopole approximation: Let us con-sider the case where the monopole term Π o is dominantin the integral expression for redshift drift (13), suchthat the contributions to the integral from the remainingterms in the series expansion (17) are small compared tothe contributions from Π o . This corresponds to the phys-ical assumption that the systematic alignment of e and d with the fluid variables such as shear and 4–accelerationis weak over the length scales of photon propagation. Inthis scenario, we have that all spatial directions of pho-ton propagation can be treated on equal footing at lowestorder, with the leading order expression for redshift drift dzdτ (cid:12)(cid:12)(cid:12) O = E E Z λ O λ E dλ Π o . (19)From (18) we see that the only potentially positive con-tributions to Π o are from the terms − u µ u ν R µν and ( D µ a µ − a µ a µ ). The other terms entering the expressionfor Π o are non-positive and in general contribute withaccumulated negative contributions to the measured red-shift drift. In general relativistic theory, negative valuesof u µ u ν R µν is equivalent to violation of the strong energycondition. For general relativistic space-times in whichintegrated values of Π are dominated by the monopolecontribution Π o for light propagation over cosmologicaldistances, the only physical mechanisms that might re-sult in the detection of positive values for redshift drift The cancelation of spatially projected traceless combinations offluid kinematic variables has been argued to be a realistic sce-nario in space-times where a notion of statistical homogeneityand isotropy is present, and where structure is slowly evolvingrelative to the timescale it takes for photons to pass an approx-imate homogeneity scale [26, 27]. This suggested cancelationhas been shown to not hold true in general, exemplified by thesystematic alignment of the propagation direction e of the nullray with the positive eigenvector of the shear tensor in a Tardisspace-time [28]. The level of accuracy of the dominant monopoleapproximation R λ O λ E dλ Π ≈ R λ O λ E dλ Π o for light propagation overcosmological distances must be tested under various model as-sumptions. are thus (i) a special 4–acceleration profile of the space-time congruence of observers yielding integrated positivevalues of D µ a µ − a µ a µ along the detected null rays. ; (ii)violation of the strong energy condition. This realisa-tion is the main result of this paper: A measured positivevalue of redshift drift indicates that the strong energy con-dition is violated.
A positive detection of redshift driftis in principle possible without such a violation, but itrequires a 4–acceleration profile of the observer congru-ence giving systematic contributions along the null raysthrough its gradient. Alternatively, contributions fromsystematic alignment of the direction variables e and d with the dynamic fluid variables Π e µ , Π d µ , Π ee µν , etc., in(18) over the length scales of light-propagation can causeinaccuracy of the approximation (19).In the monopole approximation (19), inhomogeneitiestend to act with negative contributions to the redshiftdrift signal. We might thus in general expect the redshiftdrift signal to be negative in the absence of sources vio-lating the strong energy condition – even for space-timesexhibiting globally defined acceleration of large scale cos-mological volume sections due to the ‘backreaction’ ofcosmic structures. This expectation is consistent withthe numerical findings in [20, 21] for general relativisticinhomogeneous models without a cosmological constant,where the only example of positive redshift drift signalswas obtained in an unphysical space-time scenario witha source of negative energy density violating the strongenergy condition [21]. CONCLUSION
In FLRW universe models a positive detection of red-shift drift implies a non-zero cosmological constant [8–10] and hence violation of the strong energy condition.In this paper we have considered the redshift drift signalin a general space-time, and written it in a form usefulfor examining the sign of redshift drift and its link tothe strong energy condition in general relativistic uni-verse models with no symmetry assumptions imposed onthe metric. The redshift drift signal can be written interms of a physically interpretable multipole series, wherethe coefficients are given in terms of kinematic variablesand 4–acceleration of the observer congruence along with In general relativistic perfect fluid cosmologies, the 4–acceleration field is given by a µ = − D µ ( p ) / ( ǫ + p ), where ǫ is theenergy density and p is the pressure associated with the perfectfluid description [29] (see also the generalisation in [30] to arbi-trary general relativistic space-times). Accumulated positive val-ues of D µ a µ − a µ a µ = − D µ D µ ( p ) / ( ǫ + p )+ D µ ( p ) D µ ( ǫ ) / ( ǫ + p ) along null rays might in this case occur for specific pressure andenergy density profiles. For overviews of backreaction in cosmological modelling, see, e.g.,[31, 32].
Ricci and Weyl curvature variables. The monopole con-tribution in this series represents the isotropic contribu-tion common for all directions on the sky of the observercongruence. In a Universe where this monopole contribu-tion is statistically dominant in the integral over the lightpath, and where the 4–acceleration profile of observers isnot of a special form, a measured positive value of red-shift drift is a direct signature of violation of the strongenergy condition.This work is part of a project that has received fund-ing from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and in-novation programme (grant agreement ERC advancedgrant 740021–ARTHUS, PI: Thomas Buchert). I thankThomas Buchert for his reading of the manuscript, andSofie Marie Koksbang for comments. ∗ asta.heinesen@ens–lyon.fr[1] R. Penrose, Gravitational collapse and space-time singu-larities, Phys. Rev. Lett. (1965), 57[2] S. 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