Constraining the dark fluid
aa r X i v : . [ a s t r o - ph . C O ] O c t Constraining the dark fluid
Martin Kunz, Andrew R. Liddle, David Parkinson, and Changjun Gao Astronomy Centre, University of Sussex, Brighton BN1 9QH, United Kingdom The National Astronomical Observatories, Chinese Academy of Sciences, Beijing, 100012, China (Dated: November 6, 2018)Cosmological observations are normally fit under the assumption that the dark sector can bedecomposed into dark matter and dark energy components. However, as long as the probes remainpurely gravitational, there is no unique decomposition and observations can only constrain a singledark fluid; this is known as the dark degeneracy. We use observations to directly constrain thisdark fluid in a model-independent way, demonstrating in particular that the data cannot be fit bya dark fluid with a single constant equation of state. Parameterizing the dark fluid equation ofstate by a variety of polynomials in the scale factor a , we use current kinematical data to constrainthe parameters. While the simplest interpretation of the dark fluid remains that it is comprised ofseparate dark matter and cosmological constant contributions, our results cover other model typesincluding unified dark energy/matter scenarios. PACS numbers: 95.36.x,98.80.-k
I. INTRODUCTION
The standard cosmological model appeals to two sep-arate dark components — dark matter and dark energy— and the usual application of observational constraintsplaces limits on each of these. However, at present thesetwo components have only been detected through theirgravitational influence, and these measurements do notprovide enough information to permit a unique decom-position into these components. Rather, it is a modelassumption that the two components are separate. Thispoint was first made in Ref. [1] and specific consequencesexplored in Refs. [2, 3]. Ref. [4] extended it to coupledmodels and named it the dark degeneracy.This degeneracy is extremely general. One can write G µν − κT visible µν = κT dark µν (1)where G µν is the Einstein tensor, T µν the energy–momentum tensor, and κ the gravitational coupling. Anygravitational probe of the dark sector amounts to evalu-ating the left-hand side of this equation, and interpret-ting the non-zero result as evidence for the dark sector.Having obtained the dark sector energy–momentum ten-sor this way, such observations can offer no guidance onhow that tensor might be split amongst different darkcomponents. Although the degeneracy holds even for structure for-mation probes (both linear and non-linear) of the darksector, we will focus here only on kinematical probes, i.e.those referring to the homogeneous and isotropic back-ground cosmology. A complete dark sector description is We work entirely in Einsteinian gravity. A somewhat related is-sue arises in modified gravity models, where the ‘non-Einsteinian’gravitational terms could be shifted to the right-hand side of thegeneralized Einstein equation and potentially reinterpretted asmatter terms, see e.g. Ref. [5]. then given by the total dark sector energy density and itsequation of state w dark , the latter determining the evo-lution of the former. This enables a simple analysis. Toinclude structure formation data, while retaining statisti-cal homogeneity, one would also need to consider at leastthe dark sector sound speed and perhaps also anisotropicstress [1, 4], and one might expect that structure forma-tion observations would end up mostly constraining theirform rather than imposing further upon w dark .As a simple example [1], in the standard cosmologi-cal model the redshift evolution of the total dark sectorequation of state w dark ≡ P ρ i w i P ρ i , (2)(where ‘ i ’ runs over the dark components) is given by w SCMdark ( z ) = − − Ω m , − Ω m , + (Ω m , − Ω b , ) (1 + z ) , (3) ≃ −
11 + 0 . z ) . (4)Here Ω m and Ω b are the total matter density parame-ter and the baryon density parameter, and the subscriptindicates present value. The second line follows frominserting the values Ω m , ≃ .
274 and Ω b , ≃ .
046 ob-tained from current data compilations [6].Inclusion of a single dark energy component with thisequation of state evolution would give the same observa-tional predictions as the standard cosmology. Indeed,once one allows the dark energy equation of state toevolve arbitrarily, one cannot say anything from obser-vations about the dark matter density Ω dm , as its effectscan always be reinterpretted as due to the dark energy.Analyses which appear to measure Ω dm accurately onlymanage to do so because the dark energy parameteriza-tion adopted is not general enough to be able to mimicthe form of Eq. (3).This degeneracy is perfect. All we can say in a model-independent way is that the present total dark sectordensity is about 0.95, and that its equation of state isconstrained to evolve in a particular way from an early-time value at or near zero to arrive at its present valueof w obsdark ( z = 0) ≃ − .
8. Our aim in this paper is tomore precisely quantify these constraints, by consideringgeneral dark sector models that do not include a purecold dark matter component.
II. MODELS AND DATA
Having set up this non-standard framework for analyz-ing the dark sector, our analysis procedure is standardand straightforward.
A. Models
In order to impose constraints on the dark fluid, weneed to employ a parameterization of its equation ofstate. Henceforth we drop the subscript ‘dark’, using w throughout as the total dark sector equation of state.As we wish to reach high redshift, we parameterize w as a function of scale factor a , which has the boundeddomain 0 < a ≤
1. We simply expand this as a powerseries in a at the present, which we find to be sufficient.At linear order this is the well-known CPL parameteriza-tion w = w + (1 − a ) w [7], normally applied to the darkenergy alone but here referring to the combined dark sec-tor. To general order this expression was given, again forthe dark energy alone, in Ref. [8], which in actual calcu-lation considered the linear and quadratic versions. Aswe are demanding that our parameterization describe theentire dark sector, we will explore up to cubic order. Ourmodels are hence w ( a ) = N X n =0 w n (1 − a ) n (5)for different choices of N .As we will see, the data strongly demand that w ( a ) isclose to zero as a →
0, i.e. the dark sector is requiredto behave as dark matter at early times (note that thisis true even without inclusion of any structure forma-tion data). Accordingly we will also consider the sameexpansions supplemented with the additional constraint w ( a = 0) = 0 which fixes their highest-order term asa function of the others, and with the combined con-straints w ( a = 0) = 0 and dw/da | a =0 = 0. We will callthese the ‘constrained’ and ‘doubly-constrained’ expan-sions respectively. The former can be used from linearorder upwards, and the latter from quadratic upwards.In all cases we choose priors on the expansion coeffi-cients which are wide enough that the outcome is deter-mined entirely by the data and not the prior. The standard rulers measured by the cosmic mi-crowave background (CMB) and baryon acoustic oscil-lations (BAO) are all fixed by the sound horizon at de-coupling (which we fix to be z dec = 1089), which is de-pendent on the baryon density. The standard rulers areall distance ratios independent of H , and the supernovaemake no measurement of the Hubble parameter H sinceit is degenerate with the unknown absolute normalizationover which we marginalize. As our data compilation doesnot constrain H , it is not able to give an accurate con-straint on the total dark sector energy density. Howeverthe physical baryon density Ω b h is accurately measuredto the usual value, and so if supplemented with a directdetermination of H , we would find the expected totaldark sector energy density Ω dark ≃ .
95. In our analysiswe allow Ω b h and H to vary, but constrain them us-ing other datasets, and marginalize over them, in effectmarginalizing over the total dark sector energy density.In light of the above, when we quote model parame-ter counts they are of the dark sector equation of state,and do not include the total dark sector energy density.For the general dark sector models, we quote the numberof parameters specifying w ( a ). For ΛCDM, the equiva-lent parameter is the relative amount of dark matter anddark energy at present (equivalently, the coefficient of the(1 + z ) term in Eq. (4)). B. Data
We use a fairly typical combination of kinematical datato constrain our models. Standard candle data comesfrom supernova type Ia luminosity distances, for whichwe use the cut Union supernova sample [10] (with sys-tematic errors included), and standard ruler data comesfrom the angular positions of the CMB [11] and BAOpeaks [12]. Note that Ref. [11] give constraints on thescaled distance to recombination R and the angular scaleof the sound horizon l a . These are defined to be R ≡ q Ω m H r ( z CMB ) , l a ≡ πr ( z CMB ) r s ( z CMB ) . (6)Since R is scaled by the physical matter density, andso makes assumptions about the separability of the darkmatter and dark energy, we ignore it in this work. We useonly the constraints on l a , as well as those on Ω b h andthe correlations between the two. This still assumes thatthe sound speed of the dark component at high redshiftis small, in order to avoid an early ISW effect shifting thepeak position. We also include the SHOES (Supernovaeand H for the Equation of State) measurement of theHubble parameter today, H = 74 . ± . − Mpc − [9]. TABLE I: Parameters (of the dark sector equation of state)and best-fit chi-squared for our various models. The con-strained models force w ( a = 0) = 0, and the doubly-constrained ones additionally dw/da | a =0 = 0.Model Dark sector χ parametersΛCDM 1 311.9Constant w III. RESULTS
Table I shows the number of adjustable parameters andbest-fit chi-squared for most models, including ΛCDM forcomparison. The total number of data points is 313 (308SN-Ia, 2 BAO, 2 CMB and 1 from the SHOES project),but correlations between the data points make it difficultto state the number of independent data points, and sothe number of degrees of freedom. We can say that thenumber of degrees of freedom is, at most, 311 minus thenumber of dark sector parameters, meaning that the datais an acceptable fit to all the models, except the constant w model, if the correlations are small.The corresponding w ( a ) curves of the best-fitting ver-sion of each model are shown in Fig. 1. The immediateconclusions from these are as follows:1. ΛCDM, as expected, does a good job of fitting thedata, bettered only by other models with more darksector parameters. For our data the best-fit Ω m is0 . ± .
02. This agrees well with the result of Ko-matsu et al. [6], with somewhat larger uncertaintyas less data is being used.2. Even the best-fit version of the constant w modelis a very poor fit. The dark sector has not had aconstant equation of state throughout its evolution.3. The full four-parameter cubic does not significantlyimprove the fit over the three-parameter quadratic,indicating that three-parameter models saturatethe constraining power of the data.4. All the models have w ( a = 0) at or very close tozero, enforced entirely by the data. This indicatesthat the full phenomenology can be captured usingthe constrained versions of the expansion, reducingthe variable parameter set by at least one.5. The doubly-constrained quadratic (i.e. simply w ( a ) = w a ) gives a tolerable one-parameter fit to w lcdmconstantcplcpl (constrained)quadquad (constrained)quad (dbl. constr.)cubiccubic (constrained)cubic (dbl. constr.) FIG. 1: The best-fit w ( a ) for our various models. Note thatthe approach to w = 0 at a = 0 is determined entirely by thedata in the unconstrained cases, while being enforced in theconstrained models. −0.7 −0.65 −0.6 −0.55 0.6 0.65 0.7w w w −0.7 −0.65 −0.6 −0.550.60.650.7 FIG. 2: Parameter constraints on the linear (CPL) parame-terization. The constrained models lie on the line w = − w ,shown as the (blue) line on the w , w plot. the data, though not quite as good as the ΛCDMfit. By contrast, the single-parameter constrainedlinear fit is a poor fit to the data when compared tothe ΛCDM fit which has the same number of pa-rameters (though the general CPL model can ac-ceptably fit it by overshooting to w > −1.2 −1 −0.8 −0.6 0 1 2 −0.04 0 0.04w b w −1.2 −1 −0.8 −0.600.511.522.5 w b w −1.2 −1 −0.8 −0.6−0.04−0.0200.020.04 w FIG. 3: Parameter constraints on the quadratic parameteri-zation. Here w b = w + w + w , which is the combinationwhich is held at zero in the constrained quadratic case. −1.2 −1 −0.8 −0.6 1 1.5 2 2.5w w w −1.2 −1 −0.8 −0.611.52 FIG. 4: Parameter constraints on the constrained quadraticparameterization, where w ( a = 0) is set to zero. The doubly-constrained models lie on the line w = − w , shown as the(blue) line on the w , w plot. all constant w models are ruled out, we show in Figs. 2and 3 the constraints on the parameters of the CPL andquadratic parameterizations, and in Fig. 4 we show theconstraints on the constrained quadratic model, whichforces w ( a = 0) = 0.For illustrative purposes, Fig. 5 shows 400 quadratic w ( a ) curves drawn randomly from the Markov chain.They are shaded so that the likelihood increases from w FIG. 5: An illustration of the form of the best fitting w ( a )curves for the quadratic case. A sparse sampling of 400 chainelements, colour coded by likelihood with the red (lighter)shading the highest, is shown. blue (darker) to red (lighter). We can see that the high-redshift region near a = 0 is strongly constrained andrequires w ≈
0, while the value of the total equationof state parameter today, w = w ( a = 1), is only veryweakly constrained by the data sets used in this work. Due to the integrated nature of the distance constraintson w , curves that oscillate around the best-fit incur onlya small penalty. This becomes more problematic whengoing to higher order in power law expansions since thenmore oscillations become possible. As the oscillationshave to average out, the expansion parameters are highlycorrelated and not really independent, which can alreadybe seen for w and w in the quadratic case in Fig. 3.We also carried out a different analysis where the equa-tion of state w is allowed to take different values in binnedregions of scale factor a , for simplicity taken to be lin-early spaced with 10 or 20 bins. A Principal ComponentAnalysis showed that three modes were well measured( σ < . IV. CONCLUSIONS
Due to the dark degeneracy, there is no unique splitinto dark matter and dark energy. For this reason, weconsidered in this paper the total dark sector equation ofstate. We parameterized it with polynomial expansionsin the scale factor a and used type Ia supernovae, baryon As we were completing this work, Mortonson et al. [13] arXiveda paper considering this specific point in much more detail. Thispoint had also previously been noted in Ref. [14]. acoustic oscillations, and the CMB peak position to findconstraints on the expansion parameters.The strongest constraints come at very high redshift,from the CMB and BAO measurements of the sound hori-zon at decoupling. There the dark fluid must evolve withan equation of state close to zero, to recover the correctangular scale of the acoustic oscillations. At low redshiftthe constraints from SN-Ia and BAO are weaker, but re-quire a negative pressure fluid with w dark ≃ − . non-gravitational probes of the sector, for instancedirect detection of dark matter particles at accelerators orin underground experiments [4]. Such detections wouldimmediately move such particles from the dark sector tothe ‘known’ sector, removing them from the dark sectordegeneracy. Acknowledgments
M.K., A.R.L., and D.P. were supported by STFC(UK). C.G. was supported by the National Science Foun-dation of China under the Distinguished Young ScholarGrant 10525314, the Key Project Grant 10533010, andGrant 10575004; by the Chinese Academy of Sciencesunder grant KJCX3-SYW-N2; and by the Ministry ofScience and Technology under the National Basic Sci-ences Program (973) under grant 2007CB815401. A.R.L.thanks the Institute for Astronomy, University of Hawaii,for hospitality while this work was completed. We ac-knowledge use of the CosmoMC package [15]. [1] W. Hu and D. J. Eisenstein, Phys. Rev. D , 083509(1999), arXiv:astro-ph/9809368 .[2] I. Wasserman, Phys. Rev. D , 123511 (2002), arXiv:astro-ph/0203137 .[3] C. Rubano and P. Scudellaro, Gen. Rel. and Grav. ,1931 (2002), arXiv:astro-ph/0203225 .[4] M. Kunz, arXiv:astro-ph/0702615 .[5] M. Kunz and D. Sapone, Phys. Rev. Lett. ,121301 (2007), arXiv:astro-ph/0612452 ; W. Hu, arXiv:0906.2024 [astro-ph] .[6] E. Komatsu et al , Astrophys. J. Suppl. , 330 (2009), arXiv:0803.0547 [astro-ph] .[7] A. Chevallier and D. Polarski, Int. J. Mod. Phys. D ,213 (2001), arXiv:gr-qc/0009008 ; E. V. Linder, Phys.Rev. D , 083504 (2003), arXiv:astro-ph/0304001 .[8] B. A. Bassett, P. S. Corasaniti and M. Kunz, Astrophys. J. , L1 (2004), arXiv:astro-ph/0407364 .[9] A. G. Riess et al. , Astrophys. J. (2009) 539 arXiv:0905.0695 [astro-ph.CO] .[10] M. Kowalski et al. , Astrophys. J. , 749 (2008) arXiv:0804.4142 [astro-ph] .[11] Y. Wang and P. Mukherjee, Phys. Rev. D , 103533(2007) arXiv:astro-ph/0703780 .[12] W. J. Percival, et al. , arXiv:0907.1660 [astro-ph.CO] .[13] M. J. Mortonson, W. Hu, and D. Huterer, arXiv:0908.1408 [astro-ph] .[14] P. S. Corasaniti, M. Kunz, D. Parkinson, E. J. Copeland,and B. A. Bassett, Phys. Rev. D , 083006 (2004), arXiv:astro-ph/0406608 .[15] A. Lewis and S. Bridle, Phys. Rev. D , 103511 (2002), arXiv:astro-ph/0205436arXiv:astro-ph/0205436