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On Using the Cosmic Microwave Background Shift Parameter in Tests of Models of Dark Energy A&A 471, 65–70 (2007) Astronomy DOI: 10.1051/0004-6361:20077292 & c ESO 2007 Astrophysics On using the cosmic microwave background shift parameter in tests of models of dark energy Ø. Elgarøy1 and T. Multamäki2 1 Institute of theoretical astrophysics, University of Oslo, Box 1029, 0315 Oslo, Norway e-mail: [email protected] 2 Department of Physics, University of Turku, 20014 Turku, Finland e-mail: [email protected] Received 13 February 2007 / Accepted 31 May 2007 ABSTRACT Context. The so-called shift parameter is related to the position of the first acoustic peak in the power spectrum of the temperature anisotropies of the cosmic microwave background (CMB). It is an often used quantity in simple tests of dark energy models. However, the shift parameter is not directly measurable from the cosmic microwave background, and its value is usually derived from the data assuming a spatially flat cosmology with dark matter and a cosmological constant. Aims. To evaluate the effectiveness of the shift parameter as a constraint on dark energy models. Methods. We discuss the potential pitfalls in using the shift parameter as a test of non-standard dark energy models. Results. By comparing to full CMB fits, we show that combining the shift parameter with the position of the first acoustic peak in the CMB power spectrum improves the accuracy of the test considerably. Key words. cosmology: theory – cosmological parameters 1. Introduction understanding of dark energy argues for keeping an open mind. One would like to have some means of testing both groups of Comparing cosmological models to current observational data models, incorporating as much empirical information as pos- can be cumbersome and computationally intensive. Large multi- sible, but avoiding the need to calculate the behaviour of den- dimensional parameter spaces cannot be probed by grid-based sity perturbations. In practice, this means restricting the obser- methods but more sophisticated approaches are required, for ex- vational tests to those involving the age and distance scale, in ample Monte Carlo Markov Chains (MCMC) (Gamerman & particular the luminosity distance-redshift relationship as probed Lopes 2006; Lewis & Bridle 2002). Of the most commonly used by supernovae of type Ia (SNIa). Important as the supernova data cosmological data sets available today, cosmic microwave back- are, they are still not very restrictive if one allows for e.g. non- ground (CMB) and large-scale structure (LSS) observations in zero spatial curvature or a time-varying equation of state for dark particular require computational effort in parameter estimation. energy (Riess et al. 1998; Perlmutter et al. 1999; Tonry et al. Given computing time and patience, this is not a problem, at 2003; Barris et al. 2004; Riess et al. 2004; Astier et al. 2006; least in principle, when testing models in which the evolution of Clocchiatti et al. 2006; Wood-Vasey et al. 2007; Miknaitis et al. linear density perturbations is well understood and hence calcu- 2007; Davis et al. 2007). lable. However, for several of the more imaginative dark energy To tighten up constraints on dark energy models, a common models the situation is more complicated. In cases where the approach is therefore to include additional information about the model is specified by an action, like the DGP model (Dvali et al. distance scale from the CMB in the form of the so-called shift 2000), one should in principle be able to set up the equations parameter (Efstathiou & Bond 1999) that is related to the po- for linear perturbations, but in practice this has turned out to be ffi sition of the first acoustic peak in the power spectrum of the di cult and it is only recently that progress in this direction has temperature anisotropies (Amarzguioui et al. 2006; Barger et al. been made in this particular case (Koyama & Maartens 2006; 2007; Elgarøy & Multamäki 2005; Davis et al. 2007; Fairbairn Koyama 2006; Sawicki et al. 2006; Song et al. 2006). And even & Goobar 2006; Fairbairn & Rydbeck 2007; Lazkoz et al. 2006; though the equations are known, they may be so complicated Nesseris & Perivolaropoulos 2007; Rydbeck et al. 2007; Wang to treat numerically as to make it practically impossible to ex- & Mukherjee 2006; Xia et al. 2006; Zhao et al. 2005, 2006). plore the parameter space of the model in MCMC. In addition, there are a large number of dark energy models, based on phe- Recently, after the baryon acoustic oscillations (BAO) where nomenological considerations, that lack the detail to allow one to observed in the SDSS Luminous Red Galaxy sample (Eisenstein proceed with well-defined calculations. Examples of such mod- et al. 2005), it has also become common to include the informa- els include the various proposed modifications of the Friedmann tion about the angular scale of the oscillations (Davis et al. 2007; equation, where the model is simply not specified well enough Wright 2007). What one should bear in mind, however, is that to allow the calculation of the density perturbations (see e.g. these distance scales are not directly measured quantities, but Freese & Lewis 2002; Gondolo & Freese 2003). The justifi- are derived from the observations by assuming a specific model, cation of such models may be questioned, but the state of our usually the flat ΛCDM model or a slight variation thereof. Care Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20077292 66 Ø. Elgarøy and T. Multamäki: On using the CMB shift parameter in tests of models of dark energy needs to be exercised when using these derived quantities to test The mth Doppler peak has the comoving wave number (Hu & more exotic dark energy models. Sugiyama 1996) mπ = kmrs(ar), and hence the location of the Here we consider the shift parameter in more detail by com- first peak in multipole space is approximately given by paring its predictions to those obtained from full CMB fits for different types of cosmological models. We identify the limita- dA(zr) a ≈ π , (6) tions of using such a measure and advocate using a combination rs(ar) of the acoustic peak scale along with the shift parameter as a = R 1/2 more accurate probe of the CMB power spectrum. Such a com- where dA c /H0ωm is the angular diameter distance. bination is quick and easy to implement and should be included Rewriting Eq. (6), we have in tests of dark energy models where it is either cumbersome or ⎛ ⎛ √ ⎞⎞ ⎜ ⎜ ⎟⎟−1 unknown how to calculate the full CMB and matter power spec- ⎜ ⎜ R(ar) + R(aeq) + 1 + R(ar)⎟⎟ ≈ 151πω1/2R ⎜ln ⎜ ⎟⎟ · (7) trum. a b ⎝ ⎝ ⎠⎠ 1 + R(aeq) The location of the first peak is hence a combination of the shift 2. Theory parameter and the size of the sound horizon at recombination, The use of the shift parameter as a probe of dark energy is based as is expected on physical grounds. Even though the relation R on the observation that different models will have an almost between a and is linear, the two parameters are not degen- identical CMB power spectra if all of the following criteria are erate and in fact complement each other well in constraining 2 2 satisfied (Efstathiou & Bond 1999): ωc =Ωch and ωb =Ωbh models, as is shown later. Note that the analytical expressions are equal, primordial fluctuation spectrum is unchanged, and the above were derived for the ΛCDM class of models. For more shift parameter, general models, the relevant quantities should be calculated nu- merically, although in some cases analytical results have been 1/2 derived (Grupposo & Finelli 2006). R = ωm 1/2 sinnk(ωk y), (1) The best fit value calculated from the WMAP-team provided ω1/2 k MCMC chains for the shift parameter in the standard flat ΛCDM where sinn(x) = {sin(x), x, sinh(x)} for k =+1, 0, −1 respec- model is tively, with + R = 1.71 0.03, (8) −0.03 1 da y = (2) which is in good agreement with Wang & Mukherjee (2006). + 2 + 4 + 1−3w ar ωma ωka ωΛa ωQa This result is practically equal for the ΛCDM and wCDM mod- els with or without dark energy perturbations. The acoustic peak is constant. In this original definition of the shift parameter, the position as measured by Eq. (7) calculated from the same data is universe is considered to be filled with matter (dark and bary- =Ω 2 =Ω 2 +1.1 onic), ωm mh , ωb,curvature,ωk kh , cosmological a = 303.6− . (9) 2 2 1.2 constant ωΛ =ΩΛh and a dark energy component ωQ =ΩQh with a constant equation of state w. The density parameter Ωi is One should always bear in mind the conditions for the shift pa- the ratio of the present-day density of component i to the density rameter to be applicable. If one wants to use the shift param- = 2 eter as a constraint on a dark energy model, then first of all of a spatially flat universe, ρc 3H0 /8πG,andh is the dimen- −1 −1 the distribution of the shift parameter has to be derived from sionless Hubble constant defined by H0 = 100 h km s Mpc . the CMB data. This cannot be done without assuming a model. Integration is carried out from recombination, ar, until today a = 1. Since one is only looking for a constraint on the expansion his- In a spatially flat universe (k = 0), the shift parameter tory of the universe, it is easy to forget that one is also mak- reduces to ing assumptions about the primordial power spectrum of density fluctuations, since these form the basis for calculating the CMB z r dz anisotropies.
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