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<ul><li><p>Model breaking measure for cosmological surveys</p><p>Adam Amara* and Alexandre Refregier</p><p>Department of Physics, Institute for Astronomy, ETH Zurich, Wolfgang-Pauli-Strasse 27,CH-8093 Zurich, Switzerland</p><p>(Received 22 September 2013; published 1 April 2014)</p><p>Recent observations have led to the establishment of the concordance CDM model for cosmology. Anumber of experiments are being planned to shed light on dark energy, dark matter, inflation and gravity,which are the key components of the model. To optimize and compare the reach of these surveys, severalfigures of merit have been proposed. They are based on either the forecasted precision on the CDMmodeland its expansion or on the expected ability to distinguish two models. We propose here another figure ofmerit that quantifies the capacity of future surveys to rule out the CDM model. It is based on a measureof the difference in volume of observable space that the future surveys will constrain with and withoutimposing the model. This model breaking figure of merit is easy to compute and can lead to different surveyoptimizations than other metrics. We illustrate its impact using a simple combination of supernovaeand baryon acoustic oscillation mock observations and compare the respective merit of these probes tochallenge CDM. We discuss how this approach would impact the design of future cosmologicalexperiments.</p><p>DOI: 10.1103/PhysRevD.89.083501 PACS numbers: 98.80.Es</p><p>I. INTRODUCTION</p><p>Recent progress in cosmological observations have ledto the establishment of the CDM model as the standardmodel for cosmology. This simple model is able to fit awide array of observations with about six parameters [13].In spite of its success, several key ingredients of the modelare not fully understood and have been introduced to fit thedata rather than being derived from fundamental theory.These include dark matter [4], which (if attributed toparticles) exists outside the standard model of particlephysics and dark energy [5,6]. The other ingredients of themodel are associated with inflation, which conditions theinitial state of the Universe, and Einstein gravity, which hasnot been tested on cosmological scales.Alternatives to the CDM model are numerous and</p><p>growing. However since the data is currently consistentwith the CDM model, progress in the field will likely bedriven by the acquisition of new data that can be used tofurther challenge the model. In so doing, we hope to findevidence that will lead to a deeper understanding ofphysical processes and point us towards more fundamentalalternative models. Significant amounts of current effortsin cosmology are thus focused on the design of futureexperiments that can optimally increase our cosmologicalknowledge. However, since there exists a wide array ofequally compelling alternative models, finding a suitablemetric with which to compare and optimize future experi-ments is challenging.</p><p>At present, the dominant metric for gauging the qualityof planned experiments is the dark energy task force(DETF) figure of merit (FoM) [7]. This metric consistsof expanding the simplest CDM model so that the darkenergy component is modeled as having an equation ofstate w, which is given by the ratio of pressure to density ofdark energy. This equation is assumed to evolve linearlywith scale factor a, wa w0 1 awa [8,9]. TheDETF FoM can then be derived from the determinant ofthe covariance matrix of the two dark energy parametersw0 and wa, which can be calculated using Fisher matrixmethods [10]. Since the linear expansion of the equationof state is only one of many possible extensions beyondCDM, relying solely on this optimization may lead tobiases in experiment design.An alternative approach, which was proposed by the</p><p>follow-up committee known as the DETF FoM WorkingGroup [11], is to consider a more general expression for theequation of state. This approach relies on principle com-ponent analysis (PCA) methods to find the fundamentalmodes that a given experiment can measure. In their report,the DETF FoM Working Group suggests a prescriptionwhere the equation of state is divided into 36 redshift binsout to z 10. One difficulty, however, is that Fisher matrixcalculations can be unstable. The final results, therefore,can depend on the users choice of initial basis set, whichonce again may not be well motivated and can lead tounintended selection biases [12]. The DETF FoMWorkingGroup also advocates to use alternative theoretical expan-sions that can be used to model possible deviations ofgravity from Einsteins theory [13,14].Numerous alternative metrics have been proposed in</p><p>the literature. As well as further PCA based techniques*adam.amara@phys.ethz.chalexandre.refregier@phys.ethz.ch</p><p>PHYSICAL REVIEW D 89, 083501 (2014)</p><p>1550-7998=2014=89(8)=083501(7) 083501-1 2014 American Physical Society</p><p>http://dx.doi.org/10.1103/PhysRevD.89.083501http://dx.doi.org/10.1103/PhysRevD.89.083501http://dx.doi.org/10.1103/PhysRevD.89.083501http://dx.doi.org/10.1103/PhysRevD.89.083501</p></li><li><p>[1518], other methods have been proposed that includein the determinant calculations parameters of CDMbeyond those of the equation of state. These include theintegrated parameter space optimization (IPSO) [1921],and model selection methods based on forecasting theBayes factor [2226]. The latter approach relies on com-paring two models and calculating the Bayes factor[B01 pdjM0=pdjM1], which quantifies the odds ofwhich model (M0 or M1) is preferred by the data (d). Thismethod still requires a choice of an alternative model towhich the null model can be compared.The end result can vary, depending on the FoM used.</p><p>This is ultimately due to the fact that the FoMs are beingused to ask subtly different questions. In an era where thetotal amount of data is growing, it is conceivable and fullyexpected that different FoMs will lead to similar optimi-zation. However, as experiments begin to fill the entireavailable cosmic volume, the trade-offs are likely tobecome more subtle. Hence, care should be given to focusprecisely on the questions that we want to address.In this paper, we explore the motivational question:</p><p>Which experiment is most likely to find data that will falsifyCDM? Given the success of CDM so far, the detectionof any deviation from this model would be a majordiscovery. These deviations may not necessarily emergeas a deviation from w 1. As a result, to answer themotivational question above we formulate a new figure ofmerit, building on earlier work [27], which can be readilycalculated using Gaussian approximations. In its purestform this figure of merit can be calculated using only(i) current data, (ii) the predictions from the simple CDMmodel that we wish to challenge and (iii) the expectedcovariance matrix of the data for a future experiment. Aspart of our work, we also show how robust theoreticalpriors, such as light propagation on a metric, can also beincluded in the calculation, if so desired. While the DETFFOM and the Bayes ratio approach are, respectively, relatedto model fitting and model selection, our approach isrelated to the problem of model testing.This paper is organized as follows. In Sec. II, we derive</p><p>our new FoM and show the Gaussian approximationversion of the calculation. In Sec. III, we investigate asimple cosmological toy-model example to illustrate ourmethod. In this section we also compare our calculations toan FoM derived from the determinant of the Fisher matrixof the standard CDM parameters. Finally, in Sec. IV weoffer a discussion to summarize our findings.</p><p>II. FORMALISM</p><p>The basic principle of our approach is to make compar-isons between the likely outcomes of future experiments indata space. In its purest form, this is a comparison betweenpDfjDc and pDfjDc;, where the former is theprobability of future data Df, given only current data Dcand the latter is the probability of future data given current</p><p>data and the constraint that the standard model (with para-meters) being studied (in our case standard CDM) musthold. In this empirical case, we can calculate the probabilityof future data by integrating over all possible values of thedata (see derivation in the Appendix) such that</p><p>pDfjDc Z</p><p>pDfjTpTjDcdT; (1)</p><p>where we have introduced the concept of true value Tthat corresponds to the value we obtain as the errors tend tozero. For the case where we assume a standard model holds,we can calculate the probability of future data by integrat-ing over all possible values of the model parameters, ,</p><p>pDfjDc; Z</p><p>pDfjpjDcd: (2)</p><p>In both cases, we can calculate the probabilities of theunderlying variables given todays data. For instance, in thecase of the model parameters,</p><p>pjDc pDcjp</p><p>pDc: (3)</p><p>Given two density distributions [for instance Eqs. (1) and(2)] we will need to be able to quantitatively compare them.For this, the concept of information entropy, which quan-tifies the level of uncertainty, is useful. A robust measurefor this purpose is the relative entropy, also known as theKullback-Leibler (KL) divergence [28], between the twodistributions. In this case, this can be calculated as</p><p>KLp; q Z</p><p>ln</p><p>pxqx</p><p>pxdx; (4)</p><p>where px and qx are the two probability distributionsto be compared. This measure quantifies the difference ofinformation in the two cases and provides a measure of thedifference between the two distributions.Using this measure our proposed figure of merit measure</p><p>for model breaking is simply</p><p> KLpDfjDc;; pDfjDc: (5)</p><p>A. The Gaussian case</p><p>The analysis outlined above is general and can be used tostudy probability distribution functions of arbitrary shape.However, due to the their simplicity, probability distribu-tion functions (PDFs) that are multivariate Gaussians arevery attractive cases to study. In this case, the probabilitieswould be given by</p><p>px 12k=2jCj1=2 exp1</p><p>2x TC1x </p><p>;</p><p>(6)</p><p>ADAM AMARA AND ALEXANDRE REFREGIER PHYSICAL REVIEW D 89, 083501 (2014)</p><p>083501-2</p></li><li><p>where k is the number of dimensions, is the mean (i.e.peak) of the PDF and C is the covariance matrix.The relative entropy between two multivariate Gaussian</p><p>distributions, e.g. px and qx, with the covariancematrices Cp and Cq is given by [29]</p><p>KL 12ln jCpC1q j</p><p> 12trC1p q pq pT Cq Cp: (7)</p><p>For the simplest case, where the two distributions have thesame mean, this reduces to</p><p>KL 12ln jCpC1q j </p><p>1</p><p>2trC1p Cq Cp: (8)</p><p>B. Calculating the covariance matrix</p><p>Given the covariance matrix for current data, Cc, we cancompute the covariance matrix, Cm, of the parametersof the model that need to be adhered to. This can be doneby calculating the Fisher matrix, C1m , through a matrixrotation, as</p><p>C1m YC1c YT; (9)</p><p>where Y is the Jacobian matrix of derivatives such thatYij Di=j. In principle, it is also possible to haveconstraints on the parameters for external data that willnot change in the future. To make the predictions for futureerror bars, we can then project back to the covariance in theobservables, Cx, based on existing errors</p><p>Cx YTCmY: (10)</p><p>Finally, to calculate the full covariance matrix for futuredata, C1, of pDfjDc;M, we need to account for the errorbars associated with the future experiment, given by thematrixCf. The full matrix corresponding to the operation inEq. (2) is then</p><p>C1 Cx Cf: (11)</p><p>For the case of the purely empirical predictions, wherethere is no model and data vector entries are independent ofeach other, the Jacobian matrices Y become the identitymatrix I, which greatly simplifies the equations above. Forinstance, in the case where no external data set is used, thecovariance matrix of pDfjDc;M becomes</p><p>C0 Cc Cf: (12)</p><p>In this case the model breaking figure of merit defined inEq. (5) and using (8) reduces to</p><p> 12ln jC1C10 j </p><p>1</p><p>2trC11 C0 C1: (13)</p><p>In the case where we also want to consider shifts in meanvalues one would use an analogous expression with extraterms coming from Eq. (7).The two cases above are the extreme examples: (i) one</p><p>where all the data points are correlated with all other datapoint when projected through a model and (ii) the casewhere all the data points are independent from each other.It is possible to construct an intermediate case that we call aminimal model that defines a weak correlation betweensubsets of the data. For example, in the cosmologicalsetting we could introduce a correlation between data takenat the same redshift, while making the data from twodifferent redshifts fully independent. In this case, theJacobian would be constructed using the derivatives withrespect to the data, i.e Yij Di=Dj and C0 would bemodified accordingly for the model breaking figure ofmerit in Eq. (13).</p><p>III. COSMOLOGICAL EXAMPLE</p><p>To demonstrate our approach, we briefly explore asimple cosmological example. For this we will focus ongeometrical tests, namely supernovae flux decrements andtangential and radial measurements of the baryon acousticoscillation (BAO) scales.</p><p>A. Background cosmology</p><p>Within the standard CDM concordance model, thegeometry measure can be derived from the line of sightcomoving distance, ,</p><p>a cZ</p><p>daa2Ha ; (14)</p><p>where c is the speed of light, a is the scale factor and Hais the Hubble function. The Hubble function can be easilycalculated in the CDM and in the late time Universe bythe Friedmann equation,</p><p>H2a H20ma3</p><p> ka2</p><p>; (15)</p><p>where m is the matter over density, is the dark energydensity and k is the curvature. The curvature term canbe defined through the relation m k 1.With this, it is clear that we can describe the geometrymeasures through three free parameters: h, m and ,where we use the standard approach of recasting theHubble constant,H0, as the dimensionless quantity throughh H0=100 Kms1Mpc1.Observed distance measures are typically determined</p><p>through the angular diameter distance (DA) and theluminosity distance (DL), which can be related to eachother through the scale factor</p><p>MODEL BREAKING MEASURE FOR COSMOLOGICAL SURVEYS PHYSICAL REVIEW D 89, 083501 (2014)</p><p>083501-3</p></li><li><p>DA a2DL ar; (16)</p><p>where r is the comoving angular diameter distance.The supernovae technique measures the distance modulus(DM) as a function of redshift [5], where the distancemodulus is determined from the flux ratio between theabsolute and apparent fluxes of SNe. This can then belinked to the radial comoving distance through the lumi-nosity distance DL,</p><p>DM 5 log</p><p>DL10 pc</p><p>: (17)</p><p>We also find it useful to define a new quantity,</p><p>RDM </p><p>DL10 pc</p><p>; (18)</p><p>which contains the same information in a form closer toflux ratios rather than magnitude differences.Baryon acoustic oscillation (BAO) studies rely on</p><p>using galaxy surveys to measure the same acousticpeaks that are seen in the CMB, thereby using thescale set by these peaks as a standard ruler. The mea-surements can be made perpendicular to the line ofsight (rp and along the line of sight (rpa), which canbe linked to the observed angular scale and redshiftextent z [21],</p><p> arsDA</p><p>; (19)</p><p>and</p><p>z Hrsc</p><p>; (20)</p><p>where rs is the sound horizon [1] that, for simplicity, weset to 140 Mpc in this study. In this framework, the obser-vable quantities are O fRDMa; a and zagand the model param...</p></li></ul>

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