1 Statistical Techniques for Data Analysis in Cosmology

Transcription

1 1 Statistica Techniques for Data Anaysis in Cosmoogy 1.1 Introduction Statistics is everywhere in Cosmoogy, today more than ever: cosmoogica data sets are getting ever arger, data from different experiments can be compared and combined; as the statistica error bars shrink, the effect of systematics need to be described, quantified and accounted for. As the data sets improve the parameter space we want to expore aso grows. In the ast 5 years there were more than 370 papers with statistic- in the tite! There are many exceent books on statistics: however when I started using statistics in cosmoogy I coud not find a the information I needed in the same pace. So here I have tried to put together a starter kit of statistica toos usefu for cosmoogy. This wi not be a rigorous introduction with theorems, proofs etc. The goa of these ectures is to a) be a practica manua: to give you enough knowedge to be abe to understand cosmoogica data anaysis and/or to find out more by yoursef and b) give you a bag of tricks (hopefuy) usefu for your future work. Many usefu appications are eft as exercises (often with hints) and are thus proposed in the main text. I wi start by introducing probabiity and statistics from a Cosmoogist point of view. Then I wi continue with the description of random fieds (ubiquitous in Cosmoogy), foowed by an introduction to Monte Caro methods incuding Monte-Caro error estimates and Monte Caro Markov Chains. I wi concude with the Fisher matrix technique, usefu for quicky forecasting the performance of future experiments. 1

2 2 1.2 Probabiities What s probabiity: Bayesian vs Frequentist Probabiity can be interpreted as a frequency P = n N (1.1) where n stands for the successes and N for the tota number of trias. Or it can be interpreted as a ack of information: if I knew everything, I know that an event is surey going to happen, then P = 1, if I know it is not going to happen then P = 0 but in other cases I can use my judgment and or information from frequencies to estimate P. The word is divided in Frequentists and Bayesians. In genera, Cosmoogists are Bayesians and High Energy Physicists are Frequentists. For Frequentists events are just frequencies of occurrence: probabiities are ony defined as the quantities obtained in the imit when the number of independent trias tends to infinity. Bayesians interpret probabiities as the degree of beief in a hypothesis: they use judgment, prior information, probabiity theory etc... As we do cosmoogy we wi be Bayesian Deaing with probabiities In probabiity theory, probabiity distributions are fundamenta concepts. They are used to cacuate confidence intervas, for modeing purposes etc. We first need to introduce the concept of random variabe in statistics (and in Cosmoogy). Depending on the probem at hand, the random variabe may be the face of a dice, the number of gaaxies in a voume δv of the Universe, the CMB temperature in a given pixe of a CMB map, the measured vaue of the power spectrum P (k) etc. The probabiity that x (your random variabe) can take a specific vaue is P(x) where P denotes the probabiity distribution. The properties of P are: (i) P(x) is a non negative, rea number for a rea vaues of x. (ii) P(x) is normaized so that dxp(x) = 1 (iii) For mutuay excusive events x 1 and x 2, P(x 1 +x 2 ) = P(x 1 )+P(x 2 ) the probabiity of x 1 or x 2 to happen is the sum of the individua probabiities. P(x 1 +x 2 ) is aso written as P(x 1 Ux 2 ) or P(x 1.OR.x 2 ). for discrete distribution

3 (iv) In genera: Statistica Techniques for Data Anaysis in Cosmoogy 3 P(a, b) = P(a)P(b a) ; P(b, a) = P(b)P(a b) (1.2) The probabiity of a and b to happen is the probabiity of a times the conditiona probabiity of b given a. Here we can aso make the (apparenty tautoogica) identification P(a, b) = P(b, a). For independent events then P(a, b) = P(a)P(b). Exercise: Wi it be sunny tomorrow? answer in the frequentist way and in the Bayesian way. Exercise: Produce some exampes for rue (iv) above. - Whie Frequentists ony consider distributions of events, Bayesians consider hypotheses as events, giving us Bayes theorem: P(H D) = P(H)P(D H) P(D) (1.3) where H stands for hypothesis (generay the set of parameters specifying your mode, athough many cosmoogists now aso consider mode themseves) and D stands for data. P(H D) is caed the posterior distribution. P(H) is caed the prior and P(D H) is caed ikeihood. Note that this is nothing but equation 1.2 with the apparenty tautoogica identity P(a, b) = P(b, a) and with substitutions: b H and a D. Despite its simpicity Eq. 1.3 is a reay important equation!!! The usua points of heated discussion foow: How do you chose P(H)?, Does the choice affects your fina resuts? (yes, in genera it wi). Isn t this then a bit subjective? Exercise: Consider a positive definite quantity (ike for exampe the tensor to scaar ratio r or the optica depth to the ast scattering surface τ). What prior shoud one use? a fat prior in the variabe? or a ogaritmic prior (i.e. fat prior in the og of the quantity)? for exampe CMB anaysis may use a fat prior in n r, and in Z = exp( 2τ). How is this reated to using a fat pror in r or in τ? It wi be usefu to consider the foowing: effectivey we are comparing P(x) with P(f(x)), where f denotes a function of x. For df exampe x is τ and f(x) is exp( 2τ). Reca that: P(f) = P(x(f)) dx 1. The Jacobian of the transformation appears here to conserve probabiities. These ectures were given in the Canary Isands, in other ocations answer may differ...

4 4 Exercise: Compare Fig 21 of Sperge et a (2007)[25] with figure 13 of Sperge et a 2003 [24]. Consider the WMAP ony contours. Ceary the 2007 paper uses more data than the 2003 paper, so why is that the constraints ook worst? If you suspect the prior you are correct! Find out which prior has changed, and why it makes such a difference. Exercise: Under which conditions the choice of prior does not matter? (hint: compare the WMAP papers of 2003 and 2007 for the fat LCDM case) Moments and cumuants Moments and cumuants are used to characterize the probabiity distribution. In the anguage of probabiity distribution averages are defined as foows: f(x) = dxf(x)p(x) (1.4) These can then be reated to expectation vaues (see ater). For now et us just introduce the moments: ˆµ m = x m and, of specia interest, the centra moments: µ m = (x x ) m. - Exercise: show that ˆµ 0 = 1 and that the average x = ˆµ 1. Aso show that µ 2 = x 2 x 2 Here, µ 2 is the variance, µ 3 is caed the skewness, µ 4 is reated to the kurtosis. If you dea with the statistica nature of initia conditions (i.e. primordia non-gaussianity) or non-inear evoution of Gaussian initia conditions, you wi encounter these quantities again (and again..). Up to the skewness, centra moments and cumuants coincide. For higherorder terms things become more compicated. To keep things a simpe as possibe et s just consider the Gaussian distribution (see beow) as reference. Whie moments of order higher than 3 are non-zero for both Gaussian and non-gaussian distribution, the cumuants of higher orders are zero for a Gaussian distribution. In fact, for a Gaussian distribution a moments of order higher than 2 are specified by µ 1 and µ 2. Or, in other words, the mean and the variance competey specify a Gaussian distribution. This is not the case for a non-gaussan distribution. For non-gaussian distribution, the reation between centra moments and cumuants κ for the first 6 orders

5 Statistica Techniques for Data Anaysis in Cosmoogy 5 is reported beow. µ 1 = 0 (1.5) µ 2 = κ 2 (1.6) µ 3 = κ 3 (1.7) µ 4 = κ 4 + 3(κ 2 ) 2 (1.8) µ 5 = κ κ 3 κ 2 (1.9) µ 6 = κ κ 4 κ (κ 3 ) (κ 2 ) 3 (1.10) Usefu trick: the generating function The generating function aows one, among other things, to compute quicky moments and cumuants of a distribution. Define the generating function as Z(k) = exp(ikx) = dx exp(ikx)p(x) (1.11) Which may sound famiiar as it is a sort of Fourier transform... Note that this can be written as an infinite series (by expanding the exponentia) giving (exercise) (ik) n Z(k) = ˆµ n (1.12) n! n=0 So far nothing specia, but now the neat trick is that the moments are obtained as: ˆµ n = ( i n ) dn dk n Z(k) k=0 (1.13) and the cumuants are obtained by doing the same operation on n Z. Whie this seems just a neat trick for now it wi be very usefu shorty Two usefu distributions Two distributions are widey used in Cosmoogy: distribution and the Gaussian distribution The Poisson distribution these are the Poisson The Poisson distribution describes an independent point process: photon noise, radioactive decay, gaaxy distribution for very few gaaxies, point sources... It is an exampe of a discrete probabiity distribution. For cosmoogica appications it is usefu to think of a Poisson process as foows. Consider a random process (for exampe a random distribution of gaaxies in

6 6 space) of average density ρ. Divide the space in infinitesima ces, of voume δv so sma that their occupation can ony be 0 or 1 and the probabiity of having more than one object per ce is 0. Then the probabiity of having one object in a given ce is P 1 = ρδv and the probabiity of getting no object in the ce is therefore P 0 = 1 ρδv. Thus for one ce the generating function is Z(k) = n P n exp(ikn) = 1 + ρδv (exp(ik) 1) and for a voume V with V/δV ces, we have Z(k) = (1 + ρδv (exp(ik) 1)) V/δV exp[ρv (exp(ik) 1)]. With the substitution ρv λ we obtain Z(k) = exp[λ(exp(ik) 1)] = n=0 λ n /n! exp( λ) exp(ikn). Thus the Poission probabiity distribution we recover is: P n = λn exp[ λ] (1.14) n! Exercise: show that for the Poisson distribution n = λ and that σ 2 = λ The Gaussian distribution The Gaussian distribution is extremey usefu because of the Centra Limit theorem. The Centra Limit theorem states that the sum of many independent and identicay distributed random variabes wi be approximatey Gaussiany distributed. The conditions for this to happen are quite mid: the variance of the distribution one starts off with has to be finite. The proof is remarkaby simpe. Let s take n events with probabiity distributions P(x i ) and < x i >= 0 for simpicity, and et Y be their sum. What is P(Y )? The generating function for Y is the product of the generating functions for the x i : Z Y (k) = m= m=0 [ ] n ( (ik)m µ m 1 1 k 2 < x 2 n > +...) (1.15) m! 2 n for n then Z Y (k) exp[ 1/2k 2 < x 2 >]. By recaing the definition of generating function (eq we can see that the probabiity distribution which generated this Z is [ 1 P(Y ) = 2π < x 2 > exp 1 Y 2 ] 2 < x 2 (1.16) > that is a Gaussian!

7 Statistica Techniques for Data Anaysis in Cosmoogy 7 Exercise: Verify that higher order cumuants are zero for the Gaussian distribution. Exercise: Show that the Centra imit theorem hods for the Poisson distribution. Beyond the Centra Limit theorem, the Gaussian distribution is very important in cosmoogy as we beieve that the initia conditions, the primordia perturbations generated from infation, had a distribution very very cose to Gaussian. (Athough it is crucia to test this experimentay.) We shoud aso remember that thanks to the Centra Limit theorem, when we estimate parameters in cosmoogy in many cases we approximate our data as having a Gaussian distribution, even if we know that each data point is NOT drawn from a Gaussian distribution. The Centra Limit theorem simpifies our ives every day... There are exceptions though. Let us for exampe consider N independent data points drawn from a Cauchy distribution: P(x) = [πσ(1 + [(x x)/σ] 2 ] 1. This is a proper probabiity distribution as it integrates to unity, but moments diverge. One can show that the numerica mean of a finite number N of observations is finite but the popuation mean (the one defined through the integra of equation (1.4) with f(x) = x) is not. Note aso that the scatter in the average of N data points drawn from this distribution is the same as the scatter in 1 point: the scatter never diminishes regardess of the sampe size Modeing of data and statistica inference To iustrate this et us foow the exampe from [27]. If you have an urn with N red bas and M bue bas and you draw from the urn, probabiity theory can te you what the chances are of you to pick a red ba given that you has so far drawn m bue and n red ones... However in practice what you want to do is to use probabiity to te you what is the distribution of the bas in the urn having made a few drawn from it! In other words, if you knew everything about the Universe, probabiity theory coud te you what the probabiities are to get a given outcome for an observation. However, especiay in cosmoogy, you want to make few observations and draw concusions about the Universe! With the added compication that experiments in Cosmoogy are not quite ike experiments in the ab: you can t poke the Universe and see how it reacts, and in many

8 8 cases you can t repeat the observation, and you can ony see a sma part of the Universe! Keeping this caveat in mind et s push ahead. Given a set of observations often you want to fit a mode to the data, where the mode is described by a set of parameters α. Sometimes the mode is physicay motivated (say CMB anguar power spectra etc.) or a convenient function (e.g. initia studies of arge scae structure were fitting gaaxies correation functions with power aws). Then you want to define a merit function, that measures the agreement between the data and the mode: by adjusting the parameters to maximize the agreement one obtains the best fit parameters. Of course, because of measurement errors, there wi be errors associated to the parameter determination. To be usefu a fitting procedure shoud provide a) best fit parameters b) error estimates on the parameters c) possiby a statistica measure of the goodness of fit. When c) suggests that the mode is a bad description of the data, then a) and b) make no sense. Remember at this point Bayes theorem: whie you may want to ask: What is the probabiity that a particuar set of parameters is correct?, what you can ask to a figure of merit is Given a set of parameters, what is the probabiity that that this data set coud have occurred?. This is the ikeihood. You may want to estimate parameters by maximizing the ikeihood and somehow identify the ikeihood (probabiity of the data given the parameters) with the ikeihood of the mode parameters Chisquare, goodness of fit and confidence regions Foowing Numerica recipes ([23], Chapter 15) it is easier to introduce mode fitting and parameter estimation using the east-squares exampe. Let s say that D i are our data points and y( x i α) a mode with parameters α. For exampe if the mode is a straight ine then α denotes the sope and intercept of the ine. The east squares is given by: χ 2 = i w i [D i y(x i α)] 2 (1.17) and you can show that the minimum variance weights are w i = 1/σ Exercise: if the points are correated how does this equation change? - Best fit vaue parameters are the parameters that minimize the χ 2. Note

9 Statistica Techniques for Data Anaysis in Cosmoogy 9 Fig Exampe of inear fit to the data points D i in 2D. that a numerica exporation can be avoided: by soving χ 2 / α i 0 you can find the best fit parameters Goodness of fit In particuar, if the measurement errors are Gaussiany distributed, and (as in this exampe) the mode is a inear function of the parameters, then the probabiity distribution of for different vaues of χ 2 at the minimum is the χ 2 distribution for ν n m degrees of freedom (where m is the number of parameters and n is the number of data points. The probabiity that the observed χ 2 even for a correct mode is ess than a vaue ˆχ 2 is P(χ 2 <

10 10 ˆχ 2, ν) = P(ν/2, ˆχ 2 /2) = Γ(ν/2, ˆχ 2 /2) where Γ stands for the incompete Gamma function. Its compement, Q = 1 P(ν/2, ˆχ 2 /2) is the probabiity that the observed χ 2 exceed by chance ˆχ 2 even for a correct mode. See numerica recipes [23] chapters 6.3 and 15.2 for more detais. It is common that the chi-square distribution hods even for modes that are non inear in the parameters and even in more genera cases (see an exampe ater). The computed probabiity Q gives a quantitative measure of the goodness of fit when evauated at the best fit parameters (i.e. at χ 2 min ). If Q is a very sma probabiity then a) the mode is wrong and can be rejected b) the errors are reay arger than stated or c) the measurement errors were not Gaussiany distributed. If you know the actua error distribution you may want to Monte Caro simuate synthetic data stets, subject them to your actua fitting procedure, and determine both the probabiity distribution of your χ 2 statistic and the accuracy with which mode parameters are recovered by the fit (see section on Monte Caro Methods). On the other hand Q may be too arge, if it is too near 1 then aso something s up: a) errors may have been overestimated b) the data are correated and correations were ignored in the fit. c) In principe it may be that the distribution you are deaing with is more compact than a Gaussian distribution, but this is amost never the case. So make sure you excude cases a) and b) before you invest a ot of time in exporing option c). Postscript: the Chi-by eye rue is that the minimum χ 2 shoud be roughy equa to the number of data-number of parameters (giving rise to the widespread use of the so-caed reduced chisquare). Can you possiby rigorousy justify this statement? Confidence region Rather than presenting the fu probabiity distribution of errors it is usefu to present confidence imits or confidence regions: a region in the m- dimensiona parameter space (m being the number of parameters), that contain a certain percentage of the tota probabiity distribution. Obviousy you want a suitaby compact region around the best fit vaue. It is customary to choose 68.3%, 95.4%, 99.7%... Eipsoida regions have connections with the norma (Gaussian) distribution but in genera things may be very

11 Statistica Techniques for Data Anaysis in Cosmoogy 11 different... A natura choice for the shape of confidence intervas is given by constant χ 2 boundaries. For the observed data set the vaue of parameters α 0 minimize the χ 2, denoted by χ 2 min. If we perturb α away from α 0 the χ 2 wi increase. From the properties of the χ 2 distribution it is possibe to show that there is a we defined reation between confidence intervas, forma standard errors, and χ 2. We report here the χ 2 for the conventionas 1, 2,and 3 σ as a function of the number of parameters for the joint confidence eves: p % % % In genera, et s spe out the foowing prescription. If µ is the number of fitted parameters for which you want to pot the join confidence region and p is the confidence imit desired, find the χ 2 such that the probabiity of a chi-square variabe with µ degrees of freedom being ess than χ 2 is p. For genera vaues of p this is given by Q described above (for the standard 1,2,3 σ see tabe above). P.S. Frequentists use χ 2 a ot Likeihoods One can be more sophysticated than χ 2, if P(D) (D is data) is known. Remember from the Bayes theorem (eq.1.3) the probabiity of the data given the mode (Hypothesis) is the ikeihood. If we set P(D) = 1 (after a, you got the data) and ignore the prior, by maximizing the ikeihood we find the most ikey Hypothesis, or, often, the most ikey parameters of a given mode. Note that we have ignored P(D) and the prior so in genera this technique does not give you a goodness of fit and not an absoute probabiity of the mode, ony reative probabiities. Frequentists rey on χ 2 anayses where a goodness of fit can be estabished. In many cases (thanks to the centra imit theorem) the ikeihood can be we approximated by a muti-variate Gaussian: 1 L = (2π) n/2 exp 1 (D y) detc 1/2 i Cij 1 2 (D y) j (1.18) ij

12 12 where C ij = (D i y i )(D j y j ) is the covariance matrix. Exercise: when are ikeihood anayses and χ 2 anayses the same? Confidence eves for ikeihood For Bayesian statistics, confidence regions are found as regions R in mode space such that R P( α D)d α is, say, 0.68 for 68% confidence eve and 0.95 for 95% confidence. Note that this encoses the prior information. To report resuts independenty of the prior the ikeihood ratio is used. In this case compare the ikeihood at a particuar point in mode space L( α) with the vaue of the maximum ikeihood L max. Then a mode is said acceptabe if [ ] L( α) 2 n threshod (1.19) L max Then the threshod shoud be caibrated by cacuating the distribution of the ikeihood ratio in the case where a particuar mode is the true mode. There are some cases however when the vaue of the threshod is the corresponding confidence imit for a χ 2 with m degrees of freedom, for m the number of parameters. Exercise: in what cases? Marginaization, combining different experiments Of a the mode parameters α i some of them may be uninteresting. Typica exampes of nuisance parameters are caibration factors, gaaxy bias parameter etc, but aso it may be that we are interested on constraints on ony one cosmoogica parameter at the time rather than on the joint constraints on 2 or more parameters simutaneousy. One then marginaizes over the uninteresting parameters by integrating the posterior distribution: P (α 1..α j D) = dα j+1,...dα m P ( α D) (1.20) if there are in tota m parameters and we are interested in j of them (j < m). Note that if you have two independent experiments, the combined ikeihood of the two experiments is just the product of the two ikeihoods. Soution: The data must have Gaussian errors, the mode must depend ineary on the parameters, the gradients of the mode with respect to the parameters are not degenerate and the parameters do not affect the covariance.

13 Statistica Techniques for Data Anaysis in Cosmoogy 13 (of course if the two experiments are non independent then one woud have to incude their covariance). In many cases one of the two experiments can be used as a prior. A word of caution is on order here. We can aways combine independent experiments by mutipying their ikeihoods, and if the experiments are good and sound and the mode used is a good and compete description of the data a is we. However it is aways important to: a) think about the priors one is using and to quantify their effects. b) make sure that resuts from independent experiments are consistent: by mutipying ikeihood from inconsistent experiments you can aways get some sort of resuts but it does not mean that the resut actuay makes sense... Sometimes you may be interested in pacing a prior on the uninteresting parameters before marginaization. The prior may come from a previous measurement or from your beief. Typica exampes of this are: marginaization over caibration uncertainty, over point sources ampitude or over beam errors for CMB studies. For exampe for marginaization over, say, point source ampitude, it is usefu to know of the foowing trick for Gaussian ikeihoods: P (α 1..α m 1 D) = da (2π) m 2 C 1 2 [ 1 exp 1 2πσ 2 2 e [ 1 2 (C i (Ĉi+AP i ))Σ 1 ij (C j (Ĉj+AP j ))] (1.21) ] (A Â)2 σ 2 repeated indices are summed over and C denotes the determinant. Here, A is the ampitude of, say, a point source contribution P to the C anguar power spectrum, A is the m th parameter which we want to marginaize over with a Gaussian prior with variance σ 2 around Â. The trick is to recognize that this integra can be written as: P (α 1..α m 1 D) = C 0 exp [ 12 ] C 1 2C 2 A + C 3 A 2 da (1.22) (where C denote constants) and that this kind of integra is evauated by using the substitution A A C 2 /C 3 giving something exp[ 1/2(C 1 C 2 2 /C 3)]. It is eft as an exercise to write the constants expicity An exampe Let s say you want to constrain cosmoogy by studying custers number counts as a function of redshift. Here we foow the paper of Cash 1979 [3].

14 14 The observation of a discrete number N of custers is a Poisson process, the probabiity of which is given by the product P = Π N i=1[e n i i exp( e i )/n i!] (1.23) where n i is the number of custers observed in the i th experimenta bin and e i is the expected number in that bin in a given mode: e i = I(x)δx i with i being the proportiona to the probabiity distribution. Here δx i can represent an interva in custers mass and/or redshift. Note: this is a product of Poisson distributions, thus one is assuming that these are independent processes. Custers may be custered, so when can this be used? For unbinned data (or for sma bins so that bins have ony 0 and 1 counts) we define the quantity: N C 2 n P = 2(E n I i ) (1.24) i=1 where E is the tota expected number of custers in a given mode. The quantity C between two modes with different parameters has a χ 2 distribution! (so a that was said in the χ 2 section appies, even though we started from a highy non-gaussian distribution.) 1.4 Description of random fieds Let s take a break from probabiities and consider a sighty different issue. In comparing the resuts of theoretica cacuations with the observed Universe, it woud be meaningess to hope to be abe to describe with the theory the properties of a particuar patch, i.e. to predict the density contrast of the matter δ( x) = δρ(x)/ρ at any specific point x. Instead, it is possibe to predict the average statistica properties of the mass distribution. In addition, we consider that the Universe we ive in is a random reaization of a the possibe Universes that coud have been a reaization of the true underying mode (which is known ony to Mother Nature). A the possibe reaizations of this true underying Universe make up the ensambe. In statistica inference one may sometime want to try to estimate how different our particuar reaization of the Universe coud be from the true underying one. Thinking back at the exampe of the urn with coored bas, it woud be ike considering that the particuar urn from which we are drawing the bas is ony one possibe reaization of the true underying distribution of urns. For exampe, say that the true distribution has a spit in red and A very simiar approach is taken in statistica mechanics.

15 Statistica Techniques for Data Anaysis in Cosmoogy 15 bue bas but that the urn can have ony an odd number of bas. Ceary the exact spit cannot be reaized in one particuar urn but it can be reaized in the ensambe... Foowing the cosmoogica principe (e.g. Peebes 1980 [22]), modes of the Universe have to be homogeneous on the average, therefore, in widey separated regions of the Universe (i.e. independent), the density fied must have the same statistica properties. A crucia assumption of standard cosmoogy is that the part of the Universe that we can observe is a fair sampe of the whoe. This is cosey reated to the cosmoogica principe since it impies that the statistics ike the correation functions have to be considered as averages over the ensembe. But the pecuiarity in cosmoogy is that we have just one Universe, which is just one reaization from the ensembe (quite fictitious one: it is the ensembe of a possibe Universes). The fair sampe hypothesis states that sampes from we separated part of the Universe are independent reaizations of the same physica process, and that, in the observabe part of the Universe, there are enough independent sampes to be representative of the statistica ensembe. The hypothesis of ergodicity foows: averaging over many reaizations is equivaent to averaging over a arge (enough) voume. The cosmoogica fied we are interested in, in a given voume, is taken as a reaization of the statistica process and, for the hypothesis of ergodicity, averaging over many reaizations is equivaent to averaging over a arge voume. Theories can just predict the statistica properties of δ( x) which, for the cosmoogica principe, must be a homogeneous and isotropic random fied, and our observabe Universe is a random reaization from the ensembe. In cosmoogy the scaar fied δ( x) is enough to specify the initia fuctuations fied, and we utimatey hope aso the present day distribution of gaaxies and matter. Here ies one of the big chaenges of modern cosmoogy. A fundamenta probem in the anaysis of the cosmic structures, is to find the appropriate toos to provide information on the distribution of the density fuctuations, on their initia conditions and subsequent evoution. Here we concentrate on power spectra and correation functions Gaussian random fieds Gaussian random fieds are cruciay important in cosmoogy, for different reasons: first of a it is possibe to describe their statistica properties anayticay, but aso there are strong theoretica motivations, namey infation, to assume that the primordia fuctuations that gave rise to the present-day cosmoogica structures, foow a Gaussian distribution. Without resorting

16 16 to infation, for the centra imit theorem, Gaussianity resuts from a superposition of a arge number of random processes. The distribution of density fuctuations δ defined as δ = δρ/ρ cannot be exacty Gaussian because the fied has to satisfy the constraint δ > 1, however if the ampitude of the fuctuations is sma enough, this can be a good approximation. This seems indeed to be the case: by ooking at the CMB anisotropies we can probe fuctuations when their statistica distribution shoud have been cose to its primordia one; possibe deviations from Gaussianity of the primordia density fied are sma. If δ is a Gaussian random fied with average 0, its probabiity distribution is given by: P n (δ 1,, δ n ) = DetC 1 (2π) n/2 [ exp 1 ] 2 δt C 1 δ (1.25) where δ is a vector made by the δ i, C 1 denotes the inverse of the correation matrix which eements are C ij = δ i δ j. An important property of Gaussian random fieds is that the Fourier transform of a Gaussian fied is sti Gaussian. The phases of the Fourier modes are random and the rea and imaginary part of the coefficients have Gaussian distribution and are mutuay independent. Let us denote the rea and imaginary part of δ k by Reδ k and Imδ k respectivey. Their joint probabiity distribution is the bivariate Gaussian: P (Reδ k, Imδ k )dreδ k dimδ k = 1 2πσk 2 exp [ Reδ2 k + Imδ2 k 2σ 2 k ] dreδ k dimδ k (1.26) where σk 2 is the variance in Reδ k and Imδ k and for isotropy it depends ony on the magnitude of k. Equation (1.26) can be re-written in terms of the ampitude δ k and the phase φ k : P ( δ k, φ k )d δ k dφ k = 1 2πσk 2 exp [ δ k 2 2σ 2 k ] δ k d δ k dφ k (1.27) that is δ k foows a Rayeigh distribution. From this foows that the probabiity that the ampitude is above a certain threshod X is: [ P ( δ k 2 1 > X) = X σk 2 exp δ k 2 ] [ 2σk 2 δ k d δ k = exp X ] δ k 2. (1.28) Which is an exponentia distribution. Note that δ = 0

17 Statistica Techniques for Data Anaysis in Cosmoogy 17 The fact that the phases of a Gaussian fied are random, impies that the two point correation function (or the power spectrum) competey specifies the fied. P.S. If your advisor now asks you to generate a Gaussian random fied you know how to do it. (It you are not famiiar with Fourier transforms see next section) - The observed fuctuation fied however is not Gaussian. The observed gaaxy distribution is highy non-gaussian principay due to gravitationa instabiity. To competey specify a non-gaussian distribution higher order correation functions are needed ; conversey deviations from Gaussian behavior can be characterized by the higher-order statistics of the distribution Basic toos The Fourier transform of the (fractiona) overdensity fied δ is defined as: δ k = A d 3 rδ( r) exp[ i k r] (1.29) with inverse δ( r) = B d 3 kδ k exp[i k r] (1.30) and the Dirac deta is then given by δ D ( k) = BA d 3 r exp[±i k r] (1.31) Here I chose the convention A = 1, B = 1/(2π) 3, but aways beware of the FT conventions. The two point correation function (or correation function) is defined as: ξ(x) = δ( r)δ( r + x) = < δ k δ k > exp[i k r] exp[i k ( r+ x)]d 3 kd 3 k (1.32) because of isotropy ξ( x ) (ony a function of the distance not orientation). Note that in some cases when isotropy is broken one may want to keep the orientation information (see e.g. redshift space distortions, which affect custering ony aong the ine-of sight ). For non pathoogica distributions. For a discussion see e.g. [17].

18 18 The definition of the power spectrum P (k) foows : < δ k δ k >= (2π) 3 P (k)δ D ( k + k ) (1.33) again for isotropy P (k) depends ony on the moduus of the k-vector, athough in specia cases where isotropy is broken one may want to keep the direction information. Since δ( r) is rea. we have that δ k = δ k, so < δ k δ k >= (2π) 3 d 3 xξ(x) exp[ i k x]δ d ( k k ) (1.34). The power spectrum and the correation function are Fourier transform pairs: 1 ξ(x) = (2π) 3 P (k) exp[i k r]d 3 k (1.35) P (k) = ξ(x) exp[ i k x]d 3 x (1.36) At this stage the same amount of information is encosed in P (k) as in ξ(x). From here the variance is σ 2 =< δ 2 (x) >= ξ(0) = 1 (2π) 3 P (k)d 3 k (1.37) or better σ 2 = 2 (k)d n k where 2 (k) = 1 (2π) 3 k3 P (k) (1.38) and the quantity 2 (k) is independent form the FT convention used. Now the question is: on what scae is this variance defined? Answer: in practice one needs to use fiters: the density fied is convoved with a fiter (smoothing) function. There are two typica choices: 1 f = (2π) 3/2 RG 3 exp[ 1/2x 2 /RG] 2 Gaussian f k = exp[ k 2 RG/2] 2 (1.39) 1 f = (4π)RT 3 Θ(x/R T ) TopHat f k = 3 (kr T ) 3 [sin(kr T ) kr T cos(kr T )] (1.40) roughy R T 5R G. Remember: Convoution in rea space is a mutipication in Fourier space; Mutipication in rea space is a convoution in Fourier space.

19 Statistica Techniques for Data Anaysis in Cosmoogy 19 - Exercise: consider a muti-variate Gaussian distribution: P (δ 1..δ n ) = 1 (2π) n/2 detc 1/2 exp[ 1 2 δt C 1 δ] (1.41) where C ij =< δ i δ j > is the covariance. Show that is δ i are Fourier modes then C ij is diagona. This is an idea case, of course but this is teing us that for Gaussian fieds the different k modes are independent! which is aways a nice feature. Another question for you: if you start off with a Gaussian distribution (say from Infation) and then eave this Gaussian fied δ to evove under gravity, wi it remain Gaussian forever? Hint: think about present-time Universe, and think about the dark matter density at, say, the center of a big gaaxy and in a arge void The importance of the Power spectrum The structure of the Universe on arge scaes is argey dominated by the force of gravity (which we think we know we) and no too much by compex mechanisms (baryonic physics, gaaxy formation etc.)- or at east that s the hope... Theory (see ectures on infation) give us a prediction for the primordia power spectrum: ( ) k n P (k) = A (1.42) k0 n - the spectra index is often taken to be a constant and the power spectrum is a power aw power spectrum. However there are theoretica motivations to generaize this to ( ) k n(k0 )+ 1 2 P (k) = A k0 dn d n k n(k/k 0) (1.43) as a sort of Tayor expansion of n(k) around the pivot point k 0. dn/d n k is caed the running of the spectra index. Note that different authors often use different choices of k 0 (sometimes the same author in the same paper uses different choices...) so things may get confused... so et s report expicity the conversions: ( ) k n(k0 )+1/2(dn/d n k) n(k 1 /k 0 ) A(k 1 ) = A(k 0 ) (1.44) k0

20 20 Exercise: Prove the equations above. Exercise:Show that given the above definition of the running of the spectra index, n(k) = n(k 0 ) + dn/d n k n(k/k 0 ). It can be shown that as ong as inear theory appies and ony gravity is at pay δ << 1, different Fourier modes evove independenty and the Gaussian fied remains Gaussian. In addition, P (k) changes ony in ampitude and not in shape except in the radiation to matter dominated era and when there are baryon-photon interactions and baryons-dark matter interactions (see Lectures on CMB). In detai, this is described by inear perturbation growth and by the transfer function Exampes of rea word issues Say that now you go and try to measure a P (k) from a reaistic gaaxy cataog. What are the rea word effects you may find? We have mentioned before redshift space distortions. Here we concentrate on other effects that are more genera (and not so specific to arge-scae structure anaysis) Discrete Fourier transform In the rea word when you go and take the FT of your survey or even of your simuation box you wi be using something ike a fast Fourier transform code (FFT) which is a discrete Fourier transform. If your box has side of size L, even if δ(r) in the box is continuous, δ k wi be discrete. The k-modes samped wi be given by ( ) 2π k = (i, j, k) where k = 2π (1.45) L L The discrete Fourier transform is obtained by pacing the δ(x) on a attice of N 3 grid points with spacing L/N. Then: δk DF T = 1 N 3 exp[ i k r]δ( r) (1.46) δ DF T ( r) = k r exp[i k r]δ DF T k (1.47) Beware of the mapping between r and k, some routines use a weird wrapping! There are different ways of pacing gaaxies (or partice in your simuation) on a grid: Nearest grid point, Coud in ce, tranguar shaped coud etc...

21 Statistica Techniques for Data Anaysis in Cosmoogy 21 For each of these remember(!) then to deconvove the resuting P (k) for their effect. Note that ( ) x 3 δ k N 3 δk DF T 1 2π k 3 δdf k T (1.48) and thus P (k) < δdf T 2 > ( k) 3 since δ D (k) δk ( k) 3 (1.49) The discretization introduces severa effects: The Nyquist frequency: k Ny = 2π N L 2 is that of a mode which is samped by 2 grid points. Higher frequencies cannot be propery samped and give aiasing (spurious transfer of power) effects. You shoud aways work at k < k Ny. There is aso a minimum k (argest possibe scae) that you finite box can test :k min > 2π/L. This is one of the many reason why one needs ever arger N-body simuations... In addition DFT assume periodic boundary conditions, if you do not have periodic boundary conditions then this aso introduces aiasing Window, seection function, masks etc Seection function: Gaaxy surveys are usuay magnitude imited, which means that as you ook further away you start missing some gaaxies. The seection function tes you the probabiity for a gaaxy at a given distance (or redshift z) to enter the survey. It is a mutipicative effect aong the ine of sight in rea space. Window or mask You can never observe a perfect (or even better infinite) squared box of the Universe and in CMB studies you can never have a perfect fu sky map (we ive in a gaaxy...). The mask (sky cut in CMB jargon) is a function that usuay takes vaues of 0 or 1and is defined on the pane of the sky (i.e. it is constant aong the same ine of sight). The mask is aso a rea space mutipication effect. In addition sometimes in CMB studies different pixes may need to be weighted differenty, and the mask is an extreme exampe of this where the weights are either 0 or 1. Aso this operation is a rea space mutipication effect. Let s reca that a mutipicaton in rea space (where W ( x) denotes the effects of window and seection functions) is a convoution in Fourier space: δ true ( x) δ obs ( x) = δ true ( x)w ( x) (1.50) δ true ( k) δ obs ( k) = δ true ( k) W ( k) (1.51)

22 22 Fig The importance of periodic boundary conditions the sharper W ( r) is the messier and deocaized W ( k) is. As a resut it wi coupe different k-modes even if the underying ones were not correated! Discreteness Whie the dark matter distribution is amost a continuous one the gaaxy distribution is discrete. We usuay assume that the gaaxy distribution is a samping of the dark matter distribution. The discreteness effect give the gaaxy distribution a Poisson contribution (aso caed shot noise contribution). Note that the Poisson contribution is non Gaussian: it is ony in the imit of arge number of objects (or of modes) that it approximates a Gaussian. Here it wi suffice to say that as ong as a gaaxy number density is high enough (which wi need to be quantified and checked for any practica appication) and we have enough modes, we say

23 Statistica Techniques for Data Anaysis in Cosmoogy 23 that we wi have a superposition of our random fied (say the dark matter one characterized by its P (k)) pus a white noise contribution coming from the discreteness which ampitude depends on the average number density of gaaxies (and shoud go to zero as this go to infinity), and we treat this additiona contribution as if it has the same statistica properties as the underying density fied (which is an approximation). What is the shot noise effect on the correation properties? Foowing [22] we recognize that our random fied is now given by f( x) = n( x) = n[1 + δ( x)] = i δ D ( x x i ) (1.52) where n denotes average number of gaaxies: n =< i δd ( x x i ) >. Then, as done when introducing the Poisson distribution, we divide the voume in infinitesima voume eements δv so that their occupation can ony be 0 or 1. For each of these voumes the probabiity of getting a gaaxy is δp = ρ( x)δv, the probabiity of getting no gaaxy is δp = 1 ρ( x)δv and < n i >=< n 2 i >= nδv. We then obtain a doube stochastic process with one eve of randomness coming from the underying random fied and one eve coming from the Poisson samping. The correation function is obtained as: ij δ D ( r 1 r i )δ D ( r 2 r j ) = n 2 (1 + ξ 12 ) + nδ D ( r 1 r 2 ) (1.53) thus < n 1 n 2 >= n 2 [1+ < δ 1 δ 2 > d ] where < δ 1 δ 2 > d = ξ(x 12 ) + 1 n δd ( r 1 r 2 ) (1.54) and in Fourier space ( < δ k1 δ k2 > d = (2π) 3 P (k) + 1 n ) δ d ( k 1 + k 2 ) (1.55) This is not a compete surprise: the power spectrum of a superposition of two independent processes is the sum of the two power spectra pros and cons of ξ(r) and P (k) Let us briefy recap the pros and cons of working with power spectra or correation functions. Power spectra Pros: Direct connection to theory. Modes are uncorreated (in the idea case). The average density ends up in P (k = 0) which is usuay discarted, so no accurate knowedge of the mean density is needed. There is a cear distinction between inear and non-inear scaes. Smoothing is not a probem

24 24 (just a mutipication) Cons: window and seection functions act as compicated convoutions, introducing mode couping! (this is a serious issue) Correation function Pros: no probem with window and seection function Cons: scaes are correated. covariance cacuation a rea chaenge even in the idea case. Need to know mean densities very we. No cear distinction between inear and non-inear scaes. No direct correspondence to theory and for CMB? If we can observe the fu sky the the CMB temperature fuctuation fied can be nicey expanded in spherica harmonics: where and thus T (ˆn) = >0 a m = m= a m Y m (ˆn). (1.56) dω n T (ˆn)Y m(ˆn). (1.57) < a m 2 >= a m a m = δ δ mm C (1.58) C is the anguar power spectrum and C = 1 (2 + 1) m= a m 2 (1.59) Now what happens in the presence of rea word effects such as a sky cut? Anaogousy to the rea space case: ã m = dω n T (ˆn)W (ˆn)Y m (ˆn) (1.60) where W (ˆn) is a position dependent weight that in particuar is set to 0 on the sky cut. As any CMB observation gets pixeized this is ã m = Ω p T (p)w (p)ym(p) (1.61) p where p runs over the pixes and Ω p denotes the soid ange subtended by the pixe. Ceary this can be a probem (this can be a nasty convoution), but et us initiay ignore the probem and carry on.

25 Statistica Techniques for Data Anaysis in Cosmoogy 25 The pseudo-c s (Hivon et a 2002)[15] are defined as: C = 1 (2 + 1) m= ã m 2 (1.62) Ceary C C but C = G C (1.63) where <> denotes the ensambe average. We notice aready two things: as expected the effect of the mask is to coupe otherwhise uncorreated modes. In arge scae structure studies usuay peope stop here: convove the theory with the various rea word effects incuding the mask and compare that to the observed quantities. In CMB usuay we go beyond this step and try to deconvove the rea word effects. First of a note that where W = G 1 2 = π 3 ( )W 3 W m 2 and W m = m ( ) 2 (1.64) dω n W (ˆn)Y m(ˆn) (1.65) So if you are good enough to be abe to invert G and you can say that < C > is the C you want then C = G 1 C (1.66) Not for a experiments it is viabe (possibe) to do this ast step. In adddition to this, the instrument has other effects such as noise and a finite beam Noise and beams Instrumenta noise and the finite resoution of any experiment affect the measured C. The effect of the noise is easiy found: the instrumenta noise is an independent random process with a Gaussian distribution superposed to the temperature fied. In a m space a m a signa m Whie a noise m noise bias: + a noise m. = 0, in the power spectrum this gives rise to the so-caed C measured = C signa + C noise (1.67)

26 26 where C noise = (2 + 1) m anoise m 2. As the expectation vaue of C noise is non zero, this is a biased estimator. Note that the noise bias disappears if one computes the so-caed cross C obtained as a cross-correation between different, uncorreated, detectors (say detector a and b) as a noise,a m a noise,b m = 0. One is however not getting something for nothing: when one computes the covariance (or the associated error) for auto and for cross correation C (exercise!) the covariance is the same and incudes the extra contribution of the noise. It is ony that the cross-c are unbiased estimators. Every experiment sees the CMB with a finite resoution given by the experimenta beam (simiar concept to the Point Spread Function for optica astronomy). The observed temperature fied is smoothed on the beam scaes. Smoothing is a convoution in rea space: T i = dω nt (ˆn)b( ˆn ˆn ) (1.68) where we have considered a symmetric beam for simpicity. The beam is often we approximated by a Gaussian of a given Fu Width at Haf Maximum. Remember that σ b = 0.425F W HM. Thus in harmonic space the beam effect is a mutipication: C measured and in the presence of instrumenta noise C measured = C sky e 2 σ 2 b (1.69) = C sky e 2 σb 2 + C noise (1.70) Of course, one can aways deconvove for the effects of the beam to obtain an estimate of C measured as cose as possibe to C sky : C measured = C sky + C noise e 2 σ 2 b. (1.71) That is why it is often said that the effective noise bows up at high (sma scaes) and why it is important to know the beam(s) we. Exercise: What happens if you use cross-c s? Exercise: What happens to C measured if the beam is poory reconstructed? - Note that the signa to noise of a CMB map depends on the pixe size (by smoothing the map and making arger pixes the noise per pixe wi decrease as Ω pix, Ω pix being the new pixe soid ange), on the integration

27 Statistica Techniques for Data Anaysis in Cosmoogy 27 time σ pix = s/ t where s is the detector sensitivity and t the time spent on a given pixe and on the number of detectors σ pix = s/ M where M is the number f detectors. To compare maps of different beam sizes it is usefu to have a noise measure that in independent of Ω pix : w = (σ 2 pix Ω pix) 1. - Exercise: Compute the expression for C noise given: t = observing time s = detector sensitivity (in µk/ s) n = number of detectors N = number of pixes f sky = fraction of the sky observed Assume uniform noise and observing time uniformy distributed. You may find [18] very usefu Aside: Higher orders correations From what we have earned so far we can concude that the power spectrum (or the correation function) competey characterizes the statistica properties of the density fied if it is Gaussian. But what if it is not? Higher order correations are defined as: δ 1...δ m where the detas can be in rea space giving the correation function or in Fourier space giving power spectra. At this stage, it is usefu to present here the Wick s theorem (or cumuant expansion theorem). The correation of order m can in genera be written as sum of products of unreducibe (connected) correations of order for = 1...m. For exampe for order 3 we obtain: δ 1 δ 2 δ 3 f = (1.72) δ 1 δ 2 δ 3 + (1.73) δ 1 δ 2 δ 3 + (3cyc.terms) (1.74) δ 1 δ 2 δ 3 (1.75) and for order 6 (but for a distribution of zero mean): < δ 1...δ 6 > f = (1.76) < δ 1 δ 2 >< δ 3 δ 4 >< δ 5 δ 6 > +..(15terms) (1.77) < δ 1 δ 2 >< δ 3 δ 4 δ 5 δ 6 > +..(15terms) (1.78)

28 28 < δ 1 δ 2 δ 3 >< δ 4 δ 5 δ 6 > +...(10terms) (1.79) < δ 1..δ 6 > (1.80) For computing covariances of power spectra, it is usefu to be famiiar with the above expansion of order More on Likeihoods Whie the CMB temperature distribution is Gaussian (or very cose to Gaussian) the C distribution is not. At high the Centra Limit Theorem wi ensure that the ikeihood is we approximated by a Gaussian but at ow this is not the case. L(T C th ) exp[ (T S 1 T )/2] det(s) (1.81) where T denotes a vector of the temperature map, C th denotes the C given by a theoretica mode (e.g. a cosmoogica parameters set), and S ij is the signa covariance: S ij = (2 + 1) C th 4π P ( ˆn i ˆn j ) (1.82) and P denote the Legendre poynomias. If we then expand T in spherica harmonics we obtain: ] L(T C th ) exp[ 1/2 a m 2 /C th C th Isotropy means that we can sum over m s thus: 2 n L = [ ( ) ( ) ] C th (2 + 1) n C data C data + C th 1 (1.83) (1.84) where C data = m a m 2 /(2 + 1). - Exercise: show that for an experiment with (gaussian) noise the expression is the same but with the substitution C th C th + N with N denoting the power spectrum of the noise. Exercise: show that for a partia sky experiment (that covers a fraction of sky f sky you can approximatey write: n L f sky n L (1.85)

29 Statistica Techniques for Data Anaysis in Cosmoogy 29 Hint: Think about how the number of independent modes scaes with the sky area. As an aside... you coud ask: But what do I do with poarization data?. We... if the a T m are Gaussiany distributed aso the ae m and ab m wi be. So we can generaize the approach above using a vector (a T m ae m ab m ). Let us consider a fu sky, idea experiment. Start by writing down the covariance, foow the same steps as above and show that: 2 n L = { ( ) ( C BB C T T C EE (C T E ) 2 ) (2 + 1) n + n Ĉ BB Ĉ T T Ĉ EE (ĈT E ) 2 + ĈT T where C denotes C th C EE C T T + C T T C EE Ĉ EE (C T E and Ĉ denotes C data. C T E 2ĈT E ) 2 + ĈBB C BB It is easy to show that for a noisy experiment then C XY 3 } C XY, (1.86) +N XY. (this is often the case as the sky cut for poarization may be different from that of temperature( the foregrounds are different) and in genera the cut (or the weighting) for B may need to be arger than that for E. where N denotes the noise power spectrum and X, Y = {T, E, B}. Exercise: generaize the above to partia sky coverage: for added compication take fsky T T f sky EE f bb sky Foowing [26] et us now expand in Tayor series Equation (1.84) around its maximum by writing Ĉ = C th (1 + ɛ). For a singe mutipoe, ( ) ɛ 2 2 n L = (2 + 1)[ɛ n(1 + ɛ)] (2 + 1) 2 ɛ3 3 + O(ɛ4 ). (1.87) We note that the Gaussian ikeihood approximation is equivaent to the above expression truncated at ɛ 2 : 2 n L Gauss, (2+1)/2[(Ĉ C th)/cth (2 + 1)ɛ 2 /2. Aso widey used for CMB studies is the ognorma ikeihood for the equa variance approximation (Bond et a 1998): approximation is 2 n L LN = (2 + 1) 2 [ n ( )] 2 ( ) Ĉ ɛ 2 C th (2 + 1) 2 ɛ3 2 ]2. (1.88)