Partition function based analysis of cosmic microwave background maps

to (CMB) base our analysis on the study of the partition function. to examine the maps, use of the different information embedded at different scales and function for testing the statistical distribution of the COBE –DMR data set. We conclude that no evidence of non-Gaussianity can be found by means of this method.


I N T R O D U C T I O N
In a few years the forthcoming CMB data sets from the MAP (NASA) and PLANCK (ESA) missions will offer us a much better image of the young Universe than ever before. The CMB represents a view of the Universe when it was about 0.002 per cent of its present age. CMB anisotropies provide a link between theoretical predictions and observational data. Undoubtedly these data will constrain more accurately the fundamental cosmological parameters. In recent years several groups have been very active in the study of the CMB anisotropies. Many statistical methods have been adapted to the analysis of the future CMB maps, and others are being developed.
There are many methods that can give relatively accurate values for the parameters of the cosmological models. For example, the power spectrum is considered to be the best discriminator between different models (Bond, Efstathiou & Tegmark 1997;Hinshaw et al. 1996a;Wright et al. 1996;Tegmark 1996). Related through a Legendre expansion to the power spectrum, the two-point correlation function is also a useful discriminator (Cayo Ân et al 1996;Hinshaw et al 1996b). These analyses, based on the power spectrum, are considered as classical but there are many other methods that do not make use of the power spectrum. The quality of the CMB maps demands for other statistics to supplement the power spectrum, looking for instance at morphological or topological characteristics of the data. For example, the one-dimensional analysis is a geometrical method useful for one-dimensional scans of CMB data and is based on the study of regions above or below a certain level (Gutie Ârrez et al. 1994). The peak analyses, similar to the previous one but for two-dimensional data, deal with the number of spots above a given threshold (Fabbri & Torres 1995) or with other geometrical properties like the Gaussian curvature or excentritity of the maxima (Barreiro et al. 1997). Other approaches are the genus (Smoot et al. 1994;Torres et al. 1995), Minkowski functionals, which relate several geometrical aspects at the same time (Schmalzing & Go Ârski 1997;Winitzki & Kosowsky 1997), wavelet-based techniques (Pando, Valls-Gabaud & Fang 1998;Hobson, Jones & Lasenby 1998) who have shown wavelets to be very effective at detecting non-Gaussianity in the CMB, and fractality (Pompilio et al. 1995;De Gouveia dal Pino et al. 1995;Mollerach et al. 1998).
In this paper we present an alternative method to analyse CMB maps based on the partition function. This function contains useful information about the temperature anisotropies at the different scales and moments. The method presented here is related to the one used by Smoot et al. (1994) based on moments at different smoothing angles. However, our method is more general and powerful because it works with any moment, not only with positive and integer ones.
The structure of the paper is as follows. In Section 2 we present the partition function and discuss its main characteristics. In the same section three different analyses based on that function are introduced: a likelihood analysis, a multifractal analysis and a test of Gaussianity. The likelihood analysis uses the partition function to search for the parameters (Q rmsÀPS and n) that best ®t a given data set. The multifractal analysis searches for scaling laws and fractal behaviour of the data. There are theoretical reasons (Sachs±Wolfe effect) to expect scaling in the data. Here we present the generalized fractal dimensions and the scaling exponents and also comment on the possible multifractality of the CMB sky. In a recent paper, Ferreira, Magueijo & Go Ârski (1998) found evidence of non-Gaussianity in COBE±DMR data at the 99 per cent con®dence level. We show that the partition function can be used to study this property. There is a clear relation between the partition function and the cumulant function, the last one having a speci®c form for a Gaussian signal. In Section 3 we apply the results of the previous sections to the COBE±DMR 4 years data set and we compare this with other results. We conclude in Section 4.

T H E PA R T I T I O N F U N C T I O N
Let us start directly with the de®nition, where Zq; d is the partition function. The quantity m i d is called the measure, it is a function of d which is the size or scale of the boxes used to cover the sample. The boxes are labelled by i, and N boxes d is the number of boxes (or cells) needed to cover the map when the grid with resolution d is used. The exponent q is a continuous real parameter that plays the role of the order of the moment of the measure. Let us consider a map of N pixels. Now the map is divided in boxes of size d´d pixels and the measure m i d is computed in each one of the resulting boxes. Changing both, q and d, one calculates the function Zq; d. We would like to emphasize that the calculation of Zq; d is ON.
One is free to make any choice of the measure md provided that several conditions are satis®ed, the most restrictive being m i d $ 0. There are no general rules to decide which is the best choice. For CMB maps, we use the most natural measure de®ned as follows: Thus the measure in the box i is the sum of the absolute temperatures T pix of the pixels inside the box in units of Kelvin. This is a very natural measure comparable to the measure used in the study of galaxies or clusters distribution (Martõ Ânez et al. 1990, see Borgani 1995 for a review), where the measure is taken as the total mass (or the total number of galaxies/clusters) contained in the box. The constant T Ã is a normalization constant. The measures are interpreted as probabilities and they have to be normalized, i.e i m i 1. So T Ã is simply the sum of the absolute temperatures over all pixels and therefore is a constant for all boxes and scales.
The temperature in the pixels is almost the same everywhere because of the homogeneity of the signal, and one expects that different models will behave in a very similar way, making dif®cult the task of distinguishing them. We shall show how the partition function overcomes this problem.
Alternatively, Pompilio et al. (1995), in a multifractal analysis of string-induced CMB anisotropies (one-dimensional scans) used as a measure where D j denotes the¯uctuation of the temperature in pixel j with respect to the mean. Here M is the total number of points in the data set, and i À Md=2 and i Md=2 are the lower and upper edges of the ith segment with M´d points, centred on the ith point of the scan. The scale d runs between 1=M for the smallest segment and 1 for the whole segment. However, this measure is not sensitive to the sign of the temperature¯uctuations because of the square in its de®nition. As a result of this fact the full information of thē uctuations is not conveniently considered. In addition, the generalization of this measure to 2D maps is not unique. Using the measure proposed in this paper, the differences between two-temperature data sets appear when high values of the exponent q are considered. The method is able to differentiate between two very close models with a q range of À2:5´10 5 ; 2:5´10 5 . This range for q is in agreement with the level of inhomogeneity. We are using absolute temperatures, that is, we have inhomogeneities of order 10 À5 with respect to the mean value and the signal is almost¯at. One can consider q as a powerful microscope, able to enhance the smallest differences of two very similar maps. Furthermore, q is a selective parameter. Choosing large values of q in the partition function, favours contributions from cells with relatively high values of m i d since m q i q m q j for m i > m j , if q q 0. Conversely, q p 0 favours the cells with relatively low values of the measure. This is the role played by the moments, changing q one explores the different parts of the measure probability distribution. The other parameter, d, acts like a ®lter. Choosing large values of d is similar to apply a largescale ®lter to the map. One looks at different scales when the parameter d is changed.
To summarize, Zq; d contains information at different scales and moments. The multi-scale information gives an idea of the correlations in the map, meanwhile the moments are sensitive to possible asymmetries in the data, as well as some deviations from Gaussianity. In what follows we show the power of the partition function to extract useful information from CMB data. Three different analyses are used for this purpose.

Likelihood analysis
We shall use the partition function to encode the information of a given map. We compute it both for the experimental data and for simulated ones corresponding to different models. In this process we are comparing the data and the models at several scales and using different moments. If there are some differences at some scale or moment, then the partition function should make it evident. The likelihood function will have a maximum for the best-®tting model to the data. For the CMB maps analyses, we consider models corresponding to different values of the spectral index n and the normalization Q rmsÀPS .
The likelihood is de®ned in the usual way [assuming a Gaussian distribution for ln Zq; d]. We work with Z ln Zq; d instead of Zq; d because of the large values of q, which make imposible to compute directly Zq; d, where, and hZii is the average of the Z for the N ea realizations of the model at bin i. The index i de®nes pairs of values (q,d) and runs from 1 to N q´Nd . That is, i runs from 1 to the total number of points N p where Zq; d is de®ned. Z D i is the value of Z for the experimental data at bin i. M ij is the covariance matrix calculated with Monte Carlo realizations: Z k i denotes the value of Z at bin i for the k realization.
We tried different number of realizations N ea but the results appear to be stable for N ea > 2000 per value of Q rmsÀPS and n.
We have two possibilities to perform a best ®t to the data. The ®rst one is to minimize x 2 and take the values of Q rmsÀPS and n at the minimum of the x 2 surface as the best-®tting values. The second possibility is to work with the likelihood L looking for the maximum. We tested the two possibilities using simulated CMB maps derived from a given pair of parameters (Q rmsÀPS , n) and then using these maps as the data maps. Owing to cosmic variance we obtain a set of maxima in the likelihood and of minima in the x 2 . The conclusion is that the likelihood is somewhat better than the x 2 as expected. For instance with 2000 input realizations with Q rmsÀPS 14 mK and n 1:3 the distribution of maximun likelihood values ®nds a maximum at Q rmsÀPS 13 3 À4:25 mK and n 1:2 0:8 À0:15 while the x 2 renders a minimum in Q rmsÀPS 17 5 À4 mK and n 1:0 0:45 À0:35 . The errors are marginalized at the 68 per cent con®dence level and are similar to those obtained with the standard methods based on the power spectrum (see for instance Wright et al. 1996).

Multifractal analysis
The notion of multifractal measure was ®rst introduced by Mandelbrot (Mandelbrot 1974) in order to study different aspects of the intermittency of turbulence (see also Sreenivasan & Menevau 1988). The multifractal formalism was further developed by many other authors and today it is a standard tool applied in almost all ®elds of science: molecular physics, biology, geology, astronomy, etc . In the context of the description of the large-scale structure of the Universe it was ®rst introduced by Jones et al. (1988).
Some authors (Pietronero, Montuori & Sylos-Labini 1997) suggest that the distribution of matter in the Universe is fractal with dimensionality D 2 . 2. They defend that the scaling remains up to the larger scales probed by the currently available redshift catalogues. Many other authors, however, have found enough evidence of homogeneity at large scales (Davis 1997;Guzzo 1997;Scaramella et al. 1998) in the analysis of the same data sets. One of the basic tenets of the standard cosmology is that at very large scales the distribution of matter is homogeneous. The homogeneity and isotropy of the CMB support this overwhelming evidence, indicating that there exists a continuous transition between scale invariant clustering at small scales and homogeneity at large scales Wu, Lahav & Rees 1998).
At large angular scales, the CMB anisotropies DT=T generated from a scale-free density perturbation power spectrum in a¯at Q 1 universe can be described by a fractional Brownian fractal (as shown in Mollerach et al. 1998). In particular, both in¯ationary and defect models predict an approximately scale invariant Harrison±Zel'dovich spectrum on large angular scales showing the scaling predicted by the Sachs±Wolfe effect. At small angular scales 0:2 ± # v # 1 ± the predictions of in¯ation and topological defects models are different (Durrer et al. 1997) allowing to differentiate them. It is then interesting to study the possible fractality of the CMB anisotropies, since the seeds or¯uctuations that are supposed to be the precursors of the largest structures observed today, are yet unperturbed by evolutionary phenomena. Several works follow this kind of analysis. In the paper by De Gouveia dal Pino et al. (1995), the authors based their analysis in the study of the perimeter-area relation of the isocontours of temperature at a given threshold. They used the COBE±DMR 1 year data set and only the 53-GHz channel. They found evidence for a fractal structure in the COBE±DMR data with dimension D 1:43 suggesting that the CMB could not be homogeneous. Apart from the fact that these data have a low signal-to-noise ratio, this does not necessarily mean that the CMB is not homogeneous. This dimension corresponds to the temperature isocontours, and not to the temperature itself. Other works use multifractal analysis with CMB. Pompilio et al. (1995) apply the multifractal analysis to simulated string-induced CMB scans searching for the non-Gaussian behaviour induced by cosmic strings. More recently, Mollerach et al. (1998) have applied a fractal analysis in order to study the roughness of the last scattering surface and used this technique to search for the model that best ®ts the COBE±DMR 4yr data. These authors show the capabilities of this method for the analysis of future data, in particular for those experiments with high signal-to-noise ratio. In this section we will use the partition function to study the possible multifractality of the CMB sky, using as measure the absolute temperature (see equation 2). The multifractal analysis has been presented in several versions but the most popular is from Frisch & Parisi (1985), Jensen et al. (1985) and Halsey et al. (1996), where the spectrum of singularities f a was introduced. We will give here a brief description of the multifractal approach. A presentation of the method can be found in Feder (1988), Schuster (1989), Vicsek (1989) and more formally in Falconer (1990).
The multifractal formalism has as its starting point the partition function. The generalized or Renyi dimensions are de®ned by the asymptotic behaviour (as the scale d tends to zero) of the ratio between ln Zq; d and ln d, It is easy to see that for q 0 we obtain the box-counting or capacity dimension, For q 1, D1 is the information dimension, which is obtained from equation (7) by applying L'Ho Ãpital's rule. For q 2, D2 is the correlation dimension (see Schuster 1989 for other alternative de®nitions and the relation between them). A simple fractal or monofractal is de®ned by a constant Dq. Dependence of D on q de®nes a multifractal. In most of the practical applications of the multifractal analysis, the limit in equation (7) cannot be calculated, either because we do not have information for small distances (as it happens in this case) or because below a minimum physical length no scaling can exist at all (for example the size of a galaxy in the multifractal nature of the galaxy distribution). This problem is usually overcome by ®nding a scaling range d 1 ; d 2 where a power-law can be ®tted to the behaviour of the partition function The scaling exponents tq are related with the generalized dimensions by tq q À 1Dq: 10 Other quantity, commonly used in the characterization of multifractals, is the so-called f a spectrum. If for a given box (labelled by j) the measure scales as then, the exponent a, which depends in principle on the position is known as crowding index or Ho Èlder exponent. If all the points have the same scaling, then all the exponents a will be the same and this corresponds to a monofractal. Otherwise, if we have boxes with different scaling, what we have is a mixture of monofractals. This set is known as a multifractal (each monofractal formed by the points with the same scaling and therefore with the same exponent a). The exponent a is used to label the boxes covering the set supporting a measure, thereby allowing a separate counting for each value of a. In a multifractal set a can take different values within a certain range, corresponding to the different strength of the measure (Halsey et al. 1996). The subset formed by the boxes with the same a will be denoted S a . This subset has N a d elements (boxes) and in general, for a multifractal set, this number varies with the scale d as N a d , d Àf a : 12 Comparing this expression with the de®nition of the box-counting dimension, equation (8), the quantity f a can be interpreted as the fractal dimension of the subset S a . However, this physical meaning of the function f a is not always true (Grassberger, Badii & Politi 1988;Falconer 1990). It can be shown (Halsey et al. 1996;Martõ Ânez et al. 1990) that the quantities q and tq can be related through a Legendre transformation with a and f a. These relations are: f a qaq À tq: 14 To illustrate this section we use the well-known multiplicative multifractal cascade (Meakin 1987;Martõ Ânez et al. 1990). The construction of this multifractal is as follows. A square is divided into four equal square pieces and a probability p i , i 1; . . . ; 4, such that 4 i1 p i 1, is assigned to each one. Each piece is again subdivided in four small squares, allocating again a value p i randomly permuted to each one. The measure assigned to each one of the new subsquares is the product of this value of p i and the corresponding value of its parent square. The subdivision process is continued recursively. In Fig. 1   multifractal on a grid of 256´256 pixels for the values of the probabilities p 1 0:18, p 2 0:23, p 3 0:28 and p 4 0:31. We can easily calculate the theoretical values of the multifractal functions Dq and f a for this illustrative example (Martõ Ânez et al. 1990). With the multiplicative multifractal we tested the power of the method to recover the true dimensions. In Fig. 2 we show the generalized dimensions Dq and the corresponding spectrum of fractal dimensions f a. These curves match perfectly the theoretically expected ones. Note that a single monofractal should render a straight line for Dq and a single point for f a.

Testing Gaussianity
A Gaussian distribution of CMB temperature¯uctuations is a generic prediction of in¯ation. Forthcoming high-resolution maps of the CMB will allow detailed tests of Gaussianity down to small angular scales, providing a crucial test of in¯ation. Most of the works that analyse CMB maps assume Gaussian initial¯uctuations. Kogut et al. (1996) ®nd that the genus, three-point correlation function, and two-point correlation function of temperature maxima and minima are all in good agreement with the hypothesis that the CMB anisotropy on angular scales larger than 7 ± represents a random-phase Gaussian ®eld. Other alternative methods are proposed, like the angular-Fourier transform (Lewin, Albrecht & Magueijo 1998), Minkowski functionals (Schmalzing & Go Ârsky 1998), correlation of excursion sets (Barreiro et al. 1998), and the bispectrum (Heavens 1998). In an analysis of the 4-year COBE± DMR data based on the bispectrum Ferreira et al. (1998) have found that Gaussianity is ruled out at a con®dence level in excess of 99 per cent near the multipole of order l 16.
In this section we will test the Gaussianity of the CMB data using an alternative method. The idea is to use the relation between the partition function and the generating function, the last one de®ned as, If we know that x is Gaussian distributed then, solving the integral corresponding to the mean value of the previous de®nition, results: G Gauss x t e thxi t 2 j 2 This relation between Zq; d and G m q allows us to construct the function Hq which, for a Gaussian measure m, should be zero for all q at each scale d. This is a simple way to ®nd non-Gaussian signals. The function Hq represents the contribution of all the moments larger than 2. This contribution should be zero only for a Gaussian ®eld. A plot of this function indicates directly the deviations from Gaussianity.

R E S U LT S : A P P L I C AT I O N T O C O B E ± D M R D ATA
As a practical use of the methods presented we will apply them to the 4-year COBE±DMR data.

Description of the data
We use the COBE±DMR 4-year 53 90 GHz maps combination, which is the choice with the largest signal-to-noise ratio (Bennet et al. 1996). These data are in the`Quad-Cube' pixelization with a pixel size of , 28 : 6 and the resulting number of pixels is 6144. The data in each pixel represents DT in units of mK. The dipole has already been subtracted. Assigned to each pixel there is an Partition function based analysis of CMB maps 431 q 1999 RAS, MNRAS 306, 427±436 additional information, the number of times that this part of the sky was explored by the antenna. This information is relevant for the estimation of the instrumental noise. Part of the data is contaminated by Galactic emission. There is a strip between , 6 20 ± (in galactic coordinates) in which the Galactic emission dominates the CMB signal. This strip should not be included in the analysis in order to avoid spurious signals. In addition to this strip there are two patches in the sky (one near Orion and the other one in Ophiucus) that show a strong Galactic emission at mm wavelengths (Cayo Ân & Smoot 1995), and should therefore also be removed from the analysis. When this mask is applied, the number of surviving pixels reduces to 3881 from the original 6144.

Likelihood analysis
In order to determine which are the values of the quadrupole normalization Q rmsÀPS and the spectral index n that best ®t the COBE data, we perform Monte Carlo simulations of the CMB maps for a scale-free model with a power spectrum given by Pk~k n , which has variance in the a lm multipoles given by (Bond & Efstathiou 1987): Gl n À 1=2G9 À n=2 Gl 5 À n=2G3 n=2 : 22 We consider different values for Q rmsÀPS and the n ranging from Q rmsÀPS 4 mK to Q rmsÀPS 35 mK and from n 0:3 to 2.3. We add instrumental noise based on the number of data collected by COBE±DMR at each pixel. Furthermore, there is another effect that must be taken into account, the cosmic variance. To treat conveniently this effect we perform a large number of simulations ($2000 ) for each pair of values (Q rmsÀPS , n) and then we compare the average Z ln Zq; d values of these simulations with the Z corresponding to the COBE±DMR data (the used values for q and d were q À120 000; À40 000; 72 000; 152 000 and d 3; 4; 8; 16 pixel). The choice for q and d values is based on the test described in the last part of Section 2.1. The size of the Zq; d grid, N q´Nd is not critical and what is now relevant is the q values considered. In particular, high-order moments (i.e. large q) are very sensitive to the tail of the distribution and therefore the results obtained with those high values on the parameter estimates are not stable. The combination of q and d values, was one of the combinations for which the recovered parameters Q rms and n were closer to the input parameters and with smaller error bars. As mentioned in Section 2, q should take values of order 10 5 in order to distinguish between models with temperature¯uctuations of order 10 À5 . The values of q where chosen to be asymmetric in an attempt to consider possible asymmetries that could exist between the negative and positive temperature¯uctuations. The range for d runs from 2 pixels (approximately the antenna size) to 24 pixels which is the largest box size required to have at least 8 boxes. Using a maximum likelihood method one can determine which are the best-®tting parameter values of the simulations (signal + noise) to the COBE± DMR data. In Fig. 3 we show a contour plot of the likelihood obtained for the COBE±DMR data. The maximum is at Q rmsÀPS 10 3 À2:5 mK and n 1:8 0:35 À0:65 (95 per cent marginalized errors) and the contour level at 68 per cent is compatible with the assumed standard value Q rmsÀPS 18 6 3 mK for the Einstein±de Sitter model with a scale invariant primordial spectrum of density perturbations, n 1. The various analysis of the 4-year COBE data when combined give as the best-®tting parameters Q rmsÀPS 15:3 3:8 À2:8 mK and n 1:2 6 0:3. The result presented here predicts larger values of n and smaller values of Q rmsÀPS than the result indicated above (although always inside the anticorrelation law for the two parameters). This result is in agreement with the one found by Smoot et al. (1994), using a similar approach. Smoot et al. (1994) found for the best ®t, Q rmsÀPS 13:2 6 2:5 mK and n 1:7 0:3 À0:6 . A possible explanation for the discrepancy between our results and those obtained with the standard methods could be a bias present in the likelihood estimator. In the tests of our algorithm we found a systematic bias in the marginalized likelihood functions both for Q rms and n with typical values of dn , 0:2 and q 1999 RAS, MNRAS 306, 427±436 dQ rms , À2 which could explain part of our discrepancy. The reason for this bias can be the difference between the assumed Gaussian form for the likelihood of the partition function in equation (4) and the real non-Gaussian distribution. The probability distribution of the Z at each q; d obtained from simulations is similar to a Gaussian probability distribution but with a longer tail for high values. We also think that maybe the noise can contribute to that bias. The high-order moments (large q) of the partition function are very sensitive to the tails of the distribution of the temperaturē uctuations. A low signal-to-noise ratio (as is the case for the COBE±DMR data) could raise the parameter n that best ®t the COBE±DMR data. We did some tests in this direction and apparently the noise can increase the value of n (and consequently can produce a lower value of Q rms ).

Multifractal analysis
We apply the formalism of Section 2.2 to the simulations and to the COBE±DMR data. In Fig. 4 we plot Dq and f a for the COBE± DMR and for one model (Q rmsÀPS 15 mK, n 1:2) inside the Q rmsÀPS ±n degeneration with its error bars. The Dq curve has been obtained by ®tting a power law to the partition function in the range of scales 2 # d # 24 pixels, following equations (9) and (10) . Note that the value of D0 is not 2 as it would be expected for a continuous bidimensional surface. The mask slightly lowers this value.
By means of a Legendre transform (equations 13 and 14) we have obtained the corresponding f a curve. A narrow f a curve means a very homogeneous data. If the measure associated to the data is multifractal in nature, these curves should be the same for all the scale ranges. We have found that this is not the case for the COBE± DMR data. The multifractal curves corresponding to different scale ranges do not match each other. CMB simulations without noise show the same behaviour. The reason for that lies in the fact that a scaling like that in equation (9) is not present. This can be illustrated by looking at the behaviour of the local slopes of ln Zq; d versus ln d. In Fig. 5 we show the change in the reduced slopes (tq; d=q À 1) as a function of the scale for a ®xed value of the parameter q for the COBE±DMR data. For this plot the analysis was performed only in the top and bottom faces of the Quad-Cube which are not affected by the mask. For a multifractal measure these curves should be horizontal straight lines. As we can appreciate in the left panel, this is not the case for the COBE±DMR data. The result for a simulation without noise is shown in the right panel. In both cases, we do not see a neat plateau for large absolute values of q. However it is not clear whether the¯uctuations of the local reduced slopes are just due to numerical noise related to the resolution of the maps (i.e. the limited number of pixels) or, on the contrary, these¯uctuations are intrinsic to the measure and, therefore, prove that the measure is not a multifractal. Although our result neither support nor contradict this interpretation, it seems more natural to expect fractal behaviour in the case that one is using  q 1999 RAS, MNRAS 306, 427±436 Figure 5. Reduced slopes computed between two consecutive scales for the COBE±DMR data (left) and for simulated data (without noise, right). Each curve corresponds to a ®xed value of q betwen À10 6 and 10 6 . the absolute value of the relative temperature¯uctuations DT=T as the measure (Mollerach et al. 1998). As shown in that paper, DT=T uctuations generated by the Sachs±Wolfe effect behave like a fractional brownian fractal.

Gaussianity
To test whether the COBE±DMR data are Gaussian distributed, we compare the Hq curve for COBE±DMR with those curves arising from the best-®tting CMB Gaussian models obtained in Section 3.2. In Fig. 6 we show the plots of Hq for different grid scales. For each realization, the measure is rescaled in order to have dispersion equal to one. This allows to have a small and equal range of q values for all scales. We would like to point out the deviation of the mean value from zero when q moves away from zero. This is due to the fact that we have a ®nite number of pixels (i.e. a cosmic variance effect). The predicted behaviour of equation (15) is only true when we compute the mean over in®nite values (or equivalently, solve the integral between À¥ and ¥). Otherwise Hq is not zero at large values (positive and negative) of q. Fig. 7 shows the likelihood distribution of the 1000 Gaussian realizations (with noise) and the dotted line corresponds to the COBE value. It is clear that the COBE±DMR is perfectly compatible with the Gaussian hypothesis.

D I S C U S S I O N A N D C O N C L U S I O N S
We have shown in this work the power of the partition function to describe CMB maps taking into account the information given at different scales and by different moments. We have also shown thē exibility of such a function to be used in various analyses: standard likelihood, multifractal and Gaussian. We applied these analyses to the 4-year COBE±DMR data.
Based on the likelihood function we ®nd the best-®tting parameters Q rmsÀPS 10 3 À2:5 mK and n 1:8 0:35 À0:65 . It is remarkable the agreement between our work and the one by Smoot et al. (1994).
The COBE±DMR data (and the simulations of scale-invariant power spectrum) do not show a fractal behaviour, regarding the absolute temperature map. On the other hand, recent galaxy surveys covering large scales (> 100 Mpc) do not show either a fractal behaviour (Wu et al. 1998). Both results allow to conclude that neither the mass distribution (assuming a linear bias) nor the intensity of the CMB show a fractal behaviour on large scales. The partition function analysis performed shows no evidence for non-Gaussianity in the COBE±DMR data. This is in agreement with all the previous analyses of the COBE±DMR data except the one by Ferreira et al. (1998). Simulations done at higher resolution have shown the power of this method to discriminate between Partition function based analysis of CMB maps 435 q 1999 RAS, MNRAS 306, 427±436 Gaussian and non-Gaussian signals. That analysis will be presented in a future paper. Finally, we would like to remark that the likelihood analysis based on the partition function is computationally intensive. Actually a non-optimized code applied to the COBE±DMR data takes a few days (CPU time) to run in an Alpha server 2100 5/250. Moreover, the computation of the partition function increases with the number of pixels N as ON. This rate should be compared with the most widely applied method used to compress data and to estimate cosmological parameters, the power spectrum of thē uctuations. The direct computation of the power spectrum goes like ON log N (this behaviour is a result of the FFT). Standard brute-force approaches used to estimate the power spectrum go like ON 3 . The reason for this ON 3 rate is the matrix inversion and determinant calculation whose dimension grows as the number of pixels. On the contrary, in the partition function likelihood analysis the number of bins (or matrix dimension) of the likelihood is N q´Nd , being this number usually well below one thousand (even for high-resolution maps). The number of moments q is an arbitrary parameter independent of N and the number of scales d increases as #ON 1=2 . The process of inverting the correlation matrix is clearly reduced in the case of the partition function. This point makes the method useful for forthcoming large data-sets. One can therefore consider the partition function as an alternative way to compress large data sets. Furthermore, for the general situation of non-Gaussian data sets, the partition function is clearly preferable to the power spectrum since the former contains information on several moments of the data.