A&A 457, 385-391 (2006)
A. P. A. Andrade1,2 - A. L. B. Ribeiro1 - C. A. Wuensche2
1 - Laboratório de Astrofísica Teórica e Observacional, Universidade Estadual de Santa Cruz, Brazil
2 - Divisão de Atrofísica, Instituto Nacional de Pesquisas Espaciais, Brazil
Received 10 May 2006 / Accepted 9 June 2006
Context. One possible way to investigate the nature of the primordial power spectrum fluctuations is by investigating the statistical properties of the local maximum in the density fluctuation fields.
Aims. In this work we present a study of the mean correlation function, , and the correlation function for high-amplitude fluctuations (peak-peak correlation) in a slighlty non-Gaussian context.
Methods. From the definition of the correlation excess, we computed the Gaussian two-point correlation function and, using an expansion in generalized Hermite polynomials, we estimated the correlation of high-density peaks in a non-Gaussian field with a generic distribution and power spectrum. We also applied the results to a scale-mixed distribution model, which corresponds to a nearly Gaussian model.
Results. The results reveal that, even for a small deviation from Gaussianity, we can expect high-density peaks to be much more correlated than in a Gaussian field with the same power spectrum. In addition, the calculations reveal how the amplitude of the peaks in the fluctuation field is related to the existing correlations.
Conclusions. Our results may be used as an additional tool for investigating the behavior of the N-point correlation function, to understand how non-Gaussian correlations affect the peak-peak statistics, and extract more information about the statistics of the density field.
Key words: methods: statistical - large-scale structure of Universe
Investigation of the statistical properties of cosmological density fluctuations is a very useful tool for understanding the origin of the cosmic structure. Cosmological models to describe primordial fluctuations can be roughly divided into two classes: Gaussian and non-Gaussian. The most accepted model for structure formation assumes initial quantum fluctuations created during inflation and amplified by gravitational effects. The standard inflationary models predict an uncorrelated random field, with a scale-invariant power spectrum, which follows a nearly-Gaussian distribution (Guth & Pi 1982; Gangui et al. 1994). However, non-Gaussian fluctuations are also allowed in a wide class of alternative models, such as: the multiple interactive fields (Allen et al. 1987; Salopek et al. 1989), the cosmic defects models (Kibble 1976), and the hybrid models (Magueijo & Brandenbergher 2000; Battye & Weller 2000). By discriminating between different classes of models, the statistical properties of the fluctuations field can be used to investigate the nature of cosmic structure. However, non-Gaussian models include an infinite range of possible statistics. As a consequence, performing statistical tests of this kind is not a straightforward task, since there is no adequate general test for every kind of model. To attack this problem, any effort to better understand how the statistical properties of the density fluctuation field affect the observed Universe is welcome, since it may bring extra pieces of information to the investigation of cosmic structure.
Due to the importance of characterizing non-Gaussian signatures, many statistical approaches have been used to study the distribution of fluctuations in the cosmic microwave background radiation (CMBR) (Chiang et al. 2003; Bond & Efstathiou 1987; Eriksen et al. 2005; Cabella et al. 2004; Komatsu et al. 2003) and the large-scale structure (LSS) (Frith et al. 2005; Takada & Jain 2003; Verde et al. 2001; Fry 1985). One possible way to investigate the nature of the primordial power spectrum fluctuations is by investigating the statistical properties of the local maximum in the density fluctuation fields. Since some of the peak properties, such as number, frequency, correlation, height, and extrema, are highly dependent upon the statistics of the fluctuation field, we can gather information about the statistical distribution function by studying the morphological properties of the fluctuations fields (Bardeen et al. 1986; Novikov et al. 1999; Larson & Wandelt 2005). Other useful statistical estimators applied to investigate density fluctuations are the wavelets tools (McEwen et al. 2004; Mukherjee & Wang 2004; Vielva et al. 2004), the phase correlations (Coles et al. 2004; Naselsky et al. 2005), and the most widely used estimator: the N-point correlations in phase (Lewin et al. 1999; Verde & Heavens 2001; Komatsu et al. 2005) or density spaces (Takada & Jain 2005; Coles & Jones 1991; Heavens & Sketh 1999; Heavens & Gupta 2001; Peebles 2001; Eriksen et al. 2004).
The extensive use of the two-point correlation function to characterize the statistical properties of the fluctuations field is justified by the mathematical simplicity in the Gaussian condition, since the mean correlation function can be obtained analytically, and it specifies a Gaussian distribution completely (as well as its power spectrum). However, this assumption is not true for a non-Gaussian case, where higher order correlations may make a significant contribution, despite the great effort demanded to compute a wide range of correlations. However, it is possible to detect primordial non-Gaussianity with a non-zero measure of the N-point correlation . In order to achieve a good description of non-Gaussian signatures in cosmic structure, many works have been done to estimate the three-point correlation function and the related bi or tri-spectrum for a few classes of non-Gaussian models (Verde & Heavens 2001; Komatsu et al. 2003; Eriksen et al. 2005; Komatsu et al. 2005; Gaztanaga & Wagg 2004; Hansen et al. 2004; Creminelli et al. 2005; De Troia et al. 2003). It is believed that, with the advent of the CMBR experiments and the high quality surveys in cosmology, the N-point correlation function will be the main statistical descriptor for the cosmic structure.
In this work we present a technique for extracting non-Gaussian components from a two-point correlation function. In the presence of a non-Gaussian component, and even between two points, it is possible to estimate the influence of higher order correlations for models with different statistical descriptions. By performing the calculation of the high-order correction terms for the peak-peak correlation function, we show how the amplitude of peaks is dependent on the correlations involved. We point out that for a non-Gaussian statistics the higher order correlations impose some restrictions for the amplitude of high-density peaks.
This paper is divided as follows. In Sect. 2 we give a general description of a random variable field and estimate the peak-peak correlation function for a Gaussian field. In Sect. 3 we describe the general treatment to obtain the peak-peak non-Gaussian correlation function. In Sect. 4 we apply the calculations for a slightly non-Gaussian model in two steps: first we consider an approximated solution for a non-Gaussian model with null high-order correlations and finally we estimate the complete solution for a slightly non-Gaussian field. In Sect. 5, we summarize our results and discuss the possibilities for using the peak-peak correlation function as a statistical descriptor of the density fluctuation field.
Most of the models for the early universe (i.e. inflation) actually predict that the fluctuation field is random. This requires that be treated as a random variable in the 3D-space and assumes that the universe is a random realization of a statistical ensemble of possible universes.
We define a random variable,
using the fact that, instead of knowing its
exact value, we only know how to measure various values of
which define a random variable field
under certain experimental conditions. Therefore, a random variable can only be characterized
by a certain statistical ensemble of realizations. When we say that a random variable is known,
it means that we only know the statistical sample that characterizes it. To completely describe
the statistical properties of a random variable,
we define the probability
density function, ,
which can be obtained from the Fourier transform of the
characteristic distribution function,
(Gnedenko & Kolmogorov 1968):
The main numerical indication of the correlation degree between random variables
are the N-order correlation functions. The autocorrelation (or double correlation)
for a random variable
is defined by:
For a statistical process where the correlation functions of order greater than one are null, we set the variable described by this correlation function as not random, or deterministic. In the case where the correlation functions of order greater than two are null, we have a Gaussian variable. For the case of correlation functions of order greater than two and not completely null, the variable is considered to be non-Gaussian. In this sense, we can say that a Gaussian random field is a simplified version of a general random field.
Usually, the cosmological density fluctuation field is statistically described by
the mean correlation function,
applied to a galaxy or a cluster distribution,
with two-point mean separation defined by r (Peebles 1980). For an isotropic and homogenous field, the correlation function is defined as the
excess of probability for a density field described by a Poisson distribution. Therefore, the probability of finding two points in a volume
by a distance r12 is given by:
For a Gaussian random field, the n-dimensional probability density function can
be estimated from the Fourier transform of the characteristic function
(Eqs. (1) and (2)) defined for the moment distribution of s
For a bidimensional Gaussian fluctuation field with zero mean, the correlation matrix can
be obtained and inverted, resulting in:
To find the correlation function between high density peaks, we calculate the probability of
values that are higher than the variance field
by a factor
For a Gaussian field, this probability is given by the integral (Padmanabhan 1999):
The correlation function of high density peaks in a non-Gaussian field can also be computed
using Eq. (13), except that we have to consider the non-Gaussian probability of finding
This probability can be
estimated using Eq. (9), performing the sum over terms of order higher than 2. Since the
summation is also computed for additional terms, the result can be synthesized by:
One intriguing question we could ask is how important the high-order residual terms for
slightly non-Gaussian statistics are. First, we could consider the approximated case
in which the extra calculation in Eq. (9) was avoided, so correlation terms of order (s
2) could be neglected (
and the non-Gaussian
peak-peak correlation function would be reduced to:
At this point we want to assess the robustness of the assumption
stated in the previous paragraph. For this purpose, we compute the higher order correction terms
considering a slightly non-Gaussian component, which means a very small contribution to
correlations of order greater than two. However, the non-Gaussian probability described in
Eq. (9) cannot be calculated using the Fourier transform of the characteristic function, since there is no analytical
solution for that expression. One possible way to obtain
suggested by Gnedenko & Kolmogorov (1968), is to expand it in a series of generalized Hermite polynomials, H:
In order to estimate how the high-order terms affect the correlation between high-density peaks, we estimated the for a Gaussian and a slightly non-Gaussian field, computing the approximated solutions (Eq. (21)) and the full calculation of the expansion (Eq. (22)) until order. For this comparison, we have considered a mixed probability distribution, as proposed by Ribeiro et al. (2001), hereafter (RWL).
The general procedure to create a wide class of non-Gaussian models is to admit the existence of an operator that transforms Gaussianity into non-Gaussianity according to a specific rule. An alternative approach for studing non-Gaussian fields was proposed by RWL, in which the PDF is treated as a mixture: , where is a (dominant) Gaussian PDF and is a second distribution, with . The parameter gives the absolute level of Gaussian deviation, while modulates the shape of the resultanting non-Gaussian distribution. RWL used this mixed scenario to probe the evolution of galaxy cluster abundance in the universe and found that even at a level of non-Gaussianity the mixed field can introduce significant changes in the cluster abundance rate.
The effects of such mixed models in the CMBR power spectrum, combining a Gaussian adiabatic field with a second non-Gaussian isocurvature fluctuation field to produce a positive skewness density field, was discussed by Andrade et al. (2004) (hereafter AWR04). In this approach, they adopted a scale-dependent mixture parameter and a power-law initial spectrum to simulate the CMBR temperature and polarization power spectra for a flat -CDM model, generating a large grid of cosmological-parameter combination. The choice of a scale-dependent mixture is not unjustified, since it could fit both CMB and high-z galaxy clustering in the Universe (e.g. AWR04; Avelino & Liddle 2004; Mathis et al. 2004). At the same time, in a mixed scenario, the scale-dependence acts in order to keep a continuously mixed field inside the Hubble horizon. Simulation results show how the shape and amplitude of the fluctuations in CMBR depend upon such mixed fields and how a standard adiabatic Gaussian field can be distinguished from a mixed non-Gaussian one. They also allow one to quantify the contribution of the second component. By applying a test to recent CMBR observations, the contribution of the isocurvature field was estimated by Andrade et al. (2005) (hereafter AWR05) as 0 = 0.0004 0.00030 with a 68% confidence limit. In the present work, we also investigate the predictions of scale-dependent mixed non-Gaussian cosmological density fields for the peak-peak correlation function.
To obtain the mean correlation function, ,
we have computed the Fourier transform
of the power spectrum related to a pure Gaussian field and a mixed non-Gaussian PDF.
In this sense, we rewrite Eq. (8), which is equivalent to:
In Fig. 1, we show the mean correlation function estimated for (i) a purely Gaussian field, , (ii) for a mixed PDF, , and also (iii) the individual contribution of the non-Gaussian field for . In this plot, it is possible to observe the importance of even a small contribution of the second component to the mean correlation function. For 10-3 the non-Gaussian component dominates the correlation function. This behavior illustrates the excess of power on small scales, as observed on the CMB angular power spectrum in the mixed context (AWR04).
|Figure 1: The mean two-point correlation function estimated for non-correlated or weakly correlated fields: pure Gaussian, pure log-normal and, mixed field with .|
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Inserting the mean correlation function described in Eq. (28) into the approximated expression of the correlation function for high density peaks (Eq. (21)) for a few classes of PDFs, we estimate the functions plotted in Fig. 2. In this plot, we show that the effect of the second component is still observed and that the correlation function for high density peaks is also sensitive for different non-Gaussian distributions. Comparing Figs. 1 and 2, we see how the high density peaks can be much more correlated than the mean field for a non-Gaussian case, especially on small scales, , where is nearly two orders of magnitude greater than the mean correlation.
|Figure 2: The two-point correlation function computed for density peaks in both a pure Gaussian and mixed context ( ). For this estimation, we used: ; ; ; and .|
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With the help of a program that performs algebraic and numerical calculations, we actually computed , as indicated in Eq. (23), for correlations up to the 6-th order (). This limit was set in order to keep a meaningful non-Gaussian distortion, avoiding more time-consuming calculations. We do not present the full computed expression in this section since it contains hundreds of non-linear terms in the mean correlation function and , where the coefficients are the high-order correlations (or cumulants). In fact, the is more accurately described as . In Appendix A, we show the steps in calculating the quasi-moment function, blm, and the Hermite polynomials, Hlm.
|Figure 3: The behavior of the higher order factor, , for a fixed mean correlation ( ). The curves in A) show the factor estimated for third-order correlation, , fourth, , and fifth, . In B) the sixth-order factor, , is also plotted.|
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In general, correlations of a very high order tend to zero (Gnedenko & Kolmogorov 1968), the most extreme case being the normal distribution, where all cumulants of are null. Deviations from Gaussianity are set by the increment of non-vanishing cumulants in the expansion of . A possible question that may be raised is related to the convergence and normalization of the expansion in Eq. (23) in which some terms are set as non-vanishing. However, investigation of general nearly Gaussian deviations have already been performed by the use of the Edgeworth expansion in one-dimension by setting the function to zero in higher-order terms and normalizing it appropriately (Martinez-Gonzalez et al. 2002). Following these authors, we have considered the following behavior as a working hypothesis for such coefficients: we set the terms to zero and the cumulants involved in the quasi moment function to 10-3 for .
One very interesting result is summarized in Fig. 3. In order to follow the absolute behavior of the corrections terms in the nearly Gaussian field, we set the normalization value of the mean correlation function to one, and explore the dependence of with the purpose of understanding how the amplitude of the fluctuations is related to the high-order correlations. While computing only third-order terms, the describes a gradual enhancement in the correlation function. For fourth and sixth-order terms, the and , what is observed, in the case of small Gaussian deviationis, is that the and just overlap each other, meaning that the fifth-order correlations do not contribute significantly. However, the fourth-order correlation describes peaks of maximum density allowed by a weakly correlated field at about . For higher order correlations, the shows a very large increment in the two-point correlation for densities of about and a nearly null contribution to peaks with amplitudes ( ).
We conclude that one should not expect very high density peaks for the specified, slightly non-Gaussian, correlated field. However, factors of order , are quite significant for peaks with amplitude up to and are too far from a null correction, even if we consider weakly correlated fields. Then, we cannot consider the simplified solution in Eq. (21) as a good description of the two-point correlation function for high density peaks despite our choice of considering only a small deviation from Gaussianity. It is important to note that this result is independent of the power spectrum or the mean correlation function. It only shows the influence of higher order correlations in the amplitude of the field of fluctuations.
Analyzing the relation between and , we gain some insight into the amplitude of such high-order terms, since controls the amplitude of the permitted peaks in the fluctuations field. Increasing values of correlations with s > 6 imply a high probability of very high-amplitude (very rare) peaks, which contradict the observations of large-scale structures. However, when we impose correlation levels of the order 10-3 up to sixth order, we favor the existence of peaks up to , which is very reasonable for a nearly Gaussian field.
In Fig. 4 we show the behavior of the two-point correlation function estimated for a Gaussian, ; for a mixed approximated solution, , and for the non-Gaussian complete solution, . In this plot, we set the amplitude threshold of and test the dependence on . The observed effect of is to amplify the correlations between high density peaks. This result is valid for the case of an increasing mean correlation function and does not depend on the mixed model. While non-Gaussian deviation tends to add non-vanish high-order correlations, we conclude that we can expect high-density peaks to be much more correlated even in a slightly non-Gaussian model.
|Figure 4: The behavior of the two-point correlation function is shown for three estimated cases: a pure Gaussian, (lower curve); a mixed approximated solution, with 10-3 (mid curve); and for the complete non-Gaussian solution, , estimated for a PDF of the type: (Gauss + Exp) with 10-3|
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In this work, we have estimated the two-point mean correlation function and the peak-peak correlation function for the density field. In the Gaussian case, the calculations are simplified, since the Fourier transform of the power spectrum describes the random variable completely. However, for a non-Gaussian field, the calculations are much more complicated, since high-order correlations between these two-points may not vanish and strongly contribute to the final function, even in a small deviation from the Gaussian case.
In this work, we showed that, when considering a mixed model, both the mean correlation and the peak-peak correlation functions are much more intense on small scales than in the Gaussian case. This result can be particularly relevant since it is generally accepted that galaxies form in high-density regions. In addition, we conclude that the peak-peak correlation function is quite sensitive to the PDF of the fluctuations field, especially for a mixed model. This result suggests that it is possible to use the peak-peak correlation function as a test of the nature of cosmic structures. Nevertheless, we have to be careful when approximating terms for high-order correlations, since the peak-peak correlation function is very sensitive to the correction terms. We also point out that correlations of order 2 can be a very important tool for characterizing non-Gaussian fields, and they definitely deserve deeper investigation. Estimating high order correlations allows us to investigate the behavior of the N-point correlation function, as well as gather more information about the amplitude of the expected high-density fluctuation field. As seen in Fig. 3, correlations may restrict the amplitude of the density peaks. Furthermore, from the amplitude of the peaks found in the density field (in CMB or LSS fluctuations), we are able to extract more information about the statistics of the density field. It is also good to remember that the presence of correlations of order lead to formation of structures in earlier times than would be expected for a model with the same power spectrum but with weaker spatial correlations. The results presented in this paper may be used to set new constraints in structure formation models.
One possible application of this method in investigations of primordial non-Gaussianity could be implemented in the search for maximum-amplitude fluctuations of the full sky CMB temperature maps, such those derived from WMAP (Spergel et al. 1989) and the future Planck mission (Wright 2000). However, variations in the number of density peaks and their correlation could as well be related to non-Gaussian Galactic foregrounds or other contaminants. Once such a non-Gaussian trace has detected, we have to be very careful before assigning it a primordial origin. To avoid misunderstanding, the investigation of the peak statistics in CMB datasets should be realized over several datasets and different frequencies. Indeed, one possible manner to minimize the foregrounds effects is to analyze the most sensitive and cleaned map, such as the WMAP three-year di-biased internal linear combination, WMAP-DILC map, or the WMAP coadded map, with the combination of Q+V+W frequencies. This analysis will be the next test applied to the non-Gaussian mixed model.
APAA thanks FAPESB and CNPq for the financial support under grant 1431030005400. ALBR thanks CNPq for the financial support under grants 470185/2003-1 and 306843/2004-8. CAW was partially supported by the CNPq grant 307433/2004-8.
To obtain the non-Gaussian probability for a multi-dimensional case, we have to
perform the calculation in Eq. (22) by expanding the Hermite polynomials and
the quasi-moment function. One possible way to obtain the quasi-moment functions,
bs, is by relating to the correlation function, since:
Performing the calculations for the quasi-moment function, we have
Note that the first five terms of bs are equivalent to the correlation function,
only the sixth term has additional terms. As an example, for s = 3, we obtain four different
forms of b3:
Proceeding in a similar manner, we find five terms for b4, six for b5, and seven terms for b6.
The best way to find the Hermite polynomials is using expression (Gnedenko & Kolmogorov 1968)
where we used the notation to indicate a simetrization set.
To perform the calculation above for blm and Hlm up to (l+m = 6)and perform the integration in and , we used a software for algebraic and numerical calculations.