A reassessment of the evidence of the Integrated SachsWolfe effect through the WMAPNVSS correlation
Abstract
We reassess the estimate of the crosscorrelation of the spatial distribution of the NRAO VLA Sky Survey (NVSS) radio sources with that of Cosmic Microwave Background (CMB) anisotropies from the Wilkinson Microwave Anisotropy Probe (WMAP). This reanalysis is motivated by the fact that most previous studies adopted a redshift distribution of NVSS sources inconsistent with recent data. We find that the constraints on the biasweighted redshift distribution, , of NVSS sources, set by the observed angular correlation function, , or, equivalently, by the power spectrum of their spatial distribution, strongly mitigate the effect of the choice of . If such constraints are met, even highly discrepant redshift distributions yield NVSS–WMAP crosscorrelation functions consistent with each other within statistical errors. The models favoured by recent data imply a bias factor, , decreasing with increasing , rather than constant, as assumed by most previous analyses. As a consequence, the function has more weight at , i.e. in the redshift range yielding the maximum contribution to the ISW in a standard CDM cosmology. On the whole, the NVSS turns out to be better suited for ISW studies than generally believed, even in the absence of an observational determination of the redshift distribution. The systematics introducing spurious power in the angular correlation function on scales of several degrees are strongly reduced restricting the analysis to the subsample brighter than 10 mJy. Even though this subsample comprises less than one third of the NVSS sources, it yields a slightly more significant detection of the ISW effect than the full sample ( rather than ). The NVSS–WMAP crosscorrelation function is found to be fully consistent with the prediction of the standard CDM cosmology.
keywords:
cosmic microwave background — cosmological parameters — cosmology: observations — radio continuum: galaxies.1 Introduction
As first pointed out by Crittenden & Turok (1996) a promising way of probing the (linear) Integrated Sachs & Wolfe (1967) (ISW) effect is through correlations of Cosmic Microwave Background (CMB) maps with tracers of large scale structure. Since the ISW effect shows up on large angular scales, a significant application of this test had to await until the first allsky high quality CMB temperature maps have been provided by WMAP (Bennett et al. 2003; Hinshaw et al. 2007). The WMAP data have been cross correlated with a variety of radio, IR, optical, and Xray surveys (see Aghanim et al. 2007 for a review) to look for evidences of a decay of the gravitational potential due to the influence of dark energy.
The first (marginally) significant correlations (at the level) of the WMAP first year data were found by Boughn & Crittenden (2004, 2005) with the HEAO1 A2 full sky hard Xray map (Boldt 1987) and with a map of the number density distribution of 1.4 GHz radio sources detected by the NRAO VLA Sky Survey (NVSS), covering about 80% of the sky. The cross correlation between NVSS and WMAP 1st year data was confirmed by Nolta et al. (2004), and by Vielva et al. (2006) using different techniques, in real, harmonic and wavelet spaces. Further analysis was triggered by the release of the WMAP 3 year data. Some groups exploited again the NVSS data (Pietrobon et al. 2006; McEwen et al. 2007, 2008).
Independently of the analysis technique used, the comparison of the correlations inferred from the data with model predictions require two basic ingredients: the redshift distribution and the bias parameter of sources. Since redshift measurements are available only for a tiny fraction of NVSS sources, it is necessary to resort to model estimates. All the investigations quoted above basically adopted the redshift distribution given by model RLF1 of Dunlop & Peacock (1990) that was shown by Boughn & Crittenden (2002) to reproduce the autocorrelation function of NVSS sources assuming a redshiftindependent bias parameter (see also Cress & Kamionkowsky 1998, Magliocchetti et al. 1999). However, as pointed out by Blake et al. (2004) and Negrello et al. (2006), this model overestimates the density of lowredshift radio sources (Magliocchetti et al. 2000), which was found to be consistent with Dunlop & Peacock’s (1990) pure luminosity evolution model (Magliocchetti et al. 2002). But if the adopted redshift distribution, and the associated bias model, are incorrect, the conclusions of the analysis may be flawed. In fact, the contribution to ISW signal in a CDM cosmology peaks at and rapidly declines at higher and lower redshifts (Afshordi 2004).
This situation has motivated our revisitation of the problem. In § 2 we discuss the interpretation of the autocorrelation function of NVSS sources in the light of the results by Negrello et al. (2006). In § 3 we provide a new estimate of the WMAP–NVSS crosscorrelation as a function of the angular separation and of the associated errors using both the Internal Linear Combination (ILC) temperature map built using the 3 year WMAP data (Hinshaw et al. 2007) and the foreground cleaned CMB map obtained by Bonaldi et al. (2007) using the same data and allowing for the presence of an “anomalous” dustcorrelated Galactic emission, in addition to the canonical synchrotron, freefree and thermal dust emissions. In § 4 the observed crosscorrelation function (CCF) is compared with models prediction. The main conclusions are presented and discussed in § 5.
2 The twopoint angular correlation function of NVSS sources
The NVSS (Condon et al. 1998) is the largestarea radio survey at GHz. It covers sr of the sky north of ; the source catalogue contains 1.810 sources. The twopoint angular correlation function, , of NVSS sources has been measured by Blake Wall (2002a, 2002b) and Overzier et al. (2003) for different fluxdensity thresholds between 3 mJy and 500 mJy. The overall shape of is well reproduced by a double powerlaw. On scales below , the steeper powerlaw reflects the distribution of the resolved components of single giant radio sources. On larger scales the shallower powerlaw describes the correlation between distinct radio sources. The latter provides insights on the way in which they trace the underlying dark matter distribution, and on the cosmological framework which determines the distribution of dark matter at each epoch. In the present context we are interested only on the largescale behaviour.
Earlier investigations of the ISW effect used NVSS sources down to the formal detection limit of mJy, although the completeness is only at the faintest fluxes, while it rapidly increases with increasing flux, reaching 99% at 3.4 mJy (Condon et al. 1998). Also, systematic surface density gradients, yielding spurious contributions to (or to the power spectrum), are approximately negligible only for mJy (Blake & Wall 2002a). Following Negrello et al. (2006), we will therefore include in our analysis only sources brighter than 10 mJy. The NVSS source surface density at this threshold is .
The angular correlation function of a population of extragalactic sources is related to their spatial correlation function, , and to their redshift distribution, , by Limber’s (1953) equation:
(1) 
Here, is the comoving spatial distance between two objects located at redshifts and and separated by an angle on the sky.
Equation (1) shows that the contributions to of a population of extragalactic sources are weighted by the square of their redshift distribution. The yielded by Dunlop & Peacock (1990) model RLF1, adopted in previous ISW studies, has a spike at low , which, as pointed out by Negrello et al. (2006), is inconsistent with the recent accurate determinations of local luminosity functions of NVSS sources (Magliocchetti et al. 2002; Sadler et al. 2002; Mauch & Sadler 2007). Also, Ho et al. (2008) found that such redshift distribution is inconsistent with the results of crosscorrelations of NVSS with 2MASS and SDSS samples. As stressed by Magliocchetti et al. (1999), because of the weighting of the projected spatial correlation function, the spike makes a significant contribution to , even though the visibility volume of the corresponding sources is small. Actually, such sources, that have been interpreted as the spiral and starburst galaxy populations emerging at mJy levels, would provide the dominant contribution to the clustering signal on angular scales larger than (Cress & Kamionkowski 1998; Magliocchetti et al. 1999; Blake et al. 2004). But the contribution of spiral and starburst galaxies to counts at mJy, computed using the recent determinations of their luminosity function, is very small (; Negrello et al. 2006). Yet, a statistically significant positive was measured on scales , and the interpretation of this data proved to be challenging.
On scales large enough for the clustering signal be due to galaxies residing in distinct dark matter haloes and under the assumption of a onetoone correspondence between sources and their host haloes, the spatial twopoint correlation function can be written as the product of the correlation function of dark matter, , times the square of the bias parameter, (Matarrese et al. 1997; Moscardini et al. 1998):
(2) 
where represents the effective mass of the dark matter haloes in which the sources reside. The observationally determined spatial correlation length of starforming galaxies, – (Saunders et al. 1992; Wilman et al. 2003) is consistent with an effective halo mass not exceeding . Together with the small contribution of these sources to the counts for mJy, this implies that they are negligible contributors to the observed (see Negrello et al. 2006 for more details).
The clustering properties of low AGNfuelled radio sources are much stronger and imply that they reside in dark matter haloes more massive than (Magliocchetti et al. 2004). These could then easily account for the observed on scales –. However, a generic prediction of the Cold Dark Matter paradigm of structure formation, is that the spatial correlation function of matter displays a sharp cutoff around a comoving radius of a few hundred Mpc, and becomes negative on larger scales. For typical redshifts , this translates in a cut off of on scales . On larger scales, the negative contribution of sources overwhelms the positive contributions from low sources if is redshiftindependent, as indicated by data on optically selected quasars (Porciani et al. 2004; Croom et al. 2005). To account for the data in the framework of the standard hierarchical clustering scenario it is necessary that the effective bias factor decreases with increasing , rather than strongly increase like for optical quasars. This adds weight to the contribution of low NVSS sources to the matter density fluctuations, enhancing the expected CMB–NVSS crosscorrelation.
Negrello et al. (2006) found that the observed can be reproduced if is proportional to the characteristic mass of virialized systems, , with . This implies , close to the value found for optical quasars. A possible interpretation of this result is that AGNpowered radio sources are so rare locally that the typical halo hosting one of them must include a very large number of galaxies, i.e. must correspond to a rich cluster of galaxies. The strong evolution of radio sources implies that they become relatively more abundant with increasing redshift, and the typical halo mass decreases correspondingly. This happens in such a way that the clustering of radio sources reflects that of the largest haloes that collapse at any cosmic epoch.
We have repeated the analysis using the 2 versions (MEAN and HIGH) of the luminosity/density evolution (LDE) model by Dunlop & Peacock (1990), which are also consistent with the recent data on the local luminosity function. As illustrated by Fig. 1, the requirement that the observed is reproduced (lower lefthand panel) forces (lower righthand panel) the biasweighted redshift distributions, , corresponding to the 3 models to be very close to each other in the redshift interval from which most of the contribution to the ISW signal comes, thus lessening the effect of uncertainties in the redshift distribution and on the bias factor (upper panels). Also the redshift dependence of the bias factor adds weight to the interesting redshift range, implying that the NVSS sample is very well suited for ISW studies.
3 Observational estimate of the WMAP–NVSS crosscorrelation function
The empirical crosscorrelation function was computed as:
(3) 
where is the number of NVSS sources in the ith pixel of the map of number count fluctuations, is the temperature fluctuation in the jth pixel of the CMB map, the two pixels being at an angular distance . The weights and are equal to unity for valid pixels or equal to zero if the pixels fall within a region not covered by the survey or masked; the apex and refer to the CMB and to the NVSS maps, respectively. The maps are pixelized with the Healpix (Górski et al. 2005) resolution parameter NSIDE=32, corresponding to a pixel size of about .
We used two CMB maps: the WMAP threeyear Internal Linear Combination (ILC) map (Hinshaw et al. 2007) and the foregroundcleaned map yielded by model M2 of Bonaldi et al. (2007). As for the NVSS map, our choice of a 10 mJy flux limit has dispensed us with the need of introducing corrections for the spurious density gradients in the survey noted by Boughn & Crittenden (2002) for the sample limited to 2.5 mJy. We have removed the strip at where extragalactic sources are blurred by bright Galactic sources and by small scale structure of the synchrotron and freefree emission. Including the unobserved region at , the NVSS mask covers 25% of the sky.
The CMB mask adopted is based on the Kp0 mask (Bennett et al. 2003), which excludes the regions most contaminated by Galactic foregrounds and the brightest point sources in the WMAP bands. The Kp0 mask has NSIDE=512, corresponding to a pixel size of about . We degraded this map to the lower resolution adopted for this work and we set for all pixels including at least one masked pixel. The resulting mask excludes 37% of the sky. The union of NVSS and CMB masks covers 48% of the sky.
For both CMB maps, we computed the empirical CCF [eq. (3)] for a set of angular separations, in degrees: . As discussed by Blake & Wall (2002a) the tiny angular correlations among NVSS sources on scales [] may well be due to small systematic variations in the source surface density due to calibration problems. This may also add some spurious power to the CCF; the effect is however small and the empirical CCF is indeed found to be consistent with zero on these scales. For each we considered pixel pairs whose centers have separations in the range with . To estimate the errors we simulated 1000 mock NVSS maps by randomly redistributing the unmasked pixels of the true NVSS map. For each mock NVSS map we computed the CCF with the CMB map. The error on is then estimated as the rms value of the CCFs for the mock maps. Alternatively, the significance of the signal can be estimated from the fraction of mock CCFs at or above its amplitude. We find that the significance estimated in this way is in excellent agreement with that expected from the S/N ratio for a Gaussian distribution. The results are shown in Fig. 2. The significance of the measured signal is similar for the two CMB maps used. The CCF differs from zero at the 2.1– level for each of the first 2 points, and the overall significance of the ISW signal is 99.7% or slightly higher, corresponding to or somewhat more. Boughn & Crittenden (2003) reported a somewhat less significant () detection of the NVSSCMB crosscorrelation, suggesting that the systematics affecting the distribution of the faintest NVSS sources overcome the statistical advantage of their far larger number.
4 The theoretical crosscorrelation function
Following Nolta et al. (2004) we write the crosscorrelation power spectrum between the surface density fluctuations of NVSS sources and CMB fluctuations as:
(4) 
where and are the NVSS and CMB filter functions, respectively, and is the logarithmic matter power spectrum today as a function of the wave number :
(5) 
For we have used the expression given by eq. (A3) of Eisenstein & Hu (1998), valid for a flat universe with a cosmological constant. The reference values of cosmological parameters are those given in Table 2 of Spergel et al. (2007), namely , , , , and . The constant was computed normalizing the power spectrum to ; we find . The matter transfer function, , was computed using CMBfast (Seljak & Zaldarriaga 1996).
The NVSS filter function writes:
(6) 
where is the mean number of sources per steradian with redshift within , brighter than the flux limit, is the bias factor relating the source overdensity to the mass overdensity, assumed to be scaleindependent, is the linear growth factor of mass fluctuations, is the spherical Bessel function, and is the conformal lookback time:
(7) 
with , being the curvature parameter, set to zero in this paper. For the linear growth factor, appropriate for the large scales of interest here, we adopt the approximation (Carroll et al. 1992):
(8)  
with and .
As mentioned in § 1, we restrict ourselves to NVSS sources brighter than mJy and adopt the redshift distributions given by three Dunlop & Peacock (1990) evolution models for AGN powered sources. As for starforming galaxies, we estimate the redshift distribution using the local luminosity function of Magliocchetti et al. (2002) and no evolution: evolutionary effects, if any, are irrelevant for our sample since starforming galaxies brighter of 10 mJy are found to be at . The filter function for the ISW effect is:
(9)  
Since eq. (9) holds for linear perturbations, i.e. for , the integration over of eq. (4) requires some caution. We have checked for a set of multipoles that the crosscorrelation function obtained using the power spectrum of CMB temperature perturbation given by CMBfast are well approximated by setting in eq. (4) .
The crosscorrelation function as a function of the angular separation is then obtained as:
(10) 
where are the Legendre polynomials. For the range of scales of interest here, it is enough to sum over the interval .
Figure 2 shows that, once the constraints from the observed are taken into account, the predicted CCF for the WMAP 3year values of cosmological parameters is only weakly affected by the uncertainties on the redshift distributions of NVSS sources. We find a good agreement between the predicted and empirical CCFs, somewhat at odds with the finding by Ho et al. (2008) that the WMAP 3year model predicts an ISW amplitude about below their estimate, which exploits various galaxy surveys in addition to the NVSS. The strong constraints from the angular correlation function of the latter sources are not taken into account, however.
5 Discussion and conclusions
The NVSS sample is of primary importance for the investigation of the ISW effect because of its very large area (82% of the sky), its large number of sources (almost , its depth in redshift, and its immunity to systematics related to the uncertain corrections for dust extinction. It has however drawbacks, not all of which were properly dealt with by previous analyses. On one side, spurious contributions to the power spectrum of the NVSS source distribution is introduced by known systematic effects such as small systematic variations in the source surface density due to calibration problems at low flux densities and to striping effects, particularly at negative declinations (Blake & Wall 2002a; Boughn & Crittenden 2002). To minimize this problem, we have restricted our analysis to sources with mJy (Blake & Wall 2002a,b); at this flux limit, striping effects are negligible.
The main potential problem, however, stems from the very limited direct redshift information, forcing us to resort to models. Most previous analyses (Boughn & Crittenden 2002, 2004; Nolta et al. 2004; Pietrobon et al. 2006; Vielva et al. 2006; McEwen et al. 2007) relied on model RLF1 of Dunlop & Peacock (1990), which was found by Boughn & Crittenden (2002) to reproduce the NVSS autocorrelation function. However, as pointed out by Negrello et al. (2006), this model is inconsistent with the recent determinations of the luminosity function of low NVSS sources (Magliocchetti et al. 2002; Sadler et al. 2002; Mauch & Sadler 2007). A totally independent confirmation of this conclusion has been obtained by Ho et al. (2008) by crosscorrelating the NVSS with 2MASS and SDSS samples, whose redshift distributions are known. In this way these authors obtained constraints on the biasweighted redshift distribution, , which is the relevant quantity for predicting the NVSSCMB crosscorrelation induced by the ISW effect [eq. (6)].
The large spread of redshiftdependent radio luminosity functions yielded by the original set of Dunlop & Peacock (1990) models is substantially narrowed down by the more recent data on the radio luminosity function at low redshifts, that are fully consistent with the pure luminosity evolution and by the two luminosity/density evolution models. Strong constraints on follow from the observational determination of the angular correlation function, , of NVSS sources, which depends on . In fact, the functions for three models considered here are found to be substantially closer to each other, particularly at lowredshifts, than the corresponding redshift distributions. The decrease of the bias factor with increasing implied by observational determinations of adds weight to the redshift range yielding most of the contribution to the ISW signal, making the NVSS sample even better suited to test the effects of dark energy on the growth of structure than one would guess from the estimated redshift distribution.
A concern expressed in some previous papers is that the microwave emission of radio sources themselves contribute to the observed WMAP–NVSS correlations. However, the radio source contribution to temperature fluctuations is dominated by the brightest sources, while the contribution to the source number density fluctuations, relevant for studies of the ISW effect is dominated by the faintest sources. Number density and surface brightness fluctuations are thus largely uncorrelated, implying that the source microwave emission rather weakens the correlations. We find that, if no sources are masked, the CCF estimates decrease by 6–10%. On the other hand, removing just the WMAP sources (Hinshaw et al. 2007; LopezCaniego et al. 2007) is enough to make the contamination of the ISW signal negligibly small, consistent with previous results (e.g. McEwen et al. 2007; Ho et al. 2008).
In summary, the main results of this paper are:

the constraints on the biasweighted redshift distribution, , of NVSS sources, set by the observed angular correlation function or, equivalently, by the power spectrum of their spatial distribution, strongly mitigate the effect of the large uncertainties on . Even highly discrepant redshift distributions, including instances ruled out by recent data, yield NVSS–WMAP crosscorrelation functions consistent with each other within statistical errors, although the use of the Dunlop & Peacock (1990) RLF1 model leads to a slight underestimate of the predicted ISW signal for the WMAP 3 year values of the cosmological parameters. This fact largely alleviates what was perceived as the main weakness of the exploitation of the NVSS for studies of the ISW.

The models favoured by recent data imply bias functions, , decreasing with increasing , rather than constant, as assumed by most previous analyses. As a consequence, the function has more weight at , i.e. in the redshift range yielding the maximum contribution to the ISW in a standard CDM cosmology. Again, this means that the NVSS is better suited for ISW studies than generally believed.

The systematics (striping, incompleteness, etc.) introducing spurious power in the angular correlation function on scales of several degrees are strongly reduced restricting the analysis to the subsample with a flux limit of 10 mJy. Even though this subsample comprises less than one third of the NVSS sources, it yields a slightly more significant detection of the ISW effect ( rather than ). The microwave emission of NVSS sources weakens the correlations with WMAP fluctuations, rather than increasing them as previously argued. The effect is, however, negligibly small.

The NVSS–WMAP crosscorrelation function is found to be fully consistent with the prediction of the standard CDM cosmology.
Acknowledgements
We acknowledge useful suggestions from A. Balbi and M. Liguori. Work supported in part by ASI (contracts Planck LFI Phase E2 Activity and COFIS) and MUR.
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