Spherical Mexican Hat Wavelet Research Articles

The C-Band All-Sky Survey: total intensity point-source detection over the northern sky

ABSTRACT We present a point-source detection algorithm that employs the second-order Spherical Mexican Hat wavelet filter (SMHW2), and use it on C-Band All-Sky Survey (C-BASS) northern intensity data to produce a catalogue of point sources. This catalogue allows us to cross-check the C-BASS flux-density scale against existing source surveys, and provides the basis for a source mask that will be used in subsequent C-BASS and cosmic microwave background (CMB) analyses. The SMHW2 allows us to filter the entire sky at once, avoiding complications from edge effects arising when filtering small sky patches. The algorithm is validated against a set of Monte Carlo simulations, consisting of diffuse emission, instrumental noise, and various point-source populations. The simulated source populations are successfully recovered. The SMHW2 detection algorithm is used to produce a $4.76\, \mathrm{GHz}$ northern sky source catalogue in total intensity, containing 1784 sources and covering declinations δ ≥ −10°. The C-BASS catalogue is matched with the Green Bank 6 cm (GB6) and Parkes-MIT-NRAO (PMN) catalogues over their areas of common sky coverage. From this we estimate the 90 per cent completeness level to be approximately $610\, \mathrm{mJy}$, with a corresponding reliability of 98 per cent, when masking the brightest 30 per cent of the diffuse emission in the C-BASS northern sky map. We find the C-BASS and GB6 flux-density scales to be consistent with one another to within approximately 4 per cent.

The cosmic microwave background Cold Spot anomaly: the impact of sky masking and the expected contribution from the integrated Sachs–Wolfe effect

Abstract We re-analyse the cosmic microwave background (CMB) Cold Spot (CS) anomaly with particular focus on understanding the bias a mask (contaminated by Galactic and point sources) may introduce. We measure the coldest spot, found by applying the Spherical Mexican Hat Wavelet transform on 100 000 cut-sky (masked) and full-sky CMB simulated maps. The CS itself is barely affected by the mask; we estimate a 94 per cent probability that the CS is the full-sky temperature minimum. However, ∼48 per cent (masked fraction of the mask) of full-sky minima are obscured by the mask. Since the observed minima are slightly hotter than the full-sky ensemble of minima, a cut-sky analysis would have found the CS to be significant at ∼2.2σ with a wavelet angular scale of R = 5°. None the less, comparisons to full-sky minima show the CS significance to be only ∼1.9σ and <2σ for all R. The CS on the last scattering surface may be hotter due to the integrated Sachs–Wolfe effect in the line of sight. However, our simulations show that this is on average only ∼10 per cent (about 10 μK but consistent with zero) of the CS temperature profile. This is consistent with Lambda and cold dark matter reconstructions of this effect based on observed line-of-sight voids.

The effect of the linear term on the wavelet estimator of primordial non-Gaussianity

In this work we present constraints on different shapes of primordial non-Gaussianity using the Wilkinson Microwave Anisotropy Probe (WMAP) 7-year data and the spherical Mexican hat wavelet fnl estimator including the linear term correction. In particular we focus on the local, equilateral and orthogonal shapes. We first analyse the main statistical properties of the wavelet estimator and show the conditions to reach optimality. We include the linear term correction in our estimators and compare the estimates with the values already published using only the cubic term. The estimators are tested with realistic WMAP simulations with anisotropic noise and the WMAP KQ75 sky cut. The inclusion of the linear term correction shows a negligible improvement (< 1 per cent) in the error-bar for any of the shapes considered. The results of this analysis show that, in the particular case of the wavelet estimator, the optimality for WMAP anisotropy levels is basically achieved with the mean subtraction and in practical terms there is no need of including a linear term once the mean has been subtracted. Our best estimates are now: local fnl = 39.0 +/ 21.4, equilateral fnl = -62.8 +/- 154.0 and orthogonal fnl = -159.8 +/- 115.1 (all cases 68 per cent CL). We have also computed the expected linear term correction for simulated Planck maps with anisotropic noise at 143 GHz following the Planck Sky Model and including a mask. The improvement achieved in this case for the local fnl error-bar is also negligible (0.4 per cent).

Constraints on general primordial non-Gaussianity using wavelets for the Wilkinson Microwave Anisotropy Probe 7-year data

We present constraints on the non-linear coupling parameter fnl with the Wilkinson Microwave Anisotropy Probe (WMAP) data. We use the method based on the spherical Mexican hat wavelet (SMHW) to measure the fnl parameter for three of the most interesting shapes of primordial non-Gaussianity: local, equilateral and orthogonal. Our results indicate that this parameter is compatible with a Gaussian distribution within the two sigma confidence level (CL) for the three shapes and the results are consistent with the values presented by the WMAP team. We have included in our analysis the impact on fnl due to contamination by unresolved point sources. The point sources add a positive contribution of Delta(fnl) = 2.5 \pm 3.0, Delta(fnl) = 37 \pm 18 and Delta(fnl) = 25 \pm 14 for the local, equilateral and orthogonal cases respectively. As mentioned by the WMAP team, the contribution of the point sources to the orthogonal and equilateral form is expected to be larger than the local one and thus it cannot be neglected in future constraints on these parameters. Taking into account this contamination, our best estimates for fnl are -16.0 \leq fnl \leq 76.0, -382 \leq fnl \leq 202 and -394 \leq fnl \leq 34 at 95% CL for the local, equilateral and orthogonal cases respectively. The three shapes are compatible with zero at 95% CL (2{\sigma}). Our conclusion is that the WMAP 7-year data are consistent with Gaussian primordial fluctuations within ~2{\sigma} CL. We stress however the importance of taking into account the unresolved point sources in the measurement of fnl in future works, especially when using more precise data sets such as the forthcoming Planck data.

Constraints on fNL from Wilkinson Microwave Anisotropy Probe 7-year data using a neural network classifier

We present a multi-class neural network (NN) classifier as a method to measure nonGaussianity, characterised by the local non-linear coupling parameter fNL, in maps of the cosmic microwave background (CMB) radiation. The classifier is trained on simulated non-Gaussian CMB maps with a range of known fNL values by providing it with wavelet coefficients of the maps; we consider both the HealPix (HW) wavelet and the spherical Mexican hat wavelet (SMHW). When applied to simulated test maps, the NN classfier produces results in very good agreement with those obtained using standard chi2 minimization. The standard deviations of the fNL estimates for WMAPlike simulations were {\sigma} = 22 and {\sigma} = 33 for the SMHW and the HW, respectively, which are extremely close to those obtained using classical statistical methods in Curto et al. and Casaponsa et al. Moreover, the NN classifier does not require the inversion of a large covariance matrix, thus avoiding any need to regularise the matrix when it is not directly invertible, and is considerably faster.

INTRODUCING MEXICAN NEEDLETS FOR CMB ANALYSIS: ISSUES FOR PRACTICAL APPLICATIONS AND COMPARISON WITH STANDARD NEEDLETS

Over the last few years, needlets have a emerged as a useful tool for the analysis of Cosmic Microwave Background (CMB) data. Our aim in this paper is first to introduce in the CMB literature a different form of needlets, known as Mexican needlets, first discussed in the mathematical literature by Geller and Mayeli (2009a,b). We then proceed with an extensive study of the properties of both standard and Mexican needlets; these properties depend on some parameters which can be tuned in order to optimize the performance for a given application. Our second aim in this paper is then to give practical advice on how to adjust these parameters in order to achieve the best properties for a given problem in CMB data analysis. In particular we investigate localization properties in real and harmonic spaces and propose a recipe on how to quantify the influence of galactic and point source masks on the needlet coefficients. We also show that for certain parameter values, the Mexican needlets provide a close approximation to the Spherical Mexican Hat Wavelets (whence their name), with some advantages concerning their numerical implementation and the derivation of their statistical properties.

On the optimality of the spherical Mexican hat wavelet estimator for the primordial non-Gaussianity

We study the spherical Mexican hat wavelet (SMHW) as a detector of primordial non-Gaussianity of the local type on the Cosmic Microwave Background (CMB) anisotropies. For this purpose we define third order statistics based on the wavelet coefficient maps and the original map. We find the dependence of these statistics in terms of the non-linear coupling parameter fnl and the bispectrum of this type of non-Gaussianity. We compare the analytical values for these statistics with the results obtained with non-Gaussian simulations for an ideal full-sky CMB experiment without noise. We study the power of this method to detect fnl, i. e. the variance of this parameter, and compare it with the variance obtained from the primary bispectrum for the same experiment. Finally we apply our wavelet based estimator on WMAP-like maps with incomplete sky and inhomogeneous noise and compare with the optimal bispectrum estimator. The results show that the wavelet cubic statistics are as efficient as the bispectrum as optimal detectors of this type of primordial non-Gaussianity.

A Comprehensive Overview of the Cold Spot

The report of a significant deviation of the CMB temperature anisotropies distribution from Gaussianity (soon after the public release of the WMAP data in 2003) has become one of the most solid WMAP anomalies. This detection grounds on an excess of the kurtosis of the Spherical Mexican HatWavelet coefficients at scales of around 10 degrees. At these scales, a prominent feature—located in the southern Galactic hemisphere—was highlighted from the rest of the SMHW coefficients: the Cold Spot. This paper presents a comprehensive overview related to the study of the Cold Spot, paying attention to the non‐Gaussianity detection methods, the morphological characteristics of the Cold Spot, and the possible sources studied in the literature to explain its nature. Special emphasis is made on the Cold Spot compatibility with a cosmic texture, commenting on future tests that would help to give support or discard this hypothesis.

Disks in the sky: A reassessment of the WMAP “cold spot”

We reassess the evidence that WMAP temperature maps contain a statistically significant “cold spot” by repeating the analysis using simple circular top-hat (disk) weights, as well as Gaussian weights of varying width. Contrary to previous results that used Spherical Mexican Hat Wavelets, we find no significant signal at any scale when we compare the coldest spot from our sky to ones from simulated Gaussian random, isotropic maps. We trace this apparent discrepancy to the fact that WMAP cold spot’s temperature profile just happens to favor the particular profile given by the wavelet. Since randomly generated maps typically do not exhibit this coincidence, we conclude that the original cold spot significance originated at least partly due to a fortuitous choice of using a particular basis of weight functions. We also examine significance of a more general measure that returns the most significant result among several choices of the weighting function, angular scale of the spot, and the statistics applied, and again find a null result.

IMPACT OF THE 1/fNOISE AND THE ASYMMETRIC BEAM ON NON-GAUSSIANITY SEARCHES WITHPLANCK

We study the impact of correlated instrumental noise and non-circular antenna beam patterns on primordial non-Gaussianity analysis. The two systematic effects are reproduced in the case of the Planck mission, using Planck-like realistic simulations. The non-Gaussian analysis is conducted with different approaches. First we adopt a blind approach, using the Spherical Mexican Hat Wavelet and the Minkowski functionals, and then a fNL estimator. We look respectively for false primordial non-Gaussian detections and for bias or variance increase in the estimated fNL value. Even if some slight effects are present, we can not observe any significant impact of the 1/f noise and the asymmetric beam on non-Gaussianity searches in the context of the Planck mission.

IMPROVED CONSTRAINTS ON PRIMORDIAL NON-GAUSSIANITY FOR THEWILKINSON MICROWAVE ANISOTROPY PROBE5-YEAR DATA

We present constraints on the nonlinear coupling parameter f nl with the Wilkinson Microwave Anisotropy Probe (WMAP) data. We use an updated method based on the spherical Mexican hat wavelet (SMHW) which provides improved constraints on the f nl parameter. This paper is a continuation of a previous work by Curto et al., where several third-order statistics based on the SMHW were considered. In this paper, we use all the possible third-order statistics computed from the wavelet coefficient maps evaluated at 12 angular scales. The scales are logarithmically distributed from 6.9 arcmin to 500 arcmin. Our analysis indicates that f nl is constrained to –18 < f nl < +80 at 95% confidence level (CL) for the combined V+W WMAP map. This value has been corrected by the presence of undetected point sources, which adds a positive contribution of Δf nl = 6 ± 5. Our result excludes at ~99% CL the best-fitting value f nl = 87 reported by Yadav & Wandelt. We have also constrained f nl for the Q, V, and W frequency bands separately, finding compatibility with zero at 95% CL for the Q and V bands but not for the W band. We have performed some further tests to understand the cause of this deviation which indicate that systematics associated with the W radiometers could be responsible for this result. Finally, we have performed a Galactic north-south analysis for f nl. We have not found any asymmetry, i.e., the best-fitting f nl for the northern pixels is compatible with the best-fitting f nl for the southern pixels.

Wilkinson Microwave Anisotropy Probe5-yr constraints onfnlwith wavelets

We present a Gaussianity analysis of the WMAP 5-year Cosmic Microwave Background (CMB) temperature anisotropy data maps. We use several third order estimators based on the spherical Mexican hat wavelet. We impose constraints on the local non-linear coupling parameter fnl using well motivated non-Gaussian simulations. We analyse the WMAP maps at resolution of 6.9 arcmin for the Q, V, and W frequency bands. We use the KQ75 mask recommended by the WMAP team which masks out 28% of the sky. The wavelet coefficients are evaluated at 10 different scales from 6.9 to 150 arcmin. With these coefficients we compute the third order estimators which are used to perform a chi-squared analysis. The chi-squared statistic is used to test the Gaussianity of the WMAP data as well as to constrain the fnl parameter. Our results indicate that the WMAP data are compatible with the Gaussian simulations, and the fnl parameter is constrained to -8 < fnl < +111 at 95% CL for the combined V+W map. This value has been corrected for the presence of undetected point sources, which add a positive contribution of Delta_fnl = 3+-5 in the V+W map. Our results are very similar to those obtained by Komatsu et al (2008) using the bispectrum.

Detection of the integrated Sachs-Wolfe effect and corresponding dark energy constraints made with directional spherical wavelets

Using a directional spherical wavelet analysis we detect the integrated Sachs-Wolfe (ISW) effect, indicated by a positive correlation between the first-year Wilkinson Microwave Anisotropy Probe (WMAP) and NRAO VLA Sky Survey (NVSS) data. Detections are made using both a directional extension of the spherical Mexican hat wavelet and the spherical butterfly wavelet. We examine the possibility of foreground contamination and systematics in the WMAP data and conclude that these factors are not responsible for the signal that we detect. The wavelet analysis inherently enables us to localize on the sky those regions that contribute most strongly to the correlation. On removing these localized regions the correlation that we detect is reduced in significance, as expected, but it is not eliminated, suggesting that these regions are not the sole source of correlation between the data. This finding is consistent with predictions made using the ISW effect, where one would expect weak correlations over the entire sky. In a flat universe the detection of the ISW effect provides direct and independent evidence for dark energy. We use our detection to constrain dark energy parameters by deriving a theoretical prediction for the directional wavelet covariance statistic for a given cosmological model. Comparing these predictions with the data we place constraints on the equation-of-state parameter w and the vacuum energy density Ω Λ . We also consider the case of a pure cosmological constant, that is, w = - 1. For this case we rule out a zero cosmological constant at greater than the 99.9 per cent significance level. All parameter estimates that we obtain are consistent with the standard cosmological concordance model values. Although wavelets perform very well when attempting to detect the ISW effect since one may probe only the regions where the signal is present, once all information is incorporated when computing parameter estimates, the performance of the wavelet analysis is comparable to other methods, as expected for a linear approach.

Non-Gaussianity detections in the Bianchi VIIh corrected WMAP one-year data made with directional spherical wavelets

Many of the current anomalies reported in the WMAP text (WMAP) one-year data disappear after ‘correcting’ for the best-fitting embedded Bianchi type VIIh component, albeit assuming no dark energy component. We investigate the effect of this Bianchi correction on the detections of non-Gaussianity in the WMAP data that we previously made using directional spherical wavelets. We confirm that the deviations from Gaussianity in the kurtosis of spherical Mexican hat wavelet coefficients are eliminated once the data are corrected for the Bianchi component, as previously discovered by Jaffe et al. This is due to the reduction of the cold spot at Galactic coordinates (l, b) = (209 ◦ , −57 ◦ ), which Cruz et al. claimed to be the sole source of nonGaussianity introduced in the kurtosis. Our previous detections of non-Gaussianity observed in the skewness of spherical wavelet coefficients are not reduced by the Bianchi correction. Indeed, the most significant detection of non-Gaussianity made with the spherical real Morlet wavelet at a significant level of 98.4 per cent remains (using a very conservative method to estimate the significance). Furthermore, we perform preliminary tests to determine if foregrounds or systematics are the source of this non-Gaussian signal, concluding that it is unlikely that these factors are responsible. We make our code to simulate Bianchi-induced temperature fluctuations publicly available.

The non-Gaussian cold spot in Wilkinson Microwave Anisotropy Probe: significance, morphology and foreground contribution

The non-Gaussian cold spot in the 1-yr Wilkinson Microwave Anisotropy Probe (WMAP) data, described by Vielva et al. and Cruz et al., is analysed in detail in the present paper. First of all, we perform a more rigorous calculation of the significance of the non-zero kurtosis detected in WMAP maps by Vielva et al. in wavelet space, mainly generated by the Spot. We confirm the robustness of that detection, since the probability of obtaining this deviation by chance is 0.69 per cent. Afterwards, the morphology of the Spot is studied by applying Spherical Mexican Hat Wavelets with different ellipticities. The shape of the Spot is found to be almost circular. Finally, we discuss if the observed non-Gaussianity in wavelet space can arise from bad subtracted foreground residues in the WMAP maps. We show that the flat frequency dependence of the Spot cannot be explained by a thermal Sunyaev–Zeldovich effect. Based on our present knowledge of Galactic foreground emissions, we conclude that the significance of our detection is not affected by Galactic residues in the region of the Spot. Considering different Galactic foreground estimates, the probability of finding such a big cold spot in Gaussian simulations is always below 1 per cent.

Cross-correlation of the cosmic microwave background and radio galaxies in real, harmonic and wavelet spaces: detection of the integrated Sachs-Wolfe effect and dark energy constraints

We report on the first detection of the integrated Sachs-Wolfe (ISW) effect in wavelet space, at scales in the sky around 0 7° with a significance 3.3σ, by cross-correlating the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data and the National Radio Astronomy Observatories (NRAO) Very Large Array (VLA) Sky Survey (NVSS). In addition, we present a detailed comparison of the capabilities of three different techniques for two different objectives: to detect the ISW effect and to put constraints on the nature of the dark energy. The three studied techniques are the cross-angular power spectrum (CAPS; harmonic space), the correlation function (CCF; real space) and the covariance of the spherical Mexican hat wavelet (SMHW) coefficients (CSMHW; wavelet space). We prove that the CSMHW is expected to provide a higher detection (in terms of the signal-to-noise ratio) of the ISW effect for a certain scale. However, the detection achieved by the CAPS is the lowest, being the signal-to-noise ratio dispersed among a wide multipole range. The CCF provides an intermediate detection level. This prediction has been corroborated by the analysis of the data. The SMHW analysis shows that the cross-correlation signal is caused neither by systematic effects nor foreground contamination. However, by taking into account the information encoded in all the multipoles/scales/angles, the CAPS provides slightly better constraints than the SMHW in the cosmological parameters that define the nature of the dark energy. The limits provided by the CCF are wider than for the other two methods, although the three of them give similar confidence levels (CLs). Two different cases have been studied: (i) a flat A cold dark matter universe; (ii) a flat universe with an equation of state parameter that, although it does not change with time, could take values different from - 1. In the first case, the CAPS provides (for a bias value of b = 1.6) Ω Λ = 0.73 +0.11 -0.14 (at la CL). Moreover, the CAPS rejects the range Ω Λ < 0.1 at 3.5σ, which is the highest detection of dark energy reported to date. In the second case, the CAPS gives Ω DE = 0.70 +0.12 -0.20 and w = -0.75 +0.32 -0.41 (at la CL). This is the first estimation of the equation of state of dark energy made through the cross-correlation of the cosmic microwave background (CMB) and the nearby galaxy density distribution. It also provides an independent estimation from that made by the WMAP team using the CMB and large-scale structure.

Detection of a non-Gaussian spot in WMAP

An extremely cold and big spot in the Wilkinson Microwave Anisotropy Probe (WMAP) 1-yr data is analysed. Our work is a continuation of a previous paper by Vielva et al. that reported the detection of non-Gaussianity, with a method based on the spherical Mexican hat wavelet (SMHW) technique. We study the spots at different thresholds on the SMHW coefficient maps, considering six estimators, namely the number of maxima, the number of minima, the numbers of hot and cold spots, and the number of pixels of those spots. At SMHW scales around 4° (10° on the sky), the data deviate from Gaussianity. The analysis is performed on all of the sky, the Northern and Southern hemispheres, and on four regions covering all of the sky. A cold spot at (b = -57°,l = 209°) is found to be the source of this non-Gaussian signature. We compare the spots of our data with 10 000 Gaussian simulations, and conclude that only around 0.2 per cent of them present such a cold spot. Excluding this spot, the remaining map is compatible with Gaussianity, and even the excess of kurtosis in the paper by Vielva et al. is found to be due exclusively to this spot. Finally, we study whether the spot causing the observed deviation from Gaussianity could be generated by systematics or foregrounds. None of them seem to be responsible for the non-Gaussian detection.

Wavelets andWilkinson Microwave Anisotropy ProbeNon‐Gaussianity

We study the statistical properties of the 1st year WMAP data on different scales using the spherical mexican hat wavelet transform. Consistent with the results of Vielva et al. (2003) we find a deviation from Gaussianity in the form of kurtosis of wavelet coefficients on $3-4^\circ$ scales in the southern Galactic hemisphere. This paper extends the work of Vielva et al. as follows. We find that the non-Gaussian signal shows up more strongly in the form of a larger than expected number of cold pixels and also in the form of scale-scale correlations amongst wavelet coefficients. We establish the robustness of the non-Gaussian signal under more wide-ranging assumptions regarding the Galactic mask applied to the data and the noise statistics. This signal is unlikely to be due to the usual quadratic term parametrized by the non-linearity parameter $f_{NL}$. We use the skewness of the spherical mexican hat wavelet coefficients to constrain $f_{NL}$ with the 1st year WMAP data. Our results constrain $f_{NL}$ to be $50\pm 80$ at 68% confidence, and less than 280 at 99% confidence.

Detection of Non‐Gaussianity in theWilkinson Microwave Anisotropy ProbeFirst‐Year Data Using Spherical Wavelets

A non-Gaussian detection in the Wilkinson Microwave Anisotropy Probe (WMAP) first-year data is reported. The detection has been found in the combined Q - V - W map proposed by the WMAP team after applying a wavelet technique based on the spherical Mexican hat wavelet (SMHW). The skewness and the kurtosis of the SMHW coefficients are calculated at different scales (ranging from a few arcminutes to tens of degrees). A non-Gaussian signal is detected at scales of the SMHW around 4° (size in the sky of around 10°). The right-tail probability of the detection is ≈0.4%. In addition, a study of Gaussianity is performed in each hemisphere. The northern hemisphere is compatible with Gaussianity, whereas the southern one deviates from Gaussianity with a right-tail probability of ≈0.1%. Systematics, foregrounds, and uncertainties in the estimation of the cosmological parameters are carefully studied in order to identify the possible source of non-Gaussianity. The detected deviation from Gaussianity is not found to be caused by systematic effects: (1) Each one of the Q, V, and W receivers shows the same non-Gaussianity pattern. (2) Several combinations of the different receivers at each frequency band—which highly reduce the cosmic microwave background (CMB) and the foreground emissions—do not show this non-Gaussian pattern. Similarly, Galactic foregrounds show a negligible contribution to the non-Gaussian detection: non-Gaussianity is detected in all the WMAP maps (from 23 to 94 GHz), and no frequency dependence is observed. Moreover, the expected foreground contribution to the combined WMAP map was added to CMB Gaussian simulations showing a behavior compatible with the Gaussian model. The influence of uncertainties in the CMB power spectrum estimation are also quantified. Hence, possible intrinsic temperature fluctuations (such as secondary anisotropies and primordial features) cannot be rejected as the source of this non-Gaussian detection. We remark that our result implies not only asymmetries north/south—like other previous WMAP analyses—but also a direct non-Gaussian detection.

Topology of the Universe fromCOBE-DMR - a wavelet approach

In this paper, we pursue a new technique to search for evidence of a finite universe, making use of a spherical Mexican Hat wavelet decomposition of the microwave background fluctuations. Using the information provided by the wavelet coefficients at several scales, we test whether compact orientable flat topologies are consistent with the COBE-DMR data. We consider topological sizes ranging from half to twice the horizon size. A scale-scale correlation test indicates that non-trivial topologies with appropriate topological sizes are as consistent with the COBE-DMR data as an infinite universe. Among the finite models, the data seems to prefer a universe which is about the size of the horizon for all but the hypertorus and the triple-twist torus. For the latter, the wavelet technique does not seem a good discriminator of scales for the range of topological sizes considered here, while a hypertorus has a preferred size which is 80 per cent of the horizon. This analysis allows us to find a best fit topological size for each model, although cosmic variance might limit our ability to distinguish some of the topologies.