*by Thomas Buchert, Martin J. France & Frank Steiner.*

**This challenging question touches on the initial conditions of the primordial Universe, on modeling assumptions, and statistical ensembles generating the Cosmic Microwave Background. **

**Our CQG paper explores model-independent approaches to these challenges.**

We observe only a single Universe, the one we live in. We cannot rerun cosmic history to see how actual observations might have varied. Nor can we communicate with distant aliens to build an ensemble of observations of the Universe from different vantages in space and time. The only possibility that remains is to make a model of the Universe. Running this model a large number of times, we can generate an ensemble of realizations of the Cosmic Microwave Background (CMB) sky maps. In principle, it is then possible to answer the question, whether there is a single realization of the chosen model that agrees with what is observed. Moreover, we should determine the probability of finding this single realization within the ensemble of patterns that our model allows. To do this we have to select the statistical average properties of the model and those of its initial conditions. (Compare the recent comments by Abhay Ashtekar and Brajesh Gupt, here and here, and by Massimo Giovannini.)

The efforts to understand and construct the CMB sky maps, built from raw *Planck* satellite data, are based on a large number of hypotheses, i.e., data reduction and modeling assumptions that can be tested for self-consistency.

The key-assumption (i) is that initial conditions are drawn from a Gaussian random field with a nearly scale-invariant power spectrum that is built on further assumptions about the matter fields in the model. Cosmologists trace back the Gaussian hypothesis to general predictions from the inflationary paradigm. (See, however, the critical discussion in the articles of this focus section by Anna Ijjas and Paul Steinhardt, John Ellis et al., and by Abhay Ashtekar and Aurélien Barrau.) Further essential hypotheses, typically adopted in constructing CMB sky maps, are: (ii) to idealize the surface of last scattering of the CMB radiation as an exact 2-sphere by adopting the hypothesis (iii) that the cosmological model is homogeneous and isotropic with constant spatial curvature (that is zero in the so-called concordance model of cosmology); (iv) to assume that the observed dominant dipole anisotropy is solely the result of our proper motion relative to the rest-frame of the CMB; and (v) an assumption on the global topology.

Combined with hypotheses about preparing ideal CMB maps, (e.g., recipes for masking the galactic emission and foreground contamination), we finally arrive at statistical ensembles of model maps as candidates for the comparison with the data.

In view of the large number of hypotheses, it would seem plausible that the so-called *anomalies* observed in the *Planck* maps (like the missing power on large scales, the hemispherical asymmetry, and the *Cold Spot*, see the article in this focus section by Dominik Schwarz et al.) are a result of violation of one or some of the assumptions made. A result of our CQG paper is, however, that we have to be cautious about rapidly abandoning the above hypotheses. Inspection by eye reveals “anomalous patterns” in some maps resulting from realizations of Gaussian random fields. Are such “anomalies” compatible with all of the assumptions currently made? Realizing 100,000 maps we found likelihood envelopes on various statistical characteristics of CMB maps that allow single maps in excess of 6σ from the averaged assumed Gaussian distribution. Indeed, it turns out that the characteristics of the *Planck* maps do not lie close to the average characteristics of the model ensemble: the average variance of the maps created from the standard concordance model of cosmology is bigger than the variance of the *Planck* maps. But individual members of the ensemble offer compatible values.

Despite the disclaimers made above, there are alternative hypotheses which provide “natural” explanations of some of the anomalies. As an example we mention the anomaly of missing power on large scales in the observed CMB maps. The standard concordance model assumes infinite spatial sections with perturbations on all scales up to infinity. The assumption on the global topology can be altered to closed spatial sections. A finite volume introduces an infrared cutoff (as pointed out by Leopold Infeld already in 1949), which leads to a natural explanation of missing power on large scales, since perturbations with wavelengths beyond the finite volume are missing. This issue of *cosmic topology* has attracted many scientists. A variety of closed and open topologies has been analyzed by one of us (FS) with his group at Ulm University. One could even ask whether one can determine the spatial topology of the Universe by analyzing CMB maps. The answer has not been found yet. (See a recent paper on black holes.)

Hitherto, the most conservative way to advance our understanding of the *Planck* maps has been to keep the assumptions listed above. This already provides a starting point of rich and unforeseen consequences, as our paper shows. We also attempted to go down another route, starting from the standard list of hypotheses, but aiming to critically examine further model-dependences in the analysis of the deviations from the assumed Gaussian distribution (so-called *non-Gaussianities*). It is here, where model-independent strategies can be developed and compared with current perturbative models of the CMB anisotropies. We employ discrepancy functions to measure the drifts from Gaussianity for: (i) the probability density function (PDF); and (ii) the complete set of three Minkowski Functionals. The latter method, exploiting the wealth of results of integral geometry, has been developed over the last 23 years, after its introduction into cosmology by one of us (TB) with Klaus R. Mecke and Herbert Wagner at the LMU university in Munich. Nowadays it is used by many teams worldwide including the *Planck* team. In particular, the family of Minkowski Functionals provide a robust set of morphological descriptors which supersedes the power of correlation analyses. A small number of functionals is needed to completely characterize the morphology of the CMB, a characterization that contains all orders of the correlation functions in integrated form.

In our analysis, we use Hermite expansions for the discrepancy functions and show that they hold under very general conditions, even for large deviations from a Gaussian behaviour. (Large deviations are expected, e.g., in string gas cosmologies — see the article by Robert Brandenberger — and multi-field inflation models — see the article by Jérôme Martin.) Then, the non-Gaussianities are parametrized by the Hermite expansion coefficients for which we derive explicit expressions in terms of complete Bell polynomials depending only on the cumulants respectively the moments of the PDF of the random CMB temperature field. Any drift from Gaussianity is unveiled by non-vanishing expansion coefficients respectively higher cumulants (order ≥ 3) or moments.

This general characterization of the CMB non-Gaussianities is compared with the results of the standard concordance model based on an ensemble of 100,000 CMB sky maps. Among the models of inflation many predict the so-called hierarchical ordering (HO) of the normalized cumulants. This leads to expansions in powers of the standard deviation of the CMB anisotropy (perturbative expansion) for the PDF and the Minkowski functionals. But note that the standard deviation is not a small parameter. To compare the two approaches, one has to perform numerical tests. The result of our tests are that, at comparable orders of the expansions (we had to go to fourth order in the standard deviation), the HO-expansion yields satisfactory results, but the full Hermite expansion still provides a better fit. In our paper we also found a stable signature of small non-Gaussianity in the ensemble of map realizations despite the Gaussian assumption underlying their construction. This calls for systematic further analyses to disentangle the various secondary effects that go into the production of maps.

To add a fresh remark, there are issues related to general-relativistic modeling of the CMB that have not yet been addressed. These include inhomogeneity effects — so-called cosmological backreaction — hints coming from the reality of inhomogenous expansion, as well as related problems such as the current assumption of a single universal frame for the description of all energy sources in the model. (See this comment.) To understand these effects is one of the major goals of our future work.

*“Is the Cosmological Microwave Background Gaussian?”* is thus a question that is far from being answered, but the *Planck* data will certainly reveal more information than that which has been exploited so far.

*Read the full article in Classical and Quantum Gravity:
Model-independent analyses of non-Gaussianity in Planck CMB maps using Minkowski functionals
Thomas Buchert et al 2017 Class. Quantum Grav. *

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Planck and fundamentals of cosmology focus issue

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