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# Cosmic variance

Cosmic variance is a notion related to the fact that we only observe one copy of the Universe.

The more popular extensions of the Big Bang model are the inflationary scenarios. In models such as these, the comoving spatial part of the Universe which we see is an extremely tiny fraction, much less than a billionth (1/109) of the global comoving volume of the Universe.

So, properties which we observe inside of the particle horizon of the Universe, i.e. inside the sphere which is often called the observable Universe or the observable sphere, are considered to be the results of some random processes which follow some general physical laws throughout the global volume.

This raises statistical and hence philosophical problems.

Suppose that the physical processes happen on length scales both smaller than and bigger than the horizon. A physical process (such as an amplitude of a primordial perturbation in density) that happens on the horizon scale only gives us one observable realization. A physical process on a larger scale gives us zero observable realizations. A physical process on a slightly smaller scale gives us a small number of realizations.

In the case of only one realization, any statistical statement about the underlying physical process becomes more of a statement about one's belief in levels of statistical significance rather than a conventional scientific statement.

For example, if the underlying model of a physical process implies that the observed property should occur only 1% of the time, does that really mean that the model is excluded?

Consider the physical model of the citizenship of human beings in the early 21st century, where about 30% are Indian and Chinese citizens, about 5% are USA citizens, about 1% are French citizens, and so on. For an observer who has only one observation - of his/her own citizenship- and who happens to be French and cannot make any external observations, the model can be rejected at the 99% significance level. Yet we, the external observers with more information unavailable to the first observer, know that the model is correct.

In other words, even if the bit of the Universe we observe is the result of a statistical process, we can only observe one occurrence of that process, so our observation is statistically insignificant for saying much about the model, unless we are careful to include the variance due to observing only one occurrence.

In physical cosmology, the common way of dealing with this on the horizon scale and on slightly sub-horizon scales (where the number of occurrences is greater than one but still quite small), is to explicitly include the variance of very small statistical samples (Poisson distribution) when calculating uncertainties.

This variance is called the cosmic variance and is normally plotted separately from other sources of uncertainty. Because it is necessarily a large fraction of the signal, people who interpret the quantitative results of close to horizon scale cosmology observations are often tempted to ignore cosmic variance.

Since cosmic variance is an attempt to make a scientific statement about an unrepeatable scientific experiment (over the lifetime of any 21st century scientist remaining on Earth), this lies at the borderline between physcial cosmology and philosophical cosmology.