Cosmic variance
Encyclopedia
Cosmic variance is the statistical
uncertainty
inherent in observations of the universe at extreme distances. It is based on the idea that it is only possible to observe part of the universe at one particular time, so it is difficult to make statistical statements about cosmology
on the scale of the entire universe, as the number of observations (sample size
) must be too small.
model is usually supplemented with cosmic inflation
. In inflationary models, the observer only sees a tiny fraction of the whole universe, much less than a billionth (1/109) of the volume of the universe
postulated in inflation. So the observable universe (the so-called particle horizon of the universe) is the result of processes that follow some general physical laws, including quantum mechanics
and general relativity
. Some of these processes are random: for example, the distribution of galaxies throughout the universe can only be described statistically
and cannot be derived from first principles.
problems: suppose that random 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 it is difficult to draw statistical conclusions about its significance. 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 India
n and Chinese
citizens, about 5% are American
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 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 observed is the result of a statistical process, the observer can only view one realization of that process, so our observation is statistically insignificant for saying much about the model, unless the observer is careful to include the variance
. This variance is called the cosmic variance and is separate from other sources of experimental error: a very accurate measurement of only one value drawn from a distribution
still leaves considerable uncertainty about the underlying model. Variance is normally plotted separately from other sources of uncertainty. Because it is necessarily a large fraction of the signal, workers must be very careful in interpreting the statistical significance of measurements on scales close to the horizon.
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 is important in describing the low multipoles
of the cosmic microwave background and has been the source of much controversy in the cosmology community since the COBE
and WMAP measurements.
.
Another problem of limited sample sizes in astronomy, here practical rather than essential, is in the Titius–Bode law on spacing of satellites in an orbital system. Originally observed for the solar system, the difficulty in observing other solar systems has limited data to test this.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
uncertainty
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
inherent in observations of the universe at extreme distances. It is based on the idea that it is only possible to observe part of the universe at one particular time, so it is difficult to make statistical statements about cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
on the scale of the entire universe, as the number of observations (sample size
Sample size
Sample size determination is the act of choosing the number of observations to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample...
) must be too small.
Background
The standard big bangBig Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
model is usually supplemented with cosmic inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
. In inflationary models, the observer only sees a tiny fraction of the whole universe, much less than a billionth (1/109) of the volume of the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
postulated in inflation. So the observable universe (the so-called particle horizon of the universe) is the result of processes that follow some general physical laws, including quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. Some of these processes are random: for example, the distribution of galaxies throughout the universe can only be described statistically
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
and cannot be derived from first principles.
Philosophical issues
This raises philosophicalPhilosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
problems: suppose that random 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 it is difficult to draw statistical conclusions about its significance. 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 India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
n and Chinese
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
citizens, about 5% are American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
citizens, about 1% are French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
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 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 observed is the result of a statistical process, the observer can only view one realization of that process, so our observation is statistically insignificant for saying much about the model, unless the observer is careful to include the variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
. This variance is called the cosmic variance and is separate from other sources of experimental error: a very accurate measurement of only one value drawn from a distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
still leaves considerable uncertainty about the underlying model. Variance is normally plotted separately from other sources of uncertainty. Because it is necessarily a large fraction of the signal, workers must be very careful in interpreting the statistical significance of measurements on scales close to the horizon.
In physical cosmology
Physical cosmology
Physical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...
, 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
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
of very small statistical samples (Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
) when calculating uncertainties
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
. This is important in describing the low multipoles
Spherical Harmonic
Spherical Harmonic is a science fiction novel from the Saga of the Skolian Empire by Catherine Asaro. It tells the story of Dyhianna Selei , the Ruby Pharaoh of the Skolian Imperialate, as she strives to reform her government and reunite her family in the aftermath of a devastating interstellar...
of the cosmic microwave background and has been the source of much controversy in the cosmology community since the COBE
COBE
The COsmic Background Explorer , also referred to as Explorer 66, was a satellite dedicated to cosmology. Its goals were to investigate the cosmic microwave background radiation of the universe and provide measurements that would help shape our understanding of the cosmos.This work provided...
and WMAP measurements.
Similar problems
A similar problem is faced by evolutionary biologists. Just as cosmologists have a sample size of one universe, biologists have a sample size of one fossil record. The problem is closely related to the anthropic principleAnthropic principle
In astrophysics and cosmology, the anthropic principle is the philosophical argument that observations of the physical Universe must be compatible with the conscious life that observes it. Some proponents of the argument reason that it explains why the Universe has the age and the fundamental...
.
Another problem of limited sample sizes in astronomy, here practical rather than essential, is in the Titius–Bode law on spacing of satellites in an orbital system. Originally observed for the solar system, the difficulty in observing other solar systems has limited data to test this.
Sources
- Stephen Hawking (2003). Cosmology from the Top Down. Proceedings of the Davis Meeting on Cosmic Inflation.