combining information over 13 billion years of time
September 29, 2014 Leave a comment
Post by Tom Kitching, MSSL
Combining information over 13 billion years of time
On this blog we have already talked about 3D cosmic shear and the Cosmic Microwave background, this post is about how to combine them.
Cosmic shear is the effect where galaxy images, that are (relatively!) near-by – a mere few billion light years – are distorted by gravitational lensing caused by the local matter in the Universe. We can measure this and use the data to learn about how the distribution of matter evolved over that time.
The Cosmic Microwave Background (CMB) is the ubiquitous glow of microwaves, that comes from every part of the sky, and that was emitted nearly 14 billion years ago. Analysis of the CMB allows us to learn about the early Universe, but also the nearby Universe because the local matter also gravitationally lenses the microwave photons.
In a recent paper we have shown how to combine Cosmic Shear information and CMB together in a single all-encompassing statistic. Because we see the Universe in three dimensions (2 on the sky and one in distance or look-back-time) this new statistic needed to work in three dimensions too.
What we found was the when the galaxy and microwave data are combined properly the resulting statistic is more powerful than the sum of the two previous statistics, because there is the extra information that comes from the “cross-correlation” between them. In particular we found that the extra information helps in measuring systematic effects in the cosmic shear data.
What is a “cross correlation”?
A correlation is a determined relationship between two things. The definition (that the Apple dictionary on my computer gives) is
noun
a mutual relationship or connection between two or more things
In recent cosmological literature we use this term somewhat colloquially to refer to relationship between data points in a single data set. For example one could correlate the position of galaxies separated by a particular separation – to determine if they were clustered together – or one could correlate the temperature of microwave emission from different parts of the sky (both of these have been done with much success).
The word “cross” in “cross correlation” refers to taking correlations of quantities observed from different data sets. The addition of the word “cross” seems somewhat superfluous in fact. If we have experiment A and B one can correlate the data points from A, or correlate data points from B, or correlate data points between A and B.
In the new paper we instead used a more descriptive nomenclature that refers to inter and intra datum aspects of the analysis. Intra-datum means using statistics within a single data set and inter-datum means calculating statistics between them; for example the plotting a histogram of points within a data set, compared to plotting the points from two data sets on one graph.
When should one attempted to find inter-datum correlations between any data points? In this regard there seems to be two modes of investigation that one could take, following a Popper-inspired categorisation one can define the following modes:
- Deductive mode. In this approach one has a clearly defined scientific hypothesis, for example the measurement of some parameter (or model) predicted to have a given value(s). Then one can find a statistic that maximises the signal-to-noise (or expected evidence ratio) for that parameter or model. That statistic may or may not include inter-datum aspects.
- Inductive mode. Alternatively one may simply wish to correlate everything with everything, with no regard to a hypothesis or model. In this approach the motivation would just be to explore the space of possibilities; trying to find something new. If a positive correlation is found then this may, or may not, indicate an underlying physical process or causal relation between the quantities.
The danger of an inductive approach, of course, is one can find correlations for which the underlying physical process is much more complicated than that taken at face value. To illustrate this point one can look on Google Correlate and find some interesting correlations, for example:
Which brings us to an old warning from @LegoAcademics that:
Recent Comments