There are a surprisingly large number of ways that mathematics has supported our attempts to understand the universe. If you live near a major urban area as I do, the night sky features the moon, a few bright stars and an occasional glimpse of a planet. But many years ago when I was in South Dakota, on a very clear night I could see virtually to the horizon in all directions.

This was my most personal encounter with the cosmos. There are certainly a lot of stars out there, more than books or living near New York City had given me a clue to. Perhaps one of the specks of light I saw was this object, now made more visible by mankind's ingenuity. Although it may appear that understanding the universe in which we live is far removed from mathematics, this is not true.

There are a surprisingly large number of ways that mathematics has supported our attempts to understand the universe. First, there is the question of the nature of the matter which makes up the cosmos. Mathematics has played a big role in understanding the subatomic particles which form the matter of living things, planets and stars. Second, geometers have been working with physicists and astronomers to understand the geometry of the space we live in.

Whereas mathematicians might view alternative geometries as "different games" which are fun to investigate because their properties are so varied, physicists are concerned with what is space and its nature. The mathematics involved in both of these areas gets very technical. This being Mathematics Awareness Monthwith this year's theme Mathematics and the CosmosI will try to put in perspective how mathematics has played an essential role in the way humankind has come to view the cosmos.

The history of cosmology has two important threads: the history of the geometry of space as it evolved both in the mathematics and physics communities, and the history of astronomybroadly interpreted. Let us begin with a history of astronomy. It is a sign of human concern with cosmological issues that all ancient cultures took an interest in astronomy, sometimes independently and sometimes building on earlier interest.

Interest in the stars in ancient times combined an interest in astrology and astronomy. Astrology was concerned with the issue of the relationship between configurations in the heavens and human events. It must have been indeed scary for people during pre-scientific times to have seen such strange events as meteor showers, comets, and eclipses.

Imagine having the sun suddenly disappear and having darkness descend. Curiosity about the stars was one of the earlier manifestations of early science.

A towering figure in this study was Claudius Ptolemy. Ptolemy's work had an enormous effect on the history of astronomy and cosmology. He developed a system that tried to explain the apparent motions of the planets under the assumption that the earth was the center of the universe. This required that the sun and planets' motions had to be explained in terms of the earth's being still, as it seemed to be, and that the sun and planets moved around the earth.

Xerox phaser 3610 replace drum cartridge errorAfter the decline of Greek culture, the development of interest in astronomy was kept alive in the Islamic worldwhere mathematics was used in support of observational studies.

Some of this concern with astronomy stemmed from an interest in charting the time for prayers, tracking important religious holidays using the lunar calendarand knowing the direction to Mecca.

Although even in ancient times there had been individuals who were suspicious of a cosmology which placed the earth at the center of the universe, their opinion was largely ignored. Much later Copernicus put forward the theory that the earth moved around the sun.

Srt subtitles urlThere was great turmoil between holding on to older well-established ideas and trying to prove that the new approach was correct. How has progress in our insights into the cosmos been made? Science proceeds by a delicate interplay of theory and experiment.

Sometimes an experimentalist produces careful data over a long period of time which suggests the route for codifying a new theory based on the data. Other times, theory suggests experiments which make it possible to have much greater confidence in that theory. An example of the first type of progress is given in the interaction of the great Danish astronomer Tyco Brahe.

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It is somewhat ironic that Brahe publicly remained an adherent to his own variant of an earth-centered universe. His concern with the Copernican approach was being able to achieve sufficient verification. His empirical studies laid the foundation for a great leap forward in being able to give confirmation for a heliocentric view.Bayesian inference is often used in cosmology and astrophysics to derive constraints on model parameters from observations. This approach relies on the ability to compute the likelihood of the data given a choice of model parameters.

In many practical situations, the likelihood function may however be unavailable or intractable due to non-gaussian errors, non-linear measurements processes, or complex data formats such as catalogs and maps.

In these cases, the simulation of mock data sets can often be made through forward modeling. We discuss how Approximate Bayesian Computation ABC can be used in these cases to derive an approximation to the posterior constraints using simulated data sets. This technique relies on the sampling of the parameter set, a distance metric to quantify the difference between the observation and the simulations and summary statistics to compress the information in the data.

We test the performance of the implementation using a Gaussian toy model. We then apply the ABC technique to the practical case of the calibration of image simulations for wide field cosmological surveys.

We find that the ABC analysis is able to provide reliable parameter constraints for this problem and is therefore a promising technique for other applications in cosmology and astrophysics. Comment: 19 pages, 5 figures, 1 algorithm. Accepted for publication in JCAP. Location of Repository.

OAI identifier: oai:arXiv. Provided by: arXiv. Suggested articles.Bayesian inference is often used in cosmology and astrophysics to derive constraints on model parameters from observations. This approach relies on the ability to compute the likelihood of the data given a choice of model parameters.

New year speech in gujaratiIn many practical situations, the likelihood function may however be unavailable or intractable due to non-gaussian errors, non-linear measurements processes, or complex data formats such as catalogs and maps. In these cases, the simulation of mock data sets can often be made through forward modeling. We discuss how Approximate Bayesian Computation ABC can be used in these cases to derive an approximation to the posterior constraints using simulated data sets.

This technique relies on the sampling of the parameter set, a distance metric to quantify the difference between the observation and the simulations and summary statistics to compress the information in the data.

We test the performance of the implementation using a Gaussian toy model. We then apply the ABC technique to the practical case of the calibration of image simulations for wide field cosmological surveys. Similar records in OSTI. GOV collections:. Title: Approximate Bayesian computation for forward modeling in cosmology. Full Record Other Related Research. Abstract Bayesian inference is often used in cosmology and astrophysics to derive constraints on model parameters from observations.

Approximate Bayesian computation for forward modeling in cosmology. United States: N. Copy to clipboard. United States. We find that the ABC analysis is able to provide reliable parameter constraints for this problem and is therefore a promising technique for other applications in cosmology and astrophysics.

Rivista di studi sulla sostenibilità 1/2018Other availability. Find in Google Scholar. Search WorldCat to find libraries that may hold this journal. LinkedIn Pinterest Tumblr. Similar Records.The statistics of shear peaks have been shown to provide valuable cosmological information beyond the power spectrum, and will be an important constraint of models of cosmology in forthcoming astronomical surveys.

Surveys include masked areas due to bright stars, bad pixels etc. We advocate a forward-modeling approach, where the impacts of masking and other survey artifacts are accounted for in the theoretical prediction of cosmological parameters, rather than correcting survey data to remove them.

GOV collections:. A forward modeling approach. Title: Masked areas in shear peak statistics. Full Record Other Related Research. Abstract The statistics of shear peaks have been shown to provide valuable cosmological information beyond the power spectrum, and will be an important constraint of models of cosmology in forthcoming astronomical surveys.

Authors: Bard, D. Bard, D. Masked areas in shear peak statistics. United States: N. Copy to clipboard. United States. A forward modeling approach". We also explore how this potential bias scales with survey area and evaluate how much small survey areas are impacted by the differences in cosmological structure in the data and simulated volumes, due to cosmic variance.

Free Publicly Available Full Text. Accepted Manuscript DOE. Copyright Statement.

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Other availability. Search WorldCat to find libraries that may hold this journal. Cited by: 2 works.

**Galaxy Cluster Cosmology with the South Pole Telescope by Sebastian Bocquet**

Citation information provided by Web of Science. LinkedIn Pinterest Tumblr. Similar Records.The chronology of the universe describes the history and future of the universe according to Big Bang cosmology.

The earliest stages of the universe's existence are estimated as taking place For the purposes of this summary, it is convenient to divide the chronology of the universe since it originatedinto five parts.

It is generally considered meaningless or unclear whether time existed before this chronology:. It includes the Planck epochduring which currently understood laws of physics may not apply; the emergence in stages of the four known fundamental interactions or forces —first gravitationand later the electromagneticweak and strong interactions; and the expansion of space itself and supercooling of the still immensely hot universe due to cosmic inflationwhich is believed to have been triggered by the separation of the strong and electroweak interaction.

Tiny ripples in the universe at this stage are believed to be the basis of large-scale structures that formed much later. Different stages of the very early universe are understood to different extents.

The earlier parts are beyond the grasp of practical experiments in particle physics but can be explored through other means. Lasting aroundyears. Initially, various kinds of subatomic particles are formed in stages. These particles include almost equal amounts of matter and antimatterso most of it quickly annihilates, leaving a small excess of matter in the universe.

If primordial black holes exist, they are also formed at about one second of cosmic time. By 20 minutes, the universe is no longer hot enough for nuclear fusionbut far too hot for neutral atoms to exist or photons to travel far. It is therefore an opaque plasma. At around 47, years, [2] as the universe cools, its behaviour begins to be dominated by matter rather than radiation.

At aboutyears, helium hydride is the first molecule. Much later, hydrogen and helium hydride react to form molecular hydrogen, the fuel needed for the first stars. At aboutyears, [3] the universe finally becomes cool enough for neutral atoms to form " recombination "and as a result it also became transparent for the first time.

The newly formed atoms—mainly hydrogen and helium with traces of lithium —quickly reach their lowest energy state ground state by releasing photons " photon decoupling "and these photons can still be detected today as the cosmic microwave background CMB. This is currently the oldest observation we have of the universe. Fromyears until about 1 billion years.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Astronomy Stack Exchange is a question and answer site for astronomers and astrophysicists. It only takes a minute to sign up. In my research on exoplanets, I have heard many people talk "forward modeling of exoplanet atmospheres".

I don't know what the "forward" means in "forward modeling" and how it compares with "reverse modeling", if that's even a thing. What is forward modeling, and why is it so special that it needs to be distinguished from just plain ol' regular modeling?

There are different ways to model something. From what you're asking, there are two main types of modeling: forward modeling and inverse modeling. In this type of modeling, you have a specific model that defines the "current" state of your system. In the case of exoplanet atmospheres, it'd likely be something that defines the molecular content, ionization level, density, etc.

In this setup, what you've created is a system for predicting system states from a predetermined physics model. Such an example would be someone creating their own atmosphere of an exoplanet in a model and then saying, okay what happens when I shine light through this atmosphere. What observations might I record? In some sense this is the opposite of forward modeling, albeit it doesn't really mean you're running a model to see into the past.

Instead, what happens with this setup is you know a particular state or result, and you want to construct a model of your system which can produce said state.

Essentially, you want your model to arrive at a certain state when it is done calculating. If it does, you have a reasonable confidence that your model was some indication of what your system is actually like. In this situation, you'd measure components of the atmosphere, e. If you can, then the hope is that the model accurately represents what your system is. Forward modeling is the use of a model in order to simulate an outcome.

The problem of getting the model to produce data from the input is called the forward problem. The forward model takes certain parameters and produces data that can then be compared with the actual observations.

Forward modeling seems to be in common use in the Earth sciences, referring e.

Cesare azanForward problem direct problem, normal problem : The problem of calculating what should be observed for a particular model, e. A Dictionary of Earth Sciences. The opposite procedure is called the inverse problem :. An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in computer tomography, source reconstructing in acoustics, or calculating the density of the Earth from measurements of its gravity field.

It is called an inverse problem because it starts with the results and then calculates the causes. This is the inverse of a forward problem, which starts with the causes and then calculates the results. Solving an inverse problem then means, given a set of observations, constructing a model that accounts for them. I suppose it's to be expected that exoplanet atmospheres are studied through forward modeling, because we already have adequate atmospheric models for Earth and the understanding to adjust them to other planets, while we don't yet have an adequate characterization of exoplanet atmospheres.

From math's point of view it is simple. This is straightforward. Inverse modelling is where you use features of your data to estimate a set of underlying parameters of your physical model of what is going on. Forward modelling is where you use your model to predict what you would observe and use a comparison of these predictions to your data to infer your model parameters. A simple exoplanet example. Consider a sparsely sampled radial velocity curve.

You could fit a sinusoid or an elliptical orbit solution to these data and estimate the period, radial velocity amplitude and then deduce a minimum mass for the orbiting exoplanet by plugging these numbers, along with an estimate of the stellar mass into the mass function formula.

A forward modelling approach would start with the mass of the star and planet, specify an orbital period and inclination and then predict what would be observed - including if necessary, functions that allow for imperfections and uncertainties in the measurements. Many such models are produced and compared with the observations until one can estimate probability functions for each of the model parameters.This number is too high here.

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## Forward Modelling for Supernova Cosmology

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