In this chapter we present the basics of linear perturbations in cosmology. After a general introduction of cosmological perturbation theory, we work out several cases: (i) single pressureless perfect fluid, (ii) single general perfect fluid, and (iii) two fluids: matter and radiation. We also discuss a number of topics such as the velocity field, the redshift distribution, Boltzmann equations, the matter power spectrum, and the perturbed photon propagation. These provide us with important tools when we confront dark energy models with observations of the cosmic microwave background (CMB) and large-scale structure (LSS).

In this chapter the treatment and notation are fairly standard and the topic is covered in most modern textbooks on cosmology. Readers familiar with cosmological perturbation theory may skip this chapter.

Perturbing General Relativity

In Chapter 2 we have outlined the cosmic expansion history in the homogeneous and isotropic FLRW background. However, our Universe is far richer than this simple picture. A metric that deviates from the FLRW spacetime can be written as the sum of an unperturbed FLRW part plus something else, that we can generally call “perturbed” metric. If the perturbed part is assumed to be small, in a sense to be defined later, then this splitting of the full metric into a background part and a perturbed one leads to extremely useful results. As we all know, physics is to a large extent described by a Taylor expansion to some low order, and cosmology is not an exception.

The crucial kick to dark energy research was the interpretation i n 1998 of standard candle observations in terms of cosmic acceleration in the FLRW metric. What we observe is however merely that distant sources (z> 0.3) recedes lower than we would predict in an Einstein–de Sitter Universe calibrated through “nearby” sources. That is, we observe different expansion rates at different distances rather than an increase in the expansion rate at all distances. Can this be caused by a strong in homogeneity rather than by an accelerating Universe?

We also noticed that cosmic acceleration seems to be a recent phenomenon, at least for standard dark energy models. The epoch in which dark energy begins to play a role is close to the epoch in which most of the cosmic structures formed out of the slow linear gravitational growth. We are led to ask again: can the acceleration be caused by strong in homogeneities rather than by a dark energy component?

The answer to both questions is yes, at least in principle. First, we can always interpret a homogeneous evolution H(z) a s a line-of-sight inhomogeneous rate H(r) since we observe only a long our past light cone ds2 = 0 and time and distance a re inextricably related. Second, one can always arrange matter sources so that in some region of the Universe they accelerate away from each other even if on larger scales the expansion is decelerated.

Perhaps the first recognition that the matter composing the universe may be different from the one we touch and experience every day has been put in writing by early Greek philosophers and by Aristotle in particular. In his work On the Heavens, Aristotle argues that the nature and movement of the stars and planets is so fundamentally different from Earth-like elements that a new substance is required, a “bodily substance other than the formations we know, prior to them all and more divine than they.” Later on this cosmic element came to be called quintessentia, or fifth element, and drawing on Plato's classification of the elements a dodecahedron's figure was associated with it.

More than two thousand years after, astrophysicists have begun to pile up evidence that a new form of matter pervades our Universe. This idea is based on observations that reminds one of Aristotle's thoughts: the global movement we observe in distant reaches of our cosmos is unexplainable by ordinary matter. All the matter we see on Earth, in the solar system, inside our Galaxy or in similar structures across the Universe has a small or negligible positive pressure and clumps under the influence of gravity. An expanding Universe filled with this form of matter would by necessity slow down. But in 1998, astronomers studying the global expansion by the use of supper novae found that their observed luminosities can be explained only by an accelerated expansion of the Universe.

Upon foundations of evidence, astronomers erect splendid narratives about the lives of stars, the anatomy of galaxies or the evolution of the Universe. Inaccurate or imprecise evidence weakens the foundation and imperils the astronomical story it supports. Wrong ideas and theories are vital to science, which normally works by proving many, many ideas to be incorrect until only one remains. Wrong data, on the other hand, are deadly.

As an astronomer you need to know how far to trust the data you have, or how much observing you need to do to achieve a particular level of trust. This chapter describes the formal distinction between accuracy and precision in measurement, and methods for estimating both. It then introduces the concepts of a population, a sample of a population, and the statistical descriptions of each. Any characteristic of a population (e.g. the masses of stars) can be described by a probability distribution (e.g. low-mass stars are more probable than high-mass stars) so we next will consider a few probability distributions important in astronomical measurements. Finally, armed with new statistical expertise, we revisit the question of estimating uncertainty, both in the case of an individual measurement, as well as the case in which multiple measurements combine to produce a single result.

You can discover traces of the history of astronomy scattered in the names of the objects astronomers discuss – a history that starts with the mythological interpretation of the sky echoed in constellation names, and that continues to an era when comets are named after spacecraft and quasars after radio telescopes. As discoveries accumulate, so too do the names. As the number of objects of interest has risen to the hundreds of millions, tracking their identities and aliases has grown to a daunting enterprise, made tractable only by the use of worldwide computer networks and meta-database software. In this chapter we introduce the methods for identifying a particular celestial object, but more importantly, the methods for discovering what is known about it.

Very early in the history of astronomy, as Ptolemy tells us, astronomers realized the obvious. The identities of most objects in the sky, like the identities of mountains or cities, could be safely tied to their locations. However, a difficult problem arises in our Solar System (the subject of most of The Almagest), where objects move around the sky quickly.

Astronomers have measured apparent brightness since ancient times, and, as is usual in science, technology has acutely influenced their success. Prior to the 1860s, observers estimated brightness using only their eyes, expressing the results in the uncannily persistent magnitude system that Ptolemy introduced in the second century. As Arago notes, the results were not satisfactory.

In this chapter, after a brief summary of the history of photometry, we will examine in detail the surprisingly complex process for answering the question: how bright is that object? To do so, we will first introduce the notion of a defined bandpass and its quantitative description, as well as the use of such bandpasses in the creation of standard photometric systems. Photometry is most useful if it represents the unadulterated light from the object of interest, so we will take some pain to describe how various effects might alter that light: spectrum shifts, absorption by interstellar material, and the characteristics of the observing system. We will pay particular attention, however, to the heavy burden of the ground-based photometrist: the influence of the terrestrial atmosphere and the techniques that might remove it.

There is an old joke: a lawyer, a priest, and an observational astronomer walk into a bar. The bartender turns out to be a visiting extraterrestrial who presents the trio with a complicated-looking black box. The alien first demonstrates that when a bucket ful of garbage is fed into the entrance chute of the box, a small bag of high-quality diamonds and a gallon of pure water appear at its output. Then, assuring the three that the machine is his gift to them, the bartender vanishes.

The lawyer says, “Boys, we're rich! It's the goose that lays the golden egg! We need to form a limited partnership so we can keep this thing secret and share the profits.”

The priest says, “No, no, my brothers, we need to take this to the United Nations, so it can benefit all humanity.”

“We can decide all that later,” the observational astronomer says. “Get me a screwdriver. I need to take this thing apart and see how it works.”

This text grew out of 16 years of teaching observational astronomy to undergraduates, where my intent has been partly to satisfy – but mainly to cultivate – my students' need to look inside black boxes. The text introduces the primary tools for making astronomical observations at visible and infrared wavelengths: telescopes, detectors, cameras, and spectrometers, as well as the methods for securing and understanding the quantitative measurements they make.

Chapter 1 introduced the situations that produce line and continuous spectra as summarized by Kirchhoff's laws of spectrum analysis. This chapter descends to the microscopic level to examine the interaction between photons and atoms. We show how the quantum mechanical view accounts for Kirchhoff's laws, and how atomic and molecular structure determines the line spectra of gasses.

To understand modern astronomical detectors, we also turn to a quantum mechanical account – this time of the interaction between light and matter in the solid state. The discussion assumes you have had an introduction to quantum mechanics in a beginning college physics course. We will pay particular attention to some simple configurations of solids: the metal oxide semiconductor (MOS) capacitor, the p–n junction, the photo-emissive surface, and the superconducting Josephson junction. Each of these is the physical basis for a distinct class of astronomical detector.

Beginning in 1862, Huggins used a spectroscope to probe the chemical nature of stars and nebulae. Since then, spectrometry has been the tool for the observational investigation of almost every important astrophysical question, through direct or indirect measurement of temperature, chemical abundance, gas pressure, wavelength shift, and magnetic field strength. The book by Hearnshaw (1986), from which the above quotes were taken, provides a history of astronomical spectroscopy prior to 1965. Since 1965, the importance of spectroscopy has only increased. This chapter introduces some basic ideas about spectrometer design and use. Kitchin (1995, 2009) and Schroeder (1987) give a more complete and advanced treatment, and Hearnshaw (2009) provides a history of the actual instruments.

Literally, a spectroscope is an instrument to look through visually, a spectrometer measures a spectrum in some fashion, and a spectrograph records the spectrum. Astronomers are sometimes particular about such distinctions, but very often use the terms interchangeably.

Astronomy is not for the faint of heart. Almost everything it cares for is indescribably remote, tantalizingly untouchable, and invisible in the daytime, when most sensible people do their work. Nevertheless, many – including you, brave reader – have enough curiosity and courage to go about collecting the flimsy evidence that reaches us from the universe outside our atmosphere, and to hope it may hold a message.

This chapter introduces you to astronomical evidence. Some evidence is in the form of material (like meteorites), but most is in the form of light from faraway objects. Accordingly, after a brief consideration of the material evidence, we will examine three theories for describing the behavior of light: light as a wave, light as a quantum entity called a photon, and light as a geometrical ray. The ray picture is simplest, and we use it to introduce some basic ideas like the apparent brightness of a source and how that varies with distance. Most information in astronomy, however, comes from the analysis of how brightness changes with wavelength, so we will next introduce the important idea of spectroscopy. We end with a discussion of the astronomical magnitude system. We begin, however, with a few thoughts on the nature of astronomy as an intellectual enterprise.

Astronomers normally present the output of a sensor array in the form of a digital image, a picture, but a mathematical picture. One appealing characteristic of a digital image is that the astronomer can readily subject it to mathematical manipulation, both for purposes of improving the image itself, as well as for purposes of extracting information.

Accordingly, the chapter will proceed by first presenting some general thoughts about array data, and some general algorithms for image manipulation. Because they are so useful in astronomy, we next examine some procedures for removing image flaws introduced by the observing system, as well as some operations that can combine multiple images into a single image. Finally, we look at one important method for extracting information: digital photometry, and derive the CCD equation, an expression that describes the quality you can expect from a digital photometric measurement.

Arrays

Astronomers usually use panoramic detectors to record two-dimensional images and, at optical wavelengths, they most often use a charge-coupled device (CCD). Unlike a photographic plate (until the 1980s, the panoramic detector of choice), a CCD is an array – a grid of spatially discrete but identical light-detecting elements. Although this chapter discusses the CCD specifically, most of its ideas are relevant to images from other kinds of arrays.

Where is Mars? The center of our Galaxy? The brightest X-ray source? Where, indeed, are we? Astronomers have always needed to locate objects and events in space. As our science evolves, it demands ever more exact locations. Suppose, for example, an astronomer observes with an X-ray telescope and discovers a source that flashes on and off with a curious rhythm. Is this source a planet, a star, or the core of a galaxy? It is possible that the X-ray source will appear to be quite unremarkable at other wavelengths. The exact position for the X-ray source might be the only way to identify its optical or radio counterpart. Astronomers need to know where things are.

Likewise, knowing when something happens is often as important as where it happens. The rhythms of the spinning and orbiting Earth gave astronomy an early and intimate connection to timekeeping. Because our Universe is always changing, astronomers need to know what time it is.

Astronomical detection, even more than the work of Sherlock Holmes, is an exact science. Watson, though, has an equally important point: no astronomer, not even the coldest and most unemotional, is immune to that pleasant, even romantic, thrill that comes when the detector does work, and the Universe does seem to be speaking.

An astronomical detector receives photons from a source and produces a corresponding signal. The signal characterizes the incoming photons: it may measure their rate of arrival, their energy distribution, or perhaps their wave phase or polarization. Although detecting the signal may be an exact science, its characterization of the source is rarely exact. Photons never pass directly from source to detector without some mediation. They traverse both space and the Earth's atmosphere, and in both places emissions and absorptions may modify the photon stream. A telescope and other elements of the observing system, like correcting lenses, mirrors, filters, optical fibers, and spectrograph gratings, collect and direct the photons, but also alter them. Only in the end does the detector do its work.

While I disagree with Proust about the thrill of seeing utterly new things (I'm sorry, that is an adventure), astronomers immediately come to mind if I wonder who might be obsessively concerned with the acquisition of new “eyes.” No instrument has so revolutionized a science, nor so long and thoroughly dominated its practice, as has the telescope astronomy. With the possible exception of the printing press, no instrument so simple (amateurs still make their own) has produced such a sustained transformation in humanity's understanding of the Universe.

In this chapter, we examine the basic one- and two-mirror optical layouts of the preferred modern designs, as well as the layouts of a few telescopes that use both transmitting and reflecting elements. Schroeder (1987) provides a more advanced treatment.

Space-based telescopes have some pronounced advantages, disadvantages, and special requirements compared with their ground-based cousins, and we will consider these in some detail, along with recent advances in the construction of very large telescopes. Because it is such an important technology, we will take some trouble to understand the principles of adaptive optics, and its potential for removing at least some of the natural but nasty (for astronomy) consequences of living on a planet with an atmosphere.