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Summary

From a cosmological perspective, matter is the energy component with negligible EOS parameter so that its energy density varies with expansion as εm ∝ a−3. Unlike the case with radiation, matter in the Universe comes in many forms and is only partially visible or otherwise detectable, so its gravitating density is difficult to evaluate.

The current standard model of particle physics includes two main types of fermion, hadrons and leptons. Hadrons are quark composites and are the only particles subject to the strong nuclear force. They are further divided into two types. Baryons are three-quark composites: the only stable baryons are protons and neutrons, so that atomic nuclei are made up entirely of baryons, and protons/ neutrons are thus often called nucleons. Mesons are two-quark composites and are all unstable on short time scales, so that the only stable hadrons are the nucleons. Leptons are subject to electroweak forces but not to the strong nuclear force; stable leptons are the electron and its anti-particle the positron, and neutrinos. For convenience – and since the Universe is overall electrically neutral – electrons are often included with nucleons as ‘baryonic matter’.

Baryonic matter is what is usually meant by ‘normal matter’– it is the stuff we, and all we can see around us, are made of. But our ability to detect fundamental particles is limited to masses less than those achievable by current particle accelerators, which (with the advent of the Large Hadron Collider) are limited to masses on the order of a few times 100 GeV or less (nucleons have masses of a bit less than 1 GeV). Since some exotic particle theories predict particles of even greater mass, there is probably a lot left to discover in the realms of possible forms of matter.

So it really should be of no great surprise to find cosmological evidence for currently unknown forms of matter. The gravitational evidence for ‘dark matter’ has been growing for at least 80 years, and has reached the point of near certainty. This exotic stuff appears to have no appreciable cross-section to electromagnetism so that it neither absorbs nor emits detectable radiation: hence, ‘dark’. It can only be detected and measured in terms of the influence of its gravitation on ordinary matter and light.

Summary

The Concordance Cosmological Model (CCM), aka the Standard Model of Big Bang Cosmology, is the solution to the Friedmann Equations that incorporates parameters that represent best estimates taken from a wide range of observations, and that purports to explain not only observations of universal expansion but also consequences of primordial cosmology and of structure formation in the Universe. Perhaps somewhat surprisingly, all these matters can be successfully modelled with one set of parameters for the Friedmann Equations; hence ‘Concordance’. The parameters of the CCM are the following.

Curvature From several indirect lines of evidence, the Universe appears to be geometrically flat or nearly so: Ω0 ≈ 1. The principal reason for this choice is the apparent need for a nearly flat geometry in order to account for the Hubble Relation for SN Ia supernovae (Section 15.2), and for the shape of the CMB anisotropy spectrum (Chapter 17). A (very nearly) flat geometry is also a robust prediction of inflation (Chapter 16).

Radiation energy density The well-observed CMB temperature, together with expected primordial neutrinos, implies εr, 0 = 7.01 × 10−14 J/m3.

Matter density Gravitational evidence (Chapter 12) and model fitting to diagnostics (Chapter 14) point to ρm, 0 ≈ 2.6 × 10−27 kg/m3 (corresponding to εm, 0 ≈ 2.4 × 10−10 J/m3), of which ∼ 15% is baryonic (as inferred from primordial nucleosynthesis, Section 16.3); and the remaining exotic dark matter. These densities are also consistent with details in the CMB anisotropy spectrum (Section 17.3).

Dark energy An energy component arising from the cosmological constant of εΛ,0 ≈ 6.4×10−10 J/m3 is apparently needed to (1) fit the SN Ia brightnesses to a plausible Hubble Relation, (2) provide enough energy to flatten the Universe, and (3) yield an expansion model in which the Universe is at least as old as the oldest stars in our Galaxy. The nature and density of this energy is uncertain; see Chapter 13.

Summary

General covariance has something of the character of a strategy, rather than as a fundamental principle. It probably would be possible to construct a theory of GR without resorting to tensors or other forms of general covariance, although it certainly would not be easy to do so. Equivalence, on the other hand, is absolutely fundamental to GR, constituting the connection between curvature and gravitation. As a principle underlying GR it has its basis in the observed equality of gravitational and inertial mass, so that all objects experience the same acceleration in a gravitational field, quite unlike the case with, e.g., electrical or magnetic fields. By itself, this sets gravitation apart from other forces; but Einstein extended the principle further by applying it – in some sense – to all of physics, not just gravitational dynamics. It is thus useful to distinguish between the Weak Equivalence Principle and the Strong Equivalence Principle, both of which assert that acceleration and gravitation are equivalent, but in somewhat different senses and with different consequences.

Weak Equivalence Principle

The Weak Equivalence Principle (WEP) states simply that all objects in a gravitational field experience the same acceleration. As a consequence, the accelerating effects of gravitation can be transformed away by going over to a coordinate system falling freely with the gravitational field, much like astronauts in the space station or Einstein's workmen falling from a roof. Thus, an operational definition of the WEP is:

The dynamical effects of a gravitational field can be transformed away by moving to a reference frame that is freely falling in the gravitational field.

The utility of this principle is limited by its strict applicability only to static, uniform gravitational fields, which are rare. Objects falling to the ground, for instance, experience a gravitational force that increases as they get closer to the Earth; and objects in orbit experience a gravitational field that changes direction with orbital location. Applications of the WEP are thus limited to the near locality (in space-time) of a falling object: such Locally Inertial Reference Frames are formally defined in the following section.

Even with this limitation the WEP can have interesting consequences for gravitational physics, the most important of them being the equations of motion of a freely falling object – i.e., one experiencing only gravitational forces.

Summary

All credible cosmological models are governed by decreasing energy densities and temperatures as the Universe expands. The very early Universe almost certainly was hot and dense, and probably nearly uniform as a consequence of the high rate of particle interactions likely to have prevailed in that state; and thus largely without structure. The current Universe is quite different, with matter coarsely distributed and large temperature gradients on many scales, and coherent structures of all sorts – from galaxy clusters and superclusters to individual galaxies of all sizes; to planetary systems and astronomy students. How the Universe changed so greatly in ∼ 13 billion years is the subject of much study amongst cosmologists, who are also interested in what the progress of those changes can tell us about the Universe at large.

It is useful and customary to break the Universe's history into a series of eras characterized by processes and contents. The selection of eras is largely arbitrary – it constitutes something of a Rorschach test for theoretical cosmologists. For present purposes we choose the following:

Summary

Einstein's General Theory of Relativity (GR) was motivated principally by his desire to expand his very successful theory of Special Relativity (SR) to non-inertial reference frames. SR served to reconcile the invariance of the speed of light for all observers – as predicted by Maxwell's Equations of electromagnetism, and verified by the Michelson–Morley experiment – with Einstein's Principle of Special Relativity: that the laws of physics were the same in all non-accelerating reference frames. General Relativity, as Einstein envisioned it, would require the laws of physics to be identical in all reference frames, including accelerating ones. That this extension of the relativity principle leads to a theory of gravitation – which is what GR has become – was a consequence of the observed equality of gravitational and inertial mass: since all objects fell with the same acceleration in a given gravitational field, acceleration and gravitation are, in some sense, equivalent. Note that this singling out of gravitation distinguishes it from other fundamental forces, such as electromagnetism: acceleration and gravitation are connected in a unique manner.

But the details of that connection were totally non-obvious when Einstein set out to discover them; in particular, it did not seem possible at first to write laws of mechanics in a manner that is independent of the acceleration of the reference frame. In fact, Einstein never successfully united all forms of non-inertial motion into a single theory, but he did manage to do so with gravitation so that his General Relativity theory has effectively become one of gravitation, relegating Newton's theory of gravity to that of an approximation to the full relativistic theory. In particular, it is Einstein's theory of gravity that must be employed on the scales encountered in cosmology for a successful theory of the Universe's large-scale structure and evolution to be constructed.

The fundamental concepts underlying Einstein's theory of gravitation are these three: General Covariance, which expresses the relativity principle, that the laws of physics take the same form in all reference frames; Equivalence, which embodies the equality of gravitational and inertial mass; and Space-Time Curvature, which provides the means by which gravitation controls dynamics. These are conceptually summarized in this chapter and are each the detailed subject of a separate chapter in Part II of this text.

Summary

To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour.

William Blake, Auguries of Innocence

Consider now the current state of our understanding of the expanding Universe. We appear to have a good understanding of the fundamental physics underlying its expansion, of the origin of structure in the current Universe, and of its detailed history and likely future. Our current cosmological world-view is remarkable not only for its scope but also for its coherence: everything seems to hang together in a nearly seamless picture that purports to explain nearly all of the Universe's largescale structure and evolution, from shortly after the moment of creation to the current time, and on scales dwarfing anything else in our experience or intellectual musings.

Consider, for instance:

Summary

The conceptual basis of GR is that matter and energy cause space-time to be curved, and that curvature determines the paths of freely falling objects. From the previous chapter we expect that the curvature will be reflected in the affine connection or, more generally, in the metric tensor. It is thus necessary to define appropriate measures of curvature in terms of the metric tensor components.

Now, it is pretty easy to mathematically describe the curvature of, say, the two-dimensional surface of a sphere embedded in our three-dimensional space. But it's not at all obvious how to describe the curvature of the three-dimensional space in which we presumably live, nor even to understand what curvature means in that context. The trick is to use the concept of distance or, more precisely, the metric of the space. Then curvature will reveal itself by, e.g., the circumference of a circle in terms of its radius, the sum of the three angles in a triangle, etc. To be useful in application to GR we require that the curvature be revealed without moving outside the surface or space in question. Going back to our sphere embedded in 3-space, imagine a two-dimensional bug wandering around on the surface of the sphere, taking measurements. We need to develop the tools by which those measurements can be used to quantify the curvature of the surface, then generalize to four-dimensional space-time.

Simple curvature

Descriptions of curvature of plane curves and of two-dimensional surfaces are based on circles and spheres. The curvature K of a circle is defined to be the inverse of its radius R, which is called the radius of curvature: K ≡ 1/R. For any other plane curve the curvature at a point on the curve is defined to be that of the best-fitting circle to the curve at that point. In Cartesian coordinates, the curvature of a plane curve y(x) at any chosen point is given by

where the sign is chosen according to the chosen orientation. Note the presence of the second derivative, a characteristic of measures of curvature.

Extension of these ideas to a curved, two-dimensional surface in 3-space is fairly straightforward. At any point on a surface one can define principal curvatures R1 and R2 corresponding to curves resulting from intersections with tangent and normal planes and computed by Equation (6.1).

Summary

Cosmology is the study of the Universe on the largest scales. As such it deals with structures and dynamics that are quite different from those of terrestrial physics or, indeed, of most other branches of astronomy. It is a subject in which the finite speed of light plays a major role, conflating distance with time: in observing distant galaxies we see them not as they are now, but as they were when the light left them; and that is typically long enough ago to encompass significant evolution in the Universe's contents and dynamics. On cosmological scales the dynamics are dominated by gravitation and, possibly, dark energy. The current theory of gravitation is Einstein's General Theory of Relativity (GR), a non-linear and hugely complex theory entailing subtle and largely unfamiliar mathematical and physical concepts; the nature of dark energy remains speculative as of this writing. It is thus a non-trivial matter to assemble a coherent physical model of the Universe at large, requiring careful definitions of such seemingly mundane things as distance and time. We begin the effort in this introduction with a brief description of the Universe and its contents, and an assessment of our ability to understand the Universe on large scales in both space and time.

Galaxies and friends

Figure 1 illustrates the Universe as we naively think of it: a vast, crowded collection of galaxies. But this is deceptive, for such a picture collapses three dimensions into two and amplifies brightnesses, and thus under-represents the distances between galaxies and overstates both the density of matter and the degree of illumination of the Universe at large. The Universe in actuality is quite thinly populated and only faintly illuminated.

The visible contents of the Universe at large are almost entirely in the form of galaxies, mostly as large, luminous galaxies such as ours. Large galaxies, such as our Milky Way Galaxy, contain billions of stars and much diffuse matter.

Summary

Solutions to the Friedmann Equations are models of the Universe's expansion and constitute a four-parameter family of expansion functions a (t); it is the cosmologists’ job to determine which of them is correct, or at least plausible. We approach this problem from two directions: estimations of the densities of sources of gravitation from observations, for use in solving the Friedmann Equations; and comparisons of predictions of trial solutions with observed structure and kinematics of the Universe's contents.

We begin by solving the Friedmann Equations for single-component models, and demonstrating the comparison of their predictions with observations of the real Universe. The best-fitting model at present is the Concordance Cosmological Model (CCM), containing radiation, matter, and dark energy; and discussed in detail in Chapter 15.

Summary

Space-time curvature is established by the density of gravitating matter and energy, and is reflected in the metric tensor components. The equations relating the resulting metric tensor to matter/energy density are the Einstein Field Equations (EFE) of gravitation; generically, G(g) = T(ρ, ε) where T characterizes the density of sources of gravitation and G is a curvature tensor containing metric tensor components and their derivatives.

The Newtonian equivalent of the EFE is Poisson's Equation, ∇2Φ = 4πGρ, Where Φ is the gravitational potential and ρ is matter density. This equation may be logically deduced from Newton's laws of motion and of gravitation, but Einstein's Field Equations are less secure: the relation between space-time curvature and mass/energy cannot be unambiguously deduced from fundamental principles, but must be inferred – guessed at, if you like – from broad principles and analogies that leave room open for many specific possibilities. It was arguably Einstein's greatest contribution that he came up with what appears to be the right answer: his field equations have stood the test of time (so far) and are the basis for relativistic cosmology (among other things).

Of the two halves of the EFE the curvature side is the more problematic, so we begin the discussion with the tensor representing matter and energy, which can – to a large extent – be logically deduced from familiar physics.

Sources of gravitation

Energy-momentum tensor

Possible sources of gravitation include all forms of mass/energy, including such things as kinetic energy, pressure (which carries dimensions of energy density), stress, electromagnetic field energies, etc.; in addition to ‘normal’ matter and radiation. One consequence is that the tensor describing the densities of sources of gravitation is known by many names: mass-energy tensor, energy-momentum tensor, stress-energy tensor, etc. We will stick to the most commonly used name, the energy-momentum tensor, and denote it by T.

It is not hard to guess that this tensor must be of rank 2 if it is to include directional components such as momenta, stress, and electromagnetic field potentials. The proper and most general definition of T is a variational one: T is the quantity needed for a stationary matter/radiation action.

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12 - Matter

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15 - Concordance Cosmological Model

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5 - Equivalence Principle

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V - Expansion history

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I - Conceptual foundations

2 - General Relativity

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Appendix B - Newtonian approximations

Index

References

III - Universal expansion

19 - Afterword: the new modern cosmology

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6 - Space-time curvature

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Appendix D - Symbols

1 - Newtonian cosmology

Introducing the Universe

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18 - Galaxy Era

IV - Expansion models

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Appendix A - Differential geometry

Appendix C - Useful numbers

7 - Einstein Field Equations of gravitation

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