The vielbein–spin connection formulation of general relativity is described, and this being the one that appears in supergravity. Anti-de Sitter space, as a Lorentzian version of Lobachevsky space, is described. It is a symmetric space solution for the case of a cosmological constant. Black holes, as objects with event horizons and singularities at the center, are described.

This short chapter touches on the limitations of the SM. The SM does not include gravity, and it does not explain the major components of the mass–energy budget of the universe, dark matter and dark energy, the latter being probably the cosmological constant. CP violation in the quark sector is too small to explain the matter–antimatter asymmetry of the Universe, but, if confirmed, the non-SM CP violation in the neutrino sector might be large enough. The ‘strong CP violation’ problem might be solved with the existence of a very light particle, the axion; experiments are reaching the requested sensitivity. Supersymmetric particles present in some extensions of the SM have been searched for, but not found so far.

The SM contains too many free parameters: the masses of the fermions and of the bosons, and the mixing angles. The masses of the fermions, from neutrinos to the top quark, span 13 orders of magnitude. Why such big difference? Why is mixing small in the quark sector, and large in the neutrino sector? Why do the proton and the electron have exactly equal (and opposite) charges? Why are there just three families? Are there any spatial dimensions beyond the three we know? And so on.

The Robertson–Walker metrics are presented as the simplest candidates for the models of our observed Universe. The Friedmann solutions of the Einstein equations (which follow when a R–W metric is taken as an ansatz), with and without the cosmological constant, are derived and discussed in detail. The Milne–McCrea Newtonian analogues of the Friedmann models are derived. Horizons in the R–W models are discussed following the classical Rindler paper. The conceptual basis of the inflationary models is critically reviewed.

The solutions so far have all be “in vacuum,” away from sources. In this chapter, we study gravity “in material.” For comparison, we review the continuum form of Newton’s second law and think about Newtonian gravitational predictions for, for example, hydrostatic equilibrium. Then we develop the relativistic version of those equations directly from Einstein’s equation with various source assumptions (spherical symmetry, perfect fluid) and obtain the interior Schwarzschild solution. Cosmology is another example of working “in material,” and we briefly review the Robertson–Walker starting point and solutions both with and without a cosmological constant. At the end of the chapter, spacetimes requiring exotic sources, including the Ellis wormhole and Alcubierre warp drive, are described.

In this core chapter, the one-loop effective action for Matrix theory on 3 + 1 dimensional branes is elaborated, and the Einstein–Hilbert term is obtained in the presence of fuzzy extra dimensions. Some justification for the stability of the background is given.

This concluding chapter recaps what has been learnt in the previous chapters about the Standard Model. This model is highly successful in describing particle physics phenomena. Some of its successes are briefly underlined, such as the number of light neutrino families. However, as with any model, it also has its weaknesses, which are also provided. The most important open questions of particle physics are addressed in the second part of the chapter, in particular, the matter–antimatter asymmetry, the hypothetical presence of the dark matter. Possible extensions of the Standard Model are presented to incorporate massive neutrinos.

Having developed the necessary mathematics in chapters 4 to 6, chapter 7 returns to physics Evidence for homogeneity and isotropy of the Universe at the largest cosmological scales is presented and Robertson-Walker metrics are introduced. Einstein’s equations are then used to derive the Friedmann equations, relating the cosmic scale factor to the pressure and density of matter in the Universe. The Hubble constant is discussed and an analytic form of the red-shift distance relation is derived, in terms of the matter density, the cosmological constant and the spatial curvature, and observational values of these three parameters are given. Some analytic solutions of the Friedmann equation are presented. The cosmic microwave background dominates the energy density in the early Universe and this leads to a description of the thermal history of the early Universe: the transition from matter dominated to radiation dominated dynamics and nucleosynthesis in the first 3 minutes. Finally the horizon problem and the inflationary Universe are described and the limits of applicability of Einstein's equations, when they might be expected to break down due to quantum effects, are discussed.

Einsteins field equations are derived and discussed. It is argued that the Einstein tensor is proportional to the energy-momentum tensor and the constant of proportionality is derived by demanding that Newton’s Universal Law of gravitation be recovered in the non-relativistic limit. The modification of Einstein's equations when a cosmological constant is introduced is also presented.

Here we review dark energy, the component that causes accelerated expansion of the universe. We start by reviewing the history of this fascinating discovery, describing in detail how type Ia supernovae were used to measure the expansion rate and find that the expansion is speeding up. We then outline modern evidence for the existence of dark energy, how dark energy is parametrically described, and what its phenomenological properties are. We review the cosmological-constant problem that encapsulates the tiny size of dark energy relative to expectations from particle physics. Next we introduce physical candidates for dark energy, including scalar fields and modified gravity. We end by explaining the controversial anthropic principle, and describe the possible future expansion histories of the universe dominated by dark energy.

Conservation laws and the energy–momentum–stress pseudotensor; the cosmological principle and the structure of the universe at large, the Robertson–Walker metric and the Friedman universe(s), Hubble’s law, the expansion of the universe, and the cosmological constant.

The Leaning Tower of Pisa, used by Galileo to demonstrate the simplicity of science, is also a testament to the complexity of science. Over an 800-year period, multiple attempts were made to fix the errors in the tower’s construction that caused it to lean. Often, the fixes had unanticipated consequences, necessitating additional compensating fixes. Climate models face a similar problem. The models use approximate formulas called parameterizations, with adjustable parameters, to represent processes like clouds that are too fine to be resolved by the model grids. The optimal values of these parameters that minimize simulation errors are determined by a trial-and-error process known as “model tuning.” Tuning minimizes errors in simulating current and past climates, but it cannot guarantee that the predictions of the future will be free of errors. This means that models can be confirmed, but they cannot be proven to be correct.

The observations described in Chapter 17 show our universe to be approximately homogeneous and isotropic on spatial distance scales above several hundred megaparsecs. The simplest cosmological models enforce these symmetries exactly as a first approximation. For instance, the matter in galaxies and the radiation are approximated by smooth density distributions that are exactly uniform in space. Similarly, the geometry of spacetime incorporates the homogeneity and isotropy of space exactly. These simplifying assumptions define the Friedman–Robertson–Walker (FRW) family of cosmological models, which are the subject of this chapter.

To test which of these models applies to our universe, one needs to extend redshift measurements to large distances, out to several Giga-light years. The most successful approach has been to use white-dwarf supernovae (SN type Ia) as very luminous standard candles. One of the greatest surprises of modern astronomy is that the expansion of the universe must be accelerating! This implies there must be a positive, repulsive force that pushes galaxies apart, in opposition to gravity. We dub this force “dark energy.”

This chapter explores what is known as the Cosmic Microwave Background (CMB), what it is, how it was discovered and our recent efforts to measure and map it. In general, the analysis finds remarkably good overall agreement with predictions of the now-standard “lambda CDM” model of a universe, in which there is both cold dark matter (CDM) to spur structure formation, as well as dark-energy acceleration that is well-represented by a cosmological constant, lambda. From this we can infer 13.8 Gyr for the age of the universe.

According to many physicists, several aspects of the laws of nature, the constants, and the cosmic boundary conditions are fine-tuned for life: had they been slightly different, life would not have existed. Here I review the claimed instances of fine-tuning and some of the criticism that has been levelled against the fine-tuning considerations. I also discuss in which sense, if any, fine-tuned parameters may qualify as improbable. Finally, I review the naturalness criterion of theory choice and discuss how violations of naturalness may be regarded as relevant to the discussion about fine-tuning for life.

This chapter turns to the prospects for empirically testing specific cosmological multiverse theories such as the landscape multiverse scenario or cyclic multiverse models. The most commonly pursued strategy to extract concrete empirical consequences from specific multiverse theories is to regard them as predicting what typical multiverse inhabitants observe if the theories are correct, where "“typical” is spelled out as “randomly selected from some suitably chosen reference class.” I scrutinize a proposal by Srednicki and Hartle to treat the self-sampling assumption and the reference class to which it is applied as matters of empirical fact that are themselves amenable to empirical tests. Unfortunately, this proposal turns out to be incoherent. A much better idea, which coheres well with the intuitive motivation for the self-sampling assumption, is that we should make this assumption with respect to some reference class of observers precisely if our background information is consistent with us being any of those observers and neutral between them. I call this principle the “background information constraint” (BIC) and point out that it at least formally solves the problem of selecting the appropriate observer reference class.

The first chapter contains a räsumä of the cosmology treating the homogeneous and isotropic universe. The Friedmann equations are derived and the thermal history of the Universe is discussed in some detail. Special emphasis is laid on the process of recombination and the decoupling of photons from the cosmic uid. Nucleosynthesis and cosmic in ation are also discussed.

The standard model of cosmology called LCDM has its origins in the work of great scientists including Einstein, Friedmann, Slipher, Hubble, Lemaitre, and Gamow. Lemaitre’s 1930s “Cosmic Egg” or “Primeval Nucleus” was the basis for the Big Bang model. In its new variant called LCDM, “L” represents the cosmological constant Lambda and “CDM” represents Cold Dark Matter. These two components, L and CDM, account for 95 percent of the mass–energy content of the Universe. Edwin Hubble correctly showed in the 1930s that galaxies are distributed on the largest scales in a homogeneous and isotropic way, but on a more local scale of 300 million light-years Hubble failed to recognize significant inhomogeneities. Hubble and Humason validated the velocity–distance relation for galaxies and galaxy clusters demonstrating the expansion of the Universe. They did not call out how significant the velocity–distance relationship would become in our effort to determine the 3D structure in the galaxy distribution.

With the development of general relativity, Einstein realised that he had a theory which for the first time could be used to create fully self-consistent cosmological models. In 1917, he introduced the cosmological constant to create a static closed Universe.The standard world models were discovered by Friedman in 1922 and 1924 and rediscovered by Lemaitre a few years later. The expansion of the Universe was discovered by Hubble in 1929. A key discovery was that of the cosmic microwave background radiation by Penzias and Wilson in 1965. The resulting hot big bang scenario for the large-scale structure and evolution of the Universe became the preferred cosmological model. With the development of precision cosmology through precise measurements of the cosmic microwave background radiation, it was established that the cosmological constant has a finite value and that the Universe is geometrically flat.