Zeta-function regularization is arguably the most elegant of the four major regularization methods used for quantum fields in curved spacetime, linked to the heat kernel and spectral theorems in mathematics. The only drawback is that it can only be applied to Riemannian spaces (also called Euclidean spaces), whose metrics have a ++++ signature, where the invariant operator is of the elliptic type, as opposed to the hyperbolic type in pseudo-Riemannian spaces (also called Lorentzian spaces) with a −+++ signature. Besides, the space needs to have sufficiently large symmetry that the spectrum of the invariant operator can be calculated explicitly in analytic form. In the first part we define the zeta function, showing how to calculate it in several representative spacetimes and how the zeta-function regularization scheme works. We relate it to the heat kernel and derive the effective Lagrangian from it via the Schwinger proper time formalism. In the second part we show how to obtain the correlation function of the stress-energy bitensor, also known as the noise kernel, from the second metric variation of the effective action. Noise kernel plays a central role in stochastic gravity as much as the expectation values of stress-energy tensor do for semiclassical gravity.

In this chapter we derive the full two-point quantum metric perturbations on a de Sitter background including one-loop corrections from conformal fields. We do the calculation using the CTP effective action with the 1/N expansion, and select an asymptotic initial state by a suitable prescription that defines the vacuum of the interacting theory. The decomposition of the metric perturbations into scalar, vector and tensor perturbations is reviewed, and the effective action is given in terms of that decomposition. We first compute the two-point function of the tensor perturbations, which are dynamical degrees of freedom. The relation with the intrinsic and induced fluctuations of stochastic gravity is discussed. We then compute the two-point metric perturbations for the scalar and vector modes, which are constrained degrees of freedom. The result for the full two-point metric perturbations is invariant under spatial rotations and translations as well as under a simultaneous rescaling of the spatial and conformal time coordinates. Finally, our results are extended to general conformal field theories, even strongly interacting ones, by deriving the effective action for a general conformal field theory.

In this chapter we construct the closed-time-path (CTP) two-particle-irreducible (2PI) effective action to two-loop order. The CTP formalism introduced in Chapter 3 is needed to track the dynamics of expectation values and to produce real and causal equations of motion. The composite particle or 2PI formalism introduced in Chapter 6 is needed to treat critical phenomena, because the correlation function and the mean field act as independent variables, instead of the former being a derivative of the latter, as in the 1PI formulation. The large N expansion has the advantage of yielding nonperturbative evolution equations in the regime of strong mean field and a covariantly conserved stress-energy tensor. To leading order in large N, the quantum effective action for the matter fields can be interpreted as a leading-order term in the expansion of the full matter plus gravity quantum effective action, which produces equations of motion for semiclassical gravity and, at the next-to-leading order in large N, stochastic gravity. Two types of quantum fields are treated: (a) O(N) self-interacting Phi-4 fields, and (b) Yukawa coupling between scalar and spinor fields, as an example of dealing with fermions in curved spacetime.

Distant points of light – quasars – show us the intervening universe in silhouette. The result is a map of the gas in and between galaxies, expanding with space.

Distant points of light – quasars – show us the intervening universe in silhouette. The result is a map of the gas in and between galaxies, expanding with space.

The universe is smooth on the largest scales, with roughly the same number of galaxies in every large cosmic neighbourhood. But the standard history of the universe won't allow any process to smooth out an initially smooth universe. An addition to the standard model, called cosmic inflation, aims to fill this void.

At night, the sky is dark. This familiar fact has some remarkable implications for the history and structure of the universe.

The big bang theory is not perfect. We lay out the most serious shortcomings, and the observations in the near future that will continue to test it.

Uniform light at just a few degrees above absolute zero bathes the universe, pointing to a time in the past when the universe was hotter and denser.

The farther we look, the redder the light from galaxies appears. This fact points to a remarkable feature of our universe: it is not static. It is expanding.

We explain science, both the idealised version to which scientists aspire, and the real version that involves actual human beings. If you are a cosmic revolutionary, who wants to replace the prevailing big bang theory with their own ideas, we explain the importance of mathematical models, publishing, peer review and presentation of your ideas. In particular, we show how to make scientist's human motivations work in your favour.