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Inflation and string theory, by Daniel Baumann and Liam McAllister

discovered (by chance) by Arno Penzias and Robert Wilson in 1964. Their signal was isotropic with a spectrum corresponding to thermal blackbody radiation at a temperature of 3.5 ± 1.0◦K. Nowadays, data from the Planck satellite, launched in 2009, give the temperature known much more precisely: 2.72548 ± 0.00057◦K; we also know that the radiation is not quite isotropic, with temperature fluctuations at the 10 level. Nevertheless, the very high degree of isotropy is a problem. The radiation originates from the so-called recombination era when the universe became transparent to photons. In the standard cosmology (without inflation), causal signals travel a finite distance between the initial singularity and the recombination era. However, the observed CMB anisotropies are correlated over much larger scales than this. The now widely accepted solution to this problem is that the universe went through a period of ‘inflation’, during which the scale factor a(t) in the Friedmann–Robertson– Walker metric evolved exponentially a(t) ∝ e with the Hubble ‘constant’ H ≡ ȧ∕a nearly constant. Two intertwined approaches to the theoretical description of inflation are described in this book. In an effective theory, an economical or natural description of the low-energy (long-distance) phenomena is sought without reference to the high-energy (short-distance) behaviour that characterises a more fundamental theory; for example, a useful description of fluid dynamics and thermodynamics is perfectly possible without reference to the underlying atomic structure, so long as one is only concerned with macroscopic distance scales. So one might hope for an effective field theory (EFT) of inflation that would generally not include non-renormalisable interactions suppressed by the Planck scale m P if one were interested only in energy scales E ∼ H ≪ m P . However, it turns out that EFTs that support inflation are sensitive to such Planck-suppressed terms. Indeed, in all models of inflation, the duration of the inflationary era is affected by at least some non-renormalisable terms, and in some, the so-called largefield models, it is affected by an infinite series of such terms. This sensitivity to Planck-scale physics therefore necessitates addressing such questions in the context of a quantum theory of gravity. Daniel Baumann and Liam McAllister (naturally) favour string theory for this rôle. After all, ‘Consistent theories of quantum gravity do not grow on trees’, as they observe. At the very least a string theory treatment will permit controlled computation of the quantum gravity effects; such computations can clarify and justify symmetry assumptions made in the EFT approach. Embedding the EFT in quantum gravity can also lead to additional observational signatures, typically small corrections. String theory has also inspired novel EFTs that might otherwise not have been proposed; specifically, the authors cite D-brane inflation, DBI inflation, fibre inflation and axion monodromy inflation. Further, what is considered ‘natural’ in string theory is not obviously natural in an EFT; for example, string theory abounds with ‘moduli’, massless scalar fields associated, for example, with the size (Kähler moduli) and shape (complex structure moduli) of the compactified space, whereas such a proliferation might not appear natural starting from an EFT perspective. It’s even glass was able to perform, in an analogue fashion, a complex mathematical transformation. The standard textbook for Fourier optics is generally considered to be Goodman’s ‘Introduction to Fourier Optics’, a slim volume that has largely remained relevant in spite of its age. Even so, during the almost 30 years since the publication of Goodman’s reissued book, it would be an understatement to claim that quite a bit of technological progress has occurred, providing an opportunity for a new textbook incorporating some of the latest progress–namely, the omnipresence of digital imaging systems. The book is roughly divided into three sections, the first covering mathematical preliminaries, the second is concerned with optical physics, and the final section devoted to computational imaging systems. While fast paced, the section thoroughly covers all of the ‘standard’ main results and adds many new ones. For example, the middle section covers scalar diffraction, angular spectra and coherence but also adds eigenmodes and eigenfunctions of imaging systems. Similarly, the mathematical development section covers sampling, linear systems analysis, and random processes but also adds deconvolution/denoising and use of the fast Fourier transform. The third section is nearly all ‘new’, discussing topics as varied as computational cameras, structural illumination and ghost imaging. Professor Khare’s text grew from his lecture series, and is generally suitable as a textbook geared towards advanced undergraduates or graduate students. The text should probably be supplemented with supporting texts, for example, Gaskill’s ‘Linear systems, Fourier Transforms, and Optics’ for the mathematical foundations or Vollmerhausen and Driggers’ ‘Analysis of Sampled Imaging Systems’ to cover sampling issues in-depth. Additionally, while there are a set of ‘suggested problems’ at the end of Khare’s text that are intended as homework, the instructor may need to provide additional problems. In summary, Professor Khare’s book could very well be the textbook needed to finally replace Goodman’s classic.