To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

Gravitational waves (GWs) have a great potential to probe cosmology. We review early universe sources that can lead to cosmological backgrounds of GWs. We begin by presenting proper definitions of GWs in flat space-time and in a cosmological setting (section 2). Following, we discuss the reasons why early universe GW backgrounds are of a stochastic nature, and describe the general properties of a stochastic background (section 3). We recap current observational constraints on stochastic backgrounds, and discuss the basic characteristics of present and future GW detectors, including advanced LIGO, advanced Virgo, the Einstein telescope, KAGRA, and LISA (section 4). We then review in detail early universe GW generation mechanisms, as well as the properties of the GW backgrounds they give rise to. We classify the backgrounds in five categories: GWs from quantum vacuum fluctuations during standard slow-roll inflation (section 5), GWs from processes that operate within extensions of the standard inflationary paradigm (section 6), GWs from post-inflationary preheating and related non-perturbative phenomena (section 7), GWs from first order phase transitions related or not to the electroweak symmetry breaking (section 8), and GWs from general topological defects, and from cosmic strings in particular (section 9). The phenomenology of these early universe processes is extremely rich, and some of the GW backgrounds they generate can be within the reach of near-future GW detectors. A future detection of any of these backgrounds will provide crucial information on the underlying high energy theory describing the early universe, probing energy scales well beyond the reach of particle accelerators.

http://iopscience.iop.org/article/10.1088/1361-6382/aac608/pdf

Cosmological Backgrounds of Gravitational Waves

The Inflationary Universe

We investigate the potential for the LISA space-based interferometer to detect the stochastic gravitational wave background produced from different mechanisms during inflation. Focusing on well-motivated scenarios, we study the resulting contributions from particle production during inflation, inflationary spectator fields with varying speed of sound, effective field theories of inflation with specific patterns of symmetry breaking and models leading to the formation of primordial black holes. The projected sensitivities of LISA are used in a model-independent way for various detector designs and configurations. We demonstrate that LISA is able to probe these well-motivated inflationary scenarios beyond the irreducible vacuum tensor modes expected from any inflationary background.

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Science with the space-based interferometer LISA. IV: probing inflation with gravitational waves

Soft limits of inflationary correlation functions are both observationally relevant and theoretically robust. Various theorems can be proven about them that are insensitive to detailed model-building assumptions. In this paper, we re-derive several of these theorems in a universal way. Our method makes manifest why soft limits are such an interesting probe of the spectrum of additional light fields during inflation. We illustrate these abstract results with a detailed case study of the soft limits of quasi-single-field inflation.

https://iopscience.iop.org/article/10.1088/1475-7516/2012/11/047/pdf

On soft limits of inflationary correlation functions

Single-field slow-roll inflation with a nonvacuum initial state has an enhanced bispectrum in the local limit. We numerically calculate the local-type ${f}_{\mathrm{NL}}$ signal in the cosmic microwave background (CMB) that would be measured for such models (including the full transfer function and 2D projection). The nature of the result depends on several parameters, including the occupation number ${N}_{k}$, the phase angle ${\ensuremath{\theta}}_{k}$ between the Bogoliubov parameters, and the slow-roll parameter $ϵ$. In the most conservative case, where one takes ${\ensuremath{\theta}}_{k}\ensuremath{\approx}{\ensuremath{\eta}}_{0}k$ (justified by physical reasons discussed within) and $ϵ\ensuremath{\lesssim}0.01$, we find that $0l{f}_{\mathrm{NL}}l1.52(ϵ/0.01)$, which is likely too small to be detected in the CMB. However, if one is willing to allow a constant value for the phase angle ${\ensuremath{\theta}}_{k}$ and ${N}_{k}=\mathcal{O}(1)$, ${f}_{\mathrm{NL}}$ can be much larger and/or negative (depending on the choice of ${\ensuremath{\theta}}_{k}$), e.g. ${f}_{\mathrm{NL}}\ensuremath{\approx}28(ϵ/0.01)$ or $\ensuremath{-}6.4(ϵ/0.01)$; depending on $ϵ$, these scenarios could be detected by Planck or a future satellite. While we show that these results are not actually a violation of the single-field consistency relation, they do produce a value for ${f}_{\mathrm{NL}}$ that is considerably larger than that usually predicted from single-field inflation.

Calculating the local-type f NL for slow-roll inflation with a nonvacuum initial state

We investigate the phenomenological consequences of a modification of the initial state of quantum fluctuations of a single inflationary field. While single-field inflation with the standard Bunch-Davies initial vacuum state does not generally produce a measurable three-point function (bispectrum) in the so-called squeezed triangle configuration (where one wave number, $k$, is much smaller than the other two, $k\ensuremath{\ll}{k}_{1}\ensuremath{\approx}{k}_{2}$), allowing for a nonstandard initial state produces an exception. Here, we calculate the signature of an initial state modification in single-field slow-roll inflation as it would appear in both the scale-dependent bias of the large-scale structure and $\ensuremath{\mu}$-type distortion in the black-body spectrum of the cosmic microwave background (CMB). We parametrize the initial state modifications and identify certain choices of parameters as natural, though we also note some fine-tuned choices that can yield a larger bispectrum. In both cases, we observe a distinctive ${k}^{\ensuremath{-}3}$ signature in large-scale structure (as opposed to the ${k}^{\ensuremath{-}2}$ of the so-called local-form bispectrum). As a nonzero bispectrum in the squeezed configuration correlates one long-wavelength mode with two short-wavelength modes, it induces a correlation between the CMB temperature anisotropy observed on large scales with the temperature-anisotropy-squared on very small scales; this correlation persists as the small-scale anisotropy-squared is processed into the $\ensuremath{\mu}$-type distortion of the black-body spectrum. While the correlation induced by the local-form bispectrum turns out to be too small to detect in near future, a modified initial vacuum state enhances the signal by a large factor owing to an extra factor of ${k}_{1}/k$ compared to the local form. For example, a proposed absolutely-calibrated experiment, PIXIE, is expected to detect this correlation with a signal-to-noise ratio greater than 10, for an occupation number of about 0.5 in the observable modes. Relatively-calibrated experiments such as Planck and LiteBIRD should also be able to measure this effect, provided that the relative calibration between different frequencies meets the required precision. Our study suggests that the CMB anisotropy, the distortion of the CMB black-body spectrum, and the large-scale structure of the Universe offer new ways to probe the initial state of quantum fluctuations.

Scale-dependent bias of galaxies and μ-type distortion of the cosmic microwave background spectrum from single-field inflation with a modified initial state

Axions play a central role in many realizations of large field models of inflation and in recent alternative mechanisms for generating primordial tensor modes in small field models. If these axions couple to gauge fields, the coupling produces a tachyonic instability that leads to an exponential enhancement of the gauge fields, which in turn can decay into observable scalar or tensor curvature perturbations. Thus, a fully self-consistent treatment of axions during inflation is important, and in this work we discuss the perturbative constraints on axions coupled to gauge fields. We show how the recent proposal of generating tensor modes through these alternative mechanisms is in tension with perturbation theory in the in-in formalism. Interestingly, we point out that the constraints are parametrically weaker than one would estimate based on naive power counting of propagators of the gauge field. In the case of non-Abelian gauge fields, we derive new constraints on the size of the gauge coupling, which apply also in certain models of natural large field inflation, such as alignment mechanisms.

On the validity of the perturbative description of axions during inflation

In 2004, Creminelli and Zaldarriaga proposed a consistency relation for the primordial curvature perturbation of all single-field inflation models; it related the bispectrum in the squeezed limit to the spectral tilt. We have developed a technique, based in part on the Creminelli and Zaldarriaga argument, that can greatly simplify the calculation of the squeezed-limit bispectrum using the in-in formalism; we were able to arrive at a generic formula that does not rely on a slow-roll approximation. Using our formula, we explicitly tested the consistency relation for power-law inflation and for an exactly scale-invariant model by Starobinsky; for the latter model, Creminelli and Zaldarriaga's argument predicts a vanishing bispectrum whereas our quantum calculation shows a non-zero bispectrum that approaches zero in the long-wavelength limit and for inflation with a large number of e-folds.

Jonathan Ganc

Papers

Calculating the local-type f NL for slow-roll inflation with a nonvacuum initial state

Scale-dependent bias of galaxies and μ-type distortion of the cosmic microwave background spectrum from single-field inflation with a modified initial state

On the validity of the perturbative description of axions during inflation

A new method for calculating the primordial bispectrum in the squeezed limit

On the validity of the perturbative description of axions during inflation