Anatomy of family trees in cosmological correlators

The correlators of large-scale fluctuations belong to the most important observables in modern cosmology. Recently, there have been considerable efforts in analytically understanding the cosmological correlators and the related wavefunction coefficients, which we collectively call cosmological amplitudes. In this work, we provide a set of simple rules to directly write down analytical answers for arbitrary tree-level amplitudes of conformal scalars with time-dependent interactions in power-law FRW universe. With the recently proposed family-tree decomposition method, we identify an over-complete set of multivariate hypergeometric functions, called family trees, to which all tree-level conformal scalar amplitudes can be easily reduced. Our method yields series expansions and monodromies of family trees in various kinematic limits, together with a large number of functional identities. The family trees are in a sense generalizations of polylogarithms and do reduce to polylogarithmic expressions for the cubic coupling in inflationary limit. We further show that all family trees can be decomposed into linear chains by taking shuffle products of all subfamilies, with which we find simple connection between bulk time integrals and boundary energy integrals.

Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and “twist factors” that depend on the cosmological evolution. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict — by hand — the equations for all tree graphs. While the rules of this “kinematic flow” are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in tr ϕ 3 theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.

MultiHypExp: A Mathematica package for expanding multivariate hypergeometric functions in terms of multiple polylogarithms

Cosmological correlators encode invaluable information about the wave function of the primordial Universe. In this Letter we present a duality between correlators and wave function coefficients that is valid to all orders in the loop expansion and manifests itself as a Z_{4} symmetry. To demonstrate the power of the duality, we derive a correlator-to-correlator factorization formula for the parity-odd part of cosmological correlators that relates n-point observables to lower-point ones via a series of diagrammatic cuts. These relations serve as the first example of physically testable cutting rules as they involve observables defined for arbitrary physical kinematics. We further show how the duality allows us to translate the cosmological optical theorem for wave function coefficients into statements about cosmological correlators.

We present the Mathematica package MultiHypExp that allows for the expansion of multivariate hypergeometric functions (MHFs), especially those likely to appear as solutions of multi-loop, multi-scale Feynman integrals, in the dimensional regularization parameter. The series expansion of MHFs can be carried out around integer values of parameters to express the series coefficients in terms of multiple polylogarithms. The package uses a modified version of the algorithm prescribed in [1]. In the present work, we relate a given MHF to a Taylor series expandable MHF by a differential operator. The Taylor expansion of the latter MHF is found by first finding the associated partial differential equations (PDEs) from its series representation. We then bring the PDEs to the Pfaffian system and further to the canonical form, and solve them order by order in the expansion parameter using appropriate boundary conditions. The Taylor expansion so obtained and the differential operators are used to find the series expansion of the given MHF. We provide examples to demonstrate the algorithm and to describe the usage of the package, which can be found [here](https://github.com/souvik5151/MultiHypExp)

A multivariable hypergeometric function considered recently by Niukkanen and Srivastava (1984, 1985) provides an interesting unification of the generalised hypergeometric function pFq of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions of n variables, and also of many other hypergeometric series which arise naturally in various physical and quantum chemical applications. As pointed out by Srivastava, the multivariable hypergeometric function is an obvious special case of the generalised Lauricella function of n variables, which was first introduced and studied by Srivastava and Daoust (1969). By employing such connections of this multivariable hypergeometric function with the more general multiple hypergeometric functions several interesting and useful properties of this function have been studied, many of which have not been given by Niukkanen. The author derives a number of new reduction formulae for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting summation formulae for the multivariable hypergeometric function with x1= . . . =xn=1 and x1= . . . =xn=-1 are also presented.

A multivariable hypergeometric function which was studied recently by Niukkanen (1984) and Srivastava (1985), provides an interesting and useful unification of the generalised hypergeometric pFq function of one variable (with p numerator and q denominator parameters), Appell and Kampe de Feriet's hypergeometric functions of two variables, and Lauricella's hypergeometric functions of n variables, and also of many other classes of hypergeometric series which arise naturally in various physical and quantum chemical applications. Indeed, as already observed by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalised Lauricella hypergeometric function of n variables, which was first introduced and studied systematically by Srivastava and Daoust (1969). By employing such useful connections of this function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this multivariable hypergeometric function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a number of new Neumann expansions in series of Bessel functions for the multivariable hypergeometric function from substantially more general expansions involving, for example, multiple series with essentially arbitrary terms. Some interesting special cases of the Neumann expansions presented here are also indicated.

This paper concludes the study of recursion formulas of multivariable hypergeometric functions. Earlier in [V. Sahai and A. Verma, Recursion formulas for multivariable hypergeometric functions, Asian–Eur. J. Math. 8 (2015) 50, 1550082], the authors have given the recursion formulas for three variable Lauricella functions, Srivastava’s triple hypergeometric functions and [Formula: see text]-variable Lauricella functions. Further, in [V. Sahai and A. Verma, Recursion formulas for Recursion formulas for Srivastava’s general triple hypergeometric functions, Asian–Eur. J. Math. 9 (2016) 17, 1650063], we have obtained recursion formulas for Srivastava general triple hypergeometric function [Formula: see text]. We present here the recursion formulas for generalized Kampé de Fériet series and Srivastava and Daoust multivariable hypergeometric function. Certain particular cases leading to recursion formulas of certain generalized hypergeometric function of one variable, certain Horn series, Humbert’s confluent hypergeometric series and some confluent forms of Lauricella series in [Formula: see text]-variables are also presented.

The multivariable hypergeometric function $$F_{q_0 :q_1 ;...;q_n }^{P_0 :P_1 ;...;P_n } \left( {\begin{array}{*{20}c} {x_1 } \\ \vdots \\ {x_n } \\ \end{array} } \right),$$ considered recently by A. W. Niukkanen and H.M. Srivastava, is known to provide an interesting unification of the generalized hypergeometric functionp F q of one variable, Appell and Kampe de Feriet functions of two variables, and Lauricella functions ofn variables, as also of many other hypergeometric series which arise naturally in various physical, astrophysical, and quantum chemical applications. Indeed, as already pointed out by Srivastava, this multivariable hypergeometric function is an obvious special case of the generalized Lauricella function ofn variables, which was first introduced and studied by Srivastava and M. C. Daoust. By employing such fruitful connections of this multivariable hypergeometric function with much more general multiple hypergeometric functions studied in the literature rather systematically and widely, Srivastava presented several interesting and useful properties of this function, most of which did not appear in the work of Niukkanen. The object of this sequel to Srivastava's work is to derive a further reduction formula for the multivariable hypergeometric function from substantially more general identities involving multiple series with essentially arbitrary terms. Some interesting connections of the results considered here with those given in the literature, and some indication of their applicability, are also provided.

We consider the evolution of quantum fields during inflation, and show that the total-energy singularities appearing in the perturbative expansion of the late-time Wavefunction of the Universe are purely real when the external states are massless scalars and massless gravitons. Our proof relies on the tree-level approximation, Bunch-Davies initial conditions, and exact scale invariance (IR-convergence), but without any assumptions on invariance under de Sitter boosts. We consider all n-point functions and allow for the exchange of additional states of any mass and integer spin. Our proof makes use of a decomposition of the inflationary bulk-bulk propagator of massive spinning fields which preserves UV-convergence and ensures that the time-ordered contributions are purely real after we rotate to Euclidean time. We use this reality property to show that the maximally-connected parts of wavefunction coefficients, from which total-energy singularities originate, are purely real. In a theory where all states are in the complementary series, this reality extends to the full wavefunction coefficient. We then use our reality theorem to show that parity-odd correlators (correlators that are mirror asymmetric) are factorised and do not diverge when the total-energy is conserved. We pay special attention to the parity-odd four-point function (trispectrum) of inflationary curvature perturbations and use our reality/factorisation theorems to show that this observable is factorised into a product of cubic diagrams thereby enabling us to derive exact shapes. We present examples of couplings between the inflaton and massive spin-1 and spin-2 fields, with the parity-violation in the trispectrum driven by Chern-Simons corrections to the spinning field two-point function, or from parity-violating cubic interactions which we build within the Effective Field Theory of Inflation. In addition, we present a first-of-its-kind example of a parity-violating trispectrum, generated at tree-level, that arises in a purely scalar theory where the inflaton mixes linearly with an additional massive scalar field.

Primordial perturbations in our universe are believed to have a quantum origin, and can be described by the wavefunction of the universe (or equivalently, cosmological correlators). It follows that these observables must carry the imprint of the founding principle of quantum mechanics: unitary time evolution. Indeed, it was recently discovered that unitarity implies an infinite set of relations among tree-level wavefunction coefficients, dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a systematic set of “Cosmological Cutting Rules” which constrain wavefunction coefficients for any number of fields and to any loop order. These rules fix the discontinuity of an n-loop diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in terms of tree-level diagrams with fewer external fields. Our results apply with remarkable generality, namely for arbitrary interactions of fields of any mass and any spin with a Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application, we show how one-loop corrections in the Effective Field Theory of inflation are fixed by tree-level calculations and discuss related perturbative unitarity bounds. These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.

We define a new type of multivariable multiple hypergeometric functions in this paper, which is inspired by Exton's multiple hypergeometric functions given by in [13]. Then, for these functions, we obtain some certain type linear generating functions. After that, we derive a variety classes of multilinear and multilateral generating functions for a family of the multivariable multiple hypergeometric functions. In addition, by employing the Erkus-Srivastava polynomials (see [11]) and the fourth type multivariable Horn functions (see [13]), we have also provided some of its conclusions.

Genealogy as a Heuristic Device for Franciscan Order History in the Middle Ages and Early Modernity:Texts and Trees Marianne P. Ritsema van Eck (bio) This paper explores the significance of spiritual genealogy as a historiographical device in Franciscan representations of the order's past during the medieval and early modern period. Certain visual exponents of this heuristic – murals, engravings, and manuscript paintings of Franciscan family trees – have been the subject of increasing scholarly attention. I argue that these visual family trees are only one manifestation of a broader tendency to represent and analyse Franciscan order history in genealogical terms. Other manifestations include written historiography, as well as genealogical images other than trees. The versatility of these visual and verbal genealogical representations of the Franciscan past made them into an adaptable means for communicating a variety of messages, apart from emphasizing Franciscan community in a general sense. First, I discuss the main developments in visual representations of the Franciscan family tree during the late medieval period, in tandem with closely related written perspectives on Franciscan order history, so as to point out the perennial conversation between its textual and visual manifestations. By shifting away some of the attention from the visual tree-model in favour of seeing it as part of a larger tendency in textual culture to represent order history in genealogical and/or arboreal terms, it becomes clear that late medieval Franciscan genealogical representations offer a particular, eschatological perspective on order history, associated with Spiritual and Observant Franciscan contexts. Second, my examination of the same phenomena during the sixteenth and seventeenth centuries, when family tree visualisations became much more widespread, suggests that the genealogia emerged as a particular form for organising and presenting written order histories, current among all Franciscan orders. I outline the contours of this diversified sub-set of Franciscan order historiography that employed genealogy as a versatile heuristic, connecting Franciscan communities to a shared familial past, often elaborating links or occasionally even claims to certain local territories. [End Page 135] Overall, it shall become clear that textual and visual representations of the order's past often – but not necessarily – went hand in hand, and that genealogical perspectives on Franciscan order history were a deeply-seated heuristic device that exceeded the visual rhetoric of the tree diagram. Medieval genealogical representations of Franciscan order history The late medieval exponents of the Franciscan family tree serve as the starting point of my discussion of perspectives on Franciscan genealogy. This tree image is relatively well-known because it has recently attracted the attention of several art historians, who usually connect its iconography with Lignum Vitae representations (discussed below). In what follows I complement these existing perspectives by also considering coeval written accounts of the Franciscan family tree and again other (non-genealogical) Franciscan tree visualisations. As a result, it becomes evident that the late medieval Franciscan family tree iconography is indeed only one expression of a particular (not necessarily visual) perspective on Franciscan order history, articulated by ideologies of apocalyptic renewal associated with Spiritual and later Observant Franciscan contexts. My analysis of these late medieval Franciscan tree representations thus suggests that genealogy was the stuff of history. Visualisations of the biblical tree of Jesse – widespread from the twelfth century onward – are an important step in the development toward the 'family tree' representations of the later Middle Ages. The motif is based on a prophecy in Isaiah 11:1 "And there shall come forth a rod out of the root of Jesse, and a flower shall rise up out of his root."1 This was typically interpreted as foretelling the Incarnation and as referring to the genealogy of Christ.2 Jesse is often depicted in a reclining position, with a stem growing from his abdomen that branches out to hold Old Testament ancestors of Christ and a centrally positioned Virgin (fig. 1).3 [End Page 136] Click for larger view View full resolution Figure 1. Tree of Jesse, tympanum of the Cathédrale Notre-Dame de Rouen, sculpted by Pierre des Aubeaux, c. 1512–13 (later damaged by Huguenots). Photograph by the author. During the later Middle Ages, family tree representations of religious orders eventually emerged from a complex melting...

The Bergman Kernel of Complex Ovals and Multivariable Hypergeometric Functions

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type pΦs can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).