Seminar

764 events

Seminar RIKEN Quantum Seminar
Methods for neural decoding using machine learning, deep learning, and quantum-inspired algorithms

January 17 (Wed) at 15:00 - 16:15, 2024

Kei Majima (Researcher, National Institutes for Quantum Science and Technology (QST))

Note: The format of this event has changed from hybrid to Zoom only. However, you will still be able to watch it on the screen in Room #359 of the Main Research Building. (This is a joint seminar with iTHEMS Biology group.) Recent advances in machine learning have enabled the extraction of intrinsic information from neural activities, a field known as neural decoding. In this presentation, I will introduce several machine learning methods recently developed for neural decoding analysis: 1) a method for visualizing subjective images in the human mind based on brain activity [1], 2) a supervised algorithm designed for predicting discrete ordinal variables [2], and 3) a fast classical algorithm algorithm inspired by quantum computation for approximating principal component analysis (PCA) and canonical correlation analysis (CCA), potentially allowing for the analysis of vast-dimensional neural data [3]. Following these presentations, I am eager to engage in discussions with participants at the RIKEN Quantum Seminar regarding potential collaborations.

Venue: via Zoom

Event Official Language: English
Seminar iTHEMS Seminar
Dust-driven instabilities in protoplanetary disks: toward understanding formation of planetesimals

January 17 (Wed) at 10:30 - 11:30, 2024

Ryosuke Tominaga (Special Postdoctoral Researcher, Star and Planet Formation Laboratory, RIKEN Cluster for Pioneering Research (CPR))

Planet formation starts from collisional growth of sub-micron-sized dust grains in a gas disk called a protoplanetary disk. They are expected to grow toward km-sized objects called planetesimals. The resulting planetesimals further coalesce by gravity and form planets. However, there are some barriers preventing planetesimal formation, which includes fast radial drift and collisional fragmentation of dust grains. To circumvent the barriers and to explain planetesimal formation, previous studies have proposed hydrodynamic instabilities of dusty-gas disks. The instabilities can cause dust clumping, and planetesimals form if the resulting clumps collapse self-gravitationally. We have been investigating the linear/nonlinear development of these dust-gas instabilities. We also found a new instability driven by collisional growth of dust, which can bridge a potential gap between the first dust growth and the later planetesimal formation via the previous instabilities. In this talk, I will introduce our work on the dust-driven instabilities and their impact on planetesimal formation.

Venue: Hybrid Format (3F #359 and Zoom), Main Research Building

Event Official Language: English
Seminar iTHEMS Biology Seminar
Does horizontal gene transfer stabilize cooperation in bacteria?

January 16 (Tue) at 16:00 - 17:00, 2024

Anna Dewar (Postdoctoral Researcher, Department of Biology, University of Oxford, UK)

Bacteria are highly social. Much of this sociality occurs through the production of cooperative ‘public goods’. Unlike in animals, bacterial genes are able to transfer horizontally between individuals, in addition to vertically via descendants. This widespread horizontal gene transfer has implications for the concept of relatedness and how cooperation is maintained in bacteria. It has been suggested that horizontal gene transfer, particularly via small segments of DNA called plasmids, could stabilize cooperation in bacteria. Transfer of a cooperative gene could turn non-cooperative ‘cheats’ into cooperators, preventing cheats from invading and destabilizing cooperation. We tested this with a comparative analysis across bacterial species. In contrast to the predictions of the hypothesis, we found that genes for cooperative traits were not more likely to be carried on either: (1) plasmids compared to chromosomes; or (2) plasmids that transfer at higher rates. Our results were supported by theoretical modelling which showed that, while horizontal gene transfer can help cooperative genes initially invade a population, it has less influence on the longer-term maintenance of cooperation.

Venue: via Zoom

Event Official Language: English
Seminar ABBL-iTHEMS Joint Astro Seminar
Probing structure of neutron stars through X-ray bursters

January 12 (Fri) at 14:00 - 15:15, 2024

Akira Dohi (Special Postdoctoral Researcher, Astrophysical Big Bang Laboratory, RIKEN Cluster for Pioneering Research (CPR))

Type-I X-ray bursts are rapidly brightening phenomena triggered by the nuclear burning of light elements near the surface of accreting neutron stars. Most of the X-ray bursters show irregular behavior of light curves. However, some X-ray bursters are somehow quite regular, i.e., constant recurrence time and constant shaper of light curves, and are often called Clocked bursters, which are powerful sites to probe uncertainties of many model parameters such as accretion rate, the composition of accreted matter, reaction rates, neutron star structure, and temperature. In this study, we focus on the uncertainties of the equation of states, which determines the latter two properties. Based on our numerical models covering whole areas of neutron stars, we will present their impact on X-ray burst light curves. Furthermore, we will discuss the possibility of constraining the equation of states from Clocked bursters such as GS 1826-24 and 1RXS J180408.9-342058.

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Symmetry Topological field theory for Subsystem symmetry

January 9 (Tue) at 15:00 - 16:00, 2024

Qiang Jia (Research Fellow, School of Physics, Korea Institute for Advanced Study (KIAS), Republic of Korea)

We generalize the idea of symmetry topological field theory (SymTFT) to subsystem symmetry. We propose the 2-foliated BF theory with level N in (3+1)d as subsystem SymTFT for subsystem Z_N symmetry in (2+1)d. Focusing on N=2, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem SL(2,Z_2) symmetry of the bulk foliated BF theory.

Venue: via Zoom / Seminar Room #359

Event Official Language: English
Seminar Math-Phys Seminar
Application of Modular tensor category to Lattice gauge theory

December 29 (Fri) at 10:30 - 16:00, 2023

Tomoya Hayata (Assistant Professor, Faculty of Economics, Keio University)

Inspired by the recent development in quantum computers, much efforts have been devoted to exploring their potential applications in lattice gauge theories. However, in contrast to condensed matter systems, we face many challenges in applications of quantum computations to lattice gauge theories, where one of the major obstructions lies in implementation of gauge symmetries in quantum computations. In this seminar, I talk about a possible solution to the problem based on a unitary modular tensor category, expressing the Hamiltonian of lattice gauge theories in terms of the so called F moves, and implementing the F moves on quantum computers. References: TH, Y. Hidaka, JHEP 09 (2023) 126; JHEP 09 (2023) 123.

Venue: Seminar Room #359

Event Official Language: English
Seminar iTHEMS Biology Seminar
Oscillatory data analysis using the extended Hilbert transform method

December 26 (Tue) at 16:00 - 17:00, 2023

Akari Matsuki (Postdoctoral Researcher, Department of Advanced Transdisciplinary Sciences, Hokkaido University)

Oscillatory phenomena are observed in various biological systems, such as spinal nervous systems and circadian rhythms. These macroscopic oscillatory phenomena appear as a result of synchronization of microscopic oscillators, such as pacemaker cells. The first step in the analysis of synchronization is to reconstruct the "phase" from the observed signal. The Hilbert transform method is one of the popular methods for phase reconstruction, but it is known that it can only accurately reconstruct the phase from a limited class of signals such as narrowband signals. In this study, we show that the Hilbert transform method has a low-pass filter-like effect on the phase modulation and propose an "extended Hilbert transform method" that can be applied to a wider class of signals. In this talk, I will introduce the extended Hilbert transform method, and its application to phase shift detection and coupling network inference.

Venue: via Zoom

Event Official Language: English
Seminar Quantum Matter Seminar
A symmetry principle for gauge theories with fractons

December 22 (Fri) at 17:00 - 18:15, 2023

Yuji Hirono (Program-Specific Associate Professor, Department of Physics, Division of Physics and Astronomy, Graduate School of Science, Kyoto University)

Fractonic phases are emergent quantum phases of matter that host excitations with restricted mobility. Although these phases have been considered to be of “beyond Landau” order, we show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of generalized symmetries. The corresponding symmetries are continuous higher-form symmetries whose conserved charges do not commute with spatial translations, and we refer to them as nonuniform higher-form symmetries. For a given set of nonuniform symmetries, the effective theory associated with the spontaneous breaking of them can be constructed. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu–Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. In this formulation, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions. Field: condensed matter physics Keywords: fractonic phases, higher-form symmetries, Nambu-Goldstone modes, Higgs mechanism, gauge theories

Venue: via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Inflationary Cosmology with a scalar-curvature mixing term $\frac{1}{2} \xi R \phi^2$

December 20 (Wed) at 16:00 - 17:30, 2023

Payel Sarkar (Visiting Researcher, Kyoto University)

We use the PLANCK 2018 and the WMAP data to constraint inflation models driven by a scalar field $\phi$ in the presence of the non-minimal scalar-curvature mixing term $\frac{1}{2}\xi R \phi^2$ . We consider four distinct scalar field potentials $\phi^p e^{-\lambda\phi},~(1 - \phi^{p})e^{-\lambda\phi},~(1-\lambda\phi)^p$ and $\frac{\alpha\phi^2}{1+\alpha\phi^2}$ to study inflation in the non-minimal gravity theory. We calculate the potential slow-roll parameters, predict the scalar spectral index $n_s$ , tensor-to-scalar ratio $r$ , leading and higher order non-Gaussianity parameters ( $f_{NL},~\tau_{NL}$ and $g_{NL}$ ) and the amplitude of the scalar spectrum $A_s$ in the parameter ( $\lambda, p, \alpha$ ) space of the potentials corresponding to different values of the non-minimal coupling parameter $\xi$ . We have compared our results with the ones existing in the literature, and this indicates the present status of non-minimal inflation after the release of the PLANCK 2018 data.

Venue: Hybrid Format (3F #359 and Zoom), Main Research Building (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Biology Seminar
Application of mathematical models to the COVID-19 cohort study

December 19 (Tue) at 16:00 - 17:00, 2023

Takara Nishiyama (Ph.D. Student, Graduate School of Science, Nagoya University)

The COVID-19 pandemic, which began in 2019, has caused widespread morbidity and mortality across the globe. In response, a multitude of studies focusing on SARS-CoV-2 have been undertaken. Among these, cohort studies have been particularly significant. These studies, as a key observational research method, play a crucial role in exploring the links between various factors and the onset of diseases, offering valuable insights for disease control. Mathematical model, applied within these studies, provide essential quantitative data. In my talk, I will introduce how mathematical models are instrumental in cohort studies, drawing on two of my own COVID-19 cohort studies as examples.

Venue: Hybrid Format (3F #359 and Zoom), Main Research Building

Event Official Language: English
Seminar Quantum Computation SG Seminar
Energy spectrum and time evolution of a SU(2) pure gauge lattice theory on a quantum annealer

December 18 (Mon) at 14:00 - 15:00, 2023

Emanuele Mendicelli (Postdoctoral Research Associate, Department of Mathematical Sciences, University of Liverpool, UK)

Lattice gauge theory is an indispensable tool for non-Abelian fields, such as those in quantum chromodynamics where lattice results have been of central importance for several decades. Recent studies suggest that quantum hardware could extend the reach of lattice gauge theory to inaccessible phenomena due to the need for an exponentially large amount of resources, the so-called sign problem. Among the available quantum hardware gate-based quantum computer are well know but less common quantum annealer can play a role too. In this talk we briefly report one of the first use of D-Wave quantum annealer to study the energy spectrum and the time evolution of a SU(2) pure gauge lattice theory in its Hamiltonian formulation. In particular we present how to extract the energy spectrum using the quantum Quantum Annealer Eigensolver algorithm and perform the time evolution using the Kitaev-Feynman clock states. Despite the nosy hardware, no error mitigation techniques were needed but the usability of the D-Wave hardware was extended by simply block-diagonalizing the Hamiltonian.

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Biology Seminar
Plasticity in the endogenous rhythms and the adaptation to the tidal environment in a freshwater snail

December 14 (Thu) at 16:00 - 17:00, 2023

Takumi Yokomizo (JSPS Research Fellow PD, Graduate School of Science, Chiba University)

Organisms have diverse biological clocks synchronized with environmental cycles　depending on their habitats. The change in endogenous rhythms could contribute to range expansion in a novel rhythmic environment. For example, the Anticipation of tidal changes has driven the evolution of circatidal rhythms in some marine species. I am interested in the genetic and non-genetic changes in the biological rhythms and adaptation to tidal environments in the freshwater snail, Semisulcospira reiniana. Chronobiological analyses of behavior and gene expression revealed that snails had habitat-specific endogenous rhythms: individuals in a nontidal population showed the circadian rhythm while those in a tidal population showed the circadian and circatidal rhythms. The entrainment to the simulated tidal cycles increased the strength of circatidal activity only in snails in a tidal population. Although the circatidal rhythms in the transcriptome were clearer in individuals entrained to tidal cycles, the number of circatidal rhythmic transcripts was greater in a tidal population than in a nontidal population. These results suggest biological rhythms in the snails plastically change at the molecular level, but the strength of circatidal rhythm is different between populations. Finally, transcriptome-wide population genetic analysis revealed that these two populations can be clearly distinguished genetically, though the genetic distance was very small. Thus, genetic differentiation in biological rhythms could result from the evolution of a small number of genes. These findings suggest that adaptive plasticity and genetic changes in the biological rhythms play an important role in coping with tidal environments.

Venue: via Zoom

Event Official Language: English
Seminar iTHEMS Math Seminar
Tropical geometry and period integrals

December 13 (Wed) at 14:00 - 16:30, 2023

Yuto Yamamoto (Special Postdoctoral Researcher, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))

Tropical geometry is a field of mathematics that naturally emerges when considering the limits of spaces with respect to some parameters. One of the motivations to study tropical geometry is to describe the behaviors of the spaces under the limit. In this math seminar, starting with a brief introduction to tropical geometry, we discuss its application to computation of period integrals, which are one of the most fundamental quantities of complex manifolds. The goal is to compute asymtptotics of period integrals for complex hypersurfaces in toric varieties using tropical geometry, and observe that the Riemann zeta values (or the gamma classes) appear in the result of the computation. The first half of the talk will be a brief introduction to tropical geometry for non-experts including those who are working outside mathematics, and everyone will be welcome.

Venue: Hybrid Format (3F #359 and Zoom), Main Research Building

Event Official Language: English
Seminar ABBL-iTHEMS Joint Astro Seminar
Exploring material strengths of dust aggregates in planet formation by numerical simulations

December 8 (Fri) at 14:00 - 15:15, 2023

Misako Tatsuuma (Research Scientist, RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS))

The planet formation process is the growth from sub-micrometer-sized cosmic dust grains to thousand-kilometer-sized planets. This growth process has broadly two phases: the growth from dust grains to kilometer-sized planetesimals, mainly driven by intermolecular forces like van der Waals forces and hydrogen bonds, and the subsequent growth from planetesimals to planets, governed by gravitational forces. However, the planetesimal formation process encounters various challenges, including fragmentation and bouncing resulting from collisions among dust aggregates. To gain insights into the planetesimal formation process and how to avoid these obstacles, I have been focused on measuring and formulating the material strengths of dust aggregates using grain simulations. In this talk, I will introduce my works on the material strengths of dust aggregates and their applications to kilometer-sized bodies in the solar system, such as comets and asteroids.

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Gravitational Lensing in Black Hole Spacetimes of the Plebanski-Demianski Class

December 6 (Wed) at 16:00 - 17:30, 2023

Torben Christian Frost (Postdoctoral Researcher, Kavli Institute for Astronomy and Astrophysics, Peking University, China)

Einstein's field equations allow various different black hole solutions. Among these solutions, the most famous are most likely the Schwarzschild and the Kerr spacetimes, which are both special cases of the so-called Plebanski-Demianski spacetime. Besides the Schwarzschild and Kerr spacetimes, the Plebanski-Demianski spacetime also includes other solutions as special cases, among them the C-metric and the NUT metric. They describe a linearly accelerating black hole and a black hole with gravitomagnetic charge, respectively. The question is now how we can determine if an astrophysical black hole can be described by one of these spacetimes. We will address this question using gravitational lensing for the three spacetimes with the most salient lensing features, namely the C-metric, the NUT metric, and the Kerr metric. For this purpose, we will first outline how to solve the equations of motion analytically using elementary and Jacobi's elliptic functions as well as Legendre's elliptic integrals. Then we will fix an observer in the domain of outer communication and relate the constants of motion of the lightlike geodesics to latitude-longitude coordinates on the observer's celestial sphere. We will use the analytic solutions to write down the lens equations, calculate the redshift, and the travel time. Finally, we will discuss and compare the results and comment on how we can use them to place constraints on the spin parameter, the acceleration parameter, and the gravitomagnetic charge of a black hole.

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Seminar
Cosection localization via shifted symplectic geometry

December 6 (Wed) at 10:00 - 11:30, 2023

Young-Hoon Kiem (Professor, School of Mathematics, Korea Institute for Advanced Study (KIAS), Republic of Korea)

Modern enumerative invariants are defined as integrals of cohomology classes against virtual fundamental classes constructed by Li-Tian and Behrend-Fantechi. When the obstruction sheaf admits a cosection, the virtual fundamental class is localized to the zero locus of the cosection. When the cosection is furthermore enhanced to a (-1)-shifted closed 1-form, the zero locus admits a (-2)-shifted symplectic structure and thus we have another virtual fundamental class by the Oh-Thomas construction. An obvious question is whether these two virtual fundamental classes coincide or not. In this talk, we will see that (-1)-shifted closed 1-forms arise naturally as an analogue of the Lagrange multiplier method. Furthermore, a proof of the equality of the two virtual fundamental classes and its applications will be discussed. Based on a joint work with Hyeonjun Park.

Venue: Seminar Room #359

Event Official Language: English
Seminar iTHEMS Biology Seminar
Translating between evolutionary game theory and theoretical ecology

December 5 (Tue) at 16:30 - 17:30, 2023

Arne Traulsen (Director, Department for Theoretical Biology, Max Planck Institute for Evolutionary Biology, Germany)

Both theoretical ecology and evolutionary game theory describe the dynamics of interacting populations. More than 40 years ago, Hofbauer and Sigmund established a mathematical equivalence between the Lotka-Volterra equations and the replicator dynamics from evolutionary game theory. However, this equivalence has not been exploited by empiricists so far. One of the issues is dimensionality: An ecological interaction of two species corresponds to an evolutionary game between three types. Only when we focus on a special case with identical growth rates, it is possible to translate without this trick, leading to a more direct equivalence between the frameworks. Consequently, one has to be particularly careful how to classify interactions and how to assess dynamical outcomes. For example, a ‘Prisoner's Dilemma’ interaction where the $cooperators' have a higher intrinsic growth rate than$ defectors' can result in stable coexistence of the two types and may ultimately not represent a social dilemma at all.

Venue: via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Rotating discs on the Kerr black hole background

December 5 (Tue) at 15:00 - 16:30, 2023

David Kofroň (Postdoctoral Researcher, Institute of Theoretical Physics, Charles University, Czechia)

Analytical solution of a rotating black hole surrounded by accretion disc in full GR is, so far, unknown. The Ernst equation is nonlinear. In this talk, we will provide a framework in which the solutions of linearised Ernst equations can be obtained from the linear perturbations of Kerr black hole treated in the formalism of the Debye potentials. In this way, we recover all the metric perturbations in term of a single complex scalar function (which solves the Laplace equation).

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Gravity of Accretion Discs and Black Holes

December 5 (Tue) at 14:00 - 15:00, 2023

Petr Kotlařík (Ph.D. Student, Institute of Theoretical Physics, Charles University, Czechia)

The typical black hole solutions describe only isolated black holes. However, in astrophysics, such a condition is never strictly satisfied. As matter accretes onto the black hole, disc structures are often formed. In this talk, I will summarize our recent attempts to find the gravitational field of such a nonisolated black hole. We start from the simplest case of static and axially symmetric metric. Although it is a somewhat "rough" approximation in the astrophysical context, this idealization may already help us to understand some interesting implications of the disc's gravity. Moreover, with such a simplification, we can obtain exact analytical "superpositions" of the Schwarzchild black hole and a disc. When some rotation is present, dragging effects complicate the situation dramatically. Then, one typically has to resort to numerical relativity or some approximate methods, e.g., perturbations. In the talk, I also address the stationary case and demonstrate what we can do on the level of the direct metric perturbation.

Venue: Seminar Room #359 (Main Venue) / via Zoom

Event Official Language: English
Seminar iTHEMS Theoretical Physics Seminar
Breaking down the magnonic Wiedemann-Franz law in the hydrodynamic regime

December 4 (Mon) at 15:00 - 16:30, 2023

Ryotaro Sano (Ph.D. Student, Division of Physics and Astronomy, Graduate School of Science, Kyoto University)

Quantum transport has attracted a profound growth of interest owing to its fundamental importance and many applications in condensed matter physics. Recent significant developments in experimental techniques have further boosted the study of quantum transport. Notably in ultraclean systems, strong interactions between quasi-particles drastically affect the transport properties, resulting in an emergent hydrodynamic behavior. Recent experiments on ultrapure ferromagnetic insulators have opened up new pathways for magnon hydrodynamics. Hydrodynamic magnon transport implies exhibiting extraordinary features and has a potential for innovative functionalities beyond the conventional non-interacting magnon picture. However, the direct observation of magnon fluids remains an open issue due to the lack of probes to access the time and length scales characteristics of this regime. In this work, we derive a set of coupled hydrodynamic equations for a magnon fluid and study the spin and thermal conductivities by focusing on the most dominant time scales [1]. As a hallmark of the hydrodynamic regime, we reveal that the ratio between the two conductivities shows a large deviation from the so-called magnonic WF law. We also identify an origin of the drastic breakdown of the magnonic WF law as the difference in relaxation processes between spin and heat currents, which is unique to the hydrodynamic regime. Therefore, our results will become key evidence for an emergent hydrodynamic magnon behavior and lead to the direct observation of magnon fluids.

Venue: Hybrid Format (3F #359 and Zoom), Main Research Building

Event Official Language: English

764 events

Seminar

Methods for neural decoding using machine learning, deep learning, and quantum-inspired algorithms

Dust-driven instabilities in protoplanetary disks: toward understanding formation of planetesimals

Does horizontal gene transfer stabilize cooperation in bacteria?

Probing structure of neutron stars through X-ray bursters

Symmetry Topological field theory for Subsystem symmetry

Application of Modular tensor category to Lattice gauge theory

Oscillatory data analysis using the extended Hilbert transform method

A symmetry principle for gauge theories with fractons

Inflationary Cosmology with a scalar-curvature mixing term 12ξRϕ2\frac{1}{2} \xi R \phi^2

Application of mathematical models to the COVID-19 cohort study

Energy spectrum and time evolution of a SU(2) pure gauge lattice theory on a quantum annealer

Plasticity in the endogenous rhythms and the adaptation to the tidal environment in a freshwater snail

Tropical geometry and period integrals

Exploring material strengths of dust aggregates in planet formation by numerical simulations

Gravitational Lensing in Black Hole Spacetimes of the Plebanski-Demianski Class

Cosection localization via shifted symplectic geometry

Translating between evolutionary game theory and theoretical ecology

Rotating discs on the Kerr black hole background

Gravity of Accretion Discs and Black Holes

Breaking down the magnonic Wiedemann-Franz law in the hydrodynamic regime

Inflationary Cosmology with a scalar-curvature mixing term $\frac{1}{2} \xi R \phi^2$