Hardness of Linearly Ordered 4-Colouring of 3-Colourable 3-Uniform Hypergraphs

A linearly ordered (LO) k-colouring of a hypergraph is a colouring of its vertices with colours 1, … , k such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO k-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the "linearly ordered chromatic number" of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO 3-colourable, and the case that it is not even LO 4-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opršal, Wrochna, and Živný (2023)

Hardness of linearly ordered 4-colouring of 3-colourable 3-uniform hypergraphs

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed number of colours is NP-complete (and in the special case of graphs, LO colouring coincides with the usual graph colouring). Here, we investigate the complexity of approximating the `linearly ordered chromatic number' of a hypergraph. We prove that the following promise problem is NP-complete: Given a 3-uniform hypergraph, distinguish between the case that it is LO $3$-colourable, and the case that it is not even LO $4$-colourable. We prove this result by a combination of algebraic, topological, and combinatorial methods, building on and extending a topological approach for studying approximate graph colouring introduced by Krokhin, Opr\v{s}al, Wrochna, and \v{Z}ivn\'y (2023)

Cosmology with the Laser Interferometer Space Antenna

254 pags:, 44 figs.The Laser Interferometer Space Antenna (LISA) has two scientific objectives of cosmological focus: to probe the expansion rate of the universe, and to understand stochastic gravitational-wave backgrounds and their implications for early universe and particle physics, from the MeV to the Planck scale. However, the range of potential cosmological applications of gravitational-wave observations extends well beyond these two objectives. This publication presents a summary of the state of the art in LISA cosmology, theory and methods, and identifies new opportunities to use gravitational-wave observations by LISA to probe the universe.This work is partly supported by: A.G. Leventis Foundation; Academy of Finland Grants 328958 and 345070; Alexander S. Onassis Foundation, Scholarship ID: FZO 059-1/2018-2019; Amaldi Research Center funded by the MIUR program “Dipartimento di Eccellenza” (CUP: B81I18001170001); ASI Grants No. 2016-24-H.0 and No. 2016-24-H.1-2018; Atracción de Talento Grant 2019-T1/TIC-15784; Atracción de Talento contract no. 2019-T1/TIC-13177 granted by the Comunidad de Madrid; Ayuda ‘Beatriz Galindo Senior’ by the Spanish ‘Ministerio de Universidades’, Grant BG20/00228; Basque Government Grant (IT-979-16); Belgian Francqui Foundation; Centre national d’Etudes spatiales; Ben Gurion University Kreitman Fellowship, and the Israel Academy of Sciences and Humanities (IASH) & Council for Higher Education (CHE) Excellence Fellowship Program for International Postdoctoral Researchers; Centro de Excelencia Severo Ochoa Program SEV-2016-0597; CERCA program of the Generalitat de Catalunya; Cluster of Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter” (PRISMA? EXC 2118/1); Comunidad de Madrid, Contrato de Atracción de Talento 2017-T1/TIC-5520; Czech Science Foundation GAČR, Grant No. 21-16583M; Delta ITP consortium; Department of Energy under Grant No. DE-SC0008541, DE-SC0009919 and DESC0019195; Deutsche Forschungsgemeinschaft (DFG), Project ID 438947057; Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy - EXC 2121 Quantum Universe - 390833306; European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project CoGraDS - CZ.02.1.01/0.0/0.0/15 003/0000437); European Union’s H2020 ERC Consolidator Grant “GRavity from Astrophysical to Microscopic Scales” (Grant No. GRAMS-815673); European Union’s H2020 ERC, Starting Grant Agreement No. DarkGRA-757480; European Union’s Horizon 2020 programme under the Marie Sklodowska-Curie Grant Agreement 860881 (ITN HIDDeN); European Union’s Horizon 2020 Research and Innovation Programme Grant No. 796961, “AxiBAU” (K.S.); European Union’s Horizon 2020 Research Council grant 724659 MassiveCosmo ERC-2016-COG; FCT through national funds (PTDC/FIS-PAR/31938/2017) and through project “BEYLA – BEYond LAmbda” with Ref. Number PTDC/FIS-AST/0054/2021; FEDER-Fundo Europeu de Desenvolvimento Regional through COMPETE2020 - Programa Operacional Competitividade e Internacionalização (POCI-01-0145- FEDER-031938) and research Grants UIDB/04434/2020 and UIDP/04434/2020; Fondation CFM pour la Recherche in France; Foundation for Education and European Culture in Greece; French ANR project MMUniverse (ANR-19-CE31-0020); FRIA Grant No.1.E.070.19F of the Belgian Fund for Research, F.R. S.-FNRS Fundação para a Ciência e a Tecnologia (FCT) through Contract No. DL 57/2016/CP1364/ CT0001; Fundação para a Ciência e a Tecnologia (FCT) through Grants UIDB/04434/2020, UIDP/04434/ 2020, PTDC/FIS-OUT/29048/2017, CERN/FIS-PAR/0037/2019 and “CosmoTests – Cosmological tests of gravity theories beyond General Relativity” CEECIND/00017/2018; Generalitat Valenciana Grant PROMETEO/2021/083; Grant No. 758792, project GEODESI; Government of Canada through the Department of Innovation, Science and Economic Development and Province of Ontario through the Ministry of Colleges and Universities; Grants-in-Aid for JSPS Overseas Research Fellow (No. 201960698); I?D Grant PID2020-118159GB-C41 of the Spanish Ministry of Science and Innovation; INFN iniziativa specifica TEONGRAV; Israel Science Foundation (Grant No. 2562/20); Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Nos. 20H01899 and 20H05853; IFT Centro de Excelencia Severo Ochoa Grant SEV-2; Kavli Foundation and its founder Fred Kavli; Minerva Foundation; Ministerio de Ciencia e Innovacion Grant PID2020-113644GB-I00; NASA Grant 80NSSC19K0318; NASA Hubble Fellowship grants No. HST-HF2-51452.001-A awarded by the Space Telescope Science Institute with NASA contract NAS5-26555; Netherlands Organisation for Science and Research (NWO) Grant Number 680-91-119; new faculty seed start-up grant of the Indian Institute of Science, Bangalore, the Core Research Grant CRG/2018/002200 of the Science and Engineering; NSF Grants PHY-1820675, PHY-2006645 and PHY-2011997; Polish National Science Center Grant 2018/31/D/ ST2/02048; Polish National Agency for Academic Exchange within the Polish Returns Programme under Agreement PPN/PPO/2020/1/00013/U/00001; Pró-Reitoria de Pesquisa of Universidade Federal de Minas Gerais (UFMG) under Grant No. 28359; Ramón y Cajal Fellowship contract RYC-2017-23493; Research Project PGC2018-094773-B-C32 [MINECO-FEDER]; Research Project PGC2018-094773-B-C32 [MINECO-FEDER]; ROMFORSK Grant Project. No. 302640; Royal Society Grant URF/R1/180009 and ERC StG 949572: SHADE; Shota Rustaveli National Science Foundation (SRNSF) of Georgia (Grant FR/18-1462); Simons Foundation/SFARI 560536; SNSF Ambizione grant; SNSF professorship Grant (No. 170547); Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme Grants SEV-2016- 0597 and PID2019-110058GB-C22; Spanish Ministry MCIU/AEI/FEDER Grant (PGC2018-094626-BC21); Spanish Ministry of Science and Innovation (PID2020-115845GB-I00/AEI/10.13039/ 501100011033); Spanish Proyectos de I?D via Grant PGC2018-096646-A-I00; STFC Consolidated Grant ST/T000732/1; STFC Consolidated Grants ST/P000762/1 and ST/T000791/1; STFC Grant ST/ S000550/1; STFC Grant ST/T000813/1; STFC Grants ST/P000762/1 and ST/T000791/1; STFC under the research Grant ST/P000258/1; Swiss National Science Foundation (SNSF), project The Non-Gaussian Universe and Cosmological Symmetries, Project Number: 200020-178787; Swiss National Science Foundation Professorship Grants No. 170547 and No. 191957; SwissMap National Center for Competence in Research; “The Dark Universe: A Synergic Multi-messenger Approach” Number 2017X7X85K under the MIUR program PRIN 2017; UK Space Agency; UKSA Flagship Project, Euclid.Peer reviewe

New horizons for fundamental physics with LISA

The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas

Persistent Homology as a Tool for Timeseries Analysis

Persistence is a fairly well established tool in topological data analysis used to infer geometric information form discrete data. The aim of our work is to modify the classical persistent workflow in order to study the behavior of time series. First we introduce the sliding window embedding technique in order to codify temporal regularity of the time series in geometric properties of the obtained point cloud, and we give an general overview on the historical approach to persistent homology. Secondly, we present the categorical approach to persistence in order to cover all the relevant theoretical background needed for our analysis. In order to compute the complete invariant (the barcode decomposition) of the persistent module, we used the algorithm introduced by de Silva, Morozov and Vejdemo-Johansson for persistent cohomology. Finally, we applied our methodology to time series of different nature; we studied orbits of both the logistic map and the Chirikov standard map on the torus. Finally, we test our method on a financial time series; precisely on the time series of the exchange rate BitCoin/USDollar

Gravitational-wave luminosity distance in quantum gravity

International audienceDimensional flow, the scale dependence of the dimensionality of spacetime, is a feature shared by many theories of quantum gravity (QG). We present the first study of the consequences of QG dimensional flow for the luminosity distance scaling of gravitational waves in the frequency ranges of LIGO and LISA. We find generic modifications with respect to the standard general-relativistic scaling, largely independent of specific QG proposals. We constrain these effects using two examples of multimessenger standard sirens, the binary neutron-star merger GW170817 and a simulated supermassive black-hole merger event detectable with LISA. We apply these constraints to various QG candidates, finding that the quantum geometries of group field theory, spin foams and loop quantum gravity can give rise to observable signals in the gravitational-wave spin-2 sector. Our results complement and improve GW propagation-speed bounds on modified dispersion relations. Under more model-dependent assumptions, we also show that bounds on quantum geometry can be strengthened by solar-system tests

Gravitational waves from inflation in LISA: reconstruction pipeline and physics interpretation

International audienceVarious scenarios of cosmic inflation enhance the amplitude of the stochastic gravitational wave background (SGWB) at frequencies detectable by the LISA detector. We develop tools for a template-based analysis of the SGWB and introduce a template databank to describe well-motivated signals from inflation, prototype their template-based searches, and forecast their reconstruction with LISA. Specifically, we classify seven templates based on their signal frequency shape, and we identify representative fundamental physics models leading to them. By running a template-based analysis, we forecast the accuracy with which LISA can reconstruct the template parameters of representative benchmark signals, with and without galactic and extragalactic foregrounds. We identify the parameter regions that can be probed by LISA within each template. Finally, we investigate how our signal reconstructions shed light on fundamental physics models of inflation: we discuss their impact for measurements of \emph{e.g.,}~the couplings of inflationary axions to gauge fields; the graviton mass during inflation; the fluctuation seeds of primordial black holes; the consequences of excited states during inflation, and the presence of small-scale spectral features

Testing modified gravity at cosmological distances with LISA standard sirens

Modifications of General Relativity leave their imprint both on the cosmic expansion history through a non-trivial dark energy equation of state, and on the evolution of cosmological perturbations in the scalar and in the tensor sectors. In particular, the modification in the tensor sector gives rise to a notion of gravitational-wave (GW) luminosity distance, different from the standard electromagnetic luminosity distance, that can be studied with standard sirens at GW detectors such as LISA or third-generation ground based experiments. We discuss the predictions for modified GW propagation from some of the best studied theories of modified gravity, such as Horndeski or the more general degenerate higher order scalar-tensor (DHOST) theories, non-local infrared modifications of gravity, bigravity theories and the corresponding phenomenon of GW oscillation, as well as theories with extra or varying dimensions. We show that modified GW propagation is a completely generic phenomenon in modified gravity. We then use a simple parametrization of the effect in terms of two parameters (Ξ0,n), that is shown to fit well the results from a large class of models, to study the prospects of observing modified GW propagation using supermassive black hole binaries as standard sirens with LISA . We construct mock source catalogs and perform detailed Markov Chain Monte Carlo studies of the likelihood obtained from LISA standard sirens alone, as well as by combining them with CMB, BAO and SNe data to reduce the degeneracies between cosmological parameters. We find that the combination of LISA with the other cosmological datasets allows one to measure the parameter Ξ0 that characterizes modified GW propagation to the percent level accuracy, sufficient to test several modified gravity theories. LISA standard sirens can also improve constraints on GW oscillations induced by extra field content by about three orders of magnitude relative to the current capability of ground detectors. We also update the forecasts on the accuracy on H0 and on the dark-energy equation of state using more recent estimates for the LISA sensitivity

Quantum gravity and gravitational-wave astronomy

We investigate possible signatures of quantum gravity which could be tested with current and future gravitational-wave (GW) observations. In particular, we analyze how quantum gravity can influence the GW luminosity distance, the time dependence of the effective Planck mass and the instrumental strain noise of interferometers. Using both model-dependent and model-independent formulae, we show that these quantities can encode a non-perturbative effect typical of all quantum-gravity theories, namely the way the dimension of spacetime changes with the probed scale. Effects associated with such dimensional flow might be tested with GW observations and constrained significantly in theories with a microscopically discrete spacetime geometry, more strongly than from propagation-speed constraints. Making use of public LIGO data as well as of a simulated higher-redshift LISA source, we impose the first, respectively, actual and mock constraints on quantum-gravity parameters affecting the GW luminosity distance and discuss specific theoretical examples. If also the Newtonian potential is modified but light geodesics are not, then solar-system bounds may be stronger than GW ones. Yet, for some theories GW astronomy can give unique information not available from solar-system tests