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In recent years, forecasting activities have become an important tool in designing and optimising large-scale structure surveys. To predict the performance of such surveys, the Fisher matrix formalism is frequently used as a fast and easy way to compute constraints on cosmological parameters. Among them lies the study of the properties of dark energy which is one of the main goals in modern cosmology. As so, a metric for the power of a survey to constrain dark energy is provided by the figure of merit (FoM). This is defined as the inverse of the surface contour given by the joint variance of the dark energy equation of state parameters fw0;wag in the Chevallier-Polarski-Linder parameterization, which can be evaluated from the covariance matrix of the parameters. This covariance matrix is obtained as the inverse of the Fisher matrix. The inversion of an ill-conditioned matrix can result in large errors on the covariance coe cients if the elements of the Fisher matrix are estimated with insu cient precision. The conditioning number is a metric providing a mathematical lower limit to the required precision for a reliable inversion, but it is often too stringent in practice for Fisher matrices with sizes greater than 2 2. In this paper, we propose a general numerical method to guarantee a certain precision on the inferred constraints, such as the FoM. It consists of randomly vibrating (perturbing) the Fisher matrix elements with Gaussian perturbations of a given amplitude and then evaluating the maximum amplitude that keeps the FoM within the chosen precision. The steps used in the numerical derivatives and integrals involved in the calculation of the Fisher matrix elements can then be chosen accordingly in order to keep the precision of the Fisher matrix elements below this maximum amplitude. We illustrate our approach by forecasting stage IV spectroscopic surveys cosmological constraints from the galaxy power spectrum. We infer the range of steps for which the Fisher matrix approach is numerically reliable. We explicitly check that using steps that are larger by a factor of two produce an inaccurate estimation of the constraints. We further validate our approach by comparing the Fisher matrix contours to those obtained with a Monte Carlo Markov chain (MCMC) approach – in the case where the MCMC posterior distribution is close to a Gaussian – and finding excellent agreement between the two approaches. ; MIUR through Rita Levi Montalcini project 'PROMETHEUS - Probing and Relating Observables with Multi-wavelength Experiments To Help Enlightening the Universe's Structure' Ministry of Education, Universities and Research (MIUR) National Aeronautics & Space Administration (NASA) 80NM0018D0004 Spanish Government ESP2017-89838-C3-1-R H2020 programme of the European Commission 776247 UK Research & Innovation (UKRI) MR/S016066/1 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1200171 UKRI MR/S016066/1 Funding Data Source:UKRI Appeared in source as:UKRI Total Award Amount: £1,199,608.00 GBP Grant Project Title:Exploring the Universe with radio and optical galaxy surveys Start Date (YYYY-MM-DD): 2019-04-30 End Date (YYYY-MM-DD): 2023-04-29 Grant Status:Active Principal Investigator:Alkistis Pourtsidou Unique Identifier: 0000-0001-9110-5550 Principal Investigator Institution:Queen Mary, University of London ; Versión publicada - versión final del editor

Context. In metric theories of gravity with photon number conservation, the luminosity and angular diameter distances are related via the Etherington relation, also known as the distance duality relation (DDR). A violation of this relation would rule out the standard cosmological paradigm and point to the presence of new physics.Aims. We quantify the ability of Euclid, in combination with contemporary surveys, to improve the current constraints on deviations from the DDR in the redshift range 0< z< 1.6.Methods. We start with an analysis of the latest available data, improving previously reported constraints by a factor of 2.5. We then present a detailed analysis of simulated Euclid and external data products, using both standard parametric methods (relying on phenomenological descriptions of possible DDR violations) and a machine learning reconstruction using genetic algorithms.Results. We find that for parametric methods Euclid can (in combination with external probes) improve current constraints by approximately a factor of six, while for non-parametric methods Euclid can improve current constraints by a factor of three.Conclusions. Our results highlight the importance of surveys like Euclid in accurately testing the pillars of the current cosmological paradigm and constraining physics beyond the standard cosmological model. ; La Caixa Foundation 100010434 LCF/BQ/PI19/11690015 Spanish Agencia Estatal de Investigacion through the grant "IFT Centro de Excelencia Severo Ochoa" SEV-2016-0597 FEDER -Fundo Europeu de Desenvolvimento Regional funds through the COMPETE 2020 -Operational Programme for Competitiveness and Internationalisation (POCI) Portuguese funds through FCT -Fundacao para a Ciencia e a Tecnologia POCI-01-0145-FEDER-028987 Centro de Excelencia Severo Ochoa Program SEV-2016-059 Spanish Government RYC-2014-15843 Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1200171 Spanish Ministry of Science, Innovation and Universities ESP2017-89838-C3-1-R H2020 programme of the European Commission 776247 UK Research & Innovation (UKRI) Science & Technology Facilities Council (STFC) Science and Technology Development Fund (STDF) ST/P000703/1 European Research Council (ERC) 769130 Academy of Finland European Commission Agenzia Spaziale Italiana (ASI) Belgian Federal Science Policy Office Canadian Euclid Consortium Centre National D'etudes Spatiales Helmholtz Association German Aerospace Centre (DLR) Danish Space Research Institute Portuguese Foundation for Science and Technology European Commission Spanish Government National Aeronautics & Space Administration (NASA) Netherlandse Onderzoekschool Voor Astronomie Norwegian Space Agency Romanian Space Agency State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffice (SSO) United Kingdom Space Agency PGC2018-094773-B-C32

Aims. The Euclid space telescope will measure the shapes and redshifts of galaxies to reconstruct the expansion history of the Universe and the growth of cosmic structures. The estimation of the expected performance of the experiment, in terms of predicted constraints on cosmological parameters, has so far relied on various individual methodologies and numerical implementations, which were developed for different observational probes and for the combination thereof. In this paper we present validated forecasts, which combine both theoretical and observational ingredients for different cosmological probes. This work is presented to provide the community with reliable numerical codes and methods for Euclid cosmological forecasts.Methods. We describe in detail the methods adopted for Fisher matrix forecasts, which were applied to galaxy clustering, weak lensing, and the combination thereof. We estimated the required accuracy for Euclid forecasts and outline a methodology for their development. We then compare and improve different numerical implementations, reaching uncertainties on the errors of cosmological parameters that are less than the required precision in all cases. Furthermore, we provide details on the validated implementations, some of which are made publicly available, in different programming languages, together with a reference training-set of input and output matrices for a set of specific models. These can be used by the reader to validate their own implementations if required.Results. We present new cosmological forecasts for Euclid. We find that results depend on the specific cosmological model and remaining freedom in each setting, for example flat or non-flat spatial cosmologies, or different cuts at non-linear scales. The numerical implementations are now reliable for these settings. We present the results for an optimistic and a pessimistic choice for these types of settings. We demonstrate that the impact of cross-correlations is particularly relevant for models beyond a cosmological constant and may allow us to increase the dark energy figure of merit by at least a factor of three. ; Academy of Finland European Commission Agenzia Spaziale Italiana (ASI) Belgian Federal Science Policy Office Canadian Euclid Consortium Centre National D'etudes Spatiales Deutsches Zentrum fur Luft-and Raumfahrt Danish Space Research Institute Fundacao para a Cienca e a Tecnologia Spanish Government National Aeronautics & Space Administration (NASA) 80NM0018D0004 Netherlandse Onderzoekschool Voor Astronomie Norvegian Space Center Romanian Space Agency State Secretariat for Education, Research and Innovation (SERI) at the Swiss Space O ffice (SSO) United Kingdom Space Agency Ministry of Education, Universities and Research (MIUR) Ministry of Education, Universities and Research (MIUR) L. 232/2016 European Research Council through the Darklight Advanced Research Grant 291521 Ministry of Education, Universities and Research (MIUR) Centre National D'etudes Spatiales Fonds de la Recherche Scientifique - FNRS Swiss National Science Foundation (SNSF) European Commission NASA ROSES grant 12-EUCLID12-0004 UK Science & Technology Facilities Council ST/N000668/1 ST/S000437/1 UK Space Agency ST/N00180X/1 D-ITP consortium, a program of the NWO - the OCW Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 1200171 Spanish Ministry of Science, Innovation and Universities ESP2017-89838-C3-1-R H2020 programme of the European Commission 776247 German Research Foundation (DFG) Transregio 33 International Max Planck Research School for Astronomy and Astrophysics at the University of Bonn International Max Planck Research School for Astronomy and Astrophysics at the University of Cologne Bonn-Cologne Graduate School for Physics and Astronomy Royal Society of London European Research Council (ERC) 617656