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Primordial black holes (PBHs) can be the source for all or a part of today's dark matter density. Inflation provides a mechanism for generating the seeds of PBHs in the presence of a temporal period where the velocity of an inflaton field $\phi$ rapidly decreases toward 0. We compute the primordial power spectra of curvature perturbations generated during Gauss-Bonnet (GB) corrected Higgs inflation in which the inflaton field has not only a nonminimal coupling to gravity but also a GB coupling. For a scalar-GB coupling exhibiting a rapid change during inflation, we show that curvature perturbations are sufficiently enhanced by the appearance of an effective potential $V_{\rm eff}(\phi)$ containing the structures of plateau-type, bump-type, and their intermediate type. We find that there are parameter spaces in which PBHs can constitute all dark matter for these three types of $V_{\rm eff}(\phi)$. In particular, models with bump- and intermediate-types give rise to the primordial scalar and tensor power spectra consistent with the recent Planck data on scales relevant to the observations of cosmic microwave background. This property is attributed to the fact that the number of e-foldings $\Delta N_c$ acquired around the bump region of $V_{\rm eff}(\phi)$ can be as small as a few, in contrast to the plateau-type where $\Delta N_c$ typically exceeds the order of 10., Comment: 20 pages, 7 figures

We study linear perturbations about static and spherically symmetric black holes with a time-independent background scalar field in shift-symmetric Horndeski theories, whose Lagrangian is characterized by coupling functions depending only on the kinetic term of the scalar field $X$. We clarify conditions for the absence of ghosts and Laplacian instabilities along the radial and angular directions in both odd- and even-parity perturbations. For reflection-symmetric theories described by a k-essence Lagrangian and a nonminimal derivative coupling with the Ricci scalar, we show that black holes endowed with nontrivial scalar hair are unstable around the horizon in general. This includes nonasymptotically flat black holes known to exist when the nonminimal derivative coupling to the Ricci scalar is a linear function of $X$. We also investigate several black hole solutions in nonreflection-symmetric theories. For cubic Galileons with the Einstein-Hilbert term, there exists a nonasymptotically flat hairy black hole with no ghosts/Laplacian instabilities. Also, for the scalar field linearly coupled to the Gauss-Bonnet term, asymptotically flat black hole solutions constructed perturbatively with respect to a small coupling are free of ghosts/Laplacian instabilities., 23 pages, 1 figure, 1 table, published version

In a model of spontaneous scalarization of neutron stars proposed by Damour and Esposite-Farese, a general relativistic branch becomes unstable to trigger tachyonic growth of a scalar field $\phi$ toward a scalarized branch. Applying this scenario to cosmology, there is fatal tachyonic instability of $\phi$ during inflation and matter dominance being incompatible with solar-system constraints on today's field value $\phi_0$. In the presence of a four-point coupling $g^2 \phi^2 \chi^2/2$ between $\phi$ and an inflaton field $\chi$, it was argued by Anson et al. that a positive mass squared heavier than the square of a Hubble expansion rate leads to the exponential suppression of $\phi$ during inflation and that $\phi_0$ can remain small even with the growth of $\phi$ after the radiation-dominated epoch. For several inflaton potentials approximated as $V(\chi)=m^2 \chi^2/2$ about the potential minimum, we study the dynamics of $\phi$ during reheating as well as other cosmological epochs in detail. For certain ranges of the coupling $g$, the homogeneous field $\phi$ can be amplified by parametric resonance during a coherent oscillation of the inflaton. Incorporating the backreaction of created particles under a Hartree approximation, the maximum values of $\phi$ reached during preheating are significantly smaller than those obtained without the backreaction. We also find that the minimum values of $g$ consistent with solar system bounds on $\phi$ at the end of reheating are of order $10^{-5}$ and hence there is a wide range of acceptable values of $g$. Thus, the scenario proposed by Anson et al. naturally leads to the viable cosmological evolution of $\phi$ consistent with local gravity constraints, without modifying the property of scalarized neutron stars., Comment: 15 pages, 8 figures

In a subclass of Horndeski theories with the speed of gravity equivalent to that of light, we study gravitational radiation emitted during the inspiral phase of compact binary systems. We compute the waveform of scalar perturbations under a post-Newtonian expansion of energy-momentum tensors of point-like particles that depend on a scalar field. This scalar mode not only gives rise to breathing and longitudinal polarizations of gravitational waves, but it is also responsible for scalar gravitational radiation in addition to energy loss associated with transverse and traceless tensor polarizations. We calculate the Fourier-transformed gravitational waveform of two tensor polarizations under a stationary phase approximation and show that the resulting waveform reduces to the one in a parametrized post-Einsteinian (ppE) formalism. The ppE parameters are directly related to a scalar charge in the Einstein frame, whose existence is crucial to allow the deviation from General Relativity (GR). We apply our general framework to several concrete theories and show that a new theory of spontaneous scalarization with a higher-order scalar kinetic term leaves interesting deviations from GR that can be probed by the observations of gravitational waves emitted from neutron star-black hole binaries. If the scalar mass exceeds the order of typical orbital frequencies $\omega \simeq 10^{-13}$~eV, which is the case for a recently proposed scalarized neutron star with a self-interacting potential, the gravitational waveform practically reduces to that in GR., Comment: 30 pages, 2 figures

We study static and spherically symmetric charged stars with a nontrivial profile of the scalar field $\phi$ in Einstein-Maxwell-scalar theories. The scalar field is coupled to a $U(1)$ gauge field $A_{\mu}$ with the form $-\alpha(\phi)F_{\mu \nu}F^{\mu \nu}/4$, where $F_{\mu \nu}=\partial_{\mu}A_{\nu}-\partial_{\nu} A_{\mu}$ is the field strength tensor. Analogous to the case of charged black holes, we show that this type of interaction can induce spontaneous scalarization of charged stars under the conditions $({\rm d}\alpha/{\rm d}\phi) (0)=0$ and $({\rm d}^2\alpha/{\rm d}\phi^2) (0)>0$. For the coupling $\alpha (\phi)=\exp (-\beta \phi^2/M_{\rm pl}^2)$, where $\beta~(, Comment: 14 pages, 3 figures

We use the cross-correlation power spectrum of the integrated Sachs-Wolfe (ISW) effect in the cosmic microwave background (CMB) temperature anisotropy and galaxy fluctuations to probe the physics of late-time cosmic acceleration. For this purpose, we focus on three models of dark energy that belong to a sub-class of Horndeski theories with the speed of gravity equivalent to that of light: Galileon Ghost Condensate (GGC), Generalized Cubic Covariant Galileon (GCCG), and K-mouflage. In the GGC and GCCG models, the existence of cubic-order scalar self-interactions allows a possibility for realizing negative ISW-galaxy cross-correlations, while the K-mouflage model predicts a positive correlation similar to the $\Lambda$-cold-dark-matter (LCDM) model. In our analysis, we fix the parameters of each model to their best-fit values derived from a baseline likelihood analysis with observational data from CMB, baryon acoustic oscillations, and supernovae type Ia. Then we fit those best-fit models to the ISW-galaxy cross-correlation power spectrum extracted from a collection of photometric redshift surveys. We find that both GGC and GCCG best-fit models degrade the fit to the ISW-galaxy cross-correlation data compared to LCDM best-fit model. This is attributed to the fact that, for their best-fit values constrained from the baseline likelihood, the cubic-order scalar self-interaction gives rise to suppressed ISW tails relative to $\Lambda$CDM. The K-mouflage best-fit model is largely degenerate with the LCDM best-fit model and has a positively correlated ISW-galaxy power close to that of LCDM., Comment: 25 pages, 5 figures. Published by JCAP

For ultra-light scalar particles like axions, dark matter can form a state of the Bose-Einstein condensate (BEC) with a coherent classical wave whose wavelength is of order galactic scales. In the context of an oscillating scalar field with mass $m$, this BEC description amounts to integrating out the field oscillations over the Hubble time scale $H^{-1}$ in the regime $m \gg H$. We provide a gauge-invariant general relativistic framework for studying cosmological perturbations in the presence of a self-interacting BEC associated with a complex scalar field. In particular, we explicitly show the difference of BECs from perfect fluids by taking into account cold dark matter, baryons, and radiation as a Schutz-Sorkin description of perfect fluids. We also scrutinize the accuracy of commonly used Newtonian treatment based on a quasi-static approximation for perturbations deep inside the Hubble radius. For a scalar field which starts to oscillate after matter-radiation equality, we show that, after the BEC formation, a negative self-coupling hardly leads to a Laplacian instability of the BEC density contrast. This is attributed to the fact that the Laplacian instability does not overwhelm the gravitational instability for self-interactions within the validity of the nonrelativistic BEC description. Our analysis does not accommodate the regime of parametric resonance which can potentially occur for a large field alignment during the transient epoch prior to the BEC formation., 23 pages, 3 figures

Deci-hertz Interferometer Gravitational Wave Observatory (DECIGO) is the future Japanese space mission with a frequency band of 0.1 Hz to 10 Hz. DECIGO aims at the detection of primordial gravitational waves, which could be produced during the inflationary period right after the birth of the universe. There are many other scientific objectives of DECIGO, including the direct measurement of the acceleration of the expansion of the universe, and reliable and accurate predictions of the timing and locations of neutron star/black hole binary coalescences. DECIGO consists of four clusters of observatories placed in the heliocentric orbit. Each cluster consists of three spacecraft, which form three Fabry-Perot Michelson interferometers with an arm length of 1,000 km. Three clusters of DECIGO will be placed far from each other, and the fourth cluster will be placed in the same position as one of the three clusters to obtain the correlation signals for the detection of the primordial gravitational waves. We plan to launch B-DECIGO, which is a scientific pathfinder of DECIGO, before DECIGO in the 2030s to demonstrate the technologies required for DECIGO, as well as to obtain fruitful scientific results to further expand the multi-messenger astronomy., Comment: 10 pages, 3 figures

We study the dynamics of dark energy in the presence of a 2-form field coupled to a canonical scalar field $\phi$. We consider the coupling proportional to $e^{-\mu \phi/M_{\rm pl}} H_{\alpha \beta \gamma}H^{\alpha \beta \gamma}$ and the scalar potential $V(\phi) \propto e^{-\lambda \phi/M_{\rm pl}}$, where $H_{\alpha \beta \gamma}$ is the 2-form field strength, $\mu, \lambda$ are constants, and $M_{\rm pl}$ is the reduced Planck mass. We show the existence of an anisotropic matter-dominated scaling solution followed by a stable accelerated fixed point with a non-vanishing shear. Even if $\lambda \geq {\cal O}(1)$, it is possible to realize the dark energy equation of state $w_{\rm DE}$ close to $-1$ at low redshifts for $\mu \gg \lambda$. The existence of anisotropic hair and the oscillating behavior of $w_{\rm DE}$ are key features for distinguishing our scenario from other dark energy models like quintessence., Comment: 11 pages, 5 figures

We place observational constraints on two models within a class of scenarios featuring an elastic interaction between dark energy and dark matter that only produces momentum exchange up to first order in cosmological perturbations. The first one corresponds to a perfect-fluid model of the dark components with an explicit interacting Lagrangian, where dark energy acts as a dark radiation at early times and behaves as a cosmological constant at late times. The second one is a dynamical dark energy model with a dark radiation component, where the momentum exchange covariantly modifies the conservation equations in the dark sector. Using Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations (BAO), and Supernovae type Ia (SnIa) data, we show that the Hubble tension can be alleviated due to the additional radiation, while the $\sigma_8$ tension present in the $\Lambda$-Cold-Dark-Matter model can be eased by the weaker galaxy clustering that occurs in these interacting models. Furthermore, we show that, while CMB+BAO+SnIa data put only upper bounds on the coupling strength, adding low-redshift data in the form of a constraint on the parameter $S_8$ strongly favours nonvanishing values of the interaction parameters. Our findings are in line with other results in the literature that could signal a universal trend of the momentum exchange among the dark sector., Comment: 26 pages, 11 figures

The next generation of galaxy surveys will allow us to test some fundamental aspects of the standard cosmological model, including the assumption of a minimal coupling between the components of the dark sector. In this paper, we present the Javalambre Physics of the Accelerated Universe Astrophysical Survey (J-PAS) forecasts on a class of unified models where cold dark matter interacts with a vacuum energy, considering future observations of baryon acoustic oscillations, redshift-space distortions, and the matter power spectrum. After providing a general framework to study the background and linear perturbations, we focus on a concrete interacting model without momentum exchange by taking into account the contribution of baryons. We compare the J-PAS results with those expected for DESI and Euclid surveys and show that J-PAS is competitive to them, especially at low redshifts. Indeed, the predicted errors for the interaction parameter, which measures the departure from a ΛCDM model, can be comparable to the actual errors derived from the current data of cosmic microwave background temperature anisotropies. © 2021 IOP Publishing Ltd and Sissa Medialab., MB acknowledges Istituto Nazionale di Fisica Nucleare (INFN), sezione di Napoli, iniziativa specifica QGSKY. RvM acknowledges support from the Programa de Capacitacao Institucional do Observatorio Nacional PCI/ON/MCTI. SC is supported by CNPq (Grants No. 307467/2017-1 and 420641/2018-1). JA is supported by CNPq (Grants No. 310790/20140 and 400471/2014-0) and FAPERJ (Grant No. 233906). JCF is supported by CNPq (Grant No. 304521/2015-9) and FAPES (Grant No. 78/2017). ST is supported by the Grant-inAid for Scientific Research Fund of the JSPS No. 19K03854. VM thanks CNPq (Brazil) and FAPES (Brazil) for partial financial support. This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 888258. This paper has gone through internal review by the J-PAS collaboration. Funding for the J-PAS Project has been provided by the Governments of Spain and Aragon through the Fondo de Inversion de Teruel, European FEDER funding and the MINECO and by the Brazilian agencies FINEP, FAPESP, FAPERJ and by the National Observatory of Brazil., With funding from the Spanish government through the Severo Ochoa Centre of Excellence accreditation SEV-2017-0709.

In Einstein-Aether theory, we study the stability of black holes against odd-parity perturbations on a spherically symmetric and static background. For odd-parity modes, there are two dynamical degrees of freedom arising from the tensor gravitational sector and Aether vector field. We derive general conditions under which neither ghosts nor Laplacian instabilities are present for these dynamical fields. We apply these results to concrete black hole solutions known in the literature and show that some of those solutions can be excluded by the violation of stability conditions. The exact Schwarzschild solution present for $c_{13} = c_{14} = 0$, where $c_i$'s are the four coupling constants of the theory with $c_{ij}=c_i + c_j$, is prone to Laplacian instabilities along the angular direction throughout the horizon exterior. However, we find that the odd-parity instability of high radial and angular momentum modes is absent for black hole solutions with $c_{13} = c_4 = 0$ and $c_1 \geq 0$., Comment: Revtex4-1, no tables and figures. The analysis of stability against the inclined modes is added, and some typos are further corrected. Version to appear in Phys. Rev. D

Full list of authors: Bonoli, S.; Marín-Franch, A.; Varela, J.; Vázquez Ramió, H.; Abramo, L. R.; Cenarro, A. J.; Dupke, R. A.; Vílchez, J. M.; Cristóbal-Hornillos, D.; González Delgado, R. M.; Hernández-Monteagudo, C.; López-Sanjuan, C.; Muniesa, D. J.; Civera, T.; Ederoclite, A.; Hernán-Caballero, A.; Marra, V.; Baqui, P. O.; Cortesi, A.; Cypriano, E. S.; Daflon, S.; de Amorim, A. L.; Díaz-García, L. A.; Diego, J. M.; Martínez-Solaeche, G.; Pérez, E.; Placco, V. M.; Prada, F.; Queiroz, C.; Alcaniz, J.; Alvarez-Candal, A.; Cepa, J.; Maroto, A. L.; Roig, F.; Siffert, B. B.; Taylor, K.; Benitez, N.; Moles, M.; Sodré, L.; Carneiro, S.; Mendes de Oliveira, C.; Abdalla, E.; Angulo, R. E.; Aparicio Resco, M.; Balaguera-Antolínez, A.; Ballesteros, F. J.; Brito-Silva, D.; Broadhurst, T.; Carrasco, E. R.; Castro, T.; Cid Fernandes, R.; Coelho, P.; de Melo, R. B.; Doubrawa, L.; Fernandez-Soto, A.; Ferrari, F.; Finoguenov, A.; García-Benito, R.; Iglesias-Páramo, J.; Jiménez-Teja, Y.; Kitaura, F. S.; Laur, J.; Lopes, P. A. A.; Lucatelli, G.; Martínez, V. J.; Maturi, M.; Overzier, R. A.; Pigozzo, C.; Quartin, M.; Rodríguez-Martín, J. E.; Salzano, V.; Tamm, A.; Tempel, E.; Umetsu, K.; Valdivielso, L. ; von Marttens, R.; Zitrin, A.; Díaz-Martín, M. C.; López-Alegre, G.; López-Sainz, A.; Yanes-Díaz, A.; Rueda-Teruel, F.; Rueda-Teruel, S.; Abril Ibañez, J.; L Antón Bravo, J.; Bello Ferrer, R.; Bielsa, S.; Casino, J. M.; Castillo, J.; Chueca, S.; Cuesta, L.; Garzarán Calderaro, J.; Iglesias-Marzoa, R.; Íniguez, C.; Lamadrid Gutierrez, J. L.; Lopez-Martinez, F.; Lozano-Pérez, D.; Maícas Sacristán, N.; Molina-Ibáñez, E. L.; Moreno-Signes, A.; Rodríguez Llano, S.; Royo Navarro, M.; Tilve Rua, V.; Andrade, U.; Alfaro, E. J.; Akras, S.; Arnalte-Mur, P.; Ascaso, B.; Barbosa, C. E.; Beltrán Jiménez, J.; Benetti, M.; Bengaly, C. A. P.; Bernui, A.; Blanco-Pillado, J. J.; Borges Fernandes, M.; Bregman, J. N.; Bruzual, G.; Calderone, G.; Carvano, J. M.; Casarini, L.; Chaves-Montero, J.; Chies-Santos, A. L.; Coutinho de Carvalho, G.; Dimauro, P.; Duarte Puertas, S.; Figueruelo, D.; González-Serrano, J. I.; Guerrero, M. A.; Gurung-López, S.; Herranz, D.; Huertas-Company, M.; Irwin, J. A.; Izquierdo-Villalba, D.; Kanaan, A.; Kehrig, C.; Kirkpatrick, C. C.; Lim, J.; Lopes, A. R.; Lopes de Oliveira, R.; Marcos-Caballero, A.; Martínez-Delgado, D.; Martínez-González, E.; Martínez-Somonte, G.; Oliveira, N.; Orsi, A. A.; Penna-Lima, M.; Reis, R. R. R.; Spinoso, D.; Tsujikawa, S.; Vielva, P.; Vitorelli, A. Z.; Xia, J. Q.; Yuan, H. B.; Arroyo-Polonio, A.; Dantas, M. L. L.; Galarza, C. A.; Gonçalves, D. R.; Gonçalves, R. S.; Gonzalez, J. E.; Gonzalez, A. H.; Greisel, N.; Jiménez-Esteban, F.; Landim, R. G.; Lazzaro, D.; Magris, G.; Monteiro-Oliveira, R.; Pereira, C. B.; Rebouças, M. J.; Rodriguez-Espinosa, J. M.; Santos da Costa, S.; Telles, E., The Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS) will scan thousands of square degrees of the northern sky with a unique set of 56 filters using the dedicated 2:55m Javalambre Survey Telescope (JST) at the Javalambre Astrophysical Observatory. Prior to the installation of the main camera (4:2 deg2 field-of-view with 1.2 Gpixels), the JST was equipped with the JPAS-Pathfinder, a one CCD camera with a 0:3 deg2 field-of-view and plate scale of 0.23 arcsec pixel?1. To demonstrate the scientific potential of J-PAS, the JPAS-Pathfinder camera was used to perform miniJPAS, a _1 deg2 survey of the AEGIS field (along the Extended Groth Strip). The field was observed with the 56 J-PAS filters, which include 54 narrow band (FWHM _ 145 ) and two broader filters extending to the UV and the near-infrared, complemented by the u; g; r; i SDSS broad band filters. In this miniJPAS survey overview paper, we present the miniJPAS data set (images and catalogs), as we highlight key aspects and applications of these unique spectro-photometric data and describe how to access the public data products. The data parameters reach depths of magAB ' 22?23:5 in the 54 narrow band filters and up to 24 in the broader filters (5_ in a 300 aperture). The miniJPAS primary catalog contains more than 64 000 sources detected in the r band and with matched photometry in all other bands. This catalog is 99% complete at r = 23:6 (r = 22:7) mag for point-like (extended) sources. We show that our photometric redshifts have an accuracy better than 1% for all sources up to r = 22:5, and a precision of _0:3% for a subset consisting of about half of the sample. On this basis, we outline several scientific applications of our data, including the study of spatially-resolved stellar populations of nearby galaxies, the analysis of the large scale structure up to z _ 0:9, and the detection of large numbers of clusters and groups. Sub-percent redshift precision can also be reached for quasars, allowing for the study of the large-scale structure to be pushed to z 2. The miniJPAS survey demonstrates the capability of the J-PAS filter system to accurately characterize a broad variety of sources and paves the way for the upcoming arrival of J-PAS, which will multiply this data by three orders of magnitude. © 2021 EDP Sciences. All rights reserved., Funding for OAJ, UPAD, and CEFCA has been provided by the Governments of Spain and Aragon through the Fondo de Inversiones de Teruel; the Aragon Government through the Research Groups E96, E103, and E16_17R; the Spanish Ministry of Science, Innovation and Universities (MCIU/AEI/FEDER, UE) with grant PGC2018-097585-B-C21; the Spanish Ministry of Economy and Competitiveness (MINECO/FEDER, UE) under AYA2015-66211-C2-1-P, AYA2015-66211-C2-2, AYA2012-30789, and ICTS2009-14; and European FEDER funding (FCDD10-4E-867, FCDD13-4E-2685). This work has made used of CEFCA's Scientific High Performance Computing system which has been funded by the Governments of Spain and Aragon through the Fondo de Inversiones de Teruel, and the Spanish Ministry of Economy and Competitiveness (MINECO-FEDER, grant AYA2012-30789). Funding for the J-PAS project has been provided also by the Brazilian agencies FINEP, FAPESP, FAPERJ and by the National Observatory of Brazil. Additional funding was also provided by the Tartu Observatory and by the Chinese Consortium from the Academy of Sciences SB acknowledges partial support from the project PGC2018-097585-B-C22. R.A.D. acknowledges support from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico -CNPq through BP grant 308105/2018-4, and the Financiadora de Estudos e Projetos -FINEP grants REF. 1217/13 -01.13.0279.00 and REF 0859/10 -01.10.0663.00 and also FAPERJ PRONEX grant E-26/110.566/2010 for hardware funding support for the J-PAS project through the National Observatory of Brazil and Centro Brasileiro de Pesquisas Fisicas. LRA acknowledges financial support from CNPq (306696/2018-5) and FAPESP (2015/17199-0). VMthanks CNPq (Brazil) and FAPES (Brazil) for partial financial support and the H2020 project No 888258. L.A.D.G. and K.U. acknowledge support from the Ministry of Science and Technology of Taiwan (grant MOST 106-2628-M-001-003-MY3) and from the Academia Sinica (grant AS-IA-107-M01). J.M.D. and D.H acknowledge the support of project PGC2018-101814-B-100. MQ thanks CNPq (Brazil) and FAPERJ (Brazil) for financial support. PC acknowledges financial support from FAPESP (2018/05392-8) and CNPq (310041/2018-0). AAC acknowledges support from FAPERJ (E26/203.186/2016), CNPq (304971/2016-2 and 401669/2016-5), and the Universidad de Alicante (contract UATALENTO1802). C.Q. acknowledges support from FAPESP (2015/11442-0 and 2019/067661). V.M.P. is supported by NOIRLab, which is managed by AURA under a cooperative agreement with the NSF. P.B acknowledges support from CAPES -Finance Code 001. IAA researchers acknowledge financial support from the State Agency for Research of the Spanish MCIU through the "Center of Excellence Severo Ochoa" award to the Instituto de Astrofisica de Andalucia (SEV-2017-0709). Authors acknowledge support from the Generalitat Valenciana project of excellence Prometeo/2020/085. RGD, GMS, JRM, RGB, EP acknowledge financial support from the project AyA2016-77846-P. TC is supported by the INFN INDARK PD51 and PRIN-MIUR 2015W7KAWC. MAR and ALM acknowledge support from the MINECO project FIS2016-78859P(AEI/FEDER, UE). ET, AT and JL acknowledge the support by ETAg grants IUT40-2 and by EU through the ERDF CoE grant TK133 and MOBTP86. CK, JMV, JIP acknowledge financial support from project AYA2016-79724C4-4P. PAAL thanks the support of CNPq (309398/2018-5). LC thanks CNPq for partial support. Y.J-T acknowledges financial support from the FAPERJ (E26/202.835/2016), and from the Horizon 2020 Marie Sklodowska-Curie grant agreement No 898633. DMD acknowledges financial support from the SFB 881 of the DFG and from the MINECO grant AYA2016-81065-C2-2. FP acknowledges support of the project PGC2018-101931-B-I00. JC acknowledges support of the project E AYA2017-88007-C3-1-P, and co-financed by the FEDER. JIGs acknowledges support of projects of reference AYA2017-88007-C3-3-P, and PGC2018-099705-B-I00 and co-financed by the FEDER. EMG and PV would like to acknowledge financial support from the project ESP2017-83921C2-1-R. GMS acknowleges financial support from a predoctoral contract, ref. PRE2018-085523 (MCIU/AEI/FSE, UE). S.C. is partially supported by CNPq. R.G.L. acknowledges CAPES (process 88881.162206/2017-01) and Alexander von Humboldt Foundation for the financial support. JSA acknowledges support from FAPERJ (E26/203.024/2017), CNP (310790/2014-0 and 400471/2014-0) and FINEP (1217/13 -01.13.0279.00 and Ref. 0859/10 -01.10.0663.00). RvM acknowledges support from CNPq. AFS, PAM, VJM and FJB acknowledge support from project AYA2016-81065-C2-2. PAM acknowledges support from the "Subprograma Atraccio de Talent -Contractes Postdoctorals de la Universitat de Valencia". ESC acknowledges support from CNPq (308539/20184) and FAPESP (2019/19687-2). CMdO acknowledges support from CNPq (grant 312333/2014-5) and FAPESP (grant 2009/54202-8). LSJ acknowledges support from CNPq (grant 304819/2017-4) and FAPESP (grant 2012/008004). JMC acknowledges support from CNPq (grant 310727/2016-2). C.H.-M. and N. Greisel also acknowledge the support of the European Union via the Career Integration Grant CIG-PCIG9-GA-2011-294183. JJBP and AMC would like to acknowledge the support from the grant PGC2018-094626-B-C21 and the Basque Government grant IT-979-16. AMC acknowledges the postdoctoral contract from the University of the Basque Country UPV/EHU "Especializacioon de personal investigador doctor" program. MLLD acknowledges CAPES -Finance Code 001; and CNPq (142294/2018-7). GB acknowledges financial support from the UNAM through grant DGAPA/PAPIIT IG100319, from CONACyT through grant CB2015-252364, and from FAPESP projects 2017/02375-2 and 2018/05392-8. M.J. Reboucas acknowledges the support of FAPERJ under a CNE E-26/202.864/2017 grant, and CNPq. Support by CNPq (305409/2016-6) and FAPERJ (E-26/202.841/2017) is acknowledged by DL. AB acknowledges a CNPq fellowship. C.A.G.acknowledges support from CAPES. EA acknowledges support from FAPESP (2011/18729-1). AC acknowledges support from PNPD/CAPES. ABA and FSK acknowledge the Severo Ochoa program SEV-2015-0548. FSK also thanks the AYA2017-89891-P and the RYC2015-18693 grants. DF acknowledges support from the Atraccion del Talento Cientifico en Salamanca programme and the project PGC2018-096038B-I00.

In a class of generalized Proca theories, we study the existence of neutron star solutions with a nonvanishing temporal component of the vector field $A_\mu$ approaching 0 toward spatial infinity, as they may be the endpoints of tachyonic instabilities of neutron star solutions in general relativity with $A_{\mu}=0$. Such a phenomenon is called spontaneous vectorization, which is analogous to spontaneous scalarization in scalar-tensor theories with nonminimal couplings to the curvature or matter. For the nonminimal coupling $\beta X R$, where $\beta$ is a coupling constant and $X=-A_{\mu}A^{\mu}/2$, we show that there exist both 0-node and 1-node vector-field solutions, irrespective of the choice of the equations of state of nuclear matter. The 0-node solution, which is present only for $\beta=-{\cal O}(0.1)$, may be induced by some nonlinear effects such as the selected choice of initial conditions. The 1-node solution exists for $\beta=-{\cal O}(1)$, which suddenly emerges above a critical central density of star and approaches the general relativistic branch with the increasing central density. We compute the mass $M$ and radius $r_s$ of neutron stars for some realistic equations of state and show that the $M$-$r_s$ relations of 0-node and 1-node solutions exhibit notable difference from those of scalarized solutions in scalar-tensor theories. Finally, we discuss the possible endpoints of tachyonic instabilities., Comment: 19 pages, 5 figures, published version

We study the stability of relativistic stars in scalar-tensor theories with a nonminimal coupling of the form $F(\phi)R$, where $F$ depends on a scalar field $\phi$ and $R$ is the Ricci scalar. On a spherically symmetric and static background, we incorporate a perfect fluid minimally coupled to gravity as a form of the Schutz-Sorkin action. The odd-parity perturbation for the multipoles $l \geq 2$ is ghost-free under the condition $F(\phi)>0$, with the speed of gravity equivalent to that of light. For even-parity perturbations with $l \geq 2$, there are three propagating degrees of freedom arising from the perfect-fluid, scalar-field, and gravity sectors. For $l=0, 1$, the dynamical degrees of freedom reduce to two modes. We derive no-ghost conditions and the propagation speeds of these perturbations and apply them to concrete theories of hairy relativistic stars with $F(\phi)>0$. As long as the perfect fluid satisfies a weak energy condition with a positive propagation speed squared $c_m^2$, there are neither ghost nor Laplacian instabilities for theories of spontaneous scalarization and Brans-Dicke (BD) theories with a BD parameter $\omega_{\rm BD}>-3/2$ (including $f(R)$ gravity). In these theories, provided $0, Comment: 24 pages, 2 figures

We consider a cosmological scenario where the dark sector is described by two perfect fluids that interact through a velocity-dependent coupling. This coupling gives rise to an interaction in the dark sector driven by the relative velocity of the components, thus making the background evolution oblivious to the interaction and only the perturbed Euler equations are affected at first order. We obtain the equations governing this system with the Schutz-Sorkin Lagrangian formulation for perfect fluids and derive the corresponding stability conditions to avoid ghosts and Laplacian instabilities. As a particular example, we study a model where dark energy behaves as a radiation fluid at high redshift while it effectively becomes a cosmological constant in the late Universe. Within this scenario, we show that the interaction of both dark components leads to a suppression of the dark matter clustering at late times. We also argue the possibility that this suppression of clustering together with the additional dark radiation at early times can simultaneously alleviate the $\sigma_8$ and $H_0$ tensions., Comment: 22 pages, 12 figures, fixed minor typos

We study a coupled dark energy scenario in which a massive vector field $A_{\mu}$ with broken $U(1)$ gauge symmetry interacts with the four-velocity $u_c^{\mu}$ of cold dark matter (CDM) through the scalar product $Z=-u_c^{\mu} A_{\mu}$. This new coupling corresponds to the momentum transfer, so that the background vector and CDM continuity equations do not have explicit interacting terms analogous to the energy exchange. Hence the observational preference of uncoupled generalized Proca theories over the $\Lambda$CDM model can be still maintained at the background level. Meanwhile, the same coupling strongly affects the evolution of cosmological perturbations. While the effective sound speed of CDM vanishes, the propagation speed and no-ghost condition of a longitudinal scalar of $A_{\mu}$ and the CDM no-ghost condition are subject to nontrivial modifications by the $Z$ dependence in the Lagrangian. We propose a concrete dark energy model and show that the gravitational interaction on scales relevant to the linear growth of large-scale structures can be smaller than the Newton constant at low redshifts. This leads to the suppression of growth rates of both CDM and total matter density perturbations, so our model allows an interesting possibility for reducing the tension of matter density contrast $\sigma_8$ between high- and low-redshift measurements., Comment: 25 pages, 3 figures

We investigate the possibility that axion-like particles (ALPs) with various potentials account for the isotropic birefringence recently reported by analyzing the Planck 2018 polarization data. For the quadratic and cosine potentials, we obtain lower bounds on the mass, coupling constant to photon $g$, abundance and equation of state of the ALP to produce the observed birefringence. Especially when the ALP is responsible for dark energy, it is possible to probe the tiny deviation of dark energy equation of state from $-1$ through the cosmic birefringence. We also explore ALPs working as early dark energy (EDE), which alleviates the Hubble tension problem. Since the other parameters are limited by the EDE requirements, we narrow down the ALP-photon coupling to $10^{-19}\, {\rm GeV}^{-1}\lesssim g\lesssim 10^{-16}\, {\rm GeV}^{-1}$ for the decay constant $f=M_\mathrm{pl}$. Therefore, the Hubble tension and the isotropic birefringence imply that $g$ is typically the order of $f^{-1}$, which is a non-trivial coincidence., Comment: 19pages, 7 figures, typos corrected

For a scalar field $\phi$ coupled to cold dark matter (CDM), we provide a general framework for studying the background and perturbation dynamics on the isotropic cosmological background. The dark energy sector is described by a Horndeski Lagrangian with the speed of gravitational waves equivalent to that of light, whereas CDM is dealt as a perfect fluid characterized by the number density $n_c$ and four-velocity $u_c^\mu$. For a very general interacting Lagrangian $f(n_c, \phi, X, Z)$, where $f$ depends on $n_c$, $\phi$, $X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$, and $Z=u_c^{\mu} \partial_{\mu} \phi$, we derive the full linear perturbation equations of motion without fixing any gauge conditions. To realize a vanishing CDM sound speed for the successful structure formation, the interacting function needs to be of the form $f=-f_1(\phi, X, Z)n_c+f_2(\phi, X, Z)$. Employing a quasi-static approximation for the modes deep inside the sound horizon, we obtain analytic formulas for the effective gravitational couplings of CDM and baryon density perturbations as well as gravitational and weak lensing potentials. We apply our general formulas to several interacting theories and show that, in many cases, the CDM gravitational coupling around the quasi de-Sitter background can be smaller than the Newton constant $G$ due to a momentum transfer induced by the $Z$-dependence in $f_2$., Comment: 26 pages, 1 figure, published version

We investigate a dark energy scenario in which a canonical scalar field $\phi$ is coupled to the four velocity $u_{c}^{\mu}$ of cold dark matter (CDM) through a derivative interaction $u_{c}^{\mu} \partial_{\mu} \phi$. The coupling is described by an interacting Lagrangian $f(X, Z)$, where $f$ depends on $X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$ and $Z=u_{c}^{\mu} \partial_{\mu} \phi$. We derive stability conditions of linear scalar perturbations for the wavelength deep inside the Hubble radius and show that the effective CDM sound speed is close to 0 as in the standard uncoupled case, while the scalar-field propagation speed is affected by the interacting term $f$. Under a quasi-static approximation, we also obtain a general expression of the effective gravitational coupling felt by the CDM perturbation. We study the late-time cosmological dynamics for the coupling $f \propto X^{(2-m)/2}Z^m$ and show that the gravitational coupling weaker than the Newton constant can be naturally realized for $m>0$ on scales relevant to the growth of large-scale structures. This allows the possibility for alleviating the tension of $\sigma_8$ between low- and high-redshift measurements., Comment: 15 pages, 2 figures, published version

We place observational constraints on the Galileon ghost condensate model, a dark energy proposal in cubic-order Horndeski theories consistent with the gravitational-wave event GW170817. The model extends the covariant Galileon by taking an additional higher-order field derivative $X^2$ into account. This allows for the dark energy equation of state $w_{\rm DE}$ to access the region $-2, Comment: 6 pages, 4 figures, accepted version by PRD

In $f(R)$ gravity and Brans-Dicke theory with scalar potentials, we study the structure of neutron stars on a spherically symmetric and static background for two equations of state: SLy and FPS. In massless BD theory, the presence of a scalar coupling $Q$ with matter works to change the star radius in comparison to General Relativity, while the maximum allowed mass of neutron stars is hardly modified for both SLy and FPS equations of state. In Brans-Dicke theory with the massive potential $V(\phi)=m^2 \phi^2/2$, where $m^2$ is a positive constant, we show the difficulty of realizing neutron star solutions with a stable field profile due to the existence of an exponentially growing mode outside the star. As in $f(R)$ gravity with the $R^2$ term, this property is related to the requirement of extra boundary conditions of the field at the surface of star. For the self-coupling potential $V(\phi)=\lambda \phi^4/4$, this problem can be circumvented by the fact that the second derivative $V_{,\phi \phi}=3\lambda\phi^2$ approaches 0 at spatial infinity. In this case, we numerically show the existence of neutron star solutions for both SLy and FPS equations of state and discuss how the mass-radius relation is modified as compared to General Relativity., Comment: 19 pages, 6 figures, published version

We provide a general framework for studying the evolution of background and cosmological perturbations in the presence of a vector field $A_{\mu}$ coupled to cold dark matter (CDM). We consider an interacting Lagrangian of the form $Q f(X) T_c$, where $Q$ is a coupling constant, $f$ is an arbitrary function of $X=-A_{\mu}A^{\mu}/2$, and $T_c$ is a trace of the CDM energy-momentum tensor. The matter coupling affects the no-ghost condition and sound speed of linear scalar perturbations deep inside the sound horizon, while those of tensor and vector perturbations are not subject to modifications. The existence of interactions also modifies the no-ghost condition of CDM density perturbations. We propose a concrete model of coupled vector dark energy with the tensor propagation speed equivalent to that of light. In comparison to the $Q=0$ case, we show that the decay of CDM to the vector field leads to the phantom dark energy equation of state $w_{\rm DE}$ closer to $-1$. This alleviates the problem of observational incompatibility of uncoupled models in which $w_{\rm DE}$ significantly deviates from $-1$. The maximum values of $w_{\rm DE}$ reached during the matter era are bounded from the CDM no-ghost condition of future de Sitter solutions., Comment: 18 pages, 3 figures, published version

We provide a general framework for studying the dark energy cosmology in which a scalar field $\phi$ is nonminimally and kinetically coupled to Cold Dark Matter (CDM). The scalar-graviton sector is described by the action of Horndeski theories with the speed of gravitational waves equivalent to that of light, whereas CDM is treated as a perfect fluid given by a Schutz-Sorkin action. We consider two interacting Lagrangians of the forms $f_1(\phi,X)\rho_c (n_c)$ and $f_2 (n_c, \phi,X) J_c^{\mu} \partial_{\mu}\phi$, where $X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$, $\rho_c$ and $n_c$ are the energy density and number density of CDM respectively, and $J_c^{\mu}$ is a vector field related to the CDM four velocity. We derive the scalar perturbation equations of motion without choosing any special gauges and identify conditions for the absence of ghosts and Laplacian instabilities on scales deep inside the sound horizon. Applying a quasi-static approximation in a gauge-invariant manner, we also obtain the effective gravitational couplings felt by CDM and baryons for the modes relevant to the linear growth of large-scale structures. In particular, the $n_c$ dependence in the coupling $f_2$ gives rise to an interesting possibility for realizing the gravitational coupling with CDM weaker than the Newton gravitational constant $G$., Comment: published version with corrections, 24 pages

We study observational constraints on a specific dark energy model in the framework of Gleyzes-Langlois-Piazza-Vernizzi theories, which extends the Galileon ghost condensate (GGC) to the domain of beyond Horndeski theories. In this model, we show that the Planck cosmic microwave background (CMB) data, combined with datasets of baryon acoustic oscillations, supernovae type Ia, and redshift-space distortions, give the tight upper bound $|\alpha_{\rm H}^{(0)}| \le {\cal O}(10^{-6})$ on today's beyond-Horndeski (BH) parameter $\alpha_{\rm H}$. This is mostly attributed to the shift of CMB acoustic peaks induced by the early-time changes of cosmological background and perturbations arising from the dominance of $\alpha_{\rm H}$ in the dark energy density. In comparison to the $\Lambda$-cold-dark-matter ($\Lambda$CDM) model, our BH model suppresses the large-scale integrated-Sachs-Wolfe (ISW) tail of CMB temperature anisotropies due to the existence of cubic Galileons, and it modifies the small-scale CMB power spectrum because of the different background evolution. We find that the BH model considered fits the data better than $\Lambda$CDM according to the $\chi^2$ statistics, yet the deviance information criterion (DIC) slightly favors the latter. Given the fact that our BH model with $\alpha_{\rm H}=0$ (i.e., the GGC model) is favored over $\Lambda$CDM even by the DIC, there are no particular signatures for the departure from Horndeski theories in current observations., Comment: 17 pages, 12 figures, accepted version by PRD

In dark energy models where a scalar field $\phi$ is coupled to the Ricci scalar $R$ of the form $e^{-2Q (\phi-\phi_0)/M_{\rm pl}}R$, where $Q$ is a coupling constant, $\phi_0$ is today's value of $\phi$, and $M_{\rm pl}$ is the reduced Planck mass, we study how the recent Lunar Laser Ranging (LLR) experiment places constraints on the nonminimal coupling from the time variation of gravitational coupling. Besides a potential of the light scalar responsible for cosmic acceleration, we take a cubic Galileon term into account to suppress fifth forces in over-density regions of the Universe. Even if the scalar-matter interaction is screened by the Vainshtein mechanism, the time variation of gravitational coupling induced by the cosmological background field $\phi$ survives in the solar system. For a small Galileon coupling constant $\beta_3$, there exists a kinetically driven $\phi$-matter-dominated-epoch ($\phi$MDE) prior to cosmic acceleration. In this case, we obtain the stringent upper limit $Q \le 3.4 \times 10^{-3}$ from the LLR constraint. For a large $\beta_3$ without the $\phi$MDE, the coupling $Q$ is not particularly bounded from above, but the cosmological Vainshtein screening strongly suppresses the time variation of $\phi$ such that the dark energy equation of state $w_{\rm DE}$ reaches the value close to $-1$ at high redshifts. We study the modified gravitational wave propagation induced by the nonminimal coupling to gravity and show that, under the LLR bound, the difference between the gravitational wave and luminosity distances does not exceed the order $10^{-5}$ over the redshift range $0, Comment: 18 pages, 9 figures

In cubic-order Horndeski theories where a scalar field $\phi$ is coupled to nonrelativistic matter with a field-dependent coupling $Q(\phi)$, we derive the most general Lagrangian having scaling solutions on the isotropic and homogenous cosmological background. For constant $Q$ including the case of vanishing coupling, the corresponding Lagrangian reduces to the form $L=Xg_2(Y)-g_3(Y)\square \phi$, where $X=-\partial_{\mu}\phi\partial^{\mu}\phi/2$ and $g_2, g_3$ are arbitrary functions of $Y=Xe^{\lambda \phi}$ with constant $\lambda$. We obtain the fixed points of the scaling Lagrangian for constant $Q$ and show that the $\phi$-matter-dominated-epoch ($\phi$MDE) is present for the cubic coupling $g_3(Y)$ containing inverse power-law functions of $Y$. The stability analysis around the fixed points indicates that the $\phi$MDE can be followed by a stable critical point responsible for the cosmic acceleration. We propose a concrete dark energy model allowing for such a cosmological sequence and show that the ghost and Laplacian instabilities can be avoided even in the presence of the cubic coupling., Comment: 17 pages, 3 figures, version to appear in Physical Review D

In quadratic-order degenerate higher-order scalar-tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solutions characterized by $\dot{\phi}/H^p={\rm constant}$, where $\dot{\phi}$ is the time derivative of a scalar field $\phi$, $H$ is the Hubble expansion rate, and $p$ is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian $L=c_2X-c_3X^{(p-1)/(2p)} \square \phi$, where $c_2, c_3$ are constants and $X$ is the kinetic energy of $\phi$, the DHOST interaction breaks this structure for $p \neq 1$. Even in the latter case, however, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state $w_{\phi}=-1-2p\dot{H}/(3H^2)$. The scaling solution, which corresponds to $p=1$, is the unique case in which all the terms in the field density $\rho_{\phi}$ and the pressure $P_{\phi}$ obey the scaling relation $\rho_{\phi} \propto P_{\phi} \propto H^2$. Extending the analysis to the coupled DHOST theories with the field-dependent coupling $Q(\phi)$ between the scalar field and matter, we show that the scaling solution exists for $Q(\phi)=1/(\mu_1 \phi+\mu_2)$, where $\mu_1$ and $\mu_2$ are constants. For the constant $Q$, i.e., $\mu_1=0$, we derive fixed points of the dynamical system by using the general Lagrangian with scaling solutions. This result can be applied to the model construction of late-time cosmic acceleration preceded by the scaling $\phi$-matter-dominated epoch., Comment: 12 pages

In the framework of effective field theories with prominent helicity-0 and helicity-1 fields coupled to each other via a dimension-3 operator, we study the dynamics of inflation driven by the helicity-0 mode, with a given potential energy, as well as the evolution of cosmological perturbations, influenced by the presence of a mixing term between both helicities. In this scenario, the temporal component of the helicity-1 mode is an auxiliary field and can be integrated out in terms of the time derivative of the helicity-0 mode, so that the background dynamics effectively reduces to that in single-field inflation modulated by a parameter $\beta$ associated to the coupling between helicity-0 and helicity-1 modes. We discuss the evolution of a longitudinal scalar perturbation $\psi$ and an inflaton fluctuation $\delta \phi$, and explicitly show that a particular combination of these two, which corresponds to an isocurvature mode, is subject to exponential suppression by the vector mass comparable to the Hubble expansion rate during inflation. Furthermore, we find that the effective single-field description corrected by $\beta$ also holds for the power spectrum of curvature perturbations generated during inflation. We compute the standard inflationary observables such as the scalar spectral index $n_s$ and the tensor-to-scalar ratio $r$ and confront several inflaton potentials with the recent observational data provided by Planck 2018. Our results show that the coupling between helicity-0 and helicity-1 modes can lead to a smaller value of the tensor-to-scalar ratio especially for small-field inflationary models, so our scenario exhibits even better compatibility with the current observational data., Comment: 13 pages, 2 figures. Version to be published in Physical Review D

For a theory in which a scalar field $\phi$ has a nonminimal derivative coupling to the Einstein tensor $G_{\mu \nu}$ of the form $\phi\,G_{\mu \nu}\nabla^{\mu}\nabla^{\nu} \phi$, it is known that there exists a branch of static and spherically-symmetric relativistic stars endowed with a scalar hair in their interiors. We study the stability of such hairy solutions with a radial field dependence $\phi(r)$ against odd- and even-parity perturbations. We show that, for the star compactness ${\cal C}$ smaller than $1/3$, they are prone to Laplacian instabilities of the even-parity perturbation associated with the scalar-field propagation along an angular direction. Even for ${\cal C}>1/3$, the hairy star solutions are subject to ghost instabilities. We also find that even the other branch with a vanishing background field derivative is unstable for a positive perfect-fluid pressure, due to nonstandard propagation of the field perturbation $\delta \phi$ inside the star. Thus, there are no stable star configurations in derivative coupling theory without a standard kinetic term, including both relativistic and nonrelativistic compact objects., Comment: 17 pages, 8 figures, published version

We consider higher-order derivative interactions beyond second-order generalized Proca theories that propagate only the three desired polarizations of a massive vector field besides the two tensor polarizations from gravity. These new interactions follow the similar construction criteria to those arising in the extension of scalar–tensor Horndeski theories to Gleyzes–Langlois–Piazza–Vernizzi (GLPV) theories. On the isotropic cosmological background, we show the existence of a constraint with a vanishing Hamiltonian that removes the would-be Ostrogradski ghost. We study the behavior of linear perturbations on top of the isotropic cosmological background in the presence of a matter perfect fluid and find the same number of propagating degrees of freedom as in generalized Proca theories (two tensor polarizations, two transverse vector modes, and two scalar modes). Moreover, we obtain the conditions for the avoidance of ghosts and Laplacian instabilities of tensor, vector, and scalar perturbations. We observe key differences in the scalar sound speed, which is mixed with the matter sound speed outside the domain of generalized Proca theories., Physics Letters B, 760, ISSN:0370-2693, ISSN:0031-9163, ISSN:1873-2445

We argue that the $\Lambda$CDM tensions of the Hubble-Lemaitre expansion rate $H_0$ and the clustering normalization $\sigma_8$ can be eased, at least in principle, by considering an interaction between dark energy and dark matter in such a way to induce a small and positive early effective equation of state and a weaker gravity. For a dark energy scalar field $\phi$ interacting with dark matter through an exchange of both energy and momentum, we derive a general form of the Lagrangian allowing for the presence of scaling solutions. In a subclass of such interacting theories, we show the existence of a scaling $\phi$-matter-dominated-era ($\phi$MDE) which can potentially alleviate the $H_0$ tension by generating an effective high-redshift equation of state. We also study the evolution of perturbations for a model with $\phi$MDE followed by cosmic acceleration and find that the effective gravitational coupling relevant to the linear growth of large-scale structures can be smaller than the Newton gravitational constant $G$ at low redshifts. The momentum exchange between dark energy and dark matter plays a crucial role for realizing weak gravity, while the energy transfer is also required for the existence of $\phi$MDE., Comment: 20 pages, three figures; some typos corrected; matches version accepted for publication on JCAP

The gravitational-wave event GW170817 from a binary neutron star merger together with the electromagnetic counterpart showed that the speed of gravitational waves $c_t$ is very close to that of light for the redshift $z, Comment: 51 pages, 5 figures, published version