Eur. Phys. J. C (2021) 81:544 https://doi.org/10.1140/epjc/s10052-021-09306-2 Regular Article - Theoretical Physics Constraints on interacting dark energy models through cosmic chronometers and Gaussian process Muhsin Aljaf1,2,a , Daniele Gregoris3,4,5,b , Martiros Khurshudyan6,7,8,c 1 Department of Astronomy, University of Science and Technology of China, Hefei 230026, Anhui, China 2 Department of Physics, College of Education, University of Garmian, Kalar, Kurdistan Region, Iraq 3 School of Science, Jiangsu University of Science and Technology, Zhenjiang 212003, China 4 Center for Gravitation and Cosmology, College of Physical Science and Technology, Yangzhou University, 180 Siwangting Road, Yangzhou 225002, Jiangsu, China 5 School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China 6 Institute of Physics, University of Silesia, Katowice, Poland 7 Consejo Superior de Investigaciones Cientcas, ICE/CSIC-IEEC, Campus UAB, Carrer de Can Magrans s/n, 08193 Bellaterra, Barcelona, Spain 8 International Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia Received: 6 July 2020 / Accepted: 3 June 2021 © The Author(s) 2021 Abstract In this paper, after reconstructing the redshift 1 Introduction evolution of the Hubble function by adopting Gaussian pro- cess techniques, we estimate the best-fit parameters for some Despite being introduced for addressing a galactic puzzle, flat Friedmann cosmological models based on a modified i.e., the flattening of the rotation curves, on cosmological Chaplygin gas interacting with dark matter. In fact, the expan- scales dark matter combined together with dark energy can sion history of the Universe will be investigated because account for almost the full energy budget of the Universe. passively evolving galaxies constitute cosmic chronometers. While there are still no experimental devices for confirming An estimate for the present-day values of the deceleration the existence of dark energy directly, the situation seems to parameter, adiabatic speed of sound within the dark energy be different for dark matter thanks to the model-independent fluid, effective dark energy, and dark matter equation of state study of its distribution within the Milky Way [1]. In the sim- parameters is provided. By this, we mean that the interac- plest cosmological scenario, the ColdDarkMatter (CDM) tion term between the two dark fluids, which breaks the model, dark matter is macroscopically pictured as a pressure- Bianchi symmetries, will be interpreted as an effective contri- less fluid [2]. However, different microscopic foundations for bution to the dark matter pressure similarly to the framework dark matter have been proposed linking it to some fundamen- of the “Generalized Dark Matter”. We investigate whether tal elementary particle theories like those of massive neutri- the estimates of the Hubble constant and of the present-day nos [3], sterile neutrinos [4], axions [5], axinos [6], gravitinos abundance of dark matter are sensitive to the dark matter– [7], and neutralinos [8], just to mention a few examples (for a dark energy coupling. We will also show that the cosmic review of the different proposals of dark matter modelings in chronometers data favor a cold dark matter, and that our find- terms of elementary particles beyond the standard model, and ings are in agreement with the Le Châtelier–Braun principle how they affect the possible detection methods see [9,10]). according to which dark energy should decay into dark mat- However, massive neutrinos may not explain the formation ter. of large-scale structures [11,12], while, on the other hand, sterile neutrinos and axions are consistent with the CP vio- lation [13,14]. Furthermore, a detection of dark matter con- stituted of axinos, gravitinos, or neutralinos can lead to an experimental confirmation of supersymmetric field theories [15]. Microscopically, the possible different modelings of dark matter can be classified into hot (with the massive neu- a e-mail: [email protected] trinos being one example), warm (as for sterile neutrinos), b e-mail: [email protected] (corresponding author) and cold (like for axions and neutralinos) depending on the c e-mail: [email protected] 0123456789().: V,-vol 123 544 Page 2 of 14 Eur. Phys. J. C (2021) 81:544 energy scale of the elementary particles constituting this fluid entering the energy conservation equation, and consequently [2]. affecting the dust picture of dark matter. Thus, evaluating The aforementioned ColdDarkMatter model assumes a the effective equation of state parameter for the dark mat- dark energy fluid equivalent to a cosmological constant term ter, we can discriminate between cold, warm, and hot mod- entering the Einstein field equations, and that the two dark els. fluids are separately conserved, i.e., that they do not interact From the technical point of view, we employ Gaussian with each other through any energy exchange. A consistent Process techniques for reconstructing the redshift evolution joint interpretation of Planck results and weak lensing data of the Hubble function with the purpose of selecting the best however suggests that some redshift evolution of the dark cosmological model involving energy flows between dark energy equation of state parameter may be necessary [16, matter and dark energy. The latter is modeled in the form Sect.6.3]. Furthermore, interactions between dark energy and of the modified Chaplygin gas. In particular, we use 30 data dark matter can alleviate the coincidence problem [17–20], points for H = H(z) consisting of samples deduced from and mitigate the discrepancies between the estimates of the the differential age method, allowing the Gaussian Process to Hubble constant from cosmic microwave background mea- constrain the model parameters. Our purpose is to extend and surements or large scale structures versus supernovae data as complement the analysis of [81,82] by allowing a redshift- argued in [21,22]; we refer as well to [23–40] for quantita- dependent equation of state for dark energy (for accounting tive analyses on whether energy flows between dark matter for Planck observations), and interactions in the dark sector and dark energy affect the estimates of various cosmological (for alleviating the coincidence problem). parameters. Complementary studies have investigated how Our paper is organized as follows: we introduce our cos- the growth of instabilities in interacting dark models affects mological model in Sect. 2 reviewing the physical properties the formation of astrophysical structures [41–48], such as pri- of the modified Chaplygin gas and the features of the postu- mordial black holes [49–51], and galactic halos [52–62] due lated energy exchanges between the two cosmic fluids. Then, to the fact that the density of dark matter does not dilute any in Sect. 3 we explain the importance of the cosmic chronome- longer with the cube of the scale factor of the universe. Grav- ters as model-independent observational data for the recon- itational waves has been used for constraining dark interac- struction of the Hubble function, and for constraining the tions as well [63–66]. Moreover, a coupling between the dark values of the free parameters entering our class of models. In energy field and dark matter, with the latter pictured as neutri- sect. 4 we present the reconstruction for the Hubble function nos, affects the neutrinos’ masses estimates [67–69]. From a through gaussian processes, while in Sect. 5 we describe the more mathematical point of view, specific interplays between numerical method we have adopted for the integration of the the equation of state of dark energy and the interaction term field equations. The same Sect. exhibits explicitly also our with dark matter can give rise to different types of finite-time cosmological results comparing and contrasting between the kinematic and matter density singularities [70,71]. different possible choices of the interaction term. Lastly, we For taking into account the observational requirement of conclude in Sect. 6 with some remarks about the importance an evolving equation of state of dark energy, we model the of our study in light of the current literature estimates of dark energy fluid as a modified Chaplygin gas [72,73], rather the cosmological parameters by means of various different than considering just a redshift parametrization [74,75], datasets. because of its well established physical motivation. In fact, this fluid approach belongs to the wider class of chameleon field theories in which the constant equation of state param- 2 Overview of the cosmological model eter p = wρ is promoted to an energy-dependent functional according to w → w(ρ), and therefore it exhibits a sort of In this section we will introduce the basic equations of the running [76,77]. In particular, our fluid model interpolates cosmological model under investigation. For the geometrical between an ideal fluid behavior at low energy densities and modeling of the Universe we adopt the flat Friedmann metric a generalized Chaplygin gas in the high energy limit. There- which, in a Cartesian system of coordinates, reads [83]: fore, we implement a sort of asymptotic freedom at low ener- gies because the interactions within the fluid are suppressed ds2 =−dt2 + a2(t)(dx2 + dy2 + dz2), (1) [78,79], while at high energies, we match with the Born– Infeld paradigm with our model being formulated in terms where a(t) is the time-dependent scale factor of the Uni- of the Nambu–Goto string theory [80]. verse. Moreover, we picture the matter content of the Uni- Therefore, in this paper, we test a set of dark energy–dark verse as two perfect fluids with energy density ρ(t) and matter interacting models with the purpose of enlightening pressure p(t), respectively. Their stress-energy tensors are μ the physical properties of dark matter. In fact, the interaction T ν = diag[−ρi (t), pi (t), pi (t), pi (t)] with i = de, m for term between the two fluids behaves as an effective pressure dark energy and dark matter respectively.