# Gravitational Particle Production and the Hubble Tension
## 1 Introduction  
Gravitational particle production is a generic property of quantum fields in time-dependent backgrounds such as the Friedman-Lemaitre-Robertson-Walker (FLRW) spacetimes [1; 2]. For example, the solution of the effective equation of motion of a free scalar field (namely, the mode function) at two different times, in general, is different since the equation of motion contains a time-dependent effective mass. Hence, there exist different vacua at different times (that are described by different creation and annihilation operators and different mode functions). The mode function (and the corresponding creation/annihilation operators) at a given time may be expressed in terms of the mode function (and the corresponding creation/annihilation operators) at another time by a Bogolyubov transformation. Thus, a mode function at an initial time (that describes an \"in\" state) evolves into another value at a later time that may be expanded in terms of the mode function at that time (namely, the mode function of the \"out\" state). This is the well-known gravitational particle production. Therefore, gravitational particle production is a generic process for quantum fields in FLRW spacetimes. Hence, gravitational particle production necessarily takes place in cosmology. The aim of this study is to see the degree of the impact of this process on the standard cosmology through the example of a scalar field in the era after the photon decoupling till the present.  
Hubble tension is the huge discrepancy between the direct local measurements of Hubble constant by type SN Ia supernovas calibrated by [PERSON] [3] and the measurements of Planck [4] and other non-local measurements such as baryon acoustic oscillations (BAO) [5; 6] imprinted on galaxy autocorrelation functions (that also involve effects of much earlier times and assume \(\Lambda\)CDM). The values of Hubble constant obtained from local measurements are almost certainly higher than the ones that also include the effect of higher redshifts. For example, [3] finds the Hubble constant as \((73.04\pm 1.04)\)\(\,\mathrm{km\,s^{-1}\,Mpc^{-1}}\), while [4] finds it as \((67.4\pm\,0.5)\)\(\,\mathrm{s^{-1}\,Mpc^{-1}}\). SN Ia supernova and Planck measurements differ by at least \(5\sigma\)[3; 4; 7; 8; 9]. This is called Hubble tension. There are many different approaches and models proposed as solutions of the Hubble tension problem [7; 8; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. The standard approach of the theoretical models that attempt to solve this problem is to assume the value of the Hubble constant obtained by local measurements to be the correct one, and to seek a model that makes the results of Planck (and other non-local measurements)compatible with local measurements. In this vein, they try to modify \(\Lambda\)CDM at late times (close to the present time) or early times (just before the time of recombination) or at both epochs so that the equations given below have the same result as the local measurements. In this study, a different approach is adopted. The effect of gravitational particle production of scalar particles on the Hubble constant is considered. It is shown that, depending on the value of the total mass of the scalars in the model, inclusion of the effect of gravitational particle production in the context of \(\Lambda\)CDM may ameliorate or relieve the Hubble tension.  
In the following, first, in Section 2, the basic concepts and techniques necessary for a better understanding of the present study are briefly reviewed. In Section 3, it is shown that adiabatic approximation that is used in the present study is applicable to the era after photon decoupling in \(\Lambda\)CDM for a wide range of scalar particle masses. In Section 4, the contribution of gravitational particle production to energy density is discussed. In Section 5, the implications of gravitational particle production for Hubble tension are discussed. Finally, Section 6 summarizes the main conclusions.