A note on gravitational dark matter production

Academic Editor: Andrea Lapi Received: 14 December 2024 Revised: 29 January 2025 Accepted: 31 January 2025 Published: 4 February 2025 Citation: de Haro, J.; Pan, S. A Note on Gravitational Dark Matter Production. Universe 2025,11, 49. https://doi.org/10.3390/ universe11020049 Copyright: © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/ licenses/by/4.0/). universe Article A Note on Gravitational Dark Matter Production Jaume de Haro 1,* and Supriya Pan 2,3 1Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain 2Department of Mathematics, Presidency University, 86/1 College Street, Kolkata 700073, India; [email protected] 3Institute of Systems Science, Durban University of Technology, P.O. Box 1334, Durban 4000, South Africa *Correspondence: jaime.har[email protected] Abstract: Dark matter, one of the fundamental components of the universe, has remained mysterious in modern cosmology and particle physics, and hence, this field is of utmost importance at the present moment. One of the foundational questions in this direction is the origin of dark matter, which directly links to its creation. In the present article, we study the gravitational production of dark matter in two distinct contexts: firstly, when reheating occurs through gravitational particle production, and secondly, when it is driven by decay of the inflaton field. We establish a connection between the reheating temperature and the mass of dark matter, and from the reheating bounds, we determine the range of viable dark matter mass values. Keywords: reheating; gravitational particle production; constraints; dark matter 1. Introduction Dark matter, a fundamental component of the universe, remains one of the most profound mysteries in modern cosmology and particle physics. Despite its gravitational effects being observed across a variety of astrophysical scales, from galaxies to the cosmic microwave background, its origin and nature continue to elude us. Among the proposed mechanisms for dark matter generation, the gravitational production of dark matter stands out as a particularly compelling explanation, especially within the context of the early universe, see for instance [1–20]. Gravitational dark matter production leverages the unique role of gravity, a universal interaction, as the primary mechanism for generating dark matter particles. This process requires no additional couplings or interactions with the Standard Model, relying solely on the dynamics of the expanding universe. Such production is especially relevant during the reheating phase following cosmic inflation, where the universe transits from an early inflationary epoch to a radiation-dominated era. Two primary scenarios dominate discussions of gravitational dark matter production: one in which reheating occurs through the copious production of heavy particles [ 21 ] that subsequently decay into Standard Model particles, and another in which reheating results from the decay of the inflaton field directly into Standard Model particles [ 22 ]. These pathways connect the physics of inflation, reheating, and dark matter, establishing a relationship between the reheating temperature and the mass of dark matter. This connection allows us, through reheating constraints, to identify the range of viable dark matter masses. This short note delves into the theoretical foundations of gravitational dark matter production studied in several works [ 1 – 10 , 12 – 20 ] emphasizing its dependence on inflationary reheating dynamics, its potential observational implications, and the range of viable Universe 2025,11, 49 https://doi.org/10.3390/universe11020049 Universe 2025,11, 49 2 of 16 dark matter masses within this framework. The rest of this short note is organized as follows. In Section 2, we present the gravitational reheating formulas. Section 3describes the gravitational production of dark matter in the context of gravitational reheating. In Section 4, we discuss the gravitational production of dark matter and reheating through the decay of the inflaton field. Finally, in Section 5we conclude the present note with a brief summary. Throughout the article, we work under the assumption of the spatially flat Friedmann– Lemaître–Robertson–Walker (FLRW) geometry with a(t) (hereafter a ) representing the expansion scale factor of the universe and we have used the following notations: 1. “END” denotes the end of inflation. 2. “0” denotes the present time. 3. “reh” denotes the reheating time. 4. ρA,END denotes the energy density of the produced A -particles, at the end of inflation. 5. ρB,END = 3 M2 plH2 END is the energy density of the background at the end of inflation ( Mpl is the reduced Planck mass), that is, it corresponds to the energy density of the inflaton field. 6. ρrcorresponds to the energy density of the radiation. 7. ΘA=ρA,END ρB,END is the heating efficiency of the A-particles. 8. ¯ ΘA=3Γ2 AM2 pl ρB,END is the decay efficiency of the A -particles, where ΓA is the decay rate of the A-particles. 9. Ωrh2 is the density parameter for radiation ( h=H0/ 100 km/s/Mpc in which H0 is the present day value of the Hubble constant). 10. ΩAh2is the density parameter of the A-particles. Concerning the observational constraints on some of the parameters, we have used the following values [23]: 1. ΩYh2=0.12 ±0.0012, where the Y-particles are the candidate for dark matter. 2. Ωrh2∼ =2.47 ×10−5. 3. h=0.674 ±0.005. 4. T0=2.7255 ±0.006K∼ =2.35 ×10−13 GeV ∼ =9.6 ×10−32 Mpl. 2. Gravitational Reheating Formulas This section provides a detailed review of the results recently obtained in [ 24 , 25 ] (see also [ 21 , 26 – 28 ] to find some of the recent results in the context of gravitational reheating). To begin with, we consider a potential, which, near the minimum φ= 0, behaves like φ2n ( n is a natural number). We examine heavy massive X -particles, which are produced gravitationally due to the coupling of a X -field with the Ricci scalar (see, for instance, [ 21 ], and also [ 29 , 30 ] for the foundations of quantum field theory in curved spaces and its gravitational effects), and then they decay into Standard Model (SM) particles in order to reheat the universe. During the oscillations, close to the minimum, as the potential behaves like φ2n , hence, with the use of virial theorem, the effective Equation of State (EoS) parameter, weff , becomes weff =n−1 n+1 . Note that weff is higher than 1 / 3 for n> 2. This guarantees that the inflaton’s energy density decays faster than the energy density of the produced particles as well as their decay products, because, as a function of the scale factor, the energy density of matter decays as a−3 , the one of radiation as a−4 , and for the fluid with EoS parameter weff , it decays as a−3(1+weff) , which in the case of the inflaton’s energy density becomes a−6n n+1 . 1 As a consequence, the energy density of the latter products will eventually dominate and finally this will lead to a successful reheating of the universe. We shall concentrate on the class of inflationary potentials having similar Universe 2025,11, 49 3 of 16 behavior close to the minimum, e.g., Hyperbolic Inflation, Superconformal α -Attractor E-models or Superconformal α-Attractor T-models [31–33]. In order to clearly realize the mechanism of gravitational reheating, it is essential to understand the decay process. The decay process can be described with the use of the dynamics governed by the Boltzmann equations:    dρX(t) dt +3HρX(t) = −ΓXρX(t) dρr(t) dt +4Hρr(t) = ΓXρX(t), (1) where ρX(t) stands for the energy density of the produced X -particles, ρr(t) is the energy density of the radiation (the energy density of the decay products), and ΓX is the decay rate of X -particles into SM particles and it is assumed to be a constant. For example, considering an interaction of the X -field with fermions, this leads to the decay of the X -particles with the decay rate ΓX=ˆ h2mX 8π, where ˆ his a dimensionless constant [34]. We proceed with the solution which describes the energy density of the heavy massive particles as follows ρX(t) = ρX,ENDaEND a(t)3 e−ΓX(t−tEND),t≥tEND. (2) Here, this solution represents that the decay begins at the end of inflation, as discussed in [ 11 ]. This is due to the fact that the heavy X -particles are produced at the end of inflation, when the inflaton starts to oscillate. Consequently, the decay of these particles commences immediately upon their creation. Now, inserting (2) into the second equation of (1) and considering the fact that decay starts at the end of inflation, one can obtain ρr(t) = ρX,ENDaEND a(t)4Zt tEND a(s) aEND ΓXe−ΓX(s−tEND)ds. (3) We now proceed by defining tdec as the time where decay ends, that is, when ΓX(tdec −tEND)∼1 . This quickly gives tdec ∼tEND +1 ΓX , ( ΓX= 0), and we focus on the investigation of the evolution of the decay products for t≫tdec . We presume that the background dominates and with such an assumption, we have a(s)∼ =aEND(s/tEND)n+1 3n since weff =n−1 n+1 . Now, bearing in mind that Z∞ tEND (s/tEND)n+1 3nΓXe−ΓX(s−tEND)ds ∼ =¯ Θ−n+1 6n XΓ4n+1 3n∼ =¯ Θ−n+1 6n X, (4) where Γ denotes Euler’s Gamma function, we reach the conclusion that for t>tdec , the following approximation can be made ρr(t)∼ =ρX,END ¯ Θ−n+1 6n XaEND a(t)4 , (5) where we have used the definition of the decay efficiency of X -particles. Now, since, weff =n−1 n+1 , the energy density of the background, that is, the energy density of the inflaton field, satisfies the equation ˙ ρB= (1+weff)ρB, and thus, it evolves as ρB(t) = ρB,ENDaEND a(t)6n n+1. (6) Universe 2025,11, 49 4 of 16 Therefore, since the universe becomes reheated when ρB∼ρr, we obtain ΘX=¯ Θ n+1 6n XaEND areh 2(n−2) n+1, (7) and as a result of which, the energy density of the decay products at the time of reheating is given by the following ρr,reh ∼ =ρB,END ¯ Θ−n+1 2(n−2) XΘ 3n n−2 X. (8) Now, using the Stefan–Boltzmann law Treh =30 π2greh 1/4ρ1/4 r,reh (here greh = 106.75 denotes the effective number of degrees of freedom in the SM), one obtains the following reheating temperature, Treh =90 π2greh 1/4 ¯ Θ−n+1 8(n−2) XΘ 3n 4(n−2) XqHENDMpl. (9) It is essential to calculate the range of values for ΓX . We first note that ΓX≪HEND because the decay occurs well after the end of inflation. Additionally, we have assumed that the energy density of the background dominates at the end of decay, that means, ρr,dec ≪ρB,dec ∼ =3M2 plΓ2 X. Therefore, using the relation ρB,dec ∼ =3M2 plΓ2 X, we obtain, aEND adec 4∼ =¯ Θ 2(n+1) 3n X, (10) and inserting the above relation into ρr,dec ≪3M2 plΓ2 X, one arrives at Θ n n−1 X≪p¯ ΘX≪1. (11) Moreover, combining (11) with the following bound of the reheating temperature, 5 × 10 −22Mpl ≤Treh ≤ 5 × 10 −10Mpl , which states that the reheating temperature remains in an interval consistent with the Big Bang Nucleosynthesis (BBN), which occurs at about ∼ 1 MeV scale, and attains an upper bound at about ∼ 10 9 GeV in order to mitigate the issues related to the gravitino problem [ 35 – 38 ] one obtains four distinct cases. However, the only viable case is the following Θ n n−1 X≪p¯ ΘX≤1084(n−2) n+1 HEND Mpl !2(n−2) n+1 Θ 3n n+1 X, (12) provided that the inequality 10−42(n−1) nMpl HEND n−1 n ≪ΘX≪10−28(n−2) nMpl HEND 2(n−2) 3n , (13) holds. Finally, we comment on the maximum value of the reheating temperature. The maximum reheating temperature is obtained at the epoch when the decay coincides with the end of the inflaton’s domination, which means, when ρr,reh ∼ 3 Γ2 XM2 pl . Now, considering Equation (8), one obtains p¯ ΘX∼Θ n n−1 X, (14) Universe 2025,11, 49 5 of 16 and inserting this into (9), the maximum reheating temperature becomes Tmax reh ∼ =5.4 ×10−1Θ n 2(n−1) XqHENDMpl, (15) where ΘX, due to the bounds of the reheating temperature, is constrained as follows 10−42(n−1) nMpl HEND n−1 n ≤ΘX≤10−18(n−1) nMpl HEND n−1 n . (16) 3. Gravitational Production of Dark Matter This section deals with the gravitational production of dark matter in the framework of gravitational reheating. We consider two quantum scalar fields, X and Y , which are conformally coupled to the Ricci scalar. The X -field produces heavy X -particles with mass mX , which will decay into SM particles and reheat the universe. The Y -field produces Y-particles with mass mY, and they correspond to the present-day dark matter candidate. 3.1. Maximum Reheating Temperature The case when the X -particles decay close to the onset of radiation leads to the maximum reheating temperature. We begin by considering the energy density of the dark matter at the present time which is given by ρY,0 =ρY,ENDaEND a03 =ΘYρB,ENDaEND a03 . (17) Now, with the use of the following aEND a03 =aEND areh 3areh a04a0 areh =aEND areh 3areh a04Tmax reh T0 , (18) where the adiabatic expansion of the universe after reheating has been considered, i.e., a0T0=arehTmax reh , we now calculate ρr,0a0 areh 4areh aEND 3 =ρB,rehareh aEND 3 =ρB,ENDΘX, (19) wherein we make use of aEND areh 3=Θ n+1 n−1 X [ 25 ], which arises under the assumptions of the background evolving as ρB(t) = ρB,END(aEND/a(t))6n n+1, (20) and the decay of the X -particles is at the onset of radiation. With these, we find the following relation between the energy density of the Y-particles and the maximum reheating temperature ρY,0 =ΘY ΘX ρr,0 Tmax reh T0 , (21) that is: ΩYh2=Ωrh2ΘY ΘX Tmax reh T0 . (22) Universe 2025,11, 49 6 of 16 Now, inserting the value of the maximum reheating temperature and the observational values of ΩYh2and Ωrh2, one arrives at the following bound 8.63 ×10−28sMpl HEND =ΘYΘ 2−n 2(n−1) X(23) Next, we deal with scalar particles conformally coupled to gravity (for the nonconformally coupled case, specially the minimally coupled case, see, for instance, [39,40]), when mA≪HEND , because in this case one can use the Wentzel–Kramers–Brillouin (WKB) method in the complex plane in order to analytically calculate the β -Bogoliubov coefficients, which is the key piece to finding the energy density of the produced particles. Using the results obtained in [25], we obtain ΘA=1 12π3 mA Mpl !5/2sMpl √2HEND ∼ =2.26 mA Mpl !5/2 , (24) where we have used the usual value of the Hubble rate at the end of inflation, i.e., HEND ∼10−6Mpl. Then, we obtain: mY Mpl ∼ =1.7 ×10−10Θ n−2 5(n−1) X∼ =1.7 ×10−10 2.26 mX Mpl !5/2  n−2 5(n−1) , (25) and taking into account the constraint, 5 × 10 −22Mpl ≤Tmax reh ≤ 5 × 10 −10Mpl , coming from the BBN success, after substituting (15) into it, we obtain: 10−21sMpl HEND ≤Θ n 2(n−1) X≤10−9sMpl HEND =⇒10−18 ≤Θ n 2(n−1) X≤10−6, (26) that is: 0.7 ×10−72(n−1) 5n≤mX Mpl ≪10−6, (27) where we have used that mX≪HEND ∼ = 10 −6Mpl . Finally, taking into account (15) and (24), the reheating temperature can be written as a function of the mass mX and the parameter n , as follows: Tmax reh =5.4 ×10−4 2.26 mX Mpl !5/2  n 2(n−1) Mpl. (28) Finally, we close this section with Figures 1and 2. In Figure 1, we display the dependence of mY with mX as given in Equation (25) for different values of n . This clearly exhibits a pattern for increasing n . Similarly, in Figure 2we show how the maximum reheating temperature, Tmax reh , depends on mX for different values of n . One can clearly notice that for a specific value of n , if mX increases, Tmax reh also increases. Additionally, one can further notice that for a particular value of mX , if n increases, then Tmax reh increases and it assumes the maximum value for n=∞. Universe 2025,11, 49 7 of 16 Figure 1. The dependence of mY masses with mX has been shown for different values of n . We work with the units where Mpl =1. Figure 2. The maximum reheating temperature versus the mass of the produced particles, mX for different values of nhas been depicted. We work with the units where Mpl =1. Quintessential Inflation In this section, we discuss the bounds on the masses of X and Y particles in a special cosmological scenario, namely, the Quintessential Inflation [ 21 , 26 , 27 , 41 – 56 ]—a unified cosmological model, where after inflation, the universe enters a kination phase, i.e., all the energy density is kinetic, which means that weff = 1, and thus, it is equivalent to the case n=∞ . Using the values of ΘX and ΘY , we have the following relation between the masses mY Mpl ∼ =5.64 ×10−11Mpl HEND 1/10smX Mpl , (29) which for HEND ∼ =10−6Mpl, leads to mY Mpl ∼ =2×10−10smX Mpl . (30) Conversely, the bound of the maximum reheating temperature leads to the following constraint 10−18 ≤pΘX≤10−6, (31) Universe 2025,11, 49 8 of 16 where we continue taking HEND ∼ = 10 −6Mpl . Inserting the value of ΘX∼ = 2.26 mX Mpl 5/2 , we arrive at the following 5.51 ×10−15 ≤mX Mpl ≤2.2 ×10−5, (32) which must be improved due to the fact that we have assumed mX≪HEND ∼ = 10 −6Mpl , leading to, 5.51 ×10−15 ≤mX Mpl ≪10−6. (33) Now, using this last constraint, we find the range of viable values of the mass of dark matter lies in the following region: 1.48 ×10−17 ≤mY Mpl ≪10−13. (34) 3.2. General Case: Decay Before the Onset of Radiation When the decay is before the end of the domination of the inflaton, using (7) and (8), one has, ρB,rehareh aEND 3 =ρB,ENDΘ 3(n−1) 2(n−2) X¯ Θ−n2−1 4n(n−2) X. (35) Therefore, from (17), (18) and (19), the energy density of the Yparticles becomes ρY,0 =ΘYΘ−3(n−1) 2(n−2) X¯ Θ n2−1 4n(n−2) Xρr,0 Treh T0 , (36) and taking into consideration the formula of the reheating temperature (9) one arrives at, ρY,0 =90 π2greh 1/4 ΘYΘ−3/4 X¯ Θ n+1 8n Xρr,0 qHENDMpl T0 , (37) which in terms of the density parameters, takes the form ΩYh2=90 π2greh 1/4 ΘYΘ−3/4 X¯ Θ n+1 8n XΩrh2qHENDMpl T0 . (38) This coincides with the previous case when p¯ ΘX=Θ n n−1 X . Now, inserting the observational values, one obtains, 8.67 ×10−28sMpl HEND =ΘYΘ−3/4 X¯ Θ n+1 8n X, (39) and from the expression of ΘY, we derive mY Mpl ∼ =1.7 ×10−10Θ3/10 X¯ Θ−n+1 20n X, (40) with the following constraints (see for details [25]): 10−36(n−1) n≪ΘX≪10−24(n−2) n, (41) Universe 2025,11, 49 9 of 16 and Θ n n−1 X≪p¯ ΘX≤1072(n−2) n+1Θ 3n n+1 X, (42) where we continue using HEND ∼ = 10 −6Mpl . From this last constraint, it is also possible to obtain the bound for mYas, 1.7 ×10−1010−36(n−2) 5n≤mY Mpl ≪2×10−10Θ n−2 5(n−1) X, (43) with ΘX∼ =2.26mX Mpl 5/2 and 10−72(n−1) 5n≪mX Mpl ≪min(10−48(n−2) 5n; 10−6). (44) Quintessential Inflation In the case with n=∞ , the formulas simplify as follows. The reheating temperature in this case is given by Treh =90 π2greh 1/4 ¯ Θ−1 8 XΘ 3 4 XqHENDMpl ∼ =5.4 ×10−4¯ Θ−1 8 XΘ 3 4 XqHENDMpl. (45) The relation between different heating efficiencies and the decay efficiency is given by 8.67 ×10−25 =ΘYΘ−3/4 X¯ Θ 1 8 X, (46) which leads to mY Mpl ∼ =1.7 ×10−10Θ3/10 X¯ Θ−1 20 X, (47) with the constraints: 10−36 ≪ΘX≪10−24, (48) and ΘX≪p¯ ΘX≤1072Θ3 X. (49) Finally, we have the following bound for the masses: 4×10−14 ≪mX Mpl ≪3×10−10, (50) and 1.07 ×10−17 ≤mY Mpl ≪2×10−10Θ 1 5 X, (51) with ΘX∼ = 2.26 mX Mpl 5/2 . Taking into account the bound of ΘX , one can conclude that the viable masses for dark matter are of the order (10−17 −10−16)Mpl ∼(101−102)GeV. Universe 2025,11, 49 16 of 16 46. Rosenfeld, R.; Frieman, J.A. A Simple model for quintessential inflation. J. Cosmol. Astropart. Phys. 2005,2005, 003. [CrossRef] 47. Bento, M.C.; Gonzalez Felipe, R.; Santos, N.M.C. A simple quintessential inflation model. Int. J. Mod. Phys. A 2009,24, 1639–1642. [CrossRef] 48. Lankinen, J.; Vilja, I. Gravitational Particle Creation in a Stiff Matter Dominated Universe. J. Cosmol. Astropart. Phys. 2017, 2017, 025. [CrossRef] 49. De Haro, J.; Aresté Saló, L. Reheating constraints in quintessential inflation. Phys. Rev. D 2017,95, 123501. [CrossRef] 50. Aresté Saló, L.; de Haro, J. Quintessential inflation at low reheating temperatures. Eur. Phys. J. C 2017,77, 798. [CrossRef] 51. Haro, J.; Yang, W.; Pan, S. Reheating in quintessential inflation via gravitational production of heavy massive particles: A detailed analysis. J. Cosmol. Astropart. Phys. 2019,2019, 023. [CrossRef] 52. de Haro, J.; Pan, S.; Aresté Saló, L. Understanding gravitational particle production in quintessential inflation. J. Cosmol. Astropart. Phys. 2019,2019, 056. [CrossRef] 53. Dimopoulos, K.; Karam, A.; Sánchez López, S.; Tomberg, E. Palatini R 2 quintessential inflation. J. Cosmol. Astropart. Phys. 2022, 10, 076. [CrossRef] 54. de Haro, J. Reheating formulas in quintessential inflation via gravitational particle production. Phys. Rev. D 2024,109, 023517. [CrossRef] 55. Inagaki, T.; Taniguchi, M. Quintessential Inflation in Logarithmic Cartan F(R) Gravity. arXiv 2023, arXiv:2312.11776. [CrossRef] 56. Giarè, W.; Di Valentino, E.; Linder, E.V.; Specogna, E. Testing α -attractor quintessential inflation against CMB and low-redshift data. arXiv 2024, arXiv:2402.01560. [CrossRef] 57. Kofman, L.; Linde, A.D.; Starobinsky, A.A. Towards the theory of reheating after inflation. Phys. Rev. D 1997,56, 3258–3295. [CrossRef] 58. Garcia, M.A.G.; Kaneta, K.; Mambrini, Y.; Olive, K.A. Reheating and Post-inflationary Production of Dark Matter. Phys. Rev. D 2020,101, 123507. [CrossRef] 59. Giudice, G.F.; Kolb, E.W.; Riotto, A. Largest temperature of the radiation era and its cosmological implications. Phys. Rev. D 2001, 64, 023508. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.