Kinetic Model for Dark Energy -- Dark Matter Interaction: Scenario for the Hubble Tension
Giovanni Montani Nakia Carlevaro













e1e-mail: giovanni.montani@enea.it



ENEA, Nuclear Department, C.R. Frascati, Via E. Fermi 45, 00044 Frascati (Roma), Italy 
           Physics Department, “Sapienza” University of Rome, P.le Aldo Moro 5, 00185 Roma, Italy 






Kinetic Model for Dark Energy - Dark Matter Interaction:
 Scenario for the Hubble
Tension

    Giovanni Montaniaddr1,addr2,e1
        Nakia Carlevaroaddr1 

    Received: date / Accepted: date
==========================================================================================




We analyze a model for Dark Energy - Dark Matter interaction, based on a
decaying process of the former constituents into the latter ones.
The dynamical equations are constructed following a kinetic formulation,
which separates the interacting fluctuations from equilibrium distribution of the both the species. The emerging dynamical picture consists of coupled equations, which
are specialized in the case of a Dark Energy equation of state parameter: we deal with a modified Lambda Cold Dark Matter model, which is investigated
versus a possible interpretation of the Hubble tension.
We compare our model with data corresponding to 6 points of the
expansion rate from Type Ia Supernovae. We show that, the proposed model suitably fits data according to a value of the Hubble constant compatible with the SHOES Collaboration measurement. The tension is solved because, essentially for redshift greater than one, the correction to the Lambda Cold Dark Matter model vanishes and its presence does not affect the Planck measurements.







§ INTRODUCTION


Since the original Hubble measurement, the value of the Hubble constant, H_0, has played a central role in experimental and theoretical cosmology. Only in the epoch of the so-called “precision cosmology” it has been detected with a good degree of accuracy <cit.>. However, in recent years, the reliability of the Hubble constant has been subjected to a surprising fate: clear observational evidences <cit.> suggested a possible dependence of H_0 on the average redshift of the sources adopted for its determination. This observational discrepancy, known as the “Hubble tension”, calls attention to an explanation based on a possible unrecognized redshift evolution of astrophysical sources (see, e.g. the analysis in <cit.>, or on possible “new physics” as discussed in <cit.>.


In <cit.> (see also <cit.>), it was argued that the decaying tendency of H_0 with the increasing different source redshift can also be recovered within the Type Ia Supernova (SNIa) Pantheon sample and it is well-reproduced by the functional form H_0∝(1+z)^-α, where α is a parameter of the order 10^-2. Such behavior has been properly reproduced via modified f(R)-gravity in the Jordan frame <cit.>, as investigated in <cit.>. For further studies facing the Hubble tension via a modified gravity theory, see <cit.>, while for approaches involving also evolutionary phantom energy see <cit.>. A recent model using a slow-rolling scalar dynamics and dealing with the binned SNIa data, as processed in <cit.>, see <cit.>. Finally, for a study of the creation of scalar matter via the cosmological gravitational field, see <cit.>.

However, the difficulty to reconcile the Baryon Aco-ustic Oscillation (BAO) <cit.> data to those ones due to the SH0ES Collaboration on SNIa <cit.> (which live in overlapping redshift regions) has motivated early Universe modification of the dynamics, for instance affecting the sound horizon value, in addition to late Universe models. In <cit.>, it has been suggested that the most reliable solution to the Hubble tension could require a combination of late and early Universe modified physics.

Here, to address the Hubble tension, we consider a Dark Energy (DE) - Dark Matter (DM) interaction model (for a review on this topic, see <cit.>), based on a kinetic approach to describe the decaying process of the DE constituents into DM particles. Thus, we assume that the equilibrium configurations for the species are perturbed by small fluctuations associated with a Boltzmann equation typical of a decay process from one species into another one. Actually, while the DM decay process is described via the kinetic theory, by reconstructing the macroscopic equation for the DE fluctuation, the transfer of energy to DM via particle creation is recovered by requiring the conservation of the total energy-momentum tensor for the two species. As a result of this underlying physical scenario, we get a modified Friedmann equation containing an additional evolutionary term with respect to the Lambda Cold DM (ΛCDM) model, which is the net effect of the DE decaying process. Clearly, when this additional term vanishes, the model is exactly reduced to a ΛCDM picture, in the correspondence to the instant when the DE-DM interaction starts.

We show how the proposed model is able to reconcile the measurements of the Hubble constants due to the Planck Satellite and the SHOES Collaboration. Actually, the former is unable to identify the interaction process and the corresponding modified dynamics. This is because we find that the Cosmic Microwave Background (CMB) luminosity distance is not affected by the obtained late Universe modification within the intrinsic error. The SNIa measurement is instead sensitive to the evolutionary term due to the interaction, and then the achieved value of H_0 is correspondingly enhanced. To reach this result, we use the data provided in <cit.> for the expansion rate, which give specific constraints on the range of the model free parameters.












§ THEORETICAL FRAMEWORK

We consider here a model of DE-DM interaction, based on the idea that a fraction of the DE constituents decay versus the creation of DM particles 
(this process is viewed in a kinetic formulation). 

In order to describe the decaying process of DE particles (labelled by the suffix de), we first consider (according to the Planck data analysis in <cit.>) a flat isotropic Universe, described by the line element

    ds^2 = dt^2 - a^2(t)( 
    	dx^2 + dy^2 + dz^2)
    	  ,

where t denotes the synchronous time 
(we used c=1 units), (x,y,z) are Cartesian coordinates and a(t) is the cosmic scale factor, governing the Universe expansion. 
Then, we set the DE distribution function as f=f_eq(E_de) + δ f(E_de), where E_de denotes the particle energy, f_eq the dominant equilibrium contribution and 
δ f describes the decaying process. Thus, while f_eq is responsible 
for the DE density of the Universe, i.e. ρ_de=ρ_de^0 (1 + z)^3w + 3 (ρ_de^0 being its 
present-day value and w<-1/3 the equation of state parameter), δ f is 
subjected to the following Boltzmann equation <cit.>:

    ∂_z δ f + P_de/1 + z∂_P_deδ f = Γ/(1 + z)Hδ f
    	  ,

where P_de is DE particle momentum, H ≡ a^-1da/dt is the Hubble parameter and we are using the redshift time variable  z(t) = 
a_0/a - 1 (a_0 is the today scale factor). We thus get d(...)/dt = - (1 +z )H d(...)/dz. In the equation 
above, Γ is a constant such that 1/Γ corresponds to the decaying time of the DE species.

We now recall the definition of the energy density fluctuation as:

    δρ_de≡g_de/2π^2∫ P^2_deE_deδ f dP_de  ,

where g_de is the particle degree of freedom and we used the isotropy of the momentum space. 
Hence, we get the following equation: 

    d δρ_de/dz - 
    	3(1 + w)/1 + zδρ_de = Γ/(1 + z)Hδρ_de  ,

where we used the following phenomenological relation for the DE fluctuation pressure δ p_de

    δ p_de≡g_de/2π^2∫P^4_de/3E_deδ f dP_de = w δρ_de  .

This expression, compared with Eq.(<ref>), for negative values of w (like DE), leads to particles with E^2_de<0. 
This does not mean that we deal with tachyons but, more realistically, kinetic theory would suggest that the DE constituents are not elementary free particles, while corresponding to relativistic bounded states (self-interacting micro-clusters).

The solution of Eq.(<ref>) reads

    δρ_de = δρ_de^0(1    + z)^3w + 3×
       ×exp[ Γ∫_0^z dz'/(1 + z')H(z')]
    	  ,

where δρ_de^0 = 
δρ _de(z=0). As soon as, we require that the 
sum of the DE and DM energy-momentum tensors be conserved, 
we find the following equation 
for the DM energy density fluctuation 
δρ_dm:

    d δρ_dm/dz - 3/1 + zδρ_dm = 
    	- Γ/(1 + z)Hδρ_de  .


Now, defining δ≡δρ _de + δρ_dm, from Eqs.(<ref>) and (<ref>), we get the following dynamical equation:

    dδ/dz - 3/1 + zδ = - 3|w|/1 + zδρ_de= 
            =- 3| w|δρ_de^0(1 + z)^3w + 2×
               ×exp[ 
    Γ∫_0^zdz'/(1 + z')H(z')]
      ,

where we emphasized the negative values of w for the DE contribution and we made use of the solution (<ref>). 

Finally, we observe that the baryonic component of the Universe is not involved in the particle creation process and its energy density ρ_b 
satisfies the standard equation

    dρ_b/dz - 3/1 + zρ_b = 0
    	  .

In what follows, we denote by 
ρ_m = ρ_dm + ρ_b the 
total equilibrium value of the matter component of the Universe. 












§ DYNAMICS FOR THE HUBBLE PARAMETER

The Friedmann equation corresponding to the scenario depicted above takes the following form:

    H^2(z) = χ/3( ρ_m + ρ_de + δ)
    	  ,

χ being the Einstein constant. We introduce the quantities

    Ω_m^0 ≡ρ_m^0/3H_*^2  ,    Ω^0_de≡χρ^0_de/3H_*^2  ,     Δ≡χδ/3H_*^2  ,

where ρ^0_m and ρ^0_de are the present-day value of the matter and DE energy density, respectively, while H_* is a fiducial Hubble constant. 

In this scheme, Eq.(<ref>) can be recast as

    E_*^2(z) ≡H^2(z)/H_*^2=
           = 
    Ω _m^0(1 + z)^3 
    + Ω^0_de(1 + z)^3w+3 + Δ (z)
      ,

to be coupled with Eq.(<ref>), here restated in the form

    dΔ/dz = 3/1 + zΔ   - 
    	3|w|Δ̅(1 + z)^3w + 2×
       ×exp[E̅_*∫ _0^z dz'/(1 + z')E_*(z')]
    	  ,

where we set Δ̅≡χδρ_de^0/3H_*^2 and 
E̅_* ≡Γ/H_*. Eqs.(<ref>) and (<ref>) formally describe the dynamics of the Hubble parameter.





















 §.§ Modified ΛCDM model

In order to develop a dynamical scheme close to the ΛCDM model, we study in detail the case w=-1, i.e. we set ρ_de=const. It is immediate to specialize Eqs.(<ref>) and (<ref>) to this scenario. By introducing the definition Δ≡ (1 + z)^3𝒟 and the auxiliary function F(z), the system rewrites

    E^2_*(z) = Ω _m^0(1 + z)^3( 1 + 𝒟/Ω_m^0) + 1 - Ω _m^0	 , 
       d𝒟/dz = - 3Δ̅/(1 + z)^4e^F(z) , 
       dF/dz = E̅_*/(1 + z)E_*(z)  .

where, according to the standard ΛCDM model, we assume Ω_de^0 = 1 - Ω_m^0 and, for completeness, we remark that we always coherently assume F(z=0)=0.

From Eq.(<ref>), 𝒟 is found to be a decreasing function of z. Thus, if we indicate by z_i the instant when 𝒟 vanishes (i.e. also Δ (z_i)=0), then it will increase up to the today value 𝒟(z=0)≡𝒟_0. We regard z_i as the instant at which, in the past, the decaying process of DE versus DM starts. At z≥
z_i, the process is not present and the model exactly coincides with a ΛCDM dynamics of the Universe.

If we introduce E(z) ≡ H(z)/H_0, following our formalism, it results

    E(z)=E_*(z)H_*/H_0 ,

where now 

    H_0=H_* √(1+𝒟_0) ,

and the Hubble parameter finally reads

    H^2(z)=H_0^2 Ω _m^0(1 + z)^3( 1 +𝒟/Ω_m^0) + 1 - Ω _m^0/1+𝒟_0 .

In order to reconcile the late Universe predictions with the CMB data, we set (here and in the following H is in units: km s^-1 Mpc^-1)  

    H_*=H_0^Pl=67.4 ,
       Ω_m^0=Ω_m^0Pl=0.315 ,

thus implementing the corresponding values measured by Planck experiment. In this respect, in order the satellite be unable to recognize if the addressed late Universe modification is present (i.e. the term 𝒟), the impact of such a contribution on the CMB luminosity distance must be smaller than the error in the measurement itself.

We conclude this section by discussing the physical meaning of the parameter Γ regarded as the inverse decaying time of the DE constituents. 
Since H_* is assumed the value of the Hubble constant measured by Planck, it is clear that, in order the decaying process to take place at the equilibrium (de facto being efficient), we have to require E̅_* ≳ 1. In this respect, we observe that, since DE starts to dominate at the Universe half age <cit.>, it is reasonable to require that the decaying time of DE be a fraction of the Universe age, greater than 1/2, in order to avoid a complete decaying of the DE contribution before z=0. As a consequence, it is natural to require that E̅_*≲ 2. We also observe that the quantity Δ̅ is the critical parameter for the DE fluctuations today. According to the perturbation scheme
adopted above, we have to require that this contribution is small in
comparison to the DE equilibrium component of the present
Universe. Moreover, since the DM contribution is smaller than the DE one, we
assume that the corresponding critical parameter for the DM
fluctuations be even smaller.















§ MODEL TESTING

In this section, we constrain the model above using the 6 points of the expansion rate for 0.07≤ z≤ 1.5 provided in <cit.> for SNIa. This measurements alone actually constitutes an almost identical characterization of the complete SNIa sample and they thus represent a faithfully data compression, combining the Pantheon sample with 15
distant additional sources.

According the discussion above, we first set the physical constants of the model at the following nominal representative values: E̅_*=1.5 and Δ̅=0.15. The model parameter 𝒟_0 is than fixed by the best fit of the 6 points of E(z) in the late Universe. The fit result for Eq.(<ref>) is:

    𝒟_0 = 0.1724 ± 0.0458 ,

with a χ^2=0.12 (p-value=0.998). The obtained profile of E(z) is depicted in Fig.<ref>, together with the addressed measurements. According to Eq.(<ref>), this result provides 

    H_0 = 72.98 ±1.42,

which is compatible with the SHOES measurement of the Hubble constant <cit.>: 73.6±1.1. 


Moreover, by analyzing the evolution of the function 𝒟(z), we find for this case z_i≃0.88. We recall that this corresponds to the instant  at which 𝒟=0 and for z≥ z_i, the model coincides with a flat ΛCDM dynamics. The obtained profile of the Hubble parameter in Eq.(<ref>) is finally reported in Fig.<ref>, together with the expression specified for the Planck measured values of the parameters, i.e. H_Pl(z)=H_0^Pl(Ω_m^0Pl(1+z^3)+1-Ω_m^0Pl)^1/2.

We finally remark that, as discussed above, the viability of this scenario must be guaranteed the by the undetectable character of the additional late Universe contribution on the CMB luminosity distance. In this respect, the profile of H(z) for the obtained parameters provides d_L^CMB=50.67 (center value) which results compatible with the Planck value d_LPl^CMB=51.17± 0.59.

Although the limited number of data prevents a statistically meaningful test of all the free parameter of the model, nonetheless it is remarkable that a range of the physical constants exists for which the SHOES value of the Hubble constant is reconciled to the Planck measurements. Actually the latter result insensitive to the dynamical modification of the very late Universe here considered.

























§ CONCLUDING REMARKS


We derived a kinetic model for the DE-DM interaction, which separates the equilibrium distribution function of the two species into their equilibrium component and small corrections, accounting for the decaying process of DE constituents into DM particles. This way, we arrive to a set of coupled equations, which are reduced to their simplified form when the DE parameter is taken w=-1. This choice has been justified in order to construct a modified ΛCDM dynamics, appropriate for a comparison with the SNIa data. 

The model we analyzed in detail contained three free parameters, E̅_* (corresponding to the ratio between the Universe age and the decaying time of DE constituents), Δ̅ (which fixes the value in z=0 for the DE fluctuation critical parameter) and 𝒟_0 (which is the most important quantity since it determines H(z=0)). Then, we compared the model we set up with the 6 points of the expansion rate for the SNIa, which are provided in <cit.>. The data analysis has been performed by fixing the values of E̅_* and Δ̅, while 𝒟_0 has been left free for the fitting procedure. The resulting best fit indicates that the value of H_0 is reconciled with the value provided by the SH0ES collaboration within the errors. This achievement is relevant for the Hubble tension since the modification of the ΛCDM model vanishes as z≳ 1 and, hence the Planck Satellite can not detect such a modification, because the obtained fluctuation induced on the luminosity distance is smaller than the corresponding fluctuation due to the intrinsic error of the measurement. Moreover, since we fixed E̅_*=1.5, it is worth noting that the relatively late triggering of the DE decay process is the consequence of the significantly large value of the corresponding characteristic time.

The proposed scenario has the merit to demonstrate how a suitable late Universe phenomenon is unable to significantly affect the Planck Satellite data 
(and the corresponding ΛCDM model as the best fit), 
but it is strong enough to provide a higher value of H_0,  which becomes compatible with the SH0ES data analysis. 

















10


urlstyle

2000Natur.404..955D
P. de Bernardis, P.A.R. Ade, J.J. Bock, J.R. Bond, J. Borrill, A. Boscaleri, K. Coble, B.P. Crill, G. De Gasperis, P.C. Farese, P.G. Ferreira, K. Ganga, M. Giacometti, E. Hivon, V.V. Hristov, A. Iacoangeli, A.H. Jaffe, A.E. Lange, L. Martinis, S. Masi, P.V. Mason, P.D. Mauskopf, A. Melchiorri, L. Miglio, T. Montroy, C.B. Netterfield, E. Pascale, F. Piacentini, D. Pogosyan, S. Prunet, S. Rao, G. Romeo, J.E. Ruhl, F. Scaramuzzi, D. Sforna, N. Vittorio, Nature 404(6781), 955 (2000).
10.1038/35010035

2013ApJS..208...20B
C.L. Bennett, D. Larson, J.L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K.M. Smith, R.S. Hill, B. Gold, M. Halpern, E. Komatsu, M.R. Nolta, L. Page, D.N. Spergel, E. Wollack, J. Dunkley, A. Kogut, M. Limon, S.S. Meyer, G.S. Tucker, E.L. Wright, ApJ S.R. 208(2), 20 (2013).
10.1088/0067-0049/208/2/20

Planck:2018vyg
N. Aghanim, et al., A& A 641, A6 (2020).
10.1051/0004-6361/201833910.
[Erratum: Astron.Astrophys. 652, C4 (2021)]

2024ApJ...963L..43A
R.I. Anderson, N.W. Koblischke, L. Eyer, ApJ Lett. 963(2), L43 (2024).
10.3847/2041-8213/ad284d

Scolnic_2022
D. Scolnic, D. Brout, A. Carr, A.G. Riess, T.M. Davis, A. Dwomoh, D.O. Jones, N. Ali, P. Charvu, R. Chen, E.R. Peterson, B. Popovic, B.M. Rose, C.M. Wood, P.J. Brown, K. Chambers, D.A. Coulter, K.G. Dettman, G. Dimitriadis, A.V. Filippenko, R.J. Foley, S.W. Jha, C.D. Kilpatrick, R.P. Kirshner, Y.C. Pan, A. Rest, C. Rojas-Bravo, M.R. Siebert, B.E. Stahl, W. Zheng, ApJ 938(2), 113 (2022).
10.3847/1538-4357/ac8b7a.
<https://dx.doi.org/10.3847/1538-4357/ac8b7a>

2018ApJ...859..101S
D.M. Scolnic, D.O. Jones, A. Rest, Y.C. Pan, R. Chornock, R.J. Foley, M.E. Huber, R. Kessler, G. Narayan, A.G. Riess, S. Rodney, E. Berger, D.J. Brout, P.J. Challis, M. Drout, D. Finkbeiner, R. Lunnan, R.P. Kirshner, N.E. Sanders, E. Schlafly, S. Smartt, C.W. Stubbs, J. Tonry, W.M. Wood-Vasey, M. Foley, J. Hand, E. Johnson, W.S. Burgett, K.C. Chambers, P.W. Draper, K.W. Hodapp, N. Kaiser, R.P. Kudritzki, E.A. Magnier, N. Metcalfe, F. Bresolin, E. Gall, R. Kotak, M. McCrum, K.W. Smith, ApJ 859(2), 101 (2018).
10.3847/1538-4357/aab9bb

Brout:2022vxf
D. Brout, et al., ApJ 938(2), 110 (2022).
10.3847/1538-4357/ac8e04

2021ApJ...914L..40D
M.G. Dainotti, V. Petrosian, L. Bowden, ApJ Lett. 914(2), L40 (2021).
10.3847/2041-8213/abf5e4

2023MNRAS.518.2201D
M.G. Dainotti, A.Ł. Lenart, A. Chraya, G. Sarracino, S. Nagataki, N. Fraija, S. Capozziello, M. Bogdan, Mon.Not. RAS 518(2), 2201 (2023).
10.1093/mnras/stac2752

2022PASJ...74.1095D
M.G. Dainotti, G. Sarracino, S. Capozziello, Pub. Astro. Soc. Japan 74(5), 1095 (2022).
10.1093/pasj/psac057

DiValentino:2021izs
E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D.F. Mota, A.G. Riess, J. Silk, Class. Quant. Grav. 38(15), 153001 (2021).
10.1088/1361-6382/ac086d

2024Univ...10..140C
S. Capozziello, G. Sarracino, G. De Somma, Universe 10(3), 140 (2024).
10.3390/universe10030140

2023PDU....4001201C
S. Capozziello, G. Sarracino, A.D.A.M. Spallicci, Phys. Dark Univ. 40, 101201 (2023).
10.1016/j.dark.2023.101201

apj-powerlaw
M.G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, G. Lambiase, ApJ 912(2), 150 (2021).
10.3847/1538-4357/abeb73

galaxies10010024
M.G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, G. Lambiase, M. Bogdan, S. Ugale, Galaxies 10(1) (2022).
10.3390/galaxies10010024.
<https://www.mdpi.com/2075-4434/10/1/24>

Krishnan:2020obg
C. Krishnan, E.O. Colgáin, Ruchika, A.A. Sen, M.M. Sheikh-Jabbari, T. Yang, Phys. Rev. D 102(10), 103525 (2020).
10.1103/PhysRevD.102.103525

Krishnan:2020vaf
C. Krishnan, E.O. Colgáin, M.M. Sheikh-Jabbari, T. Yang, Phys. Rev. D 103(10), 103509 (2021).
10.1103/PhysRevD.103.103509

kazantzidis
L. Kazantzidis, L. Perivolaropoulos, Phys. Rev. D 102(2), 023520 (2020).
10.1103/PhysRevD.102.023520

Sotiriou-Faraoni:2010
T.P. Sotiriou, V. Faraoni, Rev. Mod. Phys. 82, 451 (2010).
10.1103/RevModPhys.82.451.
<https://link.aps.org/doi/10.1103/RevModPhys.82.451>

NOJIRI201159
S. Nojiri, S.D. Odintsov, Physics Reports 505(2), 59 (2011).
https://doi.org/10.1016/j.physrep.2011.04.001

2011PhR...509..167C
S. Capozziello, M. de Laurentis, Phys. Rept. 509(4), 167 (2011).
10.1016/j.physrep.2011.09.003

schiavone_mnras
T. Schiavone, G. Montani, F. Bombacigno, Mon. Not. RAS 522(1), L72 (2023).
10.1093/mnrasl/slad041.
<https://doi.org/10.1093/mnrasl/slad041>

Nojiri:2022ski
S. Nojiri, S.D. Odintsov, V.K. Oikonomou, Nucl. Phys. B 980, 115850 (2022).
10.1016/j.nuclphysb.2022.115850

Odintsov:2020qzd
S.D. Odintsov, D. Sáez-Chillón Gómez, G.S. Sharov, Nucl. Phys. B 966, 115377 (2021).
10.1016/j.nuclphysb.2021.115377

deangelis-fr-mnras
G. Montani, M. De Angelis, F. Bombacigno, N. Carlevaro, Mon. Not. RAS 527(1), L156–L161 (2024).
10.1093/mnrasl/slad159

2024PhRvD.109b3527A
S. Adil, Ö. Akarsu, E. Di Valentino, R. Nunes, E. Özülker, A. Sen, E. Specogna, Phys. Rev. D 109(2), 023527 (2024).
10.1103/PhysRevD.109.023527

2024PDU....4401486M
G. Montani, N. Carlevaro, M.G. Dainotti, Phys. Dark Univ. 44, 101486 (2024).
10.1016/j.dark.2024.101486

2024arXiv240216791E
R. Erdem, arXiv e-prints arXiv:2402.16791 (2024).
10.48550/arXiv.2402.16791

2021MNRAS.505.3866E
G. Efstathiou, Mon. Not. RAS 505(3), 3866 (2021).
10.1093/mnras/stab1588

PhysRevD.98.083501
S. Vagnozzi, S. Dhawan, M. Gerbino, K. Freese, A. Goobar, O. Mena, Phys. Rev. D 98, 083501 (2018).
10.1103/PhysRevD.98.083501.
<https://link.aps.org/doi/10.1103/PhysRevD.98.083501>

2023Univ....9..393V
S. Vagnozzi, Universe 9(9), 393 (2023).
10.3390/universe9090393

2020PhRvD.102b3518V
S. Vagnozzi, Phys. Rev. D 102(2), 023518 (2020).
10.1103/PhysRevD.102.023518

2016RPPh...79i6901W
B. Wang, E. Abdalla, F. Atrio-Barandela, D. Pavón, Rep. Prog. Phys. 79(9), 096901 (2016).
10.1088/0034-4885/79/9/096901

riess18
A.G. Riess, S.A. Rodney, D.M. Scolnic, D.L. Shafer, L.G. Strolger, H.C. Ferguson, M. Postman, O. Graur, D. Maoz, S.W. Jha, B. Mobasher, S. Casertano, B. Hayden, A. Molino, J. Hjorth, P.M. Garnavich, D.O. Jones, R.P. Kirshner, A.M. Koekemoer, N.A. Grogin, G. Brammer, S. Hemmati, M. Dickinson, P.M. Challis, S. Wolff, K.I. Clubb, A.V. Filippenko, H. Nayyeri, V. U, D.C. Koo, S.M. Faber, D. Kocevski, L. Bradley, D. Coe, ApJ 853(2), 126 (2018).
10.3847/1538-4357/aaa5a9

2020MNRAS.496L..91E
G. Efstathiou, S. Gratton, Mon. Not. RAS 496(1), L91 (2020).
10.1093/mnrasl/slaa093

bib:montani-primordialcosmology
G. Montani, M.V. Battisti, R. Benini, G. Imponente, Primordial Cosmology (World Scientific, Singapore, 2011).
10.1142/7235

melchiorriDE
A. Melchiorri, L. Pagano, S. Pandolfi, Phys. Rev. D 76(4), 041301 (2007).
10.1103/PhysRevD.76.041301




