We show how this non-thermal dark matter production mechanism can generate dark radiation and solve the problem (H_0) problem. Recall that the radiation density ((rho_{rad})) is determined by the temperature of the photon (*T*) and the relativistic degrees of freedom ((g_*))that is,

$$begin{aligned} rho _{rad} = frac{pi ^2}{30}g_*T^4. end{aligned}$$

(1)

In a radiation-dominated phase of the universe in which only photons and neutrinos are ultrarelativistic, the relationship between the temperature of photons and neutrinos is ((4/11)^{1/3}). Since photons have two polarization states and neutrinos are only left-handed in the standard model (SM); therefore, we write (g_*) in the following way,

$$begin{aligned} g_* = 2 + frac{7}{4} left( frac{4}{11} right) ^{4/3}N_{eff}. end{aligned}$$

(2)

where is it (N_{eff}) is the effective number of relativistic neutrino species, where in (Lambda)CDM is (N_{eff}=3).

In a more general context there may be new species of light contributing to (N_{eff}), or some new physics interactions with neutrinos that will alter the neutrino decoupling temperature, or as in our case, some particles that mimic neutrino effects. While we are trying to increase (H_0) increasing (N_{eff}), (Delta N_{eff}) tell us how much extra radiation we are adding to the universe via our mechanism. In other words,

$$begin{aligned} Delta N_{eff} = frac{rho _{extra}}{rho _{1nu }}. end{aligned}$$

(3)

where is it ({rho _{1nu }}) is the radiation density generated by an extra neutrino species.

Thus, in principle, we can reproduce the effect of an extra neutrino species by adding any other type of radiation source. Calculation of the ratio of the density of a neutrino species to the density of cold dark matter at matter-radiation equality ((t = t_{eq})) we have,

$$begin{aligned} left. frac{rho _{1nu }}{rho _{DM}} right| _{t = t_{eq}} = frac{Omega _{nu ,0}rho _c}{3a^4_{eq}} times left( frac{Omega _{DM,0} rho _c}{a^3_{eq}}right) ^{-1} = 0.16. end{aligned}$$

(4)

where we used (Omega _{nu ,0} = 3,65 times 10^{-5}), (Omega _{DM,0} = 0,265) And (a_{eq} = 3 times 10^{-4})^{17}.

The previous equation tells us that it represents an extra neutrino species (16%) of dark matter density to matter-radiation equality. Assume (who) it is produced by two decays of the body of a parent particle (who’)where is it (who ‘ rightarrow who + nu). In (who’) rest frame, the 4-momentum of the particles is,

$$begin{aligned} p_{chi ‘} = left( m_{chi ‘}, varvec{0} right) ,\ p_{chi } = left( E(varvec{p }), varvec{p} right) ,\ p_{nu } = left( left| varvec{p} right| , -varvec{p} right) . end{aligned}$$

Thus, 4-moment conservation implies,

$$begin{aligned} E_{chi }(tau ) = m_{chi } left( frac{m_{chi ‘} }{2m_{chi }} + frac{m_{chi } }{2m_{chi ‘}} right) equiv m_{chi }gamma _{chi }(tau ), end{aligned}$$

(5)

where is it (tau) is the duration of the parent particle (who’). We emphasize that we will adopt the instantaneous decay approximation.

Using this result and the fact that the momentum of a particle is inversely proportional to the scale factor, we get,

$$begin{aligned} &E^2_{chi } – m^2_{chi } = varvec{p}^2_{chi } propto frac{1}{a^2}\& quad Rightarrow left( E^2_{chi }

(6)

In the non-relativistic regime, (m_{who }) is the dominant contribution to the energy of a particle. So, rewriting the energy of the dark matter we find,

$$begin{aligned} E_{chi } = m_{chi }left( gamma _{chi } -1 right) + m_{chi }. end{aligned}$$

So, in the ultra-relativistic regime (m_{chi }left( gamma _{chi } -1 right)) dominant. Consequently, the total energy of the dark matter particle can be written as,

$$begin{aligned} E_{DM} = N_{HDM}m_{chi }left( gamma _{chi } -1 right) + N_{CDM}m_{chi }. end{aligned}$$

Gentleman, (N_{HDM}) is the total number of relativistic dark matter particles (hot particles), while (N_{CDM}) is the total number of non-relativistic DM (cold particles). Obviously, (N_{HDM} ll N_{CDM}) be consistent with cosmological data. The ratio of the density energy of relativistic to non-relativistic dark matter is,

$$begin{aligned} frac{rho _{HDM}}{rho _{CDM}} = frac{N_{HDM}m_{chi }left( gamma _{chi } -1 right) }{N_{CDM}m_{chi }} equiv fleft( gamma _{chi } -1 right) . end{aligned}$$

(7)

consequentially, *f* is the fraction of dark matter particles produced by this non-thermal process. Like above, *f* it should be small, but we don’t have to assume a precise value, it will be on the order of 0.01. This fact will be further clear.

Using Eqs. (3) and (7), we find that the extra radiation produced via this mechanism is,

$$begin{aligned} Delta N_{eff} = lim _{t rightarrow t_{eq}} frac{fleft( gamma _{chi } -1 right) }{0.16}, end{aligned}$$

(8)

where we used eq. (4) and we wrote (rho _{CDM} = rho _{who }).

In the Regime (m_{who ‘} gg m_{who })let’s simplify,

$$begin{aligned} gamma _{chi }(t_{eq}) -1 about gamma _{chi }(t_{eq}) about frac{m_{chi ^prime } }{2m_{chi }} sqrt{frac{tau }{t_{eq}}}, end{aligned}$$

and Eq. (8) boils down to,

$$begin{aligned} Delta N_{eff} about 2.5 times 10^{-3}sqrt{frac{tau }{10^{6}s}} times ffrac{m_{ who ‘}}{m_{who }}. end{aligned}$$

(9)

with (t_{eq} about 50{,}000 ~ text {years} about 1.6 times 10^{12} ~s).

From EQ. (9), we conclude that the (Delta N_{eff} sim 1) implies in a greater relationship (f, m_{chi ^prime }/m_{chi }) for a lifetime of decay (tau sim 10^4- 10^8,s). Note that our overall results are based on two free parameters: (i) duration, (tau)and (ii) (f, m_{chi ^prime }/m_{chi }).