*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 universe phase where only photons and neutrinos are ultrarelativistic the relation between photons and neutrinos temperature is ((4/11)^{1/3}). As 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 (N_{eff}) is the effective number of relativistic neutrino species, where in the (Lambda)CDM itself (N_{eff}=3).

In a more general setting there could be new light species 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 mimicking the effects of neutrinos. As we are trying to raise (H_0) by 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 ({rho _{1nu }}) is the radiation density generated by an extra neutrino species.

Hence, in principle, we may not reproduce the effect of an extra neutrino species by adding any other type of radiation source. Calculating the ratio between one neutrino species density and cold dark matter density at the matter-radiation equation ((t = t_{eq})) we get

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

(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 above equation tells us that one extra neutrino species represents (16%) of the dark matter density at the matter-radiation equation. Assuming (chi) is produced via two body decays of a mother particle (chi ‘)where (chi ‘ rightarrow chi + nu). In (chi ‘) resting frame, the 4-momentum of particles are,

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

Therefore, the 4-momentum 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 (tau) is the lifetime of the mother particle (chi ‘). We highlight that we will adopt the instant decay approximation.

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

$$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 nonrelativistic regime, (m_{chi }) is the dominant contribution to the energy of a particle. Thus, rewriting the dark matter energy we find,

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

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

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

Here, (N_{HDM}) is the total number of relativistic dark matter particles (hot particles), whereas (N_{CDM}) is the total number of nonrelativistic DM (cold particles). Obviously, (N_{HDM} ll N_{CDM}) to be consistent with the cosmological data. The ratio between relativistic and nonrelativistic dark matter energy density 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)

Consequently, *f* is the fraction of dark matter particles which are produced via this non-thermal process. As mentioned above, *f* ought to be small, but we do not have to assume a precise value for it, but it will be of the order of 0.01. This fact will be clear further.

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 _{chi }).

In the regime (m_{chi ‘} gg m_{chi })we simplify,

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

and Eq. (8) reduces to,

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

(9)

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

From Eq. (9), we conclude that the (Delta N_{eff} sim 1) implies in a larger ratio (f, m_{chi ^prime }/m_{chi }) for a decay lifetime (tau sim 10^4- 10^8,s). Notice that our overall results rely on two free parameters: (i) the lifetime, (tau)and (ii) (f, m_{chi ^prime }/m_{chi }).