We show how this non-thermal dark matter production mechanism can generate dark radiation and the \(H_0\) issue. We recall that the radiation density \((\rho _{rad})\) is determined by the temperature of the photon (*T*) and the relativistic degrees of freedom \((g_*)\)i.e

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

(1)

In a radiation-dominated universe phase in which only photons and neutrinos are ultrarelativistic, the relationship between photons and neutrinos is temperature \((4/11)^{1/3}\). Since photons have two polarization states and neutrinos are only left-handed in the Standard Model (SM); that’s why 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 is \(N_{eff}=3\).

In a more general setting, new types of light could contribute to this \(N_{eff}\), or some new physical interactions with neutrinos that will change the decoupling temperature of neutrinos, or as in our case, some particles that mimic the effects of neutrinos. As we try to raise \(H_0\) by increasing \(N_{eff}\), \(\Delta N_{eff}\) telling us how much extra radiation we are adding to the universe through our mechanism. In other words,

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

(3)

Where \({\rho _{1\nu }}\) is the radiation density generated by an additional neutrino species.

Therefore, in principle, we can reproduce the effect of an extra neutrino species by adding any other kind of radiation source. Calculation of the ratio between the density of one neutrino species and the density of cold dark matter at the equality of matter and radiation \((t = t_{eq})\) we get,

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

(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 equation above tells us that one represents an extra neutrino species \(16\%\) of the density of dark matter at the matter-radiation equality. Assuming \(\chi\) is produced via two body decays of a parent particle \(\chi’\)Where \(\chi ‘ \right arrow \chi + \nu\). In \(\chi’\) rest frame, being the 4 momentum of particles,

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

Therefore, conservation of 4 impulses 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 parent particle \(\chi’\). We emphasize that we will use the instant decay approach.

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 non-relativistic regime, \(m_{\chi}\) is the dominant contribution to a particle’s energy. So, rewriting the dark matter energy we find,

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

So in the ultra-relativistic regime \(m_{\chi}\left( \gamma _{\chi} -1 \right)\) 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), while \(N_{CDM}\) is the total number of non-relativistic DM (cold particles). Clear, \(N_{HDM} \ll N_{CDM}\) to be consistent with the cosmological data. The ratio between relativistic and non-relativistic density energy of dark matter is,

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

(7)

Thereafter, *f* is the fraction of dark matter particles produced through this non-thermal process. As mentioned earlier, *f* should be small, but we don’t need to take an exact value for it, but will be in the order of 0.01. This fact will become clear later.

Using equations. (3) and (7), we find that the extra radiation produced through this mechanism,

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

(8)

where we cf. used. (4) and we wrote \(\rho _{CDM} = \rho _{\chi}\).

In the regimen \(m_{\chi ‘} \gg m_{\chi }\)we 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 Cf. (8) reduces to,

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

(9)

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

From Cf. (9), we conclude that the \(\Delta N_{eff} \sim 1\) implies in a larger proportion \(f\, m_{\chi ^\prime}/m_{\chi}\) for a decaying life \(\tau \sim 10^4- 10^8\,s\). Note that our overall results depend on two free parameters: (i) the lifetime, \(\tau\)and (ii) \(f\, m_{\chi ^\prime}/m_{\chi}\).