We show how this non-thermal dark matter production mechanism can generate and solve dark radiation \(H_0\) Problem. We recall that the radiance \((\rho_{rad})\) is determined by the temperature of the photon (*T*) and the relativistic degrees of freedom \((G_*)\)ie,

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

(1)

In a radiation-dominated universe phase, where only photons and neutrinos are ultrarelativistic, the relationship between photons and neutrinos is temperature \((4/11)^{1/3}\). Since photons have two states of polarization 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 help \(N_{eff}\), or some new physical interactions with neutrinos that change the neutrino decoupling temperature, or as in our case some particles that mimic the effect of neutrinos. Because we’re trying to increase \(H_0\) by increasing \(N_{eff}\), \(\Delta N_{eff}\) tell us how much additional radiation we are adding to the universe via 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 additional neutrino species by adding any other type of radiation source. Calculation of the ratio between the density of a neutrino species and the density of cold dark matter at matter-radiation equality \((t = t_{eq})\) 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 an additional neutrino species represents \(16\%\) of dark matter density at matter-radiation equality. provided \(\chi\) arises from two body decays of a mother particle \(\chi’\)Where \(\chi ‘ \rightarrow \chi + \nu\). in the \(\chi’\) rest system, the 4-momentum of the particles are,

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

Hence 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 \(\dew\) is the lifetime of the mother particle \(\chi’\). We emphasize that we will adopt the instantaneous decay approximation.

Using this result and the fact that a particle’s momentum 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 \(m_{\chi }\) is the dominant contribution to the energy of a particle. Here’s how we rewrite 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), whereas \(N_{CDM}\) is the total number of non-relativistic DM (cold particles). Apparently, \(N_{HDM} \ll N_{CDM}\) agree with the cosmological data. The ratio between relativistic and non-relativistic dark matter density energies 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)

Consequently *f* is the fraction of dark matter particles produced by this non-thermal process. As mentioned, *f* should be small, but we don’t have to assume an exact value for this, but it will be of the order of 0.01. This fact will continue to become clear.

Using Eq. (3) and (7) we find that the additional radiation generated via this mechanism

$$\begin{aligned} \Delta N_{eff} = \lim _{t \rightarrow t_{eq}} \frac{f\left( \gamma _{\chi } -1 \right) }{0{, }16}, \end{aligned}$$

(8th)

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

In 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,

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

(9)

With \(t_{eq} \approx 50.000 ~ \text {years} \approx 1.6 \times 10^{12} ~s\).

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

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