text:
\section{Introduction}\label{sec:intro}
Nowadays, the existence of dark matter has been firmly supported by numerous astrophysical and cosmological observations.
However, despite indisputable evidence for its existence, the nature of dark matter remains a mystery.
Indeed, all the observational evidence, such as the observations on the galactic rotation curve \cite{Rubin:1970zza}, the bullet cluster \cite{Clowe:2006eq} and the cosmic microwave background (CMB) \cite{Aghanim:2018eyx}, depends on the gravitational interaction of dark matter only.
Due to the lack of knowledge on its non-gravitational interaction,
the properties of dark matter, such as its mass, spin, its couplings to the Standard Model (SM), and whether it is composite or fundamental, remain unknown, nor do we know anything about its production mechanism.
Understanding the nature of dark matter would not be so urgent if it plays an insignificant role in the universe.
However, dark matter occupies roughly a quarter of the total energy density of the universe today, which is about five times the energy density of the content from the Standard Model (SM),
it is the dominating species that governs the expansion of the universe in a large portion of the cosmic history, and it seeds the formation of structure which gives rise to the galaxies and the clusters of galaxies that we observe today.
Therefore, resolving the mystery of dark matter is one of the biggest challenges facing particle physics, astrophysics and cosmology today.

Among the numerous attempts to tackle the problem of dark matter, the weakly interacting massive particle (WIMP) is perhaps the most studied dark-matter candidate.
In the standard WIMP paradigm, the mass and the interaction strength between the WIMPs and the SM particles are often assumed to be at the weak scale.
The WIMPs are assumed to be in thermal equilibrium with the radiation bath in the radiation dominated (RD) epoch, but freeze out of the thermal equilibrium when the interaction rate falls below the expansion rate.
It is often referred to as the WIMP miracle that the combination of weak scale mass and coupling strength together with the thermal freeze-out mechanism successfully gives rise to the correct dark-matter relic abundance, $\Omega_{\rm DM}~\simeq~0.26$.
However, aside from its minimality and strong motivation from many models of the physics beyond the Standard Model (BSM), such as theories of supersymmetry, decades of null result in both direct and indirect detections, as well as collider searches, have made the standard WIMP scenarios more and more constrained \cite{Roszkowski:2017nbc}.
Therefore, the current situation has invoked a lot of interest in other dark-matter scenarios.

{
Indeed, by modifying some of the defining features of the thermal WIMP paradigm, other viable dark-matter scenarios could also be constructed.
For example, dark matter need not be produced via the thermal freeze-out mechanism.
In fact, the interaction between dark matter and the SM particles could be very feeble such that the thermal equilibrium is never established between dark matter and the SM thermal bath.
An equally motivated production mechanism suitable for this type of scenarios is the freeze-in mechanism \cite{Hall:2009bx,Elahi:2014fsa,Bernal:2017kxu,Krnjaic:2017tio,Berger:2018xyd} in which dark matter is produced gradually by the injection from the visible thermal bath.
Typically, if dark matter is coupled to the thermal bath via renormalizable operators, its production is IR dominated by the temperature near its mass and thus insensitive to initial conditions.
On the other hand, for non-renormalizable interactions \cite{Elahi:2014fsa}, the production is sensitive to UV physics.
Another approach is to modify the single-particle picture of most WIMP models.
For instance, models from the non-minimal dark-sector framework often depend on the collective behavior of an entire ensemble of dark-sector particles \cite{Dienes:2011ja,Dienes:2011sa,Dienes:2016vei}.
One could also modify the standard $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology and consider dark-matter production in non-standard cosmologies, such as producing thermal dark matter during a matter dominated universe before the Big-Bang Nucleosynthesis (BBN), see \cite{Kane:2015jia, Acharya:2009zt} for reviews and \cite{Berlin:2016vnh,Berlin:2016gtr,Heurtier:2019eou} for recent progress.
By considering different possibilities, the relevant constraints and the potential signatures could be drastically different.}

In this paper, we are motivated by the possibility that the interaction between the dark matter and the SM thermal bath is very feeble such that the dark-matter particles could only be populated gradually via the annihilation of the SM particles.
To be specific, we consider a \emph{hidden sector} consisting of a dark scalar mediator $\phi$ and a fermionic dark matter $\chi$, and the hidden sector is only connected to the SM via a neutrino portal.
Typically, there are two kinds of neutrino-portal dark-matter models --- the $t$-channel models with the $\nu-\chi-\phi$ interaction \cite{Falkowski:2009yz,Macias:2015cna,Ko:2015nma,Hamze:2014wca,Yu:2016lof, Gonzalez-Macias:2016vxy,Escudero:2016tzx,Escudero:2016ksa,Batell:2017rol,Batell:2017cmf,Chianese:2018dsz,Bian:2018mkl,Blennow:2019fhy,Chianese:2019epo}, and the $s$-channel models with the $\nu-\nu-\phi$ interaction \cite{Cherry:2014xra,Berlin:2018ztp}, in the freeze-out scenarios.
While the $t$-channel models have been investigated in details in recent years,
in this paper, we consider the $s$-channel models in which the dark matter $\chi$ only interacts with the SM neutrinos through an s-channel exchange of $\phi$ in the freeze-in regime.

{The neutrino portal could naturally give rise to a tiny coupling between $\phi$ and the SM neutrinos in the low-energy effective theory, which is suitable for the hidden-sector freeze-in production of dark matter.
Although the dark matter can never thermalize with the SM thermal bath in freeze-in scenarios, we shall see that the dynamics within the dark sector might be able to significantly affect the dark-matter production.
In particular, if the interaction rate between $\chi$ and $\phi$ is negligible, the dark-matter production depends only on the energy injection from the SM thermal bath.
However, if such interaction is sufficiently rapid, the reannihilation of $\chi$ and $\phi$ could establish a period of thermal equilibrium within the dark sector \cite{Cheung:2010gj,Cheung:2010gk,Chu:2011be,Bernal:2015ova,Bernal:2017kxu,Krnjaic:2017tio,Berger:2018xyd}.
In the latter case, the dark-sector particles would eventually break away from the thermal equilibrium either by directly freezing out of the dark thermal equilibrium as their interaction rate drops below the expansion rate,
or by entering a phase called the \emph{Quasi-Static equilibrium} (QSE) \cite{Cheung:2010gj} in which the annihilation rate of $\chi$ is much larger than that of $\phi$ and is balanced with the injection from the SM sector before finally freezing out of it.
Ultimately, the relic abundance of the dark matter is sensitive to the differences in the evolution history.}

A sizable interaction strength within the dark sector would allow rapid annihilations of $\chi$ inside dark-matter halos at the present epoch.
Such late-time annihilations of $\chi$ followed by a rapid decay of $\phi$ could produce a substantial amount of neutrinos and thus would be constrained by the current neutrino observations.
In addition, a sufficiently large coupling between $\chi$ and $\phi$ could be exploited to solve problems on small scale structure at late time,
making our model a potentially viable candidate for the self-interacting dark matter (SIDM, see $e.g.$~Ref.~\cite{Tulin:2017ara} for review and the references in it).

This paper is organized as follows.
In Sec.\,\ref{sec:model}, we set up the neutrino-portal dark-matter model and discuss several examples for UV completion.
In Sec.\,\ref{sec:evo}, we study the evolution of the dark sector in different regimes and discuss the consequence of having different evolution history.
In Sec.\,\ref{sec:phe}, we discuss the phenomenological aspects of our model, including the constraints on the decay of the dark scalar $\phi$, the potential for our model to solve the small scale structure problems, and the indirect-detection constraints.
Finally, we conclude in Sec.\,\ref{sec:con}.

\section{The Neutrino Portal Model}\label{sec:model}
In the $s$-channel neutrino model, we consider a dark sector which consists of a Dirac fermionic $\chi$ and a light scalar mediator $\phi$.
While the dark-matter candidate $\chi$ is directly coupled to $\phi$ only, the mediator $\phi$ is in addition coupled to the SM neutrinos.
To be specific, the effective Lagrangian for a dark sector described above can be in general written as
{
\begin{eqnarray}
{\cal L}&=&{\cal L}_{\rm kin}+{\cal L}_{\rm int}\,\,\,,\\
{\cal L}_{\rm kin}&=&\frac{1}{2}(\partial_\mu \phi\partial^\mu \phi-m_\phi^2\phi^2)+\bar{\chi}(i\slashed{\partial} -m_{\chi})\chi\,\,\,,\\
{\cal L}_{\rm int}&=&-\phi \bar{\chi}(g_\chi+i\bar{g}_\chi\gamma^5) \chi-\phi \bar{\nu}_i(g_\nu^{ij}+i\bar{g}_\nu^{ij}\gamma^5)\nu_j \label{eq:lag_int}\,\,\,,\label{eq:lag_general}
\end{eqnarray}}
{
where $g_{\chi}$ and $\bar{g}_{\chi}$ are the couplings between the mediator $\phi$ and the dark matter $\chi$, while $g_{\nu}^{ij}$ and $\bar{g}_{\nu}^{ij}$ are the effective couplings of $\phi$ to the SM neutrinos in the flavor basis with the superscript $i$ and $j$ being the flavor index.}
In our model, we do not assume any non-vanishing vacuum expectation value (VEV) for the $\phi$ field.
Consequently,
1) $\phi$ does not mix with the SM Higgs boson and could thus evade all the collider constraints;
2) the only way for the dark sector to interact with the SM sector is through the light mediator $\phi$.
However, it is possible that the Higgs VEV might contribute to the mass of $\phi$, which we implicitly include in the $m_\phi$ term.

To produce dark matter through freeze-in in the early universe, we generically require the couplings {$g_\nu^{ij}$ and $\bar{g}_\nu^{ij}$} to be very tiny in order to avoid establishing thermal equilibrium between the dark and the SM sectors.
Such small couplings could be naturally generated through appropriate UV models.
In what follows, we shall discuss the UV completion for the general Lagrangian above in two limiting scenarios
-- the one with a pure scalar $\phi$ and only pure-scalar interactions; and the one in which $\phi$ is a pseudoscalar, and only pseudoscalar interactions exist.
We shall refer to these two scenarios as {\it Scenario I} and {\it Scenario II}.
The interacting Lagrangian in Eq.~(\ref{eq:lag_int}) can be rewritten by setting $\bar{g}_{\chi(\nu)}~=~0$ or $g_{\chi(\nu)}~=~0$ for the two scenarios, respectively:
{
\begin{eqnarray}
{\cal L}_{\rm int}&=&\left\{
\begin{array}{ll}-g_\chi\phi \bar{\chi}\chi - g_\nu^{ij} \phi \bar{\nu}_i \nu_j & ({\rm Scenario~ I}),\\
-i\bar{g}_\chi\phi \bar{\chi}\gamma^5 \chi - i\bar{g}_\nu^{ij} \phi \bar{\nu}_i \gamma^5 \nu_j & ({\rm Scenario~ II}).
\end{array} \label{eq:lag_eff}
\right.
\end{eqnarray}}

For Scenario I,
we are motivated by the type-I seesaw and introduce the right-handed neutrinos $N$ in addition to the scalar $\phi$.
We assume the masses $M_{N}~\gg~M_{\rm EW}$ with $M_{\rm EW}$ being the electroweak scale.
At high energy scale, the Lagrangian can be written as
\begin{eqnarray}
\mathcal{L}_{\rm int}~\supset~g_{H} \bar{L} \tilde{H} N+g_{\phi}\overline{N^c}N\phi + h.c.\,\,\,,
\end{eqnarray}
where $\tilde{H}\equiv i\sigma_2H^*$.
Around the electroweak scale, one can integrate out the heavy fields $N_i$ and obtain the dimension-6 operators:\footnote{We shall suppress the neutrino flavor index from now on, if not otherwise mentioned.}
\begin{eqnarray}
	\mathcal{L}_{\rm int}~\supset~\frac{1}{M_{N}^2} {\phi (\bar{L}\tilde{H}) (\tilde{H}^T L^c)} \,\,\,.
\end{eqnarray}
After electroweak spontaneous symmetry breaking, tree level interactions between the scalar $\phi$ and the SM neutrinos can be generated as follows:
\begin{eqnarray}
	\frac{1}{M_{N}^2} {\phi (\bar{L} \tilde{H}) (\tilde{H}^\dagger L)} \to g_\nu \phi \bar{\nu} \nu,\quad  {\textrm{~with~}} g_\nu \sim \frac{v^2}{M_N^2},
\end{eqnarray}
where $v$ is the SM Higgs VEV.
Note that since we assume $M_{N}~\gg~M_{\rm EW}$, $g_\nu$ is naturally suppressed.

For Scenario II, the $\phi-\nu-\nu$ interaction for a pseudo-scalar $\phi$ could be generated in the so-called minimal Majoron model\,\cite{Chikashige:1980qk,Chikashige:1980ui,Schechter:1981cv,Gelmini:1980re}.
In this type of model, we still introduce three right-handed neutrinos $N_i$ but in addition a SM-singlet complex scalar field $\Phi$ above the electroweak scale.
At high energy scale, the interacting Lagrangian can be written as
\begin{eqnarray}
\mathcal{L}_{\rm int}~\supset~g_{\Phi} \overline{N^c}N\Phi - g_H{\overline{L}} \tilde{H} N + h.c.\,\,\,.\label{eq:majoron1}
\end{eqnarray}
We then assume that $\Phi$ develops a VEV and spontaneously breaks a global $U(1)$ symmetry at a scale higher than $M_{\rm EW}$.
Therefore, we can write $\Phi=\varphi+v_\Phi+i\phi$, where $v_{\Phi}$ is the VEV, the pseudoscalar $\phi$ is the Nambu-Goldstone boson, and we assume $v_{\Phi}~\gg~v$.
We further assume that the $U(1)$ symmetry is not exact, but explicitly broken by a soft-breaking mass term in the potential of $\Phi$.
As a result, the pseudoscalar $\phi$ acquires a small mass and becomes a pseudo Nambu-Goldstone boson.
After the symmetry breaking, first term in Eq.~(\ref{eq:majoron1}) generates the Majorana mass term and the interacting Lagrangian becomes
\begin{eqnarray}
\mathcal{L}_{\rm int}~\supset~-\frac{M_N}{2}\bar{N}N-ig_\Phi \phi\bar{N}\gamma_5 N - g_H{\overline{L}} \tilde{H} N \,\,\,,
\end{eqnarray}
with $M_N~=~g_\Phi v_\Phi$, where we have transform $N$ to the four component notation for a Majorana field such that $N^c=N$.
We can then integrate out the heavy filed $N$, and the effective Lagrangian can be expressed as
\begin{eqnarray}
{\cal L}_{\rm int}~\supset~ i\frac{{g}_\Phi g_H^2}{M_N^2} \phi ({\overline{L}} \tilde{H}) \gamma^5 (\tilde{H}^T L^c ) + h.c.\,\,\,.
\end{eqnarray}
Finally, after the electroweak spontaneous symmetry breaking,
\begin{eqnarray}
{\cal L}_{\rm int}~\supset~ - i\bar{g}_\nu\phi \bar{\nu} \gamma^5 \nu\,,\quad {\text{with~~}} \bar{g}_\nu~\sim~\frac{v^2}{M_N^2}
~\sim~\frac{v^2}{g_\Phi^2v_\Phi^2}\,\,\,.
\end{eqnarray}
Again, $\bar{g}_\nu$ is naturally suppressed.
We point out that it has also been studied when the pNGB $\phi$ becomes a dark-matter candidate in Refs.\,\cite{Gelmini:1984pe,Lattanzi:2013uza,Berezinsky:1993fm,Bazzocchi:2008fh,Gu:2010ys,Abe:2020dut}.

\iffalse
\begin{table}[t]
\centering
\begin{tabular}{ccccc}\toprule[1.5pt]
\bf{(Inverse) Decays}  & $\phi\to\nu_i\nu_i$  & $\nu_i\nu_i \to\phi$ & - & - \\\midrule
\bf{Annihilations} & $\chi\chi\to\phi\phi$ & $\chi\chi\to \nu_i\nu_i$ & $\phi\phi\to\nu_i\nu_i$ & $\phi\phi\to\chi\chi$ \\\midrule
\bottomrule[1.25pt]
\end {tabular}\par
\caption{$i=e,\mu,\tau$, we consider here Dirac/Majorana DM and Majorana neutrinos.}
\bigskip
\label{tab:prod}
\end{table}
\fi
\begin{figure}\centering
\includegraphics[width=1.\textwidth]{plots/feyn-eps-converted-to.pdf}
\caption{Relevant Feynman diagrams in our study.}\label{feyndia}
\end{figure}

Having demonstrated how Scenario I and Scenario II could be generated from UV theories, in what follows, we shall simply use the effective Lagrangian in Eq.~(\ref{eq:lag_eff}) when discussing production of dark matter.
Note that, in the above discussion, we choose not to specify the coupling strength between $\chi$ and $\phi$.
As a consequence, while a highly suppressed $g_{\nu}$ or $\bar{g}_{\nu}$ could ensure a sufficiently feeble interaction between the SM sector and the dark sector during the dark-matter production, by tuning the couplings $g_{\chi}$ or $\bar{g}_{\chi}$, we are able to study interesting dark-sector dynamics and phenomenologies in several different regimes.
To conclude this section, we list all the Feynman diagrams relevant for our scenarios in FIG.\,\ref{feyndia}.

\FloatBarrier
\section{The Freeze-In Production of Dark Matter}\label{sec:evo}
As we have discussed in the previous sections, we are interested in the regime where the interaction between the dark-matter particles and the SM neutrinos is too feeble to ever establish thermal equilibrium between them, and the dark sector is mainly populated by a gradual injection of energy from the SM thermal bath, $i.e.$~via the freeze-in mechanism.
{Since the couplings in Eq.~(\ref{eq:lag_general}) are all renormalizable, our scenario falls in the category of IR freeze-in.
Similar to the freeze-out scenario, it is well know that the final yield of dark-matter particles is insensitive to the initial condition in IR freeze-in scenarios.}
Therefore, without loss of generality, we shall also assume that the initial energy density of the dark-sector particles is negligible. To be concrete, the evolution of dark-matter number density is determined by the following Boltzmann equation
\begin{eqnarray}
\frac{dn_\chi}{dt}
&=&-3Hn_\chi-\left(n_{\chi}^2-n_{\chi}^{\rm eq}(T)^2\right) \expt{\sigma_{\chi\chi\to\nu\nu} v}_{T} - n_{\chi}^2 \expt{\sigma_{\chi\chi\to\phi\phi} v}_{T_{\chi}}
+n_{\phi}^2 \expt{\sigma_{\phi\phi\to\chi\chi} v}_{T_{\phi}}\,\,\,,\nonumber{\nonumber}\\
\label{eq:Boltzmann_nchi}
\end{eqnarray}
where $T$ refers to the temperature of the SM thermal bath, and $T_{\chi,\phi}$ are the temperatures that the dark-sector particles $\chi$ and $\phi$ might have during their evolution which are not necessarily identical.
In the equation above, we have implicitly assumed that the phase-space distribution of $\chi$ and $\phi$ takes the form $f_{\chi,\phi}~\sim~e^{-(E-\mu_{\chi,\phi})/T_{\chi,\phi}}$ with their own temperature and chemical potential.
Besides, the condition of detailed balance, $f^{\rm eq}_\nu(p_{\nu_1})f^{\rm eq}_\nu(p_{\nu_2})=f^{\rm eq}_\chi(p_{\chi_1})f^{\rm eq}_\chi(p_{\chi_2})$, is applied to convert $n_{\nu}^{\rm eq}(T)^2\expt{\sigma_{\nu\nu\to\chi\chi}}_T$ --- the term that describes the production of dark matter from neutrino annihilation,
 into $n_{\chi}^{\rm eq}(T)^2\expt{\sigma_{\chi\chi\to\nu\nu}}_T$ which involves only the number density of $\chi$ at the temperature of the SM thermal bath.
In general, if the scattering rates are too slow, $\phi$ or $\chi$ might not be able to reach kinetic equilibrium ---
they might not have a thermal distribution with a well defined temperature.
Nevertheless, we shall keep using the thermally averaged cross-section assuming that it is a good approximation when the typical kinetic energy of the dark-sector particles is $T_{\chi,\phi}$.

As it is already discussed in Ref~\cite{Cheung:2010gj,Cheung:2010gk,Chu:2011be,Krnjaic:2017tio,Berger:2018xyd}, depending on $g_\chi$ and/or $\bar{g}_\chi$ and the dark-sector number densities, the production of dark-sector particles can be categorized into two different regimes:
\begin{itemize}
\item The \emph{pure freeze-in regime}, where the couplings between $\chi$ and $\phi$ are so small that interactions between them are negligible. As a result, $\chi$ and $\phi$ can never reach thermal equilibrium.\\
\item The \emph{reannihilation regime}, where the couplings $g_\chi$ and/or $\bar{g}_\chi$ are large enough that the annihilations process $\chi\chi\leftrightarrow\phi\phi$ starts to play a significant role in the evolution of $n_\chi$ and $n_\phi$. This could also allow $\chi$ and $\phi$ to reach thermal equilibrium during a certain period of time in the evolution of the universe with a common temperature denoted as $T_D$ which is smaller than the temperature of the SM thermal bath $T$.
\end{itemize}

In what follows, we describe these two regimes in detail while focusing on Scenario II in which $\phi$ is a pseudoscalar case, as the discussion can be easily carried over to Scenario I in which $\phi$ is a pure scalar. We also associate the FIG.~\ref{fg:fchart} for a summary for the different regime with the corresponding scale of the relevant couplings.

\begin{figure}\centering
\includegraphics[width=0.9\textwidth]{./plots/flowchart.png}
\caption{ The classification of production mechanism for our freeze-in dark matter model. $\bar{g}_\chi$ and $\bar{g}_\nu$ are the pseudo-scalar couplings of the mediator $\phi$ to the dark matter particle $\chi$ and the SM neutrino respectively.
The quasi-static equilibrium (QSE)\cite{Cheung:2010gj} refers to the situation in which the $\chi\chi\to\phi\phi$ process greatly exceeds its inverse process due to the injection of $\chi$ from the thermal bath and the injection of $\chi$ is balanced by its annihilation into $\phi$.
}\label{fg:fchart}
\end{figure}

\subsection{Pure Freeze-in Regime}
As mentioned above, in the pure freeze-in regime, not only the dark matter is decoupled from the SM thermal bath, the thermal equilibrium between dark-sector constituents $\chi$ and $\phi$ is also never reached.
Therefore, during the production of dark-matter particles $\chi$,
one can make the following approximations:
1) the number density $n_{\chi}$ is so small that the annihilation process $\chi\chi\rightarrow \nu\nu$ can be safely neglected;
2) the particle exchange between dark-sector species $\phi\phi \leftrightarrow \chi\chi$ is also negligible.
With these approximations, the Boltzmann equation Eq.~(\ref{eq:Boltzmann_nchi}) can be approximated as
\begin{eqnarray}
\frac{dn_\chi}{dt}
~\approx~-3Hn_\chi+ n_{\chi}^{\rm eq}(T)^2\langle \sigma_{\chi\chi\to\nu\nu} v \rangle_{T}\,\,\,,\label{eq:Boltzmann_pure_fi}
\end{eqnarray}
where $T$ represents the temperature of the SM thermal bath, and we have used the condition $n_{\chi}^{\rm eq}(T)~\gg~n_{\chi}(T)$.

To connect with the observed dark-matter relic density today, it is often convenient to use entropy conservation $\tilde{s}a^3~=~\rm const.$ where $\tilde{s}$ is the entropy density and $a$ is the scale factor,
and define the comoving number density or the \emph{yield} of dark matter \mbox{$Y_{\chi}~\equiv~n_{\chi}/\tilde{s}$} which eventually becomes a constant after the freeze-in process concludes.
Therefore, to reproduce the expected present-day dark-matter relic density, the final yield $Y_{\chi}(T_f)$ is required to be
\begin{equation}
Y_{\chi}(T_f)=Y_{\chi}(T_{\rm now})\approx \frac{\rho_{\rm crit}\Omega_{\chi}}{m_{\chi}s(t_{\rm now})}\approx4.14\times 10^{-10}\times\left(\frac{\rm GeV}{m_{\chi}}\right)\,\,\,.\label{eq:yield_purefi}
\end{equation}
The above expression can thus be used to constrain our model parameters.
In what follows, we shall estimate the yield from the pure freeze-in process.
We first note that
\begin{eqnarray}
{\sigma_{\chi\chi\rightarrow\nu\nu}v_{\text{M\o l}}}
&=& \frac{3\bar{g}_{\chi}^2\bar{g}_{\nu}^2 s^2}{64\pi [(s-m_\phi^2)^2+m_\phi^2\Gamma_\phi^2]E_{\chi_1}E_{\chi_2}} \,\,\,,\label{eq:xxphiphiv}
\end{eqnarray}
where $\Gamma_\phi$ is the decay width of $\phi$, $s$ is the Mandelstam variable, and $E_{\chi_{1}},~E_{\chi_{2}}$ represent the energy of the dark-matter particles in the annihilation process.
In the high-temperature regime ($T~\gg~m_{\chi}$), most of the annihilations take place with $s~\gg~4m_{\chi}^2$, whereas in the low-temperature regime ($T~\ll~m_{\chi}$), typically $s~\sim~4m_{\chi}^2$.
Taking these approximations, the thermally averaged cross-section can be expressed as follows assuming $m_\chi~\gg~m_\phi$:
\[
\expt{\sigma_{\chi\chi\rightarrow\nu\nu}v_{\text{M\o l}}}_T
~\approx~\left\{
\begin{array}{ll}
\displaystyle\frac{3\bar{g}_{\chi}^2\bar{g}_{\nu}^2 }{128\pi \langle E_\chi\rangle^2}\simeq 7.5\times 10^{-4}\times\frac{\bar{g}_{\chi}^2\bar{g}_{\nu}^2}{T^2}\,\,\,,&\quad T\gg m_{\chi}\\
\displaystyle\frac{3\bar{g}_{\chi}^2\bar{g}_{\nu}^2 }{64\pi \langle E_\chi\rangle^2}\simeq 0.015\times  \frac{\bar{g}_{\chi}^2\bar{g}_{\nu}^2}{m_{\chi}^2}\,\,\,, &\quad T\ll m_{\chi}
\end{array}
\right.\,,
\label{eq:chichi2phiphiv}
\]
where we have applied $\expt{E_{\chi}}~=~{\rho_{\chi}}/{n_{\chi}}~\simeq~3.15\,T$ when $T~\gg~m_{\chi}$ and $\expt{E_{\chi}}~\simeq~m_{\chi}$ when $T\ll m_{\chi}$.
Assuming that $f_{\chi}~=~\exp(-E_\chi/T)$, we further obtain the expressions for the annihilation rate:
\begin{eqnarray}
n_{\chi}^{\rm eq} \expt{\sigma_{\chi\chi\rightarrow\nu\nu} v}_T~\simeq~\left\{
\begin{array}{ll} 1.4 \times 10^{-4}~\bar{g}_{\chi}^2\bar{g}_{\nu}^2~T\,\,\,, & \quad T\gg m_{\chi} \\
1.9\times10^{-3}~\bar{g}_{\chi}^2\bar{g}_{\nu}^2~T\times e^{-m_{\chi}/T}\sqrt{\displaystyle\frac{T}{m_{\chi}}}\,\,\,, & \quad T\ll m_{\chi}
\end{array}
\right.\,.\label{eq:ann_rate}
\end{eqnarray}
In the pure freeze-in regime, since the $\nu\nu\to\chi\chi$ process is the only production channel for the dark-matter particles , the condition that the dark matter never reaches thermal equilibrium with the SM generally requires that
\begin{equation}
n_{\chi}^{\rm eq} \expt{\sigma_{\chi\chi\rightarrow\nu\nu} v}_T~<~H\,,\label{eq:condition_neq}
\end{equation}
where $H$ is the Hubble expansion rate.
In the radiation dominated epoch, we have
\begin{equation}
H~\simeq~\sqrt{\frac{\rho_R}{3M_P^2}}~=~\sqrt{\frac{\pi^2}{90}}~g_{\star}^{1/2}(T)\frac{T^2}{M_P}\,\,\,,\label{eq:H_RD}
\end{equation}
where $g_\star$ is the effective numbers of relativistic degrees of freedom for the energy density, and $M_P$ is the reduced Planck mass.
Since the number density in Eq.~(\ref{eq:ann_rate}) is always exponentially suppressed in the non-relativistic regime, we simply require Eq.~(\ref{eq:condition_neq}) to be satisfied in the relativistic regime, which implies
\begin{eqnarray}
\bar{g}_{\chi}\bar{g}_{\nu} ~\lesssim~ \mathcal{O}(10^{-8})~\left(\frac{m_{\chi}}{\rm GeV}\right)^{1/2}\,\,\,,\label{eq:fi_condition}
\end{eqnarray}
if $g_{\star}^{1/2}(T)~\sim~\mathcal{O}(10)$.

To proceed further with the estimate, we rewrite the Eq.~(\ref{eq:Boltzmann_pure_fi}) as
\begin{eqnarray}
\frac{dY_{\chi}}{dT}~\approx~-\frac{s\expt{\sigma v}}{HT}Y_{\chi}^{\rm eq}(T)^2\,\,\,,
\end{eqnarray}
which we can directly integrate from a large initial temperature $T_i~\gg~m_{\chi}$ to the present-day temperature $T_{\rm now}$:
\begin{eqnarray}
\int_{Y_{\chi}^i}^{Y_{\chi}^{\rm now}} dY_{\chi}
~\approx~-\int_{T_{\rm now}}^{T_i} dT~\frac{s\expt{\sigma v}}{HT}Y_{\chi}^{\rm eq}(T)^2 \,\,\label{eq:int_yield}
\end{eqnarray}
Using our previous discussion, the integrand
\begin{equation}
\frac{s\expt{\sigma v}}{HT}Y_{\chi}^{\rm eq}(T)^2~\propto~\left\{
\begin{aligned}
& 1/T^2\,\,, & \quad T~\gg~m_\chi \\
& e^{-2m_\chi/T}/T^3\,\,, &  \quad T~\ll~m_\chi
\end{aligned}
 \right.\,\,.
\end{equation}
Obviously, the Boltzmann suppression factor which appears in the non-relativistic regime suggests that the contribution to the entire integral from this part, $i.e.$~\mbox{$m_{\chi}>~T~\geq~T_{\rm now}$}, is always subleading.
At the same time, the $1/T^2$ behavior in the relativistic regime also indicates that the production of dark matter is very small when $T~\gg~m_{\chi}$.
In fact, most of the contribution to the integral in Eq.~(\ref{eq:int_yield}) actually comes from the part where $T~\sim~m_\chi$.
Therefore, we find as expected that the dark-matter production in the pure freeze-in regime is largely insensitive to the initial conditions.

Based on our discussions, we can then make following approximations and roughly estimate the final yield:
\begin{equation}
{Y_{\chi}^{\rm now}}~\approx~\int^{m_{\chi}}_{T_i} dT~\frac{s\expt{\sigma v}}{HT}Y_{\chi}^{\rm eq}(T)^2~\approx~2.1\times 10^{14}\times \frac{\bar{g}_{\chi}^2\bar{g}_{\nu}^2}{g_{\star}^{1/2}(m_{\chi})g_{\star,s}(m_{\chi})}\left(\frac{\rm GeV}{m_{\chi}}\right)\,\,\,,\label{eq:yield_approx}
\end{equation}
in which $g_{\star,s}$ is the effective numbers of relativistic degrees of freedom for entropy density.
For $m_{\chi}~\sim~\mathcal{O}(\rm GeV)$, the denominator $g_{\star}^{1/2}g_{\star,s}~\sim~10^3$.
Comparing with Eq.~(\ref{eq:yield_purefi}),
one generally need $\bar{g}_{\chi}\bar{g}_{\nu}~\sim~\mathcal{O}(10^{-11})$ to obtain the correct dark-matter relic abundance.

\begin{figure}\centering
\includegraphics[width=0.8\textwidth]{./plots/purefi.pdf}\\
\includegraphics[width=0.8\textwidth]{./plots/purefi_s.pdf}
\caption{Examples in the pure freeze-in regime.
The upper panels correspond to \emph{Scenario II} where $\phi$ is a pseudoscalar, and the lower panels correspond to \emph{Scenario I} where $\phi$ is a pure scalar.
In the left panels, the blue curve shows the evolution of the dark-matter production rate $n_{\chi}^{\rm eq}(T)\expt{\sigma_{\chi\chi\to\nu\nu}v}_T$, whereas the dash-dotted purple line is the decay rate of $\phi$.
The dashed red curve is the evolution of the Hubble expansion rate $H$.
In the right panels, we show the evolution of $Y_{\chi}$ as a function of $x~\equiv~m_{\chi}/T$.
The dotted horizontal line indicates the observed relic density of the dark matter today.
}\label{fg:purefi}
\end{figure}

Examples of the pure freeze-in scenario for both the pseudoscalar (upper panels) and scalar (lower panels) cases are shown in FIG.~\ref{fg:purefi}, in which we have chosen to plot against $x~\equiv~m_\chi/T$.
In the left panels, we show a comparison between the expansion rate $H$ and the dark-matter production rate $n_{\chi}^{\rm eq}(T)\expt{\sigma_{\chi\chi\to\nu\nu}v}_T$.
As expected, the dark-matter production rate is always smaller than $H$.
However, despite the fact that it is always decreasing as the SM temperature drops, it is closest to $H$ at $T~\sim~m_\chi$, which supports the approximation we use in Eq.~(\ref{eq:yield_approx}).
After the temperature drops below $m_{\chi}$, a quick down turn appears as the neutrinos no longer have enough energy to create the dark-matter particles.
As a result, we see in the right panel that the yield grows monotonically from a vanishingly small initial value, and then asymptotes to its maximum near $T~\sim~m_\chi$.
Note that the sudden jump for both $H$ and $Y_{\chi}$ around $x~\sim~5-10$ is not a numerical artifact but merely due to the sudden change in $g_{\star}$ and $g_{\star,s}$ when the temperature drops to $T~\sim~\mathcal{O}(100)~\rm MeV$, $i.e.$,~when the QCD phase transition occurs.

Comparing the pseudoscalar with the pure scalar cases, it is obvious that for couplings of the same size, the final yield in the latter is slightly smaller.
This is simply because the factor $s^2$ in the numerator of the thermally averaged cross-section in Eq.~(\ref{eq:xxphiphiv}) becomes $s(s-4m_{\chi}^2)$ in Scenario I (see Eq.~(\ref{eq:xsection_xxnunu}) for more details), which results in a suppression in the production rate as $T$ approaches $m_{\chi}$.
\FloatBarrier

\subsection{Reannihilation Regime}

In the pure freeze-in regime, the interactions between the dark-sector particles are so weak such that both $\chi$ and $\phi$ barely annihilate. As a consequence, the $\chi\chi\leftrightarrow\phi\phi$ process has little impact on the evolution of dark-matter number density.
However, it is also possible that while the entire dark sector is decoupled from the SM sector, the annihilations of $\chi$ and $\phi$ proceed with a non-negligible rate.
Indeed, since the interaction strength between $\chi$ and $\phi$ is governed by the coupling $\bar{g}_{\chi}$ in Scenario II (or $g_{\chi}$ in Scenario I),
by increasing $\bar{g}_{\chi}$ while adjusting $\bar{g}_{\nu}$ accordingly in order to prevent thermalizing with the SM sector, as required in Eq.~(\ref{eq:fi_condition}),
the dark sector could establish its own thermal equilibrium for a period of time in the history of the universe as it is populated by the SM thermal bath.
This thermal equilibrium is characterized by a common temperature $T_D$ for both $\chi$ and $\phi$.
In this situation, Eq.~(\ref{eq:Boltzmann_nchi}) can be rewritten as
\begin{eqnarray}
\frac{dn_\chi}{dt}
&\approx&-3Hn_\chi+n_{\chi}^{\rm eq}(T)^2 \expt{\sigma_{\chi\chi\to\nu\nu} v}_{T}
-\left(n_{\chi}^2-n_{\chi}^{\rm eq}(T_D)^2\right)\expt{\sigma_{\chi\chi\to\phi\phi} v}_{T_D}\,\,\,,\label{eq:Boltzmann_dseq}
\end{eqnarray}
in which the the thermally averaged cross-section for $\chi\chi\leftrightarrow\phi\phi$ is evaluated at $T_D$.

The thermal equilibrium in the dark sector requires the interaction rate between the dark-sector particles
\mbox{$n_{\chi}^{\rm eq}(T_D)\expt{\sigma_{\chi\chi\to\phi\phi}}_{T_D}>H(T)$}.
For a sizable cross-section $\sigma_{\chi\chi\to\phi\phi}$ , this condition could be immediately realized if the initial particle number density $n_{\chi}$ or $n_\phi$ is large enough.
By contrast, if the initial number density is small, the dark sector could start without thermal equilibrium but slowly build it up as the SM sector keeps populating the dark-sector particles.
Once the thermal equilibrium is established within the dark sector,
one would expect the number density $n_{\chi}$ to track $n_\chi^{\rm eq}(T_D)$, since the last term in the Boltzmann equation above always tends to bring back any deviation from the thermal equilibrium.
This is indeed true if the second term on the right-hand side of Eq.~(\ref{eq:Boltzmann_dseq}) is not large enough such that the $\chi\chi\to\phi\phi$ process can always be balanced by the $\phi\phi\to\chi\chi$ process.
However, if this SM source term is sufficiently large, $n_{\chi}$ might be able to grow significant larger than $n_\chi^{\rm eq}(T_D)$.
Therefore, the annihilation rate of $\chi$ could overwhelm that of $\phi$, and this could occur even when the latter is still larger than the Hubble rate.
In this situation, Eq.~(\ref{eq:Boltzmann_dseq}) can be approximated as
\begin{eqnarray}
\frac{dn_\chi}{dt}
~\approx~-3Hn_\chi+n_{\chi}^{\rm eq}(T)^2 \expt{\sigma_{\chi\chi\to\nu\nu} v}_{T}
 -n_{\chi}^2\expt{\sigma_{\chi\chi\to\phi\phi} v}_{T_D}\,\,\,.\label{eq:Boltzmann_qse}
\end{eqnarray}
The equation above simply means that while the SM thermal bath is always sourcing the dark matter $\chi$, the annihilation of $\chi$ always tends to deplete this particle injection.
As a result, the QSE phase often occurs in which the last two terms in Eq.~(\ref{eq:Boltzmann_dseq}) are balanced against each other \cite{Cheung:2010gj}.
To be precise, the QSE phase is defined by
\begin{eqnarray}
Y_{\chi}(T)~\approx~Y_{\chi}^{\rm QSE}(T)~\equiv~Y_{\chi}^{\rm eq}(T)\sqrt{\frac{\expt{\sigma_{\chi\chi\to\nu\nu} v}_{T}}{\expt{\sigma_{\chi\chi\to\phi\phi} v}_{T_D}}}\,\,\,.
\end{eqnarray}

The dark thermal equilibrium or the QSE persists until $n_{\chi}(T)\expt{\sigma_{\chi\chi\to\phi\phi}}_{T_D}~\lesssim~H(T)$, and then the dark matter experiences a \emph{dark freeze-out}.
However, unlike the standard thermal freeze-out scenario, the comoving number density of the dark matter might not be fixed at its value around the dark freeze-out, because the production of $\chi$ from the SM thermal bath could still make a significant contribution to the final yield even after the dark freeze-out.
In any case, differences caused by having different initial number densities for $\chi$ or $\phi$ would eventually be washed out by the thermalization within the dark sector --- the freeze-in mechanism is also independent of initial conditions in the reannihilation regime.

As mentioned above, Eq.~(\ref{eq:Boltzmann_dseq}) depends on the dark-sector temperature $T_D$, which, unlike the SM temperature, is not known \emph{a priori}.
In order to find $T_D$, we numerically solve the Boltzmann equation for the total energy density in the dark sector:
\begin{eqnarray}
\frac{d\rho_D}{dt}&=&-3H(P_D+\rho_D)+2n_\nu^{\rm eq}(T)^2\mathcal{P}_{\nu\nu\to\chi\chi}(T)+2n_\nu^{\rm eq}(T)^2\mathcal{P}_{\nu\nu\to\phi\phi}(T)
\nonumber{\nonumber}\\
&~&+n_\nu^{\rm eq}(T)^2\mathcal{P}_{\nu\nu\to\phi}(T)
-n_\phi\mathcal{P}_{\phi\to\nu\nu}(T_D)\,\,\,,
\label{eq:rhodBoltzmann}
\end{eqnarray}
where the energy density $\rho_D\equiv\rho_\chi+\rho_\phi$, the pressure $P_D\equiv P_\chi+P_\phi$,
and the thermally averaged energy transfer rate $\mathcal{P}_{a\dots\to i\dots}$ gives the energy transferred to or from the dark sector through the process $a+\dots\to i+\dots$.
We shall show its explicit form in Appendix~\ref{sec:energy_transfer}.
Note that the factor of 2 in the second and third term on the right-hand side takes into account the fact that two dark-sector particles are produced via each of the corresponding processes.
Again, we have made the assumption that the annihilation rate of the dark-sector particles into the SM neutrinos is negligible.
Such assumption is valid as long as $T_D\ll T$, which implies the number density of either $\chi$ or $\phi$ is much smaller than that of the SM neutrinos, and is also consistent with the second law of thermodynamics --- the dark sector can never be hotter than the SM thermal bath which populates it in the first place.
We shall see later that the decay and inverse decay could potentially thermalize $\phi$ with the SM neutrinos.
However, this has little impact for the yield of $\chi$, if any.

While in thermal equilibrium, the dark-sector energy density is only a function of
the dark-sector temperature $T_D$ once the masses of $\chi$ and $\phi$ are fixed.
Therefore, at every step of the numerical simulation, $T_D$ can be obtained by solving
\begin{equation}
\rho_D(T_D)~=~\xi_\chi\int\frac{d^3p}{(2\pi)^3}\frac{E_\chi }{e^{E_\chi/T_D}+1}+\xi_\phi\int\frac{d^3p}{(2\pi)^3}\frac{E_\phi }{e^{E_\phi/T_D}-1}\,\,\,,\label{eq:rhod}
\end{equation}
in which $\xi_{\chi}=2$ and $\xi_{\phi}=1$ are the number of degrees of freedom for $\chi$ and $\phi$.
After obtaining the time dependence of the dark sector temperature $T_D(t)$, one can then insert it back to Eq.~(\ref{eq:Boltzmann_dseq}) and solve for $n_{\chi}$.

\begin{figure}\centering
\includegraphics[width=0.9\textwidth]{./plots/eqfi.pdf}
\includegraphics[width=0.9\textwidth]{./plots/eqfi_s.pdf}
\includegraphics[width=0.9\textwidth]{./plots/eqfi_QSE.pdf}
\includegraphics[width=0.9\textwidth]{./plots/eqfi_QSE_s.pdf}
\caption{Benchmark results for the reannihilation regime, with the
upper (lower) two panels showing the cases without (with) the QSE phase.
The left columns shows the rate of the energy transfer from the SM sector as well as the rates of energy-density dilution due to the expansion of the universe.
In the middle column, we compare the expansion rate with the interaction rates relevant for the dark-matter number density.
In the right column, we show the evolution of $Y_{\chi}$ and $Y_{\chi}^{\rm eq}(T_D)$ --- the yield of $\chi$ if it is in thermal equilibrium with $\phi$, which we have colored to indicate the corresponding $T_D$.
The evolution of $Y_{\chi}^{\rm QSE}(T)$) is also plotted in the lower two panels for the cases with the QSE phase.
}\label{fg:reannihilation}
\end{figure}

In FIG.~\ref{fg:reannihilation}, we present our benchmark results in the reannihilation regime including the evolution of the energy-transfer rates, the particle exchange rates, as well as the dark-matter yield.
The upper two panels show the cases in which the dark sector undergoes a period of dark thermal equilibrium and then freezes out.
By contrast, the cases with the additional QSE phase before the dark freeze-out are shown in the lower panels.
Comparing with the pure freeze-in cases in FIG.~\ref{fg:purefi}, we have enhanced the coupling of $\phi$ to the dark matter, and at the same time lowered its coupling to the neutrinos in order to avoid thermalization with the SM thermal bath.
The couplings are tuned such that the cases in Scenario II could have a final yield of $\chi$ similar to that of the dark matter today.
At the same time, for cases with (or without) the QSE phase, we choose to use couplings of the same size just to demonstrate that the cases in Scenario I tend to have a smaller final yield due to the same reason that we have mentioned in the pure freeze-in cases.

In the left panels, we see for all the cases, most of the energy injected into the dark sector is from the $\nu\nu\to\chi\chi$ channel at early times (solid blue curve).
However, the contribution from this channel decreases drastically after the SM temperature drops below the mass of the dark-matter particles, as we can tell from the quick down turn of the solid blue curve for $x~\gtrsim~1$, and then the inverse decay $\nu\nu\to\phi$ (solid green curve) starts to dominate.
On the other hand, the energy transfer rate of the annihilation process $\nu\nu\to\phi\phi$ (solid orange curve), is always negligibly small.
These observations are consistent with the expectations from our scenario in which $\bar{g}_{\nu}$ (or $g_{\nu}$) is very tiny, since the amplitude of the $\nu\nu\to\phi\phi$ process is proportional to $\bar{g}_{\nu}^2$ (or $g_{\nu}^2$), whereas the amplitudes of the $\nu\nu\to\chi\chi$ and $\nu\nu\to\phi$ processes are proportional to $\bar{g}_{\nu}\bar{g}_{\phi}$ and $\bar{g}_{\nu}$ (or $g_{\nu}g_{\phi}$ and $g_{\nu}$), respectively.

Comparing the cases with and without the QSE phase, we notice that
because both the coupling of $\phi$ to the dark matter and to the neutrinos are larger in the cases with the QSE phases, more energy is injected from the SM in these cases.
As a result, in the middle and the right panels, while the upper two cases only reach thermal equilibrium around $x~\sim~10^{-3}$, the lower two cases have already established thermal equilibrium long before that, and have acquired much more dark-matter particles when the dark sector is in thermal equilibrium.
As a consequence, the evolution after reaching thermal equilibrium is also drastically different.

In the first two cases, $Y_{\chi}$ tracks $Y_{\chi}^{\rm eq}(T_D)$ until $n_{\chi}^{\rm eq}(T_D)\expt{\sigma_{\chi\chi\to\phi\phi}}_{T_D}~\lesssim~H$, which occurs around $x~\sim~0.02-0.05$.
At this time, the temperature in the dark sector $T_D$ is already smaller than $m_{\chi}$ as one can tell from the color variation in $Y_{\chi}^{\rm eq}(T_D)$.
As we have mentioned above, after the dark freeze-out, the yield $Y_{\chi}$ is not fixed yet.
Similar to the pure freeze-in scenario, the solid blue curve in the middle panel shows that the production of $\chi$ from the SM neutrinos is still significant until the SM temperature is much smaller than $m_{\chi}$.
Therefore, with the injection from the SM thermal bath continues to contribute, the yield only approaches its asymptote after $x~\gtrsim~1$.

In the bottom two cases, $Y_{\chi}$ tracks $Y_{\chi}^{\rm eq}(T_D)$ at the beginning, and can even grow larger than its final asymptotic value because of a larger $\bar{g}_{\nu}$ (or $g_\nu$).
These ``excess'' dark-matter particles are rapidly depleted by annihilating into $\phi$.
As a result, as $T_D$ approaches and goes below $m_{\chi}$, $Y_\chi$ grows increasingly slower and eventually starts to decrease (see the green part of $Y_{\chi}^{\rm eq}(T_D)$).
Note that, due to a larger coupling between $\chi$ and $\phi$, $Y_\chi$ can still track $Y_{\chi}^{\rm eq}(T_D)$ even when $\chi$ is already non-relativistic.
As $T_D$ keeps dropping, the inverse process $\phi\phi\to\chi\chi$ becomes less and less important due to kinematics.
However, since $\bar{g}_{\nu}$ (or $g_\nu$) is larger in these cases, the dark-matter annihilation rate $n_{\chi}\expt{\sigma_{\chi\chi\to\phi\phi}v}_{T_D}$ can still be larger than $H$.
This is clearly shown in the middle panels as the solid brown curve peels off from the dash-doted black curve.
Consequently, the dark matter breaks away from the thermal equilibrium and enters the QSE phase in which $Y_{\chi}$ tracks $Y_{\chi}^{\rm QSE}(T)$.
Finally, when $n_{\chi}(T)\expt{\sigma_{\chi\chi\to\phi\phi}v}_{T_D}~\lesssim~H(T)$, the QSE can no longer be maintained.
Then, $\chi$ freezes out, and the yield approaches its final value.

\subsubsection{\texorpdfstring{$T_D$}~~Evolution}
\begin{figure}[t]\centering
\includegraphics[width=0.6\textwidth]{./plots/temp.pdf}
\caption{The evolution of $T_D$ if the dark sector is in thermal equilibrium.
The yellow curve is obtained from the cases with the QSE phase, whereas the blue curve is from cases without it, and there is no visible distinction no matter $\phi$ is a scalar or pseudoscalar.
Since the temperature in the dark sector is only physical when $\chi$ and $\phi$ are in thermal equilibrium, we roughly shaded the region in which the dark thermal equilibrium is established.
As a reference, we use the red curve to show the temperature of the SM thermal bath.
}\label{fg:TD}
\end{figure}
We comment here on the evolution of dark-sector temperature $T_D$ in the the reannihilation regime.
The curves for $T_D$ which are obtained by solving Eq.~(\ref{eq:rhod}) are shown explicitly in FIG.~\ref{fg:TD}.
Note that, we only include two curves for $T_D$ because there is no visible distinction between the scalar and pseudoscalar cases.
It is worth emphasizing that, though the solution to Eq.~(\ref{eq:rhod}) is unique and exists in the entire regime we are considering, $T_D$ is only physically meaningful when the dark-sector particles $\chi$ and $\phi$ are in thermal equilibrium.
We therefore shade the region in which the dark thermal equilibrium is reached.
Within this region, we can see that slopes of both $T_D$ curves are less steep than the that of $T$.
This is because while the energy density of the SM thermal bath is only subject to the expansion of the universe and some minor impact from the change of $g_{\star}$, the dark-sector energy density is in addition always sourced by the SM.
Therefore, its temperature $T_D$ decreases slower than $T$.
After $\chi$ and $\phi$ decouple, the momentum distributions of $\chi$ and $\phi$ evolve independently.
As we can see from the left panels of FIG.~\ref{fg:reannihilation}, the inverse decay of neutrinos into $\phi$ starts to surpass the annihilation into a pair of $\chi$ around $x~\sim~0.1$, which is shortly before or after $\phi$ and $\chi$ decouple.
After that, most of the energy injection from the SM thermal bath goes into $\phi$.
However, the energy that goes into $\phi$ and cannot be rapidly converted into $\chi$ since $\chi$ is already non-relativistic when it decouples.
The energy density in the dark sector is thus dominated by $\phi$ after decoupling.
Therefore, assuming that the kinetic equilibrium is still maintained for $\phi$, and that chemical potential is negligible, the $T_D$ curves to the right of the shaded region largely reflect the evolution of the energy density of $\phi$, $i.e.$~$T_D~\approx~T_{\phi}$ since $\rho_D~\approx~\rho_{\phi}^{\rm eq}(T_D)$.
However, $T_D$ always tends to be an overestimate for $T_{\phi}$ in this region as $\rho_{\chi}~>~\rho_{\chi}^{\rm eq}(T_D)$ after $\chi$ decouple.
Eventually, when $H(T)~\sim~\Gamma_{\phi}$, $\phi$ decays rapidly into neutrinos.
For the choice of $m_{\phi}$ and $\bar{g}_{\nu}$ (or $g_{\nu}$) in the examples, the decay occurs around $x~\sim~10-20$, when $\phi$ starts to transition from relativistic to non-relativistic.
Therefore, though both $T_D$ curves eventually merge with $T$, which indicates that $\phi$ could thermalizes with the SM sector through decay and inverse decay, its energy density is already suppressed at that time --- it is only the neutrinos living on the high momentum tail of the Fermi-Dirac distribution that are able to create the $\phi$ particles, which, in turn, quickly decay and deposit its energy back to the SM thermal bath.

\FloatBarrier

\section{Model Phenomenologies}\label{sec:phe}
In this section, we discuss general considerations on the phenomenological aspects of our model.
We notice that, at early times, the existence of the dark scalar $\phi$ might contribute to a non-negligible fraction of energy density in the universe, and its decay would inject energy into the SM thermal bath.
Therefore, our model could be constrained by measurements on BBN and CMB.
At late times, the interaction within the dark sector could allow a large annihilation rate inside dark-matter halos which would produce a lot of neutrinos.
Thus our model also receives constraints from the current observations on astrophysical neutrinos.
Moreover, the self-interaction between dark-matter particles could also suppress the formation of small structures.
As a result, our model could potentially be exploited to solve the small scale structure problems.
In what follows, we discuss these aspects in turn based on the general effective Lagrangian in Eq.~(\ref{eq:lag_general}) instead of referring to any specific UV completion.

\subsection{Dark Scalar Decay}
The existence of $\phi$ and its decay might lead to measurable effects by changing the expansion rate of the universe and injecting SM neutrinos to the visible sector.

On one hand, $\phi$ could decay out of equilibrium,
$i.e.$~$\Gamma_\phi/H(m_\phi)~<~1$ (or equivalently $T_{\rm dec}~<~m_{\phi}$), where $T_{\rm dec}$ is the SM temperature when $H~=~\Gamma_\phi$.
If this occurs before BBN, it simply reheats the universe by producing entropy into the visible sector since neutrinos are still tightly coupled to the SM plasma.
Therefore, other than slightly changing the temperature evolution of the SM thermal bath and diluting the relic density of $\chi$, the decay of $\phi$ would not leave any measurable effect on BBN, even if $\phi$ has a significant amount of energy density before decaying.
However, if such out-of-equilibrium decay occurs during BBN but before the neutrino decoupling, it could potentially disturb the formation of light elements by changing the way the SM temperature evolves.
In addition, if $\phi$ decays after the neutrino decoupling, it could contribute to the effective number of neutrino species $N_{\rm eff}$ which can be constrained by CMB measurements as $\Delta N_{\rm eff}~\equiv~N_{\rm eff}-3.045$ \cite{Tanabashi:2018oca}.
Therefore, to avoid upsetting BBN and CMB, we basically need $\phi$ and its decay products to have a negligible contribution to the total energy density of the universe when $T~\lesssim~10 \rm MeV$.

On the other hand, if $\Gamma_\phi/H(m_\phi)~>~1$ (or $T_{\rm dec}~>~m_{\phi}$), the decay and inverse decay would bring $\phi$ into thermal equilibrium with the SM neutrinos.
If this occurs before neutrino decoupling, $\phi$ then becomes an additional component of the visible plasma and could contribute as much as one relativistic degree of freedom to the total energy density when it is relativistic.
This could lead to significant increase of the expansion rate during BBN, which would affect the prediction for the abundances of light elements.
If $\phi$ decays in equilibrium after neutrino decoupling, the decay products could then lead to a measurable $\Delta N_{\rm eff}$ in CMB.
It could also affect BBN if the production of light elements is not concluded yet.
Therefore, observational constraints generally require $\phi$ to be non-relativistic when BBN starts, $i.e.$~$m_\phi~\gtrsim~10~\rm MeV$.

In our model, using Eq.~(\ref{eq:width_phinunu}), the SM temperature at which the decay occurs is roughly estimated to be
\begin{eqnarray}
T_{\rm dec}&\approx& \frac{3\sqrt{\bar{g}_{\nu}^2+g_{\nu}^2}}{4\sqrt{2}\pi}\left(\frac{10}{g_{\star}(T_{\rm dec})}\right)^{1/4} \left(\frac{m_{\phi}}{\rm MeV}\right)^{1/2}\left(\frac{M_{P}}{\rm MeV}\right)^{1/2}~{\rm MeV}\nonumber{\nonumber}\\
&\approx&  10^{10} \times\sqrt{\bar{g}_{\nu}^2+g_{\nu}^2}\left(\frac{10}{g_{\star}(T_{\rm dec})}\right)^{1/4} \left(\frac{m_{\phi}}{\rm MeV}\right)^{1/2}~{\rm MeV}\,\,\,.
\end{eqnarray}
Together with the relic abundance estimate in Eq.~(\ref{eq:yield_approx}), we see for $m_\chi~\sim~\mathcal{O}({\rm GeV})$ and $g_{\chi}$ or $\bar g_\chi~\lesssim~\mathcal{O}(1)$, $T_{\rm dec}~\approx~ (m_\phi/{\rm MeV})^{1/2}~\rm MeV$.
We have estimated that in order to have an out-of-equilibrium decay before BBN i.e. $m_\phi~ >~T_{\rm dec}~>~10 {\rm MeV}$ or an in-equilibrium decay with $\phi$ becoming non-relativistic when BBN starts, $m_\phi$ roughly needs to be larger than $10$ MeV.
This explains the choice of the masses in our benchmark examples.
Among the benchmark results that we have shown, the cases in FIG.~\ref{fg:purefi} and the cases in the lower two rows of FIG.~\ref{fg:reannihilation} correspond to the in-equilibrium decay, whereas the cases in the upper two rows of FIG.~\ref{fg:reannihilation} correspond to the out-of-equilibrium decay.
In the former cases, our choice of mass $m_\phi~=~50~\rm MeV$ allows $\phi$ to be non-relativistic when BBN starts, therefore its energy density is already negligible.
For the latter cases, the fact that $T_{\rm dec}~>~10~\rm MeV$ ensures that no BBN or CMB bounds would be violated.

\subsection{Indirect Detection}
The decay of $\phi$ could also be exploited in the indirect detection of dark matter.
In the late-time universe, though the interaction rate between the dark matter $\chi$ the SM particle is extremely small, $\chi\chi\to\phi\phi$ followed by rapid decay of $\phi\to\nu\nu$ could potentially produce a signal that can be probed by astrophysical neutrino observation experiments.
These experiments usually give bounds on the dark matter annihilation cross-section $\langle\sigma v\rangle$.
In the non-relativistic regime, $\langle\sigma v\rangle$ can be expanded into powers of the relative velocity $v$:
\begin{eqnarray}
\langle\sigma v\rangle~=~a + b \langle v^2\rangle +{\cal O}(v^4)\,\,\,,
\end{eqnarray}
where the angle brackets represent the average over the local dark-matter distribution at the signal source.
In our model $a$ and $b$ can be expressed as:
\begin{eqnarray}
a &=& \frac{g_\chi^2\bar{g}_\chi^2m_\chi \sqrt{m_\chi^2-m_\phi^2}}{4\pi(2m_\chi^2-m_\phi^2)^2}\,\,\,,\\
b &=&\frac{m_\chi}{24\pi(2m_\chi^2-m_\phi^2)^4\sqrt{m_\chi^2-m_\phi^2}} \Big\{ g_\chi^4(18 m_\chi^6+20m_\phi^4m_\chi^2-34m_\phi^2 m_\chi^4 -4m_\phi^6)  \nonumber \\
&~& +\big[2\bar{g}_\chi^4(m_\chi^2-m_\phi^2)^3+3g_\chi^2 \bar{g}_\chi^2(m_\phi^6 - 8 m_\phi^4m_\chi^2 +20m_\phi^2 m_\chi^4-12 m_\chi^6)\big] \Big\}\,\,\,.
\end{eqnarray}
According to the result in Ref.~\cite{Arguelles:2019ouk}, we find that the most stringent bound on the annihilation cross-section $\langle\sigma_{\chi\chi\to \nu\nu}v\rangle$ for $m_{\chi}~\sim~1~\rm GeV$  comes from Super-Kamiokande (SK) experiment \cite{Fukuda:2002uc,Richard:2015aua}, which translates into  $a~\lesssim~10^{-24}~\rm\ cm^3\cdot s^{-1}$ and $b~\lesssim~10^{-16}~\rm\ cm^3\cdot s^{-1}$.
We plot this constraint against $\bar{g}_\chi$ and $g_\chi$ in FIG.~\ref{fg:indrect} where the gray region is excluded by the SK experiment.
Our benchmark points in FIG.\,\ref{fg:reannihilation} all seat infinitely far away along the four colored lines which cannot be shown on the $\log$-scale plot as they all have one of the neutrino couplings $g_\chi$ or $\bar{g}_\chi$ being zero.
Nevertheless, our benchmarks all survive from the indirect detection constraint.

\begin{figure}[t]\centering
\includegraphics[width=0.6\textwidth]{plots/bm3.pdf}
\caption{We plot the indirect detection constraint from the Super-Kamiokande.
The excluded region is colored in gray.
Different colored lines are related to the four benchmark points in FIG.\,\ref{fg:reannihilation}.
As the benchmark all have one of the neutrino couplings vanished, they seat at infinitely negative $g_\chi$ or $\bar{g}_\chi$ along the lines in this $\log$-scale plot.
}\label{fg:indrect}
\end{figure}

\FloatBarrier

\subsection{Possible Solution to the Small Scale Structure Problem}
\begin{figure}[t]\centering
\includegraphics[width=0.6\textwidth]{plots/bm1.pdf}
\caption{Viscosity cross-section $\sigma_V$ per dark mass $m_\chi$ versus the velocity.
Four curves are clustered in two groups, $m_\phi=10~\rm MeV$ and $m_\phi=50~\rm MeV$.
The values of $g_\chi$ are chosen such that at low velocity $\sigma_V/m_\chi$ is ${\cal O}(1)~{\rm cm^2/g}$.
The values of $\bar{g}_\chi$ are chosen to demonstrate its negligible effect on the $\sigma_V/m_\chi$.
The cyan arrow represents the constraint from the bullet cluster: $\sigma_V/m_\chi~\lesssim~0.7~{\rm cm^2/g}$ for $v~\sim~4000~{\rm km/s}$ \cite{Tulin:2017ara}.
}\label{fg:selfxsec}
\end{figure}
Finally, we investigate the potential for our model to solve problems on small scale structures.
It is well known that though the standard $\Lambda$CDM cosmology is very successful in predicting the large scale structure of the universe,
it fails to predict correct structures that are compatible with the observational data at the small scales.
The most prominent small-scale problems are the follows:
1) \textit{the core-cusp problems} --- the dark-matter density profile in the center of a galaxy predicted from simulations with collisionless cold dark matter (CDM) typically scales as $r^{-1}$ (``cuspy''), while the observations show that both cored (dark matter density $\sim~r^0$) and cuspy centers exist \cite{Kahlhoefer:2019oyt};
2) \textit{the missing satellites problem} --- the observed number of satellite galaxies in the local group is one to two orders of magnitude fewer than the prediction from CDM simulations;
\textit{the too-big-to-fail problem} --- when matching the most luminous observed satellites of the Milky Way with the most massive subhalos from CDM simulations, the subhalos appear to be too massive to host those observed galaxies\footnote{This is the too-big-to-fail problem in the Local Group.
The too-big-to-fail problem in the field refers to the fact that the rotation curves for a large fraction of the dwarf galaxies indicates that they reside in halos with masses that are smaller than what is predicted using abundance matching between the observation and the CDM simulations \cite{Papastergis:2014aba} --- implying that some of the more massive halos fail in forming their galactic counterpart.}.
These problems motivate the consideration on the self-interaction of dark-matter particles which could potentially solve the problem by lowering the density in the central region of a galaxy as compared to the CDM simulations.
An excellent review on this topic is provided by {the authors of} Ref.\ \cite{Tulin:2017ara}, where a rough bound on the dark-matter self-scattering cross-section over its mass $\sigma/m_{\chi}$ can be derived by fitting the astrophysical data.
As pointed out in Ref.\ \cite{Tulin:2017ara}, $\sigma/m_{\chi}~\sim~\mathcal{O}(1)\ {\rm cm^2/g}$ can solve the small-scale problems, and a transition in the scaling of the velocity dependence of the self-interacting cross-section around ${\cal O}(10^3)~\rm km/s$ is crucial
for reconciling the collisionless nature of the dark matter on large scales with the collisional behavior on small scales.
On the other hand, Ref.\ \cite{Tulin:2013teo} points out that the viscosity cross section
\begin{eqnarray}
\sigma_V~\equiv~2\pi \int_0^\pi \frac{d\sigma}{d\Omega}(1-\cos\theta^2) \sin\theta {\rm d}\theta \,\,\,,
\end{eqnarray}
is a better proxy than the center-of-mass cross-section $\sigma$ in characterizing the self-interacting strength of the dark matter when dealing with small scale structure problems as it emphasizes the momentum transfer in the perpendicular direction and better {describes} the speed of energy equalization.
Therefore, we calculate $\sigma_V/m_{\chi}$ in our model
and find that, at zero velocity, to leading order in the expansion with respect to $m_\phi/m_\chi$
\begin{equation}
\sigma_V~\sim~\frac{g_\chi^4 m^2_\chi}{6\pi m^4_\phi}\,\,\,.\label{eq:sigmaV_exp}
\end{equation}
Notice that it depends on $g_{\chi}$ only as the terms containing $\bar g_\chi$ are suppressed by at least $(m_\phi/m_\chi)^2$.
This makes it more difficult if not entirely impossible for scenarios with $\bar g_\chi$ alone to solve the problems on small scale structures.
Our result of $\sigma_V/m_\chi$ is shown in FIG.~\ref{fg:selfxsec}, where we take four benchmark points with fixed $m_\chi~=~1~\rm GeV$.
Two values of $m_\phi$, $10$ MeV and $50$ MeV, are selected to illustrate the difference in the velocity dependence.
For each $m_\phi$, we further select a value for $g_\chi$ to ensure that $\sigma_V/m_\chi~\sim~{\cal O}(1)\ {\rm cm^2/g}$ at low velocity in order to solve the small scale structure problem, and two extreme values for $\bar{g}_\chi$ to demonstrate its weak effect on $\sigma_V$.

Obviously, for the benchmark masses we select in FIG.\,\ref{fg:reannihilation}, ${\cal O}(1)$ value is needed for $g_{\chi}$ in order to solve the small scale problems while roughly satisfying the constraints on large scales ($e.g.$~the Bullet Cluster constraint).
However, a $g_{\chi}$ of this size would lead to a long period of dark thermal equilibrium or QSE, making the yield of dark matter too low at the dark freeze-out.
In addition, since $g_\nu$ and/or $\bar{g}_\nu$ are bounded from above by the requirement on freeze-in production, increasing $g_\nu$ and/or $\bar{g}_\nu$ could not solve the problem with our choice of mass parameters.
On the other hand, decreasing $m_\phi$ to $\sim~10~\rm MeV$ could allow a smaller $g_\chi$ and is thus able to solve the small scale problems and obtain the correct relic abundance simultaneously.
However, for suitable values of $g_\nu$ and $\bar{g}_\nu$ (roughly the same order as the values shown in FIG.\,\ref{fg:purefi} and FIG.~\ref{fg:reannihilation} since it is not sensitive to $m_\phi$) that yields the correct relic abundance, the decay of $\phi$ would violate the BBN constraint by presenting a sizable $\Delta N_{\rm eff}$.
Thus, we find the only way to make our model a potentially viable SIDM candidate while satisfying the constraints on the relic abundance and BBN is to increase $m_{\chi}$ while adjusting $g_\chi$ accordingly to maintain an appropriate $\sigma_V/m_\chi$, since a larger $m_\chi$ would enable $\chi$ to freeze out earlier before getting severely Boltzmann suppressed.

In general, we see that the conditions for solving the small scale structure problems (requiring $g_{\chi}~\sim~\mathcal{O}(1)$ and preferring lighter $m_{\phi}$) tend to act against the requirement to produce the correct dark-matter relic abundance in the freeze-in regime (favoring smaller $g_{\chi}$ to prevent a late dark freeze-out) and the bounds on the decay of $\phi$ (favoring $m_\phi~\gtrsim~10~\rm MeV$).

\FloatBarrier

\section{Conclusion}\label{sec:con}

In this paper, we study a simplified model describing the neutrino-portal dark matter $\chi$.
In this model, the dark matter $\chi$ does not couple directly to the SM particles.
Instead it interacts with the SM neutrinos via an $s$-channel exchange of the light scalar mediator $\phi$.
Such neutrino-portal interaction could naturally give rise to a tiny coupling between the dark and the SM sector by relating the smallness of the coupling to the smallness of the neutrino mass, and therefore is appropriate for the freeze-in production of dark matter which requires an extremely small interaction rate between the two sectors.
We then study the possible UV completions of our model and the production of dark matter in the early universe focusing on interesting dynamics that could arise in the dark sector.
We also investigate several potential phenomenological constraints associated with our model.
Our main results are summarized in the following:
\begin{itemize}
\item We point out two UV origins of the simplified model, namely, the type-I seesaw and the Majoron model, that can naturally induce a small pure scalar and pure pseudo-scalar coupling between $\phi$ and neutrinos, respectively.
\item We study the freeze-in production of $\chi$ in the pure freeze-in regime as well as in the reannihilation regime.
For the former, we analytically estimate the relic abundance and present two benchmark results in FIG.~\ref{fg:purefi} by solving the Boltzmann equation for the number density $n_{\chi}$ numerically.
For the latter, we study the dark-sector dynamics in detail and further identify two different scenarios --- the one with the dark thermal equilibrium only and the one with the additional QSE phase.
Typically, the one with the QSE phase requires a larger rate of injection from the SM sector (larger $g_\nu$ and/or $\bar g_\nu$).
We find that since the Boltzmann equation for the $n_\chi$ depends on the dark-sector temperature $T_D$, we need to solve the Boltzmann equation for the dark-sector energy density $\rho_D$ in order to obtain the evolution of $T_D$, and use $T_D$ to solve the Boltzmann equation for $n_\chi$.
We present the other four benchmark results which include both scenarios for both the scalar and the pseudoscalar case.
We also present and analyze the evolution of $T_D$ in these cases.
\item We find that the decay of $\phi$ could potentially be constrained by BBN and CMB.
Nevertheless, we point out that, for our benchmark results, a relatively heavy mediator with $m_\phi~\sim~50~\rm MeV$ could avoid the BBN and CMB constraints while simultaneously give the correct relic abundance.
\item The Super-Kamiokande experiment is possible to constrain our model by searching for dark matter annihilating to mediator $\phi$ followed by $\phi$ decaying to SM neutrinos.
We show in FIG.~\ref{fg:indrect} that our benchmarks are all allowed by the Super-Kamiokande experiment.
\item Since our model could allow sufficiently large self-interaction between dark-matter particles, we analyze the potential for our model to solve the small scale structure problems.
We find since there is a tension between the required properties for a successful SIDM candidate and the constraints on both BBN and the relic abundance, the choice of mass parameters that we choose in the benchmarks cannot solve the problems on small scale structures.
Nevertheless, it is possible to reconcile the problem by increasing the mass of $\chi$.
\end{itemize}

Finally, we would like to point out several caveats.
First of all, in our examples for UV completion, we have not considered the possibility that the heavy right-handed neutrino $N$ could also be a dark-matter candidate.
Such possibility in general depends on the evolution history of the universe at high scales.
We argue that, in scenarios in which the temperature of the universe has never been higher than $M_N$, or the mass of the inflaton is smaller than $M_N$, one would not expect a non-negligible population of $N$.
Our analysis thus implicitly assumed an appropriate cosmological history before the scales relevant for the production of dark matter.
Scenarios in which both $N$ and $\chi$ could play the role of dark matter are nevertheless worth studying.

Second, although we have confined ourselves within the freeze-in production of dark matter through the neutrino portal, if the couplings $g_\nu$ and/or $\bar g_\nu$ are large enough, the dark-matter particles could thermalize with the SM neutrinos and undergo thermal freeze-out from the SM thermal bath.
However, we do not consider this scenario in our paper as we are motivated by the possibility that the interactions between the dark matter and the SM particles are extremely feeble.

Third, as we have pointed out before, in the reannihilation regime, the interaction rate between the dark-sector particles depends on the dark-sector temperature $T_D$.
However, $T_D$ is only physical when the $\chi$ and $\phi$ particles are in thermal equilibrium.
In our analysis, we obtain $T_D$ from the energy density of the dark sector and use it to evaluate the dark-sector interaction rate throughout.
Therefore, our approach relies on the assumption that the thermally averaged cross-sections evaluated at $T_D$ is a good approximation for the actual interaction strength near but beyond the dark freeze-out.
Far away from the dark freeze-out, the interaction within the dark sector becomes negligible, and the evolution of the dark matter number density is dominated by the injection from the SM sector and the Hubble expansion.

{Moreover, the $\nu-\nu-\phi$ interaction is subject to constraints from supernovae as the scalar $\phi$ might be produced copiusly through inverse decay of SM neutrinos in the core of a supernova \cite{Choi:1987sd,Farzan:2002wx,Acharya:2009zt,Heurtier:2016otg,deGouvea:2019qaz}.
However, this constraint is sensitive to both the mass of the scalar and the flavor dependence of the coupling.
Therefore, in principle, such constraint could be evaded by either adjusting $m_\phi$ or the flavor structure of $g_\nu$ and/or $\bar g_\nu$.
A detailed study on this would help identify the viable parameter space for our model, but is beyond the scope of our paper.}

Furthermore, when calculating the thermally averaged cross-sections and the energy transfer rates, we have assumed that the dark-sector particles $\chi$ and $\phi$ all follow a thermal distribution.
While this is true when the dark sector is in thermal equilibrium, it might not be true if the dark sector is not, as the self-scattering rate might be too small to reach kinetic equilibrium.
Therefore, it is possible for the phase-space distributions of $\chi$ and $\phi$ to be highly nonthermal as the SM thermal bath keeps injecting particles into the dark sector while the existing dark-sector particles are constantly redshifting.
This would in principle affect the calculation of the collision terms.
The nonthermal features in the phase-space distribution could also leave observable imprints on the matter power spectrum \cite{Konig:2016dzg,Murgia:2017lwo,Dienes:2020bmn} if some of the $\chi$ particles could have non-negligible momenta during the structure formation.
A thorough study on these effects would require solving the Boltzmann equations at the level of the phase-space distribution for both $\chi$ and $\phi$ which we leave for future work.

\acknowledgments
We would like to thank J.\ Heeck for useful discussions.
YD acknowledges support from U.S. Department of Energy under contract No.\ DE-SC0011095.
FH is supported by the National Natural Science Foundation of China (NSFC) under grant
No.\ 11947302, No.\ 11690022, No.\ 11851302, No.\ 11675243 and No.\ 11761141011 and also supported by the Strategic Priority Research Program of the Chinese Academy of Sciences under grant No.\ XDB21010200 and No.\ XDB23000000.
J.\ H.\ Y.\ is supported by the National Science Foundation of China under Grants No.\ 11875003 and No.\ 11947302.
H.\ L.\ L. is supported by the National Science Foundation of China under Grants No.\ 11875003 and 2019 International Postdoctoral Exchange Fellowship Program.

redpajama_set_name: RedPajamaArXiv


text:
Home UK
Lib Dem leader Tim Farron resigns: 'It's a choice between party leadership and Christ'
Mark Woods 14 June 2017 | 7:05 PM
Tim Farron took over as Liberal Democrat leader in July 2015 after a disastrous election campaign under Nick CleggReuters
Liberal Democrat leader Tim Farron has resigned his position because he is 'torn between living as a faithful Christian and serving as a political leader'.
The resignation follows close behind that of Lib Dem Shadow Home Secretary, Lord Paddick who tweeted he had stepped down 'over concerns about the leader's views on various issues that were highlighted during GE17'.
Farron was repeatedly quizzed by interviewers about his views on homosexuality and at times appeared to struggle with the questioning before eventually saying he did not think gay sex was a sin.
I've resigned as @LibDems Shadow Home Secretary over concerns about the leader's views on various issues that were highlighted during GE17.
— Brian Paddick (@brianpaddick) June 14, 2017
In his resignation statement he says: 'From the very first day of my leadership, I have faced questions about my Christian faith. I've tried to answer with grace and patience. Sometimes my answers could have been wiser.
'At the start of this election, I found myself under scrutiny again - asked about matters to do with my faith. I felt guilty that this focus was distracting attention from our campaign, obscuring our message.
'Journalists have every right to ask what they see fit. The consequences of the focus on my faith is that I have found myself torn between living as a faithful Christian and serving as a political leader.'
Farron stresses he does not believe Christians should 'impose the tenets of faith on society', saying that view was 'not liberal and it is counterproductive when it comes to advancing the gospel'.
He says he is a 'liberal to his fingertips' and that 'liberalism means that I am passionate about defending the rights and liberties of people who believe different things to me'.
However, he says: ' Even so, I seem to be the subject of suspicion because of what I believe and who my faith is in.
'In which case we are kidding ourselves if we think we yet live in a tolerant, liberal society. That's why I have chosen to step down as leader of the Liberal Democrats.'
He concludes: 'I want to say one more thing: I joined our party when I was 16, it is in my blood, I love our history, our people, I thoroughly love my party.
'Imagine how proud I am to lead this party. And then imagine what would lead me to voluntarily relinquish that honour.
'In the words of Isaac Watts it would have to be something "so amazing, so divine, (it) demands my heart, my life, my all".'
He intends to serve until the parliamentary recess begins next month. Among possible replacements are Vince Cable and Jo Swinson, both former ministers who have reclaimed their former seats.
The director of the Liberal Democrat Christian Forum, Sarah Latham, told Christian Today: 'We are saddened by the news of Tim's resignation. He has been a great leader and inspiration to many of us at LDCF. He has turned the party around in a very short space of time, doubling the membership and increasing our number of MPs. We are thankful for him and absolutely stand with him at what is undoubtedly a difficult time.
"Sadly his resignation reflects the fact we live in a society that is still illiberal in many ways and is intolerant of political leaders having a faith. This urgently needs to change. It will only change if Christians step up and get involved in all areas of life and change the rhetoric, whether in politics, media, business, or the arts. We need to bring about a society that is truly liberal – where everyone of all faiths and none are valued and considered equal.'

redpajama_set_name: RedPajamaCommonCrawl


text:
By Rick Van Nieuwenhuyse
Special to Mining News
Ambler Copper: Part of the Problem or Part of the Green Energy Solution?
More than 11,000 pounds of copper go into the construction of each of these 5-megawatt offshore wind turbines, plus additional copper is needed to deliver the electricity to the power grid.
Governments around the world are collaborating to focus on addressing Climate Change and Global Warming. The Paris Climate Accord adopted numerous measures to "limit a global temperature rise this century below 2 degrees Celsius above pre-industrial levels". Although there are many things that can be done to address climate change, the most meaningful and obvious is to use cleaner forms of energy and transportation.
Alaska has shown the world how energy resources can be responsibly developed, spearheading Alaska's transition from the rough and tumble new state that I grew up in to one providing comprehensive quality education, healthcare and transportation services for its residents – all while being good environmental stewards. Currently, fossil fuels make up 90 percent of both Alaska's revenues, and energy production on a global scale.
World leaders are trying to change that energy mix to meaningfully address Global Warming. While much of this transition is beyond Alaska's control, one thing we can do is decide to remain part of the problem or become part of the solution.
In my opinion, if you are truly concerned about Global Warming and consider yourself an environmental steward of Planet Earth – you must Think Copper!
Regardless of which forms of non-carbon-based energy the world transitions to – solar, wind, geothermal, hydro, or nuclear – they all require more copper to generate power.
Conventional coal and diesel take 1 ton of copper to produce 1 megawatt of power, whereas wind, solar and the others require 3 to 5 tons per MW ... plus additional copper wire to connect the power generation source to the grid. For example, large off-shore wind generators like CIRI's Fire Island Wind Farm take 10 tons of copper per MW – it takes a lot of copper to make a wind turbine ... and then even more copper to connect all the turbines to one another and to the grid.
Bottom line, Green Energy production requires on average five times more copper than conventional, carbon-based energy generation.
Now let's talk cars – specifically hybrid and electric cars. A typical Internal Combustion Engine (ICE) vehicle uses 20 kilograms of copper per vehicle while an all-electric uses 80 kg of copper. In addition to copper, significant amounts of nickel, cobalt, manganese, graphite and lithium are also needed for the construction and operation of these cleaner forms of transportation.
If you want to get rid of the highly carbon dioxide (CO2) polluting ICE engine and replace them with hybrid or electric vehicles, then you need 3 to 4 times as much copper per car – even more for trucks, buses and mass transit. More electric vehicles will require more power generation and charging stations. Add it up and we need five times more copper to build electric vehicles and the infrastructure to support them.
So, if we want less CO2 going into the atmosphere, we need to produce more copper under it!
The absolutely great thing about copper is that it's 100 percent re-usable. Today, about 80 percent of the copper mined is eventually recycled. New supply chains for renewable energy, and hybrid and electric vehicles will improve this to well above 95 percent – resulting in a sustainable supply of copper and clean, green energy and transportation. There are solutions to reducing pollution and C02 in the atmosphere, but copper and a host of other base and specialty metals are required – no way around it!
We have choices to make as a society.
1. We can do nothing, and our Earth keeps warming up.
2. We can do nothing "in our backyard", leaving it to others to fill our global need for copper and cobalt knowing that it may involve child labor, corruption and human rights atrocities (see Amnesty International Report on mining in Africa: https://www.amnesty.org/en/latest/news/2017/11/industry-giants-fail-to-tackle-child-labour-allegations-in-cobalt-battery-supply-chains/), or displacing water used by farmers like mines in Chile and Peru, where much of the world copper supply comes from (see http://www.bioone.org/doi/pdf/10.1659/mrd.1039).
3. We can support mining in Alaska, a jurisdiction with a long history of responsible resource development and absolutely no history of any significant disasters.
A prime opportunity for Alaska to contribute to both this global transition, and our own economic self-determination lies in the Ambler Mining District (AMD): a high quality, well known mining district containing over 10 Billion Pounds copper, significant cobalt and a host of other metals; where the State of Alaska and NANA, an Alaska Regional Native Corporation, specifically made land selection for mining as a result of the Alaska Native Claims Settlement Act (ANCSA) and Alaska National Interest Lands Conservation Act (ANILCA) legislation; and that the Federal Government specifically granted a Right-of-Way to connect this metal-rich district with the Dalton Hwy and the rest of Alaska's infrastructure. Specifically, Section 201(4) of ANILCA states:
"Congress finds that there is a need for access for surface transportation purposes across the Western (Kobuk River) unit of the Gates of the Arctic National Preserve (from the Ambler Mining District to the Alaska Pipeline Haul Road) and the Secretary [of the Interior] shall permit such access." (Emphasis added.)
Congress made it very clear what was intended with this unique and specific language in ANILCA.
To put this into perspective, 10 Billion lb of copper from the AMD can be used to build 56 million Electric Vehicles and reduce CO2 entering the atmosphere by 250 million metric tonnes every year. Remember, copper is 100 percent re-cyclable so these C02 reductions are sustainable!
How much CO2 will be emitted by the Ambler mining complex to mine 10 B lbs of Copper that could generate these Global CO2 reductions? Using liquid natural gas (LNG), site power generation would produce roughly 75,000 tonnes CO2/year. Add in transportation and refining emissions and the total CO2 emissions would be approximately 200,000 tonnes per year. That is a huge return on investment in terms of C02 reduction.
Alaska is a state that produces a lot of oil and will for the foreseeable future. Although we need to diversify our economy as a part of this global energy transition and wean ourselves off being so dependent on oil, it will take time. It won't happen overnight, but it is something we must work towards – a diversified economy is a stronger economy.
So, there you have it – alternative forms of energy and electric vehicles can help enormously to reduce C02 emissions and provide a sustainable solution to reducing greenhouse gas emissions, but this Green Energy solution requires copper – a 100 percent recyclable metal.
The choice is ours – do we support mining copper and cobalt here in Alaska and be part of a sustainable energy solution, or do nothing and remain only part of the problem by outsourcing production of these resources to jurisdictions out of sight and out of mind?
Rick Van Nieuwenhuyse
Rick Van Nieuwenhuyse is the president and CEO of Trilogy Metals, a company dedicated to advancing exploration and development of high-grade copper-cobalt-zinc-lead-gold-silver properties in the Ambler Mining District Northwest Alaska. He grew up in Alaska and graduated from West Anchorage High School in 1972. He has more than 40 years of experience in the natural resource sector, including his role as president and CEO of NOVAGOLD, an exploration company he founded in 1997, and vice president of exploration for Placer Dome Inc. from 1990 to 1997. He received the Thayer Lindsley award in 2009 for his role in the discovery of the Donlin Gold deposit in Southwest Alaska and the Colin Spence award for his role in the discovery of the South Reef of the Bornite deposit in Northwest Alaska.
A glimmer of hope for the Arctic ... humanity
Calm before the 2019 exploration storm
Alaska mineral exploration tops $100M
The 2019 mining game is afoot in Alaska


text:
Your Guide To Agency Name Pronunciations And Origins
A cheeky guide to pronouncing the names of some of the most prominent agencies is actually the second blast of moniker-themed agency navel gazing in 2013. (The other is this infographic on such names, also included here.)
By Joe Berkowitz 1 minute Read
Have you ever looked at some of the alphabet-soup names of creative agencies and wondered whether one was meant to be read as an acronym or a string of letters? Well, if so, this short video is not the best way to figure it out.
"The Agency Pronunciation Guide" is a one-joke video, sure, but it's a pretty decent joke. Logos for most of the top agencies are displayed in the center of the screen, Sesame Street-style, before a narrator grossly mispronounces each and every one of them. TBWA/ for instance is revealed to be sounded out as "tub-wah slashie."
The point of the video, and likely identity of its creator only comes at the very end. The last of the 19 agencies is the only one that actually requires a bit of guesswork or familiarity in order to pronounce it (and it's the one that's pronounced correctly). Swedish agency Forsman & Bodenfors, which currently features the video on its Facebook page, uses a sample sentence in which you might use the name: "Forsman & Bodenfors is a Swedish agency often mispronounced."
While we're on the subject of agency names, though, do have a look at the infographic below, created by British creative team Rob & Joe (Rob Donaldson and Joe Dennett), which provides a neat taxonomy of all agency names great and small.
Joe Berkowitz is an opinion columnist at Fast Company. His latest book, American Cheese: An Indulgent Odyssey Through the Artisan Cheese World, is available from Harper Perennial.
What could a global tax on wealth over $5 million pay for?
Former SpaceX engineers raised $50 million to build a Tesla for freight trains
After 149 years, Colgate's toothpaste tubes are finally recyclable
Why it's sometimes harder to get a job you're 'overqualified' for


I would first like to express my great appreciation for all the people who are working on wpa_supplicant - you guys are doing a great job!
-	How ?experimental? is the internal TLS/crypto code?
-	Does anyone have any data regarding it?s functionality with different radius services etc (like ?eap_testing.txt?)?


text:
Q: proving an iff statement about the definition of a limit using epsilon delta Let $A$ and $x_0$ be real numbers, and let $f(x)$ be a real-valued function defined in a deleted neighborhood of $x_0$. Use the definition of limit to prove that $\lim \limits_{x \to x_0} f(x) = A $ if and only if $\lim \limits_{x \to 0} f(x_0 + x) = A$.
First we prove $\lim \limits_{x \to x_0} f(x)=A \implies \lim \limits_{x \to 0}f(x_0+x)=A$
$\lvert f(x)-A \rvert <\epsilon$, $\forall x$ with $0<\lvert x-x_0 \rvert < \delta$ or $x_0- \delta <x<x_0 + \delta$
$\lvert f(x_0+x)-A \rvert < \epsilon$,  $\forall x $ with $0<\lvert x-0 \rvert < \delta$ or $-\delta < x < \delta$
So I want to choose a $\delta$ so that $x_0$ disappears?
I'm having a hard time understanding this process, can somebody give me a walkthrough?

A: Suppose $\displaystyle\lim_{x \to x_0}f(x) = A$. Fix $\epsilon>0$. Then there is a $\delta>0$ such that $|x-x_0|<\delta$ implies $|f(x)-A|<\epsilon$.
Therefore, if $\ |(x+x_0)-x_0|<\delta$, then $|f(x+x_0)-A|<\epsilon$. (I suppose this line is the trick)
That is, if $\ |x|<\delta$, then $|f(x+x_0)-A|<\epsilon$.
This means (by definition) that $\displaystyle\lim_{x \to 0}f(x+x_0) = A$

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Meet Skye Wijkord. She is one of those "Inner Fitness trainers" whose inspiration goes beyond the ordinary. She is living proof that when we let go of the things that keep us in fear the potentials and abundance for living an extrodinary life become limitless.
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Skye is a successful fundraising coordinator and Founder of a brand called Passionate Forward. She is past President of a Panamanian non-profit called The House of Hope, which provides young girls resources for a better future. She is also the owner and facilitator of a massage and wellness school in Panamá called Body Vivo. And, she is an athlete, a coach, and coordinator of a local swim team.
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lossless coding Coding in which no information whatsoever is lost during the encoding (or decoding) process. Generally, encryption and decryption are lossless, as is channel coding. Strictly, data compaction is lossless, while data compression is not, but the latter term is often used for the former. The decoding of (i.e. recovery from) compression and compaction are lossless. But the decoding of a signal received from a channel is usually lossy (strictly not lossless) by design, since the noise at least must be lost: the message entering the channel will usually have been prepared so as to permit this loss by the use of an error-correcting code.
"lossless coding." A Dictionary of Computing. . Encyclopedia.com. 19 Apr. 2019 <https://www.encyclopedia.com>.


text:
The North Valley Christian Academy (NVCA) is a private, independent Lutheran school located in north Phoenix Arizona, United States.

NVCA is affiliated with the Lutheran Church–Missouri Synod system of schools throughout the U.S. The academy is accredited by National Lutheran School Accreditation (NLSA), Association of Christian Schools International (ACSI), and Cognia.

NVCA operates a preschool, lower school, middle school, high school, and international student program.

Nate Kretzmann was the founding Executive Director of the school.

History
Founded in 2010, NVCA is a 501(c)(3), non-profit, education organization.  The school is operated and governed independently by an elected board of directors.

References

External links

Private elementary schools in Arizona
Private middle schools in Arizona
Lutheran schools in Arizona
Educational institutions established in 2011
Schools affiliated with the Lutheran Church–Missouri Synod
Schools in Phoenix, Arizona
2011 establishments in Arizona

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RB Leipzig signing Tyler Adams backed for 'successful European career'
United States international Tyler Adams has the perfect platform for a "successful European career," RB Leipzig assistant manager Jesse Marsch has told ESPN FC.
Adams, 18, begins training with Leipzig on Friday after moving to the club from New York Red Bulls and is one of two Major League Soccer players to have moved to the Bundesliga this winter, along with Canada international Alphonso Davies who signed for Bayern Munich.
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Having arrived in Leipzig on Wednesday, Adams returns to the club he trained with as an under-19 player and reunites with Marsch, who believes Adams is entering the ideal scenario in Germany.
"If there's an opportunity for a young American to come somewhere and adapt as quickly as possible, it's this situation," Marsch, who left the NYRB dugout last summer to become an assistant coach in the Bundesliga, told ESPN FC.
He added that the similarities between the two Red Bull clubs could help Adams settle in a new country.
"He's been here before, he knows the football, he's watched the team a lot." Marsch said. "He knows the personality of it, he knows the league and now his former head coach is here as an assistant coach. I can't imagine a better situation than Tyler's going to be stepping into."
Praising Adams' "American mentality and American belief," Marsch added that the midfielder will have a good chance to establish himself in the Bundesliga and European soccer.
"He has a lot of really strong qualities to have a really successful European career," Marsch said. "He is now coming to a new experience, where, frankly, he has to compete every day to show the people that he's ready for the next step."
"I believe very strongly that he'll be ready for that and I think that his mentality along with his physical gifts will give him a really good opportunity to establish himself early in his time here."
However, the former New York Red Bulls coach added that Adams must be granted time to settle in.
"It will be important for everyone here and for him to understand that it's likely to take some time, and there needs to be an adjustment period," Marsch said.
"We'll support him in the right way to give himself the right chance to succeed and grow. Everyone here is really excited to get him into our team and get him into training and I know that he's as well.
"When he came here with the U19s, he was so excited about his time here. I think he always had it in the back of his mind that once he established himself in the U.S. that this would be the next step."


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Inspired and Humbled by Our Clients
Center for Constitutional Rights
Thanks to "Waiting for Fahd: One Family's Hope for Life beyond Guantánamo," you know something about CCR client Fahd Ghazy, detained at Guantánamo since he was 17, never charged, twice cleared for release, waiting to be reunited with his teenage daughter whom he last saw as a baby. But today we want to tell you about some of the other men held illegally and unjustly in Guantánamo. Omar Farah and Aliya Hussain spent last week at the detention facility and, in addition to Fahd, met with our clients Tariq Ba Odah, Mohammed Al Hamiri and Ghaleb Al Bihani. Like Fahd, they are all from Yemen, all cleared for release – and yet still not free. Aliya and Omar gave a live report back on their visits from Guantánamo on Thursday.
Tariq has protested his unjust detention with the only means available, his very body. He has been on a hunger strike for eight years. Tariq has not eaten since President Obama was Senator Obama, and the question for the president is, Will he finally exercise his power and release this extraordinary man before he is former President Obama? The strength of Tariq's spirit is almost incomprehensible, and the damage to his body and the cruelty he must endure every day is equally unimaginable. He has been on an unbroken hunger strike for eight years, longer than President Obama has been in office. Each day, he is forcibly extracted from his cell, strapped to a chair, and force fed through his nose, but as Omar and Aliya saw in their meeting, he continues to have a sense of humor, compassion and gratitude to all those who have shown him solidarity. Mohammed is a poetic thinker and writer, defiantly good-spirited. We've long been inspired by his words from a few years ago – "We keep walking through the tunnel in search of a shred of light hoping it would appear at the end of that tunnel… and for every couple of steps we make, this strong air pushes us one step backward, as if it is stealing one step from us. Yet we keep walking forward" – and last month we shared his agony when he said, simply, "I want to be released. I want to see my mom." Ghaleb, despite his own harsh confinement, thinks of others; he identifies with the disenfranchised and downtrodden in the U.S., expressing compassion at their suffering, and he has a keen awareness of the hypocrisy of U.S. human rights ideals. That these men are locked in cages while politicians play games with their lives, feeding fear and prejudice for their own cheap political game, is an utter outrage. Last week, President Obama publicly expressed his regret at not closing Guantánamo on his first day of office. No one regrets that misstep more than our clients, but Obama still has the power to right that wrong and end the suffering of these men and others today.
"Guantanamo Diary": A Tale of Captivity, Rendition and Torture
Mohamedou Ould Slahi's recently published Guantanamo Diary describes his brutal treatment at the US prison, where he remains, uncharged.
L. Michael Hager,
"Guantanamo of the Pacific": Australian Asylum Seekers Wage Hunger Strike at Offshore Detention Site
A massive hunger strike is underway.
Nermeen Shaikh &
Amy Goodman,
Guantanamo and Permanent War: The View From Camp X-Ray
If there's anything that serves as an appropriate metaphor for American permanent war, it's Camp X-Ray.
Adam Hudson,