Off-shell Dark Matter: \\ {A Cosmological relic of Quantum Gravity
Dark_Matter_v2.tex
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\begin{document}

\title{Off-shell Dark Matter: \\ {\it A Cosmological relic of Quantum Gravity}}

\author[a,b]{Mehdi Saravani}
\author[a,b]{, Niayesh Afshordi}
\affiliation[a]{Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, ON, N2L 2Y5, Canada}
\affiliation[b]{Department of Physics and Astronomy, University of Waterloo, Waterloo, ON, N2L 3G1, Canada}
\emailAdd{msaravani@pitp.ca}
\emailAdd{nafshordi@pitp.ca}

%\date{\today}
\abstract{

We study a  novel proposal for the origin of cosmological cold dark matter (CDM) which is rooted in the quantum nature of spacetime. In this model, off-shell modes of quantum fields can exist in asymptotic states as a result of spacetime nonlocality (expected in generic theories of quantum gravity), and play the role of CDM, which we dub off-shell dark matter (O$f$DM). However, their rate of production is suppressed by the scale of non-locality (e.g. Planck length). As a result, we show that O$f$DM is only produced in the first moments of big bang, and then effectively decouples (except through its gravitational interactions). We examine the observational predictions of this model: In the context of cosmic inflation, we show that this proposal relates the reheating temperature to the inflaton mass, which narrows down the uncertainty in the number of e-foldings of specific inflationary scenarios. We also demonstrate that O$f$DM is indeed cold, and discuss potentially observable signatures on small scale matter power spectrum. 
}
%\keywords{Dark Matter, Causal Lorentzian Evolution}
\maketitle


\section{Introduction} \label{introduction}

A vast range of observations in Astrophysics and Cosmology have now provided concrete evidence for the existence of cosmological cold dark matter (CDM), which appears to make up the majority of mass density in our universe (only second to the mysterious dark energy). Rotation curves of galaxies (e.g. \cite{Persic:1995ru}), gravitational lensing (e.g. \cite{Clowe:2006eq}), and Cosmic Microwave Background (CMB) \cite{Ade:2015xua,Hinshaw:2012aka} all indicate that General Relativity with ordinary (or known) matter is not consistent with observations. It is worth noting that, unlike dark energy, evidence for the existence of CDM ranges from cosmological to galactic (i.e. six orders of magnitude) in physical scale. 

Since all the observational evidence for CDM is through its gravitational interactions, it has been tempting to explore a modification of Einstein gravity as a substitute (e.g. \cite{Milgrom:1983ca,Bekenstein:2004ne,Moffat:2004bm,Moffat:2005si}). However, given the range of observational data matched by CDM (in particular, the precision measurements of CMB anisotropy power spectrum \cite{Ade:2015xua,Hinshaw:2012aka}) it has become nearly impossible to fit the data with any modified gravity alternative (which does not have an effective built-in dark matter component) \cite{Clifton:2011jh}.


As a result, the most popular approach has been to consider CDM as a new (beyond Standard Model) weakly interacting particle. There is strong evidence that CDM particle has to be (at most) weakly interacting with the Standard Model, as otherwise it should have been detected by now, through various astrophysical or terrestrial probes (see, e.g. \cite{Feng:2010gw}). It also has to be sufficiently cold, as there is no evidence for a thermal cut-off in the cosmological matter power spectrum, down to sub-Mpc scales \cite{Viel:2013fqw}. It is quite remarkable that a simple assumption of adding a non-relativistic (and non-interacting) dark matter is compatible with all the cosmological observations.

Here, we study a rather different approach, first proposed in \cite{Saravani:2015rva}, which we shall refer to as off-shell dark matter (O$f$DM) in this paper. In this proposal, CDM originates from considering quantum gravitational effects on the evolution of quantum fields. These effects manifest themselves through modifying the evolution law of quantum fields to a non-local evolution described by a causal non-local operator $\widetilde \Box$ which substitutes the role of D'alembertian. 

Let us outline some features of this model. First, this non-local modification results in the appearance of a new set of modes (or excitations) associated to each field. In fact, modification of a field with mass $M$ leads to two sets of modes:
\begin{enumerate}
\item Modes with mass $M$, called on-shell.
\item A continuum of massive modes with mass higher than $M$, called off-shell.
\end{enumerate}
We call the original mass of the field ($M$) ``intrinsic mass''. In other words, intrinsic mass is the mass of the on-shell modes (or the least value mass of the excitations). 

The important property that differentiates these two sets of modes and points to the direction of dark matter is the following: {\it transition rate of any scattering including one (or more) off-shell mode(s) in the initial state is zero}. 
%It means the cross section of the following scattering processes
%\bea
%O+O\rightarrow O+O \label{2onshell}\\
%O+O\rightarrow O+F \label{1onshell}\\
%O+O\rightarrow F+F \label{2offshell}
%\eea
%are generically non-zero, while these ones
%\bea
%O+F\rightarrow O+O\notag\\
%O+F\rightarrow O+F\notag\\
%F+F \rightarrow F+F\notag\\
%F+F\rightarrow O+O\notag
%\eea
%are all zero. 
This property makes off-shell modes a natural candidate for CDM, simply because they cannot be detected through non-gravitational scattering experiments \cite{Saravani:2015rva}. In fact, they can be produced by scattering of ``on-shell" particles, but they do not scatter, annihilate or decay. As such, the only way to detect these particles is through their gravitational signatures.

In the next section, we will review the important features of this model. Section \ref{Dark Matter Production} is dedicated to the production of O$f$DM in the context of inflation and reheating. We will discuss the effect of O$f$DM on matter power spectrum in Section \ref{Cold OFDM}. Finally, Section \ref{conclusion} concludes the paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Review of O$f$DM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us start this section by the following question: If off-shell modes of matter can be produced by the scattering of on-shell modes, while the reverse does not happen, shouldn't we see any signature of this in scattering experiments, for example in Large Hadron Collider (LHC)?
%\begin{wrapfigure}{L}{0.5\textwidth}
\begin{figure}
\begin{center}
\includegraphics[width=0.95\linewidth]{scattering.png}
\end{center}
\caption{A simple annihilation process (on left) and decay process (on right).}
\label{scattering}
\end{figure}
%\end{wrapfigure}
In other words, whenever we perform scattering experiments, a part of the incoming energy must transfer to off-shell modes and become undetectable. Shouldn't we have already seen this effect by now?

In order to answer this question, consider a simple annihilation or decay process (Figure \ref{scattering}). First, let us define the following quantities: 
$\sigma_{1F}$ ($\Gamma_{1F})$ is the cross-section (rate) of producing one off-shell particle and one on-shell particle and
%\footnote{In this paper, we are assuming that this process can indeed happen. It is conceivable that the structure of a specific theory forces on-shell particles to appear only in pairs.}. 
$\sigma_{O}$ ($\Gamma_{O})$ is the cross-section (rate) of producing purely on-shell particles. If we assume that the energy of the process is much higher than the intrinsic mass of the out states, $E_{\rm CM} \gg M$ (as we will see later, this is the relevant regime for dark matter production), following the results in \cite{Saravani:2015rva}, we arrive at\footnote{$\delta_+(p^2)\equiv \delta(p^2)\theta(p^0)$}
\beq
\label{production_rate}
\frac{\Gamma_{1F}}{\Gamma_O}=\frac{\sigma_{1F}}{\sigma_O}
=\frac{\int d^4p_1d^4p_2 2\pi \delta_+(p_1^2)\widetilde W(p_2)\delta^4(q-p_1-p_2)}{\int d^4p_1d^4p_2 2\pi \delta_+(p_1^2)2\pi\delta_+(p_2^2)\delta^4(q-p_1-p_2)}
\eeq
where $q$ is the incoming energy-momentum and
 $\widetilde W(p)$ is given in terms of the spectrum of non-local operator $\widetilde \Box$ 
 \bea
 \widetilde W(p)=\frac{2{\rm Im}~ B(p)}{|B(p)|^2}\theta(p^0),\\
\widetilde \Box e^{ip\cdot x}=B(p)e^{ip\cdot x}.
 \eea
Note that $\widetilde W(p)$ is the two point correlation function (or Wightman function) of the field in the momentum space (see Section 4 in \cite{Saravani:2015rva})
\beq
\langle0|\hat \psi(x)\hat \psi(y)|0\rangle=\int \frac{d^4p}{(2\pi)^4}\widetilde W(p)e^{ip\cdot (x-y)}
\eeq


%\begin{figure}
%\includegraphics[width=0.5\textwidth]{scattering.png}
%\caption{A simple annihilation process (on left) and decay process (on right).}
%\label{scattering}
%\end{figure}


Equation \eqref{production_rate} can be simplified further if we assume that the energy scale of the scattering\footnote{Throughout this paper we are using $(-+++)$ signature for the metric.} $-q^2\equiv E^2_{CM}$ is much lower than the non-locality scale $\Lambda$ defined through $\widetilde \Box$. In this regime,
\bea
B(q)&=&-q^2+\mathcal{O}\left(\frac{q^4}{\Lambda^2}\right)\\
{\rm Im}~B(q)&=&a\frac{q^4}{\Lambda^2}+\mathcal{O}\left(\frac{q^6}{\Lambda^4}\right).
\eea
For $a\neq 0~$\footnote{Another possibility would be that $a=0$. In that case, the leading term to the imaginary part of $B$ comes in $6^{th}$ order. We will not pursue this possibility in this paper.}, $\Lambda$ can be redefined to set $a=\frac{1}{2}$.

With this assumption, we can make use of the Taylor expansion of $\widetilde W$
\beq\label{wtilde}
\widetilde W(q)=\frac{1}{\Lambda^2}+\mathcal{O}\left(\frac{q^2}{\Lambda^4}\right), ~~M^2\ll-q^2\ll\Lambda^2,
\eeq
to finally get (to the leading order)
\beq
\label{production_rate_simplified}
\frac{\Gamma_{1F}}{\Gamma_O}=\frac{\sigma_{1F}}{\sigma_O}= \frac{1}{4\pi}\left(\frac{E_{\rm CM}}{\Lambda}\right)^2,
\eeq
where $E_{\rm CM} \ll \Lambda$ is the centre of mass energy of the incoming particle(s). Note that for a decay process, $E_{\rm CM}$ is the mass of the decaying particle. Although, we derived \eqref{production_rate_simplified} for simple interactions of Figure \ref{scattering}, it is generally correct (up to order one corrections) as long as $E_{\rm CM}$ is much higher than the intrinsic mass of the intermediate particle(s) in Feynman diagrams. 

Now, let us define $\sigma_{2F}~(\Gamma_{2F})$ to be the cross section (rate) of producing two off-shell particles in the out state (Figure \ref{scattering}). Then,

\bea
\label{production_rate_2}
\frac{\Gamma_{2F}}{\Gamma_O}=\frac{\sigma_{2F}}{\sigma_O}&&=\frac{\int d^4p_1d^4p_2 \widetilde W(p_1)\widetilde W(p_2)\delta^4(q-p_1-p_2)}{\int d^4p_1d^4p_2 2\pi \delta_+(p_1^2)2\pi\delta_+(p_2^2)\delta^4(q-p_1-p_2)}\notag\\
&&=\frac{1}{48\pi^2}\left(\frac{E_{CM}}{\Lambda}\right)^4
\eea

As we see, adding one more off-shell particle in the final state suppresses the cross section by another factor of $\left(\frac{E_{CM}}{\Lambda}\right)^{2}$. So, the rate of two off-shell particles production is suppressed by a factor of $\left(\frac{E_{CM}}{\Lambda}\right)^{2}$ compared to one off-shell particle production. 

%This is true as long as the production of just one off-shell particle is not forbidden by a symmetry law (for example gauge symmetry). Otherwise two off-shell out states would be the main process for dark matter production.

Before going any further, let us discuss the typical mass of the off-shell particle produced in Figure \ref{scattering}. For one off-shell particle production, the mass distribution of the produced off-shell particle is given by 
\beq
P_{1F}(m)=N\int d^4p_1d^4p_2\delta_+(p_1^2)\widetilde W(p_2)\delta^{(4)}(q-p_1-p_2)~m \delta(p_2^2+m^2),
\eeq
Where $N$ is the normalization factor. Using \eqref{wtilde} it reduces to
\beq\label{mass_density}
P_{1F}(m)=\frac{4m}{E^2_{\rm CM}}\left(1-\frac{m^2}{E^2_{\rm CM}}\right) ~~0<m<E_{\rm CM},
\eeq
assuming that the off-shell particle is intrinsically massless (or that its mass is much smaller than $E_{CM}$). For production of two off-shell particles, the mass distribution is given by
\beq
P_{2F}(m)=N'\int d^4p_1d^4p_2\widetilde W(p_1)\widetilde W(p_2)\delta^{(4)}(q-p_1-p_2)~ m \delta_(p_2^2+m^2),
\eeq
which reduces to 
\beq
P_{2F}(m)=\frac{48m}{E_{CM}^2}\bigg[\frac{1}{4}-\frac{1}{4}\left(\frac{m}{E_{CM}}\right)^4-\left(\frac{m}{E_{CM}}\right)^2\sinh^{-1}\left(\frac{E_{CM}^2-m^2}{2mE_{CM}}\right)\bigg].
\eeq
In both cases, the typical mass of the produced off-shell particles is $\sim E_{CM}/2$.

Now, we can estimate how likely it is to produce off-shell particles in LHC experiments. If we set $\Lambda \sim M_{\rm P}\equiv \frac{1}{\sqrt{8\pi G}} \sim 10^{18} ~{\rm GeV}$ and $E_{\rm CM} \sim 1~{\rm TeV}$ (LHC energy scale), we realize that the rate of producing off-shell particles in LHC is $10^{-31}$ lower than the rate of a normal scattering happening. In other words, out of $10^{31}$ scatterings in LHC, on average one results into the production of an undetectable particle (off-shell mode), explaining why O$f$DM could be well-hidden from high energy physics experiments.




However, during the cosmic history much higher energy scales can be reached, and thus off-shell dark matter production may be more efficient. In other words, through cosmological history, a part of the energy in the on-shell sector has been transferred to off-shell sector (while the reverse does not happen) and we detect this energy gravitationally as dark matter. The main purpose of this study to investigate the production of O$f$DM in the early universe and its observational consequences.  

In summary:
\begin{enumerate}

\item[$\bullet$] Whenever a scattering happens, there is a chance of producing dark matter particles which is given by \eqref{production_rate_simplified} and \eqref{production_rate_2}. Furthermore, the probability of producing two dark matter particles in one scattering is much lower than producing only one.
%unless producing only one off-shell mode is prohibited by a symmetry principle.

\item[$\bullet$] Dark matter production is much more efficient at high (center of mass) energy scatterings. Therefore, most of the dark matter is produced during the stages in the cosmological history  where the universe is dense (lots of scatterings) and hot (high energies), i.e. early universe.

\end{enumerate}

Before ending this section, let us discuss the physical range for the non-locality scale $\Lambda$. If $\Lambda$ comes from quantum gravitational effects or fundamental discreteness of spacetime \cite{Sorkin:2007qi,Aslanbeigi:2014zva,Benincasa:2010ac}, we expect it to be around Planck energy, $M_{\rm P}$. On the other hand, {\it a priori}, $\Lambda$ can be much smaller than $M_{\rm P}$, even as low as $\sim 10 $ TeV, as suggested in large extra dimension models that are constructed to address the hierarchy problem (e.g., \cite{Burgess:2004kd}), or by the cosmological non-constant problem \cite{Afshordi:2015iza}. However, in this paper we assume $\Lambda \gg H_{\rm inf}$, i.e. the non-locality scale is much larger than the Hubble scale during inflation. Otherwise, it would not be consistent to use the standard results of slow-roll inflation when  $\Lambda \lesssim H_{\rm inf}$, since the effect of non-locality on the evolution of inflaton or metric could not be neglected. 


\section{Off-shell Dark Matter Production}\label{Dark Matter Production}

What are the processes in the early universe that are relevant for O$f$DM production? First of all, we consider inflation as a starting point in the universe. Whatever happened before inflation is diluted by the exponential expansion of the universe and is not relevant for our discussion. Furthermore, the effect of non-locality on the inflationary predictions can be neglected in the $H_{\rm inf} \ll \Lambda$ regime. After inflation, we consider two major processes that produce dark matter particles: inflaton decay to standard model particles (reheating) and radiation self interaction in the universe.

% We break up the discussion into two cases:
%
%1- One off-shell production is allowed and is the major channel for dark matter production.
%
%2- One off-shell production is not allowed and two off-shell production is the major channel for dark matter production. 


\subsection{Reheating}\label{sec::reheating}
In this section, we consider the simplest reheating model: inflaton ($\phi$ field)  decays through the effective interaction $g\phi \psi \bar \psi$, where $\psi$ represents standard model fields or an intermediate field\footnote{In this case we assume that the mass of $\psi$ field is much smaller than the inflaton's.} that decays into standard model particles later.
%In addition, we assume that inflaton is allowed to decay into one off-shell and one on-shell $\psi$. This means that $\psi$ field is either neutral or charge conservation {\color{red} (or gauge invariance?)} is broken at $\Lambda$-suppressed level. 

Decay of inflaton into (on-shell) standard model particles makes the radiation fluid of the universe, given that particle energies are much larger than their masses. As we mentioned earlier, however, inflaton will not only decay into on-shell particles; it also may decay into off-shell particles, or off-shell dark matter. Based on \eqref{production_rate_simplified}, decay rate into dark matter compared to the decay rate into radiation is suppressed by a factor of 
\beq\label{fraction_reheating}
f = \frac{1}{4\pi}\left(\frac{m_{\phi}}{\Lambda}\right)^2\ll1,
\eeq
where $m_{\phi}$ is the mass of inflaton at the end of inflation. As a result, after inflation there are three major constituents of the universe:
\begin{enumerate}
\item {\it Inflaton field ($\phi$):} This field can be treated as a non-relativistic matter after inflation when $m\gg H$ \cite{Kofman:1997yn}. Inflaton energy density ($\rho_{\phi}$) is the dominant energy density of the universe after inflation and it perturbatively decays into radiation (decay rate $\Gamma$) and dark matter (decay rate $f\Gamma$). We later comment on why the coherent decay of inflaton can be ignored.

\item {\it Radiation:} This includes all (on-shell) $\psi$ particles. Since the decay rate of inflaton into radiation is much bigger than the decay rate into dark matter, radiation energy density ($\rho_r$) will dominate the energy density of the universe after the decay of inflaton field.
\item {\it Dark matter:} This includes all off-shell $\psi$ particles. As we argue later, dark matter acts as a non-relativistic matter and its energy density is the last one to become dominant. 
\end{enumerate}
This system of three fluids satisfies the following equations:
\bea
&&\dot \rho_{\phi}+3H\rho_{\phi}=-(1+f)\Gamma \rho_{\phi},\label{infdecay}\\
&&\dot \rho_r+4H\rho_r=\Gamma \rho_{\phi},\label{inftorad}\\
&&\dot \rho_{\phi\to DM}+3H\rho_{\phi \to DM}=f\Gamma \rho_{\phi},\label{inftoDM}
\eea
which can be solved along with the Friedmann equation, where $H=\frac{\dot a}{a}$ is the Hubble parameter, $a$ is the scale factor of the universe and $\rho_{\phi \to DM}$ is the contribution to dark matter energy density from inflaton decay.\footnote{Annihilation of radiation into O$f$DM barely changes the radiation energy density, which is why it has been ignored in \eqref{inftorad}.}

Let us define the fraction of total dark matter energy density from inflaton decay
\beq
x=\frac{\rho_{\phi\to DM}}{\rho_{DM}},
\eeq
where $\rho_{DM}$ is the total dark matter energy density. Solving the system of differential equations, we arrive at \cite{Erickcek:2011us}

\beq\label{reheating}
T_{rh}=x\frac{T_{eq}}{f},
\eeq
where $T_{rh}$ is the reheating temperature (temperature of radiation at the time of inflaton-radiation equality) and $T_{eq}$ is the temperature at the matter-radiation equality.



\begin{figure}
	\begin{subfigure}[t]{0.47\textwidth}
        		\includegraphics[width=\hsize]{ofdm_constraint.png}
		\centering
		\caption{Blue upper (lower) region shows the prediction of natural ($R^2$) inflation for $k=0.002~{\rm Mpc}^{-1}$ with $T_{rh}=10$~MeV-$10^{15}$GeV. Orange regions show the prediction of the same models with the constraint coming from O$f$DM model for $\Lambda=0.1M_{\rm P}-M_{\rm P}$.} 
\label{natural}	
    	\end{subfigure} %
	\hfill
	\begin{subfigure}[t]{0.47\textwidth}
       	 	%\centering
        		\includegraphics[width=\hsize]{planck_data_version2.png}
		\caption{Prediction of $n_s$ and $r$ for different inflationary potentials at $k=0.002 ~{\rm Mpc}^{-1}$. Each region represents the prediction with the assumption of O$f$DM with $\Lambda=0.1M_{\rm P}- M_{\rm P}$.  The shaded region (curve) show the 68\% (95\%) constraints from CMB observations \cite{Ade:2015lrj}.}

\label{ns_r}
    	\end{subfigure} %
	\caption{Predictions of spectral index, $n_s$, and tensor to scalar ratio, $r$, for a number of inflationary potentials with O$f$DM constraint \eqref{reheating}.}
\end{figure}



%\begin{figure}
%\includegraphics[width=0.5\textwidth]{ofdm_constraint.png}
%\caption{Blue regions show the prediction of natural and $R^2$ inflation for $k=0.002{\rm Mpc}^{-1}$ with $T_{rh}=10$MeV-$10^{15}$GeV. Orange regions show the prediction of the same models with the constraint coming from O$f$DM model for $\Lambda=0.1M_{\rm P}-M_{\rm P}$.  }
%\label{natural}
%\end{figure}

Since $T_{eq} \simeq$ 0.75 eV, Equation \eqref{reheating} fixes the reheating temperature for a given mass of inflaton and $x \approx 1$.\footnote{We will show later that $x$ is very close to 1.} This can be used, for example, to constrain spectral index, $n_s$, and tensor to scalar ratio, $r$, of a given inflationary potential by using the following equation:
\bea
N_e=67&-&\ln\left(\frac{k}{a_0H_0}\right)+\frac{1}{4}\ln\left(\frac{V}{M_{\rm P}^4}\right)+\frac{1}{4}\ln\left(\frac{V}{V_e}\right)\notag\\
&+&\frac{1}{12}\ln\left(\frac{\rho_{th}}{V_e}\right)-\frac{1}{12}\ln g_{th}
\eea
where $N_e$ is the number of e-foldings that mode $k$ is superhorizon during inflation, $V_e$ is the potential energy at the end of inflation, $\rho_{th} \sim g_{th} T^4_{rh}$ is the radiation energy density at reheating temperature, $a_0H_0$ is the present Hubble radius, $V$ is the potential energy when mode $k$ crosses the horizon during inflation, $g_{th}$ is the number of effective bosonic degrees of freedom at reheating temperature and we have assumed pressureless effective equation of state for inflaton during reheating \cite{Planck:2013jfk}.

 Figure \ref{natural} shows how the predicted regions for the Natural \cite{Freese:1990rb} and $R^2$ \cite{Starobinsky:1980te} inflations have shrunk significantly in the $(n_s,r)$ plane  as a result of fixing the reheating temperature. A similar constraint can be found for other inflationary potentials, e.g. Figure \ref{ns_r} shows the prediction of O$f$DM model for a number of inflationary models. 

%\begin{figure}
%\includegraphics[width=0.5\textwidth]{planck_data.png}
%\caption{Prediction of $n_s$ and $r$ for different inflationary potentials at $k=0.002Mpc^{-1}$. Each region represents the prediction with the assumption of O$f$DM with $\Lambda=0.1M_{\rm P}- M_{\rm P}$.}
%\label{ns_r}
%\end{figure}

We shall next review and justify the assumptions we made in the above calculations. 


\subsubsection{Coherent decay of inflaton}

The coherent decay of inflaton is negligible if the following condition is satisfied \cite{Erickcek:2011us,Kofman:1997yn}

\beq\label{preheating_condition}
\frac{\Gamma}{m_{\phi}}\ll\left(\frac{m_{\phi}}{M_{\rm P}}\right)^2.
\eeq
Using $\Gamma \sim \frac{T_{rh}^2}{M_P}$ and \eqref{reheating}, this reduces to
\beq
10^{-18}\left(\frac{\Lambda}{M_p}\right)^4\left(\frac{10^{-5}M_P}{m_{\phi}}\right)^7\ll1,
\eeq
which is generically satisfied for models of large field inflation with $m_\phi \sim 10^{-5} M_P$.

\subsubsection{Non-relativistic dark matter}\label{NR_DM}
The mass distribution of dark matter particles is given in \eqref{mass_density}. When a dark matter particle is produced, its energy is below $E_{\rm CM}$, while, according to \eqref{mass_density}, masses of the 98\% of the dark matter particles are above $0.1E_{\rm CM}$. In other words, upon production, most dark matter particles are mildly relativistic, but through the expansion of the universe they soon become non-relativistic. This justifies our earlier assumption to model dark matter particles as a non-relativistic fluid. 

%\newpage
\subsection{Radiation self-interaction}

How much dark matter is produced as a result of radiation self interaction? Here we find an upper bound on the amount of dark matter production through self interaction of radiation. Let us assume a simple annihilation process, such as in Figure \ref{scattering}, and ignore the intrinsic mass of the particles. Ignoring the intrinsic mass of the particles is consistent with finding an upper limit for the dark matter production, since we are allowing for more dark matter production by ignoring the intrinsic masses (more phase space volume to produce O$f$DM). The average mass of the produced dark matter particles is
\beq\label{average_mass}
\int dm~mP_{1F}(m)=\frac{8}{15}E_{\rm CM},
\eeq
and the cross section of producing one dark matter particle is\footnote{This is again consistent with finding the upper bound, since the cross section of two off-shell production is much smaller. }
\beq\label{1Fscattering}
\sigma_{1F}=\frac{\sigma_O}{4\pi}\left(\frac{E_{\rm CM}}{\Lambda}\right)^2=\frac{\lambda^2}{128\pi^2\Lambda^2}.
\eeq
Since this contribution to dark matter has been produced at very high energies (lower bound on reheating temperature is $T_{rh}>5$~MeV), it will be highly redshifted today. As a result, current energy density of dark matter is the same as its mass density (see Section \ref{NR_DM}). The comoving mass density of the produced dark matter particles through radiation self interaction is given by
\beq
\frac{d\rho_{rad\to DM}}{dt}=a^{3}(t)\int \frac{d^3p_1}{(2\pi)^3}\frac{d^3p_2}{(2\pi)^3}g_1n(\vec p_1)g_2n(\vec p_2) \langle m\sigma_{1F}v_{rel}\rangle,
\label{comoving_mass1}
\eeq
where $t$ is the cosmological time, $n(\vec p)=\frac{1}{e^{|\vec p|/T}\pm1}$ is the occupation number of incoming on-shell states at temperature $T$, $g$ is the degeneracy factor, $v_{rel}$ is the relative velocity of the incoming particles and $\vec p_i$'s are the momenta of the incoming particles. It is clear that \eqref{comoving_mass1} results in a bigger comoving mass density when we choose bosonic occupation number. 

Using \eqref{average_mass}-\eqref{1Fscattering}, $v_{rel}\lesssim2$ and performing the integrals over momenta in \eqref{comoving_mass1}, we arrive at
\beq\label{comoving_mass2}
\frac{d\rho_{rad\to DM}}{dt} \lesssim g_1g_2\frac{8\lambda^2}{45(2\pi)^6}\Gamma^2[3.5]\zeta^2[3.5]a^3(t)\frac{T^7}{\Lambda^2},
\eeq
where $\Gamma$ and $\zeta$ are gamma and Riemann zeta functions, respectively. 

Perturbative calculations are valid only if $\lambda<1$. If we consider this condition in \eqref{comoving_mass2} and sum over all constituent of the radiation fluid, we arrive at
\beq\label{comoving_mass}
\rho_{rad\to DM} < 4\times10^{-5}\int dt~g^2 a^3(t)\frac{T^7}{\Lambda^2},
\eeq
where $g$ is the total number of degrees of freedom in the radiation fluid. 

During reheating (by solving \ref{infdecay}-\ref{inftoDM})
\beq
t \propto a^{3/2},~~ T^4\propto \rho_{rad}\propto a^{-3/2}.
\eeq
Substituting these values back in \eqref{comoving_mass}, we realize that the annihilation of radiation into dark matter is most efficient at the end of reheating.
The same manipulation shows that the annihilation of radiation into dark matter during radiation era happens at the beginning of radiation era and is of the same order.  

Let us now work out how much dark matter will be produced in radiation era (after reheating). During radiation era
\beq
t=\sqrt{\frac{45}{2\pi^2g}}\frac{M_{\rm P}}{T^2}.
\eeq
Combining this, with Eq. (\ref{comoving_mass}), and the results of Sec. (\ref{sec::reheating}), we find:
\bea
\frac{\rho_{rad\to DM}}{\rho_{DM}} &<& 10^{-5}\times  \frac{g^{3/2} M_{\rm P} T_{rh}^2}{ T_{eq} \Lambda^2} \nonumber\\ 
&\sim& 10^{-3}\times  \frac{g^{3/2} M_{\rm P} T_{eq}  \Lambda^2}{m^4_\phi} \nonumber\\
&\sim&  10^{-7} \left(g \over 124\right)^{3/2}   \left(\Lambda \over M_{\rm P} \right)^{2}  \left(m_\phi \over 10^{-5} M_{\rm P} \right)^{-4},  
\eea
where we used $g \simeq 124$ for standard model of particle physics.


%
%Plugging in these values in \eqref{comoving_mass}, and substituting the value of $g$ from standard model of particle physics $g\leq 124$, we get
%\beq\label{comoving_mass_est}
%\rho_{rad\to DM}<M_{\rm P}T_0^3\frac{T_{rh}^2}{\Lambda^2}=(4\pi)^2M_{\rm P}T_0^3x^2\frac{T_{eq}^2\Lambda^2}{m_{\phi}^4},
%\eeq 
%where we have used \eqref{reheating}.

Therefore,  for $\Lambda \sim M_{\rm P}$ and high scale inflation  $m_{\phi}\approx 10^{-5}M_{\rm P}$, the production of O$f$DM due to radiation self-interaction is much smaller than the contribution from inflaton decay (in effect $x=1$). However, $\rho_{rad\to DM}$ can become important in scenarios with lighter inflaton, i.e. if $m_\phi \lesssim 10^{-7} (M_{\rm P} \Lambda)^{1/2}$. 


So far we have studied the predictions of this model in the context of inflation. As we showed earlier, this model effectively fixes the reheating temperature of the universe. By constraining the reheating temperature, we can narrow the predictions of $(n_s,r)$ for a given inflationary potential, by fixing the number of e-foldings. However, the predictions for $(n_s,r)$ are model dependent and vary with the inflationary potential. Conversely, one can use the observational constraints on $(n_s,r)$ as a way to fix the non-locality scale $\Lambda$, in the context of a given inflationary model.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cold O$f$DM}\label{Cold OFDM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In principle, O$f$DM particles with very low masses can be produced in scatterings. These low mass particles can behave like hot dark matter at different stages in the evolution of the universe. Let us estimate an upper bound on the fraction of hot O$f$DM particles at a given redshift. 



An off-shell dark matter particle with mass $m$ has energy $E_m=\frac{E_{\rm CM}^2+m^2}{2E_{\rm CM}}$ and momentum $p_m=\frac{E_{\rm CM}^2-m^2}{2E_{\rm CM}}$ , where $E_{\rm CM}$ is the energy of the process producing the dark matter particle.\footnote{This comes from conservation of energy-momentum in the rest frame of incoming particle(s). Here, we have ignored mass of the on-shell particle produced together with O$f$DM particle.} At redshift $z$, this particles is relativistic if $p_m\frac{1+z}{1+z_{pr}} \gtrsim m$, where $z_{pr}$ is the redshift at the time of production.

Given the mass distribution of O$f$DM particles and assuming that most of the dark matter particles are produced at the time of reheating (as we discussed in previous sections), we can find the fraction of hot dark matter particles ($\Omega_h$), which is shown in Figure \ref{hot_dark_matter}. Only a small fraction of O$f$DM is hot at $z< 1000$, which makes it a good candidate for CDM. This result is not surprising since, as we mentioned earlier, even at the time of production these particles are {\it only mildly} relativistic.

Let us work out the distribution of free streaming distance $\lambda_{fs}$. This is given by
\beq\label{free_stream}
\lambda_{fs}=u\int\frac{dt}{\sqrt{a^4+u^2 a^2}}
\eeq 
where $u=a_{pr}\frac{v}{\sqrt{1-v^2}}$ and $v=\frac{p_m}{E_m}$ is the velocity of dark matter particle with mass $m$ at the time of production. Assuming $a_{pr}=a_{rh}$, Equation \eqref{free_stream} gives the free streaming distance in terms of $m$ and $T_{rh}$. This equation can be used further to derive the probability distribution of $\lambda_{fs}$, since the probability distribution of $m$ \eqref{mass_density} is known. The result is shown in Figure \ref{free_stream_distribution}. 
\begin{figure}
	\begin{subfigure}[t]{0.46\textwidth}
        		\includegraphics[width=\hsize]{hot_dark_matter2.pdf}
		\caption{The fraction of off-shell dark matter particles, produced at the time of reheating, that remain relativistic down to a given redshift.}
\label{hot_dark_matter}
    	\end{subfigure} %
	\hfill
	\begin{subfigure}[t]{0.52\textwidth}
       	 	%\centering
        		\includegraphics[width=\hsize]{free_stream_distribution.pdf}
\caption{Distribution of free streaming distance of O$f$DM for different reheating temperatures. The top axis shows the characteristic halo mass associated with the free streaming scale, in units of Earth mass.}
\label{free_stream_distribution}
    	\end{subfigure} %
	\caption{}
\end{figure}
Since the velocity distribution of O$f$DM particles is different from Maxwell-Boltzmann distribution, probability distribution of $\lambda_{fs}$  in this model is different from ordinary thermal WIMP scenario. In particular, it has a much shallower power-law (rather than gaussian) cut-off at large $\lambda_{fs}$'s. This leads to a different matter power spectrum (on small-scales) which can, in principle, be a way to distinguish these two models. Figure \ref{Transfer_function} shows the matter transfer function $T(k)$.

 In Figure \ref{Transfer_function} two effects has been considered: Growth in matter fluctuations due to an early era of matter domination (inflaton dominated era) and free streaming effect.
Early matter era result into amplification of matter fluctuations for modes that enter the horizon during reheating. This amplification is roughly $\propto \frac{k^2}{\ln(k)}$ \cite{Erickcek:2011us}. On the other hand, free streaming effect result into the decrease in the matter power spectrum on small scales $\propto k^{-2}$. The combination of the two effects is seen in Figure \ref{Transfer_function}. On small scales, transfer function drops as $(\ln k)^{-1}$ which is to be contrasted with a much steeper gaussian cut-off in thermal scenarios.

Future gravitational probes of dark matter structure on small scales can potentially test this prediction for matter power spectrum on $10^{-1}-10^{-3}$ pc scales  \cite{Erickcek:2010fc,Baghram:2011is,Rahvar:2013xya}.  



\begin{figure}
\includegraphics[width=0.9\hsize]{Transfer_function.pdf}
\caption{Matter transfer function due to the growth in early matter era and free streaming effect. Instead of an exponential cut-off for large $k$ in thermal scenarios, there is $\propto (\ln k)^{-1}$ drop in O$f$DM scenario.}
\label{Transfer_function}
\end{figure} %


%\begin{figure}
%\includegraphics[width=0.5\textwidth]{hot_dark_matter.pdf}
%\caption{The fraction of hot dark matter particles, produced at the time of reheating at a given redshift.}
%\label{hot_dark_matter}
%\end{figure}


%\begin{figure}
%\includegraphics[width=0.5\textwidth]{free_stream_m.pdf}
%\caption{Free streaming distance ($\lambda_{fs}$) for a given mass of dark matter particle.}
%\label{free_stream_mass}
%\end{figure}



%\begin{figure}
%\includegraphics[width=0.5\textwidth]{free_stream_distribution.pdf}
%\caption{Distribution of free streaming distance of O$f$DM for different reheating temperatures.}
%\label{free_stream_distribution}
%\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Gravitational detection of dark matter particles?}\label{Gravitational Detection}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%So far, we reviewed a number of features/predictions of O$f$DM. But how about the detection of O$f$DM particles?
%
%As we mentioned in Section \ref{introduction}, there will be no non-gravitational detection of O$f$DM. This property, while being consistent with all the experiments looking for direct/indirect detection of dark matter, makes it hard to confirm this model. In what follows, we propose a way to search for O$f$DM particles. 
%
%Mass distribution of the produced dark matter particles is given in \eqref{mass_density}. The typical mass of these particles are determined by the energy of the process in which they have been produced. This means that for the dark matter particles produced by the decay of inflaton, the typical mass of these particles is fixed by the mass of inflaton ($\sim m_{\phi}/2$). For high scale inflationary scenarios (like $R^2$ inflation), the amplitude of fluctuations effectively fixes $m_{\phi}$ to be $\sim 10^{13}\rm{GeV}$. This means that the typical mass of dark matter particles is around the same value. Note that this prediction is independent of a specific model for inflationary potential and only uses the mass of inflaton. 
%
%Consequently, if inflaton happens to have this mass, there is a distribution of collisionless particles with the mass of $10^{13}\rm{GeV}\approx 10^{-10}\rm{gr}$ around us. Since the individual particles are quite heavy, there is a chance of detecting individual dark matter particles through their gravitational interaction. One possibility would be to measure the change in the gravitational field. Figure \ref{gravitational_field} shows the fluctuations in the gravitational field due to the existence of these particles. Current experiments cannot detect the change in the gravitational field with enough precision to detect these particles. However, this can define a direction of research to possibly confirm/rule out this model.
%
%\begin{figure}
%\includegraphics[width=\textwidth]{Local_DM_gravity.pdf}
%\caption{Gravitational field fluctuations due to the existence of dark matter particles with mass $10^{-10}\rm{gr}$ on different scales.}
%\label{gravitational_field}
%\end{figure}
%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Future Prospect}\label{conclusion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper, we laid out the phenomenological implications of the off-shell dark matter (O$f$DM) model. This model is motivated by considering the effect of Planck scale nonlocality on the evolution of quantum fields which manifests itself by introducing a new set of excitations. The new excitations, dubbed off-shell modes, cannot be detected through scattering experiments, making them a natural candidate for dark matter. So, if O$f$DM makes up the majority of the observed cosmological dark matter, we would not be able to detect dark matter particles directly.

However, O$f$DM particles can be produced in scattering experiments and this is one way to indirectly confirm their existence by detecting missing energy in scatterings. The probability of missing energy is given by \eqref{production_rate_simplified} and \eqref{production_rate_2}. High energy collider experiments with enough  sensitivity to detect this missing energy could be  a possible way to test this model, albeit not the most practical one.

We also discussed predictions of O$f$DM model in the context of cosmology and showed that it is intertwined with the physics of inflation and reheating. For a very simple reheating model, we showed that O$f$DM particles are generically produced in the era of reheating and through the decay of inflaton. Since O$f$DM particles do not interact with other particles (or each other), they do not reach a thermal distribution. We calculated O$f$DM distribution function in our simple reheating model and  showed that it leads to much shallower suppression of matter power spectrum on small scales compared to a gaussian cutoff of thermal dark matter candidates. This, in principle, could be another way to test the model via the observations probing matter power spectrum in sub-pc scales. 

We end this paper by noting the following theoretical aspects of O$f$DM which are yet to be explored: %could be the subject of future research:
\begin{enumerate}
\item Throughout this paper we assumed that off-shell modes of a nonlocal field gravitate like ordinary (on-shell) matter, i.e. an off-shell mode with mass $m$  gravitates  like a normal particle with the same mass. This assumption, which seems reasonable, is yet to be verified through a consistent coupling of nonlocal quantum field theories to gravity.
\item So far, the quantization of this type of nonlocal field theory has only been done only scalars. But how about spinor or gauge fields?
This is especially important in the case of gauge theories which govern all interactions in the standard model of particle physics. There are (at least) two obvious ways to proceed here: 
\begin{enumerate}
\item One can define a nonlocal version of gauge transformations to keep gauge invariance. This presumably implies that scattering processes have to include pairs of on-shell modes, or otherwise charge conservation would be violated. In the case of our phenomenological reheating model in Section \ref{sec::reheating}, it means that the inflaton field has to first decay into a neutral field which later decays into standard model particles, otherwise Equation \eqref{fraction_reheating} is not applicable. 

\item Gauge invariance is broken at a Planck suppressed level, similarly to the violation of diffeomorphism invariance in Horava-Lifhsitz gravity \cite{Horava:2009uw}.  In this case, one should look for (possibly dangerous) physical consequences of breaking gauge invariance.
\end{enumerate}
\item Off-shell modes of a nonlocal field cannot be detected in realistic collider experiments. But how about other types of experiments? Scatterings are just a subset of experiments that can be done in labs. Is there a way of observing off-shell modes in laboratory directly?

\end{enumerate}

%
%discuss:
%
%only scalar fields
%
%charge conservation and Planck suppressed broken gauge invariance 
%
%gravitational interaction

\acknowledgments
This research is supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through
Industry Canada and by the Province of Ontario through the Ministry of Research and
Innovation.

%Typical mass of O$f$DM particles are high (close to inflaton mass) and this opens up the possibility of detecting their gravitational interaction (e.g. through the change in the gravitational field) in lab experiments. 


%For simplicity, consider a massless scalar field $\phi$ with $\frac{\lambda}{4!}\phi^4$ interaction. We will show later that off-shell production is dominated at temperatures high enough that mass of the particles can be ignored.
%On-shell modes of $\phi$ are massless particles with positive energy; and off-shell modes are massive particles with positive energy. The cross section of $p_1p_2\rightarrow q_1q_2$ scattering in this theory is given by [?]
%\beq\label{diffsigma}
%\frac{d\sigma}{d^4q_1d^4q_2}= \frac{\lambda^2}{4(2\pi)^4|p_1.p_2|}\tilde W(q_1)\tilde W(q_2) \delta^{(4)}(p_1+p_2-q_1-q_2),
%\eeq
%when $p_i$'s are on-shell, i.e. $p_i^2=0$. 
%
%$\tilde W(q)$ is a positive distribution containing the information about non-local evolution; it is a function of $q^2$, has a delta function for massless momenta and it is non-zero only when $q^0>0$ and $q^2\le 0$. In fact it can be written in the following form 
%\beq\label{tildeW}
%\tilde W(q)=2\pi\delta(q^2)\theta(q^0)+2\pi \tilde \rho(-q^2)\theta (q^0)
%\eeq
%where $\tilde \rho$ is a finite function and zero for negative entry.
%
%Now, let's calculate the cross section of the scattering processes (\ref{2onshell}-\ref{2offshell}). The cross section of \eqref{2onshell} is given by integrating \eqref{diffsigma} only over massless momenta. As a result,
%\beq\label{sigmao}
%\sigma_{O}=\frac{\lambda^2}{8\pi E_{\rm CM}^2}
%\eeq
%where $E_{\rm CM}$ is the centre of mass energy of the incoming particles. Note that \eqref{sigmao} is the usual cross section in local quantum field theory.
%
%The cross section of \eqref{1onshell} is given by integrating \eqref{diffsigma} over one massless momenta and one time-like momenta. The result is the following
%\beq
%\sigma_{1F}=\frac{\lambda^2}{4\pi E_{\rm CM}^2}\int_0^{\frac{E_{\rm CM}}{2}}dq~ q~\tilde \rho(E_{\rm CM}^2-2E_{\rm CM}q)
%\eeq
%Finally, the cross section of \eqref{2offshell}, $\sigma_{2F}$, is given by integrating \eqref{diffsigma} over time-like momenta.
%%\bea
%%\sigma_{2F}&=&\frac{\lambda^2}{2(2\pi)^2E_{\rm CM}^2}\int_0^{E_{\rm CM}}dq^0\notag\\
%%&~&\int d^3q \tilde \rho\left((q^0)^2-|\vec q |^2\right)\tilde \rho\left((E_{\rm CM}-q^0)^2-|\vec q|^2\right).
%%\eea
%
%$\sigma_{1F}$ and $\sigma_{2F}$ generally depend on the function $\tilde \rho$. However, if we assume that the scattering happens at energies much lower than non-locality energy scale, we get much simpler result. Non-locality energy scale, namely $E_{NL}$, is implicitly defined through $\tilde \rho$. A simple observation shows that $\tilde \rho$ for small momenta can be approximated as follows:
%\beq\label{approxrho}
%\tilde \rho(-q^2)\approx \frac{1}{E_{NL}^2}\left(-\frac{q^2}{E_{NL}^2}\right)^n ,~~~0<-q^2 \ll E_{NL}^2
%\eeq
%where $n\in\{0,1,2,\cdots\}$.
%Using \eqref{approxrho}, we get
%\bea
%\sigma_{1F}&=&\frac{\lambda^2}{16\pi E_{\rm CM}^2}\frac{1}{(n+1)(n+2)}\left(\frac{E_{\rm CM}^2}{E_{NL}^2}\right)^{n+1}\notag\\
%&=&\sigma_O\frac{1}{2(n+1)(n+2)}\left(\frac{E_{\rm CM}^2}{E_{NL}^2}\right)^{n+1}.
%\eea
%This shows that only a small fraction of scatterings at energies $E_{\rm CM}\ll E_{NL}$ produces off-shell modes. Also, a similar calculation shows that $\sigma_{2F}\ll \sigma_{1F}$, i.e. \eqref{1onshell} is the most efficient process of producing off-shell particles. 
%
%In order to find energy density of off-shell particles, we need to find the mass distribution of produced off-shell particle in \eqref{1onshell}. It can be obtained from \eqref{diffsigma} by using \eqref{approxrho}:
%\beq
%\frac{d\sigma_{1F}}{dm^2}=\frac{\lambda^2}{16\pi E_{\rm CM}^2E_{NL}^2}\left(\frac{m^2}{E_{NL}^2}\right)^n\frac{E_{\rm CM}^2-m^2}{E_{\rm CM}^2}, ~~ 0<m<E_{\rm CM}.
%\eeq
%It shows that the typical mass of the off-shell particle is at the order of $E_{\rm CM}$.
%
%%Now, we can move on to the production of off-shell particles in cosmology. Inflation can be treated as a starting point. Presumably whatever happened before inflation does not matter because everything will be diluted after inflation. 
%
%%For simplicity, we make the following assumptions:
%%\begin{enumerate}
%%
%%\item We assume that after inflation (and reheating?) universe is filled with thermalized on-shell Higgs particles with temperature $T_R$ (together with other particles) and the transition to thermal state is sudden. 
%%\item As universe expands, it remains thermal. Specifically, on-shell Higgs modes remain thermal. 
%%\item Higgs quartic coupling $\lambda$ is only a function of the temperature of the universe at the given time. 
%%\end{enumerate}
%Consider FRW universe contains on-shell $\phi$ particles at temperature $T$. The co-moving mass density in the off-shell sector ($\rho_{co})$ is given by
%\beq\label{drhodt}
%\frac{d\rho_{co}}{dt}=a^{3}(t)\int \frac{d^3p_1}{(2\pi)^3}\frac{d^3p_2}{(2\pi)^3}n(\vec p_1)n(\vec p_2)\int dm~m\frac{d\sigma_{1F}}{dm}v_{rel},
%\eeq
%where $t$ is the cosmological time, $a(t)$ is the scale factor and $n(\vec p)=(e^{\beta |\vec p|}-1)^{-1}$ is the on-shell occupation number at temperature $T=1/\beta$. The last term represents the expectation value of off-shell particle's mass where $v_{rel}=\frac{|p_1\cdot p_2|}{|\vec p_1|\vec p_2|}$.
%
%As we will show later, off-shell production is efficient at very high energies. Therefore, off-shell particles are highly massive and they become highly redshifted till today. As a result, their co-moving energy density and mass density is the same.
%
%Assuming that the effective coupling constant $\lambda$ is a function of temperature $T$, we get
%\beq\label{drhodt2}
%\frac{d\rho_{co}}{dt}=a^{3}(t)\alpha_n\lambda^2(T) T^5 \left(\frac{T}{E_{NL}}\right)^{2n+2},
%\eeq
%%this equation is only valid when a small fraction of on-shell converts to off-shell otherwise the assumption of thermal equilibrium of on-shell does not remain valid for a long time.
%where $\alpha_n=\frac{2^{2n+3}\pi \zeta^2(n+7/2)\Gamma^2(n+7/2)}{(2\pi)^6(n+1)(n+2)(2n+5)}$.
%
%Here is where the theory of inflation and reheating becomes important. Basically inflation acts as a starting point. Whatever happened before inflation does not matter, because everything dilutes during inflation. Therefore, we can assume off-shell energy density after inflation is effectively zero and its production starts after inflation
%\beq\label{rho1}
%\rho_{co}=\alpha_n\int_{t_f}^{t_0}dt~a^{3}(t)\lambda^2(T)T^5\left(\frac{T}{E_{NL}}\right)^{2n+2},
%\eeq
%where $t_0$ is the current cosmic time and $t_f$ is the time at the end of inflation. 
%Specifying the theory of inflation and reheating, and the running of parameter $\lambda$ with temperature fixes the dark matter energy density in this model. 
%
%As an example, we consider a simple case of instant reheating. In other words after inflation ends, inflation decays into standard model particles very fast and universe fills with radiation at temperature $T_{max}$. Furthermore, if we assume $\lambda$ runs logarithmically with energy, \eqref{rho1} can be approximated to
%\beq\label{rho2}
%\rho_{co}=\alpha_n\lambda^2(T_{max})\int_0^{T_{max}}dT~\left|\frac{dt}{dT}\right|a^3(T)T^5\left(\frac{T}{E_{NL}}\right)^{2n+2}.
%\eeq
%It is clear from \eqref{rho2} that most of the contribution to $\rho_{co}$ comes from $T \approx T_{max}$ and this justifies ignoring the mass of $\phi$. 
%
%% 
%%\bea
%%&a^3=\frac{g_0}{g_*}\frac{T_0^3}{T^3}\\
%%&t=\frac{\sqrt{45}M_{\rm P}}{4\pi\sqrt{\pi}\sqrt{g_*}T^2}\\
%%&T_0=2.73K=2*10^{-4}ev\\
%%&g_0=2+7/8*6=7.25,~ g_*=124\\
%%&\rho_{crit}=1.4*10^{-10}(ev)^4,~ \Omega_c=0.27
%%\eea
%
%\bea
%T_r=E_{NL}\left(\frac{1}{\beta_n}\frac{4(\pi g_*)^{3/2}}{\lambda^2g_0}\frac{\rho_{DM}}{M_{\rm P}T_0^3}\right)^{\frac{1}{2n+2}}\\
%\beta_n=\sqrt{45}\frac{2^{2n+3}\pi \zeta^2(n+7/2)\Gamma^2(n+7/2)}{(2\pi)^6(n+1)^2(n+2)(2n+5)}
%\eea
%
%Predictions and advantages:
%\begin{enumerate}
%\item no new field. just changing the evolution law with the current SM particles.
%\item it connects very different areas of physics: dark matter, top and higgs mass, inflation and quantum gravity. better measurement in each area could help to get better predictions in other areas. for example, better measurement of top and higgs mass leads to better prediction for reheating temperature.
%\item DM in this scenario is super cold. 
%\item CLE changes the renormalization group. why isn't it important?
%\end{enumerate}
%
%
%
%
%
%\bibliographystyle{jhep}
\bibliography{Dark_Matter}
\bibliographystyle{JHEP}












\end{document}


Dark_Matter_v2.bbl
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\end{thebibliography}\endgroup
