text:
\section{Introduction}
\label{sec:1}

In a given system, entanglement entropy quantifies how closely one of the subsystems would be related to another. The entanglement entropy (EE) is defined as the von Neumann entropy of the reduced density matrix $\rho_A$ of the subsystem A where the observer can only get access to:
\begin{equation}
S_A=-tr\rho_A \log\rho_A\,.
\end{equation}
The EE is also related to the degrees of freedom in a system.

Traditionally, in order to calculate the EE, we will use the Renyi entropy and then apply the replica trick. In \cite{1,2} the authors propose a powerful method to compute the EE holographically. The EE between two subsystems A and its complement B is proportional to the boundary separates them which is known as the area law:
\begin{equation}
S_A=\frac{Area(\gamma_A)}{4G_N^{d+2}}\,.
\end{equation}
Notice that the result of the EE is divergent, and it is not invariant under a UV cut-off rescaling.

In this paper, we will follow the procedure of \cite{3} in which the authors compare the difference of EE related to connected and disconnected minimal surface respectively and find the smaller one. If the difference of EE changes sign, it means that there exists confinement/deconfinement phase transition. In addition, by comparing the difference of EE, one can find the UV cut-off independent part of the EE.

In this paper, we will consider $SU(N_c)$ $\mathcal{N} =4$ SYM coupled with $N_f$ massless flavors in the Veneziano limit at non-zero temperature. After the introduction of $N_f\gg1$ massless flavors, the smearing procedure is applied. The background metric will be calculated up to first order of $\epsilon_h$, $\epsilon_h\sim\lambda_hN_f/N_c$, where $\lambda_h$ is the 't Hooft coupling at $r_h$ which is proportional to the temperature of the system \cite{4}. By dividing the $d$ dimensional space into two complementary subspaces with a strip of length $l$, we will compute the EE between these two regions. By comparing the difference of the EE with respect to the connected and disconnected minimal surface, we find that near the event horizon the EE related to disconnected minimal surface is smaller than the connected one. When the radial coordinate $r$ gets larger, the EE of connected minimal surface is then smaller than that of the disconnected one. The changes of different shape of the minimal surface implies that there exists phase transition related to confinement/deconfinement. Notice that the EE is given by the minimization of the area action, so we will always choose the smaller EE.

To get back the phase transition in the system of $AdS$, one can consider the hard and soft wall models which introduce hard-wall and soft-wall cut-off in the IR region (see the review of \cite{5}). In this paper, we get back the phase transition
by considering the non-zero temperature of the system and find that after introducing the non-zero temperature we get back the phase transition near the event horizon.

The paper is structured as follows. In section \ref{sec:2}, we will introduce the work of \cite{3} and calculate the $AdS_5 \times S^5$ at non-zero temperature as an example. We can find that there exists phase transition phenomenon near the event horizon. In section \ref{sec:3}, we will introduce the work of \cite{4} which will be served as the background metric. We choose to study the near horizon behavior of the system and find that there exists phase transition near the event horizon since the difference of EE related to connected and disconnected minimal surface changes sign. In section \ref{sec:4}, we will make conclusion on the results and show some possible work in the future.

\section{Entanglement entropy computation}
\label{sec:2}
\subsection{Calculation of entanglement entropy}
\label{sec:2.1}

In this subsection, we will give a summarization of the work in \cite{3}. In that paper, the authors generalize the work of \cite{1,2} to large $N_c$ theories.

If a system is divided into two complementary subsystems, and an observer is located in a subsystem A and gets no access to its complement B, the EE between two subsystems is the quantity that measures the entropy the observer can get from the entanglement between region A and B. In the paper of \cite{1,2}, the authors propose a convenient method to compute the EE which is known as the area law:
\begin{equation}
S_A=\frac{Area(\gamma_A)}{4G_N^{d+2}}\,.
\end{equation}
In \cite{3}, the authors give a proposal of a more generalized method of EE computation:
\begin{equation}
S_A=\frac{1}{4G^{(10)}_{N}}\int d^8\sigma e^{-2\Phi}\sqrt{G^{(8)}_{ind}}\,.
\end{equation}
Notice that we compute the EE of two complementary regions by minimizing the area action above.

The background metric we will consider is:
\begin{equation}
{ds}^{2}_{10}=\alpha(r)(\beta(r){dr}^2+{dx}^{\mu}{dx}_{\mu})+g_{ij}{dy}^i{dy}^j\,,
\end{equation}
where $r$ is the radial coordinate ranging from $r_0$ to $\infty$, $x^\mu (\mu=0,1,\cdots,d)$ is the parameters of $\mathbb{R}^{d+1}$, $y^i (i=d+2,\cdots, 9)$ is the parameters of $8-d$ internal manifold.
The volume of the internal manifold is given by:
\begin{equation}
V_{int}=\int\prod^{8-d}_{i=1}dy^i\sqrt{\det g}\,.
\end{equation}
We will introduce a new function:
\begin{equation}
H(r)=e^{-4\Phi}V^{2}_{int}\alpha^d\,.
\end{equation}

The shape of subsystem A we will consider is the straight belt with length $l$.
The calculation of EE is similar to the original paper \cite{1,2}:
\begin{equation}
\frac{S_A}{V_{d-1}}=\frac{1}{4G^{(10)}_{N}}\int^{\frac{l}{2}}_{-\frac{l}{2}}dx\sqrt{H(r)}\sqrt{1+\beta(r)(\partial_xr)^2}\,.
\end{equation}
The possible minimal surfaces have two options, and we will choose the one with smaller EE: one is a disconnected surface which consists of two cigar-like surfaces that extend in $\mathbb{R}^{d-1}$ and separate in the remaining direction of  $\mathbb{R}^{d}$ by the distance $l$; the other is like a tube which connects the two cigar-like surfaces.

The entanglement length $l$ with respect to the connected surface is:
\begin{equation}
\label{eq:2.7}
l(\tilde{r})=2\sqrt{H(\tilde{r})}\int^{\infty}_{\tilde{r}}\frac{dr\sqrt{\beta(r)}}{\sqrt{H(r)-H(\tilde{r})}}\,,
\end{equation}
where $\tilde{r}$ is the minimal value of $r$ related the turning point of the minimal surface.
The EE of the connected entanglement surface is:
\begin{equation}
\label{eq:2.8}
\frac{S_C(\tilde{r})}{V_{d-1}}=\frac{1}{2G^{(10)}_{N}}\int^{\infty}_{\tilde{r}}dr\frac{\sqrt{\beta(r)}H(r)}{\sqrt{H(r)-H(\tilde{r})}}\,.
\end{equation}
The EE of the disconnected one is:
\begin{equation}
\label{eq:2.9}
\frac{S_D}{V_{d-1}}=\frac{1}{2G^{(10)}_{N}}\int^{\infty}_{r_0}dr\sqrt{\beta(r)H(r)}\,.
\end{equation}
Both of EE of the connected and disconnected surface are UV divergent and dependent of the UV cut-off, but the difference between them is finite and UV cut-off independent:
\begin{equation}
\label{eq:2.10}
\frac{S_C-S_D}{V_{d-1}}=\frac{1}{2G^{(10)}_{N}}\int^{\infty}_{\tilde{r}}dr\sqrt{\beta(r)H(r)}\left(\frac{1}{\sqrt{1-\frac{H(\tilde{r})}{H(r)}}}\right)-\int^{\tilde{r}}_{r_0}dr\sqrt{\beta(r)H(r)}\,.
\end{equation}
Notice that the result of (\ref{eq:2.10}) may change sign which indicates a phase transition at the critical point. That means the preferred shape of entanglement surface changes between disconnected and connected. We will focus on the quantity of the difference of EE between the connected and disconnected entanglement surface in the remaining of this paper.

\subsection{Entanglement entropy of $AdS_5\times S^5$ at non-zero temperature}
\label{sec:2.2}

In this subsection we will compute the entanglement length and EE of $AdS_5\times S^5$ at non-zero temperature.

The metric of $AdS_5\times S^5$ at non-zero temperature is:
\begin{equation}
{ds}^2=\frac{R^2}{r^2}\frac{{dr}^2}{f(r)}+\frac{r^2}{R^2}(-f(r){dt}^2+{\vec{dx}}^{2}_{3})+R^2{d\Omega}^{2}_{5}\,,
\end{equation}
where $f(r)=1-r^{4}_{h}/r^4$. $r_h$ is set as the horizon radius and is proportional to the Hawking temperature. Sending $r_h$ to zero, we recover the pure $AdS_5\times S^5$ at zero temperature.

The relevant functions to be used later are listed as follows:
\begin{equation}
\label{eq:2.12}
\beta(U)=\frac{R^4}{r^4}\frac{1}{f(r)},\quad H(r)=\left(\frac{8\pi^2}{3}\right)^2R^4r^6\,,
\end{equation}
where $r$ ranges from $r_h$ to $\infty$.

The entanglement length $l$ is:
\begin{equation}
\label{eq:2.13}
l(\tilde{r})=2R^2{\tilde{r}}^{3}\int^{\infty}_{\tilde{r}}\frac{1}{\sqrt{(r^4-r^{4}_{h})(r^6-{\tilde{r}}^{6})}}\,,
\end{equation}
where $\tilde{r}$ is the minimal value of $r$ related to the connected surface and we have the following relation: $0<r_h\leq {\tilde{r}}\leq r<+\infty$. Notice that $l(r)$ is divergent at $r=r_h$ and is a monotonically decreasing function which goes to zero at infinity. This can be seen easily from (\ref{eq:2.13}) that the leading order of $l(\tilde{r})$ is $1/\tilde{r}$. Figure \ref{fig:1} shows the plot of $l$ as a function of $\tilde{r}$. Notice that we have set the scale of horizon $r_h$ and radius $R$ to 1.
\begin{figure}[bhtp]
\centering
\includegraphics[width=.45\textwidth,clip]{img1.pdf}
\caption{\label{fig:1} The entanglement length as the function of $\tilde{r}$}
\end{figure}\\
The EE of connected minimal surface is:
\begin{equation}
\begin{split}
S_C(r)&\sim\int^{+\infty}_{\tilde{r}}dr\frac{\sqrt{\beta(r)}H(r)}{\sqrt{H(r)-H({\tilde{r}})}}\\
&=R^4(\frac{8{\pi}^2}{3})\int^{+\infty}_{\tilde{r}}dr\frac{r^6}{\sqrt{(r^6-{\tilde{r}}^{6})(r^4-{{r}}^{4}_{h})}}\,.
\end{split}
\end{equation}
The EE of disconnected minimal surface is:
\begin{equation}
\begin{split}
S_D&\sim\int^{+\infty}_{r_h}dr\sqrt{\beta(r)H(r)}\\
&=R^4(\frac{8{\pi}^2}{3})\int^{+\infty}_{r_h}dr\sqrt{\frac{r^6}{r^4-{{r}}^{4}_{h}}}\,.
\end{split}
\end{equation}
The difference of them is:
\begin{equation}
\begin{split}
\label{eq:2.16}
S_C-S_D&\sim\int^{+\infty}_{\tilde{r}}dr\frac{r^6}{\sqrt{(r^6-{\tilde{r}}^{6})(r^4-{{r}}^{4}_{h})}}-\int^{+\infty}_{r_h}dr\sqrt{\frac{r^6}{r^4-{{r}}^{4}_{h}}}\\
&=\int^{+\infty}_{\tilde{r}}dr\frac{r^3}{\sqrt{r^4-r^{4}_{h}}}\frac{r^3}{\sqrt{r^6-\tilde{r}^{6}}}-\int^{+\infty}_{r_h}dr\frac{r^3}{\sqrt{r^4-r^{4}_{h}}}\,.
\end{split}
\end{equation}
Notice that when $r\rightarrow+\infty$, $\frac{r^3}{\sqrt{r^6-{\tilde{r}}^{6}}}\rightarrow1$.
When $r\rightarrow+\infty$, although the first term and second term in (\ref{eq:2.16}) both diverge, the divergent terms are the same and thus cancel. This coincides with the result of \cite{3} that the divergent part cancels at the boundary, hence we get a result independent of the UV cut-off.
To be more precise,
\begin{equation*}
\int^{+\infty}_{r_h}dr\sqrt{\frac{r^6}{r^4-{r}^{4}_{h}}}=\frac{\sqrt{r^{4}-r^{4}_{h}}}{2}{\Bigg|}^{+\infty}_{r_h}
\end{equation*}
is the shared divergent term.

In the following analysis, we will take the scale $r_h$ related to the temperature equal to 1 to simplify our computation.

The next step we will take is to compare the difference of $S_C$ and $S_D$.

Let us consider the case when $\tilde{r}\rightarrow1(=r_h)$ first. This is the near horizon case. (\ref{eq:2.16}) becomes:
\begin{equation}
\label{eq:2.17}
\int^{+\infty}_{1}dr\frac{r^3}{\sqrt{r^4-1}}\frac{r^3}{\sqrt{r^6-1}}-\int^{+\infty}_{1}dr\frac{r^3}{\sqrt{r^4-1}}\,.
\end{equation}
Since $\frac{r^3}{\sqrt{r^6-1}}>1$, the result of (\ref{eq:2.17}) is larger than zero which means that the EE of connected minimal surface is larger than that of the disconnected one at near horizon. Notice that we always choose the minimal surface with respect to a smaller EE, hence in this case the disconnected minimal surface is preferred.

The second case is when $\tilde{r}\rightarrow+\infty$. This is the near boundary case. (\ref{eq:2.16}) becomes:
\begin{equation}
\label{eq:2.18}
\int^{+\infty}_{\tilde{r}}dr\frac{r^3}{\sqrt{r^4-1}}(\frac{r^3}{\sqrt{r^6-{\tilde{r}}^6}}-1)-\int^{\tilde{r}}_{1}dr\frac{r^3}{\sqrt{r^4-1}}\,.
\end{equation}
The first term in (\ref{eq:2.18}) becomes zero, while the second term is divergent. The result of (\ref{eq:2.18}) is smaller than zero which means that the EE of connected minimal surface is smaller than that of the disconnected one, hence the connected minimal surface is preferred.

From the analysis above, we can see that as $\tilde{r}$ going from $r_h$ to $\infty$, the difference of EE of connected and disconnected minimal surface changes sign. This implies that a phase transition occurs in the bulk of black $AdS_5 \times S^5$. When $r_h$ the minimal value of $r$ is close to the horizon, $S_C-S_D>0$, $S_D$ is smaller than $S_D$ and thus preferred. In this case, $r_h\rightarrow1$ is correspondent to large $l$. When $r_h$ is closed to boundary (Actually, the phase transition occurs at near horizon, far away from the boundary. We consider the near boundary case just for convenience), $S_C-S_D>0$, $S_C$ is smaller than $S_D$ and thus preferred. In this case, $r_h\rightarrow+\infty$ is correspondent to small $l$.

In pure $AdS_5 \times S^5$, $S_C$ is always smaller than $S_D$ and thus preferred, thus we find no phase transition there \cite{5}. As we introduce the temperature into our system, the difference of $S_C$ and $S_D$ changes sign in the bulk space and we get back our phase transition.

\section{$AdS_5 \times S^5$ with massless flavors at non-zero temperature}
\label{sec:3}
\subsection{$AdS_5 \times S^5$ with massless flavors at non-zero temperature}
\label{sec:3.1}

In this subsection, we will give a review of the work in \cite{4} (see also in \cite{6}).

Let us start with $AdS_5 \times S^5$ which is dual to $\mathcal{N}=4$ SYM. The $N_c$ D3 branes are placed at the tip of a Calabi-Yau cone over a Sasaki-Einstein manifold $S^5$. Under $U(1)$ fibration, the K\"{a}hler-Einstein
base of $S^5$ is $CP^2$, so the metric of this 5d Sasaki-Einstein manifold can be written as:
\begin{equation}
{ds}^2_{X_5}={ds}^2_{KE}+(A_{KE})^2\,,
\end{equation}
where $A_{KE}$  is the connection one form. The metric of $CP^2$ reads
\begin{equation}
{ds}^2_{{CP}^2}=\frac{1}{4}{d\chi}^2+\frac{1}{4}{\cos}^2{\frac{\chi}{2}}({d\theta}^2+{\sin}^2\theta{d\varphi}^2)+\frac{1}{4}{\cos}^2{\frac{\chi}{2}}{\sin}^2{\frac{\chi}{2}}(d\psi+{\cos\theta}d\varphi)^2\,,
\end{equation}
\begin{equation*}
A_{{CP}^2}=\frac{1}{2}{\cos}^2\frac{\chi}{2}(d\psi+{\cos\theta}d\varphi)\,,
\end{equation*}
\begin{equation*}
0\leq\chi,\theta\leq\pi,0\leq\varphi,\tau<2\pi,0\leq\psi<4\pi\,.
\end{equation*}

D7 branes are introduced as matter. The D7 branes locate along the radial direction wrapping an $S^3$ inside $S^5$ and spacetime filling in the UV region. Since we will consider the Veneziano limit, which means $N_f, N_c \rightarrow \infty$, $N_f/N_c$ is fixed. The D7 brans are smeared over the transverse space. This smearing procedure is necessary since we need to avoid the difficulty of the $\delta$ function when computing the integral of DBI action \cite{7}. After the introduction of $N_f$ D7 branes, the system preserves $\mathcal{N}=1$ SYM at zero temperature.

The DBI action is:
\begin{equation}
S=S_ {\uppercase\expandafter{\romannumeral2}B}+S_{fl}\,.
\end{equation}
The action of type ${\uppercase\expandafter{\romannumeral2}B}$ supergravity is:
\begin{equation*}
S_ {\uppercase\expandafter{\romannumeral2}B}=\frac{1}{2\kappa^{2}_{10}}\int{d^{10} x}\sqrt{-g_{10}}[R-\frac{1}{2}\partial_M\Phi\partial^M\Phi-\frac{1}{2}e^{2\Phi}F^{2}_{(1)}-\frac{1}{2}\frac{1}{5!}F^{2}_{(5)}]\,,
\end{equation*}
The action of D7 flavor branes is:
\begin{equation*}
S_{fl}=-T_7\sum_{N_f}\left(\int{d^8x}e^\Phi\sqrt{-g_8}-\int C_8\right)\,,
\end{equation*}
with the gravitational constant of
\begin{equation*}
\frac{1}{\kappa_{10}^{2}}=\frac{T_7}{g_s}=\frac{1}{(2\pi)^2g_{s}^{2}\alpha^{\prime 4}}\,.
\end{equation*}

The metric ansatz of the background is:
\begin{equation}
{ds}^{2}_{10}=h^{-\frac{1}{2}}(-b {dt}^2+{d\vec{x}}^{2}_{3})+h^{\frac{1}{2}}[bS^8F^2{d\sigma}^2+S^2{ds}^{2}_{{CP}^2}+F^2(d\tau+A^{2}_{{CP}^2})]\,.
\end{equation}
When $b=1$, we recover the zero temperature result. Notice that after the smearing procedure, the functions $h$, $b$, $S$, $F$ are only dependent on the radial coordinate $\sigma$.

The equations of motion are:
\begin{equation}
\begin{split}
\partial^{2}_{\sigma}(\log b)=&0\,,
\\
\partial^{2}_{\sigma}(\log h)=&-Q^{2}_{c}\frac{b}{h^2}\,,
\\
\partial^{2}_{\sigma}(\log S)=&-2bF^4S^4+6bF^2S^6-Q_fe^{\Phi}bF^2S^6\,,
\\
\partial^{2}_{\sigma}(\log F)=&4bF^4S^4-\frac{Q^{2}_{f}}{2}e^{2\Phi}bS^8\,,
\\
\partial^{2}_{\sigma}\Phi=&Q^{2}_{f}e^{2\Phi}bS^8+4Q_fbe^{\Phi}S^6F^2\,.
\end{split}
\end{equation}
When $b\neq1$ which is dual to non-zero temperature, the functions of $b$ and $h$ can be solved first with an integration constant $r_h$:
\begin{equation}
\begin{split}
b&=e^{4r^{4}_{h}\sigma}\,,
\\
h&=\frac{Q_c}{4r^{4}_{h}}(1-e^{4r^{4}_{h}\sigma})\,,
\end{split}
\end{equation}
where $\sigma$ ranges from $-\infty$ to 0.
The next step is to define a new variable $r$ by an implying coordinate transformation:
\begin{equation}
\begin{split}
e^{4r^{4}_{h}\sigma}&=1-\frac{r^{4}_{h}}{r^4}=b\,,
\\
h&=\frac{Q_c}{4r^4}=\frac{R^4}{r^4}\,,
\end{split}
\end{equation}
where $r$ ranges from $r_h$ to $+\infty$.

The metric of $AdS_5 \times S^5$ with massless flavors at non-zero temperature is:
\begin{equation}
\label{eq:3.8}
{ds}^{2}_{10}=-\frac{r^2}{R^2}(1-\frac{r^{4}_{h}}{r^4}){dt}^2+\frac{r^2}{R^2}d{\vec{x}}^{2}_{3}+\frac{R^2{\tilde{S}}^8{\tilde{F}}^2}{r^2}\frac{dr^2}{(1-\frac{r^{4}_{h}}{r^4})}+R^2{\tilde{S}}^2ds^{2}_{KE}+R^2{\tilde{F}}^2(d\tau+A_{KE})\,,
\end{equation}
where
\begin{equation}
\begin{split}
\tilde{S}&=\frac{S}{r}\,,
\\
\tilde{F}&=\frac{F}{r}\,.
\end{split}
\end{equation}
We will expand the solutions of the functions of $\tilde{F}$ and $\tilde{S}$ up to first order in $\epsilon_{\ast}$ which is a expansion parameter whose meaning is given in \cite{4}:
\begin{equation}
\begin{split}
\label{eq:3.10}
\tilde{F}&=1-\frac{\epsilon_{\ast}}{24}(1+\frac{2r^4-r^{4}_{h}}{6r^{4}_{\ast}-3r^{4}_{h}})+O(\epsilon^{2}_{\ast})\,,
\\
\tilde{S}&=1+\frac{\epsilon_{\ast}}{24}(1+\frac{2r^4-r^{4}_{h}}{6r^{4}_{\ast}-3r^{4}_{h}})+O(\epsilon^{2}_{\ast})\,.
\end{split}
\end{equation}
Notice that when $\epsilon_{\ast}=0$, we recover the flavorless case.

\subsection{Entanglement entropy computation}
\label{sec:3.2}

We will introduce anther expansion parameter $\epsilon_h$ related to the temperature:
\begin{equation}
\epsilon_{h}=\frac{Vol(X_3)}{16\pi Vol(X_5)}\frac{N_f}{N_c}\lambda_{h}=\frac{\lambda_h}{8{\pi}^2}\frac{N_f}{N_c}\,,
\end{equation}
where $\lambda_h$ is the 't Hooft coupling at the horizon $r_h$.
Since $\epsilon_{\ast}=\frac{\lambda_{\ast}}{8{\pi}^2}\frac{N_f}{N_c}$, we have \begin{equation}
\epsilon_h=\epsilon_{\ast}\frac{e^{\Phi_h}}{e^{\Phi_{\ast}}}=\epsilon_{\ast}+O({\epsilon}^{2}_{\ast})\,.
\end{equation}
The solutions in (\ref{eq:3.10}) are also dependent on an arbitrary UV cut-off scale $r_{\ast}$. The expression of $\tilde{F}$ and $\tilde{S}$ will be used only when $r<r_{\ast}$. $r_{\ast}$ is related to the UV Landau pole. To decouple the UV Landau pole, we need to take $r_{\ast}\rightarrow+\infty$. We have the relations of scale as follows:
\begin{equation}
\begin{split}
0<r_h\leq r<r_{\ast}<+\infty\,,
\qquad
r_h\ll r_{\ast}\,.
\end{split}
\end{equation}

We will compute the entanglement and EE of the flavored $AdS_5\times S^5$ with massless flavors with the shape of a strip in this section. From (\ref{eq:3.8}), we get:
\begin{equation}
\label{eq:3.14}
\beta(r)=hF^2S^8\frac{1}{r^{10}}\frac{1}{f(r)},\quad H(r)=hF^2S^8\,,
\end{equation}
where $f(r)=1-r^{4}_{h}/r^4$.
$F$ and $S$ satisfy:
\begin{equation}
\begin{split}
\label{eq:3.15}
F&=r(1-\frac{\epsilon_{h}}{24}(1+\frac{2r^4-r^{4}_{h}}{6r^{4}_{\ast}-3r^{4}_{h}})+O(\epsilon^{2}_{h}))\,,
\\
S&=r(1+\frac{\epsilon_{h}}{24}(1+\frac{2r^4-r^{4}_{h}}{6r^{4}_{\ast}-3r^{4}_{h}})+O(\epsilon^{2}_{h}))\,.
\end{split}
\end{equation}
Notice that we have changed the expansion parameter $\epsilon_{\ast}$ to $\epsilon_h$ which is related to the temperature.

Let us consider the UV boundary case first. When $r\rightarrow+\infty$, since $r<r_{\ast}$ and $r_{\ast}\gg r_h$, we have $r\sim r_{\ast}$ and thus:
\begin{equation}
\frac{2r^4-r^{4}_{h}}{6r^{4}_{\ast}-3r^{4}_{h}}\rightarrow1/3\,.
\end{equation}
From (\ref{eq:3.15}), we have $F\sim r$ and $S\sim r$. Using (\ref{eq:3.14}) and compared with (\ref{eq:2.12}), we have $\beta(r)\sim\frac{1}{r^4-r^{4}_{h}}$ and $H(r)\sim r^6$. Hence $l(\tilde{r})$, $S_C(\tilde{r})$ and $S_D$ have the similar behavior as the flavorless case.

The next case we will consider is the near horizon case, where the phase transition phenomenon is expected. Notice that the extra scale $r_{\ast}$ is related to the UV scale. To decouple the UV pole, we need to take $r_{\ast}\rightarrow+\infty$. Since $r_{\ast}$ is regarded as the UV cut-off and we want to focus on the IR behavior of the system, $r\sim r_h$, the UV cut-off $r_{\ast}$ can be thought of as the UV completion of the system. It can be discarded since the correction from the UV has mild effect on the IR behavior in the spirit of effective theory. Hence the terms with $r^{4}_{h}/r^{4}_{\ast}$ in (\ref{eq:3.15}) are neglected when we concentrate on the IR behavior. Similar discussions can be found in \cite{6} (see also in \cite{4}).

The equations in (\ref{eq:3.15}) become (in the IR region):
\begin{equation}
\begin{split}
F=r(1-\frac{\epsilon_{h}}{24}+O(\epsilon^{2}_{h}))\,,
\qquad
S=r(1+\frac{\epsilon_{h}}{24}+O(\epsilon^{2}_{h}))\,.
\end{split}
\end{equation}
\begin{figure}[bhtp]
\centering
\includegraphics[width=.45\textwidth,clip]{img2.pdf}
\caption{\label{fig:2} The entanglement entropy as the function of $\tilde{r}$}
\end{figure}\\
Figure \ref{fig:2} shows the plot of $S_C$ and $S_D$ in the near horizon region. To be compared to the result of \cite{4}, we have set the value of $\epsilon_h$ to $1/4$ and $r_h$ to 1. Notice that we have the similar result as in section \ref{sec:2.2}. The EE of connected minimal surface is larger than the EE of the disconnected one when $r\rightarrow1(=r_h)$. When $r$ grows, the EE of connected surface is smaller than the disconnected one. The phase transition occurs near horizon as expected.

\section{Conclusion}
\label{sec:4}

In this paper, we have studied the entanglement entropy of $AdS_5 \times S^5$ with massless flavors in the Veneziano limit at non-zero temperature.  Dividing the $d$ dimensional space into two complementary subspaces with a strip of length $l$, we calculate the EE between them with the method of \cite{3}. We focus on the value of $S_C-S_D$ and find that there exists phase transition at near horizon $r_h$ which is related to the temperature.
In this paper, we bring back the phase transition with the introduction of non-zero temperature.

In section \ref{sec:2}, we calculate the entanglement length and EE of $AdS_5 \times S^5$ at finite temperature. The entanglement length is monotonically decreasing and divergent at the scale of horizon $r_h$. The EE of connected minimal surface is larger than that of the disconnected one at the horizon, while the EE of connected surface is smaller than the disconnected one at the boundary, hence phase transition occurs in the bulk space which means the different choice of the shape minimal surface in the spirit of the minimization of the area action. In section \ref{sec:3}, we calculate the entanglement length and EE of $AdS_5 \times S^5$ coupled with $N_f$ massless flavors at finite temperature, finding that it shares the similar behavior with the flavorless case. The phase transition happens near the horizon. When $r_h=\tilde{r}$, $S_C>S_D$, hence the disconnected minimal surface is the preferred shape. When $r_h$ grows, the phase transition happens near the horizon, the shape of minimal surface is connected from then on.

Let us close this section by discussion about some possible future work. In  Klebanov-Strassler model, there exists phase transition in the bulk with and without flavors. In \cite{8}, the authors calculate the critical exponents using the method of Wilson loop. In \cite{9}, the authors calculate the critical exponents with the method of EE, and find that they belong to a different universality class compared to that with the method of Wilson loop. It would be interesting to find out the underlying meaning of that difference.

\acknowledgments

This work is supported by the Wu Wen-Tsun Key Laboratory of Mathematics at USTC of Chinese Academy of Sciences. The work is partially supported by NSFC under the contract nunber of 11571336.

redpajama_set_name: RedPajamaArXiv


text:
\section{Introduction}

In the last decade an unprecedented amount of precise observations has led to the consolidation of the so called standard cosmological model, with a geometry given by a spatially flat Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) metric and an energy content formed by around $5\%$ of baryons, $23\%$ of cold dark matter and $72\%$ associated to a cosmological constant $\Lambda$, the component responsible for the present acceleration of the expansion \citep{ParticleDataGroup}. The $\Lambda$CDM model, together with the standard description of the early hot phase of universe's evolution and the inflationary paradigm for the generation of primordial density fluctuations, has been shown robust when tested against the more precise observations: i) the spectrum of anisotropies of the CMB temperature \citep{komatsu}; ii) the mass power spectrum of LSS distribution \citep{montesano}; iii) the Hubble diagram for supernovas of type Ia \citep{SNUNION,hicken,SNSNLS}; and (in a still lower precision level) the distance scale to the BAO peak \citep{wiggleZ}. The two later are background tests of the redshift-distance relation predicted by the model, and involve relatively low redshifts (up to $z\approx 2$). The two former also test predictions at the perturbative level and up to the redshift of last scattering and beyond. For example, the matter density at the time of matter-radiation equality (around $z\approx 3300$ for the $\Lambda$CDM model) is essential to determine the turn-over of the mass power spectrum and hence its present shape. In the case of the CMB anisotropy spectrum, many free parameters are available in the data fitting. If we are just interested in fixing background parameters like the present relative matter density for example, a simplified test is given by fitting the position of the first acoustic peak in the spectrum.

In spite of the relative success of the standard model in getting a good concordance between those observations, some theoretical and observational problems have stimulated the study of alternative models of dark energy. Among the theoretical problems usually mentioned we can cite the absence of a microphysical explanation for the tiny value of the cosmological constant and the approximate coincidence between $\Lambda$ and the present matter density. On the observational level, the celebrated concordance is actually dependent on the sample of SNe Ia used in the joint analysis \citep{tension,tension2}. Among the recent compilations, the Union2 \citep{SNUNION} is one of the biggest samples and, alone, predicts for the $\Lambda$CDM a present matter density parameter $\Omega_{m0} \approx 0.27$. However, we should have in mind that the Union2 dataset is calibrated with the Salt2 fitter, which makes use of a fiducial $\Lambda$CDM model for including high-z supernovas in the calibration. Therefore, that sample is not model-independent and the test should be viewed as rather a test of consistence.

Model-independent SNe samples can be obtained by using instead the MLCS2k2 fitter, which uses only low and intermediate redshifts in the calibration \citep{MLCS2K2}. Although this may lead to other kinds of limitations, it is a must if we want to test the robustness of the concordance. By using, for instance, the Constitution and SDSS samples calibrated with that fitter, we obtain $\Omega_{m0} \approx 0.33$ and $\Omega_{m0} \approx 0.40$, respectively (see e.g. Table 2 of reference \citep{jailson2011}), values in clear tension with that obtained from LSS. For example, the best-fit value obtained from the analysis of the 2dFGRS data for the mass power spectrum is $\Omega_{m0} \approx 0.23$ \citep{2dFGRS}. A similar result, $\Omega_{m0} \approx 0.22$, is obtained from the analysis of the shape of the SDSS DR5 galaxy power spectrum when only large scales ($0.01<k<0.06$ $h$Mpc$^{-1}$) are included \citep{Percival}. This tension is also clear when we look the reduced $\chi^2$'s of the joint analysis including the four tests referred above (see Table 1 below).

Such a tension may be just the result of systematics in the samples or in the calibration. But it may also have some physics behind it. As we have commented above, the mass power spectrum is particularly sensitive to the matter density at the time of matter-radiation equality. While the SNe Ia distance relations are tested for relatively low redshifts. Therefore, the mentioned tension may be suggesting a late-time process of particle creation from the vacuum, in such a way that we have at early times the matter density needed to fit the LSS observations, but a present density above that expected if no matter has been created.

In order to verify this hypothesis, the data must be fitted by a model with particle creation. Since the radiation epoch must remain unaffected, no photons should be produced. In the same way, production of baryons is not allowed if we want to maintain the main features of CMB. Therefore, only dark particles will be created in this model. Furthermore, since the observed dark matter is cold, the created matter is pressureless. The simpler way of realizing this phenomenological recipe is with a constant rate of creation. As we will see, the corresponding model is uniquely defined, has the same number of free parameters as the standard model and is not reducible to the later. The data analysis with this model leads to a good concordance. With the Union2 sample the fit is as good as in the standard case, while the use of the Constitution and SDSS samples leads to a very good agreement with the value derived from the LSS analysis in this model, $\Omega_{m0} \approx 0.45$. We will also see that the backreaction associated to the matter creation implies a time-dependent vacuum density, linearly proportional to the Hubble parameter, in order to respect the conservation of total energy.

In addition to the main observational tests referred above, several others can be done, despite the less precision involved. Among them we can cite, for example, weak gravitational lensing, gas mass fraction in clusters, redshift-distance relations for compact x-ray sources, the redshift-age relation for high-$z$ objects, and the peculiar velocity field of galaxies. For elucidating the problem we are discussing here, i.e., the tension between early and late-time predictions for the matter density, an interesting complementary test is provided by gamma ray bursts (GRB) \citep{GRB1}, since they can extend the Hubble diagram up to redshifts $z \approx 8$ \citep{Schaefer:2007,WH}. Actually, the use of GRBs as standard candles is quite controversial, as we will discuss in the Section III. In this paper we do assume that GRBs can be used as cosmological probes, following references \citep{Cuesta,GRB2,GRB3,Velten,MB}. Once more, any strong model-dependence should be avoided and, for this goal, new GRB samples will be generated in this letter, calibrated with help of the SDSS (MLCS2k2) SNe Ia compilation discussed above.

\section{The model}

Consider the possibility of creation of non-relativistic dark particles at a constant rate $\Gamma$. The balance equation for this process (some times called the Boltzmann equation) is given by
\begin{equation} \label{Boltzman}
\frac{1}{a^3}\frac{d}{dt}\left(a^3n\right)= \Gamma n,
\end{equation}
where $n$ is the particle number density at a given time. By multiplying by the created particle mass $M$, it can be rewritten as
\begin{equation} \label{conservacao}
\dot{\rho}_m + 3H\rho_m = \Gamma \rho_m,
\end{equation}
where $\rho_m = nM$  is the matter density.

In addition, let us take the Friedmann equation for the spatially flat case,
\begin{equation} \label{Friedmann}
\rho = \rho_m + \Lambda = 3H^2,
\end{equation}
where $\rho_m$ is the matter density and $\Lambda$ is the energy density associated to a vacuum term. Combining the two above equations and using for the vacuum term the equation of state $p_{\Lambda} = - \Lambda$, we can obtain
\begin{equation} \label{conservacao2}
\dot{\rho} + 3H(\rho +p) = 0
\end{equation}
(where $\rho$ and $p$ refer to the total density and pressure, respectively), provided we take $\Lambda = 2\Gamma H + \lambda_0$, where $\lambda_0$ is a constant of integration. Since there is no microphysical scale defining such a constant, we will take $\lambda_0 = 0$. Therefore, we have\footnote{Strictly speaking, this result is only exact if we dismiss the conserved baryons in the balance equations. Since baryons represent only about $5\%$ of the total energy content, relation (\ref{linear}) can be considered a good approximation.}
\begin{equation} \label{linear}
\Lambda = 2\Gamma H,
\end{equation}
and Eq. (\ref{conservacao2}) can be rewritten as
\begin{equation} \label{conservacao3}
\dot{\rho}_m + 3H\rho_m = -\dot{\Lambda}.
\end{equation}
We can see that the creation of pressureless matter is only possible with a concomitant time-variation of the vacuum density, which acts as a source of particles in the conservation equation, Eq.  (\ref{conservacao3}), decaying linearly with the Hubble parameter $H$.

Dividing Eq. (\ref{linear}) by $3H^2$ and using $\Lambda/(3H^2) \equiv 1 - \Omega_m$, we obtain
\begin{equation} \label{rate}
\Gamma = \frac{3}{2} \left( 1 - \Omega_{m} \right) H,
\end{equation}
which is valid for any (late) time. For the present time (with $\Omega_{m0} \approx 1/3$) we have $\Gamma \approx H_0$. On the other hand, in the future de Sitter limit ($\Omega_m \rightarrow 0$), it follows that $\Gamma \approx (3/2)H_{dS}$. That is, apart from a constant factor, the creation rate is equal to the Gibbons-Hawking temperature associated to the de Sitter horizon \citep{Gibbons,Gibbons2}.

With $\Lambda = 2\Gamma H$ it is easy to obtain the Hubble function \citep{Borges,jailson2006}
\begin{equation}\label{Hgeral}
\frac{H}{H_0} = 1 - \Omega_{m0} + \Omega_{m0} (1
+ z)^{3/2},
\end{equation}
where $\Omega_{m0}$ and $H_0$ (the present values of the relative matter density and of the Hubble parameter, respectively) are, like in the standard model, the only free parameters to be adjusted. In the asymptotic limit $z \rightarrow -1$ it leads to a constant $H$, that is, to a de Sitter universe. On the other hand, for early times (high $z$), we have $H\approx H_0 \Omega_{m0} z^{3/2}$, which, through the Friedmann equation, leads to $\rho_m = 3H_0^2\Omega_{m0}^2 z^3$. Therefore, at early times the energy density scales with $z^3$, that is, with $a^{-3}$, as in the standard model. However, in the later we would have $\rho_m = 3H_0^2\Omega_{m0}z^3$. The extra factor $\Omega_{m0}$ ($< 1$) in the former is related to the late-time matter creation. In order to have the same amount of matter today, we should have less matter at higher redshifts. Reversely, if we want the same amount of matter at high $z$ (in order to preserve the observed features of CMB and LSS), we expect a present matter density higher than in the standard model.

This is confirmed when we perform precise observational tests with the present model, at both the background and perturbative levels. In previous publications \citep{Borges,jailson2006,jailson2008,jailson2011,Fabris,Zimdahl,PLB} the model has been tested against observations of the Hubble diagram for type Ia supernovas, the position of the first peak in the CMB spectrum of anisotropies, the power spectrum of large scale structures and the distance to baryon acoustic oscillations. The concordance is quite good and, depending on the sample of supernovas used, it is actually better than in the $\Lambda$CDM case, as shown in Table 1 (with $H_0$ marginalized).

\begin{table*}
\begin{center}
\caption{Limits to $\Omega_{m0}$ (SNe+ CMB + BAO+LSS) \citep{PLB}.}
\begin{tabular}{r@{~~}c@{~~}c@{~~}c@{~~}c@{~~}}
\hline \hline \\
\multicolumn{1}{c}{ } & \multicolumn{2}{c}{$\Lambda(t)$CDM } & \multicolumn{2}{c}{$\Lambda$CDM } \\
\multicolumn{1}{c}{Test}&
\multicolumn{1}{c}{$\Omega_{m0}$\footnote{Error bars stand for $2\sigma$}}&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}&
\multicolumn{1}{c}{$\Omega_{m0}$ $^a$}&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}\\ \hline \\
Union2 (SALT2)......&$0.420^{+0.009}_{-0.010}$ & 1.063 & $0.235\pm 0.011$ & 1.027 \\
SDSS (MLCS2k2).......& $0.450^{+0.014}_{-0.010}$ & 0.842 & $0.260^{+0.013}_{-0.016}$ & 1.231 \\
Constitution (MLCS2k2[17]).......& $0.450^{+0.008}_{-0.014} $ &1.057 & $0.270\pm 0.013$ & 1.384\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}

We can see in the table that, when the Union2 compilation is used in the joint analysis, similar reduced $\chi^2$'s are obtained for both models. However, Union2 data are calibrated with the Salt2 fitter by making use of a fiducial $\Lambda$CDM model in the calibration process. A model-independent analysis is possible if we consider instead the MLCS2k2 fitter in the calibration, as is the case of the SDSS and Constitution samples in Table 1. With these compilations we obtain a better concordance with the present model. As anticipated above, the present matter density is higher than in the standard case, with $\Omega_{m0} \approx 0.45$.

\section{The GRB samples}

The luminosity of GRBs appears to be correlated with their temporal and spectral properties and, although these correlations are not yet fully understood from first principles, their existence has naturally suggested the use of GRBs as distance indicators, offering a possible route to probe the expansion history of the Universe up to redshifts $z \approx 8$ \citep{WH}.

The typical spectrum of the prompt emission of GRBs can be expressed as an exponentially connected broken power-law, the so-called Band function \citep{Band:1993}. Then we can determine the spectral peak energy $E_p$, the photon energy at which
the $\nu F_{\nu}$ is brightest.

The two most used correlations are the Amati relation $E_{p}-E_{iso}$, relating the rest frame energy of the spectra $E_{p}$ and the emitted isotropic energy $E_{iso}$, and the Ghirlanda  relation $E_{\gamma}-E_{p}$, relating $E_{p}$ with $E_{\gamma}=E_{iso} (1 - \cos{\theta_{jet}})$. The latter correlation is the tightest of the GRBs calibration relations. However, to be included in this relation the GRB afterglow must have an observed jet break in its light curve, thus only a fraction of the observed events can contribute for establishing this relation. Therefore, the Amati relation is more statistically reliable as it can be calibrated with a greater number of events.

We shall apply here Amati's power law expression \citep{Amati:2002,Amati:2008}, relating $E_p=E_{p, obs}(1+z)$ with the isotropic equivalent radiated energy $E_{iso}$, defined as
\begin{equation}
E_{iso}=4\pi d^2_L S_{bolo} (1+z)^{-1},
\label{Eq:Eiso}
\end{equation}
where $S_{bolo}$ is the bolometric fluence of gamma rays in the GRB at a redshift $z$ and $d_L$ is the luminosity distance to it. Similarly to \citep{Schaefer:2007}, we will rewrite the Amati relation as
\begin{equation}
\log \frac{E_{iso}}{\mathrm{erg}}=\lambda + b\log \frac{E_{p}}{300\mathrm{keV}}.
\label{Eq:Amati}
\end{equation}

GRBs can be considered as distances indicators if they can be calibrated at low redshifts. However, a few data are available at low redshifts. To mend this lack of data most calibrations take for granted a particular cosmological model. In order to avoid this circularity problem here we follow the method proposed in \citep{Liang:2008}, which consists in using SNe Ia as a distance ladder to calibrate the GRBs. Note that with the well-known relation
\begin{equation}
 \mu= 5\log \frac{d_L}{\mathrm{Mpc}}+25,
\label{Eq:mu}
\end{equation}
one can convert the distance moduli $\mu$ into luminosity distances $d_L$ (in units of Mpc).

Since the distance moduli for the SNe Ia are known, a cubic interpolation is performed to determine the parameters $\lambda$ and $b$ in the Amati relation for the low-redshift GRBs ($z < 1.4$). Then, the distance moduli for the high-$z$ GRBs are obtained and they can be used as standard candles without the circularity problem. It is worth mentioning that the calibration of GRBs is still a quite controversial subject. Indeed, even the Amati relation has been contested by some authors (see, for instance, \citep{Collazzi:2011}, where it is argued that the Amati relation could be an artifact of selection effects in both the burst population and the detector). With this cautionary remark, we proceed with the calibration.

We perform the cubic interpolation by two methods: splines and Hermite. A spline is a polynomial between each pair of table points, but one whose coefficients are determined ``slightly" nonlocally. The nonlocality is designed to guarantee global smoothness in the interpolated function up to some order of derivatives. The cubic spline interpolation produces an interpolated function that is continuous through the second derivative \citep{NumericalRecipes}. On the other side, Hermite interpolation is also a method of interpolating data-points as a polynomial function. The generated Hermite polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences. Hermite interpolation (first-derivative continuous) requires four points so that it can achieve a higher degree of continuity \citep{Stoer:1993}.

So, by using at low redshifts the $288$ SDSS SNe Ia data \citep{Kessler:2009}, we derived the distance moduli for the $50$ low-redshift ($z<1.4$) GRBs by cubic-splines and cubic-Hermite interpolations. From Eq. (\ref{Eq:Eiso}) with the corresponding $S_{bolo}$ and luminosity distances $d_L$, we then derived $E_{iso}$ for these $50$ GRBs. Furthermore, with the corresponding $E_{p}$ we found the best fit for the Amati relation, Eq.  (\ref{Eq:Amati}):
\begin{equation}
\lambda=52.8076 \pm 0.0683, \quad b=1.6909 \pm 0.1156,
\label{Eq:CubicSplines}
\end{equation}
for the cubic-splines interpolation, and
\begin{equation}
\lambda=52.8960 \pm 0.0874, \quad b=1.7851 \pm 0.1480,
\label{Eq:CubicHermite}
\end{equation}
for the cubic-Hermite one.
Extrapolating these calibrated Amati relations to the $59$ high-redshift ($z>1.4$) GRBs given in Table 2 of \citep{WH}, the distance moduli $\mu$ are obtained. Only these new 59 datapoints are then used in order to test the cosmological model \footnote{Both samples are available under request.}.

\section{Results and Discussion}

In the last section, assuming the SDSS (MLCS2k2) SNe Ia sample as our calibration source at low redshifts, we have produced two new GRB data sets based on the Hermite and the spline interpolation methods. Before proceeding with a statistical analysis with our new samples we consider three previous GRB data sets: i) the NPZ sample \citep{Liang:2011} with $27$ GRBs at $z<1.4$ and the remaining $42$ objects in the range $1.4<z<6.6$; ii) the Hymnium sample \citep{WH}, with $50$ low-redshift data points and $59$ GRBs in the range $1.4<z<8.2$; iii) the MB sample \citep{MB} that makes use of the same GRB compilation as the Hymnium data, but using a different interpolation for the $50$ low-redshift GRBs, leading to a different extrapolated Hubble diagram at high $z$. These samples share the same SNe Ia calibrating source, the Union2 data. A direct comparison of the $\Lambda$(t)CDM model with such samples retains information from the Union2 data set and, consequently, an imprint of the $\Lambda$CDM model. Nevertheless, for the sake of comparison, we will perform a statistical analysis with all the above GRB samples.

Assuming a set of free parameters ($\Omega_{m0}, h$), where $H_0=100h~$Km s$^{-1}$Mpc$^{-1}$, the agreement between theory and observation is measured by minimizing the $\chi^2$,
\begin{equation}
\chi^2\equiv\chi^2(\Omega_{m0}, h)=\sum^{N}_{i=1}\frac{\left(\mu^{th}_i-\mu^{obs}_i\right)^2}{\sigma^2_i},
\end{equation}
where $\mu^{th}$ and $\mu^{obs}$ are the theoretical and observed values of the distance moduli, respectively, and $\sigma$ means the error for each one of the $N$ data points. The $68.3\%$, $95.4\%$ and $99.73\%$ confidence levels (C.L.) are determined by $\Delta\chi^2\equiv \chi^2 - \chi^2_{min} \leq 2.3$, $6.17$ and $11.8$, respectively.

Fig. \ref{fig1} shows the constraints on the free parameters ($\Omega_{m0},h$) for the $\Lambda$(t)CDM model using the above mentioned NPZ (left panel), Hymnium (central panel) and MB (right panel) samples. The contours for the Union2 SNe Ia sample are also shown. In spite of the large dispersion observed in the GRB samples one can infer from their best fits (NPZ: ($0.463, 0.727$), Hymnium: ($0.422, 0.708$), MB: ($0.439, 0.757$)) that they agree with larger values for the amount of dark matter and the Hubble constant as compared with the Union2 best fit ($\Omega_{m0}=0.337, h=0.697$). A similar analysis for the $\Lambda$CDM model is displayed in Fig. \ref{fig2}. Note that in this case very large values for the Hubble constant are predicted. We summarize the results of Figs. \ref{fig1} and \ref{fig2} in Table 2.

\begin{figure*}
\vspace{.2in}
\centerline{\psfig{figure=NPZ.eps,width=2.2truein,height=2.2truein}
\psfig{figure=Hymnium.eps,width=2.2truein,height=2.2truein}
\psfig{figure=MB.eps,width=2.2truein,height=2.2truein}}
\caption{Constraints on the free parameters of the $\Lambda$(t)CDM model. The black dot is the best fit for each GRB sample. All GRB samples have been calibrated with the Union2 SNe Ia data set (closed elliptical contours).}
\label{fig1}
\end{figure*}

\begin{figure*}
\vspace{.2in}
\centerline{\psfig{figure=NPZLambda.eps,width=2.2truein,height=2.2truein}
\psfig{figure=HymniumLambda.eps,width=2.2truein,height=2.2truein}
\psfig{figure=MBLambda.eps,width=2.2truein,height=2.2truein}}
\caption{Constraints on the free parameters of the $\Lambda$CDM model. The black dot is the best fit for each GRB sample. All GRB samples have been calibrated with the Union2 SNe Ia data set (closed elliptical contours).}
\label{fig2}
\end{figure*}

\begin{figure*}
\vspace{.2in}
\centerline{
\psfig{figure=GRBHermiteSplineLtCDM.eps,width=2.9truein,height=2.75truein}\hspace{.1in}
\psfig{figure=GRBHermiteSplineLCDM.eps,width=2.9truein,height=2.75truein}}
\caption{Contraints on the free parameters of the $\Lambda$(t)CDM ($\Lambda$CDM) model (left (right) panel). The $59$ GRBs at $z>1.4$ were calibrated with the SDSS (MLCS2k2) SNe Ia data set using cubic interpolations with hermite and spline methods. The GRB contours correspond to the $68.3\%$ C.L. (with best fit at the black dot) for each calibration. The elipses correspond to $68.3\%, 95.4\%$ and $99.73\%$ C.L. constraints from the SDSS SNe Ia sample.}
\label{fig3}
\end{figure*}

\begin{table*}
\begin{center}
\caption{Limits to $\Omega_{m0}$ and $h$ obtained from Figs. \ref{fig1} and \ref{fig2}.}
\begin{tabular}{r@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}}
\hline \hline \\
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{$\Lambda(t)$CDM } & \multicolumn{3}{c}{$\Lambda$CDM } \\
\multicolumn{1}{c}{Test}&
\multicolumn{1}{c}{h}&
\multicolumn{1}{c}{$\Omega_{m0}$}&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}&
\multicolumn{1}{c}{h}&
\multicolumn{1}{c}{$\Omega_{m0}$ }&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}\\ \hline \\
Union2 (SALT2)......&0.697&0.338&0.978&0.700&0.269& 0.975 \\
NPZ......&0.728&0.463&0.868&0.814& 0.219&0.868\\
Hymnium......&0.708&0.422&0.405&0.776&0.206&0.405\\
MB......&0.757&0.440&0.630&0.834&0.215&0.631\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}

With the new GRB samples of the last section we obtain the results shown in Fig. \ref{fig3} for the $\Lambda$(t)CDM (left panel) and $\Lambda$CDM (right panel) models. The $68.3\%$ C.L. (with best fit at the black dot) are shown for each calibration. Solid (dashed) open contours correspond to the cubic Hermite (spline) interpolation method. Table 3 shows the results corresponding to Fig. \ref{fig3}. Again, the large dispersion does not allow a definite conclusion, except that both models are in accordance with data within $1\sigma$ level. However, a look at the best-fit points is suggestive, specially if combined with the better posed results discussed in the Introduction. In the $\Lambda$CDM case the matter density parameter is in agreement with that resulted from the LSS analysis and in conflict with the larger value obtained with SDSS alone. If we consider instead the scenario with particle creation, we have again a very good concordance between the GRB and SDSS predictions for the present matter density, with both, Hermite and spline methods. If we also take into consideration the results for the Hubble constant, the concordance is particularly good with the Hermite interpolation. In none of the models the best-fit values for the matter density change when $h$ is marginalized.

\begin{table*}
\begin{center}
\caption{Limits to $\Omega_{m0}$ and $h$ obtained from Fig. \ref{fig3}.}
\begin{tabular}{r@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}c@{~~}}
\hline \hline \\
\multicolumn{1}{c}{ } & \multicolumn{3}{c}{$\Lambda(t)$CDM } & \multicolumn{3}{c}{$\Lambda$CDM } \\
\multicolumn{1}{c}{Test}&
\multicolumn{1}{c}{h}&
\multicolumn{1}{c}{$\Omega_{m0}$}&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}&
\multicolumn{1}{c}{h}&
\multicolumn{1}{c}{$\Omega_{m0}$ }&
\multicolumn{1}{c}{$\chi^2_{min}/\nu$}\\ \hline \\
SDSS......&0.634&0.484&0.841&0.639&0.400&0.839  \\
GRB (Hermite)......&0.635&0.405&0.495&0.700&0.192&0.496\\
GRB (Spline)......&0.713&0.427&0.606&0.785&0.206&0.607\\
\hline \hline
\end{tabular}
\end{center}
\end{table*}

\begin{figure*}
\vspace{.2in}
\centerline{
\psfig{figure=CCD.eps,width=2.9truein,height=2.75truein}}
\caption{Contraints on the free parameters of the $\Lambda$(t)CDM (solid lines) and the $\Lambda$CDM (dashed lines) models. In this plot we have considered the CCD sample. The GRB contours correspond to $68.3\%, 95.4\%$ and $99.73\%$ C.L. (with best fit at the gray-filled and black dots, respectively).}
\label{fig4}
\end{figure*}

\begin{figure*}
\vspace{.2in}
\centerline{
\psfig{figure=OmegaMatterCCD.eps,width=2.9truein,height=2.75truein}}
\caption{Probability distribution function for the matter density parameter $\Omega_{m}$ in the $\Lambda$(t)CDM (solid line) and the $\Lambda$CDM (dashed line) models. In this plot we have considered the CCD sample with prior $h = 0.72 \pm 0.08$.}
\label{fig5}
\end{figure*}

In our numerical analysis we have also included a distinct sample compiled by \cite{GRB2} (we will call it CCD sample from now on). This sample, being made of $69$ GRBs in the range $0.17<z<6.6$, is very particular because it makes use of a ``local regression calibration'' method (see \citep{GRB2} for details). The results are shown in Fig. \ref{fig4}. The high $\Omega_m$ values obtained here are in complete disagreement with the expected ones. This points out that the method and results of \cite{GRB2} are strongly dependent on the adopted priors. This is clear when we use, for the Hubble constant, the (rather conservative) prior $H_0 = 72 \pm 8$ km/s.Mpc \citep{Freedman}. With this prior, the probability distribution function for the matter density parameter $\Omega_{m}$ is given in Fig. \ref{fig5}, now in better accordance with our previous results.

However, \cite{GRB2} provides an interesting discussion about future contraints obtained with forthcoming GRB data, as expected by current missions like {\it Swift} and {\it Fermi}. Such analysis includes the computation of the Fisher matrix, leading to a prediction about the future error in estimating cosmological parameters. Concerning the comparison between $\Lambda(t)$CDM and $\Lambda$CDM models, the most important parameter is $\Omega_m$. From the results of \cite{GRB2}, the complementarity of future GRB and SN samples is able to reduce the uncertainty on the parameter $\Omega_m$ to $\sigma(\Omega_m)=0.019$ (for the $\Lambda$CDM model). With such an uncertainty level, $\delta\Omega_m \approx 0.02$, we will be surely able to place strong constraints on the dividing line between $\Lambda(t)$ and $\Lambda$ cosmologies.

In conclusion, we can see that the creation of particles from vacuum, something allowed by and generally expected from quantum field theories in curved spacetimes, is not only in agreement with current cosmological observations but is indeed favored by them when model-independent datasets are used. In the present model, where pressureless dark matter is produced with a constant rate, the vacuum density scales linearly with $H$ in order to conserve the total energy. Interestingly enough, this time-dependence for $\Lambda$ is precisely that derived when we estimate the renormalized density of the QCD vacuum condensate in the FLRW spacetime (see, for instance, \citep{PLB} and references therein). In this case the creation rate is $\Gamma \approx m^3$, where $m \approx 150$ MeV is the energy scale of the QCD chiral phase transition. This value for $\Gamma$ leads to a remarkable coincidence between $\Lambda$ and the observed dark energy density. Furthermore, as discussed in \cite{PLB}, the energy flux from vacuum to the matter sector also alleviates the cosmic coincidence problem. We think that all these theoretical and observational positive results are enough reasons to consider that scenario and to test it against a wider set of observations.

\section*{Acknowledgements}
This work was supported by CNPq (Brazil). HV acknowledges support from the DFG within the Research Training Group 1620 ``Models of Gravity''. AM acknowledges financial support from Conacyt-M\'exico, through a PhD grant. We are thankful to C. Pigozzo for his help with the SDSS statistical analysis.


text:
\section{Introduction}
\label{sec:intro}

The purpose of this article is to give a complete classification of the connected components of sub-Shimura varieties of the Shimura varieties associated to quadratic forms of signature $(2,n)$.
These sub-Shimura varieties are important algebraic cycles in the Chow ring and often have important applications. For example, Heegner cycles have an important role in the theory of Borcherds lifts (See for example \cite{Borcherd_inv}, \cite{Brunier_BP}, or \cite{KudlaCycles}) and the various reinterpretations of the theory of singular moduli (See for example \cite{grosszagier}, \cite{BruinierYang1}, \cite{bruinierkudlayang}, \cite{kudlarapoport} \cite{AndreattaGorenetal}).

This sort of problem has been looked at for other Shimura varieties. In particular, the work of \cite{SatakeHolomorphic} and \cite{AbdulaliTwists} looked at the sub-Shimura varieties of the Shimura varieties associated to Symplectic groups.

The main result of this paper is Theorem \ref{thm:mainresult} which completely characterizes the possible sub-Shimura varieties of the Shimura varieties attached to orthogonal groups of signature $(2,n)$ up to considerations of component groups.

This paper is organized as follows:
\begin{itemize}
\item In Section \ref{sec:sym} we use the theory of Hermitian symmetric spaces and real lie groups to deduce information about the possible sub-Shimura varieties.
\item In Section \ref{sec:Shim} we apply the theory of algebraic groups over $\mathbb{Q}$ to obtain further information about the groups associated to the sub-Shimura varieties and prove the main theorem.
\item In Section \ref{sec:Res} we state a natural generalization of our result to certain base changes from totally real fields.
\item In Section \ref{sec:Conc} we briefly discuss several problems that this work leaves open.
\end{itemize}

\section{Hermitian Symmetric Spaces}
\label{sec:sym}

In this section we will be working over $\mathbb{R}$ and $\mathbb{C}$ using the connection between Shimura varieties and Hermitian symmetric domains. We refer the reader to \cite{Helgason} and \cite{Milne_shimura} for the relevant background on the theory of Hermitian symmetric spaces and the connection to Shimura varieties. The key feature of this connection we shall use is that maps of Shimura varieties induce maps of Hermitian symmetric domains and that in both cases, the maps come from a map between associated algebraic groups.

The key feature of the case of orthogonal Hermitian symmetric spaces we shall use is that the relevant groups have real rank at most $2$ and that the vast majority of irreducible representations of simple non-compact real lie groups cannot land in an orthogonal group with such small real rank.

Simple Hermitian symmetric spaces of the non-compact type are completely classified.
From the following classification we can already rule out sub-Shimura varieties of most types based simply on the real rank of the associated group. Ultimately we hope to determine for which $H$ below there can exist a map $\rho:H\rightarrow \SO(2,n)$ which induces a non-trivial map of Hermitian symmetric spaces.
\begin{center}
List of simple Hermitian symmetric spaces of non-compact type

\setlength\tabcolsep{2.5pt}.
\begin{tabular}{ | c | c | c | c }
\hline
	Group & Conditions & Real Rank  \\
\hline
     $\SU(p,q)$ & $1 \leq p \leq q$  &  $ p$ \\
     $\SO(2,n)$ & $n > 4 $  &   $2$ \\
     $\SO^\ast(n)$ & $n > 4$ &  $\floor{n/2}$ \\
     $\SP(2n)$&   &  $n \ge 1$ \\
     $E_6$ & & $2$  \\
     $E_7$ &  & $3$ \\
\hline
\end{tabular}
\end{center}

\begin{rmk}
We have the following remarks about the classification:
\begin{itemize}
\item The group $\SO^\ast(1)$ is isomorphic to $\SO(2)$ which is compact.
\item $\SO^\ast(2)$ is not simple, the spin cover is associated to the Shimura curve over a totally real quadratic extension which is definite at one place and indefinite at the other.
\item The spin cover of $\SO^\ast(3)$ is $\SU(1,3)$, and these give the same symmetric spaces.
\item The spin cover of $\SO^\ast(4)$ is the same as that of $\SO(2,6)$, and gives the same symmetric space.

\item We have that $\SU(2,2) \sim \SO(2,4)$ give the same symmetric space.
\item We have that $\SP(4) \sim \SO(2,3)$ give the same symmetric space.
\item We have that $\SO(2,2) \sim \SP(2)\times \SP(2)$ is not simple, and give the same symmetric space.
\item We have that $\SP(2) = \SU(1,1) \sim \SO(2,1)$ all give the same symmetric space.
\end{itemize}
\end{rmk}

The following Lemma gives us a stronger criterion than rank to characterize which real orthogonal groups a given representation can factor through.
\begin{lemma}
Let $\chi:\mathbb{G}_m \rightarrow H$ be a cocharacter, let $\rho : H\hookrightarrow \Gl(V)$ be any representation.

Let $V^\infty$ be the subspace such that for $v\in V^\infty$ we have
\[ \lim_{t\rightarrow \infty} \rho \circ \chi(t) (v)= 0. \]
If there exists a quadratic form on $V$ whose orthogonal group contains the image of $H$ its rank is at least $\lambda(\rho,\chi) = \dim(V^{\infty})$
\end{lemma}

The following Lie theoretic description of the quantity $\lambda(\rho,\chi)$ will provide a simpler method for establishing lower bounds on it.
\begin{prop}\label{prop:LieT}
Let $H$ be the a real form, with real rank at least one, of a simple lie group with complex root datum $(X,X^\vee,\Phi,\Phi^\vee)$.
Let $\tilde{\chi}$ be the element of the cocharacter lattice $X^\vee$ associated to map $\chi:\mathbb{G}_m \rightarrow H$ defined over $\mathbb{R}$.
Let $\rho$ be any representation of $H$ defined over $\mathbb{R}$.

Then $\lambda(\rho,\chi)$ is precisely the number of weights $\alpha$ (counted with multiplicity) appearing in $\rho$ for which $(\alpha,\chi) > 0$.
\end{prop}

Denote by $\lambda(\rho)$ the maximum value of $\lambda(\rho,\chi)$ as we run over all $\chi$.

In light of Proposition \ref{prop:LieT}, we have the following trivial lower bound on $\lambda(\rho)$.
\begin{prop}
If the group $H$ is simple then the quantity $\lambda(\rho)$ is bounded below by the number of non-trivial $\mathbb{C}$-irreducible factors in the representation $\rho$ multiplied by the real rank of the group.
\end{prop}

From the above, we immediately obtain also the strictly stronger lower bounds:
\begin{cor}
If $H' \subset H$ are simple, then the quantity $\lambda(\rho)$ is bounded below by the number of non-trivial $\mathbb{C}$-irreducible factors in the representation $\rho|_{H'}$ times the real rank of $H'$.
\end{cor}

We shall now use the above Corollary to rule out most representations of the groups listed above.

The following propositions follow immediately from the well known classification of representations of special unitary groups as well as the well known branching rules, see for example \cite{FH_rep}.
\begin{prop}\label{prop:branchSU2}
Consider the group $H=\SU(2,q)$, where $q\ge 2$ and the subgroup $H'=\SU(2,q-1)$
.
For every representation of $H$ except the standard representation, $\rho|_{H'}$ has at least two non-trivial $\mathbb{R}$-irreducible factors.
\end{prop}

\begin{prop}\label{prop:standardSU2}
the standard representation for $\SU(p,q)$ over $\mathbb{R}$ is reducible over $\mathbb{C}$ and has two $\mathbb{C}$-irreducible factors.
\end{prop}

\begin{cor}
The only non-trivial maps from a group $\SU(2,q)$ to a group $\SO(2,n)$ come from the isogeny $\SU(2,2)\sim \SO(2,4)$.
\end{cor}
\begin{proof}
Proposition \ref{prop:branchSU2} immediately rules out the existence of maps $\SU(2,q) \rightarrow \SO(2,n)$ not arising from the standard representation of $\SU(2,q)$ for $q>2$. (Note that for $q=2$, the rank of $\SU(2,1)$ is only $1$ and hence this is not an obstruction).

Proposition \ref{prop:standardSU2} immediately rules out the existence of maps $\SU(2,q) \rightarrow \SO(2,n)$ which do arise from the standard representation of $\SU(2,q)$ for all $q\ge 2$.

A simple check shows that $\SU(2,2)$ has two representations whose restriction to $\SU(2,1)$ is the standard representation, the second is the isogeny $\SU(2,2)\sim \SO(2,4)$ and that this is the only non-trivial representation of $\SU(2,2)$ landing in an $\SO(2,n)$ group.
\end{proof}

For simplicity we will consider $\SU(2,2)$ via its isogenous form $\SO(2,4)$.

The following proposition is again an immediate consequence of the classification of representations of special unitary groups and their branching rules.
\begin{prop}\label{prop:branchSU1}
Consider $H=\SU(1,q)$ where $q\ge 2$ and $H'=\SU(1,1)$.
For every representation of $H$ except the standard representation, $\rho|_{H'}$ has at least one copy of the standard representation and one other non-trivial representation, hence at least three $\mathbb{C}$-irreducible factors.
\end{prop}

The following corollary is an immediate consequence.
\begin{cor}
The only non-trivial maps from a group $\SU(1,q)$ to a group $\SO(2,n)$ for $q\ge 2$ come from the standard representation of $\SU(1,q)$.
\end{cor}

The following proposition is again an immediate consequence of the classification of representations of special orthogonal groups and their branching rules.
\begin{prop}
\begin{itemize}
\item
Consider $H=\SO(2,2n+1)$ with $n\ge2$ and $H'=\SO(2,2n)$.
For every representation other than the standard representation $\rho|_{H'}$ has at least two $\mathbb{R}$-irreducible factors.
\item
Consider $H=\SO(2,2n+2)$ with $n\ge2$ and $H'=\SO(2,2n)$.
For every representation other than the standard representation $\rho|_{H'}$ has at least two $\mathbb{R}$-irreducible factors.
\end{itemize}
Note that $\SO(2,2)$ has a second representation into $\SO(2,4)$ via the map $\SO(2,2) \sim \SO(1,2)\times\SO(1,2)$. The inclusion $\SO(1,2)\times\SO(1,2) \hookrightarrow \SO(2,4)$ does not give a map of symmetric spaces.
\end{prop}
The following corollary is an immediate consequence.
\begin{cor}
The only non-trivial maps from a group $\SO(2,\ell)$ to a group $\SO(2,n)$ inducing a map of symmetric spaces for $\ell > 1$ come from the standard representation of $\SO(2,\ell)$.
\end{cor}

The following follows from an explicit check on the representations of $\Sl_2$.
\begin{prop}
The group $\SU(1,1) = \Sl(2) \sim \SO(2,1)$ admits three maps into orthogonal groups of rank $2$, namely:
\begin{itemize}
\item The inclusion of $\SU(1,1)$ into $\SO(2,2)$.
\item The symmetric squares representation to $\SO(2,1)$.
\item Two copies of the symmetric squares representation to $\SO(2,1)\times\SO(2,1) \hookrightarrow  \SO(2,4)$; this final map does not give a map of symmetric spaces,
\end{itemize}
\end{prop}

The following is a consequence of work of Satake (see \cite{SatakeHolomorphic}) which gives a classification of possible sub-symmetric spaces for $\SP(2n)$.
\begin{prop}
There is no map from the Hermitian symmetric space associated to $E_6$ into any Hermitian symmetric space associated to $\SP(2n)$ for any $n$.

There is a map from the Hermitian symmetric space for $\SO(2,\ell)$ into one associated to $\SP(2n)$ for some $n$.

It follows that there is no map from the Hermitian symmetric space associated to $E_6$ into any Hermitian symmetric space associated to $\SO(2,\ell)$.
\end{prop}

The various lemmas and propositions above allow us to conclude:
\begin{lemma}
We can characterize those simple non-compact Hermitian symmetric spaces which embed into the Hermitian symmetric space associated to an orthogonal group of type $(2,n)$ as follows, they are up to isogeny:
\begin{itemize}
\item $\SU(1,q)$ or $\UU(1,q)$ where the map is given by the standard representation.

Note that the map $\UU(1,q) \rightarrow \UU(1,q) \times \SO(2)$ is up to isogeny of symmetric spaces equivalent to considering $\SU(1,q)\times \SO(2)$ or $\UU(1,q)\times \SO(2)$ and we have chosen to ignore compact factors.
\item $\SO(2,\ell)$ where the map is given by the standard representation.
\end{itemize}

Moreover, the only possible product of simple non-compact type Hermitian symmetric spaces which can embed into such a Hermitian symmetric space come from the natural inclusion of $SO(2,2)$ into $SO(2,n)$.
\end{lemma}

\begin{thm}
If $H\rightarrow \SO(2,n)$ gives rise to a map of Hermitian symmetric spaces of the non-compact type then up to isogeny:
\[ H \sim (H^{nc} \times H^{c}) \]
where  $H^c$ is compact, $H^{nc}$ is non-compact, and, up to isogeny, $H^{nc}$ is one of:
\[ \SO(2,\ell) \qquad \text{or} \qquad \SU(1,\ell) \qquad \text{or} \qquad  \UU(1,\ell). \]

Moreover, the map respects an orthogonal decomposition of the vector space into $V^{nc} \perp V^c$ where $V^{nc}$ has signature $(2,r)$ and $V^c$ has signature $(0,s)$ with $s+r=n$.
\end{thm}

\section{Shimura Varieties}
\label{sec:Shim}

In this section we will exploit the properties of algebraic groups over $\mathbb{Q}$ to obtain conditions on the possible $\mathbb{Q}$-structure of the groups whose Shimura varieties will admit maps into the Shimura varieties attached to orthogonal groups of signature $(2,n)$.

The key idea of this section is the following.
If $V$ is the vector space on which $\SO(2,n)$ has its standard representation we associate to any inclusion $\rho: H\rightarrow \SO(2,n)$ a sub-algebra of $\End(V)$.
We shall use in this section the connection between algebras with involution and algebraic groups to conclude that $\rho$ factors through a certain class of subgroup.
We finally exploit the results of the previous section to conclude that $\rho$ surjects onto our constructed group.

\begin{nota}
Let $(V,q)$ be a quadratic space of signature $(2,n)$ over $\mathbb{Q}$.

Let $\rho: H\hookrightarrow \SO(V,q)$ be an inclusion of groups giving rise to a map of Shimura varieties.

Denote by $V^\rho$ the subspace of $\rho(H)$-fixed vectors.
\end{nota}

\begin{rmk}
We are assuming that this map is an inclusion.
This both limits the component group of the Shimura variety associated to $H$ but also has some impact on existence of compact $\mathbb{Q}$-simple factors.
\end{rmk}

The following lemma gives a useful first step in the reduction.
\begin{lemma}
There exists a decomposition of $V=V^\rho \oplus (V^\rho)^\perp$ where $V^\rho$ denotes the $\rho$-fixed vectors of $V$.
Moreover, $V^\rho$ has signature $(0,m)$ for some $m$ and the map $\rho$ factors through:
\[ H \rightarrow \SO( (V^\rho)^\perp ) \rightarrow \SO(V). \]
\end{lemma}
From the above, we may without any loss of generality suppose that $V^\rho = \{ 0 \}$ and work inside the Heegner cycle associated to $V^\rho$.

\begin{cons}
Denote by $A$ the algebra generated by $\rho(H(\mathbb{Q}))$ inside $\End(V)$, denote by $E$ the center of this algebra.

The adjoint involution $\sigma$ associated  to $q$  on $\End(V)$ stabilizes both $A$ and $E$ and hence these are both algebras with involution.

Let $F=E^\sigma$ be the sub-algebra of $\sigma$-fixed elements and $B$ be the centralizer of $F$. The algebra $B$ is again an algebra with involution, its center is $F$.

Decompose $F=\oplus F_i$ into a product of fields.
The idempotents for the $F_i$ decompose $V = \oplus V_i$, $E=\oplus E_i$, $A=\oplus A_i$ and $B=\oplus B_i$.

Then each $V_i$ is an $F_i$, $E_i$, $A_i$ and $B_i$-module.
Moreover, we have that $B_i = \End_{F_i}(V_i)$ is an algebra with involution and hence there exists an $F_i$-valued quadratic form $q_i$ on $V_i$ inducing the involution $\sigma|_{B_i}$.
By uniqueness, and polarization we can rescale $q_i$ so that $q|_{V_i} = \Tr_{F_i/\mathbb{Q}}(q_i)$.

Packaging these together we find:
 \[ q = \Tr_{F/\mathbb{Q}}(\oplus_i q_i) \]
and that the inclusion of $H$ into $SO(V)$ factors through the inclusion:
\[ \Res_{F/\mathbb{Q}}( \times_i \Orth(q_i)) \rightarrow \times_i \Orth( \Tr_{F_i/\mathbb{Q}}(q_i) ). \]
\end{cons}

The following claim can be checked after base change to $\mathbb{R}$.
\begin{claim}
The algebra $F$ must be totally real, and $F$ has exactly one factor for which $\Tr_{F_i/\mathbb{Q}}(q_i)$ is indefinite.
\end{claim}
\begin{proof}
Notice that the inclusion
\[ \times_i \SO( \Tr_{F_i/\mathbb{Q}}(q_i) )  \rightarrow \SO(q) \]
gives a map of symmetric spaces through which the inclusion of the Hermitian symmetric space for $H$ must factor.
It follows that each map $H(\mathbb{R}) \rightarrow  \Orth( \Tr_{F_i/\mathbb{Q}}(q_i) )(\mathbb{R})$ induces a map of symmetric spaces.

Thus, by the results of the previous section we can conclude that $H(\mathbb{R})$ has non-compact image in at most one of $\Orth( \Tr_{F_i/\mathbb{Q}}(q_i) )(\mathbb{R})$. It follows that there is at most one (and hence exactly one) of $\Orth( \Tr_{F_i/\mathbb{Q}}(q_i) )(\mathbb{R})$ which is non-compact, for if $H(\mathbb{R})$ had compact image in a non-compact $\Orth( \Tr_{F_i/\mathbb{Q}}(q_i) )(\mathbb{R})$ this would induce a refinement of $F$ by virtue of the structure of the maximal compact subgroup of  $\Orth( \Tr_{F_i/\mathbb{Q}}(q_i) )(\mathbb{R})$.

Next we must now rule out the possibility that this non-compact factor comes from a complex factor $F_j$ of $F$. Indeed the only possibilities for:
\[ \Res_{\mathbb{C}/\mathbb{R}}(\Orth(q_j)) \hookrightarrow \Orth( \Tr_{\mathbb{C}/\mathbb{R}}(q_j) ) \]
which give quadratic forms with real rank at most $2$ are when $V_j$ has dimension $1$ or $2$ over $\mathbb{C}$.

Suppose $V_j$ has dimension $1$ over $\mathbb{C}=F_j$, so it has dimension $2$ over $\mathbb{R}$. However, $H(\mathbb{R})$ factors through $\Res_{\mathbb{C}/\mathbb{R}}(\Orth(q_j))(\mathbb{R}) = \{ \pm1 \} $. We can then readily check that regardless of whether the image of $H(\mathbb{R})$ is $\{1\}$ or $\{\pm1\}$ we will not have that $\mathbb{C}$ is contained in the center of the algebra generated of the image.

In the second case, when $V_j$ has complex dimension $2$ over $\mathbb{C}$, so it has dimension $4$ over $\mathbb{R}$, we find that $\Res_{\mathbb{C}/\mathbb{R}}(\Orth(q_j))(\mathbb{R})  \simeq \{ \pm 1 \} \ltimes \mathbb{C}^\times$ and the only maps of Hermitian symmetric domains which factor through this are discrete. It follows that $H(\mathbb{R})$ is compact and thus the image inside $\Orth( \Tr_{F_j/\mathbb{Q}}(q_j) )(\mathbb{R})$ is the torus associated to a CM-algebra.
The algebra it generates inside $\End(V_j)$ can thus not be $\mathbb{C} \times \mathbb{C}$ with the exchange involution.

In particular, we have shown that $F_j\neq \mathbb{C}$, so that there are no complex factors of $F$.
\end{proof}

We now consider two cases for the structure of the image of $H$ in the non-compact factor over $F_j$. The $2$ cases to consider are $E_j=F_j$, or $E_j$ a non-trivial $CM$-extension of $F_j$.

\begin{cons}
If $E_j$ is a non-trivial $CM$-extension of $F_j$.
In this case, the form $q_j$ on $V_j$ is the restriction of an $E_j$-valued Hermitian form on $V_j$, which we will denote $q_j'$.

If $E_j=F_j$, we find that through base change to $\mathbb{R}$, we have:
\[ H(\mathbb{R}) \rightarrow \SO(2,\ell)\times \SO(2+\ell) \times \cdots \times \SO(2+\ell) \]
and by the classification result of the first section, the projection onto the first factor $\SO(2,\ell)$ is surjective.

We conclude that $H = \Res_{F_j/\mathbb{Q}}(\SO(q_j))$.

Similarly, in the second case we find that through base change to $\mathbb{R}$, we have:
\[ H(\mathbb{R}) \rightarrow U(1,\ell)\times U(1+\ell) \times \cdots \times U(1+\ell) \]
and by the classification result of the first section, the projection onto at least the $\SU(1,\ell)$ part of the first factor is surjective.

We conclude that $\Res_{F_j/\mathbb{Q}}(\SU(q_j')) \subset H \subset \Res_{F_j/\mathbb{Q}}(U(q_j'))$.
\end{cons}

We now consider the following modifications to our group $H$.

Consider the simply connected cover $\tilde{H}$ of $H$, decompose $\tilde{H} = \tilde{H}^{nc} \times \tilde{H}^c$ over $\mathbb{Q}$ into a compact part and a non-compact part.
Denote by $H'$ the image in $H$ of $\tilde{H}^{nc}$.

\begin{claim}
Up to consideration of component groups the Shimura varietes associated to $H'$ and $H$ are isomorphic as are theier inclusions into the Shimura variety associated to $\SO(2,n)$.
\end{claim}

Because of the structure of $H'$, it admits no maps into $\mathbb{Q}$-compact factors  $\Res_{\mathbb{C}/\mathbb{R}}(\Orth(q_j)) $.
In particular if we replace $H$ by $H'$, and repeat all of the above constructions, we obtain the same map of Shimura varieties (up to consideration of component groups) and we may assume that $F$ is a field.

The following theorem is now an immediate consequence of the various proceeding claims and propositions.
\begin{thm}\label{thm:mainresult}
Up to consideration of component groups, the Sub-Shimura varieties of $O(2,n)$ type Shimura varieties are of the following sort:

They arise from the restriction of scalars from a totally real field $F$ of
\begin{itemize}
\item the Shimura variety attached to a quadratic form $q'$ which has signature $(2,\ell)$ at one place, and is negative definite at all other places.
\item the Shimura variety attached to a Hermitian form $q'$ associated to $CM$-extension $E$ of $F$ which has signature $(1,\ell)$ at one place of $F$, and is negative definite at all other places.
\end{itemize}
The only condition on the existence of these sub-Shimura varieties is that we can write:
\[ q = \Tr_{F/\mathbb{Q}}(q')  \oplus q^{\rho} \qquad \text{or} \qquad q = \Tr_{E/\mathbb{Q}}(q')  \oplus q^{\rho}. \]
\end{thm}

\begin{rmk}
The isomorphism class of the cycles described above depends on the choice of $F$ (respectively $E$) and $q'$, the isomorphism class of these (together with that of $q$) will determine that of $q^{\rho}$. Such a $q^{\rho}$ need not exist for any given pair $F,q'$ (respectively $E,q'$)  when the codimension is small.
Moreover, the isomorphism class need not determine the rational conjugacy class.

The ideas needed to study these questions are entirely different from those we are using here, and thus the questions are outside the scope of the current paper. We refer the interested reader to several places where work on this has been done.

Heegner cycles (and generalized Heegner cycles) correspond to taking $E=\mathbb{Q}$. Various features of this case are studied in the work of \cite{KudlaCycles}.

Special points correspond to taking $E$ a $CM$-field and $q'$ a one dimensional quadratic space. A more thorough study of this case was undertaken in \cite{Fiori1}. Note that this is the only case we consider where the group $H(\mathbb{R})$ is compact.

The authors Thesis \cite{Fiori_PHD} and the preprint \cite{Fiori3} includes some discussion of how to compute the invariants of these restrictions of trace forms.
\end{rmk}

\begin{rmk}
We note that the Shimura varieties considered above (both the cycles we are considering and the variety in which we are embedding them) are generally disconnected.

We note however, that by composition of maps every sub-Shimura variety of $\GSpin(2,n)$ leads to a sub-Shimura variety for $\SO(2,n)$ and conversely, if
$H\rightarrow \SO(2,n)$ gives a sub-Shimura variety  for $\SO(2,n)$ then $H\times_{\SO(2,n)} \GSpin(2,n) \rightarrow \GSpin(2,n) $ gives a sub-Shimura variety for $\GSpin(2,n)$.
Up to possibly subtle considerations of the component groups, we obtain essentially the same sub-Shimura varieties in each case.
Again, a study of these component groups, though a very interesting question both in terms of the action of the Galois group on them and even the simpler question of the possible sizes, requires entirely different techniques and is outside the scope of this paper.
\end{rmk}

\section{Restrictions of Scalars Shimura Varieties}
\label{sec:Res}

The following two theorems are essentially corollaries of Theorem \ref{thm:mainresult}, the proofs are left to the reader.

\begin{thm}
Let $F$ be a totally real field, let $q$ be an $F$ valued quadratic form which has signature $(2,\ell)$ at one place, and is negative definite at all other places.
The sub-Shimura varieties of the associated Shimura variety are of the following sort:

They arise from the restriction of scalars from a totally real field $F'$ which is an extension of $F$ of
\begin{itemize}
\item the Shimura variety attached to a quadratic form $q'$ which has signature $(2,\ell)$ at one place, and is negative definite at all other places.
\item the Shimura variety attached to a Hermitian form $q'$ associated to $CM$-extension $E$ of $F'$ which has signature $(1,\ell)$ at one place of $F'$, and is negative definite at all other places.
\end{itemize}
The only condition on the existence of these sub-Shimura varieties is that we can write:
\[ q = \Tr_{F'/F}(q')  \oplus q^{\rho} \qquad \text{or} \qquad q = \Tr_{E/F}(q')  \oplus q^{\rho}. \]

The isomorphism class of such a cycle depends on the choice of $F'$ (respectively $E$) and $q'$; the isomorphism class of these with that of $q$ will determine that of $q^{\rho}$. When the codimension is small there is no guarentee that given $q$ and a pair $F',q'$ (respectively $E,q'$) that any form $q^\rho$ can actually be found.
\end{thm}

\begin{thm}
Let $E$ be a CM-field, let $q$ be a Hermitian form which has signature $(1,q)$ at one place, and is negative definite at all other places.
The sub-Shimura varieties of the associated Shimura variety are of the following sort:

They arise from the restriction of scalars from a CM-field $E'$, an extension of $E$, of
 the Shimura variety associated to a Hermitian form $q'$ which has signature $(1,\ell)$ at one place of $E'$, and is negative definite at all other places.

The only condition on the existence of these sub-Shimura varieties is that we can write:
\[ q = \Tr_{E'/E}(q')  \oplus q^{\rho}. \]

The isomorphism class of such a cycle depends on the choice of $E'$ and $q'$; the isomorphism class of these with that of $q$ will determine that of $q^{\rho}$. Given $q$, $E'$ and $q'$ such a form $q^{\rho}$ need not exist when the codimension is small.
\end{thm}

\section{Further Questions}
\label{sec:Conc}

For $\SU(p,q)$ with $p>1$ and $\SO^\ast(n)$ with $n>4$, the idea of using branching rules and rank as was done here are adaptable to this setting. However, the number of representations that must be considered grows considerably; it is likely a more detailed study of the representations would be necessary and it is not immediately clear what results one can even hope for.

A careful inspection of the component group considerations which we have ignored would be worthwhile. The techniques we have applied here are not well suited for studying the effect of non-$\mathbb{Q}$-simple compact factors. In particular all representations of compact groups can map into a sufficiently large orthogonal group.

As discussed in the paper, determining the precise local/global conditions under which a cycle of a given rational isomorphism class will embed into the Shimura variety with a specified rational isomorphism class is worthy of some study, as is the more refined question of describing precisely the rational (or even integral) conjugacy classes of these embeddings.

\section*{Acknowledgments}

The author would like to thank Prof. Eyal Goren for suggesting the problem studied in this paper. They would also like to thank Majid Shahabi for their help in proofreading this article.

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