first part

$\displaystyle S_{2,1}(t)$
$\displaystyle:=\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k\in
E\\\
\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{k}\end{subarray}}}\mathbb{E}\Big{[}(B_{k}^{c})^{2}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{k,\Gamma_{3},\Gamma_{4}}\cdot e^{tW}\Big{]}$
contains all terms in which there is $k=\ell$. In the second part, in which
there is $k\neq\ell$, we decompose
$e^{tW}=e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}$
and we use the Taylor expansion on
$e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}$.
This yields the three terms $\displaystyle S_{2,2}(t)$
$\displaystyle:=\frac{1}{\sigma^{4}}\cdot\;\;\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\end{subarray}}}\;\mathbb{E}\Big{[}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Big{]},$
$\displaystyle S_{2,3}(t)$
$\displaystyle:=\frac{1}{\sigma^{4}}\cdot\frac{t}{\sigma}\cdot\;\;\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;\mathbb{E}\Bigg{[}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}^{c}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]},$
$\displaystyle S_{2,4}(t)$
$\displaystyle:=\frac{1}{\sigma^{4}}\cdot\frac{t^{2}}{2\sigma^{2}}\cdot\;\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E\,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;\mathbb{E}\Bigg{[}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}^{c}B_{\Gamma_{6}}^{c}\cdot
r_{2}\Big{(}\frac{t}{\sigma}\;\;\;\;\sum_{\mathclap{\Gamma\in\mathcal{M}_{\\{k,\ell\\}}}}B_{\Gamma}^{c}\Big{)}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]},$

so that

$\displaystyle\mathbb{E}\Big{[}\Big{|}\sum_{k\in E}Y_{k}U_{k}\Big{|}\cdot
e^{tW}\Big{]}$
$\displaystyle\leq\big{(}S_{2,1}(t)+S_{2,2}(t)+S_{2,3}(t)+S_{2,4}(t)\big{)}^{\frac{1}{2}}\cdot\mathbb{E}[e^{tW}]^{\frac{1}{2}}.$

To derive a bound for $S_{2,1}$, we first note that
$V_{k,\Gamma_{1},\Gamma_{2}}$ and $V_{k,\Gamma_{3},\Gamma_{4}}$ are non-
negative according to (5.4), so that $S_{2,1}(t)\geq 0$. An upper bound for
$S_{2,1}(t)$ is derived by application of the first inequality from Lemma 5.6.
We then use that $V_{k,\Gamma_{1},\Gamma_{2}}$ and
$V_{k,\Gamma_{3},\Gamma_{4}}$ are independent from $B_{k}$ according to (5.4),
and we finish by applying Lemma 5.3, Remark 5.2, and Lemma 5.5. This yields

$\displaystyle S_{2,1}(t)$
$\displaystyle\leq\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k\in
E\\\
\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{k}\end{subarray}}}\mathbb{E}\Big{[}(B_{k}^{c})^{2}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{k,\Gamma_{3},\Gamma_{4}}\Big{]}\cdot
e^{\frac{t}{\sigma}|\mathcal{M}_{\\{k\\}\cup\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}}|}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k\in
E\\\
\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{k}\end{subarray}}}\mathbb{E}\big{[}(B_{k}^{c})^{2}\big{]}\cdot\mathbb{E}\big{[}V_{k,\Gamma_{1},\Gamma_{2}}V_{k,\Gamma_{3},\Gamma_{4}}\big{]}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k\in
E\\\
\Gamma_{1},\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{k}\end{subarray}}}pq\cdot\;\;\;\sum_{\begin{subarray}{c}\\{k\\}\subset\alpha_{2}\subset\Gamma_{2}\\\
\\{k\\}\subset\alpha_{4}\subset\Gamma_{4}\end{subarray}}\frac{\mathbb{E}[B_{\Gamma_{2}\backslash\alpha_{2}}]}{|\Gamma_{2}|\binom{|\Gamma_{2}|-1}{|\alpha_{2}|-1}}\frac{\mathbb{E}[B_{\Gamma_{4}\backslash\alpha_{4}}]}{|\Gamma_{4}|\binom{|\Gamma_{4}|-1}{|\alpha_{4}|-1}}\mathbb{E}\Big{[}B_{(\Gamma_{1}\cup\alpha_{2})\backslash\\{k\\}}B_{(\Gamma_{3}\cup\alpha_{4})\backslash\\{k\\}}\Big{]}e^{\frac{4Dt}{\sigma}}\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}\Gamma_{1}\in\mathcal{M}\\\
\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\Gamma_{1}}\end{subarray}}}\;\;\;\;\;\sum_{\mathclap{k\in\mathcal{M}_{\Gamma_{1}}}}pq\cdot
p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}|-1}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{q}{\sigma^{4}}\cdot
e_{{G_{0}}}\cdot\frac{(v_{{G_{0}}}!)^{3}\cdot
2^{6e_{{G_{0}}}}}{a_{{G_{0}}}^{4}}\cdot\frac{n^{4v_{{G_{0}}}}\cdot
p^{4e_{{G_{0}}}}}{\Psi_{\min}^{3}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\leq
c_{2,1}\cdot\frac{1}{q\Psi_{\min}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$

with $\displaystyle
c_{2,1}:=\frac{2^{2+6e_{{G_{0}}}}\cdot(v_{{G_{0}}}!)^{5}\cdot
e_{{G_{0}}}}{a_{{G_{0}}}^{2}}$. For the last step we again used the lower
bound for $\sigma$ presented in (5.1).

Regarding $S_{2,2}$, we first focus on the random variables inside the
expectation. Let $k,\ell\in E$ with $k\neq\ell$,
$\Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}$ and
$\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}$. $V_{k,\Gamma_{1},\Gamma_{2}}$
and
$e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}$
are independent of $B_{k}^{c}$. If $k\not\in\Gamma_{3}\cup\Gamma_{4}$, then
$V_{\ell,\Gamma_{3},\Gamma_{4}}$ is independent of $B_{k}^{c}$, too. In this
case, we could split the expectation and obtain $\mathbb{E}[B_{k}^{c}]=0$ as a
factor. In the opposite case, in which $k\in\Gamma_{3}\cup\Gamma_{4}$, we only
have to regard those summands of $V_{\ell,\Gamma_{3},\Gamma_{4}}$ that contain
$B_{k}$ as a factor. However, then there is $B_{k}^{c}\cdot B_{k}=q\cdot
B_{k}$. Hence, $S_{2,2}(t)\geq 0$. Further, application of our main technical
Lemmas 5.6, 5.3, 5.5, as well as inequality (5.1) leads to

$\displaystyle S_{2,2}(t)$
$\displaystyle\leq\frac{1}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\end{subarray}}}\;\mathbb{E}\Big{[}q^{2}\cdot
B_{k}B_{\ell}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Big{]}\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\right\\}}\mathbbl{1}_{\left\\{\ell\in\Gamma_{1}\cup\Gamma_{2}\right\\}}$
$\displaystyle\leq\frac{q^{2}}{\sigma^{4}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\end{subarray}}}\;\mathbb{E}\Big{[}B_{k}B_{\ell}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{k,\Gamma_{3},\Gamma_{4}}\Big{]}\cdot
e^{\frac{t}{\sigma}|\mathcal{M}_{\\{k,\ell\\}\cup\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}}|}\cdot\mathbb{E}[e^{tW}]\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\right\\}}\mathbbl{1}_{\left\\{\ell\in\Gamma_{1}\cup\Gamma_{2}\right\\}}$
$\displaystyle\leq\frac{q^{2}}{\sigma^{4}}\cdot
2^{2}\sum_{\mathclap{\begin{subarray}{c}k,\ell\in E,k\neq\ell\\\
\Gamma_{1},\Gamma_{3}\in\mathcal{M}_{k}\cap\mathcal{M}_{\ell}\\\
\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{4}\in\mathcal{M}_{\ell}\end{subarray}}}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}|}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{q^{2}}{\sigma^{4}}\cdot
4\sum_{\mathclap{\begin{subarray}{c}\Gamma_{1}\in\mathcal{M}\\\
\Gamma_{2},\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\Gamma_{1}}\end{subarray}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\sum_{\mathclap{\begin{subarray}{c}k\in\Gamma_{1}\cap\Gamma_{2}\cap\Gamma_{3}\\\
\ell\in\Gamma_{1}\cap\Gamma_{3}\cap\Gamma_{4}\end{subarray}}}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}|}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{q^{2}}{\sigma^{4}}\cdot 4\cdot
e_{{G_{0}}}^{2}\cdot\frac{(v_{{G_{0}}}!)^{3}\cdot
2^{6e_{{G_{0}}}}}{a_{{G_{0}}}^{4}}\cdot\frac{n^{4v_{{G_{0}}}}\cdot
p^{4e_{{G_{0}}}}}{\Psi_{\min}^{3}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\leq
c_{2,2}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$

with $\displaystyle
c_{2,2}:=\frac{2^{4+6e_{{G_{0}}}}\cdot(v_{{G_{0}}}!)^{5}\cdot
e_{{G_{0}}}^{2}}{a_{{G_{0}}}^{2}}$.

Similar calculations yield

$\displaystyle|S_{2,3}(t)|$ $\displaystyle\leq\frac{t}{\sigma^{5}}\cdot
2\sum_{\mathclap{\begin{subarray}{c}k,\ell\in E,k\neq\ell\\\
\Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5}\in\mathcal{M}_{\ell}\end{subarray}}}\;\;\;\Bigg{(}\Bigg{|}\mathbb{E}\Bigg{[}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]}\Bigg{|}$
$\displaystyle\hskip
71.13188pt+\Bigg{|}\mathbb{E}\Bigg{[}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot\mathbb{E}[B_{\Gamma_{5}}]\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]}\Bigg{|}\Bigg{)}$
$\displaystyle\leq\frac{t}{\sigma^{5}}\cdot
2\sum_{\mathclap{\begin{subarray}{c}k,\ell\in E,k\neq\ell\\\
\Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4},\Gamma_{5}\in\mathcal{M}_{\ell}\end{subarray}}}\;\;\;\Bigg{(}\Bigg{|}\mathbb{E}\Bigg{[}q^{2}\cdot
B_{k}B_{\ell}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]}\Bigg{|}\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}\right\\}}$
$\displaystyle\hskip 71.13188pt+\Bigg{|}\mathbb{E}\Bigg{[}q^{2}\cdot
B_{k}B_{\ell}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot\mathbb{E}[B_{\Gamma_{5}}]\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]}\Bigg{|}\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\right\\}}\Bigg{)}$
$\displaystyle\leq\frac{q^{2}t}{\sigma^{5}}\cdot
2\sum_{\mathclap{\begin{subarray}{c}k,\ell\in E,k\neq\ell\\\
\Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4},\Gamma_{5}\in\mathcal{M}_{\ell}\end{subarray}}}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}|}\cdot\Big{(}e^{\frac{5Dt}{\sigma}}\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}\right\\}}+e^{\frac{4Dt}{\sigma}}\cdot\mathbbl{1}_{\left\\{k\in\Gamma_{3}\cup\Gamma_{4}\right\\}}\Big{)}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{q^{2}t}{\sigma^{5}}\cdot 2\cdot
5\sum_{\mathclap{\begin{subarray}{c}\Gamma_{1}\in\mathcal{M}\\\
\Gamma_{2},\Gamma_{3}\in\mathcal{M}_{\Gamma_{1}}\\\
\Gamma_{4},\Gamma_{5}\in\mathcal{M}_{\Gamma_{3}}\end{subarray}}}\;\;\;\;\;\;\;\;\;\;\;\sum_{\mathclap{\begin{subarray}{c}k\in\Gamma_{1}\cap\Gamma_{2}\cap\Gamma_{3}\\\
\ell\in\Gamma_{3}\cap\Gamma_{4}\cap\Gamma_{5}\end{subarray}}}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}|}\cdot
e^{\frac{5Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{10q^{2}t}{\sigma^{5}}\cdot
e_{{G_{0}}}^{2}\cdot\frac{(v_{{G_{0}}}!)^{4}\cdot
2^{10e_{{G_{0}}}}}{a_{{G_{0}}}^{5}}\cdot\frac{n^{5v_{{G_{0}}}}\cdot
p^{5e_{{G_{0}}}}}{\Psi_{\min}^{4}}\cdot
e^{\frac{5Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\leq
c_{2,3}\cdot\frac{t}{\sqrt{q\Psi_{\min}}}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{5Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$

with $\displaystyle c_{2,3}:=\frac{5\cdot
2^{\frac{7}{2}+10e_{{G_{0}}}}\cdot(v_{{G_{0}}}!)^{\frac{13}{2}}\cdot
e_{{G_{0}}}^{2}}{a_{{G_{0}}}^{\frac{5}{2}}}$, and

$\displaystyle|S_{2,4}(t)|$
$\displaystyle\leq\frac{t^{2}}{2\sigma^{6}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E\,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;\mathbb{E}\Bigg{[}\big{|}B_{k}^{c}B_{\ell}^{c}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}^{c}B_{\Gamma_{6}}^{c}\big{|}\cdot e^{\frac{2Dt}{\sigma}}\cdot
e^{\frac{t}{\sigma}\sum_{\Gamma\in\mathcal{M}\backslash\mathcal{M}_{\\{k,\ell\\}}}B_{\Gamma}^{c}}\Bigg{]}$
$\displaystyle\leq\frac{t^{2}}{2\sigma^{6}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E\,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;\mathbb{E}\Bigg{[}\big{|}B_{k}^{c}B_{\ell}^{c}\big{|}\cdot
V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
2B_{\Gamma_{5}}\cdot 2B_{\Gamma_{6}}\Bigg{]}\cdot e^{\frac{2Dt}{\sigma}}\cdot
e^{\frac{6Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{2t^{2}}{\sigma^{6}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E\,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;\mathbb{E}\big{[}\big{|}B_{k}^{c}\big{|}\big{]}\mathbb{E}\big{[}\big{|}B_{\ell}^{c}\big{|}\big{]}\mathbb{E}\Big{[}V_{k,\Gamma_{1},\Gamma_{2}}V_{\ell,\Gamma_{3},\Gamma_{4}}\cdot
B_{\Gamma_{5}}B_{\Gamma_{6}}\Big{|}B_{k}=B_{\ell}=1\Big{]}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{2t^{2}}{\sigma^{6}}\cdot\sum_{\mathclap{\begin{subarray}{c}k,\ell\in
E\,k\neq\ell\\\ \Gamma_{1},\Gamma_{2}\in\mathcal{M}_{k}\\\
\Gamma_{3},\Gamma_{4}\in\mathcal{M}_{\ell}\\\
\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\\{k,\ell\\}}\end{subarray}}}\;2pq\cdot
2pq\cdot
p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}\cup\Gamma_{6}|-2}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{8q^{2}t^{2}}{\sigma^{6}}\cdot\Bigg{(}\;\;2\sum_{\mathclap{\begin{subarray}{c}\Gamma_{1},\Gamma_{3}\in\mathcal{M}\\\
\Gamma_{2},\Gamma_{5}\in\mathcal{M}_{\Gamma_{1}}\\\
\Gamma_{4},\Gamma_{6}\in\mathcal{M}_{\Gamma_{3}}\end{subarray}}}e_{{G_{0}}}^{2}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}\cup\Gamma_{6}|}+2\sum_{\mathclap{\begin{subarray}{c}\Gamma_{1},\Gamma_{3}\in\mathcal{M}\\\
\Gamma_{2},\Gamma_{5},\Gamma_{6}\in\mathcal{M}_{\Gamma_{1}}\\\
\Gamma_{4}\in\mathcal{M}_{\Gamma_{3}}\end{subarray}}}e_{{G_{0}}}^{2}p^{|\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}\cup\Gamma_{5}\cup\Gamma_{6}|}\Bigg{)}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\frac{32q^{2}t^{2}}{\sigma^{6}}\cdot
e_{{G_{0}}}^{2}\cdot\frac{(v_{{G_{0}}}!)^{5}\cdot
2^{15e_{{G_{0}}}}}{a_{{G_{0}}}^{6}}\cdot\frac{n^{6v_{{G_{0}}}}\cdot
p^{6e_{{G_{0}}}}}{\Psi_{\min}^{4}\cdot\min\\{\Psi_{\min},1\\}}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\leq
c_{2,4}\cdot\frac{t^{2}}{q\min\\{\Psi_{\min},1\\}}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$

with $\displaystyle
c_{2,4}:=\frac{2^{8+15e_{{G_{0}}}}\cdot(v_{{G_{0}}}!)^{8}\cdot
e_{{G_{0}}}^{2}}{a_{{G_{0}}}^{3}}$.

Note that $c_{2,4}=\max\\{c_{2,1},c_{2,2},c_{2,3},c_{2,4}\\}$. Bringing
together the bounds for $S_{2,1}$, $S_{2,2}$, $S_{2,3}$, and $S_{2,4}$ results
in

$\displaystyle\mathbb{E}\Big{[}\Big{|}\sum_{k\in E}Y_{k}U_{k}\Big{|}\cdot
e^{tW}\Big{]}$
$\displaystyle\leq\big{(}S_{2,1}(t)+S_{2,2}(t)+S_{2,3}(t)+S_{2,4}(t)\big{)}^{\frac{1}{2}}\cdot\mathbb{E}[e^{tW}]^{\frac{1}{2}}$
$\displaystyle\leq\bigg{(}c_{2,1}\cdot\frac{1}{q\Psi_{\min}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\hskip
10.00002pt+c_{2,2}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\hskip
10.00002pt+c_{2,3}\cdot\frac{t}{\sqrt{q\Psi_{\min}}}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{5Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$ $\displaystyle\hskip
10.00002pt+c_{2,4}\cdot\frac{t^{2}}{q\min\\{\Psi_{\min},1\\}}\cdot\frac{1}{\Psi_{\min}}\cdot
e^{\frac{8Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]\bigg{)}^{\frac{1}{2}}\cdot\mathbb{E}[e^{tW}]^{\frac{1}{2}}$
$\displaystyle\leq\sqrt{c_{2,4}}\cdot\bigg{(}2+\frac{t}{\sqrt{\Psi_{\min}}}+\frac{t^{2}}{\min\\{\Psi_{\min},1\\}}\bigg{)}^{\frac{1}{2}}\frac{1}{\sqrt{q\Psi_{\min}}}\cdot
e^{\frac{4Dt}{\sigma}}\cdot\mathbb{E}[e^{tW}]$
$\displaystyle\leq\gamma_{2}(t)\cdot\mathbb{E}[e^{tW}]$

with

$\displaystyle\gamma_{2}(t):={}$ $\displaystyle
c_{A2}\cdot\bigg{(}1+\frac{t}{\min\\{\sqrt{\Psi_{\min}},1\\}}\bigg{)}\frac{1}{\sqrt{q\Psi_{\min}}}\cdot
e^{\frac{4Dt}{\sigma}},$ (5.6)

where $\displaystyle
c_{A2}:=\sqrt{2c_{2,4}}=\frac{2^{\frac{9}{2}+\frac{15}{2}e_{{G_{0}}}}\cdot(v_{{G_{0}}}!)^{4}\cdot
e_{{G_{0}}}}{a_{{G_{0}}}^{\frac{3}{2}}}=c_{A1}$ for all $t\geq 0$.

Conclusion: We have constructed $\gamma_{1}(t)$ and $\gamma_{2}(t)$ for $t\geq
0$, see (5.5) and (5.6), so that the assumptions (A1’) and (A2’) are
fulfilled. Finally, note that

$\displaystyle\gamma_{1}(t)+\gamma_{2}(t)\leq 2c_{G_{0}}\cdot s(t)$

with $s(t)$ and $c_{G_{0}}$ as defined in the statement of Theorem 1.4.
Theorem 3.2 yields the desired result. ∎

###### Proof of Corollary 1.5.

If we restrict $t$ to be smaller than
$c_{1}\cdot\frac{n^{2}p^{e_{{G_{0}}}}\sqrt{q}}{\sqrt{\Psi_{\min}}}$, where
$c_{1}>0$ is an arbitrary positive number, we can further simplify our
results. Under this restriction, due to $n^{2}p^{e_{{G_{0}}}}\leq\Psi_{\min}$,
there is $t\leq
c_{1}\cdot\frac{n^{2}p^{e_{{G_{0}}}}\sqrt{q}}{\sqrt{\Psi_{\min}}}\leq
c_{1}\cdot\sqrt{q\Psi_{\min}}$ and hence

$\displaystyle\frac{t}{\min\\{\sqrt{\Psi_{\min}},1\\}}$
$\displaystyle\leq\max\Big{\\{}c_{1}\cdot\sqrt{q}\,,\,t\Big{\\}}\leq
c_{1}\cdot\sqrt{q}+t.$

On the other hand, since we are working in the case of $n\geq
4v_{{G_{0}}}^{2}$, we can use explicit bounds for $D$ and $\sigma$ from Lemma
4.2 in [8] to show that
$\frac{D}{\sigma}\leq\hat{c}_{G_{0}}\cdot\frac{\sqrt{\Psi_{\min}}}{n^{2}p^{e_{{G_{0}}}}\sqrt{q}}$,
where $\hat{c}_{G_{0}}=\sqrt{2}\cdot\frac{\sqrt{v_{{G_{0}}}!}\cdot
v_{{G_{0}}}^{2}\cdot e_{{G_{0}}}}{\sqrt{a_{{G_{0}}}}}$ is a constant that only
depends on ${G_{0}}$. This implies that $e^{\frac{5Dt}{\sigma}}\leq
e^{5c_{1}\hat{c}_{G_{0}}}$ for $0\leq t\leq
c_{1}\cdot\frac{n^{2}p^{e_{{G_{0}}}}\sqrt{q}}{\sqrt{\Psi_{\min}}}$.

Additionally, in the case we focus on, there is $e^{c_{G_{0}}\cdot t^{2}\cdot
s(t)}\leq e^{c_{G_{0}}c_{2}}$.

Application of these bounds to the result of Theorem 1.4 leads to

$\displaystyle\bigg{|}\frac{\mathbb{P}(W>t)}{1-\Phi(t)}-1\bigg{|}$
$\displaystyle\leq 50c_{G_{0}}\cdot
e^{c_{G_{0}}c_{2}}\cdot(1+t^{2})\cdot(1+c_{1}\sqrt{q}+t)\cdot\frac{1}{\sqrt{q\Psi_{\min}}}\cdot
e^{5c_{1}\hat{c}_{G_{0}}}$ $\displaystyle\leq 50c_{G_{0}}\cdot
e^{5c_{1}\hat{c}_{G_{0}}+c_{2}c_{G_{0}}}\cdot(1+c_{1}\sqrt{q})\cdot(1+t^{2})\cdot(1+t)\cdot\frac{1}{\sqrt{q\Psi_{\min}}},$

To finish the proof, note that $(1+t^{2})(1+t)\leq 2(1+t^{3})$. ∎

##  6\. Acknowledgement

This research has been supported in part by the German Research Foundation
(DFG) under project number 459731056.

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# Sub-percentage measure of distances to redshift of 0.1 by a new cosmic ruler

Yong Shi,1,2 Yanmei Chen,1,2 Shude Mao,3,4 Qiusheng Gu,1,2 Tao Wang,1,2
Xiaoyang Xia,5 Zhi-Yu Zhang.1,2
1School of Astronomy and Space Science, Nanjing University, Nanjing 210093,
China.
2Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University),
Ministry of Education, Nanjing 210093, China.
3Department of Astronomy, Tsinghua University, Beijing 100084, China.
4National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun
Road, Chaoyang District, Beijing 100012, China.
5Tianjin Astrophysics Center, Tianjin Normal University, Tianjin 300387,
People’s Republic of China.
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Distance-redshift diagrams probe expansion history of the Universe. We show
that the stellar mass-binding energy (massE) relation of galaxies proposed in
our previous study offers a new distance ruler at cosmic scales. By using
elliptical galaxies in the main galaxy sample of the Sloan Digital Sky Survey
Data Release 7, we construct a distance-redshift diagram over the redshift
range from 0.05 to 0.2 with the massE ruler. The best-fit dark energy density
is 0.675$\pm$0.079 for flat $\Lambda$CDM, consistent with those by other
probes. At the median redshift of 0.11, the median distance is estimated to
have a fractional error of 0.34%, much lower than those by supernova (SN) Ia
and baryonic acoustic oscillation (BAO) and even exceeding their future
capability at this redshift. The above low-$z$ measurement is useful for
probing dark energy that dominates at the late Universe. For a flat dark
energy equation of state model (flat $w$CDM), the massE alone constrains $w$
to an error that is only a factor of 2.2, 1.7 and 1.3 times larger than those
by BAO, SN Ia, and cosmic microwave background (CMB), respectively.

###### keywords:

galaxies: general; galaxies: kinematics and dynamics; cosmology: observations;
cosmology: cosmological parameters

††pubyear: 2021††pagerange: Sub-percentage measure of distances to redshift of
0.1 by a new cosmic ruler–A

##  1 Introduction

A distance-redshift diagram is a powerful probe of the cosmic expansion
history. The measurement by E. Hubble and G. Lemaître revealed the expansion
of the Universe (Hubble, 1929; Lemaître, 1927), laying down the foundation of
the big bang theory. In the past decades, two independent distance rulers at
cosmic scales with appreciable redshifts, supernova Ia (SN Ia) and baryonic
acoustic oscillation (BAO), demonstrated unambiguously the acceleration of the
cosmic expansion, implying the existence of dark energy (Riess et al., 1998;
Perlmutter et al., 1999; Eisenstein et al., 2005).

It is essential to have independent rulers to measure the distance vs.
redshift diagram. Physically, different rulers probe the cosmic geometry from
different aspects, thus examining a concordant cosmological model.
Technically, each ruler has “nuisance” parameters that are unrelated to the
cosmological parameters but can introduce systematic errors (e.g. Zuntz et
al., 2015). Joint analysis by combining different rulers can significantly
alleviate the degeneracy among cosmological parameters and improve their
accuracy.

Uncertainties of current distance measurements using SN Ia and BAO are around
a few percent and vary with redshift, mainly limited by the sample size and
survey volume (Ross et al., 2015; Alam et al., 2017; Scolnic et al., 2018). In
the forthcoming 5-10 years, large imaging and spectroscopic surveys will reach
precision from a few sub-percent to one percent per redshift bin of 0.1 (Feng
et al., 2014; DESI Collaboration et al., 2016). In this letter, we show that
the relation between stellar masses and binding energies of galaxies as
proposed in Shi et al. (2021) offers distance rulers at cosmic scales with
significant statistical power. Through the study we will refer this relation
as massE where “mass” stands for stellar masses and “E” for binding energies.

##  2 The massE cosmic ruler

Shi et al. (2021) demonstrated a tight relation between stellar mass and
binding energy of galaxies, which covers nine orders of magnitude in stellar
masses over a large range of galaxy types and redshift. The overall
relationship is described by double power laws with the transition galaxy
stellar mass of $\sim$ 107-108 M⊙. As illustrated in their Fig. 2 and 3, the
dependence on redshift is absent up to $z$ $\sim$ 2.5, so does the dependence
on galaxy properties such as galaxy sizes, surface densities and star
formation rates. The physical origin of the massE relation may be related to
the self-regulation of a galaxy between its binding energy and the
accumulative feedback energy from its stellar populations. Although the massE
relation holds for various types of galaxies, in this study we only use
elliptical galaxies so that the velocity dispersion can be measured through
single fibers.

For galaxies with stellar masses well above 108 M⊙, the massE relation can be
written as a single power law:

$\sigma_{\rm e}R_{\rm e}^{0.25}=AM_{\rm star}^{\beta},$ (1)

where $R_{\rm e}$ is the effective radius that encloses half of stellar light,
$M_{\rm star}$ is the total stellar masses of galaxies, $\sigma_{\rm e}$ is
the velocity dispersion within $R_{\rm e}$ for dispersion-dominated galaxies,
and $\beta$=0.4036$\pm$0.0016 (see § 3). $\sigma_{\rm e}R_{\rm e}^{0.25}$
represents the fourth root of the galaxy binding energies within $R_{\rm e}$.

In Equation 1, $\sigma_{\rm e}$ is independent of redshift, $\frac{R_{\rm
e}}{\rm kpc}$=$\frac{R_{\rm e,as}}{206.27}$$\frac{D_{\rm A}}{\rm Mpc}$ and
$M_{\rm star}$=$M_{\rm star,1Mpc}$$(\frac{D_{\rm L}}{\rm 1Mpc})^{2}$, where
$R_{\rm e,as}$ is the apparent effective radius in arcsec, $M_{\rm star,1Mpc}$
is the galaxy stellar mass placed at a distance of 1 Mpc, the angular diameter
distance $D_{\rm A}$ is related to the luminosity distance $D_{\rm L}$ in
$D_{\rm A}$=$D_{\rm L}$/(1+$z$)2. $M_{\rm star,1Mpc}$ can be obtained by first
redshifting the observed spectral energy distribution (SED) to the rest-frame
and then fit with stellar population synthesis models to obtain the mass-to-
light ratio and the corresponding mass at a distance of 1 Mpc. $M_{\rm
star,1Mpc}$ is essentially a flux-like quantity and its measurement is
independent of cosmological models.

As a result, the corresponding angular diameter distance calculator is:

$\displaystyle\frac{D_{\rm A}}{\rm Mpc}$ $\displaystyle=$ $\displaystyle
S_{D_{0}}\frac{D_{0}}{\rm Mpc}\left[(1+z)^{-4\beta}\frac{\sigma_{\rm e}}{\rm
km\,s^{-1}}(\frac{M_{\rm star,1Mpc}}{M_{\odot}})^{-\beta}\right.$ (2)
$\displaystyle\left.(\frac{R_{\rm e,as}}{\rm
arcsec})^{0.25}\right]^{1/(2\beta-0.25)},$

with $D_{0}$=($\frac{0.264~{}{\rm
km~{}s^{-1}~{}kpc^{0.25}~{}M_{\odot}^{-1}}}{A})^{1/(2\beta-0.25)}$ in Mpc.
When applying Equation 2 to estimate distances for cosmological purpose, the
exact $D_{0}$ always degenerates with the local Hubble constant $H_{0}$,
similar to the case of the absolute magnitude of SN Ia. As a result, a
nuisance parameter $S_{D_{0}}$ is included as the scaling factor for a sample
to reproduce the local Hubble constant, and is related to the absolute
calibration of three observables ($\sigma_{\rm e}$, $M_{\rm star,1Mpc}$,
$R_{\rm e,as}$). The second nuisance parameter $\beta$ can be estimated from
local samples as detailed in § 3.

Not that although the massE relation invokes the same three observables as the
fundamental plane, they are different both physically and quantitatively. As
shown in Shi et al. (2021), the massE is a correlation between galaxy stellar
masses and binding energies while the fundamental plane is between galaxy
stellar mass and dynamical masses. For galaxies above 108 $M_{\odot}$, the
massE relation only has two free parameters that are $A$ and $\beta$ in
Equation 1, while the fundamental plane has three free parameters that are
$a$, $b$ and $c$ in Equation 5 of Shi et al. (2021). Furthermore, it has been
known that the fundamental plane needs a fourth hidden systematic parameter to
account for the dependence on galaxy properties or local environments (e.g.
Bernardi et al., 2003; Magoulas et al., 2012; Howlett et al., 2022). As a
result, the fundamental plane is mainly limited to z $<$ 0.05-0.1 for peculiar
velocity studies (e.g. Howlett et al., 2022).

The following strategy is proposed to measure the distance vs. redshift
diagram using the massE cosmic ruler. For a selected sample in a given
redshift bin, its median redshift, if ignoring the peculiar velocity,
corresponds exactly to its median luminosity distance, because the luminosity
distance increases monotonically with redshift, independent of cosmological
models. As a result, we measure the median of the redshift distribution and
the median of the luminosity distance distribution in individual redshift bins
to obtain the distance vs. redshift diagram. With this strategy, the sample
incompleteness does not matter, i.e., objects rejected by the selection
function does not affect at all the correspondence between the median distance
and median redshift. But the Malmquist bias does matter in a way that, e.g., a
sample with a magnitude cut is intrinsically fainter so that their stellar
mass is over-estimated.

Figure 1:  The local sample compiled in Shi et al. (2021) that is used to
derive the intrinsic massE relationship. $V_{\rm e}$ is rotation velocity and
velocity dispersion for rotation dominated and dispersion dominated galaxies,
respectively. For details, see Shi et al. (2021). The solid line indicates the
derived intrinsic relationship after considering errors on both X and Y axes
(see § 3).

Figure 2:  (a) The distance vs. redshift diagram that is derived from
elliptical galaxies in the SDSS main galaxy sample using the massE ruler.
$D_{\rm A}$ is the angular-diameter distance in Mpc and $S_{D_{0}}$ is the
scale factor that calibrates the absolute distance and degenerates with local
Hubble constant. The error bars indicate the square root of the diagonal
elements of the covariance matrix shown in in Figure 3 (a). The solid line
represents the best-fit flat $\Lambda$CDM with
$\Omega_{\Lambda}$=0.675$\pm$0.079. The dashed line is the one without dark
energy normalized at two lowest-redshift bins. (b) The residual of the best
fit. As shown in § 6.2, the scatter in the residual is consistent with the
covariance matrix in Figure 3 (a). (c) The confidence ranges of dark matter
density ($\Omega_{\rm m}$) and the product $h_{0}S_{D_{0}}$ at 68% and 95%,
where $h_{0}$ is the local Hubble constant in 100 km/s/Mpc. The plot was
produced through ChainConsumer (Hinton, 2016).

Figure 3:  (a) The covariance matrix of the distance measurements at different
redshift bins. (b) The distribution of the derived angular diameter distances
of elliptical galaxies in the SDSS main galaxy sample (blue filled) within the
redshift bin of [0.17, 0.18]. The red open histogram shows the simulated
distributions based on the median errors of three observables in the galaxy
catalog, and the orange filled histogram is the simulation that only considers
the observed redshift distribution plus random peculiar velocities of 1000
km/s (orange filled). Note that the two simulated distributions are rescaled
to have the same median and histogram peak as the observed one.

##  3 The massE relationship of the local sample

Before applying the massE to the SDSS sample, we first need to estimate the
intrinsic massE relationship, especially $\beta$ in Equation 1. We thus
perform the fitting to the sample compiled and homogenized in Shi et al.
(2021), while excluding luminous infrared galaxies and ultra diffuse galaxies
because of their large observational errors as well as high-$z$ sources in
order to remove the dependence of the relationship on the cosmological model.
This local sample contains 589 objects as shown in Fig. 1. Measurements of
galaxy sizes, velocity dispersions and stellar masses of the sample in general
have high signal to noise. We thus add additional 0.1 dex, 10% and 1%
systematic errors quadratically to $M_{\rm star}$, $R_{\rm e}$ and
$\sigma_{\rm e}$, respectively. The systematic error for $M_{\rm star}$ is
mainly caused by uncertainties in population synthetic modeling, which can be
illustrated by comparing the MPA-JHU and CIGALE stellar masses of SDSS
galaxies in § 6.1. The uncertainty for $R_{\rm e}$ is from background
subtraction, morphology irregularity, galaxy center determination etc. We
estimated this by selecting a subsample of SDSS galaxies with high $R_{\rm e}$
signal-to-noise ratio $>$ 100 and comparing measurements from different data
releases. The systematic error for $\sigma_{\rm e}$ is mainly caused by
absolute wavelength calibration and 1% is a reasonable estimate for modern
CCD-based spectra (Law et al., 2021). If increasing the errors by a small
amount, e.g., to 0.15 dex for $M_{\rm star}$, to 15% for $R_{\rm e}$ and to 2%
for $\sigma_{\rm e}$, respectively, the difference in the derived slope
$\beta$ is within one $\sigma$.

We performed PyMC3 (Salvatier et al., 2016) by considering errors on both
$M_{\rm star}$ and $\sigma_{\rm e}R_{\rm e}^{0.25}$ to obtain the intrinsic
relationship. The best fit gives $\beta$=0.4036$\pm$0.0016 and the
corresponding $D_{0}$ co-varies with $\beta$ through:

${\rm ln}({D_{0}}/{\rm Mpc})={(26.06\beta-5.54)\pm 0.005}.$ (3)

The derived slope $\beta$ is similar to that in Shi et al. (2021) but with a
smaller error, because in Shi et al. (2021) the fitting was done without
weighting by observational errors, which thus gives the result including the
observational error instead of the intrinsic relationship. Distances of this
local sample are mostly below 40 Mpc, with the remaining 17% of the sample
distributes between 40 and 300 Mpc. They are mainly based on cosmological-
independent measurements for those below $\sim$ 20 Mpc, but are homogenized to
a flat $\Lambda$CDM with $H_{0}$=73 km/s/Mpc, and $\Omega_{\Lambda}$=0.73 for
those above 20 Mpc. However, the best-fit $\beta$ changes only by a small
fraction of its 1-$\sigma$ error if changing to $\Omega_{\Lambda}$=1.0 or
$\Omega_{\Lambda}$=0.4, respectively. This is because the associated change in
$R_{\rm e}$ and $M_{\rm star}$ is much smaller than their observational
errors, plus the fact that both X and Y-axes vary simultaneously to partially
cancel out the effect.

We also carry out additional fitting to the massE relationship by including
105 high-$z$ objects compiled in Shi et al. (2021). The derived
$\beta$=0.4045$\pm$0.0016 is essentially the same as the one with only the
local sample. This suggests that $\beta$ should not evolve strongly with the
redshift.

##  4 Elliptical galaxies in the main galaxy sample of Sloan Digital Sky
Survey (SDSS)

Although all types of galaxies can be used for the massE ruler, elliptical
galaxies are the most suitable ones given their low extinction, regular
morphologies, low current star formation rates and dispersion dominated
kinematics, which will facilitate measurements of three observables
($\sigma_{\rm e}$, $M_{\rm star,1Mpc}$, $R_{\rm e,as}$). Unlike spiral
galaxies whose velocity requires spatially-resolved kinematic maps,
ellipticals’ can rely on single-fiber measurements (Cappellari et al., 2006;
Shi et al., 2021). We build our elliptical galaxy sample from the value-added
MPA-JHU catalog (Kauffmann et al., 2003; Brinchmann et al., 2004) of the SDSS
data release 7 111https://wwwmpa.mpa-garching.mpg.de/SDSS/DR7/. A sample of
about 28,0000 elliptical galaxies were selected from the SDSS main galaxy
sample with the following criteria: (1) no redshift warning as indicated by
Z_WARNING=0; (2) the main galaxy sample as defined by PRIMTARGET $\geq$ 64 but
$<$ 128 and Petrosion magnitude in $r$ $\leq$ 17.77; (3) elliptical galaxies
as defined by FRACDEV $\geq$ 0.8. Only the second criterion will introduce the
Malmquist bias while others are not related to the three observables
($\sigma_{\rm e}$, $R_{\rm e,as}$, $M_{\rm star,1Mpc}$).

To apply Equation 2, we adopt the Petrosian radius in the $r$ band
(PETROR50_R) to represent $R_{\rm e}$. The total stellar mass $M_{\rm star}$
adopts MSTAR_TOT that is derived from fitting to Petrosian magnitudes in four
SDSS optical bands. The advantage of the Petrosian radius and magnitude is
that there is no bias in redshift due to the surface dimming (Blanton et al.,
2001), which is the key for the distance vs. redshift measurement. $M_{\rm
star}$ is converted to $M_{\rm star,1Mpc}$ simply through the adopted
cosmology for the MPA-JHU catalog.

We carry out our own measurements of the velocity dispersion, because those in
the MPA-JHU catalog were derived by using the median instrumental resolution,
which will introduce redshift-dependent bias as the true instrumental
resolution is a function of the wavelength. The spectrum and associated
wavelength-dependent instrumental resolution of each object were available in
the SDSS
archive222http://data.sdss.org/sas/dr12/sdss/spectro/redux/26/spectra/. The
fitting was done using the pPXF code (Cappellari, 2017) by adopting the HR-
PYPOPSTAR theoretical stellar populations with 50,000 resolving power at 5000Å
(Millán-Irigoyen et al., 2021). During the fitting, we further exclude
5800-6100Å range where blue and red channels of the spectrograph with
different instrumental resolutions overlap. The derived velocity dispersion
within a 3′′ fiber was then corrected to that at PETROR50_R by the following
equation (Cappellari et al., 2006):

$\sigma_{\rm e}=\sigma_{\rm fiber}(R_{\rm e}/R_{\rm fiber})^{0.066\pm 0.035}.$
(4)

Here the error on the power-law index is implemented through
numpy.random.normal.

Figure 4:  The fractional errors of distance measurements at different
redshifts by different methods, including massE with SDSS main galaxies (this
work), BAO with SDSS main galaxy survey (MGS) (Ross et al., 2015), BAO with
SDSS BOSS data (Alam et al., 2017), BAO prediction with DESI (DESI
Collaboration et al., 2016) and SN Ia prediction with Rubbin/Roman (Feng et
al., 2014).  Figure 5:  The confidence range of the dark energy equation of
state ($w$) and the dark matter density ($\Omega_{\rm m}$) for the flat $w$CDM
model by CMB, BAO, SN Ia and massE, respectively. The lower-left cut for the
CMB is caused by the $H_{0}$ prior that is set to be $<$ 100 km/s/Mpc. The
plot was produced through ChainConsumer (Hinton, 2016).

##  5 The distance vs. redshift diagram of the SDSS elliptical sample.

Table 1:  The angular diameter distance as derived by the massE ruler. ID | $z_{\rm med}$ | ln($D_{A}(z_{\rm med})/S_{D_{0}}$)
---|---|---
|  | ln(Mpc)
1 | 0.0549 | 5.5840
2 | 0.0652 | 5.7371
3 | 0.0752 | 5.8830
4 | 0.0847 | 5.9884
5 | 0.0950 | 6.0851
6 | 0.1051 | 6.1746
7 | 0.1150 | 6.2391
8 | 0.1251 | 6.2892
9 | 0.1348 | 6.3628
10 | 0.1449 | 6.4589
11 | 0.1546 | 6.5070
12 | 0.1649 | 6.5449
13 | 0.1749 | 6.6054
14 | 0.1846 | 6.6676
15 | 0.1948 | 6.6847

Table 2:  The covariance matrix of the angular diameter distance
ln($D_{A}(z_{\rm med})/S_{D_{0}}$).

82.38 | 0.49 | 0.83 | 0.96 | 1.24 | 1.45 | 1.64 | 1.79 | 2.01 | 2.34 | 2.49 | 2.69 | 3.01 | 3.16 | 3.27
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
0.49 | 51.29 | 1.28 | 1.47 | 1.88 | 2.23 | 2.48 | 2.71 | 3.01 | 3.52 | 3.72 | 3.96 | 4.47 | 4.69 | 4.87
0.83 | 1.28 | 36.32 | 2.48 | 3.20 | 3.76 | 4.22 | 4.62 | 5.18 | 6.05 | 6.42 | 6.92 | 7.77 | 8.12 | 8.44
0.96 | 1.47 | 2.48 | 36.24 | 3.75 | 4.40 | 4.93 | 5.41 | 6.05 | 7.08 | 7.52 | 8.13 | 9.10 | 9.49 | 9.88
1.24 | 1.88 | 3.20 | 3.75 | 45.49 | 5.69 | 6.38 | 7.00 | 7.84 | 9.16 | 9.73 | 10.53 | 11.80 | 12.30 | 12.80
1.45 | 2.23 | 3.76 | 4.40 | 5.69 | 45.02 | 7.49 | 8.21 | 9.20 | 10.76 | 11.41 | 12.34 | 13.85 | 14.44 | 15.02
1.64 | 2.48 | 4.22 | 4.93 | 6.38 | 7.49 | 37.99 | 9.21 | 10.32 | 12.06 | 12.80 | 13.85 | 15.53 | 16.20 | 16.85
1.79 | 2.71 | 4.62 | 5.41 | 7.00 | 8.21 | 9.21 | 45.24 | 11.34 | 13.25 | 14.07 | 15.22 | 17.07 | 17.80 | 18.51
2.01 | 3.01 | 5.18 | 6.05 | 7.84 | 9.20 | 10.32 | 11.34 | 45.28 | 14.86 | 15.79 | 17.10 | 19.17 | 19.99 | 20.79
2.34 | 3.52 | 6.05 | 7.08 | 9.16 | 10.76 | 12.06 | 13.25 | 14.86 | 53.22 | 18.44 | 19.98 | 22.40 | 23.36 | 24.29
2.49 | 3.72 | 6.42 | 7.52 | 9.73 | 11.41 | 12.80 | 14.07 | 15.79 | 18.44 | 55.99 | 21.24 | 23.79 | 24.81 | 25.81
2.69 | 3.96 | 6.92 | 8.13 | 10.53 | 12.34 | 13.85 | 15.22 | 17.10 | 19.98 | 21.24 | 67.13 | 25.81 | 26.89 | 27.98
3.01 | 4.47 | 7.77 | 9.10 | 11.80 | 13.85 | 15.53 | 17.07 | 19.17 | 22.40 | 23.79 | 25.81 | 70.35 | 30.15 | 31.36
3.16 | 4.69 | 8.12 | 9.49 | 12.30 | 14.44 | 16.20 | 17.80 | 19.99 | 23.36 | 24.81 | 26.89 | 30.15 | 79.61 | 32.69
3.27 | 4.87 | 8.44 | 9.88 | 12.80 | 15.02 | 16.85 | 18.51 | 20.79 | 24.29 | 25.81 | 27.98 | 31.36 | 32.69 | 88.80

All numbers are in 10-6 ln(Mpc)2. Each number gives the covariance between
$i$-th and $j$-th redshift bins as listed in Table 1.

The redshift distribution of the SDSS elliptical sample peaks around 0.11. We
limit the redshift range to [0.05, 0.2] and construct the distance vs.
redshift diagram with a redshift bin size of 0.01. The number of galaxies in
individual redshift bins result in negligible fractional errors of their
median redshift, which is $<$ 0.06% for a random peculiar velocity of 1000
km/s. The upper range at $z$=0.2 is also necessary to avoid the Malmquist bias
caused by the magnitude cut of $r$ $<$ 17.77. Although the massE ruler is not
subject to the sample completeness, objects that are selected into the sample
can systematically bias the stellar mass if their luminosity is overestimated
due to the Malmquist bias. To estimate it roughly, we use the $r$-band
luminosity function (Blanton et al., 2003) to construct a mock $r$-band
photometric sample down to the SDSS 5-$\sigma$ depth ($r$=22.7) at different
redshifts, and add the photometric error based on the median and standard
deviation of apparent magnitude errors as a function of magnitude from the
SDSS sample. By assuming a constant mass-to-light ratio in the $r$-band, it is
found that the median distance is under-estimated by 1% at
($z$,$\Delta$$z$)=(0.2, 0.01) while by as much as 7% at
($z$,$\Delta$$z$)=(0.25, 0.01) for the sub-sample with $r$ $<$ 17.77. As a
result, we limit the $z$ range to be below 0.2 so that the Malmquist bias is a
small fraction of the typical 1-$\sigma$ uncertainty in individual redshift
bins that are around 1.3% including both observational errors and systematic
errors in $\beta$ (see § 6.2).

As stated above, the massE ruler has two nuisance parameters: $S_{D_{0}}$
calibrates the absolute distance, which degenerates with the local Hubble
constant, and $\beta$ determines the relative distances among galaxies with
different masses. The latter is transformed to the curvature of the distance-
redshift diagram, because for a flux limited sample galaxies at higher
redshifts have higher masses. The massE ruler would be a viable new probe of
cosmic expansion history, only if for a fixed $\beta$ the massE ruler gives a
smooth function of the distance vs. redshift with small fluctuations,
otherwise systematic errors unrelated to $\beta$, e.g., some hidden
parameters, may be too large for the massE ruler to be feasible. The latter is
in fact the case for other galaxy scaling laws. As a sanity check, we let
$\beta$ uniformly distribute over a very large range from 0 to 1, and carry
out the fitting with flat $\Lambda$CDM through cosmosis (Zuntz et al., 2015)
and emcee sampler (Foreman-Mackey et al., 2013). As listed in Table 3,
$\Omega_{\Lambda}$ converges to be a reasonable value, i.e., 0.54$\pm$0.28,
along with $\beta$=0.41$\pm$0.09. This indicates that once $\beta$ is
estimated from the local relationship, the performance of the massE ruler can
further improve.

Figure 2 (a) shows the derived distance-redshift diagram in terms of
ln($D_{\rm A}/S_{D_{0}}$) as a function of redshift, which is also listed in
Table 1. As shown in Figure 3 (a) and listed in Table 2, the covariance matrix
of ln($D_{\rm A}/S_{D_{0}}$) includes two parts. The first one is caused by
the observational errors of $\sigma_{\rm e}$, $R_{\rm e,as}$ and $M_{\rm
star}$. In the case of no observational errors, the derived $D_{\rm A}$
distribution should be the redshift distribution convolved with peculiar
velocity. As an example shown in Figure 3 (b) for $z$=[0.17, 0.18], the
derived ln($D_{\rm A}$) distribution (blue filled) scatters much more broadly
than that implied by the redshift distribution (orange filled). The
distribution can be approximated with a central Gaussian core plus a broad
wing. The standard deviation of the central core is 0.29, which is consistent
with but slightly larger than the square root of the quadratic sum of quoted
median errors of three observables in the catalog (red open symbols) through
Equation 2, which gives 0.21. This indicates that the quoted errors of
observables are slightly under-estimated. The broad wing is due to outliers of
measurements. As a result, we adopt the standard deviation of the whole
distribution divided by the square root of the number of objects as the error.
It is about two times larger than the one derived by the bootstrap method.
Since each redshift bin is independent, all non-diagonal elements of the
covariance matrix are set to zero for this first part. The second part is due
to the variation in $\beta$. We adopt its measurement and associated error
from the local sample to calculate the corresponding covariance matrix of
ln($D_{\rm A}/S_{D_{0}}$). Note that all uncertainties, e.g., the systematic
uncertainties in the stellar initial mass function, which shifts all distances
by the same amount, are taken into account by $S_{D_{0}}$.

Table 3:  The best-fit cosmological parameters.

$\beta$ priors† |  $M_{\rm star}$ for massE | rulers | $\beta$ | $\Omega_{\rm DE}$ | $w$
---|---|---|---|---|---
flat $\Lambda$CDM
U(0, 1) | MPA-JHU | massE | 0.41$\pm$0.09 | 0.54$\pm$0.28 | –
fixed to N (0.4036$\pm$0.0016) | MPA-JHU | massE | – | 0.675$\pm$0.079 | –
fixed to N (0.4036$\pm$0.0016) | CIGALE | massE | – | 0.678$\pm$0.078 | –
flat $w$CDM
|  | SN Ia | – | 0.68${}^{+0.07}_{-0.08}$ | -1.06${}^{+0.23}_{-0.19}$
|  | BAO | – | 0.72${}^{+0.02}_{-0.03}$ | -0.73${}^{+0.17}_{-0.16}$
|  | CMB | – | 0.80${}^{+0.06}_{-0.03}$ | -1.49${}^{+0.26}_{-0.31}$
|  | massE | – | 0.67${}^{+0.27}_{-0.17}$ | -0.97${}^{+0.42}_{-0.30}$

†U stands for a uniform distribution and N is for a normal distribution. All
other priors are included in Table 4.

##  6 Constraints on cosmological parameters

###  6.1 Constraints on the flat $\Lambda$CDM

We carry out the fitting to the distance vs. redshift diagram in Figure 2 as
obtained by the massE ruler with a flat $\Lambda$CDM. The fitting was done
through cosmosis (Zuntz et al., 2015) with the emcee sampler (Foreman-Mackey
et al., 2013), whose priors are listed in Table 4 where both $h_{0}$ and
$S_{D_{0}}$ are set to be free. As shown in Figure 2 (c), the convergent
results have $\Omega_{\Lambda}$=0.675$\pm$0.079, which is 1-$\sigma$
consistent with constraints by BAO (Ross et al., 2015; Alam et al., 2017), SN
Ia (Scolnic et al., 2018) and CMB (Planck Collaboration et al., 2020). The
massE ruler thus proves the acceleration of the universe at 8.5-$\sigma$ with
elliptical galaxies in the SDSS main galaxy sample under flat $\Lambda$CDM.

Among three observables, $M_{\rm star,1Mpc}$ relies on the stellar population
synthetic model, while the other two are straightforward measurements with
weak model dependence. We thus carry out additional sets of stellar mass
measurements and compared to the result of the MPA-JHU catalog. It is found
that with multi-band broad SED, the systematic effect of $M_{\rm star}$
measurements on the derived cosmological parameters is small as long as the
parameter space of the population synthetic model is reasonably sampled. We
use CIGALE (Boquien et al., 2019) to construct a two exponential decaying star
formation history with the second one for the late-epoch burst. The fraction
of models that experiences late bursts is 20%, smaller than 50% in MPA-JHU
(Kauffmann et al., 2003). The metallicity is uniformly distributed among four
discrete values that are 0.2, 0.4, 1.0 and 2.5 solar, while in MPA-JHU the
metallicity is interpolated to a higher resolution. The interstellar medium
extinction is uniform between 0.0 and 1.5, while the MPA-JHU catalog extends
to higher values. Although the standard deviation of the difference between
MPA-JHU’ and CICALE’ stellar masses reaches 20%, the systematic offset is much
smaller. As listed in Table 3, the best-fit $\Omega_{\rm\Lambda}$ is
0.678$\pm$0.078 for the flat $\Lambda$CDM, which is essentially the same as
the one (0.675$\pm$0.079) with the MPA-JHU mass.

###  6.2 The statistical power in distance measure

To illustrate the statistical advantage in distance measure with the massE
ruler, Figure 4 compares the fractional error of the massE-based distance to
those by SN Ia and BAO. We sum the covariance matrix to obtain this error. In
our redshift range of [0.05, 0.2] with a median redshift $z_{\rm med}$=0.11,
the corresponding angular diameter distance is ln($D_{\rm A}(z_{\rm
med})/S_{D_{0}}$[Mpc])=6.1987$\pm$0.0034. This fractional distance error of
0.34% is a quadratic sum of 0.17% by errors of three observables and 0.3% by
$\beta$ uncertainties. A double check of this estimate can be obtained by
looking at the residual of the best fit as shown in Figure 2 (b). For each
bin, the deviation from the true distance is caused by the statistical error
of three observables plus one realization of the intrinsic massE relationship
that is independent from other bins. As a result, the fluctuation in the
residual should converge to the result by the covariance matrix if the number
of redshift bins is large enough. An error of 0.35% is derived from the
standard deviation of the residual for the median distance of the above
redshift range, which is close to the one from the covariance matrix. We
further check whether the Malmquist bias can affect this error estimate by
calculating the bias for each redshift bin and treating it as additional
independent error. It is found that the final fractional distance error
increases negligibly from 0.35% to 0.36%.

As shown in the figure, the measurement with massE gives about one order of
magnitude smaller fractional errors than existing measurements at similar
redshift with SN Ia and BAO (Ross et al., 2015; Alam et al., 2017; Scolnic et
al., 2018). As compared to next-generation surveys as offered by BAO from DESI
and SN Ia from Rubin/Roman (Feng et al., 2014; DESI Collaboration et al.,
2016), the massE ruler still offers 2-4 times smaller fractional error at $z$
$\sim$ 0.1. This demonstrates the statistical power of the massE ruler as a
new probe of cosmic geometry.

The statistical power of the massE ruler can be understood in the following
way. Although each SN Ia offers a fractional distance error of $\sim$ 5% as
compared to $\sim$ 25% of the massE, the number of elliptical galaxies is far
more numerous than the SN Ia. On the other hand, BAO is essentially volume-
limited at low-$z$.

###  6.3 Constraints on the dark energy equation of state CDM ($w$CDM)

The low-$z$ distance measurements at high accuracy by massE is particularly
useful to probe the deviation from the constant of the dark energy density
that dominates at the low-$z$ universe. As shown in Figure 5, we illustrate
this by carrying out constraints on the $w$CDM model with the massE data and
compare to those with the Pantheon SN Ia data (Scolnic et al., 2018), the
MGS/BOSS/eBOSS BAO data (Ross et al., 2015; Alam et al., 2017, 2021) and the
Planck 2018 CMB TT,TE,EE+lowE data (Planck Collaboration et al., 2020),
respectively. All fittings have been done with cosmosis (Zuntz et al., 2015)
and emcee sampler (Foreman-Mackey et al., 2013). As listed in Table 3, the
massE alone constrains $w$ to an error that is only a factor of 2.2, 1.7 and
1.3 larger than the current one by BAO only, SN Ia only and CMB only,
respectively.

##  7 Conclusion

In this study we have shown that the massE relationship of galaxies could be a
new cosmic ruler to probe the distance-redshift diagram with advantage of its
statistical power. It has two nuisance parameters with three observables for
ellipticals that are galaxy sizes, velocity dispersions and stellar masses.
With the SDSS MGS sample, the distance at $z$=0.11 with the massE ruler is
constrained to a fractional error of 0.34%, with the best-fit dark energy
density of 0.675$\pm$0.079 for flat $\Lambda$CDM. In principle the massE can
be applied to higher $z$ by combining large-area space-based imaging survey
and ground-based deep spectroscopic survey.

The cosmosis modules including the input parameter files and data files for
the massE can be obtained through the link available on arXiv of this
manuscript.

## Acknowledgements

We thank the referee for constructive comments that help improve the paper. We
thank Pengjie Zhang for helpful comments. Y.S. acknowledges the support from
the National Key Research and Development Program of China (No.
2018YFA0404502), the National Natural Science Foundation of China (NSFC grants
11825302, 12141301, 12121003, 11733002), the science research grants from the
China Manned Space Project with NO. CMS-CSST-2021-B02, and the Tencent
Foundation through the XPLORER PRIZE. S.M. is partly supported by the National
Key Research and Development Program of China (No. 2018YFA0404501), by the
National Natural Science Foundation of China (11821303, 11761131004 and
11761141012) and by Tsinghua University Initiative Scientific Research Program
ID 2019Z07L02017. Z.Y.Z acknowledges the support of the National Natural
Science Foundation of China (NSFC) under grants No. 12041305, 12173016, the
Program for Innovative Talents, Entrepreneur in Jiangsu, and the science
research grants from the China Manned Space Project with NO. CMS-
CSST-2021-A08, CMS-CSST-2021-A07.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/. The SDSS is managed by the Astrophysical Research
Consortium for the Participating Institutions. The Participating Institutions
are the American Museum of Natural History, Astrophysical Institute Potsdam,
University of Basel, University of Cambridge, Case Western Reserve University,
University of Chicago, Drexel University, Fermilab, the Institute for Advanced
Study, the Japan Participation Group, Johns Hopkins University, the Joint
Institute for Nuclear Astrophysics, the Kavli Institute for Particle
Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of
Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute
for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New
Mexico State University, Ohio State University, University of Pittsburgh,
University of Portsmouth, Princeton University, the United States Naval
Observatory, and the University of Washington.

## Data Availability

All the data used here are available upon reasonable request.

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##  Appendix A Priors of cosmological parameters

The priors of cosmological parameters are detailed in Table 4. All fittings
have been done with cosmosis (Zuntz et al., 2015) and emcee sampler (Foreman-
Mackey et al., 2013). Each fitting has 4$\times$106 samples, with the Gelman-
Rubin (G-R) convergence value of $<$ 0.01, except for CMB that has
15$\times$106 samples in order to have G-R value below 0.01.

Table 4:  Priors of cosmological parameters.

parameters | priors | models
---|---|---
massE only
$\Omega_{\rm m}$ | U(0.0, 1.0) | flat $\Lambda$CDM, flat $w$CMD
$h_{0}$ | U(0.2, 1.0) | flat $\Lambda$CDM, flat $w$CMD
$S_{D_{o}}$ | U(0.3, 2.0) | flat $\Lambda$CDM, flat $w$CMD
$w$ | U(-3.0, 0.0) | flat $w$CMD
SN Ia only
$\Omega_{\rm m}$ | U(0.0, 1.0) | flat $w$CMD
$h_{0}$ | U(0.2, 1.0) | flat $w$CMD
$M$ | U(-23, -15) | flat $w$CMD
$w$ | U(-3.0, 0.0) | flat $w$CMD
BAO only
$\Omega_{\rm m}$ | U(0.0, 1.0) | flat $w$CMD
$h_{0}$ | U(0.2, 1.0) | flat $w$CMD
$\Omega_{\rm b}$ | U(0.001, 0.3) | flat $w$CMD
$w$ | U(-3.0, 0.0) | flat $w$CMD
CMB only
$\Omega_{\rm c}$ | U(0.0, 0.9) | flat $w$CMD
$\Omega_{\rm b}$ | U(0.0, 0.3) | flat $w$CMD
$h_{0}$ | U(0.2, 1.0) | flat $w$CMD
$w$ | U(-3.0, 0.0) | flat $w$CMD
$n_{s}$ | U(0.8, 1.2) | flat $w$CMD
$10^{9}A_{s}$ | U(1.48, 5.45) | flat $w$CMD
$k_{s}$ | 0.05 | flat $w$CMD
$\tau$ | U(0.01, 0.8) | flat $w$CMD

U stands for a uniform distribution.