#  $2$-capability and $2$-nilpotent multiplier of finite dimensional nilpotent
Lie algebras

Peyman Niroomand  School of Mathematics and Computer Science
Damghan University, Damghan, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Mohsen Parvizi  Department of Pure Mathematics,
Ferdowsi University of Mashhad, Mashhad, Iran<EMAIL_ADDRESS>
###### Abstract.

In the present context, we investigate to obtain some more results about
$2$-nilpotent multiplier $\mathcal{M}^{(2)}(L)$ of a finite dimensional
nilpotent Lie algebra $L$. For instance, we characterize the structure of
$\mathcal{M}^{(2)}(H)$ when $H$ is a Heisenberg Lie algebra. Moreover, we give
some inequalities on $\mathrm{dim}~{}\mathcal{M}^{(2)}(L)$ to reduce a well
known upper bound on $2$-nilpotent multiplier as much as possible. Finally, we
show that $H(m)$ is 2-capable if and only if m=1.

###### Key words and phrases:

2-nilpotent multiplier; Schur multiplier; Heisenberg algebras; derived
subalgebra; 2-capable Lie algebra,

Mathematics Subject Classification 2010. Primary 17B30 Secondary 17B60, 17B99.

##  1\. Introduction

For a finite group $G$, let $G$ be the quotient of a free group $F$ by a
normal subgroup $R$, then the $c$-nilpotent multiplier $\mathcal{M}^{(c)}(G)$
is defined as

$R\cap\gamma_{c+1}(F)/\gamma_{c+1}[R,F],$

in which $\gamma_{c+1}[R,F]=[\gamma_{c}[R,F],F]$ for $c\geq 1$. It is an
especial case of the Baer invariant [3] with respect to the variety of
nilpotent groups of class at most $c$. When $c=1$, the abelian group
$\mathcal{M}(G)=\mathcal{M}^{(1)}(G)$ is more known as the Schur multiplier of
$G$ and it is much more studied, for instance in [11, 14, 18].

Since determining the $c$-nilpotent multiplier of groups can be used for the
classification of group into isoclinism classes$($see [2]$)$, there are
multiple papers concerning this subject.

Recently, several authors investigated to develop some results on the group
theory case to Lie algebra. In [22], analogues to the $c$-nilpotent multiplier
of groups, for a given Lie algebra $L$, the $c$-nilpotent multiplier of $L$ is
defined as

$\mathcal{M}^{(c)}(L)=R\cap F^{c+1}/[R,F]^{c+1},$

in which $L$ presented as the quotient of a free Lie algebra $F$ by an ideal
$R$, $F^{c+1}=\gamma_{c+1}(F)$ and $[R,F]^{c+1}=\gamma_{c+1}[R,F]$. Similarly,
for the case $c=1$, the abelian Lie algebra
$\mathcal{M}(L)=\mathcal{M}^{(1)}(L)$ is more studied by the first author and
the others $($see for instance [4, 5, 6, 8, 9, 10, 15, 16, 17, 24]$)$.

The $c$-nilpotent multiplier of a finite dimensional nilpotent Lie algebra $L$
is a new field of interest in literature. The present context is involving the
$2$-nilpotent multiplier of a finite dimensional nilpotent Lie algebra $L$.
The aim of the current paper is divided into several steps. In [22, Corollary
2.8], by a parallel result to the group theory result, showed for every finite
nilpotent Lie algebra $L$, we have

(1.1)
$\mathrm{dim}(\mathcal{M}^{(2)}(L))+\mathrm{dim}(L^{3})\leq\frac{1}{3}n(n-1)(n+1).$

Here we prove that abelian Lie algebras just attain the bound 1.1. It shows
that always $\mathrm{Ker}~{}\theta=0$ in [22, Corollary 2.8 (ii)a].

Since Heisenberg algebras $H(m)$ $($a Lie algebra of dimension $2m+1$ with
$L^{2}=Z(L)$ and $\mathrm{dim}~{}(L^{2})=1)$ have interest in several areas of
Lie algebra, similar to the result of [5, Example 3] and [13, Theorem 24], by
a quite different way, we give explicit structure of $2$-nilpotent multiplier
of these algebras. Among the other results since the Lie algebra which
attained the upper bound 1.1 completely described in Lemma 2.6 $($they are
just abelian Lie algebras$)$, by obtaining some new inequalities on dimension
$\mathcal{M}^{(2)}(L)$, we reduce bound 1.1 for non abelian Lie algebras as
much as possible.

Finally, among the class of Heisenberg algebras, we show that which of them is
$2$-capable. It means which of them is isomorphic to $H/Z_{2}(H)$ for a Lie
algebra $H$. For more information about the capability of Lie algebras see
[18, 21]. These generalized the recently results for the group theory case in
[19].

##  2\. Further investigation on $2$-nilpotent multiplier of finite
dimensional nilpotent Lie algebra

The present section illustrates to obtain further results on $2$-nilpotent
multiplier of finite dimensional nilpotent Lie algebra. At first we give basic
definitions and known results for the seek of convenience the reader.

Let $F$ be a free Lie algebra on an arbitrary totaly ordered set $X$. Recall
from [25], the basic commutator on the set $X$, which is defined as follows,
is a basis of $F$.

The elements of $X$ are basic commutators of length one and ordered relative
to the total order previously chosen. Suppose all the basic commutators
$a_{i}$ of length less than $k\geq 1$ have been defined and ordered. Then the
basic commutators of length $k$ to be defined as all commutators of the form
$[a_{i},a_{j}]$ such that the sum of lengths of $a_{i}$ and $a_{j}$ is $k$,
$a_{i}>a_{j}$, and if $a_{i}=[a_{s},a_{t}]$, then $a_{j}\geq a_{t}$. Also the
number of basic commutators on $X$ of length $n$, namely $l_{d}(n)$, is

$\frac{1}{n}\sum_{m|n}\mu(m)d^{\frac{n}{m}},$

where $\mu$ is the Möbius function.

From [8], let $F$ be a fixed field, $L,K$ be two Lie algebras and $[\ ,\ ]$
denote the Lie bracket. By an action of $L$ on $K$ we mean an $F$-bilinear map

$(l,k)\in L\times K\mapsto~{}^{l}k\in K~{}\text{satisfying}~{}$
${}^{[l,l^{\prime}]}k=~{}^{l}(~{}^{l^{\prime}}k)-~{}^{l^{\prime}}(~{}^{l}k)~{}\text{and}~{}^{l}[k,k^{\prime}]=[~{}^{l}k,k^{\prime}]+[k,~{}^{l}k^{\prime}],~{}\text{for
all}~{}c\in F,l,l^{\prime}\in L,k,k^{\prime}\in K.$

When $L$ is a subalgebra of a Lie algebra $P$ and $K$ is an ideal in $P$, then
$L$ acts on $K$ by Lie multiplications $~{}^{l}k=[l,k]$. A crossed module is a
Lie homomorphism $\sigma:K\rightarrow L$ together with an action of $L$ on $K$
such that

$\sigma(^{l}k)=[l,\sigma(k)]~{}\text{and}~{}^{\sigma(k)}k^{\prime}=[k,k^{\prime}]~{}\text{for
all}~{}k,k^{\prime}\in K~{}\text{and}~{}l\in L.$

Let $\sigma:L\rightarrow M$ and $\eta:K\rightarrow M$ be two crossed modules,
$L$ and $K$ act on each other and on themselves by Lie. Then these actions are
called compatible provided that

$~{}^{~{}^{k}l}k^{\prime}=~{}^{k^{\prime}}(~{}^{l}k)~{}\text{and}~{}^{~{}^{l}k}l^{\prime}=~{}^{l^{\prime}}(~{}^{k}l).$

The non-abelian tensor product $L\otimes K$ of $L$ and $K$ is the Lie algebra
generated by the symbols $l\otimes k$ with defining relations

$c(l\otimes k)=cl\otimes k=l\otimes ck,(l+l^{\prime})\otimes k=l\otimes
k+l^{\prime}\otimes k,$ $l\otimes(k+k^{\prime})=l\otimes k+l\otimes
k^{\prime},~{}^{l}l^{\prime}\otimes
k=l\otimes~{}^{l^{\prime}}k-l^{\prime}\otimes~{}^{l}k,~{}l\otimes~{}^{k}k^{\prime}=~{}^{k^{\prime}}l\otimes
k-~{}^{k}l\otimes k^{\prime},$ $[l\otimes k,l^{\prime}\otimes
k^{\prime}]=-~{}^{k}l\otimes~{}^{l^{\prime}}k^{\prime},~{}\text{for
all}~{}c\in F,l,l^{\prime}\in L,k,k^{\prime}\in K.$

The non-abelian tensor square of $L$ is a special case of tensor product
$L\otimes K$ when $K=L$. Note that we denote the usual abelian tensor product
$L\otimes_{\mathbb{Z}}K$, when $L$ and $K$ are abelian and the actions are
trivial.

Let $L\square K$ be the submodule of $L\otimes K$ generated by the elements
$l\otimes k$ such that $\sigma(l)=\eta(k)$. The factor Lie algebra $L\wedge
K\cong L\otimes K/L\square K$ is called the exterior product of $L$ and $K$,
and the image of $l\otimes k$ is denoted by $l\wedge k$ for all $l\in L,k\in
K$. Throughout the paper $\Gamma$ is denoted the universal quadratic functor
$($see [8]$)$.

Recall from [18], the exterior centre of a Lie algebra $Z^{\wedge}(L)=\\{l\in
L~{}|~{}l\wedge l^{\prime}=1_{L\wedge L},~{}\forall~{}l^{\prime}\in L\\}$ of
$L$. It is shown that in [18] the exterior centre $L$ is a central ideal of
$L$ which allows us to decide when Lie algebra $L$ is capable, that is,
whether $L\cong H/Z(H)$ for a Lie algebra $H$.

The following Lemma is a consequence of [18, Lemma 3.1].

######  Lemma 2.1.

Let $L$ be a finite dimensional Lie algebra, $L$ is capable if and only if
$Z^{\wedge}(L)=0$.

The next two lemmas are special cases of [22, Proposition 2.1 (i)] when $c=2$
and that is useful for proving the next theorem.

######  Lemma 2.2.

Let $I$ be an ideal in a Lie algebra $L$. Then the following sequences are
exact.

  * (i)

$\mathrm{Ker}(\mu_{I}^{2})\rightarrow\mathcal{M}^{(2)}(L)\rightarrow\mathcal{M}^{(2)}(L/I)\rightarrow\frac{I\cap
L^{3}}{[[I,L],L]}\rightarrow 0.$

  * (ii)

$(I\wedge L/L^{3})\wedge
L/L^{3}\rightarrow\mathcal{M}^{(2)}(L)\rightarrow\mathcal{M}^{(2)}(L/I)\rightarrow
I\cap L^{3}\rightarrow 0,$ when $[[I,L],L]=0$.

######  Lemma 2.3.

Let $I$ be an ideal of $L$, and put $K=L/I$. Then

  * (i)

$\mathrm{dim}~{}\mathcal{M}^{(2)}(K)\leq\mathrm{dim}~{}\mathcal{M}^{(2)}(L)+\displaystyle\mathrm{dim}~{}\frac{I\cap
L^{3}}{[[I,L],L]}.$

  * (ii)

Moreover, if $I$ is a $2$-central subalgebra. Then

$(a).$ $(I\wedge L)\wedge
L\rightarrow\mathcal{M}^{(2)}(L)\rightarrow~{}\mathcal{M}^{(2)}(K)\rightarrow\mathrm{dim}~{}I\cap
L^{3}\rightarrow 0$.

$(b).$ $\mathrm{dim}~{}\mathcal{M}^{(2)}(L)+\mathrm{dim}~{}I\cap
L^{3}\leq\mathrm{dim}~{}\mathcal{M}^{(2)}(K)+\mathrm{dim}~{}(I\otimes
L/L^{3})\otimes L/L^{3}$.

###### Proof.

$(i)$. Using Lemma 2.2 (i).

$(ii)(a)$. Since $[I,L]\subseteq Z(L)$, $\mathrm{Ker}~{}\mu^{2}_{I}=(I\wedge
L)\wedge L$ and $[[I,L],L]=0$ by Lemma 2.2. It follows the result.

$(ii)(b)$. Since there is a natural epimorphism $(I\otimes L/L^{3})\otimes
L/L^{3}\rightarrow(I\wedge L/L^{3})\wedge L/L^{3}$, the result deduces from
Lemma 2.2 (ii). ∎

The following theorem gives the explicit structure of the Schur multiplier of
all Heisenberg algebra.

######  Theorem 2.4.

[5, Example 3] $\mathrm{and}$ [13, Theorem 24] Let $H(m)$ be Heisenberg
algebra of dimension $2m+1$. Then

  * (i)

$\mathcal{M}(H(1))\cong A(2)$.

  * (ii)

$\mathcal{M}(H(m))=A(2m^{2}-m-1)$ for all $m\geq 2$.

The following result comes from [20, Theorem 2.8] and shows the behavior of
2-nilpotent multiplier respect to the direct sum of two Lie algebras.

######  Theorem 2.5.

Let $A$ and $B$ be finite dimensional Lie algebras. Then

$\begin{array}[]{lcl}\mathcal{M}^{(2)}(A\oplus
B))&\cong&\mathcal{M}^{(2)}(A)\oplus~{}\mathcal{M}^{(2)}(B)\oplus\big{(}(A/A^{2}\otimes_{\mathbb{Z}}A/A^{2})\otimes_{\mathbb{Z}}B/B^{2}\big{)}\vspace{.3cm}\\\
&\oplus&\big{(}(B/B^{2}\otimes_{\mathbb{Z}}B/B^{2})\otimes_{\mathbb{Z}}A/A^{2}\big{)}.\end{array}$

The following theorem is proved in [22] and will be used in the next
contribution. At this point, we may give a short proof with a quite different
way of [22, Proposition 1.2] as follows.

######  Theorem 2.6.

Let $L=A(n)$ be an abelian Lie algebra of dimension $L$. Then
$\mathcal{M}^{(2)}(L)\cong A(\frac{1}{3}n(n-1)(n+1))$.

###### Proof.

We perform induction on $n$. Assume $n=2$. Then Theorem 2.5 allows us to
conclude that

$\begin{array}[]{lcl}\mathcal{M}^{(2)}(L)&\cong&\mathcal{M}^{(2)}(A(1))\oplus\mathcal{M}^{(2)}(A(1))\oplus\big{(}A(1)\otimes_{\mathbb{Z}}A(1)\otimes_{\mathbb{Z}}A(1)\big{)}\vspace{.3cm}\\\
&\oplus&\big{(}A(1)\otimes_{\mathbb{Z}}A(1))\otimes_{\mathbb{Z}}A(1)\big{)}\cong
A(1)\oplus A(1)\cong A(2).\end{array}$

Now assume that $L\cong A(n)\cong A(n-1)\oplus A(1)$. By using induction
hypothesis and Theorem 2.5, we have

$\begin{array}[]{lcl}\mathcal{M}^{(2)}(A(n-1)\oplus
A(1))&\cong&\mathcal{M}^{(2)}(A(n-1))\oplus\big{(}A(n-1)\otimes_{\mathbb{Z}}A(n-1)\otimes_{\mathbb{Z}}A(1)\big{)}\vspace{.3cm}\\\
&\oplus&\big{(}A(1)\otimes_{\mathbb{Z}}A(1)\otimes_{\mathbb{Z}}A(n-1)\big{)}\vspace{.3cm}\\\
&\cong&A(\frac{1}{3}n(n-1)(n-2))\oplus A((n-1)^{2})\oplus
A(n-1)\vspace{.3cm}\\\ &\cong&A(\frac{1}{3}n(n-1)(n+1)).\end{array}$

∎

The main strategy, in the next contribution, is to give the similar argument
of Theorem 2.4 for the $2$-nilpotent multiplier. In the first theorem, we
obtain the structure of $\mathcal{M}^{(2)}(L)$ when $L$ is non-capable
Heisenberg algebra.

######  Theorem 2.7.

Let $L=H(m)$ be a non-capable Heisenberg algebra. Then

$\mathcal{M}^{(2)}(H(m))\cong A(\frac{8m^{3}-2m}{3}).$

###### Proof.

Since $L$ is non-capable, Lemma 2.1 implies $Z^{\wedge}(L)=L^{2}=Z(L)$.
Invoking Lemma 2.3 by putting $I=Z^{\wedge}(L)$, we have
$\mathcal{M}^{(2)}(H(m))\cong\mathcal{M}^{(2)}(H(m)/H(m)^{2})$. Now result
follows from Theorem 2.6. ∎

The following theorem from [18, Theorem 3.4] shows in the class of all
Heisenberg algebras which one is capable.

######  Theorem 2.8.

$H(m)$ is capable if and only if $m=1$.

######  Corollary 2.9.

$H(m)$ is not $2$-capable for all $m\geq 2$.

###### Proof.

Since every $2$-capable Lie algebra is capable, the result follows from
Theorem 2.8. ∎

Since $H(m)$ for all $m\geq 2$ is not $2$-capable, we just need to discus
about the $2$-capability of $H(1)$. Here, we obtain 2-nilpotent multiplier of
$H(1)$ and in the next section we show $H(1)$ is $2$-capable.

######  Theorem 2.10.

Let $L=H(1)$. Then

$\mathcal{M}^{(2)}(H(1))\cong A(5).$

###### Proof.

We know that $H(1)$ is in fact the free nilpotent Lie algebra of rank 2 and
class 2. That is $H(1)\cong F/F^{3}$ in which $F$ is the free Lie algebra on 2
letters $x,y$. The second nilpotent multiplier of $H(1)$ is $F^{4}\cap
F^{3}/[F^{3},F,F]$ which is isomorphic to $F^{3}/F^{5}$ ant the latter is the
abelian Lie algebra on the set of all basic commutators of weights 3 and 4
which is the set $\\{[y,x,x],[y,x,y],[y,x,x,x],[y,x,x,y],[y,x,y,y]\\}$. So the
result holds. ∎

We summarize our result as below

######  Theorem 2.11.

Let $H(m)$ be Heisenberg algebra of dimension $2m+1$. Then

  * (i)

$\mathcal{M}^{(2)}(H(1))\cong A(5)$.

  * (ii)

$\mathcal{M}^{(2)}(H(m))=A(\frac{8m^{3}-2m}{3})$ for all $m\geq 2$.

The following Lemma lets us to obtain the structure of the $2$-nilpotent
multiplier of all nilpotent Lie algebras with $\mathrm{dim}~{}L^{2}=1$.

######  Lemma 2.12.

[15, Lemma 3.3] Let $L$ be an $n$-dimensional Lie algebra and
$\mathrm{dim}~{}L^{2}=1$. Then

$L\cong H(m)\oplus A(n-2m-1).$

######  Theorem 2.13.

Let $L$ be an $n$-dimensional Lie algebra with $\mathrm{dim}~{}L^{2}=1$. Then

$\mathcal{M}^{(2)}(L)\cong\left\\{\begin{array}[]{lcl}A(\frac{1}{3}n(n-1)(n-2))&if\
m>1,\\\ \\\ A(\frac{1}{3}n(n-1)(n-2)+3)&if\ m=1.\end{array}\right.$

###### Proof.

By using Lemma 2.12, we have $L\cong H(m)\oplus A(n-2m-1)$. Using the behavior
of $2$-nilpotent multiplier respect to direct sum

$\begin{array}[]{lcl}\mathcal{M}^{(2)}(L)&\cong&\mathcal{M}^{(2)}(H(m))\oplus~{}\mathcal{M}^{(2)}(A(n-2m-1))\vspace{.3cm}\\\
&\oplus&\big{(}(H(m)/H(m)^{2}\otimes_{\mathbb{Z}}H(m)/H(m)^{2})\otimes_{\mathbb{Z}}A(n-2m-1)\big{)}\vspace{.3cm}\\\
&\oplus&\big{(}(A(n-2m-1)\otimes_{\mathbb{Z}}A(n-2m-1))\otimes_{\mathbb{Z}}H(m)/H(m)^{2}\big{)}.\end{array}$

First assume that $m=1$, then by virtue of Theorems 2.6 and 2.11

$\mathcal{M}^{(2)}(H(1))\cong
A(5)~{}\text{and}~{}\mathcal{M}^{(2)}(A(n-3))\cong
A(\frac{1}{3}(n-2)(n-3)(n-4)).$

Thus

$\begin{array}[]{lcl}\mathcal{M}^{(2)}(L)&\cong&A(5)\oplus~{}A(\frac{1}{3}(n-2)(n-3)(n-4))\vspace{.3cm}\\\
&\oplus&\big{(}A(2)\otimes_{\mathbb{Z}}A(2))\otimes_{\mathbb{Z}}A(n-3)\big{)}\vspace{.3cm}\\\
&\oplus&\big{(}(A(n-3)\otimes_{\mathbb{Z}}A(n-3))\otimes_{\mathbb{Z}}A(2)\big{)}\vspace{.3cm}\\\
&\cong&A(\frac{1}{3}n(n-1)(n-2)+3).\end{array}$

The case $m\geq 1$ is obtained by a similar fashion. ∎

######  Theorem 2.14.

Let $L$ be a $n$-dimensional nilpotent Lie algebra such that
$\mathrm{dim}~{}L^{2}=m(m\geq 1)$. Then

$\mathrm{dim}~{}\mathcal{M}^{(2)}(L)\leq\frac{1}{3}(n-m)\big{(}(n+2m-2)(n-m-1)+3(m-1)\big{)}+3.$

In particular,
$\mathrm{dim}~{}\mathcal{M}^{(2)}(L)\leq\frac{1}{3}n(n-1)(n-2)+3$. The
equality holds in last inequality if and only if $L\cong H(1)\oplus A(n-3)$.

###### Proof.

We do induction on $m$. For $m=1$, the result follows from Theorem 2.13. Let
$m\geq 2$, and taking $I$ a $1$-dimensional central ideal of $L$. Since $I$
and $L/L^{3}$ act to each other trivially we have $(I\otimes L/L^{3})\otimes
L/L^{3}\cong(I\otimes_{\mathbb{Z}}\frac{L/L^{3}}{(L/L^{3})^{2}})\otimes_{\mathbb{Z}}\frac{L/L^{3}}{(L/L^{3})^{2}})$.
Thus by Lemma 2.3 $(ii)(b)$

$\mathrm{dim}~{}\mathcal{M}^{(2)}(L)+\mathrm{dim}~{}I\cap
L^{3}\leq\mathrm{dim}~{}\mathcal{M}^{(2)}(L/I)+\mathrm{dim}~{}(I\otimes_{\mathbb{Z}}\frac{L/L^{3}}{(L/L^{3})^{2}})\otimes_{\mathbb{Z}}\frac{L/L^{3}}{(L/L^{3})^{2}}).$

Since

$\mathrm{dim}~{}\mathcal{M}^{(2)}(L/I)\leq\frac{1}{3}(n-m)\big{(}(n+2m-5)(n-m-1)+3(m-2)\big{)},$

we have

$\begin{array}[]{lcl}\mathrm{dim}~{}\mathcal{M}^{(2)}(L)&\leq&\frac{1}{3}(n-m)\big{(}(n+2m-5)(n-m-1)+3(m-2)\big{)}+3+(n-m)^{2}\vspace{.3cm}\\\
&=&\frac{1}{3}(n-m)\big{(}(n+2m-2)(n-m-1)+3(m-1)\big{)}+3,\end{array}$

as required. ∎

The following corollary shows that the converse of [22, Proposition 1.2] for
$c=2$ is also true. In fact it proves always $\mathrm{Ker}~{}\theta=0$ in [22,
Corollary 2.8 (ii)a].

######  Corollary 2.15.

Let $L$ be a $n$-dimensional nilpotent Lie algebra. If
$\mathrm{dim}~{}\mathcal{M}^{(2)}(L)=\frac{1}{3}n(n-1)(n+1)$, then $L\cong
A(n)$.

##  3\. 2-capability of Lie algebras

Following the terminology of [7] for groups, a Lie algebra $L$ is said to be
$2$-capable provided that $L\cong H/Z_{2}(H)$ for a Lie algebra $H$. The
concept $Z_{2}^{*}(L)$ was defined in [23] and it was proved that if
$\pi:F/[R,F,F]\rightarrow F/R$ be a natural Lie epimorphism then

$Z^{*}_{2}(L)=\pi(Z_{2}(F/[[R,F],F])),~{}\text{for}~{}c\geq 0$

The following proposition gives the close relation between $2$-capability and
$Z^{*}_{2}(L)$.

######  Proposition 3.1.

A Lie algebra $L$ is $2$-capable if and only if $Z^{*}_{2}(L)=0.$

###### Proof.

Let $L$ has a free presentation $F/R$, and $Z^{*}_{2}(L)=0$. Consider the
natural epimorphism $\pi:F/[[R,F],F]\twoheadrightarrow F/R$. Obviously

$\mathrm{Ker}~{}\pi=R/[[R,F],F]=Z_{2}(F/[[R,F],F]),$

and hence $L\cong F/[[R,F],F]/Z_{2}(F/[[R,F],F])$, which is a $2$-capable.

Conversely, let $L$ is $2$-capable and so $H/Z_{2}(H)\cong L$ for a Lie
algebra $H$. Put $F/R\cong H$ and $Z_{2}(H)\cong S/R$. There is natural
epimorphism $\eta:F/[[S,F],F]\twoheadrightarrow F/S\cong L$. Since
$Z_{2}(F/[[R,F],F])\subseteq\mathrm{Ker}~{}\eta$, $Z^{*}_{2}(L)=0$, as
required. ∎

The following Theorem gives an instrumental tools to present the main.

######  Theorem 3.2.

Let $I$ be an ideal subalgebra of $L$ such that $I\subseteq Z^{*}_{2}(L)$.
Then the natural Lie homomorphism
$\mathcal{M}^{(2)}(L)\rightarrow\mathcal{M}^{(2)}(L/I)$ is a monomorphism.

###### Proof.

Let $S/R$ and $F/R$ be two free presentations of $L$ and $I$, respectively.
Looking the natural homomorphism

$\phi:\mathcal{M}^{(2)}(L)\cong R\cap
F^{2}/[[R,F],F]\rightarrow\mathcal{M}^{(2)}(L/I)\cong R\cap S^{2}/[[S,F],F]$

and the fact that $S/R\subseteq Z_{2}(F/R)$ show $\phi$ has trivial kernel.
The result follows. ∎

######  Theorem 3.3.

A Heisenberg Lie algebra $H(m)$ is $2$-capable if and only if $m=1$.

###### Proof.

Let $m\geq 2$, by Corollary 2.9 $H(m)$ is not capable so it is not $2$-capable
as well. Hence we may assume that $L\cong H(1)$. Let $I$ be an ideal of $L$ od
dimension 1. Then $L/I$ is abelian of dimension 2, and hence
$\mathrm{dim}~{}\mathcal{M}^{(2)}(L)=2$. On the other hands, Theorem 2.11
implies $\mathrm{dim}~{}\mathcal{M}^{(2)}(L)=5$, and Theorem 3.2 deduces
$\mathcal{M}^{(2)}(L)\rightarrow\mathcal{M}^{(2)}(L/I)$ can not be a
monomorphism, as required. ∎

## References

  * [1] R. Baer, Representations of groups as quotient groups, I, II, and III. Trans. Amer. Math. Soc. 58 (1945) 295-419.
  * [2] F.R. Beyl, J. Tappe, Group Extensions, Representations and the Schur Multiplicator, Lecture Notes in Math., vol. 958, Springer- Verlag, Berlin, 1982.
  * [3] P. Batten, Multipliers and covers of Lie algebras, dissertation, North Carolina State University (1993).
  * [4] P. Batten, K. Moneyhun, and E. Stitzinger, On characterizing nilpotent Lie algebras by their multipliers, Comm. Algebra, 24 (1996), 4319–4330.
  * [5] P. Batten and E. Stitzinger, On covers of Lie algebras, Comm. Algebra, 24 (1996) 4301–4317.
  * [6] L. Bosko. On Schur Multipliers of Lie algebras and groups of maximal class, Internat. J. Algebra Comput., 20 (2010), 807-821.
  * [7] J. Burns, G. Ellis, On the nilpotent multipliers of a group, Math. Z. 226 (1997) 405-428.
  * [8] G. Ellis, A non abelian tensor product of Lie algebras, Glasgow Math. J., 33 (1991) 101- 120.
  * [9] P. Hardy, On characterizing nilpotent Lie algebras by their multipliers III, Comm. Algebra, 33 (2005) 4205-4210.
  * [10] P. Hardy and E. Stitzinger, On characterizing nilpotent Lie algebras by their multiplierst $t(L)=3,4,5,6$, Comm. Algebra, (26) 1(1998), 3527-3539.
  * [11] G. Karpilovsky, The Schur multiplier, Oxford University Press, Oxford, 1987.
  * [12] M. R. R. Moghaddam, The Baer invariant of a direct product, Arch. Math. 33 (1980) 504-511.
  * [13] Moneyhun K., Isoclinisms in Lie algebras, Algebras Groups Geom., 11,(1994), 9-22.
  * [14] P. Niroomand, On the order of Schur multiplier of non abelian $p$–groups, J. Algebra, 322 (2009), 4479–4482.
  * [15] P. Niroomand, On dimension of the Schur multiplier of nilpotent Lie algebras, Cent. Eur. J. Math., 9 (2011), 57-64.
  * [16] P. Niroomand and F. G. Russo. A note on the Schur multiplier of a nilpotent Lie algebra, Comm. Algebra, 39, 4,(2011) 1293-1297.
  * [17] P. Niroomand, F. G. Russo, A restriction on the Schur multiplier of nilpotent Lie algebras, Electron. J. Linear Algebra, 22 (2011),1-9.
  * [18] P. Niroomand, M. Parvizi, F.G. Russo, Some criteria for detecting capable Lie algebras, J. Algebra, 384 (2013), 36-44.
  * [19] P. Niroomand, M. Parvizi, On the 2-nilpotent multiplier of finite $p$-groups, Glasgow Math J. 1, 57, (2015), 201-210.
  * [20] P. Niroomand and M. Parvizi, 2-Nilpotent multipliers of a direct product of Lie algebras, to appear in Rend. Circ. Mat. Palermo.
  * [21] A.R. Salemkar, V. Alamian, H. Mohammadzadeh, Some properties of the Schur multiplier and covers of Lie Algebras, Comm. Algebra 36 (2008) 697-707.
  * [22] A. R. Salemkar, B. Edalatzadeh and M. Araskhan, Some inequalities for the dimension of the $c$-nilpotent multiplier of Lie algebras. J. Algebra 322 (2009), no. 5, 1575-1585.
  * [23] A. R. Salemkar and Z. Riyahi, Some properties of the $c$-nilpotent multiplier of Lie algebras, J. Algebra 370 (2012), 320-325.
  * [24] B. Yankoski. On the multiplier of a Lie Algebra, Journal of Lie Theory, 13 (2003) 1-6.
  * [25] A.I. Shirshov, On bases of free Lie algebras, Algebra Logika 1 (1) (1962) 14-19.


# The Hubble Tension Bites the Dust:
Sensitivity of the Hubble Constant Determination to Cepheid Color Calibration

Edvard Mörtsell  Oskar Klein Centre, Department of Physics, Stockholm
University
Albanova University Center
106 91 Stockholm, Sweden  Ariel Goobar  Oskar Klein Centre, Department of
Physics, Stockholm University
Albanova University Center
106 91 Stockholm, Sweden  Joel Johansson  Oskar Klein Centre, Department of
Physics, Stockholm University
Albanova University Center
106 91 Stockholm, Sweden  Suhail Dhawan  Institute of Astronomy
University of Cambridge Madingley Road
Cambridge CB3 0HA
United Kingdom

###### Abstract

Motivated by the large observed diversity in the properties of extra-galactic
extinction by dust, we re-analyse the Cepheid calibration used to infer the
local value of the Hubble constant, $H_{0}$, from Type Ia supernovae. Unlike
the SH0ES team, we do not enforce a universal color-luminosity relation to
correct the near-IR Cepheid magnitudes. Instead, we focus on a data driven
method, where the measured colors of the Cepheids are used to derive a color-
luminosity relation for each galaxy individually. We present two different
analyses, one based on Wesenheit magnitudes, a common practice in the field
that attempts to combine corrections from both extinction and variations in
intrinsic colors, resulting in $H_{0}=66.9\pm 2.5$ km/s/Mpc, in agreement with
the Planck value. In the second approach, we calibrate using color excesses
with respect to derived average intrinsic colors, yielding $H_{0}=71.8\pm 1.6$
km/s/Mpc, a $2.7\,\sigma$ tension with the value inferred from the cosmic
microwave background.

Hence, we argue that systematic uncertainties related to the choice of Cepheid
color-luminosity calibration method currently inhibits us from measuring
$H_{0}$ to the precision required to claim a substantial tension with Planck
data.

Cepheid distance (217), Hubble constant (758), Type Ia supernovae (1728),
Interstellar dust extinction (837)

##  1 Introduction

As is well known, there is a tension between the value of the Hubble constant
as inferred from small and large distance measurements, most significantly
between the values inferred from Type Ia supernova (SNIa) distances to
redshifts $z\sim 0.1$ calibrated by Cepheid observations, as measured by the
SH0ES team, and the distance to the cosmic microwave background (CMB)
decoupling surface at $z\sim 1090$, as measured by the Planck satellite. The
former yields $H_{0}=73.2\pm 1.3$ (in units of km/s/Mpc used from now on)
(Riess et al., 2021) and the latter $H_{0}=67.4\pm 0.5$ (Aghanim et al.,
2020); a $4.1\,\sigma$ tension.

The tension between other measurements is not as significant: Calibrating the
absolute SNIa magnitude using the tip of the red giant branch (TRGB)
observations gives $H_{0}=69.6\pm 1.6$ (Freedman et al., 2019), right between
the Cepheid calibrated SNIa and the CMB inferred values.

Another local estimate of the expansion rate of the Universe is derived from
the amplitude of the gravitational wave signal GW170817, the merger of a
binary neutron-star system located to the galaxy NGC 4993 at $z=0.010$ through
the electromagnetic counterpart, AT2017gfo, yielding a Hubble constant with
relatively large uncertainties of $H_{0}=70.0^{+12.0}_{-8.0}$ (Abbott et al.,
2017).

A different route involves gravitational lensing. Time delays in the TDCOSMO
sample of seven lensed quasars yield $H_{0}=74.5^{+5.6}_{-6.1}$ (Birrer et
al., 2020). Constraining the galaxy lens mass profiles, using kinematics
observations of an independent set of gravitational lenses in the Sloan Lens
ACS sample (SLACS), lowers the value to $H_{0}=67.4^{+4.1}_{-3.2}$, assuming
that the TDCOSMO and SLACS galaxies are drawn from the same parent population.
These results are illustrated in Figure 1, from which is evident that only the
Cepheid calibrated SNIa distance scale from SH0ES is in definite tension with
the CMB inferred distance.

Figure 1: The Hubble constant as inferred from distances measured to the CMB,
strongly lensed quasars, SNIa calibrated with Cepheids (SH0ES) and the TRGB,
and the gravitational wave signal GW170817. The redshift corresponds to the
mean redshift of the distances employed in the method, slightly shifted where
needed to avoid overlap. The major tension is between the values inferred from
Cepheid calibrated SNIa by the SH0ES team and CMB observations.

The value inferred from the CMB depends on the entire expansion history of the
Universe, whereas the SNIa measurement only depends on the local expansion
rate it sets out to measure. On the other hand, the inference from SNe Ia
depends on a combination of a larger number of astrophysical probes.
Therefore, attempts to modify the CMB inferred $H_{0}$ usually rely on
modifications of the cosmological model, whereas the SNIa value is usually
studied with emphasis on possible systematic effects concerning the local
distance measurements.

At least in principle, the CMB value can be increased by various departures
from the concordance cosmological constant and cold dark matter, $\Lambda$CDM
model, see e.g. Mörtsell & Dhawan (2018); Knox & Millea (2020). Options
include decreasing the physical size of the sound horizon used to measure the
distance to $z=1090$. This can be accomplished by adding sources of energy
present before CMB photon decoupling, e.g., new thermal relativistic species
or early dark energy, or by reducing the sound speed. However, such
modifications are severely constrained when taking the full CMB power spectrum
into account. Attempts to shift the CMB value also involve changing the
expansion history at redshifts $z<1090$, with modest success since the
expansion rate is tightly constrained by SNIa and baryonic acoustic
oscillation (BAO) observations.

Given that the proposals mentioned above require substantial modifications of
the current concordance cosmological model, and still fail in relieving the
full tension, we investigate the Cepheid-SNIa value and its uncertainties, see
also Follin & Knox (2018); Efstathiou (2020). In particular, we concentrate on
dust extinction, affecting all astronomical observations in the optical and
near infrared (NIR) regime. We focus on a a very specific assumption made by
the SH0ES team throughout their series of publications, namely that there is a
universal reddening law in all galaxies.

##  2 Revisiting dust extinction corrections

In spite of the critical importance for precision cosmology, the current
understanding of light attenuation in the interstellar medium (ISM) of
galaxies remains very limited. In comparison, the Milky Way (MW) ISM has been
studied in great detail, including the properties of dust grains responsible
for dimming of light(see Draine, 2003, for a review). In particular, several
MW reddening laws have been devised, among these Cardelli et al. (1989) (CCM),
O’Donnell (1994) and Fitzpatrick (1999) (F99). They have in common the use of
a single parameter, the total to selective extinction coefficient, $R_{\rm
V}^{\rm BV}$, as a proxy for the grain composition and size distribution,
where the attenuation in the optical ${\rm V}$-band relates to the color
excess $E({\rm B}-{\rm V})\equiv A_{\rm B}-A_{\rm V}$ as $A_{\rm V}=R_{\rm
V}^{\rm BV}\,E({\rm B}-{\rm V})$, here referred to as the ”CCM-relationship”.
Low values of $R_{\rm V}^{\rm BV}$ indicate a steep wavelength dependence,
while large $R_{\rm V}^{\rm BV}$ correspond to gray extinction.

While an average $\langle R_{\rm V}^{\rm BV}\rangle=3.1$ for the MW is found
in most studies, significant variations are found in individual lines of sight
in the galaxy, ranging from $R_{\rm V}^{\rm BV}\approx 2$ in some diffuse
sight lines, to $R_{\rm V}^{\rm BV}\approx 6$ in dense molecular clouds
(Fitzpatrick, 1999). The diversity in the MW has been confirmed by Nataf et
al. (2016), who find significantly lower values of $R_{\rm V}^{\rm BV}$ in the
Galactic bulge. Moving the scope outside the MW, a study by Gordon et al.
(2003) of the extinction in the Magellanic Clouds found that a small number of
LMC extinction curves are consistent with the CCM relationship, but the
majority of the LMC and all the SMC curves are not. Fausnaugh et al. (2015)
report a gray extinction law for NGC 4258 in the line of sight of the
Cepheids, $R_{\rm V}^{\rm BV}\sim 4.9$, although they caution that this could
be the result of unresolved systematics. For more distant galaxies, observed
SNIa colors highlight the observed diversity in extinction properties, ranging
from $R_{\rm V}^{\rm BV}\sim 1$ to values consistent with the MW average (see
Krisciunas et al., 2006; Nobili & Goobar, 2008; Goobar et al., 2014; Amanullah
et al., 2014, 2015; Burns et al., 2018, and references therein). For the SNe
Ia in the Hubble flow, color corrections are based on the SALT2 lightcurve
fitter (Guy et al., 2007), which again differ from the CCM parameterization,
but are most consistent with values of $R_{\rm V}^{\rm BV}\sim 2.5$ (see e.g.,
Biswas et al., 2021, and references therein), although dust extinction
differences between host galaxy environments has been suggested as an
explanation for a systematic ”mass step” in the derived distances (Brout &
Scolnic, 2021; Johansson et al., 2021).

To minimize the impact from extinction correction uncertainties, the SH0ES
team use flux measurements in the NIR H-band, centered at 1.6 $\mu m$, where
extinction by dust, gauged using the observed color ${\rm V}-{\rm I}$, is
significantly smaller. Adopting the CCM-like relationship from F99 and the
extinction correction $A_{\rm H}=R_{\rm H}^{\rm VI}\,E({\rm V}-{\rm I})$, the
value corresponding to the MW average is $R_{\rm H}^{\rm VI}\sim 0.4$.
However, there is no theoretical, nor any empirical studies of extinction
suggesting that a universal value of $R_{\rm H}^{\rm VI}$ can be assumed. On
the contrary, a recent study by Fitzpatrick et al. (2019) finds considerable
variations between lines of sight for the extinction curves in the NIR in the
Milky-Way. Based on a parameterization fitting extinction laws from
ultraviolet to NIR for 72 well-measured stars, a very wide range in $R_{\rm
H}^{\rm VI}$ can be inferred, as shown in Figure 2. Hence, assuming a narrow
range in $R_{\rm H}^{\rm VI}$ for the anchor and Cepheid hosts in not
warranted by current observations. In this paper, we investigate to what
degree relaxing this assumption affects the inferred value of $H_{0}$ and its
corresponding uncertainties.

Figure 2: $R_{\rm H}^{\rm VI}$ vs $R_{\rm V}^{\rm BV}$ relation and 1$\sigma$
scatter from the extinction law derived by Fitzpatrick et al. (2019) (FTZ19),
extrapolated to $R_{\rm V}^{\rm BV}>6$ and $R_{\rm V}^{\rm BV}<2$. Such low
values have been inferred from SNIa colors (Burns et al., 2018, B18). Also
shown are derived $R_{\rm V}^{\rm BV}$ values from a MW stellar sample in
Schlafly et al. (2017) (SH17), two samples in the LMC (Gordon et al., 2003,
LMC1 and LMC2) and in the water mega-maser anchor NGC 4258 (Fausnaugh et al.,
2015, FA15).

##  3 Difference in methodology

The local distance ladder uses the difference between Cepheid magnitudes in
SNIa hosts and anchor galaxies. Therefore, it is relatively insensitive to
changes in the global properties of the extinction law. In Riess et al.
(2016), a global change of the parameter $R_{\rm W}$ parameterizing a Cepheid
color-luminosity (C-L) correction with respect to the observed color $R_{\rm
W}\,({\rm V}-{\rm I})$ of $R_{\rm W}=0.39\rightarrow 0.35$ changed the Hubble
constant by $\delta H_{0}\sim 0.5$. When multiplied with the observed color
${\rm V}-{\rm I}$, the parameter $R_{\rm W}$ is argued to also partly correct
for an intrinsic Cepheid C-L relation.

In Follin & Knox (2018), a slightly different C-L correction with respect to
an estimated color excess of $R_{\rm E}\,\hat{E}\,({\rm V}-{\rm I})$ was
employed, where $\hat{E}\,({\rm V}-{\rm I})\equiv({\rm V}-{\rm I})-\langle{\rm
V}-{\rm I}\rangle_{0}$ with $\langle{\rm V}-{\rm I}\rangle_{0}$ an estimate of
the mean intrinsic Cepheid color. Here, $R_{\rm E}$ is interpreted as the dust
total to selective extinction ratio, $R_{\rm H}^{\rm VI}$. Imposing a prior of
$R_{\rm E}=0.39\pm 0.1$, a value of $H_{0}=73.3\pm 1.7$ was derived when
allowing $R_{\rm E}$ to vary between galaxies, in good agreement with
$H_{0}=73.2\pm 1.3$ from Riess et al. (2021). At face value, this result seems
to suggest that the method used for calibrating the C-L relation and/or the
possibility of varying this calibration between galaxies have a small impact
on the inferred Hubble constant.

As argued below, regardless of whether calibrating with respect to the
observed color or an estimated color excess, the correction should apply for
both dust extinction and the intrinsic Cepheid C-L relation. Since both
contributions are uncertain at NIR wavelengths, and at least dust extinction
properties are known to vary between galaxies, we investigate the effect of
allowing for $R_{\rm W}$ and $R_{\rm E}$ to do the same. Given the lack of
solid independent constraints on $R_{\rm E}$, we employ less restrictive
priors than Follin & Knox (2018). Any systematic difference in dust properties
or the intrinsic C-L between SNIa hosts and anchor galaxies will shift the
inferred value of $H_{0}$. We will show that such a difference is supported by
the Cepheid and SNIa data.

##  4 Method and Data

The apparent magnitude of a source at redshift $z$ with absolute magnitude $M$
is given by

$m=5\log D(z)+M+25,$ (1)

where $D(z)$ is the luminosity distance in units of Mpc. By combining observed
magnitudes of Cepheids in SNIa host and anchor galaxies, $m_{\rm Ceph}^{\rm
host}$ and $m_{\rm Ceph}^{\rm anch}$ with SNIa magnitudes in host galaxies and
in the Hubble flow, $m_{\rm SN}^{\rm host}$ and $m_{\rm SN}^{\rm flow}$ , we
can derive

$5\log H_{0}=5\log r(z)-5\log D^{\rm anch}+\Delta m_{\rm SN}-\Delta m_{\rm
Ceph},$ (2)

where $r(z)\equiv H_{0}D(z)$ can be approximated by $r(z)\approx cz$ in the
close Hubble flow and we have defined

$\displaystyle\Delta m_{\rm SN}$ $\displaystyle\equiv m_{\rm SN}^{\rm
host}-m_{\rm SN}^{\rm flow},$ (3) $\displaystyle\Delta m_{\rm Ceph}$
$\displaystyle\equiv m_{\rm Ceph}^{\rm host}-m_{\rm Ceph}^{\rm anch}.$ (4)

Apart from getting the SNIa redshifts and anchor distances right, we thus need
to make sure there are no systematic offsets in the Cepheid and SNIa
magnitudes between host, anchor and cosmic flow galaxies. Ignoring the weak
cosmology dependence of $r(z)$ (Dhawan et al., 2020), the inferred value of
$H_{0}$ will decrease (increase) if we:

  1. 1.

Increase (decrease) the independent anchor distances, $D^{\rm anch}$.

  2. 2.

Decrease (increase) $\Delta m_{\rm SN}$.

  3. 3.

Increase (decrease) $\Delta m_{\rm Ceph}$.

Here, we focus on option 3. With regards to option 2, $\Delta m_{\rm SN}$ will
increase if SNIa in Cepheid host galaxies are systematically made brighter
than in the Hubble flow, e.g., if there is additional dust extinction not
accounted for in the host galaxies, or if the effect that SNIa in high mass
hosts are systematically brighter than in low mass galaxies, such as Cepheid
hosts, have been underestimated (see Rigault et al., 2020, and references
therein).

In terms of option 3, if there is additional dust extinction not accounted for
in the anchor galaxies, or if we have over-corrected for dust extinction in
the host galaxies, the inferred value of $H_{0}$ will decrease, and vice
versa. The fractional shift in the Hubble constant is

$\frac{\delta H_{0}}{H_{0}}=\frac{\delta r(z)}{r(z)}-\frac{\delta D^{\rm
anch}}{D^{\rm anch}}+\frac{\ln 10}{5}\left[\delta(\Delta m_{\rm
SN})-\delta(\Delta m_{\rm Ceph})\right],$ (5)

and a lower limit to the precision in $H_{0}$ is set by the precision of the
anchor distance measurements. Shifting $\delta(\Delta m_{\rm Ceph})=\pm 0.1$
will shift the Hubble constant by $\delta H_{0}\mp 4.6\,\%$.

###  4.1 Cepheid calibration

We use the Hubble Space Telescope (HST) flux in the NIR filter (${\rm H}={\rm
F160W}$) band, color calibrated using optical (${\rm V}={\rm F555W}$ and ${\rm
I}={\rm F814W}$) data, to derive Wesenheit magnitudes

$m_{\rm H}^{\rm W}\equiv m_{\rm H}-R_{\rm W}\,({\rm V}-{\rm I})=m_{\rm
H}-R_{\rm W}\,E({\rm V}-{\rm I})-R_{\rm W}\,({\rm V}-{\rm I})_{0},$ (6)

where the color excess $E\,({\rm V}-{\rm I})\equiv A_{\rm V}-A_{\rm I}=({\rm
V}-{\rm I})-({\rm V}-{\rm I})_{0}$, with $({\rm V}-{\rm I})_{0}$ the intrinsic
Cepheid color. In the last step, we see that $m_{\rm H}^{\rm W}$ is corrected
both for dust extinction, identifying the first $R_{\rm W}$ with the total to
selective extinction ratio $R_{\rm H}^{\rm VI}\equiv A_{\rm H}/(A_{\rm
V}-A_{\rm I})$, and for a possible intrinsic C-L relation, identifying the
second $R_{\rm W}$ with $\beta_{\rm H}^{\rm VI}$, as parameterized in e.g.
Madore (1982) and Madore et al. (2017). The term $\beta_{\rm H}^{\rm
VI}\,({\rm V}-{\rm I})_{0}$ corresponds to the intrinsic magnitude-color
relation at a fixed Cepheid period, whereas the correlation between the
intrinsic color with period is included in the period-luminosity (P-L)
calibration parameterized by $b_{\rm W}$ in equation 9 below. As an
alternative approach, also employed in Follin & Knox (2018), we calibrate the
C-L relation using

$m_{\rm H}^{\rm W}\equiv m_{\rm H}-R_{\rm E}\,\hat{E}\,({\rm V}-{\rm I}).$ (7)

Here, $\hat{E}\,({\rm V}-{\rm I})$ represents a proxy for the color excess
obtained by subtracting an estimate of the mean intrinsic colors, $\langle{\rm
V}-{\rm I}\rangle_{0}$ from the observed colors111Note that if $\langle{\rm
V}-{\rm I}\rangle_{0}$ is assumed to depend on the Cepheid period, the fitted
$b_{\rm W}$ parameterizing the P-L relation will shift accordingly.,

$\hat{E}\,({\rm V}-{\rm I})\equiv({\rm V}-{\rm I})-\langle{\rm V}-{\rm
I}\rangle_{0}.$ (8)

The estimated color excess $\hat{E}\,({\rm V}-{\rm I})$ also represents a
combination of dust extinction and intrinsic color, since the mean intrinsic
colors, $\langle{\rm V}-{\rm I}\rangle_{0}$ does not take into account
individual variations in Cepheid temperature along the width of the Cepheid
instability strip, see e.g. Pejcha & Kochanek (2012).

Using multi wavelength data, one can in principle attempt to distinguish the
contribution from dust and intrinsic color variations, see e.g. Pejcha &
Kochanek (2012); Madore et al. (2017) and calibrate them separately. In the
following, we will follow standard practice and assume that $R_{\rm W}$ and
$R_{\rm E}$ effectively corrects both for dust and intrinsic color variations,
but remain agnostic whether this is most effectively done using the Wesenheit
calibration using observed colors or with respect to estimated color excesses.
Taking an empirical approach, in both cases, we will fit for the values of
$R_{\rm W}$ and $R_{\rm E}$ that minimize the scatter in $m_{\rm H}^{\rm W}$.

We model the Wesenheit magnitude of the $j$th Cepheid in the $i$th SNIa host
as

$m_{{\rm H},i,j}^{\rm W}=\mu_{i}+M_{\rm H}^{\rm W}+b_{\rm W}[{\rm
P}]_{i,j}+Z_{\rm W}[{\rm M}/{\rm H}]_{i,j},$ (9)

where $[{\rm M}/{\rm H}]_{i,j}$ is a measure of the metallicity of the
Cepheid, $[{\rm P}]_{i,j}\equiv\log P_{i,j}-1$ where $P_{i,j}$ is the period
measured in days, $M_{\rm H}^{\rm W}$ the absolute Cepheid magnitude
normalized to a period of $P=10$ days and Solar metallicity and $\mu_{i}$ the
distance modulus to the $i$th galaxy. In what follows, we will allow for
separate P-L relations for short and long period Cepheids using

$b_{\rm W}[{\rm P}]_{i,j}\rightarrow b^{\rm s}_{\rm W}[{\rm P}]^{\rm
s}_{i,j}+b^{\rm l}_{\rm W}[{\rm P}]^{\rm l}_{i,j},$ (10)

where $[{\rm P}]^{\rm s}_{i,j}=0$ for Cepheids with periods $>10$ days and
$[{\rm P}]^{\rm l}_{i,j}=0$ for Cepheids with periods $<10$ days, see Section
A.

Similarly for the $j$th Cepheid in the $k$th anchor galaxy, here MW, NGC 4258
and the Large Magellanic Cloud (LMC),

$m_{{\rm H},k,j}^{\rm W}=\mu_{k}+m_{\rm H}^{\rm W}+b_{\rm W}[{\rm
P}]_{k,j}+Z_{\rm W}[{\rm M}/{\rm H}]_{k,j}.$ (11)

###  4.2 Milky Way Cepheids

Trigonometric parallaxes potentially provide the most direct calibration of
the Cepheid absolute magnitude, $M_{\rm H}^{\rm W}$. We use data from Riess et
al. (2021), with 75 MW Cepheids, out of which 68 have reliable GAIA
parallaxes. For the $j$th Cepheid in the MW,

$m_{{\rm H},j}^{\rm W}=\mu_{j}+M_{\rm H}^{\rm W}+b_{\rm W}[{\rm P}]_{j}+Z_{\rm
W}[{\rm M}/{\rm H}]_{j}.$ (12)

where the distance modulii for each Cepheid is estimated using GAIA
parallaxes, $\pi$, according to

$\pi_{j}+zp=10^{-0.2(\mu_{j}-10)},$ (13)

where $zp$ is a residual parallax calibration offset that we fit for together
with $M_{\rm H}^{\rm W},b_{\rm W}$ and $Z_{\rm W}$. In Riess et al. (2021),
$M_{\rm H}^{\rm W}$ and $zp$ are fit for using only MW data setting $b_{\rm
W}=-3.26$ and $Z_{\rm W}=-0.17$ (as fitted to all Cepheids), finding
$zp=-14\pm 6\,\mu{\rm as}$. Since we want to fit for $zp$ simultaneously with
all parameter, we write

$\displaystyle\mu_{j}$ $\displaystyle=10-\frac{5}{\ln
10}\left[\ln\pi+\ln\left(1+\frac{zp}{\pi}\right)\right]$
$\displaystyle=10-\frac{5}{\ln
10}\left[\ln\pi+\frac{zp}{\pi}+\mathcal{O}\left(\frac{zp}{\pi}\right)^{2}\right],$
(14)

effectively transforming $zp$ into a linear parameter, and

$\displaystyle m_{{\rm H},j}^{\rm W}-10+\frac{5}{\ln 10}\ln\pi$
$\displaystyle=m_{\rm H}^{\rm W}+b_{\rm W}[{\rm P}]_{j}$ $\displaystyle+Z_{\rm
W}[{\rm M}/{\rm H}]_{j}-\frac{5}{\ln 10}\frac{zp}{\pi}.$ (15)

Higher order terms, $\mathcal{O}(zp/\pi)^{2}$, are small and corrected for in
an iterative manner.

###  4.3 Type Ia Supernovae

The calibrated SNIa B-band peak magnitude in the $i$th host is modelled by

$m_{{\rm B},i}=\mu_{i}+M_{\rm B}.$ (16)

The SNIa peak apparent magnitudes need to be corrected for the width-
luminosity and C-L relations. There are several lightcurve fitting algorithms
for deriving the SNIa peak magnitude, lightcurve shape and color, the most
widely used for cosmology being the SALT2 model (Guy et al., 2007). The
derived lightcurve widths and colors are used to correct the peak magnitude
$m_{\rm B}$ in equation 16. The errors on the corrected peak magnitude include
the fitting error and a 0.1 mag term from the SNIa model added in quadrature.

###  4.4 Data

For the extra-galactic (M31 and beyond) Cepheids, including Cepheids in the
anchor galaxy NGC 4258, we use the data set from Table 4 in Riess et al.
(2016). This table is restricted to Cepheids passing a best-fit, global
$2.7\,\sigma$ outlier rejection, the impact of which is claimed to be small in
Riess et al. (2016), but not possible to confirm independently by us.

For Cepheids in the LMC, we use data in Table 2 in Riess et al. (2019). Data
for MW Cepheids, including GAIA parallax measurements are from Table 1 in
Riess et al. (2021).

Double eclipsing binaries (DEBs) provide a means to measure distances by
determining the physical sizes of the member stars via their radial velocities
and light curves (Paczynski, 1996). 20 DEBs observed using long-baseline near-
infrared interferometry give a distance to the LMC of $\mu_{\rm LMC}=18.477\pm
0.0263$ (Pietrzyński et al., 2019; Riess et al., 2019). We use the updated
distance to NGC 4258 of $\mu_{\rm LMC}=29.397\pm 0.032$ (Reid et al., 2019),
using observations of mega-masers in Keplerian motion around its central super
massive black hole.

Type Ia SN B-band magnitudes are from Table 5 in Riess et al. (2016), derived
using version 2.4 of SALT II (Betoule et al., 2014).

###  4.5 Parameter Fitting

Given the observed Cepheid magnitudes $m_{\rm H}$, colors ${\rm V}-{\rm I}$,
periods $[{\rm P}]$, metallicities $[{\rm M}/{\rm H}]$, together with the SNIa
magnitudes $m_{\rm B}$, the anchor distances $\mu_{k}$ and the MW Cepheid
parallaxes $\pi$, we can fit simultaneously for $R_{\rm W}$ or $R_{\rm E}$,
$b_{\rm W}$, $Z_{\rm W}$, the host galaxy distances $\mu_{i}$, the anchor
distances $\mu_{k}$, the GAIA parallax offset $zp$, the Cepheid absolute
magnitude $M_{\rm H}^{\rm W}$ and the SNIa absolute magnitude $M_{\rm B}$.
Since the system of equations is linear, the fit can be made analytically as
described in Section A.

Given $M_{\rm B}$, the Hubble constant is calculated as

$H_{0}=10^{M_{\rm B}/5+a_{\rm B}+5}$ (17)

where $a_{\rm B}$ is the intercept of the SNIa magnitude-redshift relation

$\displaystyle 10^{a_{\rm B}+m_{\rm B}/5}$
$\displaystyle=cz\bigg{\\{}1+\frac{1}{2}\left[1-q_{0}\right]{z}$
$\displaystyle-\frac{1}{6}\left[1-q_{0}-3q_{0}^{2}+j_{0}\right]z^{2}+O(z^{3})\bigg{\\}}$
(18)

measured with $q_{0}=-0.55$ and $j_{0}=1$ to $a_{\rm B}=0.71273\pm 0.00176$
(Riess et al., 2016).

##  5 Results for the Wesenheit calibration $R_{\rm W}\,({\rm V}-{\rm I})$

In Riess et al. (2021), a value of $H_{0}=73.2\pm 1.3$ is derived combing
anchor distances from the MW, NGC 4258 and LMC.

Using the same data with $R_{\rm W}=0.386$, using double P-L relations, doing
a full count-rate non-linearity correction following Riess et al. (2019),
identifying $[{\rm O}/{\rm H}]=[{\rm Fe}/{\rm H}]$ using $Z_{\odot}=8.824$,
and fitting for the residual GAIA offset $zp$ simultaneously with all other
parameters, we obtain $H_{0}=73.1\pm 1.3$. Here, we have added a scatter in
the P-L relation of $\sigma(m_{\rm H}^{\rm W})=0.0682$, to give a
$\chi^{2}/{\rm dof}=1$. Despite slight differences in the analysis method,
this value is in very good agreement with the value in Riess et al. (2021),
and in $4.1\,\sigma$ tension with the Planck value of $H_{0}=67.4\pm 0.5$
(Aghanim et al., 2020).

###  5.1 Systematic Shift of $R_{\rm W}$

The inferred value of $H_{0}$ will shift if there is a systematic offset in
$R_{\rm W}$ between anchor and SNIa host galaxies. We derive values for
$H_{0}$ when varying the value of $R_{\rm W}$ in the anchor(s) and the SNIa
hosts, see Figure 3. For a common $R_{\rm W}$ (indicated by the dotted line),
the inferred value of $H_{0}$ decreases when $R_{\rm W}$ is increased. Also,
$H_{0}$ is decreased when $R_{\rm W}$ is larger in anchor than in host
galaxies. Given the result in Figure 3, one could argue that a simple solution
to the Hubble tension, would be a systematic shift of $R_{\rm W}$ between
anchor and host galaxies of $\Delta R_{\rm W}\sim 0.15$, bringing $H_{0}$ as
inferred from SNIa down to the Planck value. This could be argued for, e.g.,
if dust properties in SNIa host galaxies have a systematically steeper
extinction law than the anchor galaxies. In lack of solid independent evidence
for such a systematic shift, or the value of $R_{\rm W}$ and possible
variations of it, we will next include $R_{\rm W}$ as a model parameter to be
constrained by the available data.

Figure 3: $H_{0}$ as a function of $R_{\rm W}$, as used in the Wesenheit
calibration with respect to $R_{\rm W}\,({\rm V}-{\rm I})$, in SNIa hosts and
anchor galaxies.

###  5.2 Fitting for $R_{\rm W}$

The fact that dust extinction extrapolated to the H-band is very uncertain
naturally opens up for the option of fitting for the value of $R_{\rm W}$,
either a global value common for all galaxies or individually for all
galaxies. The first option gives $R_{\rm W}=0.31\pm 0.02$ with $H_{0}=73.7\pm
1.3$ with Planck tension $4.4\,\sigma$.

If we allow for $R_{\rm W}$ to vary between galaxies we obtain the result in
Figure 4, with host galaxies systematically favoring lower values for $R_{\rm
W}$, and $H_{0}=66.9\pm 2.5$ in agreement with the Planck value.

Figure 4: Fitting individual luminosity-color relations $R_{\rm W}\,({\rm
V}-{\rm I})$ to Cepheid data. Anchor galaxies are denoted in brown and SNIa
host galaxies in petrol. The dotted line corresponds to $R_{\rm W}=0.386$ as
commonly assumed for Cepheid calibration.

With a prior constraint on the individual values of $R_{\rm W}$, centered on
$R_{\rm W}=0.386$, the inferred $H_{0}$ will transition smoothly from
$H_{0}=66.7\pm 2.7$ (for an infinite prior width) to $H_{0}=73.1\pm 1.3$ (for
zero prior width). Imposing weak prior constraints of $R_{\rm W}=0.45\pm
0.35$, guided by the discussion in Section 2, we obtain $H_{0}=69.4\pm 2.6$
with Planck tension $<1\,\sigma$.

###  5.3 Individual P-L relations

So far, we have assumed that all Cepheids can be described by a global P-L
relation, described by $b_{\rm W}^{\rm s}$ and $b_{\rm W}^{\rm l}$. However,
since there is evidence that the P-L relation can vary between galaxies
(Tammann et al., 2011; Efstathiou, 2020), in a similar spirit to our approach
of allowing $R_{\rm W}$ to vary between galaxies, we investigate to what
extent relaxing this assumption will affect the inferred Hubble constant. We
will allow for individual galactic values of $b_{{\rm W},i}$ to be fitted for
simultaneously with all other parameters, in this case restricting to the same
P-L relation for short and long period Cepheids. For a fixed global value of
$R_{\rm W}=0.386$, the resulting $b_{{\rm W},i}$ are shown in Figure 5. The
fact that the fitted $b_{{\rm W},i}$ are systematically higher in SNIa host
galaxies compared to hosts, the inferred Hubble constant is increased from
$H_{0}=73.1\pm 1.3$ to $H_{0}=76.5\pm 1.9$. If we allow for both individual
P-L and C-L relations, the result is again shifted down to $H_{0}=66.1\pm
2.7$.

Figure 5: Fitting individual P-L relations $b_{\rm W}$ to Cepheid data. Anchor
galaxies are denoted in brown and SNIa host galaxies in petrol. The dotted
line corresponds to $b_{\rm W}=-3.26$, indicating the value obtained assuming
a global P-L relation.

##  6 Results for color excess calibration $R_{\rm E}\,\hat{E}({\rm V}-{\rm
I})$

We next compare with results derived when color calibrating the Cepheid sample
with respect to the estimated color excess. We derive $\hat{E}\,({\rm V}-{\rm
I})$ subtracting mean intrinsic colors as estimated in Tammann et al. (2011),
including the quoted uncertainties and a $0.075$ dispersion in the mean
intrinsic Cepheid color between galaxies (inferred from the difference between
LMC and MW Cepheids). These colors are in good agreement with results in
Pejcha & Kochanek (2012). The intrinsic color uncertainties are included
generating random Monte Carlo samples. The mean color excess range from
$\hat{E}({\rm V}-{\rm I})=0.28$ in the LMC to $\hat{E}({\rm V}-{\rm I})=0.69$
in the MW, see also Figure 10. Calibrating using $R_{\rm E}\,\hat{E}({\rm
V}-{\rm I})$, for a fixed value of $R_{\rm E}=0.386$, we obtain $H_{0}=73.0\pm
1.3$, showing the insensitivity of calibration method for fix values of
$R_{\rm W}$ and $R_{\rm E}$.

The derived value of $H_{0}$ depends on the assumed value for $R_{\rm E}$ in
the anchor(s) and the SNIa hosts as depicted in Figure 6. We note a weaker
dependence on the values of $R_{\rm E}$ when compared to the sensitivity of
$H_{0}$ to $R_{\rm W}$ in the Wesenheit calibration, as shown in Figure 3.

Figure 6: $H_{0}$ as a function of $R_{\rm E}$ in the SNIa hosts and the
anchor galaxies when color calibrating Cepheids with respect to the estimated
color excess $R_{\rm E}\,\hat{E}({\rm V}-{\rm I})$.

###  6.1 Fitting for $R_{\rm E}$

Fitting for a global value of $R_{\rm E}$, common for all galaxies, gives
$R_{\rm E}=0.31\pm 0.02$ with $H_{0}=73.6\pm 1.3$ increasing the Planck
tension to $4.4\,\sigma$.

We next allow for $R_{\rm E}$ to vary between galaxies. Since $R_{\rm E}$
represents a combination of corrections to both dust and intrinsic colors,
both with large uncertainties in the NIR, we only impose weak prior
constraints on their values $R_{\rm E}=0.45\pm 0.35$ motivated by observations
presented in Section 2, obtaining $H_{0}=71.8\pm 1.6$ km/s/Mpc, in
$2.7\,\sigma$ tension with the Planck value, see Figure 7. The priors keep the
$1\sigma$ upper limit of the fitted values well within $R_{\rm E}>0$.
Comparing with the results for $R_{\rm W}$ in Figure 4, it is evident that we
get very similar results for $R_{\rm E}$ with increased error bars. The one
exception is LMC for which the preferred value for $R_{\rm E}$ is
significantly larger than for $R_{\rm W}$. Not imposing any priors on $R_{\rm
E}$ yields $H_{0}=70.9\pm 1.7$ km/s/Mpc, in which case for N3370 only the
$2\sigma$ upper limit has $R_{\rm E}>0$. Allowing also for individual galactic
P-L relations, we obtain $H_{0}=73.5\pm 2.0$ ($3.0\,\sigma$ tension), assuming
priors on $R_{\rm E}$.

Figure 7: The result of fitting individual galactic values for $R_{\rm E}$
using the color excess Cepheid calibration, imposing prior constraints $R_{\rm
E}=0.45\pm 0.35$.Anchor galaxies are denoted in brown and SNIa host galaxies
in petrol. The dotted line corresponds to $R_{\rm E}=0.386$.

##  7 Summary and Discussion

Figure 8:  Results for $H_{0}$ for the two main analyses employed in this
paper. The solid black line is for individual galactic values of $R_{\rm W}$
using Wesenheit magnitudes. The dark petrol lines are for individual $R_{\rm
E}$ using color excesses, with the solid line assuming prior values $R_{\rm
E}=0.45\pm 0.35$. The dashed petrol line is fitted using the Wesenheit
calibration with $R_{\rm W}=0.386$ as in Riess et al. (2021) and the dashed
brown region indicates the $1\,\sigma$ region from Planck (Aghanim et al.,
2020).  Figure 9: Comparing results for $R_{\rm V}^{\rm BV}$ as fitted using
the color excess Cepheid calibration (dark petrol) and independently from SNIa
colors in Burns et al. (2018) (light petrol). The dotted line corresponds to
$R_{\rm V}^{\rm BV}=3.1$. Shown to the right of the plot are the total
probability distribution functions for $R_{\rm V}^{\rm BV}$ for the SNIa
hosts, as derived from Cepheid and SNIa colors, respectively.

We have argued that the value inferred for the Hubble constant from SNIa
distance measurements is quite sensitive to the choice of Cepheid calibration
method. Specifically, we have compared results when color calibrating the
Cepheid magnitudes with respect to observed colors and estimated color
excesses. In both cases, guided by the lack of evidence for global dust
extinction properties at NIR wavelengths, we have allowed for individual color
calibrations in each galaxy. Regardless of whether we calibrate with respect
to observed colors or estimated color excesses, the correction terms $R_{\rm
W}\,({\rm V}-{\rm I})$ and $R_{\rm E}\,\hat{E}\,({\rm V}-{\rm I})$ only have
clear interpretations if $\\{R_{\rm W},R_{\rm E}\\}=R_{\rm H}^{\rm
VI}=\beta_{\rm H}^{\rm VI}$, correcting for both dust extinction and intrinsic
color variations. Both approaches could thus be regarded as partly
phenomenological, and partly as a physical model parameterization.

For the color excess calibration, we derive $H_{0}=71.8\pm 1.6$ in
$2.7\,\sigma$ tension with the value inferred from Planck when using a weak
prior of $R_{\rm E}=0.45\pm 0.35$. With no prior, $H_{0}=70.9\pm 1.7$
($2.0\,\sigma$ tension). Calibrating with respect to observed colors yields
$H_{0}=66.9\pm 2.5$ in agreement with the Planck value, see Figure 8. These
results are stable to the possibility of also allowing for individual galactic
P-L relations. Results for individual anchors are presented in Section C. The
lower $H_{0}$ compared to the case of a fixed global color calibration is
driven by the preferred lower values of $R_{\rm W}$ and $R_{\rm E}$ in SNIa
hosts. Comparing the SNIa host dust properties as inferred here from Cepheid
data using color excesses with independent constraints from the SNIa color
data from Burns et al. (2018) (derived using the CCM parameterization in
Cardelli et al., 1989) shows a reassuring agreement, see Figure 9, with the
one exception of SNIa 2011fe in M101.

Results for different calibration choices are summarized in Table 1 in Section
B. From the quality of the fits, there is no clear preference for any of the
calibration methods.

Regardless of calibration method, since Cepheid colors and periods are
correlated (Tammann et al., 2011), so will the inferred P-L and C-L relations,
i.e., the parameters $b_{\rm W}$ and $R_{\rm W}$ or $R_{\rm E}$. This may be
of importance if there are color selection effects related to the fact that
longer period Cepheids are brighter, see Figure 10. We have tested the
possible impact of such an effect by imposing cuts on the observed color ${\rm
V}-{\rm I}$. In Figure 11, we show the fitted $H_{0}$ as a function of the cut
in $({\rm V}-{\rm I})$ we apply, when calibrating using color excesses for a
fixed $R_{\rm E}=0.386$. The trend of obtaining a lower $H_{0}$ when cutting
out redder Cepheid is a common feature for all calibration methods we have
tested.

Finally, we note that when estimating distances using the ${\rm I}$-band for
which color corrections are larger, unsurprisingly, the sensitivity of the
result to the choice of calibration method is even larger. For example, with
the Wesenheit calibration, assuming a fixed value of $R_{\rm W,I}^{\rm
VI}=1.27$ (corresponding to $R_{\rm H}^{\rm VI}=0.386$), we obtain
$H_{0}=70.4\pm 1.5$, in $1.9\,\sigma$ tension with the CMB value, whereas
allowing for individual galactic values of $R_{\rm W,I}^{\rm VI}$ yields
$H_{0}=57.3\pm 2.7$, in $3.7\,\sigma$ tension with Planck. With the color
excess calibration, for $R_{\rm E,I}^{\rm VI}=1.27$, we obtain $H_{0}=69.6\pm
1.5$, in $1.4\,\sigma$ tension with the CMB value, whereas allowing for
individual galactic values of $R_{\rm E,I}^{\rm VI}$ yields $H_{0}=66.9\pm
1.3$, in agreement with Planck.

Figure 10: Observed Cepheid colors ${\rm V}-{\rm I}$ versus the periods
$[P]=\log P-1$. The dotted lines correspond to the upper and lower limit of
the mean intrinsic MW Cepheid color as estimated in Tammann et al. (2011).
Figure 11: Fitted $H_{0}$ as a function of the cut in $({\rm V}-{\rm I})$ for
a fixed value of $R_{\rm E}=0.386$ when calibrating using estimated color
excesses.

With the limited information at hand regarding dust extinction and intrinsic
C-L relations for Cepheids at NIR wavelengths, and our current inability to
distinguish between them in most cases, the color calibration of Cepheid
magnitudes introduces large uncertainties in the local distance ladder.
Neither approach employed in this paper is in one-to-one correspondence with
an underlying physical model and there is no clear evidence in the data for
any of them. Given that results differ by $\delta H_{0}\sim 5$, similar to the
difference between the SH0Es and the Planck value and exceeding individual
statistical errors by a factor $\gtrsim 2$, we conclude that systematic
uncertainties related to the choice of Cepheid color-luminosity calibration
method have the potential to resolve the current Hubble tension.

EM acknowledges support from the Swedish Research Council under Dnr VR
2020-03384. AG acknowledges support from the Swedish Research Council under
Dnr VR 2020-03444, and the Swedish National Space Board, grant 110-18.

##  Appendix A System of Equations

Following Riess et al. (2016), we collect all data points and their
corresponding uncertainties and possible correlations in the matrices ${\bf
Y}$ and ${\bf C}$. This includes the Wesenheit magnitudes of all Cepheids,
$m^{\rm W}_{{\rm H},i,j}$, including the anchors. The exception is the Milky
Way Cepheids for which we use $m_{\pi,j}\equiv m_{{\rm H},j}^{\rm
W}-10+\frac{5}{\ln 10}\ln\pi$. Next, we have the measured anchor distances,
$\mu_{\rm N4258},\mu_{\rm LMC}$ and possibly $\mu_{\rm M31}$. Finally, data
points include the B-band SNIa magnitudes in the Cepheid hosts, $m_{{\rm
B},i}$:

${\bf Y}=\begin{bmatrix}m_{\pi,j}\\\ m^{\rm W}_{{\rm H},1,j}\\\ \vdots\\\
m^{\rm W}_{{\rm H},19,j}\\\ m^{\rm W}_{{\rm H,N}4258,j}\\\ m^{\rm W}_{{\rm
H,LMC},j}\\\ \mu_{\rm N4258}\\\ \mu_{\rm LMC}\\\ m_{{\rm B},1}\\\ \vdots\\\
m_{{\rm B},19}\end{bmatrix},\quad{\bf
C}=\begin{bmatrix}\sigma^{2}(m_{\pi,j})&0&\ldots&&&&&0\\\ 0&\sigma^{2}(m^{\rm
W}_{\rm H})&0&\ldots&&&&\vdots\\\ \vdots&&\ddots&&&&&\\\
&&&\sigma^{2}(\mu_{\rm N4258})&&&&\\\ &&&&\sigma^{2}(\mu_{\rm LMC})&&&\\\
&&&&&\sigma^{2}(m_{{\rm B},1})&&\\\ &&&&&&\ddots&\\\
0&&&&&&&\sigma^{2}(m_{{\rm B},19})\end{bmatrix}$ (A1)

Collecting the model parameters in the matrix ${\bf X}$

${\bf X}=\begin{bmatrix}\mu_{0,1}\\\ \vdots\\\ \mu_{0,19}\\\ \mu_{\rm
N4258}\\\ \mu_{\rm LMC}\\\ m_{\rm H}^{\rm W}\\\ b_{\rm W}^{\rm s}\\\ b_{\rm
W}^{\rm l}\\\ Z_{\rm W}\\\ zp\\\ M_{\rm B}\end{bmatrix},$ (A2)

we can relate data and parameters through ${\bf Y}={\bf A}{\bf X}$ where in
schematic form

${\bf A}=\setcounter{MaxMatrixCols}{13}\begin{bmatrix}0&\ldots&&&&&&1&[{\rm
P}]^{\rm s}_{{\rm MW},1}&[{\rm P}]^{\rm l}_{{\rm MW},1}&[{\rm{\rm M}/{\rm
H}}]_{1}&\frac{-5\pi_{1}^{-1}}{\ln 10}&0\\\ \vdots&&&&&&&&&&&&\\\
0&&&&&&&1&[{\rm P}]^{\rm s}_{{\rm MW},N}&[{\rm P}]^{\rm l}_{{\rm
MW},N}&[{\rm{\rm M}/{\rm H}}]_{N}&\frac{-5\pi_{N}^{-1}}{\ln 10}&0\\\
1&0&\ldots&&&&&1&[{\rm P}]^{\rm s}&[{\rm P}]^{\rm l}&[{\rm{\rm M}/{\rm
H}}]&0&\vdots\\\ \vdots&1&0&\ldots&&&&1&\vdots&\vdots&\vdots&\vdots&\\\
&\ldots&&&&1&0&0&&&&&\\\ &&&&&&1&0&&&&&\\\ 1&0&\ldots&&&&&&&&&&1\\\
0&1&0&\ldots&&&&&&&&&1\\\ \vdots&&&&&&&&&&&&\vdots\\\
0&\ldots&&&1&0&\ldots&&&&&&1\end{bmatrix}.$ (A3)

Note that the first $N$ rows correspond to equation 4.2 and so forth, so that
${\bf A}$ is a $n_{\rm param}\times n_{\rm data}$ matrix. We can solve for the
parameter matrix ${\bf X}$ and its covariance matrix ${\bf\Sigma}$
analytically

${\bf\Sigma}=\left[{\bf A}^{T}{\bf C}^{-1}{\bf A}\right]^{-1},\quad{\bf
X}={\bf\Sigma}\left[{\bf A}^{T}{\bf C}^{-1}{\bf Y}\right].$ (A4)

Note that the formalism can easily extended to fit also for, e.g., $R_{\rm W}$
and $R_{\rm E}$, and that parameter priors are incorporated as additional data
points.

##  Appendix B Model selection

The choice of C-L and P-L calibration method can have a large impact on the
inferred $H_{0}$. Unfortunately, from data alone, it is not obvious which of
calibration models is preferred. Specifically, there is no clear preference in
data for color calibrating with respect to observed colors or estimated color
excesses. As expected, allowing for more freedom in the model, the fit will
improve222When comparing the quality of the fit for different models, we add
the same additional scatter to the P-L relation. When constraining parameters
within a given model, the scatter is adjusted to give a $\chi^{2}/{\rm
dof}=1$.. In terms of model selection, results are ambiguous, see Table 1.
Here, the Akaike Information Criterion (AIC) and the Bayesian Information
Criterion (BIC) are defined by

$\displaystyle{\rm AIC}$ $\displaystyle\equiv 2n_{\rm par}+\chi^{2},$
$\displaystyle{\rm BIC}$ $\displaystyle\equiv n_{\rm par}\ln n_{\rm
dat}+\chi^{2},$ (B1)

with the latter more penalizing for models with extra parameters for $n_{\rm
dat}>8$. In terms of the AIC, fitting for individual values of $R_{\rm W}$ or
$R_{\rm E}$ and $b_{\rm W}$ is the preferred model, whereas in terms of BIC, a
global fitted value yields the best result. As it stands, it is unclear to
what degree these results provide guidance on the choice of calibration model.

Table 1: Resulting $H_{0}$ and quality of fit for different color calibration models for Wesenheit calibration, $R_{\rm W}\,({\rm V}-{\rm I})$, and color excess calibration, $R_{\rm E}\,\hat{E}\,({\rm V}-{\rm I})$. The prior for the color excess calibration refer to $R_{\rm E}=0.45\pm 0.35$. Wesenheit cal. |  $H_{0}$ (Planck tension) | $\chi^{2}_{\rm min}$ | AIC | BIC
---|---|---|---|---
$R_{\rm W}=0.386$ |  $73.1\pm 1.3$ $(4.1\,\sigma)$ | 1616.6 | 1672.6 | 1823.9
Global $R_{\rm W}$ |  $73.7\pm 1.3$ $(4.4\,\sigma)$ | 1602.4 | 1660.4 | 1817.1
Individual $R_{\rm W}$ |  $66.9\pm 2.5$ $(0.2\,\sigma)$ | 1549.2 | 1651.2 | 1926.9
Individual $b_{\rm W}$ |  $76.5\pm 1.9$ $(4.5\,\sigma)$ | 1566.0 | 1664.0 | 1928.9
Individual $R_{\rm W}$ and $b_{\rm W}$ |  $66.4\pm 2.8$ $(0.4\,\sigma)$ | 1497.4 | 1641.4 | 2030.6
Color excess cal. |  $H_{0}$ (Planck tension) | $\chi^{2}_{\rm min}$ | AIC | BIC
$R_{\rm E}=0.386$ |  $73.0\pm 1.3$ $(4.0\,\sigma)$ | 1612.7 | 1668.7 | 1820.0
Global $R_{\rm E}$ |  $73.6\pm 1.3$ $(4.4\,\sigma)$ | 1598.9 | 1656.9 | 1813.7
Individual $R_{\rm E}$ (w prior) |  $71.8\pm 1.6$ $(2.7\,\sigma)$ | 1610.2 | 1712.2 | 1988.6
Individual $R_{\rm E}$ (wo prior) |  $70.9\pm 1.7$ $(2.0\,\sigma)$ | 1533.9 | 1635.9 | 1911.6
Individual $b_{\rm W}$ |  $76.1\pm 1.9$ $(4.4\,\sigma)$ | 1563.5 | 1661.5 | 1926.3
Individual $R_{\rm E}$ and $b_{\rm W}$ (w prior) |  $73.5\pm 2.0$ $(3.0\,\sigma)$ | 1504.8 | 1648.8 | 2039.0
Individual $R_{\rm E}$ and $b_{\rm W}$ (wo prior) |  $72.4\pm 1.9$ $(2.5\,\sigma)$ | 1487.0 | 1631.0 | 2020.2

##  Appendix C Results for Individual Anchors

In Figure 12, we show the inferred Hubble constant from each individual
anchor. Here, we have used individually fitted $R_{\rm W}$ and $R_{\rm E}$ for
each galaxy but assumed a global P-L relation. Also allowing for $b_{\rm W}$
to vary between galaxies yields similar results. Dotted lines are for a global
value $R_{\rm W}=0.386$ using the Wesenheit calibration, closest resembling
the case in Riess et al. (2021). The solid lines are for the case of fitting
for individual $R_{\rm W}$ and $R_{\rm E}$. The main difference in both cases,
besides broadening the likelihoods, is the shift of LMC anchor result.

Figure 12: Results for $H_{0}$ for different anchor distances. Dotted lines
are fitted using the Wesenheit calibration with $R_{\rm W}=0.386$ as in Riess
et al. (2021). Left panel: Solid lines are fitted for individual galactic
values of $R_{\rm W}$ using Wesenheit magnitudes. Right panel: Solid lines are
fitted for individual galactic values of $R_{\rm E}$ using estimated color
excesses.

## References

  * Abbott et al. (2017) Abbott, B. P., et al. 2017, Nature, 551, 85, doi: 10.1038/nature24471
  * Aghanim et al. (2020) Aghanim, N., Akrami, Y., Ashdown, M., et al. 2020, A&A, 641, A6, doi: 10.1051/0004-6361/201833910
  * Amanullah et al. (2014) Amanullah, R., Goobar, A., Johansson, J., et al. 2014, ApJ, 788, L21, doi: 10.1088/2041-8205/788/2/L21
  * Amanullah et al. (2015) Amanullah, R., Johansson, J., Goobar, A., et al. 2015, MNRAS, 453, 3300, doi: 10.1093/mnras/stv1505
  * Betoule et al. (2014) Betoule, M., Kessler, R., Guy, J., et al. 2014, A&A, 568, A22, doi: 10.1051/0004-6361/201423413
  * Birrer et al. (2020) Birrer, S., Shajib, A. J., Galan, A., et al. 2020, A&A, 643, A165, doi: 10.1051/0004-6361/202038861
  * Biswas et al. (2021) Biswas, R., Goobar, A., Dhawan, S., et al. 2021, arXiv e-prints, arXiv:2103.16978.  https://arxiv.org/abs/2103.16978
  * Brout & Scolnic (2021) Brout, D., & Scolnic, D. 2021, ApJ, 909, 26, doi: 10.3847/1538-4357/abd69b
  * Burns et al. (2018) Burns, C. R., Parent, E., Phillips, M. M., et al. 2018, ApJ, 869, 56, doi: 10.3847/1538-4357/aae51c
  * Burns et al. (2018) Burns, C. R., et al. 2018, ApJ, 869, 56, doi: 10.3847/1538-4357/aae51c
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245, doi: 10.1086/167900
  * Dhawan et al. (2020) Dhawan, S., Brout, D., Scolnic, D., et al. 2020, ApJ, 894, 54, doi: 10.3847/1538-4357/ab7fb0
  * Draine (2003) Draine, B. T. 2003, ARA&A, 41, 241, doi: 10.1146/annurev.astro.41.011802.094840
  * Efstathiou (2020) Efstathiou, G. 2020, A Lockdown Perspective on the Hubble Tension (with comments from the SH0ES team).  https://arxiv.org/abs/2007.10716
  * Fausnaugh et al. (2015) Fausnaugh, M. M., Kochanek, C. S., Gerke, J. R., et al. 2015, MNRAS, 450, 3597, doi: 10.1093/mnras/stv881
  * Fitzpatrick (1999) Fitzpatrick, E. L. 1999, PASP, 111, 63, doi: 10.1086/316293
  * Fitzpatrick et al. (2019) Fitzpatrick, E. L., Massa, D., Gordon, K. D., Bohlin, R., & Clayton, G. C. 2019, ApJ, 886, 108, doi: 10.3847/1538-4357/ab4c3a
  * Follin & Knox (2018) Follin, B., & Knox, L. 2018, MNRAS, 477, 4534–4542, doi: 10.1093/mnras/sty720
  * Freedman et al. (2019) Freedman, W. L., Madore, B. F., Hatt, D., et al. 2019, ApJ, 882, 34, doi: 10.3847/1538-4357/ab2f73
  * Goobar et al. (2014) Goobar, A., Johansson, J., Amanullah, R., et al. 2014, ApJ, 784, L12, doi: 10.1088/2041-8205/784/1/L12
  * Gordon et al. (2003) Gordon, K. D., Clayton, G. C., Misselt, K. A., Landolt, A. U., & Wolff, M. J. 2003, ApJ, 594, 279, doi: 10.1086/376774
  * Guy et al. (2007) Guy, J., Astier, P., Baumont, S., et al. 2007, A&A, 466, 11, doi: 10.1051/0004-6361:20066930
  * Johansson et al. (2021) Johansson, J., Cenko, S. B., Fox, O. D., et al. 2021, arXiv e-prints, arXiv:2105.06236.  https://arxiv.org/abs/2105.06236
  * Knox & Millea (2020) Knox, L., & Millea, M. 2020, Phys. Rev. D, 101, doi: 10.1103/physrevd.101.043533
  * Krisciunas et al. (2006) Krisciunas, K., Prieto, J. L., Garnavich, P. M., et al. 2006, AJ, 131, 1639, doi: 10.1086/499523
  * Madore (1982) Madore, B. F. 1982, ApJ, 253, 575, doi: 10.1086/159659
  * Madore et al. (2017) Madore, B. F., Freedman, W. L., & Moak, S. 2017, ApJ, 842, 42, doi: 10.3847/1538-4357/aa6e4d
  * Mörtsell & Dhawan (2018) Mörtsell, E., & Dhawan, S. 2018, J. Cosmology Astropart. Phys, 2018, 025–025, doi: 10.1088/1475-7516/2018/09/025
  * Nataf et al. (2016) Nataf, D. M., Gonzalez, O. A., Casagrande, L., et al. 2016, MNRAS, 456, 2692, doi: 10.1093/mnras/stv2843
  * Nobili & Goobar (2008) Nobili, S., & Goobar, A. 2008, A&A, 487, 19, doi: 10.1051/0004-6361:20079292
  * O’Donnell (1994) O’Donnell, J. E. 1994, ApJ, 422, 158, doi: 10.1086/173713
  * Paczynski (1996) Paczynski, B. 1996.  https://arxiv.org/abs/astro-ph/9608094
  * Pejcha & Kochanek (2012) Pejcha, O., & Kochanek, C. S. 2012, ApJ, 748, 107, doi: 10.1088/0004-637x/748/2/107
  * Pietrzyński et al. (2019) Pietrzyński, G., Graczyk, D., Gallenne, A., et al. 2019, Nature, 567, 200–203, doi: 10.1038/s41586-019-0999-4
  * Reid et al. (2019) Reid, M. J., Pesce, D. W., & Riess, A. G. 2019, ApJ, 886, L27, doi: 10.3847/2041-8213/ab552d
  * Riess et al. (2021) Riess, A. G., Casertano, S., Yuan, W., et al. 2021, ApJ, 908, L6, doi: 10.3847/2041-8213/abdbaf
  * Riess et al. (2019) Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85, doi: 10.3847/1538-4357/ab1422
  * Riess et al. (2019) Riess, A. G., Narayan, G., & Calamida, A. 2019, Calibration of the WFC3-IR Count-rate Nonlinearity, Sub-percent Accuracy for a Factor of a Million in Flux, Space Telescope WFC Instrument Science Report
  * Riess et al. (2016) Riess, A. G., et al. 2016, ApJ, 826, 56, doi: 10.3847/0004-637X/826/1/56
  * Rigault et al. (2020) Rigault, M., Brinnel, V., Aldering, G., et al. 2020, A&A, 644, A176, doi: 10.1051/0004-6361/201730404
  * Schlafly et al. (2017) Schlafly, E. F., Peek, J. E. G., Finkbeiner, D. P., & Green, G. M. 2017, ApJ, 838, 36, doi: 10.3847/1538-4357/aa619d
  * Tammann et al. (2011) Tammann, G. A., Reindl, B., & Sandage, A. 2011, A&A, 531, A134, doi: 10.1051/0004-6361/201016382