# On the solvability of graded Novikov algebras

###### Abstract.

We show that the right ideal of a Novikov algebra generated by the square of a
right nilpotent subalgebra is nilpotent. We also prove that a $G$-graded
Novikov algebra $N$ over a field $K$ with solvable $0$-component $N_{0}$ is
solvable, where $G$ is a finite additive abelean group and the characteristic
of $K$ does not divide the order of the group $G$. We also show that any
Novikov algebra $N$ with a finite solvable group of automorphisms $G$ is
solvable if the algebra of invariants $N^{G}$ is solvable.

Ualbai Umirbaev111Department of Mathematics, Wayne State University, Detroit,
MI 48202, USA; Department of Mathematics, Al-Farabi Kazakh National
University, Almaty, 050040, Kazakhstan; and Institute of Mathematics and
Mathematical Modeling, Almaty, 050010, Kazakhstan, e-mail<EMAIL_ADDRESS>and Viktor Zhelyabin222Institute of Mathematics of the SB of RAS, Novosibirsk,
630090, Russia, e-mail<EMAIL_ADDRESS>
Mathematics Subject Classification (2010): 17D25, 17B30, 17B70

Key words: Novikov algebra, graded algebra, solvability, nilpotency,
automorphism, the ring of invariants

##  1\. Introduction

A nonassociative algebra $N$ over a field $K$ is called a Novikov algebra [17]
if it satisfies the following identities:

(1) $\displaystyle(x,y,z)=(y,x,z)\,\text{ (left symmetry)},$ (2)
$\displaystyle(xy)z=(xz)y\,\text{ (right commutativity)},$

where $(x,y,z)=(xy)z-x(yz)$ is the associator of elements $x,y,z$.

The defining identities of a Novikov algebra first appeared in the study of
Hamiltonian operators in the formal calculus of variations by I.M. Gelfand and
I.Ya. Dorfman [8]. These identities played a crucial role in the
classification of linear Poisson brackets of hydrodynamical type by A.A.
Balinskii and S.P. Novikov [1].

In 1987 E.I. Zelmanov [25] proved that all finite-dimensional simple Novikov
algebras over a field $K$ of characteristic $0$ are one-dimensional. V.T.
Filippov [6] constructed a wide class of simple Novikov algebras of
characteristic $p\geq 0$. J.M. Osborn [17, 18, 19] and X. Xu [23, 24]
continued the study of simple finite dimensional algebras over fields of
positive characteristic and simple infinite dimensional algebras over fields
of characteristic zero. A complete classification of finite dimensional simple
Novikov algebras over algebraically closed fields of characteristic $p>2$ is
given in [23].

E.I. Zelmanov also proved that if $N$ is a finite dimensional right nilpotent
Novikov algebra then $N^{2}$ is nilpotent [25]. In 2001 V.T. Filippov [7]
proved that any left-nil Novikov algebra of bounded index over a field of
characteristic zero is nilpotent. A.S. Dzhumadildaev and K.M. Tulenbaev [5]
proved that any right-nil Novikov algebra of bounded index $n$ is right
nilpotent if the characteristic $p$ of the field $K$ is $0$ or $p>n$. In 2020
I. Shestakov and Z. Zhang proved [21] that for any Novikov algebra $N$ over a
field the following conditions are equivalent:

$(i)$ $N$ is solvable;

$(ii)$ $N^{2}$ is nilpotent;

$(iii)$ $N$ is right nilpotent.

The Freiheitssatz for Novikov algebras over fields of characteristic $0$ was
proven by L. Makar-Limanov and U. Umirbaev [15]. L.A. Bokut, Y. Chen, and Z.
Zhang [3] proved that every Novikov algebra is a subalgebra of a Novikov
algebra obtained from some differential algebra by Gelfand-Dorfman
construction [8].

This paper is devoted to the study of solvable, nilpotent, and right nilpotent
Novikov algebras and graded Novikov algebras. Notice that an algebra $A$ over
a field containing all $n$th roots of unity admits an automorphism of order
$n$ if and only if $A$ admits a $\mathbb{Z}_{n}$-grading. For this reason the
study of graded algebras is related to the study of actions of finite groups.
First we recall some definitions and classical results.

Let $R$ be an algebra over a field $K$. For any automorphism $\phi$ of $R$ the
set of fixed elements

$\displaystyle R^{\phi}=\\{x\in R|\phi(x)=x\\}$

is a subalgebra of $R$ and is called the subalgebra of invariants of $\phi$.
An automorphism $\phi$ is called regular if $R^{\phi}=0$. For any group $G$ of
automorphisms of $R$ the subalgebra of invariants

$\displaystyle R^{G}=\\{x\in R|\phi(x)=x\text{ for all }\phi\in G\\}$

is defined similarly.

In 1957 G. Higman [10] published a classical result on Lie algebras which says
that if a Lie algebra $L$ has a regular automorphism $\phi$ of prime order
$p$, then $L$ is nilpotent. It was also shown that the index of nilpotency
$h(p)$ of $L$ depends only on $p$. An explicit estimation of the function
$h(p)$ was found by A.I. Kostrikin and V.A. Kreknin [12] in 1963. A little
later, V.A. Kreknin proved [13] that a finite dimensional Lie algebra with a
regular automorphism of an arbitrary finite order is solvable. In 2005 N. Yu.
Makarenko [14] proved that if a Lie algebra $L$ admits an automorphism of
prime order $p$ with a finite-dimensional fixed subalgebra of dimension $t$,
then $L$ has a nilpotent ideal of finite codimension with the index of
nilpotency bounded in terms of $p$ and the codimension bounded in terms of $t$
and $p$.

In 1973 G. Bergman and I. Isaacs [2] published a classical result on the
actions of finite groups on associative algebras. Let $G$ be a finite group of
automorphisms of an associative algebra $R$ and suppose that $R$ has no
$|G|$-torsion. If the subalgebra of invariants $R^{G}$ is nilpotent then the
Bergman-Isaacs Theorem [2] states that $R$ is also nilpotent. Since then a
very large number of papers have been devoted to the study of automorphisms of
associative rings. The central problem of these studies was to identify the
properties of rings that can be transformed from the ring of invariants to the
whole ring. In 1974 V. K. Kharchenko [11] proved if $R^{G}$ is a PI-ring then
$R$ is a PI-ring under the conditions of the Bergman-Isaacs Theorem.

The Bergman-Isaacs Theorem was partially generalized by W.S. Martindale and S.
Montgomery [16] in 1977 to the case of a finite group of Jordan automorphisms,
that is a finite group of automorphisms of the adjoint Jordan algebra
$R^{(+)}$.

An analogue of Kharchenko’s result for Jordan algebras was proven by A. P.
Semenov [20] in 1991. In particular, A. P. Semenov proved that if $J^{G}$ is a
solvable algebra over a field of characteristic zero, then so is the Jordan
algebra $J$. His proof uses a deep result by E.I. Zel’manov [26] which says
that every Jordan nil-algebra of bounded index over a field of characteristic
zero is solvable. If a Jordan algebra $J$ over a field of characteristic not
equal to $2,3$ admits an automorphism $\phi$ of order $2$ with solvable
$J^{\phi}$, then $J$ is solvable [27].

In the case of alternative algebras one cannot expect that nilpotency of the
invariant subalgebra implies the nilpotency of the whole algebra. There is an
example (see [4, 30]) of a solvable non-nilpotent alternative algebra with an
automorphism of order two such that its subalgebra of invariants is nilpotent.
A combination of Semenov’s result [20] and Zhevlakov’s theorem [29] gives
that, for an alternative algebra $A$ over a field of characteristic zero, the
solvability of the algebra of invariants $A^{G}$ for a finite group $G$
implies the solvability of $A$. It is also known [22] that if $A$ is an
alternative algebra over a field of characteristic not equal to $2$ with an
automorphism $\phi$ of order two, then the solvability of the algebra of
invariants $A^{\phi}$ implies the solvability of $A$. In [9] M. Goncharov
proved that an alternative $\mathbb{Z}_{3}$\- graded algebra $A=A_{0}\oplus
A_{1}\oplus A_{2}$ over a field of characteristic not equal to $2,3,5$ is
solvable if $A_{0}$ is solvable.

It was shown in [28] for every $n$ of the form $n=2^{k}3^{l}$ that a
$\mathbb{Z}_{n}$\- graded Novikov

$\displaystyle N=N_{0}\oplus\ldots\oplus N_{n-1}$

over a field of characteristic not equal to $2,3$ is solvable if $N_{0}$ is
solvable.

In this paper we first prove that if $L$ is a right nilpotent subalgebra of a
Novikov algebra $N$ then the right ideal of $N$ generated by $L^{2}$ is right
nilpotent (Theorem 1). This result gives a deeper explanation of the results
on the nilpotency of $N^{2}$ mentioned above. The main result of the paper
(Theorem 2) says that if $N$ is a $G$-graded Novikov algebra with solvable
$0$-component $N_{0}$, where $G$ is a finite additive abelian group, then $N$
is solvable. This result allows us to prove (Theorem 3) that if $N$ is a
Novikov algebra with solvable algebra of invariants $N^{G}$, where $G$ is a
finite solvable group of automorphisms of $N$, then $N$ is solvable. Theorems
2 and 3 are formulated for fields of characteristic $0$ or positive
characteristic $p$ that does not divide $|G|$. Notice that the solvability and
the right nilpotency of Novikov algebras are equivalent by the result of I.
Shestakov and Z. Zhang mentioned above.

The paper is organized as follows. In Section 2 we prove some identities and
Theorem 1. Sections 3–5 are devoted to the study of the structure of
$\mathbb{Z}_{n}$-graded Novikov algebras. Theorems 2 and 3 are formulated and
proven in Section 6.

##  2\. Right nilpotent subalgebras

The identities (1) and (2) easily imply the identities

(3) $\displaystyle(xy,z,t)=(x,z,t)y$

and

(4) $\displaystyle(x,yz,t)=(x,y,t)z.$

Let $A$ be an arbitrary algebra. The powers of $A$ are defined inductively by
$A^{1}=A$ and

$\displaystyle A^{m}=\sum_{i=1}^{m-1}A^{i}A^{m-i}$

for all positive integers $m\geq 2$. The algebra $A$ is called nilpotent if
$A^{m}=0$ for some positive integer $m$.

The right powers of $A$ are defined inductively by $A^{[1]}=A$ and
$A^{[m+1]}=A^{[m]}A$ for all integers $m\geq 1$. The algebra $A$ is called
right nilpotent if there exists a positive integer $m$ such that $N^{[m]}=0$.
In general, the right nilpotency of an algebra does not imply its nilpotency.
This is also true in the case of Novikov algebras.

Example 1. [25] Let $N=Fa+Fb$ be a vector space of dimension 2. The product on
$N$ is defined as

$ab=b,a^{2}=b^{2}=ba=0.$

It is easy to check that $N$ is a right nilpotent Novikov algebra, but not
nilpotent.

The derived powers of $A$ are defined by $A^{(0)}=A$, $A^{(1)}=A^{2}$, and
$A^{(m)}=A^{(m-1)}A^{(m-1)}$ for all positive integers $m\geq 2$. The algebra
$A$ is called solvable if $A^{(m)}=0$ for some positive integer $m$. Every
right nilpotent algebra is solvable, and, in general, the converse is not
true. But every solvable Novikov algebra is right nilpotent [21].

It is well known that if $I$ and $J$ are ideals of a Novikov algebra $N$, then
$IJ$ is also an ideal of $N$. Consequently, if $N$ is a Novikov algebra then
$N^{m}$, $N^{[m]}$, and $N^{(m)}$ are ideals of $N$. If $S$ is a subset of a
Novikov algebra $N$, then denote by $\langle S\rangle$ the right ideal of $N$
generated by $S$. Notice that if $I$ is a right ideal of $N$, then $IS$ is a
right ideal of $N$ for any subset $S\subseteq N$ by (2).

In any algebra we denote by $x_{1}x_{2}\ldots x_{k}$ the right normed product
$(\ldots(x_{1}x_{2})\ldots)x_{k}$ of elements $x_{1},x_{2},\ldots,x_{k}$. For
any $x,y$ denote by $x\circ y=xy+yx$ the Jordan product.

######  Lemma 1.

Any Novikov algebra satisfies the following identities:

(5) $\displaystyle a(bx_{1}\ldots x_{t})=abx_{1}x_{2}\ldots
x_{t}-\sum_{i=1}^{t}(a,b,x_{i})x_{1}\ldots x_{i-1}x_{i+1}\ldots x_{t}$

for each positive integer $t\geq 1$,

(6) $\displaystyle(ax_{1}\ldots x_{s})\circ(bx_{s+1}\ldots x_{t})=(a\circ
b)x_{1}x_{2}\ldots x_{t}-\sum_{i=1}^{k}(a,b,x_{i})x_{1}\ldots
x_{i-1}x_{i+1}\ldots x_{t}$

for all nonnegative integers $0\leq s<t$, and

(7) $\displaystyle(ax_{1}\ldots x_{s})\circ(bx_{s+1}\ldots
x_{t})=a\circ(bx_{1}\ldots x_{t}).$

Proof. We prove (5) by induction on $t$. If $t=1$, then (5) is true by the
definition of the associator. By (4), we have

$\displaystyle a(bx_{1}\ldots x_{t})=a(bx_{1}\ldots
x_{t-1})x_{t}-(a,bx_{1}\ldots x_{t-1},x_{t})$ $\displaystyle=a(bx_{1}\ldots
x_{t-1})x_{t}-(a,b,x_{t})x_{1}\ldots x_{t-1}.$

Using this and the induction proposition, we get

$\displaystyle a(bx_{1}\ldots x_{t})=(abx_{1}x_{2}\ldots
x_{t-1}-\sum_{i=1}^{t-1}(a,b,x_{i})x_{1}\ldots x_{i-1}x_{i+1}\ldots
x_{t-1})x_{t}$ $\displaystyle-(a,b,x_{t})x_{1}\ldots
x_{t-1}=abx_{1}x_{2}\ldots x_{t}-\sum_{i=1}^{t}(a,b,x_{i})x_{1}\ldots
x_{i-1}x_{i+1}\ldots x_{t}.$

By (2), (3), and (5), we get

$\displaystyle(ax_{1}\ldots x_{s})(bx_{s+1}\ldots x_{t})=(ab)x_{1}x_{2}\ldots
x_{t}-\sum_{i=s+1}^{t}(a,b,x_{i})x_{1}\ldots x_{i-1}x_{i+1}\ldots x_{t}.$

This implies (6). The identity (7) is a direct consequence of (2), (5), and
(6). $\Box$

Let $N$ be a Novikov algebra and let $L$ be a subalgebra of $N$. Set $L_{0}=N$
and $L_{k}=\langle L^{[k]}\rangle$ for each positive integer $k$.

Consider the descending sequence of right ideals

$\displaystyle N=L_{0}\supseteq L_{1}\supseteq L_{2}\supseteq\ldots\supseteq
L_{k}\supseteq\ldots$

of the algebra $N$.

######  Lemma 2.

$L_{s}L_{t}\subseteq L_{s+t-1}$ for all positive integers $s,t$.

Proof. We prove the lemma by induction on $t$. It is true for $t=1$ by the
definition of $L_{s}$. Notice that

$\displaystyle L_{s}=L_{1}\underbrace{L\ldots L}_{s-1}$

for each $s\geq 1$ by (2).

Suppose that $t\geq 2$ and let $x\in L_{s}$ and $y=za_{1}\ldots a_{t-1}\in
L_{t}$, where $z\in L_{1}$ and $a_{1},\ldots,a_{t-1}\in L$. By (5), we get

$\displaystyle xy=xza_{1}\ldots
a_{t-1}-\sum_{i=1}^{t-1}(x,z,a_{i})a_{1}\ldots\widehat{a_{i}}\ldots a_{t-1}$

where $\widehat{a_{i}}$ means that $a_{i}$ is absent. Notice that $xz\in
L_{s}$ and

$\displaystyle xza_{1}\ldots a_{t-1}\in L_{s}\underbrace{L\ldots
L}_{t-1}=L_{s+t-1}.$

Moreover, $(x,z,a_{i})$ belongs to the right ideal generated by
$(L^{[s]},L,a_{i})$ by (3) and (4). Consequently, $(x,z,a_{i})\in L_{s+1}$ and
$(x,z,a_{i})a_{1}\ldots\widehat{a_{i}}\ldots a_{t-1}\in L_{s+t-1}$. $\Box$

In general, $L_{1}$ is not an ideal of $L_{0}=N$.

Example 2. Let $K[x,y]$ be the polynomial algebra over $K$ in the variables
$x,y$. Define a new product $\cdot$ on $K[x,y]$ by

$f\cdot g=f\frac{\partial g}{\partial x},\,f,g\in K[x,y].$

Then $N=(K[x,y],\cdot)$ is a Novikov algebra.

Let $L=Kx$. Then $L$ is a subalgebra of $N$ since $x\cdot x=x$. Let
$L_{1}=\langle L\rangle$. It is clear that $L_{1}\subseteq xK[x,y]$. Hence,

$y\cdot x=y\frac{\partial x}{\partial x}=y\not\in L_{1}.$

Consequently, $L_{1}$ is not an ideal of $L_{0}=N$.

But for each $r\geq 2$ the right ideal $L_{r}$ is an ideal of $L_{1}$ by Lemma
2.

######  Corollary 1.

$L_{2}^{n}\subseteq L_{n+1}$ for all $n\geq 1$.

Proof. It is trivial for $n=1$ and true for $n=2$ by Lemma 2. If
$L_{2}^{i}\subseteq L_{2+i-1}$ and $L_{2}^{j}\subseteq L_{2+j-1}$, then
$L_{2}^{i}L_{2}^{j}\subseteq L_{i+1}L_{j+1}\subseteq L_{i+j+1}$. Leading an
induction on $n$ we get

$\displaystyle L_{2}^{n}=\sum_{i+j=n,i,j\geq 1}L_{2}^{i}L_{2}^{j}\subseteq
L_{n+1}.\ \ \ \Box$

######  Theorem 1.

Let $L$ be a right nilpotent subalgebra of a Novikov algebra $N$ over a field
$K$. Then the right ideal $L_{2}=\langle L^{2}\rangle$ of $N$ generated by
$L^{2}$ is nilpotent.

Proof. Suppose that $L^{[n]}=0$ for some $n\geq 2$. Then $L_{n}=0$. By
Corollary 1, we have $L_{2}^{n-1}\subseteq L_{n}=0$. This means that $L_{2}$
is nilpotent. $\Box$

##  3\. $\mathbb{Z}_{n}$-graded Novikov algebras

Let $\mathbb{Z}_{n}=\mathbb{Z}/n\mathbb{Z}$ be the additive cyclic group of
order $n$. Let

(8) $\displaystyle N=N_{0}\oplus N_{1}\oplus N_{2}\oplus\ldots\oplus N_{n-1},\
\ N_{i}N_{j}\subseteq N_{i+j},\ i,j\in\mathbb{Z}_{n},$

be a $\mathbb{Z}_{n}$-graded Novikov algebra over $K$.

If $f\in N_{i}$ then we say that $f$ is a homogeneous element of degree $i$.
Notice that $i$ is an element of $\mathbb{Z}_{n}$. Sometimes we consider the
subscripts $i$ of $N_{i}$ as integers satisfying the condition $0\leq i\leq
n-1$.

Obviously, $A=N_{0}$ is a subalgebra of $N$. Recall that $A^{[r]}$ is the
right $r$th power of $A$.

######  Lemma 3.

Let $i_{1},i_{2},\ldots,i_{k}\in\mathbb{Z}_{n}$ and
$i_{1}+i_{2}+\ldots+i_{k}=0$. Then

$\displaystyle A^{[r]}N_{i_{1}}N_{i_{2}}\ldots N_{i_{k}}\subseteq A^{[r]}.$

Proof. By the definition of a $\mathbb{Z}_{n}$-graded algebra, we have

$\displaystyle AN_{i_{1}}N_{i_{2}}\ldots N_{i_{k}}\subseteq A.$

Using this and (2), we get

$\displaystyle A^{[r]}N_{i_{1}}N_{i_{2}}\ldots
N_{i_{k}}=AN_{i_{1}}N_{i_{2}}\ldots N_{i_{k}}\underbrace{A\ldots
A}_{r-1}\subseteq A\underbrace{A\ldots A}_{r-1}=A^{[r]}.\ \ \Box$

Set $A^{\\{0\\}}=N$ and for any integer $r\geq 1$ denote by
$A^{\\{r\\}}=\langle A^{[r]}\rangle$ the right ideal of $N$ generated by
$A^{[r]}$. Obviously, $A^{\\{r\\}}$ is a $\mathbb{Z}_{n}$-graded algebra,
i.e.,

$\displaystyle A^{\\{r\\}}=A^{\\{r\\}}_{0}\oplus A^{\\{r\\}}_{1}\oplus
A^{\\{r\\}}_{2}\oplus\ldots\oplus A^{\\{r\\}}_{n-1}.$

######  Corollary 2.

If $r\geq 1$ and $0\leq i\leq n-1$, then

$\displaystyle
A^{\\{r\\}}_{i}=\sum_{i_{1},i_{2},\ldots,i_{k}}A^{[r]}N_{i_{1}}N_{i_{2}}\ldots
N_{i_{k}},$

where $0\leq i_{1},i_{2},\ldots,i_{k}\leq n-1$,
$i_{1}+i_{2}+\ldots+i_{k}\equiv i(\mathrm{mod}\ n)$ and
$i_{1}+i_{2}+\ldots+i_{k}<n$.

In particular, $A^{\\{r\\}}_{0}=A^{[r]}$.

Consider the descending sequence of right ideals

(9) $\displaystyle N=A^{\\{0\\}}\supseteq A^{\\{1\\}}\supseteq\ldots\supseteq
A^{\\{r\\}}\supseteq\ldots$

of the algebra $N$ and the quotient algebra

(10) $\displaystyle B=A^{\\{1\\}}/A^{\\{2\\}}=B_{0}\oplus B_{1}\oplus
B_{2}\oplus\ldots\oplus B_{n-1},\ \ B_{i}B_{j}\subseteq B_{i+j},\
i,j\in\mathbb{Z}_{n}.$

Notice that $B$ is a right $N$-module. We establish some properties of the
algebra $B$.

######  Lemma 4.

Let $B$ be the Novikov algebra defined by (10). Then

$(i)$ $B_{0}=A/A^{2}$;

$(ii)$ $BB_{0}=0$;

$(iii)$ $x\circ y=xy+yx=0$ for any $x\in B_{i}$ and $y\in B_{n-i}$.

Proof. The statement (i) is true since $A^{\\{r\\}}_{0}=A^{[r]}$ by Corollary
2. The statement (ii) is a direct corollary of the inclusion
$A^{\\{r\\}}A\subseteq A^{\\{r+1\\}}$.

Let $x=ax_{1}x_{2}\ldots x_{s}\in A^{\\{1\\}}_{i}$ and
$y=bx_{s+1}x_{s+2}\ldots x_{t}\in A^{\\{1\\}}_{n-i}$, where $a,b\in A$ and
$x_{r}\in N_{k_{r}}$ for all $1\leq r\leq t$. If $i=0$, then $A_{i}=A_{n-i}=A$
and $xy,yx\in A^{2}$. Suppose that $i,n-i\neq 0$. Then
$\Sigma_{r=1}^{s}k_{r}=i\neq 0$, $\Sigma_{r=s+1}^{t}l_{r}=n-i\neq 0$, and
$\Sigma_{r=1}^{t}k_{r}=0$. In particular, $t>s\geq 1$. By (7), we have

$\displaystyle x\circ y=(ax_{1}\ldots x_{s})\circ(bx_{s+1}\ldots
x_{t})=a\circ(bx_{1}\ldots x_{t}).$

The condition $\Sigma_{r=1}^{t}k_{r}=0$ implies that $bx_{1}\ldots x_{t}\in
A$. Consequently, $x\circ y\in A^{2}$. This proves $(iii)$. $\Box$

##  4\. Right nilpotency modulo $A^{\\{1\\}}$

In this section we show that if the $0$-component $A=N_{0}$ of a
$\mathbb{Z}_{n}$-graded Novikov algebra $N$ of the form (8) is right nlpotent,
then $N^{[m]}\subseteq A^{\\{1\\}}$ for some positive integer $m$.

######  Lemma 5.

Let $N$ be an arbitrary Novikov algebra and let $V$ be a subspace of $N$. Then
for any $r\geq 1$ we have

$\displaystyle NV^{[r]}V\subseteq\langle V^{[r]}\rangle+NV^{[r+1]}.$

Proof. By (1), we get

$\displaystyle(NV^{[r]})V\subseteq(N,V^{[r]},V)+NV^{[r+1]}\subseteq(V^{[r]},N,V)+NV^{[r+1]}\subseteq\langle
V^{[r]}\rangle+NV^{[r+1]}.\ \ \Box$

######  Corollary 3.

If $r\geq 1$, then

$\displaystyle N\underbrace{V\ldots V}_{r+1}\subseteq\langle
V^{[r]}\rangle+NV^{[r+1]}.$

Proof. It is true for $r=1$ by Lemma 5. If it is true for some $r\geq 1$, then
we get

$\displaystyle N\underbrace{V\ldots V}_{r+2}\subseteq\langle V^{[r]}\rangle
V+NV^{[r+1]}V\subseteq\langle V^{[r+1]}\rangle+NV^{[r+2]}$

by (2) and Lemma 5. $\Box$

######  Lemma 6.

Let $N$ be an arbitrary $\mathbb{Z}_{n}$-graded Novikov algebra $N$ from (8)
and suppose that the $0$-component $A=N_{0}$ of $N$ is right nilpotent. Then
there exists a positive integer $m$ such that $N^{[m]}\subseteq A^{\\{1\\}}$.

Proof. Suppose that $A^{[r]}=0$ for some positive integer $r$. By Corollary 3,

$\displaystyle N\underbrace{A\ldots A}_{r+1}\subseteq\langle
A^{[r]}\rangle+NA^{[r+1]}=0.$

Again, by Corollary 3, we get

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{n}\subseteq\langle
N_{i}^{[n-1]}\rangle+NN_{i}^{[n]}.$

Notice that $N_{i}^{[n]}\subseteq A$. Consequently,

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{n+1}\subseteq(\langle
N_{i}^{[n-1]}\rangle+NA)N_{i}\subseteq\langle
N_{i}^{[n]}\rangle+NN_{i}A\subseteq A^{\\{1\\}}+NN_{i}A.$

Using this, we can easily show by induction on $s$ that

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{sn+1}\subseteq
A^{\\{1\\}}+NN_{i}\underbrace{A\ldots A}_{s}.$

Cosequently,

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{(r+1)n+1}\subseteq
A^{\\{1\\}}+NN_{i}\underbrace{A\ldots A}_{r+1}\subseteq A^{\\{1\\}}$

since $N\underbrace{A\ldots A}_{r+1}=0$.

Thus, every $N_{i}$ acts on $N$ nilpotently modulo $A^{\\{1\\}}$ from the
right hand side. Moreover, by (2), this action is commutative. This easily
implies the existence of an integer $m$ such that $N^{[m]}\subseteq
A^{\\{1\\}}$. $\Box$

##  5\. Right nilpotency of $B$

In this section we prove that any $\mathbb{Z}_{n}$-graded Novikov algebra $B$
defined by (10) is right nilpotent if the characteristic of $K$ does not
divide $n$. Suppose that $N$ is a $\mathbb{Z}_{n}$-graded Novikov algebra of
the form (8) satisfying the conditions

$(a)$ $NA=0$ and

$(b)$ $x\circ y=xy+yx=0$ for any $x\in N_{i}$ and $y\in N_{n-i}$ and for any
$i\in\mathbb{Z}_{n}$.

All statements in this section are formulated for the algebra $N$.

First we prove the following lemma.

######  Lemma 7.

Let $x\in N_{n-i}$, $u\in N_{i}^{[k]}$, $i\in\mathbb{Z}_{n}$, and $k\geq 1$.
Then $xu=-kux$.

Proof. We prove the statement of the lemma by induction on $k$. If $k=1$, then
it is true by $(b)$. Suppose that $k>1$ and $u=vy$, where $v\in N_{i}^{[k-1]}$
and $y\in N_{i}$. Using (1), (2), and the induction proposition, we get

$\displaystyle xu=x(vy)=-(x,v,y)+(xv)y=-(v,x,y)-(k-1)(vx)y$
$\displaystyle=-(vx)y+v(xy)-(k-1)(vx)y=-k(vy)x+v(xy)=-kux+v(xy).$

Notice that $xy\in N_{n-i}N_{i}\subseteq A$ and $v(xy)=0$ by the condition
$(a)$. Consequently, $xu=-kux$. $\Box$

######  Corollary 4.

If the characteristic of the field $K$ does not divide $n$, then
$N_{i}^{[n]}N_{n-i}=0$ for any $i\in\mathbb{Z}_{n}$.

Proof. Notice that $N_{i}^{[n]}\subseteq A$ and $N_{n-i}N_{i}^{[n]}=0$ by the
condition $(a)$. Then Lemma 7 gives that $nN_{i}^{[n]}N_{n-i}=0$. If the
characteristic of $K$ does not divide $n$, then this gives
$N_{i}^{[n]}N_{n-i}=0$. $\Box$

######  Lemma 8.

If the characteristic of the field $K$ does not divide $n$, then

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{2n}=0$

for any $i\in\mathbb{Z}_{n}$.

Proof. Corollary 3 and the condition $(a)$ give that

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{n}\subseteq\langle
N_{i}^{[n-1]}\rangle+NN_{i}^{[n]}\subseteq\langle N_{i}^{[n-1]}\rangle$

since $N_{i}^{[n]}\subseteq A$. Notice that $i(n-1)=-i=n-i$ in
$\mathbb{Z}_{n}$. This means $N_{i}^{[n-1]}\subseteq N_{n-i}$. Cosequently,

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{n}\subseteq\langle
N_{n-i}\rangle.$

Using (2) and $(a)$, we get

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{n+1}\subseteq\langle
N_{n-i}\rangle N_{i}=\langle N_{n-i}N_{i}\rangle=\langle N_{i}N_{n-i}\rangle.$

Then

$\displaystyle N\underbrace{N_{i}\ldots N_{i}}_{2n}\subseteq=\langle
N_{i}N_{n-i}\rangle\underbrace{N_{i}\ldots N_{i}}_{n-1}=\langle
N_{i}^{[n]}N_{n-i}\rangle.$

Corollary 4 implies the statement of the lemma. $\Box$

######  Proposition 1.

Let $N$ be a $\mathbb{Z}_{n}$-graded Novikov algebra of the form (8)
satisfying the conditions

$(a)$ $NA=0$ and

$(b)$ $x\circ y=xy+yx=0$ for any $x\in N_{i}$ and $y\in N_{n-i}$ and for any
$i\in\mathbb{Z}_{n}$.

If the characteristic of the field $K$ does not divide $n$, then $N$ is right
nilpotent.

Proof. By Lemma 8, every $N_{i}$ acts nilpotently on the right $N$-module $N$.
Moreover, this action is commutative by (2). Consequently, $N$ acts
nilpotently on $N$. $\Box$

##  6\. Solvability and right nilpotency

The solvability and the right nilpotency of Novikov algebras are equivalent
[21]. In this section we use these notions as synonyms.

######  Proposition 2.

Let $N$ be a $\mathbb{Z}_{n}$-graded Novikov algebra of the form (8) such that
$A=N_{0}$ is solvable. If the characteristic of the field $K$ does not divide
$n$, then $N$ is solvable.

Proof. Consider the descending sequence of right ideals (9). By Lemma 6 there
exists a positive integer $m$ such that $N^{[m]}\subseteq A^{\\{1\\}}$. The
algebra $B$ from (10) satisfies all conditions of Proposition 1 by Lemma 4. By
Proposition 1 there exists a positive integer $t$ such that $B^{[t]}=0$. This
means that $(A^{\\{1\\}})^{[t]}\subseteq A^{\\{2\\}}$. By Theorem 1, the
algebra $A^{\\{2\\}}$ is nilpotent. Consequently, $A^{\\{1\\}}$ and $N$ are
both solvable. $\Box$

Let $G$ be an additive abelian group. We say that

$\displaystyle N=\bigoplus_{g\in G}N_{g}$

is a $G$-graded algebra if $N_{g}N_{h}\subseteq N_{g+h}$ for all $g,h\in G$.

######  Theorem 2.

Let $G$ be a finite additive abelian group and let $N$ be a $G$-graded Novikov
algebra with solvable $0$-component $N_{0}$. If the characteristic of the
field $K$ does not divide the order of the group $G$, then $N$ is solvable.

Proof. We prove the statement of the theorem by induction on the order $|G|$
of $G$. If $G=\mathbb{Z}_{n}$, then $N$ is solvable by Proposition 2.

Every finite abelian group is a direct sum of cyclic subgroups. Suppose that
$G=\mathbb{Z}_{n_{1}}\oplus\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{k}}$,
where $n_{i}>1$ for all $i$ and $k\geq 2$. Then $G=\mathbb{Z}_{n_{1}}\oplus
G_{1}$, where $G_{1}=\mathbb{Z}_{n_{2}}\oplus\ldots\oplus\mathbb{Z}_{n_{k}}$.
Denote by $\mathrm{pr}$ the projection of $G$ onto the group
$\mathbb{Z}_{n_{1}}$. Set

$\displaystyle N_{i}^{\prime}=\sum_{g\in G,pr(g)=i}N_{g},$

where $i=0,1,\ldots,n_{1}-1$. It is easy to show that

$\displaystyle N=N^{\prime}_{0}\oplus\ldots N^{\prime}_{n_{1}-1}$

and $N$ is a $\mathbb{Z}_{n_{1}}$-graded algebra.

It is also clear that

$\displaystyle N_{0}^{\prime}=\sum_{g\in G,pr(g)=0}N_{g}$

is a $G_{1}$-graded algebra and the $0$-component of $N_{0}^{\prime}$ is
$N_{0}$. Since $|G_{1}|<|G|$ it follows that $N_{0}^{\prime}$ is solvable by
the induction proposition. Now we can apply Proposition 2 to the
$\mathbb{Z}_{n_{1}}$-graded algebra $N$. Hence $N$ is solvable. $\Box$

The statement of the next lemma is well known.

######  Lemma 9.

Let $G$ be a group of automorphisms of an arbitrary algebra $A$ and let $H$ be
a normal subgroup of $G$. Then $A^{H}$ is $G$-invariant, the quotient group
$G/H$ acts on $A^{H}$ by automorphisms, and $(A^{H})^{G/H}=A^{G}$.

Proof. Let $a\in A^{H}$ and let $g\in G$. Then $ghg^{-1}\in H$ for any $h\in
H$. Therefore, $a^{ghg^{-1}}=a$ and

$(a^{g})^{h}=a^{gh}=a^{ghg^{-1}g}=(a^{ghg^{-1}})^{g}=a^{g}.$

Consequently, the algebra $A^{H}$ is $G$-invariant. Let $g\in G$ and let
$\overline{g}$ be the image of $g$ in $G/H$. Then $\overline{g}$ defines an
automorphism of the algebra $A$ by the rule $a^{\overline{g}}=a^{g}$. This
action is well defined. Hence the quotient group $G/H$ acts on $A^{H}$. It is
easy to check that $(A^{H})^{G/H}=A^{G}$. $\Box$

######  Corollary 5.

Let $N$ be a Novikov algebra and let $G$ be a finite abelian group of
automorphisms of $N$. If the algebra $N^{G}$ is solvable and the
characteristic of the field $K$ does not divide the order of the group $G$,
then $N$ is solvable.

Proof. We may assume that $K$ is algebraically closed. We prove the statement
of the corollary by induction on the order $|G|$ of $G$. If $G$ is a simple
group, then $G\cong\mathbb{Z}_{p}$, where $p$ is a prime number. Let $\phi$ be
a generating element of the group $G$. Then $\phi^{p}=e$, where $e$ is the
identity element of $G$. Let $\epsilon$ be a primitive $p$th root of unity and
let $N_{i}=\ker(\phi-\epsilon^{i})$ for all $0\leq i\leq p-1$. The indexes $i$
may be considered as elements of $\mathbb{Z}_{p}$ since $\epsilon^{p}=1$.
Obviously,

$\displaystyle N=N_{0}\oplus\ldots\oplus N_{p-1}$

and it is easy to check that $N_{i}N_{j}\subseteq N_{i+j}$ for all
$i,j\in\mathbb{Z}_{p}$, i.e., $N$ is a $\mathbb{Z}_{p}$-graded algebra.
Moreover, $N_{0}=N^{G}$. By Proposition 2, $N$ is solvable.

Let $H$ be a proper subgroup of $G$. Then, by Lemma 9, the quotient group
$G/H$ acts on $N^{H}$ by automorphisms and $(N^{H})^{G/H}=N^{G}$. We get that
$N^{H}$ is solvable by the induction proposition since $|G/H|<|G|$. Now we can
apply the induction proposition to the group $H$ and get that $N$ is solvable.
$\Box$

######  Theorem 3.

Let $N$ be a Novikov algebra and let $G$ be a finite solvable group of
automorphisms of $N$. If the algebra $N^{G}$ is solvable and the
characteristic of the field $K$ does not divide the order of the group $G$,
then $N$ is solvable.

Proof. We prove the statement of the theorem by induction on $|G|$. The case
of abelian groups is considered in Corollary 5. Suppose that $G$ is not
abelian. Then the commutator subgroup $G^{\prime}$ of the solvable finite
group $G$ is a proper normal subgroup.

By Lemma 9, $(N^{G^{\prime}})^{G/G^{\prime}}=N^{G}$. Then the algebra
$N^{G^{\prime}}$ is solvable by the induction proposition since
$|G/G^{\prime}|<|G|$. Applying the induction proposition to $G^{\prime}$, we
get that $N$ is solvable. $\Box$

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# Improving cosmological constraints from galaxy cluster number counts with
CMB-cluster-lensing data: Results from the SPT-SZ survey and forecasts for the
future

P. S. Chaubal  School of Physics, University of Melbourne, Parkville, VIC
3010, Australia  C. L. Reichardt  School of Physics, University of Melbourne,
Parkville, VIC 3010, Australia  N. Gupta  School of Physics, University of
Melbourne, Parkville, VIC 3010, Australia  CSIRO Astronomy and Space Science,
PO Box 1130, Bentley WA 6102, Australia  B. Ansarinejad  School of Physics,
University of Melbourne, Parkville, VIC 3010, Australia  K. Aylor  Department
of Physics, University of California, Davis, CA, USA 95616  L. Balkenhol
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia  E.
J. Baxter  Center for Particle Cosmology, Department of Physics and Astronomy,
University of Pennsylvania, Philadelphia, PA, USA 19104  Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL,
USA 60637  F. Bianchini  Dept. of Physics, Stanford University, 382 Via Pueblo
Mall, Stanford, CA 94305  B. A. Benson  Fermi National Accelerator Laboratory,
MS209, P.O. Box 500, Batavia, IL 60510  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, IL, USA 60637  Department of
Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637  L.
E. Bleem  High Energy Physics Division, Argonne National Laboratory, Argonne,
IL, USA 60439  Kavli Institute for Cosmological Physics, University of
Chicago, Chicago, IL, USA 60637  S. Bocquet  Faculty of Physics, Ludwig-
Maximilians-Universität, 81679 München, Germany  J. E. Carlstrom  Kavli
Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA
60637  Department of Physics, University of Chicago, Chicago, IL, USA 60637
High Energy Physics Division, Argonne National Laboratory, Argonne, IL, USA
60439  Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL, USA 60637  Enrico Fermi Institute, University of Chicago,
Chicago, IL, USA 60637  C. L. Chang  High Energy Physics Division, Argonne
National Laboratory, Argonne, IL, USA 60439  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, IL, USA 60637  Department of
Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637  T.
M. Crawford  Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL, USA 60637  Department of Astronomy and Astrophysics, University
of Chicago, Chicago, IL, USA 60637  A. T. Crites  Kavli Institute for
Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL,
USA 60637  California Institute of Technology, Pasadena, CA, USA 91125  T. de
Haan  Department of Physics and McGill Space Institute, McGill University,
Montreal, Quebec H3A 2T8, Canada  Department of Physics, University of
California, Berkeley, CA, USA 94720  M. A. Dobbs  Department of Physics and
McGill Space Institute, McGill University, Montreal, Quebec H3A 2T8, Canada
Canadian Institute for Advanced Research, CIFAR Program in Cosmology and
Gravity, Toronto, ON, M5G 1Z8, Canada  W. B. Everett  Center for Astrophysics
and Space Astronomy, Department of Astrophysical and Planetary Sciences,
University of Colorado, Boulder, CO, 80309  B. Floyd  Department of Physics
and Astronomy, University of Missouri-Kansas City, 5110 Rockhill Road, Kansas
City, MO 64110, USA  E. M. George  Department of Physics, University of
California, Berkeley, CA, USA 94720  European Southern Observatory, Karl-
Schwarzschild-Straße 2, 85748 Garching, Germany  N. W. Halverson  Center for
Astrophysics and Space Astronomy, Department of Astrophysical and Planetary
Sciences, University of Colorado, Boulder, CO, 80309  Department of Physics,
University of Colorado, Boulder, CO, 80309  W. L. Holzapfel  Department of
Physics, University of California, Berkeley, CA, USA 94720  J. D. Hrubes
University of Chicago, Chicago, IL, USA 60637  L. Knox  Department of Physics,
University of California, Davis, CA, USA 95616  A. T. Lee  Department of
Physics, University of California, Berkeley, CA, USA 94720  Physics Division,
Lawrence Berkeley National Laboratory, Berkeley, CA, USA 94720  D. Luong-Van
University of Chicago, Chicago, IL, USA 60637  J. J. McMahon  Department of
Physics, University of Michigan, Ann Arbor, MI, USA 48109  S. S. Meyer  Kavli
Institute for Cosmological Physics, University of Chicago, Chicago, IL, USA
60637  Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL, USA 60637  Enrico Fermi Institute, University of Chicago,
Chicago, IL, USA 60637  Department of Physics, University of Chicago, Chicago,
IL, USA 60637  L. M. Mocanu  Kavli Institute for Cosmological Physics,
University of Chicago, Chicago, IL, USA 60637  Department of Astronomy and
Astrophysics, University of Chicago, Chicago, IL, USA 60637  J. J. Mohr
Faculty of Physics, Ludwig-Maximilians-Universität, 81679 München, Germany
Excellence Cluster Universe, 85748 Garching, Germany  Max-Planck-Institut für
extraterrestrische Physik, 85748 Garching, Germany  T. Natoli  Kavli Institute
for Cosmological Physics, University of Chicago, Chicago, IL, USA 60637
Department of Physics, University of Chicago, Chicago, IL, USA 60637  Dunlap
Institute for Astronomy & Astrophysics, University of Toronto, 50 St George
St, Toronto, ON, M5S 3H4, Canada  S. Padin  Kavli Institute for Cosmological
Physics, University of Chicago, Chicago, IL, USA 60637  Department of
Astronomy and Astrophysics, University of Chicago, Chicago, IL, USA 60637  C.
Pryke  Department of Physics, University of Minnesota, Minneapolis, MN, USA
55455  J. E. Ruhl  Physics Department, Center for Education and Research in
Cosmology and Astrophysics, Case Western Reserve University,Cleveland, OH, USA
44106  F. Ruppin  Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  L. Salvati  INAF-Osservatorio Astronomico di Trieste, via G. B.
Tiepolo 11, I-34143 Trieste, Italy  IFPU - Institute for Fundamental Physics
of the Universe, via Beirut 2, 34014 Trieste, Italy  Universite Paris-Saclay,
CNRS, Institut d’Astrophysique Spatiale, 91405, Orsay, France  A. Saro
Astronomy Unit, Department of Physics, University of Trieste, via Tiepolo 11,
I-3413 Trieste, Italy  INAF-Osservatorio Astronomico di Trieste, via G. B.
Tiepolo 11, I-34143 Trieste, Italy  IFPU - Institute for Fundamental Physics
of the Universe, via Beirut 2, 34014 Trieste, Italy  INFN-Sezione di Trieste,
Trieste, Italy  K. K. Schaffer  Kavli Institute for Cosmological Physics,
University of Chicago, Chicago, IL, USA 60637  Enrico Fermi Institute,
University of Chicago, Chicago, IL, USA 60637  Liberal Arts Department, School
of the Art Institute of Chicago, Chicago, IL, USA 60603  E. Shirokoff
Department of Physics, University of California, Berkeley, CA, USA 94720
Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL,
USA 60637  Department of Astronomy and Astrophysics, University of Chicago,
Chicago, IL, USA 60637  Z. Staniszewski  Physics Department, Center for
Education and Research in Cosmology and Astrophysics, Case Western Reserve
University,Cleveland, OH, USA 44106  Jet Propulsion Laboratory, California
Institute of Technology, Pasadena, CA 91109, USA  A. A. Stark  Harvard-
Smithsonian Center for Astrophysics, Cambridge, MA, USA 02138  J. D. Vieira
Astronomy Department, University of Illinois at Urbana-Champaign, 1002 W.
Green Street, Urbana, IL 61801, USA  Department of Physics, University of
Illinois Urbana-Champaign, 1110 W. Green Street, Urbana, IL 61801, USA  R.
Williamson  Kavli Institute for Cosmological Physics, University of Chicago,
Chicago, IL, USA 60637  Department of Astronomy and Astrophysics, University
of Chicago, Chicago, IL, USA 60637

###### Abstract

We show the improvement to cosmological constraints from galaxy cluster
surveys with the addition of CMB-cluster lensing data. We explore the
cosmological implications of adding mass information from the 3.1 $\sigma$
detection of gravitational lensing of the cosmic microwave background (CMB) by
galaxy clusters to the Sunyaev-Zel’dovich (SZ) selected galaxy cluster sample
from the 2500 deg2 SPT-SZ survey and targeted optical and X-ray followup data.
In the $\Lambda\mathrm{CDM}$ model, the combination of the cluster sample with
the Planck power spectrum measurements prefers
$\sigma_{8}\left(\Omega_{m}/0.3\right)^{0.5}=0.831\pm 0.020$. Adding the
cluster data reduces the uncertainty on this quantity by a factor of $1.4$,
which is unchanged whether or not the 3.1 $\sigma$ CMB-cluster lensing
measurement is included. We then forecast the impact of CMB-cluster lensing
measurements with future cluster catalogs. Adding CMB-cluster lensing
measurements to the SZ cluster catalog of the on-going SPT-3G survey is
expected to improve the expected constraint on the dark energy equation of
state $w$ by a factor of $1.3$ to $\sigma(w)=0.19$. We find the largest
improvements from CMB-cluster lensing measurements to be for $\sigma_{8}$,
where adding CMB-cluster lensing data to the cluster number counts reduces the
expected uncertainty on $\sigma_{8}$ by factors of $2.4$ and $3.6$ for SPT-3G
and CMB-S4 respectively.

cosmological parameters — cosmology:observations — cluster cosmology large
scale structure — CMB — cluster lensing

##  1 Introduction

Galaxy clusters are the largest gravitationally collapsed structures and a key
testing ground of cosmological models of structure growth (Allen et al.,
2011). The number density of galaxy clusters depends sensitively upon
cosmological parameters, particularly those that affect late-time structure
growth such as the sum of the neutrino masses, the dark energy equation of
state, and matter density (Wang & Steinhardt, 1998; Haiman et al., 2001;
Weller et al., 2002; Weller & Battye, 2003; Holder, 2006; Shimon et al.,
2011). Upcoming surveys such as eROSITA (Merloni et al., 2012), LSST (LSST
Science Collaboration et al., 2009; The LSST Dark Energy Science Collaboration
et al., 2018) and CMB-S4 (CMB-S4 Collaboration, 2019) are expected to detect
tens of thousands of galaxy clusters at different wavelengths, and will
dramatically improve the cosmological constraints from cluster cosmology.

Galaxy clusters already yield interesting constraints on the matter density
$\Omega_{\mathrm{m}}$ and the amplitude of density fluctuations $\sigma_{8}$
(Bocquet et al., 2019; Zubeldia & Challinor, 2019; To et al., 2020). The
cosmological constraints are limited, however, by the uncertainty on the
masses of galaxy clusters and can be biased if the cluster mass-observable
scaling relations are mis-estimated. Current cluster mass estimates are
typically based on assuming a power-law scaling relationship between observed
quantities (such as the X-ray observable $Y_{\mathrm{X}}$) and cluster masses.
Observationally expensive optical weak lensing measurements are used to
normalize the scaling relation (e.g., Dietrich et al., 2019). These optical
weak lensing mass measurements should substantially improve with surveys like
LSST and Euclid (The LSST Dark Energy Science Collaboration et al., 2018;
Euclid Collaboration et al., 2019). At higher redshifts ($z\gtrsim 1$),
optical weak lensing becomes increasingly difficult due to a dearth of
background galaxies and difficulties in measuring their shape with blending
and lower signal to noise. High-redshift mass information is important as
there are suggestions that scaling relations calibrated at lower redshifts may
mis-estimate the masses at higher redshifts (Zohren et al., 2019; Salvati et
al., 2018, 2019).

Galaxy clusters also gravitationally lens the cosmic microwave background
(CMB), an effect referred to as CMB-cluster lensing and first considered by
Seljak & Zaldarriaga (2000). While useful as an independent cross-check on
optical weak lensing cluster masses at low redshift, CMB-cluster lensing is
particularly useful at higher redshifts. Since all CMB photons originate at
the same extremely high redshift, $z\simeq 1100$, the signal-to-noise of CMB-
cluster lensing does not drop as the cluster redshift increases (Melin &
Bartlett, 2015). This also simplifies the measurement (and eliminates related
uncertainties), as one does not need to calculate intrinsic alignments, boost
factors, or the redshift distribution to background sources. The problem of
estimating the masses of clusters from their CMB lensing signals has been
extensively considered (Seljak & Zaldarriaga, 2000; Holder & Kosowsky, 2004;
Vale & Ostriker, 2004; Dodelson, 2004; Lewis & Challinor, 2006; Lewis & King,
2006; Hu et al., 2007; Raghunathan et al., 2017, 2019a; Gupta & Reichardt,
2020). Actual measurements of the CMB-cluster lensing signal have followed as
CMB surveys have advanced, from the first detections in 2015 (Madhavacheril et
al., 2015; Baxter et al., 2015; Planck Collaboration et al., 2016) to
$\sim$15% mass measurements of different cluster samples today (Baxter et al.,
2018; Raghunathan et al., 2019b).

In this work, we present the first cosmological analysis of the SPT-SZ galaxy
cluster sample that includes CMB-cluster lensing information. The SPT-SZ
survey detected galaxy clusters from the imprint of thermal SZ (tSZ)
signatures on the background primary CMB anisotropies (Bleem et al., 2015).
Bocquet et al. (2019, hereafter B19) presented cosmological constraints from
this sample along with X-ray observations and optical weak-lensing
measurements. We add the CMB-cluster lensing mass measurement of Baxter et al.
(2015, hereafter B15) to that dataset, and look at the implications for the
combined dataset on the $\Lambda\mathrm{CDM}$ and $w\mathrm{CDM}$ cosmological
models. We follow this by presenting forecasts for the cosmological
constraints from future CMB-cluster lensing measurements with SPT-3G (Benson
et al., 2014) and CMB-S4 (CMB-S4 Collaboration, 2019). We find that CMB-
cluster lensing mass measurements substantially improve the predicted
constraints on the dark energy equation of state parameter $w$ from future
cluster catalogs.

The paper is organized as follows. In §2, we review the datasets used in this
analysis. We describe the analysis methods in §3. In §4, we present the
cosmological constraints from the current CMB-cluster lensing measurement. In
§5, we forecast the constraints expected from the ongoing SPT-3G and future
CMB-S4 surveys. Finally, we conclude in §6. Throughout this work, we report
galaxy cluster masses in terms of either $M_{\rm 200}$ or $M_{\rm 500}$, the
mass contained within the radius where the mean density is 200 (500) times the
critical density of the Universe.

##  2 The Cluster catalog from the 2500d SPT-SZ survey

The main dataset in this work is the galaxy cluster sample from the 2500d SPT-
SZ survey (Bleem et al., 2015), which provides a measure of the SZ detection
significance and redshift for each cluster in the sample. As in the previous
cosmological analysis by B19, we supplement the SZ cluster catalog with
follow-up X-ray and optical weak-lensing observations. The new addition in
this work is that we add the $3.1\sigma{}$ CMB-cluster lensing mass
measurement from B15 for a stack of 513 of galaxy clusters in the sample. This
sub-sample of 513 clusters is chosen by selecting only those clusters from the
2500d SPT-SZ catalog which have measured optical redshifts. We refer to the
combination of SPT number counts, X-ray and weak lensing follow up, and CMB
cluster lensing datasets as SPT clusters. We briefly describe these datasets
in the following subsections.

For some parameter fits, we also include measurements of the CMB TT, TE and EE
power spectra from the 2018 data release of the Planck satellite (Planck
Collaboration et al., 2020). We refer to this dataset as ‘Planck’ throughout
rest of the work. The Planck CMB data allow us to demonstrate where clusters
and CMB-cluster lensing add the most information.

###  2.1 SZ detection significance and cluster redshift

The SZ detection significance and cluster redshift (or lower limit on
redshift) are reported for all cluster candidates in the Bleem et al. (2015)
catalog and were later updated in B19. The reported significance is the
maximum across a set of matched filters (to allow for variations in the
cluster angular radius with redshift and mass), and therefore is biased high
on average. To avoid this biasing in the mass estimates, we follow B19 in
using the unbiased significance $\zeta=\sqrt{(\xi^{2}-3)}$ as a mass proxy. A
detailed discussion on the validity of this approach can be found in
Vanderlinde et al. (2010). As in B19, we model the relationship between the
unbiased significance $\zeta$ and cluster mass $M_{\mathrm{500}}$ as:

$\displaystyle\zeta=\ $
$\displaystyle{A_{\mathrm{SZ}}}\left(\frac{M_{500}h_{70}}{4.3\times
10^{14}M_{\odot}}\right)^{B_{\mathrm{SZ}}}\left(\frac{E\left(z\right)}{E\left(0.6\right)}\right)^{C_{\mathrm{SZ}}}\
,$ (1)

where ${A_{\mathrm{SZ}}}$, ${B_{\mathrm{SZ}}}$, and ${C_{\mathrm{SZ}}\ }$are
free parameters in the model fits (see Table 1) and $E(z)$ is the
dimensionless Hubble parameter. Here $h_{70}$ is the Hubble constant divided
by 70 km s-1 Mpc-1, and $z$ is the cluster redshift. The intrinsic scatter in
$\mathrm{ln}\zeta$ at a fixed mass and redshift, is modeled as a Gaussian
scatter with width ${\sigma_{\ln\zeta}}$ and is also left as a free parameter
of the model.

###  2.2 Weak-lensing shear profiles

Thirty-two clusters have optical weak lensing shear profiles, with 13 from the
Hubble Space Telescope and 19 from ground-based Megacam/Magellan imaging
(Schrabback et al., 2018; Dietrich et al., 2019). The shear profiles of these
clusters are compared to the expected weak lensing shear profiles under the
assumption of a Navarro-Frenk-White (NFW) profile (Navarro et al., 1997) for
the cluster density. We allow for a systematic bias $b_{WL}$ between the halo
mass $M_{\rm halo}$ and inferred lensing mass $M_{WL}$,

$M_{WL}=b_{WL}M_{\rm halo}\ .$ (2)

We refer the reader to Eqn. 9 in B19 for the breakdown of $b_{WL}$ into
different sources of uncertainty in the weak lensing observations. The priors
on these uncertainties are included in Table 1 under the _WL modeling_
section. The weak-lensing model is described in more detail by B19.

###  2.3 X-ray $Y_{\mathrm{X}}$ data

As in B19, we use X-ray observations of 89 galaxy clusters taken through a
Chandra X-ray visionary project (McDonald et al., 2013, 2017). The X-ray data
is used to estimate $Y_{\mathrm{X}}$ (the product of the gas mass and X-ray
temperature) within $r_{500}$ for each cluster. We assume a scaling relation
between $Y_{\mathrm{X}}$ and the cluster mass $M_{500}$ of the form:

$\displaystyle\mathrm{ln}\left(\frac{M_{500}h_{70}}{8.37\times
10^{13}M_{\odot}}\right)=\ $
$\displaystyle\mathrm{ln}A_{Y_{\mathrm{X}}}+B_{Y_{\mathrm{X}}}\langle\mathrm{ln}Y_{\mathrm{X}}\rangle$
(3) $\displaystyle+B_{Y_{\mathrm{X}}}\mathrm{ln}\
\left(\frac{h_{70}^{5/2}}{3\times 10^{14}M_{\odot}\mathrm{keV}}\right)$
$\displaystyle+C_{Y_{\mathrm{X}}}\mathrm{ln}\ E\left(z\right)\ .$

The intrinsic scatter in $\mathrm{ln}\,Y_{\mathrm{X}}$ at fixed mass and
redshift is modeled as a normal distribution with width ${\sigma_{\ln
Y_{\mathrm{X}}}}$.

###  2.4 CMB-Cluster lensing measurement

CMB photons are deflected by the gravitational pull of galaxy clusters. This
deflection remaps the CMB anisotropy, and introduces a dipole-like signal
aligned with the local gradient in the primary CMB anisotropy (Lewis &
Challinor, 2006). B15 extracted this CMB-cluster lensing signal from the SPT-
SZ survey data at the positions of clusters in the SPT-SZ sample. To avoid
being biased by the cluster’s own tSZ signal, B15 used a linear combination of
the 90, 150 and 220 GHz maps from the SPT-SZ survey to make a tSZ-free map for
the analysis. We refer the reader to B15 for further details on the
measurement.

For the SPT-SZ catalog sub-sample described in §2, B15 found the mean mass of
the stacked clusters to be $\bar{M}_{200}=(5.1\pm 2.1)\times
10^{14}M_{\odot}$. We convert $M_{200}$ to $M_{\rm 500}$ by assuming a
concentration parameter $c=3$ and the same flat $\Lambda\mathrm{CDM}$
cosmological parameters used in B15 ($\Omega_{\mathrm{m}}{}=0.3$, $h=0.7$) for
the redshift of $z=0.7$. This gives us a value of $M_{\rm 500}=\left(3.49\pm
0.74\right)\times 10^{14}M_{\odot}$ which we use in our analysis. We note that
converting the mean mass of the stack from $M_{200}$ to $M_{500}$ is not
equivalent to converting individual cluster masses before stacking as the
concentration-mass relation is redshift dependent. For this sample, this
approximation results in a $\sim$ 2% systematic error, which is negligible at
the current statistical uncertainty, although the approximation may be
inadequate for future high-S/N mass measurements.

##  3 Likelihood

As in past SPT-SZ cluster analyses (Reichardt et al., 2013; de Haan et al.,
2016; Bocquet et al., 2019), we derive cosmological constraints from galaxy
clusters by using the Cash statistic (Cash, 1979) to compare the expected
number of clusters with the observed number as a function of the SZ signal and
redshift. The number density of clusters is predicted from the matter power
spectrum and mass-observable scaling relations for each set of model
parameters. We briefly review the likelihood111
https://github.com/SebastianBocquet/SPT_SZ_cluster_likelihood here, which is
presented in more detail by B19, before describing how we incorporate the new
CMB-cluster lensing information.

We choose to express the likelihood function in three parts: cluster
abundances ($\mathcal{L}_{\mathrm{abund}}$), mass calibration from the weak
lensing and X-ray observations ($\mathcal{L}_{\mathrm{fol}}$), and mass
calibration from the CMB-cluster lensing observation
($\mathcal{L}_{\mathrm{CL}}$). The abundance part (which is unchanged from
B19) calculates the chance of finding a catalog of clusters with the specified
redshifts and SZ significances as a function of the cosmology and scaling
relations. As in B19, the X-ray and weak-lensing mass calibration likelihood
is expressed as:

$\begin{split}\mathcal{L}_{\mathrm{fol}}\equiv
P(Y_{\mathrm{X}}^{\mathrm{obs}},&g_{\mathrm{t}}^{\mathrm{obs}}|\xi,z,\mbox{\boldmath$p$})=\\\
&\iiiint dM\,d\zeta\,dY_{\mathrm{X}}\,dM_{\mathrm{WL}}\,\left[\right.\\\
&P(Y_{\mathrm{X}}^{\mathrm{obs}}|Y_{\mathrm{X}})P(g_{\mathrm{t}}^{\mathrm{obs}}|M_{\mathrm{WL}})P(\xi|\zeta)\\\
&P(\zeta,Y_{\mathrm{X}},M_{\mathrm{WL}}|M,z,\mbox{\boldmath$p$})P(M|z,\mbox{\boldmath$p$})\left.\right]\
.\end{split}$ (4)

This equation gives the likelihood of observing the follow-up X-ray,
$Y_{\mathrm{X}}^{\mathrm{obs}}$, and weak lensing,
$g_{\mathrm{t}}^{\mathrm{obs}}$, observables for a cluster detected with SZ
significance $\xi$. Here, $p$ represents cosmological and scaling relation
parameters. We assume the systematics in the CMB-cluster lensing measurement
to be uncorrelated with other observations. The notation adopted for other
variables is identical to that of B19.

While we could exactly mirror the approach used for including weak lensing
data, the CMB-cluster lensing signal from individual clusters is too weak to
justify the computational complexity. Instead, we take the observed mean mass
from CMB-cluster lensing $\bar{M}_{200}=(5.1\pm 2.1)\times 10^{14}M_{\odot}$
as a prior on the modeled mean mass of the sample, $\bar{M}$:

$\bar{M}=\frac{1}{N}\sum_{i}\iint d\xi_{i}dz_{i}\
M_{i}P(M_{i}|\xi_{i},z_{i})P(\xi_{i},z_{i}|\mbox{\boldmath$p$})\ .$ (5)

Given the number of clusters in the sample, we approximate the integral by
taking the mass at the peak of the posterior for each cluster in the sample.

Table 1:  Parameter priors

Parameter | Prior
---|---
SZ scaling relation
${A_{\mathrm{SZ}}}$ | $\mathcal{U}(1,10)$
${B_{\mathrm{SZ}}}$ | $\mathcal{U}(1,2.0)$
${C_{\mathrm{SZ}}\ }$ | $\mathcal{U}(-1,2)$
${\sigma_{\ln\zeta}}$ | $\mathcal{U}(0.01,2.0)$
Priors for the SPT-SZ cluster catalog
X-ray $Y_{\mathrm{X}}$ scaling relation
${A_{Y_{\mathrm{X}}}}$ | $\mathcal{U}(3,10)$
${B_{Y_{\mathrm{X}}}}$ | $\mathcal{U}(0.3,0.9)$
${C_{Y_{\mathrm{X}}}}$ | $\mathcal{U}(-1,0.5)$
${\sigma_{\ln Y_{\mathrm{X}}}}$ | $\mathcal{U}(0.01,0.5)$
$d\ln M_{\mathrm{g}}/d\ln r$ | $\mathcal{N}(1.12,0.23^{2})$
WL modeling
$\delta_{\mathrm{WL,bias}}$ | $\mathcal{N}(0,1)$
$\delta_{\mathrm{Megacam}}$ | $\mathcal{N}(0,1)$
$\delta_{\mathrm{HST}}$ | $\mathcal{N}(0,1)$
$\delta_{\mathrm{WL,scatter}}$ | $\mathcal{N}(0,1)$
$\delta_{\mathrm{WL,LSS}_{\mathrm{Megacam}}}$ | $\mathcal{N}(0,1)$
$\delta_{\mathrm{WL,LSS}_{\mathrm{HST}}}$ | $\mathcal{N}(0,1)$
Correlated scatter
$\rho_{\mathrm{SZ-WL}}$ | $\mathcal{U}(-1,1)$
$\rho_{\mathrm{SZ-X}}$ | $\mathcal{U}(-1,1)$
$\rho_{\mathrm{X-WL}}$ | $\mathcal{U}(-1,1)$
| $\det(\mbox{\boldmath$\Sigma$}_{\text{multi-obs}})>0$
Priors on cluster-only chains
$\Omega_{b}h^{2}$ | $\mathcal{N}(0.02212,0.00022^{2})$
$\tau$ | $\mathcal{N}(0.0522,0.0080^{2})$
$10^{9}A_{s}$ | $\mathcal{N}(2.092,0.033^{2})$
$n_{s}$ | $\mathcal{N}(0.9626,0.0057^{2})$

Note. — The parameter priors used in this analysis are listed here. The symbol
$\mathcal{U}$ denotes a uniform prior over the given range while
$\mathcal{N}(\mu,\sigma^{2})$ denotes a Gaussian prior centered at $\mu$ with
variance $\sigma^{2}$. The SZ scaling relation priors are used for all results
in this work that include cluster data, while the cluster-only priors listed
in the bottom section are only used in cluster-only-MCMCs. The priors in the
X-ray, WL modeling and Correlated scatter section are used for the SPT-SZ
cluster data, but not in forecasts for future experiments.

##  4 Parameter constraints

We now turn to the cosmological implications of the CMB-cluster lensing
measurement and cluster catalog described in §2 using the likelihood function
described in §3. All MCMC analyses use the same priors for the scaling
relations, which are listed in Table 1.

We infer cosmological constraints using the publicly available COSMOSIS
parameter estimation code (Zuntz et al., 2015), running the Boltzmann code
package CAMB (Lewis et al., 2000). We use the _Multinest_ or _emcee_ samplers
(Feroz et al., 2009; Foreman-Mackey et al., 2013) as implemented by COSMOSIS.
Multinest is run with 250 live points with a tolerance value of 0.1. We look
at two cosmological models: the standard six-parameter $\Lambda\mathrm{CDM}$
model with fixed $\sum m_{\nu}=0.06$ eV, and a well-motivated extension to
$\Lambda\mathrm{CDM}$ where the dark energy equation of state, $w$, is allowed
to vary.

###  4.1 $\Lambda\mathrm{CDM}$ Cosmology

Galaxy cluster number counts are very sensitive to the growth of matter
perturbations. Previous works have found galaxy clusters constrain best the
parameter combination $S_{8}=\sigma_{8}\left(\Omega_{m}/0.3\right)^{0.5}$. We
find for the SPT cluster sample with Planck power spectrum measurement:

$S_{8}=0.831\pm 0.020\ .$ (6)

The uncertainty is larger than Planck-only by a factor of $1.4$ , due to the
tension between the Planck data favoring $S_{8}=0.834\pm 0.016$ and cluster
data favoring a lower $S_{8}=0.794\pm 0.049$. The result is similar to what
was found in B19 so we do not attribute it to CMB-cluster lensing. The
similarity is understandable since the S/N on the CMB-cluster lensing is low
compared to optical weak-lensing. For instance, changing the mass
normalization ${A_{\mathrm{SZ}}}$ from 4.4 to 5.5, the weak-lensing log-
likelihood changes by $\Delta\rm{ln}\mathcal{L}_{WL}=-5.8$, 15 times greater
than the change in the CMB-cluster lensing log-likelihood of
$\Delta\rm{ln}\mathcal{L}_{CMBcl}=-0.38$ for the same shift. As noted above
for $S_{8}$, the modest tension between the cluster and Planck data leads to
slightly wider constraints for the combined dataset on $\Omega_{\mathrm{m}}$
and $\sigma_{8}$:

$\displaystyle\Omega_{\mathrm{m}}$ $\displaystyle=$ $\displaystyle 0.316\pm
0.011\ ,$ (7) $\displaystyle\sigma_{8}$ $\displaystyle=$ $\displaystyle
0.8081\pm 0.0079\ .$ (8)

We report the parameter constraints on selected cosmological and scaling
relation parameters in Table 2.

Figure 1:  Constraints on $\Omega_{\mathrm{m}}$ and $w$ in the $w\mathrm{CDM}$
model from the SPT-SZ cluster dataset (blue contours) and the Planck TTTEEE
power spectra (green contours). The SPT-SZ cluster count constraints are
obtained using CMB-cluster lensing information along with information from
follow-up datasets. The cluster data help break the degeneracy between
$\Omega_{\mathrm{m}}$ and $w$ that exists in the CMB power spectra alone.
Table 2:  Parameter Constraints for the Planck and SPT-SZ surveys

Parameter | $\Lambda\mathrm{CDM}$ | $w\mathrm{CDM}$
---|---|---
| Planck | SPT Clusters | Planck | SPT Clusters
$\Omega_{\mathrm{m}}$ | $0.3165\pm 0.0084$ | $0.352\pm 0.047$ | $0.184\pm 0.045$ | $0.279\pm 0.042$
$\sigma_{8}$ | $0.8118\pm 0.0072$ | $0.737\pm 0.033$ | $0.985\pm 0.077$ | $0.772\pm 0.037$
$S_{8}$ | $0.834\pm 0.016$ | $0.794\pm 0.049$ | $0.774\pm 0.031$ | $0.743\pm 0.048$
$w$ | – | – | $-1.63\pm 0.28$ | $-1.07\pm 0.20$
${A_{\mathrm{SZ}}}$ | – | $5.3\pm 1.1$ | – | $5.1\pm 1.2$
${B_{\mathrm{SZ}}}$ | – | $1.668\pm 0.068$ | – | $1.631\pm 0.068$
${C_{\mathrm{SZ}}\ }$ | – | $1.09\pm 0.30$ | – | $0.73\pm 0.24$
${\sigma_{\ln\zeta}}$ | – | $0.168\pm 0.076$ | – | $0.176\pm 0.071$

Note. — Summary of constraints obtained from including cluster data in our
analysis for $\Lambda\mathrm{CDM}$ and $w\mathrm{CDM}$ cosmological models.
Constraints obtained from using Planck only dataset are given for comparison.

###  4.2 $w\mathrm{CDM}$

Clusters are an important probe of the late time Universe when dark energy
dominates the energy budget. We therefore consider the impact of the cluster
abundance and CMB-cluster lensing measurement on the dark energy equation of
state parameter $w$. The cluster data favors

$w=-1.07\pm 0.20\ ,$ (9)

consistent with a cosmological constant. As shown in Fig. 1, the cluster
abundance data prefers a higher value of the dark energy equation of state as
the matter density increases. The detection significance of the B15 CMB-
cluster lensing measurement is as yet too low to significantly tighten the
allowed parameter volume. While this uncertainty on $w$ is modestly tighter
than that inferred from Planck power spectra alone
($w=-1.56^{+0.19}_{-0.39}$), combining the cluster abundance and Planck CMB
data significantly reduces the allowed region to:

$w=-1.30\pm 0.10\ .$ (10)

##  5 Forecasts

We now examine the expected impact of CMB-cluster lensing on the cosmological
constraints from upcoming galaxy cluster surveys. Using the likelihood
framework from §3, we forecast the results from two surveys: the on-going
SPT-3G survey, and the planned CMB-S4 survey. We assume that SPT-3G will
survey 1500 ${\rm deg}^{2}$ with a temperature map noise level of 2.5 $\mu{\rm
K\mathchar 45\relax arcmin}$ (polarization map noise level a factor of
$\sqrt{2}$ higher) at 150 GHz (Sobrin et al., 2021) and produce a catalog of
$\sim$3600 clusters above a signal-to-noise of 4.5. After galactic cuts, we
assume the CMB-S4 survey will cover 60% of the sky with a map noise level of
1.0 $\mu{\rm K\mathchar 45\relax arcmin}$ (polarization map noise level a
factor of $\sqrt{2}$ higher) at 150 GHz (CMB-S4 Collaboration, 2019) and
produce a catalog of $\sim$135,000 clusters above a signal-to-noise of 4.5.
CMB-S4 will survey 3% of the sky to even lower noise levels, which is expected
to add a further 17,000 clusters. Catalogs from both CMB-S4 surveys are used
in the forecasts in this work. We look at the results for the cluster
abundances alone, and in combination with mass information from optical weak
lensing or CMB-cluster lensing. The redshift bins and the uncertainties for
SPT-3G and CMB-S4 surveys are described below.

For the full SPT-3G survey, we expect CMB-cluster lensing to lead to a 4.6%
mass measurement across the entire cluster sample (Raghunathan et al., 2017).
Given the high detection significance, we choose to subdivide the cluster
catalog into four redshift bins to better constrain any redshift evolution in
the relationship between SZ flux and mass. The four redshift bins are
$[0.25,0.55),[0.55,0.78),[0.78,1.06)$, and $[1.06,2.]$, which achieves a
roughly equal number of clusters and lensing detection significance in each
bin. The uncertainty on the average mass of the clusters in each of the four
bins is taken to be 9.2%. For simplicity, we assume equal constraining power
in each of the bins. We do not include the effect of systematic uncertainties,
such as from tSZ contamination or errors in the assumed mass profile, but
point interested readers towards Raghunathan et al. (2017) for a discussion of
potential systematic errors and their magnitude. The potential systematic
biases are expected to be correctable to better than the mass uncertainties
assumed in this work. We conservatively assume a 5% mass calibration from
optical weak lensing at $z<0.8$, again implemented as four 10% mass
constraints on redshift bins running $[0.25,0.39),[0.39,0.53),[0.53,.67),$ and
$[0.67,0.8]$, such as might be achieved from the final DES results (McClintock
et al., 2019).

The CMB-S4 survey is expected to start in the second half of this decade. As
such, we assume substantially improved optical weak lensing mass measurements
will be available from, for instance, LSST or Euclid, and provide either a 2%
(conservative) or 1% (goal) mass calibration (Grandis et al., 2019). As
before, we implement this as either a 4% or 2% mass calibration in each of
four redshift bins that cover the redshift range from z = 0.25 to 0.8. The
lower noise CMB maps will also enable tighter mass constraints from CMB-
cluster lensing. From Raghunathan et al. (2017), we estimate that the CMB-S4
wide survey will yield a 3% mass calibration in each of the four redshift
bins, while the deep survey will yield a weaker (due to fewer clusters) 5%
mass calibration in each redshift bin. As with SPT-3G, we do not include the
effect of systematic errors.

As shown in Table 3 and Fig. 2, we find that adding the mass information from
optical weak lensing and CMB-cluster lensing substantially improves
cosmological constraints from galaxy cluster abundances with SPT-3G and
CMB-S4. Assuming that that posteriors are approximately Gaussian, we calculate
the allowed parameter volume as the square root of the determinant of the
covariance matrix. The allowed parameter volume from the cluster abundance
data for the 7 parameters of the $w$CDM model is reduced by a factor of $4.1$
for SPT-3G and $6.1$ for CMB-S4 by adding the CMB-cluster and optical lensing
measurements. While the absolute mass calibration is similar between the
optical and CMB lensing channels ($\sim$ 5% for SPT-3G and $\sim$ 2-3% for
CMB-S4), the higher redshift lever arm in the CMB-cluster lensing measurement
has advantages for the SZ cluster catalogs with their high median redshifts
($\sim$ 0.8 for both the SPT-3G and CMB-S4 surveys). For the SPT-3G cluster
sample, adding only the CMB-cluster lensing measurement reduces the parameter
volume by a factor of $2.8$. Adding both CMB-cluster lensing and optical weak
lensing improves the parameter volume by a factor of $4.1$, as stated above.
This translates to an improvement on $w$ from $\sigma(w)=0.19$ for cluster
counts to $\sigma(w)=0.15$ with CMB-cluster lensing and $\sigma(w)=0.14$ with
CMB-cluster lensing and optical weak lensing information (the latter two
uncertainties are consistent given the number of samples in the MCMC). The
expected constraint on $\sigma_{8}$ shows an even larger improvement,
tightening from $\sigma(\sigma_{8})=0.039$ for cluster counts to
$\sigma(\sigma_{8})=0.016$ with CMB-cluster lensing and
$\sigma(\sigma_{8})=0.014$ with CMB-cluster lensing and optical weak lensing
information. The story is similar for CMB-S4. The 7-parameter volume is
reduced by a factor of $4.8$ ($6.1$) by adding CMB-cluster lensing (both CMB-
cluster lensing and a 2% optical weak lensing measurement). Adding both the
optical weak lensing and CMB-cluster lensing information brings
$\sigma(w)=0.028$ down to $\sigma(w)=0.023$ for a 2% mass calibration
($\sigma(w)=0.020$ for a 1% mass calibration), a factor of $1.2$ ($1.4$)
improvement over the cluster counts alone. The CMB-cluster lensing information
substantially improves the constraint on $\sigma_{8}$ from the CMB-S4 cluster
catalog by more than a factor of three, from $\sigma(\sigma_{8})=0.016$ to
$\sigma(\sigma_{8})=0.0044$. Adding a 1% (2%) optical weak lensing mass
measurement yields consistent results (within the sampling error) of
$\sigma(\sigma_{8})=0.0046{}~{}(0.0040)$. CMB-cluster lensing cluster mass
measurements will be important to achieve the full potential of cluster
cosmology over this decade.

Figure 2:  The 1 and 2 $\sigma$ contours for $\sigma_{8}$ and $w$ in the
$w\mathrm{CDM}$ model for the SPT-SZ (left panel), SPT-3G (middle panel) and
CMB-S4 (right panel) surveys. The SPT-3G and CMB-S4 contours are forecasts
from simulated cluster catalogs created for $\sigma_{8}=0.8126$ and $w=-1$.
Parameter posterior distributions from the Planck CMB data are shown in green,
while the posteriors from cluster number counts are shown in blue. The
posterior distributions from cluster number counts and CMB-cluster lensing are
shown in orange. Adding CMB-cluster lensing information significantly improves
the constraints on equation of dark energy parameter $w$ and $\sigma_{8}$.
Table 3:  Forecasts for Parameter Constraints for Upcoming Surveys

Survey | Data | $\Omega_{m}$ | $h$ | $w$ | $\sigma_{8}$ | $S_{8}$
---|---|---|---|---|---|---
Planck | CMB TTTEEE power spectra | 0.045 | 0.10 | 0.28 | 0.077 | 0.031
SPT-3G | Number counts | $0.026$ | $0.030$ | $0.19$ | $0.039$ | $0.051$
\+ CMB-cluster lensing | $0.025$ | $0.028$ | $0.15$ | $0.016$ | $0.025$
\+ CMB-cluster and optical weak lensing | $0.024$ | $0.024$ | $0.14$ | $0.014$ | $0.023$
CMB-S4 | Number counts | $0.0063$ | $0.012$ | $0.028$ | $0.016$ | $0.016$
\+ CMB-cluster lensing | $0.0057$ | $0.0092$ | $0.029$ | $0.0044$ | $0.0059$
\+ CMB-cluster and 2% optical weak lensing | $0.0052$ | $0.0071$ | $0.023$ | $0.0040$ | $0.0059$
\+ CMB-cluster and 1% optical weak lensing | $0.0050$ | $0.0072$ | $0.020$ | $0.0046$ | $0.0059$

Note. — Cluster counts from SPT-3G and CMB-S4 can significantly improve
cosmological constraints. We report here forecasted constraints in the
7-parameter $w\mathrm{CDM}$ model for $w$, $\sigma_{8}$, and
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/0.3}$. The second row has current
uncertainties from the Planck 2018 TTTEEE data shown for comparison. The third
through fifth rows have, in order, the expected uncertainties with the SPT-3G
cluster counts , with the SPT-3G cluster counts and CMB-cluster lensing mass
measurement, with the SPT-3G cluster counts and a DES-like optical weak
lensing mass measurement, and with both the optical and CMB-cluster lensing
mass measurements. The sixth through ninth rows are the same except for CMB-S4
and two options for an LSST-like optical survey that yields either a 1% or 2%
mass measurement. Adding the optical weak lensing mass measurements to the
CMB-S4 catalog does not improve estimates of large scale structure today (i.e.
$\sigma_{8}$) but does noticeably improve the constraints on the dark energy
equation of state.

##  6 Conclusions and Outlook

We present the first cosmological parameter constraints incorporating CMB-
cluster lensing mass estimates from the South Pole Telescope. While the CMB-
cluster lensing mass information does not yet substantively improve
cosmological constraints as compared to B19, this work serves as a
demonstration for the method which will be important for the next generation
of large galaxy cluster surveys.

We show that adding CMB-cluster lensing mass measurements should significantly
improve cosmological constraints from on-going cluster surveys such as SPT-3G.
In the 7-parameter $w$CDM cosmological model, we find that adding CMB-cluster
lensing mass estimates to cluster number counts leads to a factor of $1.3$
reduction in the uncertainty of $w$ and a factor of $2.4$ on $\sigma_{8}$.

CMB-cluster lensing data remains significant for the larger galaxy cluster
catalog expected for CMB-S4. For CMB-S4, we find the CMB-cluster lensing data
reduces the uncertainty on $\sigma_{8}$ by a factor of $3.6$. CMB-cluster
lensing has the potential to significantly expand the cosmological information
we can extract from galaxy cluster surveys.

The South Pole Telescope program is supported by the National Science
Foundation (NSF) through award OPP-1852617. Argonne National Laboratory’s work
was supported by the U.S. Department of Energy, Office of High Energy Physics,
under contract DE-AC02-06CH11357. We also acknowledge support from the Argonne
Center for Nanoscale Materials. The Melbourne group acknowledges support from
the Australian Research Council’s Discovery Projects scheme (DP200101068). AAS
acknowledges support by U.S. National Science Foundation grant AST-1814719. AS
is supported by the FARE-MIUR grant ’ClustersXEuclid’ R165SBKTMA, INFN InDark,
and by the ERC-StG ‘ClustersXCosmo’ grant agreement 716762. The data analysis
pipeline also uses the scientific python stack (Hunter, 2007; Jones et al.,
2001; van der Walt et al., 2011). We acknowledge the use of the Spartan, a
high performance computing facility at the University of Melbourne (Lafayette
et al., 2016).

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