# On conjugacy of additive actions in the affine Cremona group

Ivan Arzhantsev  Faculty of Computer Science, HSE University, Pokrovsky
Boulevard 11, Moscow, 109028 Russia<EMAIL_ADDRESS>
###### Abstract.

An additive action on an irreducible algebraic variety $X$ is an effective
action ${\mathbb{G}}_{a}^{n}\times X\to X$ with an open orbit of the vector
group ${\mathbb{G}}_{a}^{n}$. Any two additive actions on $X$ are conjugate by
a birational automorphism of $X$. We prove that, if $X$ is the projective
space, the conjugating element can be chosen in the affine Cremona group and
it is given by so-called basic polynomials of the corresponding local algebra.

###### Key words and phrases:

Algebraic variety, projective space, automorphism, birational isomorphism,
Cremona group, algebraic group action, local algebra, Hassett-Tschinkel
correspondence

###### 2010 Mathematics Subject Classification:

Primary 14L30, 14R10; Secondary 13E10, 14E07, 14J50

Supported by the Russian Science Foundation grant 23-21-00472

##  1\. Introduction

Let ${\mathbb{K}}$ be an algebraically closed field of characteristic zero and
${\mathbb{G}}_{a}=({\mathbb{K}},+)$ be the additive group of the field
${\mathbb{K}}$. It is well known that any commutative unipotent linear
algebraic group over ${\mathbb{K}}$ is isomorphic to the vector group
${\mathbb{G}}_{a}^{n}$.

An additive action on an irreducible algebraic variety $X$ is an effective
action ${\mathbb{G}}_{a}^{n}\times X\to X$ with an open orbit. In the study of
additive actions it is natural to look for analogies with toric geometry.
Namely, we consider the multiplicative group ${\mathbb{G}}_{m}$ of the field
${\mathbb{K}}$ and an algebraic torus $T={\mathbb{G}}_{m}^{n}$. Let us recall
that a normal variety $X$ is toric if it admits a regular action $T\times X\to
X$ with an open orbit.

It is known that an effective torus action with an open orbit on a toric
variety $X$ is unique up to automorphism of $X$. Namely, if $X$ is a complete
toric variety, it is shown in [8] that the automorphism group
$\operatorname{Aut}(X)$ is a linear algebraic group, and the claim follows
from the fact that any two maximal tori in a linear algebraic group are
conjugate. For an arbitrary toric variety more specific arguments are needed;
see [6, Theorem 4.1].

At the same time, many varieties $X$ admit several non-equivalent additive
actions. For the first time, this effect was observed by Hassett and Tschinkel
in the case of projective spaces. More precisely, in [14] a correspondence
between commutative associative unital local algebras $A$ with $\dim A=n+1$
and additive actions on ${\mathbb{P}}^{n}$ was established; see also [15].
Using a classification of local algebras of small dimensions, one can conclude
that starting from $n=2$ an additive action on ${\mathbb{P}}^{n}$ is not
unique, and starting from $n=6$ there are infinitely many equivalence classes
of such actions.

During last decades, many results on additive actions on complete algebraic
varieties were obtained. In particular, results on uniqueness of an additive
action on non-degenerate quadrics and, more generally, arbitrary non-
degenerate projective hypersurfaces, flag varieties, and other types of
varieties can be found in [4, 9, 10, 12, 13, 21]. On the other hand, examples
of non-equivalent additive actions on complete varieties are given in [1, 2,
3, 5, 11, 17, 20]. For a recent survey of these results, see [4].

Returning to linear algebraic groups, it is well known that in any such group
all maximal unipotent subgroups are conjugate. At the same time, every
additive action on ${\mathbb{P}}^{n}$ gives rise to a separate conjugacy class
of maximal commutative unipotent subgroups in ${\operatorname{PGL}(n+1)}$.
Moreover, one can construct commutative unipotent subgroups in
$\operatorname{PGL}(n+1)$ whose dimension is greater than $n$.

This note originates from attempts to analyze these effects. It is easy to see
that any two additive actions on a variety $X$ are birationally equivalent or,
in other words, they are conjugate in the group of birational automorphisms of
$X$. More precisely, if two additive actions on $X$ share the same open orbit
$U$, they are conjugate in the automorphism group $\operatorname{Aut}(U)$; see
Proposition 1. Since the orbit $U$ is isomorphic to an affine space, the group
$\operatorname{Aut}(U)$ is the affine Cremona group.

The aim of this note is to show that in the case of the projective space
${\mathbb{P}}^{n}$ an element that conjugates a given additive action to the
standard one has a remarkable interpretation in terms of the Hassett-Tschinkel
correspondence. Namely, any local algebra $A$ with $\dim A=n+1$ defines a so-
called basic subspace $V$ in the polynomial algebra
${\mathbb{K}}[x_{1},\ldots,x_{n}]$. This subspace can be defined in terms of
exponents of elements in $A$; see Section 3 for details. The subspace $V$ is
$(n+1)$-dimensional, invariant under all translations
${(x_{1}+c_{1},\ldots,x_{n}+c_{n})}$, $c_{i}\in{\mathbb{K}}$, and generates
the algebra ${\mathbb{K}}[x_{1},\ldots,x_{n}]$. Moreover, equivalence classes
of such subspaces are in natural bijection with isomorphism classes of
$(n+1)$-dimensional local algebras; see [14, Theorem 2.14] and [4, Theorem
1.48].

We prove that a basis $1,f_{1},\ldots,f_{n}$ of the basic subspace $V$ defines
an automorphism $x_{1}\mapsto f_{1},\ldots,x_{n}\mapsto f_{n}$ from the affine
Cremona group that conjugates the additive action corresponding to the algebra
$A$ to the standard additive action; see Theorem 1. This result shows that one
conjugacy class of ${\mathbb{G}}_{a}^{n}$-subgroups in the affine Cremona
group intersects the subgroups of the group $\operatorname{PGL}(n+1)$ in
several $\operatorname{PGL}(n+1)$-conjugacy classes. Such an intersection is
called in [7] the trace on $\operatorname{PGL}(n+1)$ of a conjugacy class in
$\operatorname{Aut}(U)$; see [7] and references therein for results on an
interplay of conjugacy classes of elements in the groups we are dealing with.
Also Theorem 1 provides a new characterization of basic subspaces in the
polynomial algebra.

The author is grateful to Ivan Beldiev and Yulia Zaitseva for useful comments.

##  2\. General results on conjugation

We denote by $\operatorname{Aut}(X)$ the group of regular automorphisms and by
$\operatorname{Bir}(X)$ the group of birational automorphisms of an
irreducible algebraic variety $X$. Let $\operatorname{Aut}(n,{\mathbb{K}})$ be
the group of ${\mathbb{K}}$-automorphisms of the polynomial algebra
${\mathbb{K}}[x_{1},\ldots,x_{n}]$ and $\operatorname{Bir}(n,{\mathbb{K}})$ be
the group of ${\mathbb{K}}$-automorphisms of its field of fractions
${\mathbb{K}}(x_{1},\ldots,x_{n})$. The group
$\operatorname{Bir}(n,{\mathbb{K}})$ is the Cremona group; it can be
identified with $\operatorname{Bir}({\mathbb{A}}^{n})$, with
$\operatorname{Bir}({\mathbb{P}}^{n})$, and with $\operatorname{Bir}(X)$ for
any rational variety $X$. In particular, the group
$\operatorname{Bir}(n,{\mathbb{K}})$ coincides with $\operatorname{Bir}(X)$
for any irreducible variety $X$ admitting an additive action.

The group $\operatorname{Aut}(n,{\mathbb{K}})$ is called the affine Cremona
group; it can be identified with the group
$\operatorname{Aut}({\mathbb{A}}^{n})$. The group
$\operatorname{Aut}({\mathbb{P}}^{n})$ is much smaller, it is isomorphic to
$\operatorname{PGL}(n+1)$.

######  Proposition 1.

Let $X$ be an irreducible algebraic variety. Then

  1. (1)

any two additive actions on $X$ are conjugate in the group
$\operatorname{Bir}(X)$;

  2. (2)

if two additive actions have the same open orbit $U$ in $X$, then these
actions are conjugate in the affine Cremona group
$\operatorname{Aut}(U)\subseteq\operatorname{Bir}(X)$.

###### Proof.

Let $U_{1}$ and $U_{2}$ be the open orbits of two given additive actions on
$X$. Then an isomorphism of varieties $\psi\colon U_{1}\to U_{2}$ defines a
birational automorphism of $X$, and conjugating by this automorphism, we may
assume that the two additive actions have the same open orbit $U$.

So it suffices to prove claim (2). By definition of an additive action, the
transitive action ${\mathbb{G}}_{a}^{n}\times U\to U$ is effective, so it is
free. Fixing a point $x_{0}\in U$, the orbit map ${\mathbb{G}}_{a}^{n}\to U$,
$g\mapsto g\cdot x_{0}$ defines an isomorphism ${\mathbb{G}}_{a}^{n}\cong U$.
Having two such isomorphisms corresponding to the two given additive actions,
we obtain two structures of the affine space on the open subset $U$. The
passage from one such structure to another gives rise to an element of the
group $\operatorname{Aut}(U)$ that conjugates the first additive action to the
second one. ∎

######  Remark 1.

Proposition 1 (1) reflects a more general and rather obvious fact: any two
actions of an algebraic group $G$ with an open orbit on an irreducible variety
$X$ which have the same generic stabilizers are birationally equivalent.

######  Remark 2.

It follows from the description of additive actions on projective spaces that
in the case $X={\mathbb{P}}^{n}$ the map $\psi$ from the proof above is an
automorphism of the variety $X$. We do not know whether it is the case in
general.

We finish this section with one more general observation. Let
$\alpha\colon{\mathbb{G}}_{a}^{n}\times X\to X$ be an additive action on an
irreducible variety $X$. Denote by
$\beta\colon{\mathbb{G}}_{a}^{n}\to\operatorname{Aut}(X)$ the corresponding
injective homomorphism and consider the subgroup
$H=\beta({\mathbb{G}}_{a}^{n})$ in $\operatorname{Aut}(X)$.

######  Proposition 2.

The subgroup $H$ is a maximal connected commutative algebraic subgroup both in
$\operatorname{Aut}(X)$ and $\operatorname{Bir}(X)$.

###### Proof.

It suffices to prove the claim for the group $\operatorname{Bir}(X)$. Assume
that $G$ is a connected commutative algebraic subgroup in
$\operatorname{Bir}(X)$ that contains the subgroup $H$ properly. Then $\dim
G>n$. On the other hand, by the Regularization Theorem of André Weil [23] (see
[16] for a modern proof) there exist a variety $Y$ with a regular action of
$G$ and a $G$-equivariant birational map $X\to Y$. Since the group $H$ acts on
$X$ with an open orbit, the field of rational invariants ${\mathbb{K}}(X)^{H}$
coincides with ${\mathbb{K}}$. But the group $G$ contains $H$, so we have
${\mathbb{K}}(X)^{G}={\mathbb{K}}(Y)^{G}={\mathbb{K}}$. We conclude that the
effective regular action $G\times Y\to Y$ has an open orbit. This contradicts
the condition $\dim G>n=\dim Y$. ∎

At the same time, for $n\geqslant 2$ the groups
$\operatorname{Aut}(n,{\mathbb{K}})$ and $\operatorname{Bir}(n,{\mathbb{K}})$
contain commutative unipotent algebraic subgroups of arbitrary dimensions: one
can take all transformations of the form

$(x_{1}+f(x_{2}),x_{2},\ldots,x_{n}),$

where $f$ runs through all polynomials of degree less than $d$ for some
positive integer $d$.

##  3\. Hassett-Tschinkel correspondence

In [14], a bijective correspondence between additive actions on the projective
space ${\mathbb{P}}^{n}$ and commutative associative unital local algebras $A$
with $\dim A=n+1$ is established; see also [15]. Let us fix a basis
$s_{1},\ldots,s_{n}$ in the maximal ideal $\mathfrak{m}$ of the algebra $A$.
Under this correspondence, the space ${\mathbb{P}}^{n}$ is identified with the
projectivization ${\mathbb{P}}(A)$, and an element
$y=(y_{1},\ldots,y_{n})\in{\mathbb{G}}_{a}^{n}$ acts on ${\mathbb{P}}(A)$ by
multiplication by an element $\exp(y_{1}s_{1}+\ldots+y_{n}s_{n})$.

A classification of local algebras up to dimension 6 is given in [14] with a
reference to [22]; see also [4, Table 1]. This classification was obtained
independently in [18]; one more approach can be found in [19]. As a result of
the classification, we have the following number of isomorphism classes of
local algebras of dimension $n+1$:

$\begin{array}[]{c|c|c|c|c|c|c|c}n+1&1&2&3&4&5&6&\geqslant 7\\\
\hline\cr&1&1&2&4&9&25&\infty\end{array}$ $None$

With any local algebra $A$ one associates polynomials
$f_{1},\ldots,f_{n}\in{\mathbb{K}}[x_{1},\ldots,x_{n}]$ defined be the formula

$\exp(x_{1}s_{1}+\ldots+x_{n}s_{n})=1+f_{1}(x)s_{1}+\ldots+f_{n}(x)s_{n}.$

Choosing a basis $s_{1},\ldots,s_{n}$ in $\mathfrak{m}$ compatible with the
filtration

$\mathfrak{m}\supseteq\mathfrak{m}^{2}\supseteq\ldots\supseteq\mathfrak{m}^{d-1}\supseteq\mathfrak{m}^{d}=0,$

we observe that each polynomial $f_{i}$ has the form
$x_{i}+h_{i}(x_{1},\ldots,x_{i-1})$. So the endomorphism of the algebra
${\mathbb{K}}[x_{1},\ldots,x_{n}]$ given by $x_{1}\to f_{1}(x),\ldots,x_{n}\to
f_{n}(x)$ is a triangular automorphism.

Let us give three examples illustrating the concepts discussed above. In all
cases we realize a local algebra $A$ as a factor of the polynomial algebra
${\mathbb{K}}[S_{1},\ldots,S_{m}]$, and denote by $s_{i}$ the image of $S_{i}$
in $A$.

######  Example 1.

Take a local algebra
$A={\mathbb{K}}[S_{1},\ldots,S_{n}]/(S_{i}S_{j},1\leqslant i,j\leqslant n)$.
We have

$\exp(x_{1}s_{1}+\ldots x_{n}s_{n})=1+x_{1}s_{1}+\ldots x_{n}s_{n}.$

It corresponds to an additive action on ${\mathbb{P}}^{n}$ given by

$(y_{1},\ldots,y_{n})\circ[z_{0}:z_{1}:\ldots:z_{n}]=[z_{0}:z_{1}+y_{1}z_{0}:\ldots:z_{n}+y_{n}z_{0}].$

We call this additive action standard. Here the basic polynomials are
$f_{1}=x_{1},\ldots,f_{n}=x_{n}$.

######  Example 2.

Take $A={\mathbb{K}}[S_{1}]/(S_{1}^{3})$ with the basis
$s_{1},s_{2}=s_{1}^{2}$ in $\mathfrak{m}$. We have

$\exp(x_{1}s_{1}+x_{2}s_{2})=1+x_{1}s_{1}+(x_{2}+\frac{x_{1}^{2}}{2})s_{2}.$

It corresponds to an additive action on ${\mathbb{P}}^{2}$ given by

$(y_{1},y_{2})*[z_{0}:z_{1}:z_{2}]=[z_{0}:z_{1}+y_{1}z_{0}:z_{2}+y_{1}z_{1}+(y_{2}+\frac{y_{1}^{2}}{2})z_{0}].$

In this case the basic polynomials are $f_{1}=x_{1}$ and
$f_{2}=x_{2}+\frac{x_{1}^{2}}{2}$.

######  Example 3.

Take a local algebra
$A={\mathbb{K}}[S_{1},S_{2}]/(S_{1}S_{2},S_{1}^{3}-S_{2}^{2})$ with the basis

$s_{1},\quad s_{2},\quad s_{3}=s_{1}^{2},\quad s_{4}=s_{1}^{3}=s_{2}^{2}$

in $\mathfrak{m}$. We have

$\exp(x_{1}s_{1}+x_{2}s_{2}+x_{3}s_{3}+x_{4}s_{4})=1+x_{1}s_{1}+x_{2}s_{2}+(x_{3}+\frac{x_{1}^{2}}{2})s_{3}+(x_{4}+x_{1}x_{3}+\frac{x_{2}^{2}}{2}+\frac{x_{1}^{3}}{6})s_{4}.$

It corresponds to an additive action on ${\mathbb{P}}^{4}$ given by

$(y_{1},y_{2},y_{3},y_{4})*[z_{0}:z_{1}:z_{2}:z_{3}:z_{4}]=[z_{0}:z_{1}+y_{1}z_{0}:z_{2}+y_{2}z_{0}:$
$:z_{3}+y_{1}z_{1}+(y_{3}+\frac{y_{1}^{2}}{2})z_{0}:z_{4}+y_{1}z_{3}+y_{2}z_{2}+(y_{3}+\frac{y_{1}^{2}}{2})z_{1}+(y_{4}+y_{1}y_{3}+\frac{y_{1}^{2}}{2}+\frac{y_{1}^{3}}{6})z_{0}].$

In this case the basic polynomials are

$f_{1}=x_{1},\quad f_{2}=x_{2},\quad f_{3}=x_{3}+\frac{x_{1}^{2}}{2},\quad
f_{4}=x_{4}+x_{1}x_{3}+\frac{x_{2}^{2}}{2}+\frac{x_{1}^{3}}{6}.$

It is shown in [4, Lemma 1.40] that the linear span $V=\langle
1,f_{1},\ldots,f_{n}\rangle$ is an $(n+1)$-dimensional subspace that generates
the algebra ${\mathbb{K}}[x_{1},\ldots,x_{n}]$ and is invariant under all
translations $(x_{1}+c_{1},\ldots,x_{n}+c_{n})$, $c_{i}\in{\mathbb{K}}$.
Subspaces with these properties are called basic subspaces of the algebra
${\mathbb{K}}[x_{1},\ldots,x_{n}]$. It is proved in [14, Theorem 2.14] that
$(n+1)$-dimensional local algebras are in bijection with basic subspaces in
${\mathbb{K}}[x_{1},\ldots,x_{n}]$; see also [4, Theorem 1.48]. In particular,
the number of equivalence classes of basic subspaces in
${\mathbb{K}}[x_{1},\ldots,x_{n}]$ is indicated in the table in (1).

##  4\. The main result

We are ready to formulate the main result of this note.

######  Theorem 1.

The additive action on ${\mathbb{P}}^{n}$ corresponding to a local algebra $A$
is conjugate to the standard additive action by the automorphism given by
basic polynomials of the algebra $A$.

###### Proof.

Let ${\bf x}=x_{1}s_{1}+\ldots+x_{n}s_{n}$ and ${\bf
y}=y_{1}s_{1}+\ldots+y_{n}s_{n}$ be two elements of the maximal ideal
$\mathfrak{m}$ of the algebra $A$. By definition of basic polynomials, we have

$\exp({\bf x})=1+f_{1}(x)s_{1}+\ldots+f_{n}(x)s_{n}.$

Let us consider the map $\varphi\colon 1+\mathfrak{m}\to 1+\mathfrak{m}$ given
by $\varphi(1+{\bf x})=\exp({\bf x})$ as an automorphism of the affine space
$1+\mathfrak{m}$. The inverse map is given by $\varphi^{-1}(1+{\bf
x})=1+\ln(1+{\bf x})$.

Let

${\mathbb{G}}_{a}^{n}\times{\mathbb{P}}^{n}\to{\mathbb{P}}^{n},\quad(y,[z_{0},\ldots,z_{n}])\mapsto
y\circ[z_{0}:\ldots:z_{n}]$

be the standard additive action on ${\mathbb{P}}^{n}$ and

${\mathbb{G}}_{a}^{n}\times{\mathbb{P}}^{n}\to{\mathbb{P}}^{n},\quad(y,[z_{0},\ldots,z_{n}])\mapsto
y*[z_{0}:\ldots:z_{n}]$

be the additive action on ${\mathbb{P}}^{n}$ corresponding to the local
algebra $A$.

Clearly, an additive action on ${\mathbb{P}}(A)$ is uniquely determined by its
restriction to the invariant affine chart $1+\mathfrak{m}$, which is the open
orbit of this action.

So the claim of Theorem 1 can be reformulated in the following form.

######  Lemma 1.

In the notation introduced above, we have

$\varphi(y\circ(1+{\bf x}))=y*\varphi(1+{\bf x})$

for all $y\in{\mathbb{G}}_{a}^{n}$ and ${\bf x}\in\mathfrak{m}$.

###### Proof.

By definition of the standard additive action, we have $y\circ(1+{\bf
x})=1+{\bf x}+{\bf y}$. So, we obtain

$\varphi(y\circ(1+{\bf x}))=\varphi(1+{\bf x}+{\bf y})=\exp({\bf x}+{\bf
y})=\exp({\bf y})\exp({\bf x})=y*\varphi(1+{\bf x}).$

∎

The proof of Theorem 1 is completed. ∎

######  Remark 3.

It is easy to see that the group $A^{\times}$ of invertible elements of a
local algebra $A$ with $\dim A=n+1$ is isomorphic to
${\mathbb{G}}_{m}\times{\mathbb{G}}_{a}^{n}$. At the same time, for non-
isomorphic local algebras $(A,+,\circ)$ and $(A^{\prime},+,*)$ we can not
establish a polynomial bijection $\varphi\colon A\to A^{\prime}$ such that
$\varphi(a\circ b)=\varphi(a)*\varphi(b)$ for all $a,b\in A$. Indeed, it is
proved in [1, Lemma 7] that for any two non-isomorphic finite-dimensional
associative (not necessarily commutative) unital algebras $A$ and $A^{\prime}$
their multiplicative monoids $(A,\circ)$ and $(A,*)$ are not isomorphic as
affine algebraic monoids.

Summing up, Theorem 1 shows that one $\operatorname{Aut}(U)$-conjugacy class
of ${\mathbb{G}}_{a}^{n}$-subgroups in $\operatorname{Bir}({\mathbb{P}}^{n})$
corresponding to additive actions on ${\mathbb{P}}^{n}$ intersects the set of
subgroups of the group ${\operatorname{PGL}(n+1)}$ in several
${\operatorname{PGL}(n+1)}$-conjugacy classes, and the number of such classes
is indicated in the table in (1).

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#  Primordial Clocks within Stochastic Gravitational Wave Anisotropies

Arushi Bodas<EMAIL_ADDRESS>Raman Sundrum<EMAIL_ADDRESS>Maryland
Center for Fundamental Physics, University of Maryland, College Park, MD 20742

###### Abstract

A first order phase transition in the early universe can give an observable
stochastic gravitational background (SGWB), which will necessarily have
primordial anisotropies across the sky. In multi-field inflationary scenarios,
these anisotropies may have a significant isocurvature component very
different from adiabatic fluctuations, providing an alternate discovery
channel for high energy physics at inflationary scales. Here, we consider
classically oscillating heavy fields during inflation that can imprint
distinctive scale-invariance-breaking features in the power spectrum of
primordial anisotropies. While such features are highly constrained in the
cosmic microwave background, we show that their amplitude can be observably
large in isocurvature SGWB, despite both probing a similar period of
inflation. Measuring SGWB multipoles at the required level, $\ell\sim{\cal
O}(10-100)$, will be technologically challenging. However, we expect that
early detection of a strong isotropic SGWB, and the guarantee of anisotropies,
would spur development of next generation detectors with sufficient
sensitivity, angular resolution, and foreground discrimination.

Detection of gravitational waves (GW) by LIGO/Virgo from mergers of black-
holes and neutron stars [1, 2] has ushered in an era of GW astronomy. Apart
from localised sources, future GW experiments could also detect diffuse
stochastic gravitational wave backgrounds (SGWB). There are various well-
motivated sources of SGWB in the early universe, such as inflationary
fluctuations [3], particle production [4], (p)reheating [5], cosmic strings
[6], and first order phase transitions (PT) [7, 8, 9, 10, 11, 12, 13], many of
which can be of detectable strength. While the universe becomes transparent to
light only after recombination, the universe is “transparent” to GW throughout
its history. In this way, GW detectors may play a critical role in the
observational cosmology of the very early universe.

In particular, many extensions of the Standard Model (SM) motivated by the
Electroweak Hierarchy Problem would have PTs with critical temperatures
$T_{c}\sim(1-1000)$ TeV. The GW spectrum from such a PT would then be
redshifted to frequencies $\sim$ mHz - Hz today, which fortuitously is the
target frequency range of proposed GW experiments such as LISA [14], MAGIS
[15], TianQin [16], Taiji [17], BBO [18], DECIGO [19]. The GW frequency
spectrum is sensitive to the subhorizon particle physics properties of the PT
plasma. There will necessarily also be anisotropies $>10^{-5}$ in the SGWB
which will reflect inflationary superhorizon fluctuations [20] analogous to
those of the cosmic microwave background (CMB) (see [21] for a review of
inflation and CMB).

There is however a striking physical difference between CMB and PT-SGWB
anisotropies when it comes to isolating the primordial inflationary
fluctuations, due to the very different times of production. In the CMB,
multipole $\ell_{\rm CMB}\sim 30$ divides subhorizon modes that have re-
entered the horizon by the time of recombination, $\ell>\ell_{\rm CMB}$, and
superhorizon modes that have not, $\ell<\ell_{\rm CMB}$. Subhorizon modes are
strongly modified by causal physics such as baryon acoustic oscillations and
Silk damping, and potentially any unknown non-standard physics, while the
superhorizon modes suffer from significant cosmic variance $\propto
1/\sqrt{\ell}$. But because a SGWB is produced much earlier than the CMB, the
corresponding dividing line is at a much larger multipole, $\ell_{\rm
GW}\sim(T_{c}/T_{\rm CMB})\,\ell_{\rm CMB}\sim 10^{14}$ for a TeV-scale PT.
Thus, the large-$\ell$ SGWB anisotropies can simultaneously have small cosmic
variance and be relatively unmodified by subhorizon physics. This, in
principle, makes them a pristine and precise map of primordial fluctuations.

On the other hand, measuring such a high-precision SGWB map faces significant
practical challenges in terms of detector sensitivity, angular resolution, and
subtraction of astrophysical foregrounds. Various strategies have been
proposed to make progress on these fronts [22, 23, 24, 25, 26, 27, 28, 29,
30]. If the monopole moment of any SGWB is discovered in relatively near-term
detectors such as LISA, there will be strong motivation to build next-
generation detectors capable of extracting the guaranteed anisotropies. In
this way, we foresee a multi-stage discovery of SGWB anisotropies, analogous
to the history of CMB exploration.

We can then ask what new physics might be revealed within an SGWB map. Ref.
[20] pointed out that the SGWB anisotropy map could have a significant
isocurvature component, unlike the adiabatic perturbations seen in the CMB and
large scale structure (LSS), which is plausible within multi-field inflation
and reheating scenarios. Ref. [31] demonstrated that such an isocurvature
component of the SGWB could even contain significantly Non-Gaussian (NG)
correlations, reflecting sizeable interactions among the isocurvature fields
during inflation. Another exciting possibility is the detection of very heavy
particles/fields, with masses of inflationary energy scales, potentially as
high as $10^{15}$ GeV, generally known as “cosmological collider physics” [32,
33, 34, 35, 36, 37, 38]. In this paper, we study the simplest such phenomenon
in the context of SGWB maps.

If the classical inflationary background has (approximate) de Sitter geometry,

$\displaystyle ds^{2}=-dt^{2}+a^{2}(t)\,d\vec{x}^{2}\,;\quad a(t)=e^{Ht}\,,$
(1)

its isometries include “scale invariance”,
$\vec{x}\rightarrow\lambda\vec{x},Ht\rightarrow Ht-\ln\lambda$, where $H$ is
the inflationary Hubble scale. This relates heavy particle propagation during
inflation to spatial correlations at the end of inflation,
$e^{-iM(t-t^{\prime})}\rightarrow(k^{\prime}/k)^{-i(M/H)}$, where $\vec{k}$
and $\vec{k}^{\prime}$ are co-moving wave-vectors. Such a striking non-
analytic signature, sensitive to a heavy mass $M$ can only be seen within NG
(greater than two-point) correlators, so that there are at least two
independent wave-vectors given the spatial translation invariance.

However, the requirement of NG can be evaded if the classical background
contains a small breaking of scale invariance, which can occur if the heavy
field is excited classically, instead of quantum mechanically as above. Sudden
excitation of the heavy field at time $t_{0}$ breaks scale invariance, and the
subsequent decay of this field into light quanta allows these scale-
invariance-breaking features to be imprinted in two-point correlators, with a
single co-moving wave-vector [39]. The heavy oscillations then give $\sim
e^{-iM(t-t_{0})}\rightarrow(k_{0}/k)^{-i(M/H)}$, where $k_{0}$ is related to
the co-moving mode that is exiting the horizon at $t_{0}$. Furthermore, while
quantum production of heavy particles in NG is typically
“Boltzmann”-suppressed $\sim e^{-\pi(M/H)}$ (with some special mechanisms that
can mitigate this to a limited extent [40, 41, 42, 43, 44, 45, 46, 47]), this
is not the case for heavy classical excitations. Such an oscillating heavy
field has been dubbed a “primordial clock” (PC) in the context of testing
inflation and alternate scenarios [48]. Here, we will restrict to the
inflationary paradigm (with $\approx$ de Sitter geometry).

Such PCs have not been detected in the precision measurements of CMB by Planck
[49, 50], constraining the couplings of heavy fields to the inflaton (or the
field sourcing adiabatic perturbations), and almost saturating the theoretical
lower limit from cosmic variance in the low-multipole regime. Couplings to
isocurvature fields, however, could be significantly larger, giving large PC
features in the isocurvature component of SGWB anisotropies. A SGWB map thus
provides a possibility of an alternate channel for detecting heavy field
dynamics during inflation. We construct a model illustrating this idea.

##  I PC in single-field inflation

We first review the effects of a PC field $\sigma$ on adiabatic density
perturbations seeded by the quantum fluctuations $\delta\phi$ of the inflaton
field, as proposed in [39]. The initial condition for the classical motion is
that it is restricted to slow-roll in the $\phi$ direction of field space,
while the massive $\sigma$ field is stuck at its origin. It is assumed that
there is a feature in the scalar potential that leads to a sudden transfer of
energy to excite $\sigma$ at time $t_{0}$, after which $\phi$, however,
continues to slow-roll. The classical dynamics of $\sigma$ after $t_{0}$ is
governed by the homogeneous equation of motion,

$\displaystyle\partial^{2}_{t}\sigma+3H\,\partial_{t}\sigma+M^{2}\sigma=0\,,$
(2)

where $M$ is the mass of the $\sigma$ field. In the limit $M\gg H$, the
solution has an oscillatory form,

$\displaystyle\sigma_{\rm
osc}(t)\simeq\theta(t-t_{0})\,\sigma_{0}\left(\frac{a(t)}{a(t_{0})}\right)^{-3/2}\cos[M(t-t_{0})]\,.$
(3)

The oscillating field energy in an inflationary background imparts a small
oscillatory correction to the Hubble constant,

$\displaystyle\Delta H_{\rm
osc}\approx\theta(t-t_{0})\frac{\sigma_{0}^{2}M}{8M_{\rm
Pl}^{2}}\left(\frac{a(t)}{a(t_{0})}\right)^{-3}\cos[2M(t-t_{0})]\,,$ (4)

where $M_{\rm Pl}$ is the Planck scale. This in turn imparts an oscillatory
component to the slow-roll parameters, $\Delta\epsilon_{\rm
osc}=-\partial_{t}(\Delta H_{\rm osc})/H^{2}$ and $\Delta\eta_{\rm
osc}\approx\partial_{t}(\Delta\epsilon_{\rm osc})/(H\epsilon)$. Here $H$ and
$\epsilon$ are the unperturbed parameters in the absence of $\sigma$
oscillations.

These oscillatory components then introduce scale-invariance-breaking features
in the power spectrum of adiabatic perturbations, ${\cal P}_{\zeta_{\phi}}$,
defined as

$\displaystyle\langle\,\zeta_{\phi}(\vec{k})\zeta_{\phi}(\vec{k}^{\prime})\,\rangle\equiv(2\pi)^{3}\delta(\vec{k}+\vec{k}^{\prime}){\cal
P}_{\zeta_{\phi}}(k)\,,$ (5)

where $\zeta_{\phi}$ are the gauge-invariant perturbations capturing
$\delta\phi$, as measured in the primordial fluctuations of the CMB and LSS.
This irreducible contribution is predominantly given by

$\displaystyle\left(\frac{\Delta{\cal P}_{\zeta_{\phi}}}{{\cal
P}^{(0)}_{\zeta_{\phi}}}\right)_{\Delta H_{\rm osc}}\simeq$
$\displaystyle\,\,\theta(k-k_{g})\frac{\sqrt{\pi}}{2}\frac{(M^{2}\sigma_{0}^{2})}{\,\dot{\phi}_{0}^{2}}\left(\frac{M}{H}\right)^{1/2}$
$\displaystyle\quad\times\left(\frac{k}{k_{g}}\right)^{-3}\cos\left[\frac{2M}{H}\log\frac{k}{k_{g}}\right]\,.$
(6)

where ${\cal P}^{(0)}$ denotes the unperturbed scale-invariant spectrum
$\propto 1/k^{3}$. We can understand the key features of this calculation
intuitively, by noting that ordinarily ${\cal P}_{\zeta_{\phi}}\propto
1/\epsilon$, so that we expect roughly $\Delta{\cal P}_{\zeta_{\phi}}/{\cal
P}^{(0)}_{\zeta_{\phi}}\propto\Delta\epsilon_{\rm osc}/\epsilon$. Now,
$\Delta\epsilon_{\rm osc}(t)$ represents a high-frequency ($2M$ by Eq. (4))
gravitational source for production of inflaton pairs, so their physical
momentum on production, $ke^{-Ht}\approx M$, by the local Minkowski energy
conservation. The softest co-moving momentum $k_{g}$ produced from this
gravitational source is given by $k_{g}e^{-Ht_{0}}\approx M$. Solving for
$t,t_{0}$ in terms of $k,k_{g}$ results in the dependence on the latter seen
in Eq. (I).

Similarly, $\Delta\eta_{\rm osc}$ gives the dominant irreducible PC
contribution to the 3-point correlators, which can be written in a following
normalized form:

$\displaystyle
F\equiv\frac{5}{6}\frac{\langle\,\zeta_{\phi}(\vec{k}_{1})\zeta_{\phi}(\vec{k}_{2})\zeta_{\phi}(\vec{k}_{3})\,\rangle}{{\cal
P}_{\zeta_{\phi}}(k_{1}){\cal P}_{\zeta_{\phi}}(k_{2})+(k_{1}\rightarrow
k_{3})+(k_{2}\rightarrow k_{3})}\,.$ (7)

In the equilateral triangle configuration $k_{1}\sim k_{2}\sim k_{3}$,

$\displaystyle F_{\Delta H_{\rm osc}}$
$\displaystyle\simeq\theta(K-2k_{g})\,f_{\rm NL}^{\rm PC}$
$\displaystyle\qquad\times\left(\frac{K}{2k_{g}}\right)^{-3}\cos\left[2\frac{M}{H}\log\frac{K}{2k_{g}}\right]\,,$
(8)

where $K=\sum_{i=1}^{3}|\vec{k}_{i}|$, and the amplitude of NG can be
evaluated to be $f_{\rm NL}^{\rm
PC}\sim\frac{10}{9}\frac{\sqrt{\pi}}{4}\frac{(M^{2}\sigma_{0}^{2})}{\dot{\phi}_{0}^{2}}\left(\frac{M}{H}\right)^{5/2}$.

Apart from these irreducible contributions, the coherent $\sigma$ field may
also directly decay to inflaton fluctuations. Consider the lowest dimensional
interaction consistent with an (approximate) inflaton shift-symmetry, ${\cal
L}_{\rm
int}\sim(\sigma\,\partial_{\mu}\phi\partial^{\mu}\phi)/\Lambda_{\phi}$. The PC
contribution to ${\cal P}_{\zeta_{\phi}}$ from the decay of $\sigma_{\rm osc}$
through this interaction can be evaluated using the in-in formalism [51, 52]
(see [53] for a review of the in-in formalism),

$\displaystyle\left(\frac{\Delta{\cal P}_{\zeta_{\phi}}}{{\cal
P}^{(0)}_{\zeta_{\phi}}}\right)_{\rm dec}\simeq$
$\displaystyle\,\,\theta(k-k_{d})\sqrt{2\pi}\frac{\sigma_{0}}{\Lambda_{\phi}}\left(\frac{M}{H}\right)^{1/2}$
$\displaystyle\quad\times\left(\frac{k}{k_{d}}\right)^{-3/2}\cos\left[\frac{M}{H}\log\frac{k}{k_{d}}\right]\,.$
(9)

Here, $\sigma_{\rm osc}$ directly acts as a high-frequency ($M$ from Eq. (3))
source for inflaton production instead of the irreducible gravitational source
$\Delta H_{\rm osc}$. Therefore, the $k$-dependence can be obtained using a
similar argument given for Eq. (I). With this source, a co-moving mode $k$ is
produced when $ke^{-Ht}\approx M/2$. The softest such mode produced through
direct decay is then $k_{d}e^{-Ht_{0}}=M/2$. Since the interaction is linear
in $\sigma$, the contribution in Eq. (I) matter-dilutes $\propto k^{-3/2}$, as
compared to the $\propto k^{-3}$ dependence in Eq. (I).

Such PCs have been searched for in the Planck CMB data (see e.g. analysis in
[51, 50]). While there are potential candidates, none are statistically
significant. In the low to medium $k$-range, the constraints on the amplitudes
of PC features in adiabatic perturbations are close to the ultimate cosmic
variance limit. While future probes like CMB polarization, LSS, and 21-cm
tomography will improve sensitivity, they are typically effective for PCs at
high $k$ (small scales) [50, 54, 55]. The PCs could be undetectable if
$\sigma_{0}$ is small, or, for the direct decay contribution, if
$\Lambda_{\phi}$ is large. Considering the valuable information encoded in the
PC features, we ask if they can be probed by other cosmological
observables.111 Refs. [56, 57] have also studied PC-like scale-invariance-
breaking features in scalar-induced SGWB. There are however important
qualitative differences: 1) They consider features that appear at very small
scales, which can not be probed by CMB as they are highly Silk-damped. In our
work, PC occurs on scales at which the CMB is undamped, but are too small
there for detection. 2) Their features appear in the frequency spectrum of
SGWB, unlike in the anisotropies in our case.

##  II PC in multi-field inflation

In the following, we study one such possibility where the excited $\sigma$
field decays dominantly to another light “spectator” field, $\chi$, present
during inflation. If $m_{\chi}^{2}\ll H^{2}$, it will develop an
(approximately) scale-invariant spectrum, ${\cal P}^{(0)}_{{\cal
S}_{\chi}}\propto 1/k^{3}$, similar to that of the inflaton. Here, ${\cal
S}_{\chi}\equiv\zeta_{\chi}-\zeta_{\phi}$ is the isocurvature perturbation of
$\chi$, independent of the fluctuations of $\zeta_{\phi}$ when the energy
density in isocurvature is subdominant. To naturally maintain the lightness of
$\chi$, we again consider a shift-symmetric effective interaction of the form
$\mathcal{L}_{\rm
int}\sim(\sigma\,\partial_{\mu}\chi\partial^{\mu}\chi)/\Lambda_{\chi}$,
$\Lambda_{\chi}\ll\Lambda_{\phi}$. This gives a PC feature in $P_{{\cal
S}_{\chi}}$ completely analogous to Eq. (I),222The contribution from $\Delta
H_{\rm osc}$ is small, $(\Delta{\cal P}_{{\cal S}_{\chi}}/{\cal
P}^{(0)}_{{\cal S}_{\chi}})\propto(M^{2}\sigma_{0}^{2})/(H^{2}M_{\rm
Pl}^{2})$, and hence we only focus on the direct decay of $\sigma$ to
$\delta\chi$.

$\displaystyle\frac{\Delta{\cal P}_{{\cal S}_{\chi}}}{{\cal P}^{(0)}_{{\cal
S}_{\chi}}}$
$\displaystyle\simeq\theta(k-k_{d})\alpha\left(\frac{k}{k_{d}}\right)^{-3/2}\cos\left[\frac{M}{H}\log\frac{k}{k_{d}}\right]$
$\displaystyle\text{where}\quad\alpha$ $\displaystyle=\sqrt{\frac{2\pi
M}{H}}\frac{\sigma_{0}}{\Lambda_{\chi}}\,.$ (10)

There are some constraints on $\Lambda_{\chi}$ to ensure effective field
theory (EFT) control. We require $(\sigma_{0}/\Lambda_{\chi})<1$ to keep the
correction to the kinetic energy of $\chi$ small, and
$(\dot{\chi}_{0}/\Lambda_{\chi}^{2}),\,M/\Lambda_{\chi}<1$ to ensure that no
energy scales exceed the scale of non-renormalizability at which the EFT must
break down. The perturbative control of the in-in diagrams requires the PC
feature to be small compared to the scale-invariant spectrum, giving a
constraint $\alpha<1$. We see qualitatively that it is possible to have small
$\sigma_{0}$ and large $\Lambda_{\phi}$ so that PC feature is small in ${\cal
P}_{\zeta_{\phi}}$, while $\sigma_{0}/\Lambda_{\chi}$ gives a sizeable PC
contribution in ${\cal P}_{{\cal S}_{\chi}}$.

##  III Non-minimal reheating scenario

We now illustrate how the primordial fluctuations of $\chi$, including
sizeable PC, may be probed by a potentially observable SGWB.

Figure 1: A schematic of the model with modified post-inflationary history
showing energy densities in various sectors as a function of the scale factor
(time).

We will employ the cosmological model developed in Ref. [20] which showed how
the SGWB can have a large isocurvature component, making it uncorrelated with
the CMB. We will add the new features of the $\sigma$ oscillations, back-
reaction on the inflationary expansion and decays into $\chi$, perturbatively.
The model is shown schematically in figure 1. After the end of inflation,
$\chi$ reheats the Visible Sector (VS), which is presumed to undergo a strong
first order PT, and whose remnants are just the SM and dark matter (DM). If
the PT occurs as the universe is passing through temperatures of order
(multi-)TeV, this would result in a SGWB at frequencies observable at proposed
GW detectors. The inflaton on the other hand is taken to reheat a Hidden
Sector of particle physics (HS) that interacts very weakly with the VS. Some
time after the PT, the HS particles decay to the VS, thermalizing with it
before Big Bang Nucleosynthesis. This leaves the SGWB as the only source of
isocurvature. The constraints from isocurvature and $N_{\rm eff}$ can be cast
in terms of the ratio of energy density to critical density today as
$\delta\Omega_{\rm GW}h^{2}\lesssim 2\times 10^{-11}$ and $\Omega_{\rm
GW}h^{2}\lesssim 2\times 10^{-6}$, respectively. We will later show that our
benchmarks easily satisfy these constraints.

The GW are sourced by sub-horizon physics at the time of PT, and hence the
fluctuations $\delta\rho_{\rm GW}$ on super-horizon scales are directly
related to the initial fluctuations in the VS after it is reheated by $\chi$
(but before the HS decays into it),

$\displaystyle\delta_{\rm GW}\sim(\delta_{\rm VS})_{\rm
initial}\sim\frac{\delta\chi}{\chi_{0}}\,,$ (11)

where $\delta\equiv\delta\rho/\rho$ and $\chi_{0}$ is the original homogeneous
VEV of the $\chi$ field. More precisely, the GW anisotropy can be written as
[31, 58]

$\displaystyle\delta_{\rm GW}=-\frac{4}{3}\zeta_{\phi}+4{\cal
S}_{\chi}\left(1-\frac{4}{3}f_{\rm VS}\right)\approx 4{\cal S}_{\chi}\,,$ (12)

where we consider $f_{\rm VS}\ll 1$ and ${\cal S}_{\chi}\gtrsim\zeta_{\phi}$.

The CMB anisotropies receive two independent contributions since the final
fluctuations in the VS are a combination of the decay products of the HS and
the initial VS,

$\displaystyle\delta_{\gamma}\sim\left(\delta_{\rm VS}\right)_{\rm
final}\sim\left(\delta_{\rm HS}\right)_{\rm initial}+f_{\rm
VS}\left(\delta_{\rm VS}\right)_{\rm initial}\,.$ (13)

Here, $f_{\rm VS}\equiv(\rho_{\rm VS})_{\rm initial}/\rho_{\rm total}$. In
terms of the independent gauge-invariant fluctuations,

$\displaystyle\delta_{\gamma}\approx-\frac{4}{5}\zeta_{\phi}-\frac{4}{5}f_{\rm
VS}{\cal S}_{\chi}.$ (14)

From Eq. (14), we see that $f_{\rm VS}\ll 1$ can ensure a subdominant
contribution of ${\cal S}_{\chi}$ to CMB anisotropies. The GW map in this case
will be (mostly) uncorrelated with the CMB since the ${\cal S}_{\chi}$ is
independent of $\zeta_{\phi}$. An important subtlety of the multi-field
reheating is that the early Sachs-Wolfe (SW) effect suppresses CMB
anisotropies but not the GW anisotropies, $(\delta_{\rm
GW}/\delta_{\gamma})\approx 5({\cal S}_{\chi}/\zeta_{\phi})$, as can be seen
from Eqs. (12) and (14). This is because the SW effect is anti-correlated with
the dominant adiabatic density fluctuations from $\phi$, leading to reduced
anisotropy in the observed CMB [59]. On the other hand, the SW contribution,
which is predominantly sourced by $\rho_{\rm HS}$, is uncorrelated with $\chi$
fluctuations, and hence there is no analogous cancellation in GW anisotropies.

Considering the high precision of CMB measurements, we must consider the PC
feature induced by ${\cal S}_{\chi}$ into the CMB after the HS decay into the
VS, Eq. (14), as well as the irreducible PC feature, Eqs. (I) and (I), despite
their subdominance. From Eq. (14), ${\cal S}_{\chi}$ induces

$\displaystyle\left(\frac{\Delta{\cal P}_{\gamma}}{{\cal
P}_{\gamma}}\right)_{\rm ind}\approx f^{2}_{\rm
VS}\left(\frac{A_{\chi}}{A_{\phi}}\right)\left(\frac{\Delta{\cal P}_{{\cal
S}_{\chi}}}{{\cal P}_{{\cal S}_{\chi}}}\right)\,,$ (15)

where ${\cal P}^{(0)}_{j}(k)=2\pi^{2}A_{j}k^{-3}$. The amplitude
$A_{\phi}\approx 2\times 10^{-9}$ is fixed by the CMB. An analogous relation
holds for PC in LSS.

The anisotropies for a 2D map (CMB or PT-SGWB) are expressed in the multipole
basis,

$\displaystyle\langle\delta(\hat{n})\delta(\hat{n}^{\prime})\rangle=\sum_{\ell}\frac{(2\ell+1)}{4\pi}C_{\ell}P_{\ell}(\cos\theta)\,,$
(16)

where $\hat{n}\cdot\hat{n}^{\prime}=\cos\theta$ and $P_{\ell}$ are the
Legendre polynomials. For higher multipoles $\ell\gtrsim O(100)$ (which will
be of interest to us later), the expression for $C_{\ell}$ can be simplified
using flat-sky approximation (see for example [60]):

$\displaystyle
C_{\ell}\approx\frac{1}{r^{2}}\int\frac{dk_{\parallel}}{2\pi}{\cal
P}_{\delta}\left(k_{\parallel},k_{\perp}\sim\frac{\ell}{r}\right)\,,$ (17)

where ${\cal P}_{\delta}$ is the power spectrum of the density perturbations,
$k_{\parallel(\perp)}$ is the component of $\vec{k}$ parallel (perpendicular)
to the line-of-sight, and $r$ is the co-moving radius from us to the surface
of the map that we see today. The correlation can be separated into a nearly
scale-invariant part and a small PC correction,
$C_{\ell}=C^{(0)}_{\ell}+\Delta C_{\ell}$, where
$C^{(0)}_{\ell}[\ell(\ell+1)]\equiv A$ is approximately constant. From eqs.
(12) and (17), $A_{\rm GW}\simeq 32\pi A_{\chi}$. The PC feature in SGWB can
be computed using eqs. (II) and (17),

$\displaystyle\frac{\Delta C^{\rm GW}_{\ell}}{C^{(0){\rm GW}}_{\ell}}\approx$
$\displaystyle\,\,\theta(\ell-\ell_{d})\,\tilde{\alpha}\left(\frac{\ell}{\ell_{d}}\right)^{-3/2}\cos\left[\frac{M}{H}\log\frac{\ell}{\ell_{d}}\right]\,,$
(18)

where the amplitude of the PC feature in the multipole basis,
$\tilde{\alpha}=\sqrt{2\pi H/M}\,\alpha$, is smaller than $\alpha$ due to the
2-dimensional nature the map. $\ell_{d}\approx k_{d}r_{\rm GW}$ corresponds to
the multipole where the PC feature appears in the GW map. We can similarly
evaluate the amplitude of PCs in CMB using eqs. (I), (I), and (15) instead.

With sufficient experimental sensitivity, the detection of the PC feature is
ultimately limited by cosmic variance. If we search in the parameter regime
$H\ell_{d}/M\gg 1$, then our signal oscillates slowly as a function of $\ell$,
in which case we can mitigate cosmic variance by averaging the data over a bin
of size $1<\Delta\ell<\pi H\ell_{d}/M$ around each $\ell$. Thus the signal
adds coherently while the binned cosmic variance is

$\displaystyle\frac{\delta
C_{\ell}^{bin}}{C_{\ell}}\approx\sqrt{\frac{2}{(2\ell+1)\Delta\ell}}\,.$ (19)

We determine the observability of the PC feature by comparing its largest
amplitude (at $\ell_{d}$) with the smallest compatible binned cosmic variance,

$\displaystyle\tilde{\alpha}>\sqrt{\frac{2(M/H)}{\pi(2\ell_{d}+1)\ell_{d}}}\,.$
(20)

PCs with smaller amplitudes can be observable if they appears at larger
$\ell_{d}$, as the cosmic variance drops. It is however experimentally
challenging to probe high $\ell$ anisotropies. Keeping these issues in mind,
we will consider $\ell_{d}\sim 100$ in this work.

We also want to estimate the strength of GW signal in our model. During PT,
processes like bubble collisions, sounds waves, and turbulence give rise to GW
[61]. After production, GW decouple and redshift as radiation until today. An
estimate for the peak energy density today as a fraction of critical energy
density [62] is 333We have only considered bubble wall collisions here, but
the contribution from sound waves is generally comparable and even dominates
at larger $\beta$ as it scales linearly with ($H_{\rm PT}/\beta$).

$\displaystyle\Omega_{\rm GW}h^{2}\sim 1.3\times 10^{-6}\left(\frac{H_{\rm
PT}}{\beta}\right)^{2}f_{\rm VS}^{2}\,.$ (21)

Here $\beta$ is the rate of bubble nucleation, $H_{\rm PT}$ is the Hubble
expansion rate during PT, and we are considering a strong PT where most of the
latent heat goes into accelerating the bubble walls. The factor of $f_{\rm
VS}$ appears since the PT is occurring in the VS sector that is subdominant in
the energy density during PT.

The energy density fluctuation in the $\ell^{th}$ multipole is given by

$\displaystyle\delta\Omega_{\rm GW}h^{2}(\ell)$ $\displaystyle\sim\Omega_{\rm
GW}h^{2}\times\frac{\sqrt{A_{\rm GW}}}{\ell}$ (22)

The factor of $1/\ell$ appears because
$C^{(0)}_{\ell}\propto[\ell(\ell+1)]^{-1}$ by scale-invariance. The quantity
of interest to us is the deviation from this scale-invariant spectrum due to
the PC feature, $\Delta C_{\ell}^{\rm GW}$. From Eq. (18),

$\displaystyle\delta\Omega_{\rm GW}^{\rm PC}h^{2}(\ell_{d})\sim$
$\displaystyle 1.3\times 10^{-6}\left(\frac{H_{\rm
PT}}{\beta}\right)^{2}f_{\rm VS}^{2}\frac{\sqrt{A_{\rm
GW}}}{\ell_{d}}\left(\frac{\tilde{\alpha}}{2}\right).$ (23)

##  IV Benchmarks

We will now illustrate our mechanism with two benchmarks. Consider the
following set of parameters,

$\displaystyle\sigma_{0}\simeq 2H,\quad\Lambda_{\chi}\simeq 40H,\quad
M=10H,\quad\ell_{d}=100$ (24)

Let us first consider the PC signal in SGWB. From eqs. (II) and (18), the
amplitude of the PC feature is $\tilde{\alpha}=0.31$. This easily satisfies
the condition of observability over cosmic variance, $\tilde{\alpha}>0.018$,
given in Eq. (20). A realization of SGWB map containing this PC feature and a
random sample of cosmic variance is shown in Fig. (2) (a) without binning and
(b) with binning using $\Delta\ell=6$. We see that even with a modest bin
size, the observability of signal is significantly improved, with the signal-
to-noise ratio $\sim 7.6$.

There are also irreducible PC corrections to the maps of adiabatic
perturbations as discussed before. Currently, the strongest constraints on
these corrections come from the CMB [50]. Therefore, in our first benchmark,
we consider a PC whose contributions in the 2D map of CMB are completely
hidden under cosmic variance, and yet are observable in the SGWB. In the
future, 3D maps such as LSS and 21-cm are expected to achieve higher
sensitivity as they can access more modes than a 2D map, and hence have lower
cosmic variance. It is then interesting to ask whether there are PCs which can
not be detected even in a perfect 3D map of adiabatic perturbations, making
isocurvature SGWB possibly the only channel of discovery. We demonstrate this
possibility in the second benchmark.

Consider the first benchmark along with Eq. (24),

$\displaystyle(1)\quad f_{\rm
VS}=0.1,\,\frac{A_{\chi}}{A_{\phi}}=3,\,\frac{\beta}{H_{\rm PT}}=5\,.$ (25)

The PC corrections in CMB also appear around $\ell_{d}$ and $\ell_{g}$, as the
co-moving radii to the surface of recombination and PT are approximately the
same. We take $\Lambda_{\phi}\gg H$ such that the contribution in Eq. (I) is
negligible. Eq. (I) along with $\dot{\phi}_{0}\simeq(60H)^{2}$ [49] gives a
correction $\left(\Delta
C^{\gamma}_{\ell_{g}}/C^{(0)\gamma}_{\ell_{g}}\right)_{\Delta H_{\rm
osc}}\simeq 6\times 10^{-5}$, and Eq. (15) gives $\left(\Delta
C_{\ell_{d}}^{\gamma}/C_{\ell_{d}}^{(0)\gamma}\right)_{\rm ind}\simeq 0.01$.
This is below the cosmic variance as can be checked with Eq. (20) (even after
accounting for a smaller cosmic variance at $\ell_{g}\approx 200$). This
benchmark is actually conservative in that the current constraints on the PC
features from Planck data are even weaker [50], only constraining the
amplitudes of PC features to be $\lesssim 0.08$. Access to higher multipoles
$\ell\sim{\cal O}(1000)$ in the CMB is not really an advantage as the
$\ell^{-3/2}$ (or $\ell^{-3}$) dependence in the PC features makes them fairly
localised near $\ell_{d}$ (or $\ell_{g}$).

We now estimate the experimental sensitivity required to probe this PC in
SGWB. The strength of SGWB depends on the details of the PT. Here, we consider
a relatively strong PT (see [61] for models that could undergo strong PTs).
From Eqs. (21) and (23), we get the isotropic component, $\Omega_{\rm
GW}h^{2}\simeq 5\times 10^{-10}$, and the small PC variation in anisotropies,
$\delta\Omega_{\rm GW}^{\rm PC}h^{2}(\ell_{d})\simeq 4\times 10^{-15}$. Note
that the isocurvature and $N_{\rm eff}$ bounds mentioned earlier are easily
satisfied. While the isotropic SGWB could be within LISA sensitivity,
$\Omega_{\rm GW}^{\text{\tiny LISA}}h^{2}\gtrsim 10^{-14}$ [12], the
anisotropies require more advanced future space-based detectors. For
reference, the projected sensitivity of the proposed BBO mission for large
angular scales is $\Omega_{\rm GW}^{\text{\tiny BBO}}h^{2}\gtrsim 10^{-17}$
[24, 63] and that of ultimate-DECIGO is $\Omega_{\rm GW}^{\text{\tiny
DECI}}h^{2}\gtrsim 10^{-19}$ [25]. Detection of the PC will require a detector
that maintains such a high sensitivity out to $\ell\sim O(100)$.

Now consider a second benchmark with smaller $f_{\rm VS}$,

$\displaystyle(2)\quad f_{\rm
VS}=0.01,\,\frac{A_{\chi}}{A_{\phi}}=3,\,\frac{\beta}{H_{\rm PT}}=5\,.$ (26)

The PC corrections in this case from eqs. (I) and (15) are respectively
$\left(\Delta{\cal P}_{k_{g}}/{\cal P}_{k_{g}}^{(0)}\right)_{\Delta H_{\rm
osc}}\simeq 8\times 10^{-5}$ and $\left(\Delta{\cal P}_{k_{d}}/{\cal
P}_{k_{d}}^{(0)}\right)_{\rm ind}\simeq 10^{-4}$. 3D maps of adiabatic
perturbations like LSS and 21-cm are expected to have lower cosmic variance as
they can access more modes than the 2D CMB. Cosmic variance for an idealised
3D map is $\left(\delta{\cal P}/{\cal P}^{(0)}\right)\sim\sqrt{(2k_{\rm
min}^{3})/(4\pi k^{2}\Delta k)}$, where $k_{\rm min}$ in the largest co-moving
mode that can be probes today. Then, using the largest possible bin size,
$\Delta k\sim\pi kH/M$, the cosmic variance at $k_{d}\approx 100k_{\rm min}$
is $\left(\delta{\cal P}/{\cal P}^{(0)}\right)\sim 7\times 10^{-4}$, and at
$k_{g}\approx 200k_{\rm min}$ is $\left(\delta{\cal P}/{\cal
P}^{(0)}\right)\sim 2\times 10^{-4}$. We see that the PC correction in the
second case can not be detected even with an ideal 3D map.

The strength of SGWB in this benchmark is smaller, with monopole, $\Omega_{\rm
GW}h^{2}\simeq 5\times 10^{-12}$, and the PC feature, $\delta\Omega_{\rm
GW}^{\rm PC}h^{2}(\ell_{d})\simeq 4\times 10^{-17}$. While this case is more
challenging experimentally, the impossibility of detection even in a perfect
adiabatic map makes SGWB a powerful (and possibly the only) probe of such a
PC.

The final check is the NG contribution to adiabatic perturbations as shown in
Eq. (I). In both benchmarks, the imparted NG is $f_{\rm NL}^{\rm PC}\sim
5\times 10^{-3}$, which is well below the CMB constraint $f_{\rm NL}<{\cal
O}(5-50)$ (the exact constraint depends on the shape) [64]. A conservative
estimate of the cosmic variance for a 3D map gives $\delta f_{\rm NL}^{\rm
PC}\sim 1/(\sqrt{N_{k_{g}}}\zeta_{\phi})\sim 10^{-2}$, where $N_{k_{g}}$ is
the number of 3-point configurations satisfying $k_{1}+k_{2}+k_{3}=2k_{g}$ and
$\vec{k_{1}}+\vec{k_{2}}+\vec{k_{3}}=0$, and the anisotropy $\zeta_{\phi}\sim
10^{-5}$. Again, the imparted NG in our benchmarks is safely below cosmic
variance.

(a) Unbinned (b) Binned, $\Delta\ell=6$

Figure 2: Primordial spectrum $C^{\rm GW}_{\ell}$ with the PC feature using
benchmark parameters from Eq. (24). The yellow line is the expected signal
where the onset of PC feature is happening at $\ell_{d}=100$ and the green
dots incorporate a realization of cosmic variance. Figures (a) and (b)
correspond to GW map without and with binning ($\Delta\ell=6$) respectively.

(a) $\ell_{d}=100$ (b) $\ell_{d}=1000$

Figure 3:  Analysis of the $\alpha-M$ parameter space showing excluded regions
from following considerations: binned cosmic variance (blue), CMB constraints
from contribution to $\Delta{\cal P}_{\gamma}$ (yellow) and $f_{\rm NL}$
(green). The dashed lines show contours of PC signal strength
$\delta\Omega_{\rm GW}^{PC}h^{2}(\ell_{d})$ as given in Eq. (23) such that the
parameter regions above these lines can be probed with corresponding
experimental sensitivities. The analysis is carried out for (a)
$\ell_{d}=100$, as in the benchmark cases, and (b) $\ell_{d}=1000$, where the
cosmic variance uncertainty is smaller. See the text for more details.

##  V broader analysis of the parameter space

Let us now broaden the exploration of the parameter space of PC in SGWBs
beyond the specific benchmarks above. We continue to fix the start of the PC
feature at $\ell_{d}=100$, as in the benchmark cases, and analyze
observability of PC in $\alpha-M$ space, taking both of these as our
independent variables. The requirement of $\alpha<1$ and $M>H$ ensures
$(\sigma_{0}/\Lambda_{\chi})<1$, satisfying the EFT perturbativity constraint.

Requiring that the (envelope) of the PC feature is larger than the binned
cosmic variance excludes the blue region shown in figure (3). Requiring Eqs.
(I) and (I) to lie below current CMB sensitivity, $(\Delta{\cal P}/{\cal
P}^{(0)})_{\gamma}<0.01$ and $f_{\rm NL}<10$, along with the restriction
$M<\Lambda_{\chi}$ needed for EFT control, exclude the yellow and green
regions respectively in figure 3. The contribution from HS decay in Eq. (15)
can be kept below CMB exclusion with a conservative choice of $f_{\rm
VS}^{2}=3\times 10^{-3}$ (for $A_{\chi}/A_{\phi}=3$), while $f_{\rm VS}$ can
in principle be larger for smaller $\alpha$. The contours of different signal
strengths, $\Omega^{\rm PC}_{\rm GW}h^{2}$, at given $\ell_{d}$ are also shown
in figure 3 with dashed lines.

We have assumed that the decay rate
$\Gamma(\sigma\rightarrow\delta\chi\,\delta\chi)<3H$ such that the dilution of
the classical $\sigma$ field is primarily due to the Hubble expansion,
$\propto k^{-3/2}$ dependence in the PC signal. For the interaction
considered,
$\Gamma(\sigma\rightarrow\delta\chi\delta\chi)\approx\frac{M^{3}}{32\pi\Lambda_{\chi}^{2}}$
[65]. For the largest mass $M\simeq\Lambda_{\chi}$, we see that $\Gamma<3H$
for $M\lesssim 300H$. Also, the PC oscillation would be unobservable if it is
faster than the integer discretization of $\ell$. Since the oscillation
frequency is fastest near $\ell_{d}$, we require $(M/H)<\pi\ell_{d}$. For
$\ell_{d}=100$, we again get a similar upper limit $M\lesssim 300H$, although
CMB constraints become dominant well before reaching this limit as can be seen
in figure 3 (a).

At least in principle, PC features with smaller amplitudes can be observable
if they appear at larger $\ell_{d}$, as the cosmic variance becomes smaller at
higher multipoles. An example is given in figure 3 (b) for $\ell_{d}=1000$.
This highlights the gains that can be achieved with higher angular resolution
and sensitivity in GW experiments.

##  VI Oscillating heavy field from a polynomial potential

The above analysis considered an unspecified “sharp feature” in the potential
[39, 50] to excite the heavy field, leading to the simplest form for the PC
features. Here we construct a smoother, more familiar polynomial potential for
$\sigma,\phi$, inspired by Hybrid Inflation [66, 67]. In this model, the
rolling of the inflaton field triggers a “waterfall” for the $\sigma$ field.
The waterfall however does not end inflation, but leads to a second phase of
inflation.

$\displaystyle V(\phi,\sigma)=V_{\rm
inf}(\phi)-\kappa\sigma+\frac{1}{2}(g\phi^{2}-\mu^{2})\sigma^{2}+\frac{\lambda}{4}\sigma^{4}\,.$
(27)

We will take $V_{\rm inf}(\phi)$ to dominate the energy density and hence
drive inflation. $\sigma$ has a time-dependent mass as $\phi$ rolls towards 0,

$\displaystyle
M^{2}_{\sigma}(t)=g\phi^{2}(t)-\mu^{2}+3\lambda\langle\,\sigma\,\rangle^{2}\,.$
(28)

Initially, $g\phi^{2}\gg\mu^{2}$ and $\langle\,\sigma\,\rangle\approx 0$,
$V\approx V_{\rm inf}$. As $\phi$ rolls, $\sigma$ becomes tachyonic as
$g\phi^{2}<\mu^{2}$. The transition occurs at $t_{0}$ where
$\phi(t_{0})=\phi_{c}$ and

$\displaystyle g\phi_{c}^{2}=\mu^{2}\,.$ (29)

From here, $\sigma$ starts rolling rapidly towards the new minimum in about
$\Delta t\sim H^{-1}$ when the kinetic energy is large enough to overcome
Hubble friction. When $\sigma$ becomes large enough, the last term of Eq. (28)
ensures $\sigma$ is non-tachyonic again and oscillates around the adiabatic
minimum $M_{\sigma}(t)/\sqrt{2\lambda}$, which approaches a maximum value
$\mu/\sqrt{\lambda}$ at late time. Here we are taking $\kappa$ to be small
such that its role in the $\sigma$ dynamics after $t_{0}$ is subdominant.
Earlier, it breaks the degeneracy of the two $\sigma$ minima when $\sigma$
becomes tachyonic. This ensures selection of a unique vacuum
($\langle\,\sigma\,\rangle>0$ for $\kappa>0$) in the entire observable
universe, avoiding the domain wall problem [66]444In [67], this is done by
considering $\sigma$ to be the radial direction in the quasi-single field
model of inflation. Then the centrifugal force due to the turning trajectory
acts as the source of explicit symmetry breaking..

Consider the following model parameters: $\mu=50H$, $g=1.9\times 10^{-8}$,
$\phi_{c}=3.6\times 10^{5}H$, $\lambda=0.1$, and $\kappa=0.1H^{3}$ (with
$H/M_{\rm pl}=10^{-5}$ and $\dot{\phi}_{0}\simeq 3.6\times 10^{3}H^{2}$ [49]).
Let us first check that the back-reaction of $\sigma$ dynamics on $\phi$ is
small. The maximum energy density in $\sigma$ is subdominant to the
inflationary potential energy, $\mu^{4}/\lambda\sim 10^{7}H^{4}\ll V_{\rm
inf}\simeq 3H^{2}M_{\rm pl}^{2}\sim 10^{10}H^{4}$. The kinetic energy of
$\phi$ is given by
$3H\dot{\phi}_{0}\approx-\partial_{\phi}V(\phi,\sigma)=-\partial_{\phi}V_{\rm
inf}-g\langle\,\sigma\,\rangle^{2}\phi$. The correction
$g\langle\,\sigma\,\rangle^{2}\phi\lesssim O(10)H^{3}\ll\partial_{\phi}V_{\rm
inf}\sim 10^{4}H^{3}$. This ensures slow roll of the inflaton even after the
tachyonic transition in $\sigma$.

Figure 4: The classical dynamics of $\sigma$ field after the tachyonic
transition at $t_{0}$ (blue) and the minimum $M_{\sigma}(t)/\sqrt{2\lambda}$
(yellow). The full dynamics shows oscillations about the minimum. Figure 5:
The PC feature in ${\cal P}_{{\cal S}_{\chi}}$ in the model with polynomial
potential. The blue dots correspond to the numerical evaluation of
$\Delta{\cal P}_{{\cal S}_{\chi}}$, where the overall shift is the result of
$\langle\,\sigma\,\rangle\neq 0$. For reference, the yellow dashed line shows
a matching signal from a simplified scenario where $\sigma$ is instead excited
suddenly, resulting in a PC with constant amplitude $\alpha\approx 0.125$ and
mass $M\approx 14.8H$.

In figure 4, the full $\langle\,\sigma(t)\,\rangle$ dynamics is shown in blue
along with the expected minimum at $M_{\sigma}(t)/\sqrt{2\lambda}$ shown in
yellow. We can see that the dynamics is given by an oscillatory component
superimposed on a slowly shifting minimum. We consider the same direct
coupling to $\chi$ as before, $\mathcal{L}_{\rm
int}\sim(\sigma\,\partial_{\mu}\chi\partial^{\mu}\chi)/\Lambda_{\chi}$. The
slowly shifting minimum induces a tilt in the otherwise scale-invariant
spectrum, while the oscillatory component adds a PC feature. We take
$\Lambda_{\chi}\sim 300H$ so as to ensure
$(\langle\,\sigma\,\rangle/\Lambda_{\chi})<0.5$.

Figure 5 shows the fractional correction to ${\cal P}_{{\cal S}_{\chi}}$
evaluated numerically (blue) and a matching signal (yellow) of the type
considered in the previous sections arising from a more sharply featured
potential with $\alpha\approx 0.125$ and $M\approx 14.8H$. The overall shift
in ${\cal P}_{{\cal S}_{\chi}}$ is the result of
$\langle\,\sigma(t)\,\rangle\neq 0$. The largest amplitude of PC feature is
around $k_{d}\sim 100k_{\rm min}$, corresponding to $\ell_{d}\sim 100$ as
expected. The PC is within the observable parameter space, as can be seen from
figure 3. In this model, the irreducible PC contributions to the CMB from eqs.
(I) and (I) are $(\Delta{\cal P}/{\cal P}^{(0)})_{\gamma}\simeq 0.001$ and
$f_{\rm NL}\simeq 0.1$ respectively, which are below CMB sensitivity.
Comparing to the yellow signal due to sudden excitation of the $\sigma$ field,
we see that our waterfall model gives a PC that dilutes slightly slower and
oscillates with higher frequency as $M_{\sigma}$ increases with time.

##  VII Conclusion

We have shown that the power spectrum of SGWB superhorizon anisotropies could
reveal striking oscillations (see figure (2)), which could only arise from
extremely heavy fields during inflation. However, any such new physics
introduces irreducible (gravitational) corrections to the spectrum of
adiabatic perturbations. Remarkably, it is possible for the SGWB oscillations
to be clearly visible while the CMB and LSS remain insensitive to the
irreducible corrections, despite their high precision and statistics. While we
have considered specific PC features, the mechanism can be extended to other
localised features and to non-Gaussian correlators. We illustrated a model in
which the oscillations in anisotropy appeared at $\ell\sim{\cal O}(100)$ in
order to maintain better theoretical control and minimize cosmic variance.
Similar features could appear at $\ell\sim{\cal O}(10)$, but would require
stronger couplings and a more rigorous data analysis. However, lower
experimental angular resolution would then suffice. In our smoother model, the
primordial clock “beats” irregularly, but its non-trivial frequency $M(k)$
seen in figure 5 sheds light on the smooth potential and mechanism exciting
$\sigma$.

Acknowledgements: We would like to thank Soubhik Kumar, Yuhsin Tsai, Naren
Manjunath, and Peter Shawhan for useful discussions. This work was supported
by NSF grant PHY-1914731 and the Maryland Center for Fundamental Physics.

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