# Bounds on the gaps in Kronecker sequences
(and a little bit more)

Seungki Kim

###### Abstract.

We provide bounds on the sizes of the gaps — defined broadly — in the set
$\\{k_{1}\boldsymbol{\beta}_{1}+\ldots+k_{n}\boldsymbol{\beta}_{n}\mbox{ (mod
1)}:k_{i}\in\mathbb{Z}\cap(0,Q^{\frac{1}{n}}]\\}$ for generic
$\boldsymbol{\beta}_{1},\ldots,\boldsymbol{\beta}_{n}\in\mathbb{R}^{m}$ and
all sufficiently large $Q$. We also introduce a related problem in Diophantine
approximation, which we believe is of independent interest.

##  1\. Introduction and the statements of results

###  1.1. A brief introduction

The three-gap theorem states that for any $\alpha\in\mathbb{R}$ and $Q>0$, the
set

$K(\alpha;Q)=\\{k\alpha\mbox{ (mod 1)}:k\in\mathbb{Z}\cap(0,Q]\\}$

partitions $\mathbb{R}/\mathbb{Z}$ into intervals of at most three distinct
lengths. This simple and attractive statement has caught the attention of a
number of mathematicians for several decades, and continues to stimulate new
research to this day (see e.g. [7] and the references therein; see also [2],
[3]).

The purpose of the present paper is to propose and explore the following
simple, natural curiosity: what can we say about those lengths? More
precisely, can we prove any asymptotics or estimates as $Q\rightarrow\infty$?

The part of the literature most closely related to this question seems to be
that of the discrepancy function of a Kronecker sequence. Given
$\boldsymbol{\beta}\in\mathbb{R}^{m}$ and the associated Kronecker sequence

$K(\boldsymbol{\beta};Q)=\\{k\boldsymbol{\beta}\mbox{ (mod
1)}:k\in\mathbb{Z}\cap(0,Q]\\},$

the _discrepancy function_ $\Delta$ is defined to be

$\Delta(\boldsymbol{\beta};Q)=\max_{1\leq q\leq Q\atop 0\leq
x_{1},\ldots,x_{m}\leq 1}\Big{|}qx_{1}\cdots
x_{m}-\big{|}K(\boldsymbol{\beta};q)\cap\prod_{1\leq i\leq
m}[0,x_{i})\big{|}\Big{|}.$

Regarding the size of $\Delta$, there is a beautiful theorem of Beck [1,
Theorem 1] stating that, for any positive increasing function
$\varphi:\mathbb{R}\rightarrow\mathbb{R}$,

(1.1) $\Delta(\boldsymbol{\beta};Q)\ll(\log Q)^{m}\cdot\varphi(\log\log
Q)\Leftrightarrow\sum_{j=1}^{\infty}\frac{1}{\varphi(j)}<\infty$

for almost every $\boldsymbol{\beta}$. As a crude application of this result,
it is possible to prove, for example, that the maximal size of the “gap” — let
us provisionally define it as the volume of a hypercube in
$\mathbb{R}^{m}/\mathbb{Z}^{m}$ not intersecting $K(\boldsymbol{\beta};Q)$ —
is at most of order

$Q^{-1}\cdot(\log Q)^{m}\cdot\varphi(\log\log Q),$

for any positive increasing $\varphi$ satisfying the right-hand side of (1.1).
It seems this is also the best bound obtainable this way, since (1.1) does not
yield meaningful information when $Qx_{1}\cdots x_{m}=o\left((\log
Q)^{m}\cdot\varphi(\log\log Q)\right)$.

In this paper, we show, among other things, that the above bound can be
improved to

$Q^{-1}\cdot\log Q\cdot\varphi(\log\log Q).$

In fact, we consider our topic in a much broader generality, in terms of both
the sequences $K$ and the notion of the gap. We provide the details in the
next section.

###  1.2. A bit of setup, and the main results

Fix integers $m,n>0$. For an $m\times n$ matrix

$B=\begin{pmatrix}|&&|\\\
\boldsymbol{\beta}_{1}&\cdots&\boldsymbol{\beta}_{n}\\\ |&&|\end{pmatrix}$

and a positive real number $Q$, write

$K(B;Q)=\\{B\mathbf{k}\mbox{ (mod
1)}:\mathbf{k}\in\mathbb{Z}^{n}\cap(0,Q^{\frac{1}{n}}]^{n}\\}.$

Choose also $p\in[1,\infty]$ and an open subset $S$ of the unit sphere in
$\mathbb{R}^{m}$. The variables $m,n,p,S$ may be considered fixed throughout
this paper.

Let $d=m+n$ and $\mathrm{cone}(S)=\\{c\mathbf{u}:\mathbf{u}\in S,c>0\\}$. For
$T\subseteq\mathbb{R}^{m}$, write

${\inf}_{p}^{S}T:=\inf\\{\|\mathbf{y}\|_{p}:\mathbf{y}\in
T\cap\mathrm{cone}(S)\\},$

the infimum of the $L^{p}$-norm of the vectors in $T$ in the direction of $S$.

For $\mathbf{k}\in\mathbb{Z}^{n}\cap(0,Q^{\frac{1}{n}}]^{n}$, we define
$g_{(B;Q)}(\mathbf{k})$, the nearest gap in $K(B;Q)$ from $B\mathbf{k}\mbox{
(mod $1$)}$ in the direction of $S$ with respect to the $L^{p}$-norm, by

(1.2)
$g_{(B;Q)}(\mathbf{k})={\inf}_{p}^{S}\\{B(\boldsymbol{\ell}-\mathbf{k})+\mathbf{m}\in\mathbb{R}^{m}:(\boldsymbol{\ell},\mathbf{m})\in\mathbb{Z}^{d},\boldsymbol{\ell}\in(0,Q^{\frac{1}{n}}]^{n}\\}.$

Our question is then whether we can give any size estimate on $g_{(B;Q)}$ as
$Q\rightarrow\infty$. We are able to prove the following bounds:

######  Theorem 1.1.

Suppose $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ is an increasing
function such that

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty.$

Then, for almost all $m\times n$ matrix $B$, and all sufficiently large $Q>0$,

$\max\\{g_{(B;Q)}(\mathbf{k}):\mathbf{k}\in\mathbb{Z}^{n}\cap(0,Q^{\frac{1}{n}}]^{n}\\}<Q^{-\frac{1}{m}}\left(f(\log
Q)\right)^{\frac{1}{m}}.$

If we further assume that $f(x)=O(x^{1+\varepsilon})$ for some
$0<\varepsilon<m/n$, then it also holds that

$\min\\{g_{(B;Q)}(\mathbf{k}):\mathbf{k}\in\mathbb{Z}^{n}\cap(0,Q^{\frac{1}{n}}]^{n}\\}>Q^{-\frac{1}{m}}\left(f(\log(Q\log
Q))\right)^{-\frac{1}{m}}.$

We also introduce the following alternative notion of gap, that is, the
distance from _any_ $\mathbf{v}\in\mathbb{R}^{m}$ to the nearest point in
$K(B;Q)$, in the direction of $S$ with respect to the $L^{p}$-norm, by

(1.3)
$\gamma_{(B;Q)}(\mathbf{v})={\inf}_{p}^{S}\\{B\mathbf{n}+\mathbf{m}-\mathbf{v}\in\mathbb{R}^{m}:(\mathbf{n},\mathbf{m})\in\mathbb{Z}^{d},\mathbf{n}\in(0,Q^{\frac{1}{n}}]^{n}\\}.$

Regarding $\gamma_{(B;Q)}$, we have the following result.

######  Theorem 1.2.

Suppose $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ is an increasing
function such that

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty.$

Then, for almost all $m\times n$ matrix $B$, and all sufficiently large $Q>0$,

$\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})<Q^{-\frac{1}{m}}\left(f(\log
Q)\right)^{\frac{1}{m}}.$

If we further assume that $f(x)=O(x^{1+\varepsilon})$ for some
$0<\varepsilon<dm/n-1$, then also

$\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})>Q^{-\frac{1}{m}}\left(f(\log(Q\log
Q))\right)^{-\frac{1}{m^{2}}}.$

Theorem 1.2 has an application to the following problem in Diophantine
approximation, which we believe is of independent interest. In fact, this was
the problem we started investigating at first, but we subsequently found that
it is closely related to the gap sizes in $K(B;Q)$, which led us to Theorems
1.1 and 1.2.

######  Corollary.

Let $B$ be an $m\times n$ real-valued matrix, and $N>0$ be an integer.
Consider the inequalities

(1.4) $\|B\mathbf{n}-\mathbf{m}\|_{\infty}^{m}<N^{m}\varphi(Q)\mbox{ and
}\|\mathbf{n}\|_{\infty}^{n}<(N/2)^{n}Q,$

where $\mathbf{m}\in\mathbb{Z}^{m},\mathbf{n}\in\mathbb{Z}^{n}$, and
$\varphi:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ is a decreasing function.

Then for almost every $B$, and for all $N$ and all $Q\gg_{B,N}0$, (1.4) has a
solution $(\mathbf{m},\mathbf{n})\in\mathbb{Z}^{d}$ in _every_ congruence
class of $\mathbb{Z}^{d}$ modulo $N$, if

$\varphi(Q)\geq Q^{-1}f(\log Q)$

for $f$ in the first statement of Theorem 1.2.

On the other hand, if

$\varphi(Q)\leq Q^{-1}\left(f(\log(Q\log Q))\right)^{-\frac{1}{m}}$

for some $f$ as in the second statement of Theorem 1.2, then for $Q\gg_{B,f}0$
and $N\gg_{Q}0$, (1.4) fails to have a solution
$(\mathbf{m},\mathbf{n})\in\mathbb{Z}^{d}$ in some congruence class of
$\mathbb{Z}^{d}$ modulo $N$.

The proofs of Theorems 1.1 and 1.2 are nearly identical. Both start by
translating the problem to the space of lattices à la Marklof-Strömbergsson
[7]. Then methods in the geometry of numbers show that the sizes of
$g_{(B;Q)}$ and $\gamma_{(B;Q)}$ are controlled by the successive minima of
some $d$-dimensional lattices. The growth and decay of the successive minima
are then in turn controlled by the zero-one law due to Kleinbock-Margulis [5]
(see also [4, Proposition 1.11]).

######  Remark.

The majority of the existing dynamical studies on Diophantine approximations
depends solely on the behavior of the first successive minimum $\lambda_{1}$
of a lattice. Hence it may be amusing to note that the upper bounds we gave on
both $g_{(B;Q)}$ and $\gamma_{(B;Q)}$ are controlled by $\lambda_{d}$, and the
lower bound on $\gamma_{(B;Q)}$ is controlled by $\lambda_{m}$. Thus the
corollary to Theorem 1.2 appears to be the first written problem in
Diophantine approximation for which all successive minima are relevant.

###  1.3. Organization

We collect the results needed for later arguments in Section 2. Section 3
proves Theorem 1.1, and Section 4 proves Theorem 1.2 and its corollary.

##  2\. Preliminaries

###  2.1. Setup

We continue to use the notations defined in Section 1.2 throughout this paper.
In this section, we adopt the idea of Marklof-Strömbergsson [7] to phrase our
gap problem in terms of a geodesic flow on the space of lattices.

Write

$U_{B}=\begin{pmatrix}\mathrm{Id}_{n}&\\\
B&\mathrm{Id}_{m}\end{pmatrix},A_{Q}=\begin{pmatrix}Q^{-\frac{1}{n}}\mathrm{Id}_{n}&\\\
&Q^{\frac{1}{m}}\mathrm{Id}_{m}\end{pmatrix}.$

Then $g_{(B;Q)}$ defined in (1.2) can be reformulated as

$\displaystyle g_{(B;Q)}(\mathbf{k})$
$\displaystyle={\min}_{p}^{S}\\{B\mathbf{n}+\mathbf{m}\in\mathbb{R}^{m}:(\mathbf{n},\mathbf{m})\in\mathbb{Z}^{d},\mathbf{n}\in(0,Q^{\frac{1}{n}}]^{n}-\mathbf{k}\\}$
$\displaystyle={\min}_{p}^{S}\\{\mathbf{y}\in\mathbb{R}^{m}:(\mathbf{x},\mathbf{y})\in
U_{B}\mathbb{Z}^{d},\mathbf{x}\in(0,Q^{\frac{1}{n}}]^{n}-\mathbf{k}\\}$
$\displaystyle=Q^{-\frac{1}{m}}{\min}_{p}^{S}\\{\mathbf{y}\in\mathbb{R}^{m}:(\mathbf{x},\mathbf{y})\in
A_{Q}U_{B}\mathbb{Z}^{d},\mathbf{x}\in(0,1]^{n}-Q^{-\frac{1}{n}}\mathbf{k}\\}$
$\displaystyle=Q^{-\frac{1}{m}}F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k}),$

where

$F(M,\mathbf{v})={\min}_{p}^{S}\\{\mathbf{y}\in\mathbb{R}^{m}:(\mathbf{x},\mathbf{y})\in
M\mathbb{Z}^{d},\mathbf{x}\in(0,1]^{n}-\mathbf{v}\\}$

for $M\in\mathrm{SL}(d,\mathbb{R})$ and $\mathbf{v}\in(0,1]^{n}$. We leave
open the possibility that $F(M,\mathbf{v})=\infty$.

By a similar maneuver, $\gamma_{(B;Q)}$ defined in (1.3) can be reformulated
as

$\gamma_{(B;Q)}(\mathbf{v})=Q^{-\frac{1}{m}}\Phi(A_{Q}U_{B},Q^{\frac{1}{m}}\mathbf{v}),$

where

$\Phi(M,\mathbf{w})={\min}_{p}^{S}\\{\mathbf{y}-\mathbf{w}\in\mathbb{R}^{m}:(\mathbf{x},\mathbf{y})\in
M\mathbb{Z}^{d},\mathbf{x}\in(0,1]^{n}\\}$

for $\mathbf{w}\in\mathbb{R}^{m}$.

###  2.2. Results from the geometry of numbers

By the word _lattice_ we mean a finitely generated discrete
$\mathbb{Z}$-submodule of a Euclidean space. Unless we mention otherwise, we
assume the lattice is of the full rank, that is, its rank is the same as the
dimension of its ambient space. For a lattice $L\subseteq\mathbb{R}^{d}$ and
$i=1,2,\ldots,d$, recall that its _$i$ -th successive minimum_
$\lambda_{i}(L)$ is the smallest real number such that the elements of $L$ of
length (with respect to the $L^{2}$-norm) less than or equal to
$\lambda_{i}(L)$ span a subspace of dimension at least $i$. For a $d\times d$
matrix $M$, let us write $\lambda_{i}(M)$ for the $i$-th minimum of the
lattice generated by the columns of $M$.

The following theorem plays a crucial role in bounding $g_{(B;Q)}$ and
$\gamma_{(B;Q)}$ from above. It and the two lemmas that follow are likely
well-known in some form, though we were unable to find a reference with the
exact statements.

######  Theorem 2.1.

Let $L\subseteq\mathbb{R}^{d}$ be a (full-rank) lattice. Then any set
containing a closed ball of diameter (with respect to the $L^{2}$-norm)
$\sqrt{d}\lambda_{d}(L)$ contains a point of $L$.

This is an immediate consequence of the two lemmas below.

######  Lemma 2.2.

Every lattice $L\subseteq\mathbb{R}^{d}$ admits a covering by rectangles of
diameter $\sqrt{d}\lambda_{d}(L)$.

###### Proof.

Choose linearly independent $\mathbf{v}_{1},\ldots\mathbf{v}_{d}\in L$ such
that $\|\mathbf{v}_{i}\|_{2}=\lambda_{i}(L)$. The $\mathbf{v}_{i}$’s span a
sublattice $L^{\prime}$ of $L$ of finite index. We will demonstrate a tiling
of $\mathbb{R}^{d}$ by the $L^{\prime}$-translates of the rectangles of
diameter at most $\sqrt{d}\lambda_{d}(L)$, which will prove the lemma.

It is a well-known fact that the lattice generated by the column vectors of a
$d\times d$ upper-triangular unipotent matrix

$\begin{pmatrix}1&*&*&\cdots&*\\\ &1&*&\cdots&*\\\ &&\ddots&\vdots&\vdots\\\
&&&1&*\\\ &&&&1\end{pmatrix}$

admits a tiling of $\mathbb{R}^{d}$ by the $d$-dimensional unit cube
$[-1/2,1/2]^{d}$. One quick way to see this is as follows: denote by
$\mathbf{c}_{i}$ the $i$-th column vector of the above matrix, by $G$ the
lattice generated by $\\{\mathbf{c}_{1},\ldots,\mathbf{c}_{d}\\}$, and by
$G^{\prime}$ the (corank $1$) sublattice of $G$ generated by
$\\{\mathbf{c}_{1},\ldots,\mathbf{c}_{d-1}\\}$. Then since

$G=\coprod_{k\in\mathbb{Z}}(G^{\prime}+k\mathbf{c}_{d}),$

and the last coordinate of $\mathbf{c}_{d}$ is $1$, the problem reduces to
tiling $\mathbb{R}^{d-1}$ by the $G^{\prime}$-translates of the
$(d-1)$-dimensional unit cube. One can now invoke induction to prove the
claim.

Back to the lattice $L^{\prime}$ generated by the $\mathbf{v}_{i}$’s, the
matrix whose $i$-th column is $\mathbf{v}_{i}$ has the Iwasawa decomposition
$KAN$, where $K\in\mathrm{O}(d,\mathbb{R})$,
$A=\mathrm{diag}(a_{1},\ldots,a_{d})$ with
$0<a_{i}\leq\|\mathbf{v}_{i}\|=\lambda_{i}(L)$, and $N$ is upper-triangular
unipotent. By the above discussion, the lattice generated by the columns of
$N$ admits a tiling by the unit cube. Applying $KA$ to it, we see that
$L^{\prime}$ admits a tiling by a rectangle of diameter
$\sqrt{a_{1}^{2}+\ldots+a_{d}^{2}}\leq\sqrt{d}\lambda_{d}(L)$, as desired. ∎

######  Lemma 2.3.

Let $L\subseteq\mathbb{R}^{d}$ be a lattice, and $C\subseteq\mathbb{R}^{d}$ be
a convex, centrally symmetric closed set such that its $L$-translates cover
$\mathbb{R}^{d}$. Then any $D\subseteq\mathbb{R}^{d}$ containing a closed ball
of diameter $\operatorname{diam}(C)$ contains a point of $L$.

###### Proof.

It suffices to assume $D$ is a closed ball of diameter
$\operatorname{diam}(C)$. Suppose $L\cap D$ is empty. Then there exists an
$\varepsilon>0$ such that every point of $L$ is at least
$\frac{1}{2}\operatorname{diam}(C)+\varepsilon$ units of distance away from
the center of $D$. But this contradicts the assumption that the translates of
$C$ by the points of $L$ cover $\mathbb{R}^{d}$. ∎

###### Proof of Theorem 2.1.

By Lemma 2.2, $\mathbb{R}^{d}$ is covered by the $L$-translates of a rectangle
$C$ of diameter $\sqrt{d}\lambda_{d}(L)$. Applying Lemma 2.3 with this $C$
completes the proof. ∎

The two lemmas below will be used in Section 4 to give a lower bound on
$\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})$.

######  Lemma 2.4.

Let $L\subseteq\mathbb{R}^{d}$ be a lattice. Then there exists a subspace
$S_{m}\subseteq\mathbb{R}^{d}$ of dimension $m-1$ such that $L$ is contained
in a disjoint union of the translates of $S_{m}$. Furthermore, the
($L^{2}$-)distance between any two of those translates is at least
$c_{1}(d)\lambda_{m}(L)$, for a constant $c_{1}(d)>0$ depending only on $d$.

###### Proof.

We need the reduction theory on the space of lattices, for which we adopt the
language and the results of [6]; it is also an excellent reference for readers
seeking more details on the subject.

Take a reduced basis $\\{\mathbf{b}_{1},\ldots,\mathbf{b}_{d}\\}$ of $L$,
whose Gram-Schmidt orthogonalization is given by
$\\{\mathbf{b}_{1}^{*},\ldots,\mathbf{b}_{d}^{*}\\}$ (see [6, pp.516] for both
notions). We let
$S_{m}=\mathrm{span}\\{\mathbf{b}_{1},\ldots,\mathbf{b}_{m-1}\\}$. It is clear
that $L$ is contained in the union of the translates of $S_{m}$ by the integer
linear combinations of $\mathbf{b}_{m},\ldots,\mathbf{b}_{d}$, and that any
two translates are at least $\min_{m\leq i\leq d}\|\mathbf{b}_{i}^{*}\|_{2}$
apart. But by [6, (1.7)] and the remark following [6, Proposition 1.12],
$\min_{m\leq i\leq d}\|\mathbf{b}_{i}^{*}\|_{2}$ is bounded from below by
$c_{1}(d)\lambda_{m}(L)$ for some $c_{1}(d)>0$. ∎

######  Lemma 2.5.

Continue with the settings of Lemma 2.4. Then there exists $c_{2}(d)>0$ and
$\mathbf{y}\in\mathbb{R}^{m}$ such that
$[0,c_{2}(d)\lambda_{m}(L)]^{d}+(\mathbf{0}_{n},\mathbf{y})$ is disjoint from
any translate of $S_{m}$ for which $S_{m}\cap L\neq\phi$.

###### Proof.

Let us take $c_{2}(d)=(1-(2n)^{-1})\sqrt{d}^{-1}c_{1}(d)$, say. Since $S_{m}$
is $(m-1)$-dimensional, it is possible to choose $\mathbf{y}$ so that
$[0,c_{2}(d)\lambda_{m}(L)]^{d}+(\mathbf{0}_{n},\mathbf{y})$ is disjoint from
any fixed translate of $S_{m}$. If it is impossible to choose $\mathbf{y}$
such that it is disjoint from any translate of $S_{m}$, then for some
$\mathbf{y}$ it must intersect with two distinct such translates. But this
contradicts Lemma 2.4, which states that they are at least
$c_{1}(d)\lambda_{m}(L)$ apart in distance. ∎

###  2.3. Zero-one laws on diagonal flows

Theorems 1.7 and 8.7, and Proposition 7.1 of Kleinbock and Margulis [5] imply
the following Borel-Cantelli type statement, paraphrased in the form
convenient for our later use.

######  Theorem 2.6 (Kleinbock-Margulis [5], see also Kim-Skenderi [4,
Proposition 1.11]).

Suppose $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ is an increasing
function, and $U\in\mathrm{SL}(d,\mathbb{R})$. Then for $1\leq i\leq d-1$,

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty\Leftrightarrow\mbox{ for almost
every $U$, }\lambda_{i}(A_{R}U)>(f(\log R))^{-\frac{1}{di}}\mbox{ for
$R\gg_{U}0$}.$

Also, for $2\leq j\leq d$,

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty\Leftrightarrow\mbox{ for almost
every $U$, }\lambda_{j}(A_{R}U)<(f(\log R))^{\frac{1}{d(d-j+1)}}\mbox{ for
$R\gg_{U}0$}.$

The same statements continue to hold, if $U$ is replaced by $U_{B}$ for an
$m\times n$ matrix $B$, and “almost every $U$” is replaced by “almost every
$B$.”

##  3\. Proof of Theorem 1.1

###  3.1. Upper bound on the longest gap

Take $U\in\mathrm{SL}(d,\mathbb{R})$ and $R\in\mathbb{R}_{>0}$, and let

$P=\left(\left(1+n^{-1}\right)\sqrt{d}\lambda_{d}(A_{R}U)\right)^{n}$

(any choice slightly greater than $(\sqrt{d}\lambda_{d}(A_{R}U))^{n}$ will
do). We determine $Q$ by $Q=PR$.

By Theorem 2.1, an arbitrary translate of
$H=(0,P^{\frac{1}{n}}]^{d}\subseteq\mathbb{R}^{d}$ contains a point of
$A_{R}U$. This implies that an arbitrary translate of
$A_{P}H=(0,1]^{n}\times(0,P^{\frac{d}{mn}}]^{m}$ contains a point of
$A_{P}A_{R}U=A_{Q}U$. With this in mind, consider the set

$\left((0,1]^{n}-\mathbf{v}\right)\times\left(B_{p}^{m}(rP^{\frac{d}{mn}})\cap\mathrm{cone}(S)\right)$

for some $\mathbf{v}\in(0,1]^{n}$ and $r>0$, where
$B_{p}^{m}(t)\subseteq\mathbb{R}^{m}$ here denotes the $L^{p}$-ball of radius
$t$ at origin. If $r$ is sufficiently large, depending on $p$ and $S$, then
this set contains a translate of $A_{P}H$ — for instance, if $p=\infty$ and
$S$ is the entire unit sphere, then we can take $r=1/2$. Therefore

(3.1) $F(A_{Q}U,\mathbf{v})<rP^{\frac{d}{mn}}$

for any $\mathbf{v}\in(0,1]^{n}$.

We specialize to the case $U=U_{B}$ and
$\mathbf{v}=Q^{-\frac{1}{n}}\mathbf{k}$. By Theorem 2.6, for every increasing
function $f$ satisfying

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty,$

it follows that, for almost every $B$ and all sufficiently large $R$,

$F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k})<rP^{\frac{d}{mn}}<f(\log
R)^{\frac{1}{m}},$

by applying Theorem 2.6 with an appropriate constant multiple of $f$ if
necessary.

At this point, observe that $P\geq 1$, since $\lambda_{d}(L)\geq 1$ for any
lattice $L\subseteq\mathbb{R}^{d}$. It follows that $R\rightarrow\infty$ if
and only if $Q\rightarrow\infty$, since $Q$ is bounded from both sides by
functions increasing in $R$, and that

$f(\log R)=f(\log Q-\log P)\leq f(\log Q).$

Summarizing the discussion so far, we conclude that

$F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k})<f(\log Q)^{\frac{1}{m}},$

or equivalently

$g_{(B;Q)}(\mathbf{k})<Q^{-\frac{1}{m}}\left(f(\log Q)\right)^{\frac{1}{m}},$

for all sufficiently large $Q$, depending on $p,S$, and $B$.

###  3.2. Lower bound on the shortest gap

The argument in this case is only slightly different from the earlier one. We
again take $R\in\mathbb{R}_{>0}$ and write $R=QP^{-1}$, but this time define

$P=\left((1-(2n)^{-1})\sqrt{d}^{-1}\lambda_{1}(A_{R}U_{B})\right)^{n}.$

By construction, $A_{R}U_{B}$ has no nonzero vector inside the box
$H=\\{\mathbf{x}\in\mathbb{R}^{d}:\|\mathbf{x}\|_{\infty}\leq
P^{\frac{1}{n}}\\}$. Therefore, $A_{Q}U$ has no nonzero vectors inside
$A_{P}H=[-1,1]^{n}\times[-P^{\frac{d}{mn}},P^{\frac{d}{mn}}]^{m}$. This
implies that

$F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k})>rP^{\frac{d}{mn}}$

for any $\mathbf{k}\in(0,Q^{\frac{1}{n}}]^{n}$, and some $r>0$ depending on
$p$.

Invoking Theorem 2.6, for every increasing function $f$ satisfying

$\sum_{k=1}^{\infty}\frac{1}{f(k)}<\infty,$

it holds that, for almost every $B$ and all sufficiently large $R$,

$F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k})>rP^{\frac{d}{mn}}>(f(\log
R))^{-\frac{1}{m}}.$

Let us further assume that $f(x)=O(x^{1+\varepsilon})$ for some
$0<\varepsilon<m/n$. By increasing $\varepsilon$ a little if necessary, we may
instead assume that $f(x)\leq x^{1+\varepsilon}$ for all sufficiently large
$x$. Then for $R$ large,

(3.2) $(\log R)^{-(1+\varepsilon)n/d}\leq P\leq(1-(2n)^{-1})^{n}$

holds — the upper bound comes from Minkowski’s bound
$\lambda_{1}(L)\leq\sqrt{d}$ for any lattice $L\subseteq\mathbb{R}^{d}$.
Therefore, $Q=RP$ is again bounded from both sides by increasing functions in
$R$, and thus $R\rightarrow\infty$ if and only if $Q\rightarrow\infty$. In
addition, we have $R(\log R)^{-\delta_{1}}\leq Q$ and $R^{\delta_{2}}\leq Q$
for some $0<\delta_{1},\delta_{2}<1$ sufficiently close to $1$ and all
sufficiently large $Q$ and $R$. It follows that

$\displaystyle\log R$ $\displaystyle\leq\log Q+\delta_{1}\log\log R$
$\displaystyle\leq\log Q+\delta_{1}\log\log Q^{1/\delta_{2}}$
$\displaystyle\leq\log(Q\log Q).$

The last inequality holds provided $Q$ is large enough. We deduce that

$g_{(B;Q)}(\mathbf{k})=Q^{-\frac{1}{m}}F(A_{Q}U_{B},Q^{-\frac{1}{n}}\mathbf{k})>Q^{-\frac{1}{m}}\left(f(\log(Q\log
Q))\right)^{-\frac{1}{m}}$

for all sufficiently large $Q$, depending on $p$ and $B$. This completes the
proof of Theorem 1.1.

##  4\. Proof of Theorem 1.2

###  4.1. Upper bound on $\gamma_{(B;Q)}$

In this section, we prove that, for all $\mathbf{v}\in\mathbb{R}^{m}$, and for
almost every $B$ and sufficiently large $Q$ depending on $p,S,$ and $B$,

(4.1) $\gamma_{(B;Q)}(\mathbf{v})<Q^{-\frac{1}{m}}\left(f(\log
Q)\right)^{\frac{1}{m}}$

for any increasing $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ for which
$\sum_{k=1}^{\infty}(f(k))^{-1}$ converges. The argument is nearly identical
to that in Section 3.1 earlier. We will take the notations there, and show
that, for all sufficiently large $r$ depending on $p$ and $S$,

(4.2) $\Phi(A_{Q}U,\mathbf{w})<rP^{\frac{d}{mn}}$

for any $\mathbf{w}\in\mathbb{R}^{m}$, analogously to (3.1); then replicating
the argument in the rest of Section 3.1 will yield (4.1).

Recall from Section 3.1 that any translate of $A_{P}H$ contains a point of
$A_{Q}U\mathbb{Z}^{d}$. This is equivalent to saying that any translate of
$A_{P}H$ contains a point of any translate of $A_{Q}U\mathbb{Z}^{d}$. Now for
$r$ large enough, the set

$(0,1]^{n}\times\left(B_{p}^{m}(rP^{\frac{d}{mn}})\cap\mathrm{cone}(S)\right)$

contains a translate of $A_{P}H$, and thus contains a point of
$A_{Q}U_{B}\mathbb{Z}^{d}+(\mathbf{0}_{n},\mathbf{w})$ for any
$\mathbf{w}\in\mathbb{R}^{m}$, proving (4.2) and thus (4.1), as claimed.

###  4.2. Lower bound

Of course, $\gamma_{(B;Q)}(\mathbf{v})$ can be arbitrarily small, depending on
$\mathbf{v}$; we will provide a lower bound on
$\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})$ for almost
every $B$ and sufficiently large $Q$.

As earlier, set $R=QP^{-1}\in\mathbb{R}_{>0}$, but this time define
$P=\left(c_{2}(d)\lambda_{m}(A_{R}U_{B})\right)^{n}$, where $c_{2}(d)$ is as
in Lemma 2.5. Lemma 2.5 implies that $A_{R}U_{B}$ has no vectors inside
$(0,P^{\frac{1}{n}}]^{d}+(\mathbf{0}_{n},\mathbf{y})$ for some
$\mathbf{y}\in\mathbb{R}^{m}$, which implies in turn that $A_{Q}U_{B}$ has no
vectors inside
$(0,1]^{n}\times(0,P^{\frac{d}{mn}}]^{m}+(\mathbf{0}_{n},P^{\frac{1}{m}}\mathbf{y})$.
This implies that

$\Phi(A_{Q}U_{B},\mathbf{w})>rP^{\frac{d}{mn}}$

for some $\mathbf{w}\in\mathbb{R}^{m}$ and $r>0$ depending on $p$. Now,
repeating the same argument as in Section 3.2 (for the upper bound on $P$ in
case $m\neq 1$, one may appeal to Theorem 2.6, for instance), we conclude that

$\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})>Q^{-\frac{1}{m}}\left(f(\log(Q\log
Q))\right)^{-\frac{1}{m^{2}}}.$

This completes the proof of Theorem 1.2.

###  4.3. Proof of Corollary

The condition (1.4) is equivalent to

$\begin{pmatrix}\mathbf{n}\\\
\mathbf{m}\end{pmatrix}\in\begin{pmatrix}\mathrm{Id}_{n}&\\\
-B&\mathrm{Id}_{m}\end{pmatrix}\begin{pmatrix}NQ^{\frac{1}{n}}/2&\\\
&N(\varphi(Q))^{\frac{1}{m}}\end{pmatrix}\cdot[-1,1]^{d}.$

Therefore, a necessary and sufficient condition for (1.4) to have a solution
in every congruence class of $\mathbb{Z}^{d}$ modulo $N$ is that

$\begin{pmatrix}\mathrm{Id}_{n}&\\\
-B&\mathrm{Id}_{m}\end{pmatrix}\begin{pmatrix}NQ^{\frac{1}{n}}/2&\\\
&N(\varphi(Q))^{\frac{1}{m}}\end{pmatrix}\cdot[-1,1]^{d}\mbox{ (mod
$N\mathbb{Z}^{d}$)}$

contains every representative of $\mathbb{Z}^{d}/N\mathbb{Z}^{d}$.
Normalizing, this is equivalent to the statement that

(4.3) $\begin{pmatrix}\mathrm{Id}_{n}&\\\
-B&\mathrm{Id}_{m}\end{pmatrix}\begin{pmatrix}Q^{\frac{1}{n}}/2&\\\
&(\varphi(Q))^{\frac{1}{m}}\end{pmatrix}\cdot[-1,1]^{d}\mbox{ (mod
$\mathbb{Z}^{d}$)}$

contains every representative of $(N^{-1}\mathbb{Z})^{d}/\mathbb{Z}^{d}$. We
claim the following.

######  Proposition 4.1.

(4.3) contains every representative of $(N^{-1}\mathbb{Z})^{d}/\mathbb{Z}^{d}$
for every integer $N>0$ if and only if

(4.4)
$B\left((0,Q^{\frac{1}{n}}]^{n}\cap\mathbb{Z}^{n}\right)+[(-\varphi(Q))^{\frac{1}{m}},\varphi(Q))^{\frac{1}{m}}]^{m}\mbox{
(mod $\mathbb{Z}^{m}$)}$

covers all of $\mathbb{R}^{m}/\mathbb{Z}^{m}$. Furthermore, if (4.4) does not
cover $\mathbb{R}^{m}/\mathbb{Z}^{m}$, then (4.3) fails to contain every
representative of $(N^{-1}\mathbb{Z})^{d}/\mathbb{Z}^{d}$ for any sufficiently
large $N$.

###### Proof.

Suppose first that (4.4) covers all of $\mathbb{R}^{m}/\mathbb{Z}^{m}$. We
claim that (4.3) covers $\mathbb{R}^{d}/\mathbb{Z}^{d}$, which proves the “if”
part. Fix a representative $\mathbf{x}\in\mathbb{R}^{n}/\mathbb{Z}^{n}$. Then
the “fiber” $\mathcal{F}_{\mathbf{x}}$ of (4.3) over $\mathbf{x}$, that is,
the set of all $\mathbf{y}\in\mathbb{R}^{m}/\mathbb{Z}^{m}$ such that
$(\mathbf{x},\mathbf{y})$ is contained in $\eqref{eq:cdn}$ is given by

$\mathcal{F}_{\mathbf{x}}=-B\left([-Q^{\frac{1}{n}}/2,Q^{\frac{1}{n}}/2]^{n}\cap(\mathbf{x}+\mathbb{Z}^{n})\right)+[(-\varphi(Q))^{\frac{1}{m}},\varphi(Q))^{\frac{1}{m}}]^{m}\mbox{
(mod $\mathbb{Z}^{m}$)}.$

On the other hand,

$\displaystyle(0,Q^{\frac{1}{n}}]^{n}\cap\mathbb{Z}^{n}$
$\displaystyle=(-Q^{\frac{1}{n}}/2,Q^{\frac{1}{n}}/2]^{n}\cap(\mathbb{Z}-Q^{\frac{1}{n}}/2)^{n}+(Q^{\frac{1}{n}}/2,\ldots,Q^{\frac{1}{n}}/2)$
$\displaystyle\subseteq(-Q^{\frac{1}{n}}/2,Q^{\frac{1}{n}}/2]^{n}\cap(\mathbf{x}+\mathbb{Z}^{n})+(Q^{\frac{1}{n}}/2,\ldots,Q^{\frac{1}{n}}/2)-\mathbf{x}^{\prime},$

where $\mathbf{x}^{\prime}$ is the representative of
$(Q^{\frac{1}{n}}/2,\ldots,Q^{\frac{1}{n}}/2)+\mathbf{x}\mbox{ (mod $1$)}$ in
$(-1,0]^{n}$. Hence $-\mathcal{F}_{\mathbf{x}}$ contains a translate of (4.4),
proving the claim. In fact, they are identical provided
$\mathbf{x}^{\prime}\in(-1+Q^{\frac{1}{n}}/2-\lfloor
Q^{\frac{1}{n}}/2\rfloor,0)^{n}$ — this will be used to prove the “only if”
part below.

Thus suppose this time that (4.4) does not cover all of
$\mathbb{R}^{m}/\mathbb{Z}^{m}$; the uncovered part must have nonempty
interior. Moreover, by the remark just made earlier, there exists an open
subset $U\subseteq\mathbb{R}^{n}$ such that, for all $\mathbf{x}\in U$,
$\mathcal{F}_{\mathbf{x}}$ is a translate of (4.4) by a vector depending
continuously on $\mathbf{x}$. This implies that (4.3) leaves an open subset of
$\mathbb{R}^{d}/\mathbb{Z}^{d}$ uncovered. Thus for all sufficiently large
$N$, it leaves some elements of $(N^{-1}\mathbb{Z})^{d}/\mathbb{Z}^{d}$
uncontained. This completes the proof. ∎

It is clear that (4.4) covers $\mathbb{R}^{m}/\mathbb{Z}^{m}$ if and only if
$\varphi(Q)^{\frac{1}{m}}\geq\sup_{\mathbf{v}\in\mathbb{R}^{m}}\gamma_{(B;Q)}(\mathbf{v})$
for $p=\infty$ and $S\subseteq\mathbb{R}^{m}$ the entire unit sphere.
Therefore Theorem 1.2 and Proposition 4.1 together imply the Corollary.

## References

  * [1] J. Beck. Probabilistic Diophantine approximation: I. Kronecker sequences. Ann. Math. 140 (1994), 451-502.
  * [2] A. Das, A. Haynes. A three gap theorem for adeles. Ramanujan J. 60 (2023), 13-25.
  * [3] A. Haynes, J. Marklof. A five distance theorem for Kronecker sequences. Int. Math. Res. Not. 24 (2022), 19747-19789.
  * [4] S. Kim, M. Skenderi. Higher-rank pointwise discrepancy bounds and logarithm laws for generic lattices. Acta Arith. 205 (2022), 227-249.
  * [5] D. Kleinbock, G. Margulis. Logarithm laws for flows on homogeneous spaces. Inv. Math. 138 (1999), 451-494.
  * [6] A. Lenstra, H. Lenstra, L. Lovász. Factoring polynomials with rational coefficients. Math. Ann. 261 (1982), no. 4, 515-534.
  * [7] J. Marklof, A. Strömbergsson. The three gap theorem and the space of lattices. Amer. Math. Monthly 124 (2017), no. 8, 741-745.


The scaling of primordial gauge fields

Massimo Giovannini 111e-mail address<EMAIL_ADDRESS>
Department of Physics, CERN, 1211 Geneva 23, Switzerland
INFN, Section of Milan-Bicocca, 20126 Milan, Italy

Abstract

The large-scale magnetic fields arising from the quantum mechanical
fluctuations of the hypercharge are investigated when the evolution of the
gauge coupling is combined with a sufficiently long inflationary stage. In
this framework the travelling waves associated with the quantum mechanical
initial conditions turn asymptotically into standing waves which are the gauge
analog of the Sakharov oscillations. Even if the rate of dilution of the
hypermagnetic and hyperelectric fields seems to be superficially smaller than
expected from the covariant conservation of the energy-momentum tensor, the
standard evolution for wavelengths larger than the Hubble radius fully
accounts for this anomalous scaling which is anyway unable to increase the
amplitude of the magnetic power spectra after symmetry breaking. An effective
amplification of the gauge power spectra may instead occur when the post-
inflationary expansion rate is slower than radiation. We stress that the
modulations of the gauge power spectra freeze as soon as the relevant
wavelengths reenter the Hubble radius and not at the end of inflation. After
the Mpc scale crosses the comoving Hubble radius the scaling of the magnetic
power spectrum follows from the dominance of the conductivity. From these two
observations the late-time values of the magnetic power spectra are accurately
computed in the case of a nearly scale-invariant slope and contrasted with the
situation where the phases of Sakharov oscillations are not evaluated at
horizon crossing but at the end of inflation, i.e. when all the wavelengths
relevant for magnetogenesis are still larger than the comoving horizon.

The problem of the origin of large-scale magnetism has dates back to the
pioneering contributions of Hoyle, Zeldovich, Harrison [2, 3, 4] and many
others. As correctly emphasized by Hoyle [2] one of the most mysterious
aspects of the problem is related to the largeness of the correlation scale
associated with the magnetic fields at the onset of the gravitational collapse
of the protogalaxy. This comoving scale should be as large as
${\mathcal{O}}(\mathrm{Mpc})$ and this means that it crossed the comoving
horizon at a relatively late time, i.e. much later than electron-positron
annihilation and just before matter-radiation equality. In the past it has
been argued that if we combine the correlation scale of the protogalactic
field with the inflationary dynamics we might hope that the gauge fields are
amplified from the inflationary vacuum. The problem with this assumption is
that, besides the invariance under local gauge transformations, the Weyl [5]
and the duality [6, 7] symmetries determine the evolution of the gauge fields
in general relativity. When the governing equations of a given quantum field
are invariant under Weyl rescaling, the normal modes are not excited by the
evolution of the geometry [8, 9, 10, 11] and the corresponding particles are
not produced. Consequently in conformally flat background geometries
$\overline{G}_{\mu\nu}=a^{2}(\tau)\eta_{\mu\nu}$ (where $a(\tau)$ is the scale
factor, $\eta_{\mu\nu}$ is the Minkowski metric and $\tau$ is the conformal
time coordinate) the energy density of the hypercharge fields
$\rho_{Y}=u_{\mu}\,u_{\nu}T_{Y}^{\mu\,\nu}$ scales as $a^{-4}$ and it is
progressively diluted as the Universe expands.

If the Weyl invariance is broken because of the coupling either to the
inflaton or to some other spectator field [11, 12, 13, 14, 15, 16, 17, 18] the
gauge modes can be amplified so that the quantum mechanical initial data
(corresponding to travelling waves) turn asymptotically into standing waves
whose phases only depend on the evolution of the gauge coupling. These
modulations of the power spectra are the gauge analog of the so-called
Sakharov oscillations [19, 20, 21] and their correct evaluation determines the
amplitude of the power spectra. The purpose of this analysis is twofold. In
the first place we want to demonstrate that the late-time values of the
magnetic power spectrum (and of the corresponding Sakharov oscillations) are
not fixed at radiation dominance (and independently of the wavenumber) but
they are instead determined by the moment when the relevant wavelengths
(comparable with the Mpc) get of the order of the Hubble radius before
equality. The second point involves the post-inflationary evolution since any
deviation from the radiation dominance affects the late-time amplitude of the
magnetic power spectrum at the gravitational collapse of the protogalaxy.
These two themes are non-trivially combined and their discussion is also quite
confusing in the current literature. Even if these two problems can be
analyzed in even more general terms, from now on the attention will be focused
on the simplest possible action, namely

$S_{gauge}=-\frac{1}{4}\int
d^{4}x\,\,\frac{\sqrt{-G}}{g^{2}}\,\,Y_{\mu\nu}\,Y^{\mu\nu},\qquad\qquad
g=\sqrt{\frac{4\pi}{\lambda}},$ (1)

where $g$ denotes throughout the gauge coupling while $G$ is the determinant
of the four-dimensional metric with signature mostly minus. Equation (1) may
be complemented by a pseudo-scalar coupling with relevant implications [12,
13] that are however not central to the present discussion. We stress however
that the action (1) involves hypercharge fields so that the electromagnetic
degrees of freedom will have to be related to $Y_{\mu\nu}$ by the correct
projections involving the Weinberg angle. For the record the energy-momentum
tensor following from Eq. (1) is

$T_{Y}^{\mu\nu}=\frac{\lambda}{4\pi}\biggl{(}-Y^{\mu\alpha}\,Y^{\nu}_{\,\,\,\,\alpha}+\frac{G^{\mu\nu}}{4}\,\,Y^{\alpha\beta}\,\,Y_{\alpha\beta}\biggr{)},$
(2)

where $\lambda$ is the susceptibility which is related to the gauge coupling
as already mentioned in Eq. (1). In spite of the amplification mechanism Eq.
(2) suggests that energy density still scales $a^{-4}$ in terms of the
comoving fields $\vec{E}$ and $\vec{B}$:

$\rho_{Y}=u_{\mu}\,u_{\nu}T_{Y}^{\mu\,\nu}=\frac{1}{8\pi
a^{4}}\biggl{(}E^{2}+B^{2}\biggr{)}=\frac{\lambda}{8\pi}\biggl{(}b^{2}+e^{2}\biggr{)}.$
(3)

In Eq. (3) the comoving fields $\vec{E}=a^{2}\,\,\sqrt{\lambda}\vec{e}$ and
$\vec{B}=a^{2}\sqrt{\lambda}\,\,\vec{b}$ have been introduced in terms of the
the physical fields $\vec{e}$ and $\vec{b}$. We note that in the conformally
flat background considered here the physical fields $\vec{e}$ and $\vec{b}$
are directly related to the components of the gauge field strengths that are
defined as $Y^{i\,0}=e^{i}/a^{2}$ and
$Y^{i\,j}=\,-\,\epsilon^{i\,j\,k}b_{k}/a^{2}$.

If the background geometry is conformally flat (as it is the case during an
inflationary stage of expansion) the evolution of the comoving fields only
depends on the rate of variation of $g$ (or of $\sqrt{\lambda}$). In the
absence of sources the fields $\vec{E}$ and $\vec{B}$ are both solenoidal and
their dynamics follows from:

$\vec{E}^{\,\prime}+{\mathcal{F}}\,\vec{E}=\vec{\nabla}\times\vec{B},\qquad\qquad\vec{B}^{\,\prime}-{\mathcal{F}}\,\vec{B}=-\vec{\nabla}\times\vec{E},$
(4)

where the prime denotes a derivation with respect to the conformal time
coordinate $\tau$ and
${\mathcal{F}}=\sqrt{\lambda}^{\prime}/\sqrt{\lambda}=-g^{\prime}/g$ is the
rate of variation of $\sqrt{\lambda}$ (or of $g$). The duality symmetry [6, 7]
transforms the hyperelectric into the hypermagnetic fields and vice-
versa222The duality symmetry implies that for $\sqrt{\lambda}\to
1/\sqrt{\lambda}$ (i.e. ${\mathcal{F}}\to-{\mathcal{F}}$) the comoving fields
change as $\vec{E}\to\vec{B}$ and $\vec{B}\to-\vec{E}$ and the two expressions
of Eq. (4) are transformed one into the other. We stress that the duality
symmetry [6] is explicit in terms of the comoving normal modes of the system
[7] and this is why their use is recommended for the class of problems
discussed in this paper.. Since
${\mathcal{F}}=\sqrt{\lambda}^{\prime}/\sqrt{\lambda}\equiv-g^{\prime}/g$ we
have that ${\mathcal{F}}<0$ corresponds to the situation where the gauge
coupling increases (while the susceptibility decreases at the same rate );
conversely when ${\mathcal{F}}>0$ the gauge coupling decreases (and the
susceptibility increases at the same rate).

According to general arguments [22, 23] the inflationary stage must be long
enough to iron any preexisting inhomogeneity but also sufficiently short to
exclude a Planckian regime of initial curvatures. For the present purposes the
total number of $e$-folds will then be taken between $60$ and $80$ so that the
quantum mechanical initial conditions can be safely imposed for all the modes
of the gauge field spectrum. In this framework the comoving fields of Eq. (4)
are promoted to the status of field operators:

$\displaystyle\hat{E}_{i}(\vec{x},\tau)$ $\displaystyle=$
$\displaystyle-\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\alpha}\hat{e}^{(\alpha)}_{i}(\hat{k})\,\biggl{[}g_{k}(\tau)\,\hat{a}_{\vec{k},\,\alpha}e^{-i\vec{k}\cdot\vec{x}}+\mathrm{H}.\mathrm{c}.\biggr{]},$
(5) $\displaystyle\hat{B}_{k}(\vec{x},\tau)$ $\displaystyle=$
$\displaystyle-\epsilon_{i\,j\,k}\int\frac{d^{3}k}{(2\pi)^{3/2}}\sum_{\alpha}k_{i}\,\,\hat{e}^{(\alpha)}_{j}(\hat{k})\,\biggl{[}f_{k}(\tau)\,\hat{a}_{\vec{k},\,\alpha}e^{-i\vec{k}\cdot\vec{x}}-\mathrm{H}.\mathrm{c}.\biggr{]},$
(6)

where
$[\hat{a}_{\vec{k},\alpha},\hat{a}_{\vec{p},\,\beta}^{\dagger}]=\delta^{(3)}(\vec{k}-\vec{p})$
and $e^{(\alpha)}_{i}(\hat{k})$ (with $\alpha=\oplus,\,\,\otimes$) accounts
for the two polarizations which are orthogonal to the direction of propagation
$\hat{k}=\vec{k}/k$ (i.e. $\hat{e}^{\oplus}\times\hat{e}^{\otimes}=\hat{k}$).
The quantum mechanical normalization demands that the Wronskian of the mode
functions is time-independent and normalized as
$f_{k}(\tau)g_{k}^{\ast}(\tau)-f_{k}^{\ast}(\tau)g_{k}(\tau)=i$; in this way
the canonical commutation relations between field operators are preserved. All
in all, provided the inflationary phase is sufficiently long, the gauge fields
are initially in the vacuum and the expectation values of the field operators
in Fourier space can be computed from Eqs. (5)–(6):

$\displaystyle\langle\hat{E}_{i}(\vec{k},\tau)\,\hat{E}_{j}(\vec{p},\tau)\rangle=\frac{2\pi^{2}}{k^{3}}\,p_{i\,j}(\hat{k})\,\,P_{E}(k,\tau)\,\,\delta^{(3)}(\vec{k}+\vec{p}),$
(7)
$\displaystyle\langle\hat{B}_{i}(\vec{k},\tau)\,\hat{B}_{j}(\vec{p},\tau)\rangle=\frac{2\pi^{2}}{k^{3}}\,p_{i\,j}(\hat{k})\,\,P_{B}(k,\tau)\,\,\delta^{(3)}(\vec{k}+\vec{p}),$
(8)

where $p_{i\,j}(\hat{k})=(\delta_{i\,j}-\hat{k}_{i}\hat{k}_{j})$. We stress
that the hyperelectric and hypermagnetic power spectra (i.e. $P_{E}(k,\tau)$
and $P_{B}(k,\tau)$) appearing in Eqs. (5)–(8) have the dimensions of an
energy density and their explicit expressions are:

$P_{E}(k,\tau)=\frac{k^{3}}{2\pi^{2}}\bigl{|}g_{k}(\tau)\bigr{|}^{2},\qquad\qquad
P_{B}(k,\tau)=\frac{k^{5}}{2\pi^{2}}\bigl{|}f_{k}(\tau)\bigr{|}^{2}.$ (9)

The evolution of the mode functions entering Eq. (9) follows by inserting Eqs.
(5)–(6) into Eq. (4) and by imposing the appropriate Wronskian normalization.
The gauge power spectra only depend on the evolution of ${\mathcal{F}}$, i.e.
the rate of variation of the gauge coupling (or of the susceptibility). During
inflation $a\,H={\mathcal{H}}=-1/[\tau\,(1-\epsilon)]$ where
$\epsilon=-\dot{H}/H^{2}\ll 1$ is the standard slow-roll parameter333 For
quantitative estimates the validity of the consistency relations is also
assumed and this means that $r_{T}=16\epsilon$; since according to current
bounds [24, 25, 26] $r_{T}<0.06$ (or even $r_{T}<0.036$), the upper limit on
$\epsilon$ ranges between $3.7\times 10^{-3}$ and $2.2\times 10^{-3}$..

We could now consider a rather general evolution of the gauge coupling;
however there are two (dual) inflationary backgrounds where the gauge power
spectra acquire a particularly concise expression i.e. ${\mathcal{F}}=\pm
2{\mathcal{H}}$ [27]. Furthermore, in these two cases the rôle played by the
Sakharov oscillations is particularly transparent. If the gauge coupling
increases during an inflationary stage and then flattens out we are led to
consider the situation where, during inflation,
${\mathcal{F}}=-2{\mathcal{H}}$ while, after the end of inflation,
$|{\mathcal{F}}/{\mathcal{H}}|\ll 1$. It is relevant that the transition
across the inflationary boundary occurs in a way that $a$, $g$,
${\mathcal{H}}$ and ${\mathcal{F}}$ are all continuous; this can be achieved
by keeping ${\mathcal{F}}/{\mathcal{H}}$ arbitrarily small (but always
finite). If we impose the quantum mechanical initial data the mode functions
behave asymptotically for $\tau\to-\infty$ as travelling waves

$f_{k}=e^{-i\,k\tau}/\sqrt{2k},\qquad\qquad
g_{k}(\tau)=-i\,e^{-i\,k\tau}\,\sqrt{k/2},$ (10)

and using then the continuity of the mode functions across the inflationary
boundary the gauge power spectra after the end of inflation follow from Eq.
(9):

$P_{E}(k,\tau)=\frac{9H_{1}^{4}\,a_{1}^{4}}{4\pi^{2}}\cos^{2}{k\tau},\qquad\qquad
P_{B}(k,\tau)=\frac{9H_{1}^{4}\,a_{1}^{4}}{4\pi^{2}}\sin^{2}{k\tau}.$ (11)

Concerning Eq. (11) the following two comments are in order:

  * •

$H_{1}$ is fixed by the value of tensor to scalar ratio and by the amplitude
of the power spectrum of curvature inhomogeneities (i.e.
${\mathcal{A}}_{{\mathcal{R}}}={\mathcal{O}}(2.41)\times 10^{-9}$) according
to $(H_{1}/M_{P})=\sqrt{\pi\,{\mathcal{A}}_{{\mathcal{R}}}\,r_{T}}/4$;

  * •

even if the power spectra of Eq. (11) have been determined by positing that
the gauge coupling $g\propto 1/\sqrt{\lambda}$ increases during inflation and
then flattens out later on, by using the duality symmetry of the problem [6,
7] (see Eq. (4) and discussion thereafter) we can deduce the power spectra in
the dual situation where the gauge coupling first decreases as
${\mathcal{F}}=2{\mathcal{H}}$ during inflation and then flattens out; as
expected from duality [7] the spectra of Eq. (11) are therefore interchanged
as:

$\overline{P}_{E}(k,\tau)=\frac{9H_{1}^{4}\,a_{1}^{4}}{4\pi^{2}}\sin^{2}{k\tau},\qquad\qquad\overline{P}_{B}(k,\tau)=\frac{9H_{1}^{4}\,a_{1}^{4}}{4\pi^{2}}\cos^{2}{k\tau};$
(12)

  * •

in what follows we regard as more physical the case of increasing gauge
coupling given in Eq. (11) however the same considerations can also be applied
to Eq. (12).

The modulations appearing in Eqs. (11)–(12) are the gauge analog of Sakharov
oscillations [19, 20]; this analogy can be made even more concrete but and
this discussion can be found in [21]. It is interesting to stress, in this
respect, that the duality symmetry interchanges the phases of the Sakharov
oscillations since $P_{E}(k,\tau)\to\overline{P}_{B}(k,\tau)$ and
$P_{B}(k,\tau)\to\overline{P}_{E}(k,\tau)$. The Sakharov phases are generic
since they only depend on the rate of variation of the gauge coupling during
inflation and by the requirement that, after the end of inflation, the gauge
coupling flattens out (i.e. $|{\mathcal{F}}/{\mathcal{H}}|\ll 1$). If the
gauge coupling does not freeze we could have the situation where
$|{\mathcal{F}}/{\mathcal{H}}|{\mathcal{O}}(1)$ after the end of inflation; in
this case the Sakharov phases are always present and their analytical
expressions are given in terms of the appropriate Bessel functions [21] whose
trigonometric limit of Eqs. (11)–(12) is realized exactly when
$|{\mathcal{F}}/{\mathcal{H}}|\ll 1$.

We are now going to ask the following question: given the expression of the
comoving power spectra of Eq. (11) what is the value of the physical power
spectra at late time? The answer follows from the relation between ($\vec{e}$,
$\vec{b}$) and ($\vec{E}$, $\vec{B}$) introduced in Eq. (3). In particular,
from the general form of $P_{E}(k,\tau)$ and $P_{B}(k,\tau)$ (see Eq. (9)) the
relation between the physical and the comoving spectra is therefore given by:

${\mathcal{P}}_{X}(k,\tau)=\frac{P_{X}(k,\tau)}{\lambda(\tau)\,a^{4}(\tau)}=\frac{g^{2}(\tau)}{4\,\pi\,a^{4}(\tau)}\,\,P_{X}(k,\tau),\qquad\qquad
X=E,\,B,$ (13)

where the explicit connection between the gauge coupling and the
susceptibility (see Eq. (1) and discussion thereafter) has been used.
Consequently, if Eq. (11) is inserted into Eq. (13) the corresponding physical
spectra become:

${\mathcal{P}}_{E}(k,\tau)=\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}\,\,\cos^{2}{k\tau},\qquad\qquad{\mathcal{P}}_{B}(k,\tau)=\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}\,\,\sin^{2}{k\tau}.$
(14)

In Eq. (14) $g_{1}$ is the asymptotic value of the gauge coupling after the
end of inflation since for $\tau\gg-\tau_{1}$ the evolution of $g(\tau)$ has
been parametrized as $g(\tau)\simeq g_{1}(\tau/\tau_{1})^{\delta}$ (with
$\delta\ll 1$). Moreover for $\tau<-\tau_{1}$ we have that
$aH=-1/[(1-\epsilon)\tau]$ while for $\tau\gg-\tau_{1}$ we may consider that
the background is dominated by radiation444This choice is not mandatory and is
relaxed later on. However, already at this stage it is useful to stress that
the typical phases of the standing oscillations do not depend on the post-
inflationary evolution of the scale factor but are determined by the
asymptotic behaviour of the gauge coupling.. Equation (14) only applies for
typical time-scales shorter than the crossing time
$\tau_{k}={\mathcal{O}}(1/k)$ (i.e. in the range $-\tau_{1}<\tau\leq\tau_{k}$)
so that when $\tau<\tau_{k}$ the standing oscillations can be expanded in
their small argument limit:

$\displaystyle{\mathcal{P}}_{E}(k,\tau)$ $\displaystyle=$
$\displaystyle\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}\,\,\biggl{[}1+{\mathcal{O}}\biggl{(}\frac{k^{2}}{{\mathcal{H}}^{2}}\biggr{)}\biggr{]},\qquad\qquad\tau<\tau_{k},$
(15) $\displaystyle{\mathcal{P}}_{B}(k,\tau)$ $\displaystyle=$
$\displaystyle\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}\,\,\biggl{(}\frac{k}{{\mathcal{H}}}\biggr{)}^{2}\biggl{[}1+{\mathcal{O}}\biggl{(}\frac{k^{2}}{{\mathcal{H}}^{2}}\biggr{)}\biggr{]},\qquad\qquad\tau<\tau_{k}.$
(16)

We can now rephrase the leading order result of Eq. (16) by recalling that
${\mathcal{H}}=a\,H$. The approximate scaling of the magnetic power spectra is
then given by the following pair of (fully equivalent) expressions

$\displaystyle{\mathcal{P}}_{B}(k,\tau)$ $\displaystyle=$
$\displaystyle\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{k}{a_{1}\,H_{1}}\biggr{)}^{2}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{6}\biggl{(}\frac{H_{1}}{H}\biggr{)}^{2},\qquad\qquad\tau<\tau_{k},$
(17) $\displaystyle=$
$\displaystyle\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}|k\,\tau|^{2},\qquad\qquad
k\tau<1.$

Even if the two interchangeable results of Eq. (17) coincide, by looking at
the first line we could naively infer that between $\tau_{1}$ and the present
time (be it $\tau_{0}$) the physical power spectrum does not scale as as
$a^{-4}$ but rather as $a^{-6}H^{-2}$. This would mean, for instance, that
during radiation (where $H\propto a^{-2}$) the power spectrum, as a whole,
scales as ${\mathcal{P}}_{B}(k,\tau)\propto a^{-2}$. This conclusion is
however misleading in the light of the second form of Eq. (17) suggesting
instead that there is no gain in the amplitude and to clarify this point even
further we now specifically focus on the first expression of Eq. (17) and note
that the potential increase of the physical power spectrum is controlled by
the last pair of terms:

$\biggl{(}\frac{k}{a_{1}\,H_{1}}\biggr{)}^{2}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{6}\biggl{(}\frac{H_{1}}{H}\biggr{)}^{2}=|k\tau_{1}|^{2}\biggl{(}\frac{a_{1}}{a}\biggr{)}^{6}\biggl{(}\frac{H_{1}}{H}\biggr{)}^{2}.$
(18)

Even if, for some reason, $(a_{1}/a)^{6}(H_{1}/H)^{2}$ gets much larger than
$a^{-4}$, $|k\tau_{1}|$ is always so minute to jeopardize any potential growth
of the first term. In particular we have that for the fiducial values of the
parameters (determined in the framework of the concordance paradigm)
$|k\tau_{1}|^{2}={\mathcal{O}}(10^{-46})$ (see also, in this respect, Eq. (27)
and discussion thereafter). In other words, in spite of any gain possibly
coming from $(a_{1}/a)^{6}(H_{1}/H)^{2}$ the potential advantages of the
modified redshift are compensated by the smallness of $|k\tau_{1}|^{2}$.
Moreover, since by definition $k\tau<1$, the physical power spectrum of Eq.
(17) for $\tau<\tau_{k}$ is always suppressed in comparison to its value at
$\tau_{k}$.:

${\mathcal{P}}_{B}(k,\tau)\ll\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\biggl{(}\frac{a_{1}}{a_{k}}\biggr{)}^{4},\qquad\qquad\tau\leq\tau_{k}.$
(19)

At most the whole expression in Eq. (18) is ${\mathcal{O}}(1)$ and this
happens when $\tau=\tau_{k}$. All in all the situation can be summarized as
follows:

  * •

the hypermagnetic power spectrum, for any given time-scale $\tau<\tau_{k}$, is
always smaller than its value for $\tau\to\tau_{k}$, i.e. at horizon crossing;

  * •

taking into account this effect the authors of Ref. [28] suggest a correction
of previous estimates by $37$ orders of magnitude that would solve most of the
problems of magnetogenesis since the rate of dilution of the magnetic fields
would be much smaller than previously considered; this argument has been
further amplified later on [29, 30];

  * •

in the light of Eqs. (19)–(20) it is however difficult to see how the
anomalous scaling can affect the physical value of the magnetic field for
$\tau>\tau_{k}$ where the potential dilution is not effective [15, 16, 17, 18]
(see also [31, 32, 33, 34]);

  * •

finally, neither Eq. (16) nor Eq. (17) apply for $\tau>\tau_{k}$: indeed as
long as $k\tau>1$, the electric fields are strongly damped in comparison with
the magnetic ones which are also suppressed for typical wavenumbers shorter
than the so-called magnetic diffusivity scale [35, 36, 37].

The discussion of the previous paragraph boils down to the following pair of
complementary questions: where should we evaluate the phases of the (magnetic)
Sakharov oscillation appearing in Eq. (14)? At $\tau_{k}$ (i.e. when the Mpc
scale crosses the Hubble radius) or at $\tau_{1}$ (i.e. at the end of
inflation)? In what follows we shall consider separately the two cases and
then conclude that the purported dilution effect only applies beyond the
Hubble radius and has no appreciable effect on the final value of the magnetic
field. To achieve this program it is useful to remark that the freezing time
of the Sakharov phases cannot be selected at wish: the result of Eq. (14)
applies for $\tau\leq\tau_{k}$ since, after $\tau_{k}$, the conductivity
breaks the duality symmetry [7] and the relevant evolution equations are not
given anymore by Eq. (4). This is why the results of Eqs. (14)–(16) evaluated
for $\tau={\mathcal{O}}(\tau_{k})$

${\mathcal{P}}^{(em)}_{B}(k,\tau_{k})=\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\cos^{2}{\theta_{W}}\,\,\biggl{(}\frac{a_{1}}{a_{k}}\biggr{)}^{4}\,\,\sin^{2}{k\tau_{k}},$
(20)

are subsequently scaled down to the epoch of the gravitational collapse of the
protogalaxy. For the sake of accuracy, in Eq. (20) we added the cosine squared
of the Weinberg angle (coming from the projection of the hypercharge field on
the electromagnetic field after symmetry breaking) so that
${\mathcal{P}}^{(em)}_{B}(k,\tau_{k})$ really estimates the magnetic power
spectrum at $\tau_{k}$. The specific value of $\tau_{k}$ depends on the bunch
of wavelengths relevant for magnetogenesis (i.e.
$k={\mathcal{O}}(\mathrm{Mpc}^{-1})$); this means that we can estimate the
crossing time as $\tau_{k}={\mathcal{O}}(10^{-2})\,\,\tau_{eq}$ or, more
precisely555As usual $\Omega_{M0}$ is the present matter fraction in critical
units while $\Omega_{R0}$ is the radiation fraction; $h_{0}$, as usual, is the
Hubble rate in units $\mathrm{km}\,\,\mathrm{Hz}/\mathrm{Mpc}$.

$\frac{\tau_{k}}{\tau_{eq}}=1.01\times
10^{-2}\biggl{(}\frac{k}{\mathrm{Mpc}^{-1}}\biggr{)}^{-1}\biggl{(}\frac{h_{0}^{2}\Omega_{M0}}{0.1386}\biggr{)}\biggl{(}\frac{h_{0}^{2}\Omega_{R0}}{4.15\times
10^{-5}}\biggr{)}^{-1/2}.$ (21)

where $\tau_{eq}$ is the equality time. When the comoving wavelength
${\mathcal{O}}(\mathrm{Mpc})$ is inside the Hubble radius for $\tau>\tau_{k}$
the duality symmetry relating Eqs. (13)–(14) is broken by the presence of the
conductivity that suppresses both the electric and the magnetic fields [6, 7]
even if the electric fields are comparatively far more suppressed. If
$\vec{{\mathcal{B}}}$ denotes the (ordinary) magnetic field in the conducting
plasma prior to equality, the Ohmic electric field is
$\vec{{\mathcal{E}}}=\vec{\nabla}\times\vec{{\mathcal{B}}}/\sigma_{c}$ where
$\sigma_{c}$ is the (comoving) conductivity [35, 36, 37]. This remark implies
that the electric power spectrum is suppressed by a factor
$(k/\sigma_{c})^{2}$ in comparison with its magnetic counterpart for
$\tau\geq\tau_{k}$ and, in the same regime, the magnetic field is practically
not suppressed for typical wavenumbers smaller than the magnetic diffusivity
scale $k_{\sigma}=\sqrt{\sigma_{c}\,{\mathcal{H}}}$. All in all, at the
present time and for large wavelengths (i.e. $k<k_{\sigma}$), the magnetic
power spectrum is given by:

${\mathcal{P}}^{(em)}_{B}(k,\tau)={\mathcal{P}}^{(em)}_{B}(k,\tau_{k})\,\,\biggl{(}\frac{a_{k}}{a}\biggr{)}^{4},\qquad\qquad\tau>\tau_{k}.$
(22)

If we now insert Eq. (19) into Eq. (21) and evaluate the spectrum at the
present time we obtain from Eq. (22):

${\mathcal{P}}^{(em)}_{B}(k,\tau_{0})=\frac{9\,H_{1}^{4}\,g_{1}^{2}}{16\pi^{3}}\,\,\cos^{2}{\theta_{W}}\,\,\biggl{(}\frac{a_{1}}{a_{0}}\biggr{)}^{4}\sin^{2}{k\,\tau_{k}},$
(23)

where, by definition, $k\tau_{k}={\mathcal{O}}(1)$. The estimate of the term
$(a_{1}/a_{0})$ appearing in Eq. (23) depends on the post-inflationary
expansion history and, in the simplest case, we have
$(a_{1}/a_{0})=(2\,\Omega_{R0})^{1/4}\sqrt{H_{0}/H_{1}}$ where $\Omega_{R0}$
is the present fraction of relativistic species and $H_{0}$ is the Hubble
rate. We may then recall that the overall normalization of the temperature and
polarization anisotropies fixes the value of $H_{1}$ in terms of the tensor to
scalar ratio $r_{T}$ [24, 25, 26]. Equation (23) then takes the general form

${\mathcal{P}}^{(em)}_{B}(k,\tau_{0})=\frac{9\,\,g_{1}^{2}\,\,\cos^{2}{\theta_{W}}}{128\,\pi^{2}}\,\,\Omega_{R0}\,\,H_{0}^{2}\,M_{P}^{2}\,{\mathcal{A}}_{{\mathcal{R}}}\,\,r_{T}\,\,\sin^{2}{k\tau_{k}}.$
(24)

If we now recall the fiducial values of the quantities appearing in Eq. (24)
we can deduce the following estimate of the magnetic power spectrum:

$\frac{{\mathcal{P}}^{(em)}_{B}(k,\tau_{0})}{\mathrm{nG}^{2}}\,=3.39\times
10^{-8}\,\,\biggl{(}\frac{r_{T}}{0.06}\biggr{)}\,\,\biggl{(}\frac{g_{1}}{0.1}\biggr{)}^{2}\,\,\biggl{(}\frac{{\mathcal{A}}_{{\mathcal{R}}}}{2.41\times
10^{-9}}\biggr{)}\,\,\biggl{(}\frac{h_{0}^{2}\,\Omega_{R\,0}}{4.15\times
10^{-5}}\biggr{)},$ (25)

where $1\,\,\mathrm{nG}=10^{-9}\mathrm{G}$; we also remind that , in natural
units, the relation between the Planck mass and the Gauss is given by
$M_{P}^{2}=10^{57.33}\,\,\mathrm{G}$. Finally in Eq. (25 we took into account
that the phase of the standing oscillation must be evaluated for
$\tau\to\tau_{k}$ so that $\sin{k\,\tau_{k}}={\mathcal{O}}(0.84)$ since
$k\,\tau_{k}={\mathcal{O}}(1)$.

Equation (25) partially answers the question formulated prior to Eq.(20) and
estimates the late-time value of the magnetic power spectrum. For the record
let us now see what happens if Eq. (17) is not evaluated for
$\tau={\mathcal{O}}(\tau_{k})$ (i.e. when the relevant scale crosses the
comoving Hubble radius) but as soon as radiation dominates i.e. for
$\tau={\mathcal{O}}(\tau_{1})$; obviously the power spectrum will not be
larger but always much smaller than the one of Eq. (25) not because of the
finite value of the conductivity (that only operates for $\tau>\tau_{k}$) but
since $k\tau_{1}\ll 1$ when the relevant wavelength is larger than the Hubble
radius. Let us then evaluate Eqs. (20)–(22) for $\tau\to\tau_{1}$; the
magnetic power spectrum, in this case, is given by:

${\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})=\frac{9\,\,g_{1}^{2}\,\,\cos^{2}{\theta_{W}}}{128\,\pi^{2}}\,\,\Omega_{R0}\,\,H_{0}^{2}\,M_{P}^{2}\,{\mathcal{A}}_{{\mathcal{R}}}\,\,r_{T}\,\,\sin^{2}{k\tau_{1}},$
(26)

where ${\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})$ now denotes the magnetic power
spectrum obtained by evaluating the phases of oscillations not for
$\tau={\mathcal{O}}(\tau_{k})$ but rather for $\tau={\mathcal{O}}(\tau_{1})$.
We actually use this different notation just for comparison since we regard as
arbitrary the chain of arguments leading to
${\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})$; for the record Eq. (26) corresponds to
the strategy of Ref. [28]. The result of Eq. (26) is approximately,
$|k\,\tau_{1}|^{2}={\mathcal{O}}(10^{-46})$ smaller than the one of Eq. (25)
since, as already pointed out, $k\tau_{1}={\mathcal{O}}(10^{-23})$; more
specifically for the fiducial set of values of the concordance scenario we
have [24, 25, 26]

$k\tau_{1}=5.75\times
10^{-24}\biggl{(}\frac{k}{\mathrm{Mpc}^{-1}}\biggr{)}\biggl{(}\frac{r_{T}}{0.06}\biggr{)}^{-1/4}\,\biggl{(}\frac{{\mathcal{A}}_{{\mathcal{R}}}}{2.41\times
10^{-9}}\biggr{)}^{-1/4}\biggl{(}\frac{h_{0}^{2}\,\Omega_{R\,0}}{4.15\times
10^{-5}}\biggr{)}^{-1/4}.$ (27)

Inserting now Eq. (27) into Eq. (26) the explicit estimate of the magnetic
power spectrum becomes

$\frac{{\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})}{\mathrm{nG}^{2}}=1.12\times
10^{-54}\biggl{(}\frac{r_{T}}{0.06}\biggr{)}^{1/2}\,\,\biggl{(}\frac{k}{\mathrm{Mpc}^{-1}}\biggr{)}^{2}\,\,\biggl{(}\frac{g_{1}}{0.1}\biggr{)}^{2}\,\,\biggl{(}\frac{{\mathcal{A}}_{{\mathcal{R}}}}{2.41\times
10^{-9}}\biggr{)}^{1/2}\,\,\biggl{(}\frac{h_{0}^{2}\,\Omega_{R\,0}}{4.15\times
10^{-5}}\biggr{)}^{1/2}.$ (28)

Equation (28) is analog to the one of Ref. [28]; in that case the authors get
$10^{-67}\mathrm{G}^{2}$ while we rather get (using the same argument based on
Eq. (26))
${\mathcal{Q}}^{em}_{B}(k,\tau_{0})={\mathcal{O}}(10^{-72})\,\,\mathrm{G}^{2}$.
This quantitative disagreement comes from $\cos{\theta_{W}}^{2}$, from the
value of $g_{1}$ and from a number of specific numerical factors (neglected in
Ref. [28]) that further suppress the final magnetic field by roughly $5$
orders of magnitude.

Since the estimates of Eqs. (24)–(25) depend on the post-inflationary
expansion history we now consider how the scaling of the gauge fields is
affected by the presence of an intermediate stage of expansion not necessarily
associated with the radiation epoch. It has been actually suggested long ago
that, under some very specific conditions, the presence of a post-inflationary
stage may increase the magnetic power spectrum [18]. Even though a number of
different possibilities can be considered [21], for the sake of conciseness we
now illustrate the case of a single intermediate phase ranging between the end
of inflation and the onset of radiation dominance. During the post-
inflationary evolution the expansion rate is controlled by a barotropic index
$w$ and, for the fiducial values of Eq. (25), the final result gets modified
as:

${\mathcal{P}}^{(em)}_{B}(k,\tau_{0})\,=3.39\times
10^{-8}\,\,\biggl{(}\frac{H_{r}}{H_{1}}\biggr{)}^{\alpha(w)}\mathrm{nG}^{2},\qquad\qquad\alpha(w)=\frac{2(1-3w)}{3(1+w)},$
(29)

where $H_{r}$ denotes the Hubble rate at the radiation dominance and, as
usual, $H_{1}$ marks the end of the inflationary stage; by definition
$H_{r}/H_{1}<1$ since the curvature scale at the beginning of inflation must
always exceed the curvature scale at radiation dominance. The scale $H_{r}$
cannot be arbitrarily reduced since the intermediate stage(s) must not
jeopardize the abundances of the light elements and, for this reason, we can
generally require that $H_{r}>10^{-44}\,M_{P}$ [18]. According to Eq. (29)
there are three complementary physical possibilities characterized by the
different ranges of $w$. When $w>1/3$, the power spectrum gets larger than in
the case where $w\to 1/3$ which is the one already discussed in Eq. (25): in
this situation there is in fact a gain in the amplitude [18]. If $w<1/3$ the
magnetic power spectrum is instead further suppressed in comparison with the
case $w\to 1/3$. Finally when $w=1/3$ the result of Eq. (25) recovered and
this is what must happen if the whole approach is consistent. The result of
Eq. (29) tells that the magnetic power spectrum is less diluted when the
dominant component of the background redshifts slower than radiation while it
is more suppressed in the opposite case. As before the result of Eq. (29) can
be compared with the expression derived from Eq. (28). We can then posit, by
fiat, that in Eq. (26) the oscillating contribution should be evaluated in
$\tau_{r}$ (and not in $\tau_{k}$). For the typical values of the parameters
previously introduced we then have $k\tau_{r}=10^{-23.24}(H_{1}/H_{r})$ and
this result implies that ${\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})$ is

${\mathcal{Q}}^{(em)}_{B}(k,\tau_{0})=1.12\times
10^{-54}\biggl{(}\frac{H_{r}}{H_{1}}\biggr{)}^{\beta(w)}\,\,\,\mathrm{nG}^{2},\qquad\qquad\beta(w)=-\frac{1+9w}{3(1+w)}.$
(30)

Equation (30) stipulates that the final value of the magnetic power spectrum
gets enhanced in spite of the value of the barotropic index $w$; this is
rather bizarre especially because the enhancement even occurs when $w\to 1/3$
while we should expect that, in this situation, the result of a pure radiation
phase is fully recovered. The ultimate reason of the result of Eq. (30) (which
is incidentally the one suggested by Ref. [28]) is that the gauge power
spectra have not been evaluated at the crossing time $\tau_{k}$ but much
earlier without taking into account that, for $\tau\geq\tau_{k}$ the correct
form of the physical power spectrum follows from Eq. (22).

For completeness in Fig. 1 we illustrated the results of Eqs. (29)–(30) for a
fiducial choice of the parameters. The common logarithm of the square roots of
the magnetic power spectra (in nG units) is reported on the different contours
of both plots.

Figure 1: The common logarithms of the square roots of the magnetic power
spectra are illustrated in the plane $(w,H_{r}/H_{1})$ for $r_{T}=0.06$ and
for $k=1\,\mathrm{Mpc}^{-1}$; note furthermore that $g_{1}=0.001$. In the plot
at the right we illustrate the result of Eq. (29) while at the left we
considered the estimate Eq. (30).

In the shaded region (which is relevant for magnetogenesis considerations) the
power spectrum (29) is systematically larger than Eq. (30). Furthermore in the
right plot the power spectrum is suppressed for $w<1/3$ while it is amplified
when $w>1/3$. In spite of the previous comments (see Eq. (30) and discussion
thereafter) it turns out that the results of the left plot of Fig. 1 are only
marginally relevant for the magnetogenesis considerations. We actually know
that the galaxy rotated roughly $30$ times since its origin and if we assume
that the dynamo mechanism amplified the magnetic field of roughly one $e$-fold
for each galactic rotation we would get that at the time of the gravitational
collapse of the protogalaxy the initial magnetic field had to be, roughly,
$10^{-19}$ G [37]. This number could be further reduced by taking into account
the compressional amplification that would bring down the value by $4$ or $5$
orders of magnitude. Even assuming the most optimistic coincidences we should
have that, at least, ${\mathcal{P}}_{B}^{(em)}\geq 10^{-32}\,\mathrm{nG}$. A
more realistic requirement would imply ${\mathcal{P}}_{B}^{(em)}\geq
10^{-22}\,\mathrm{nG}$. Both bounds are safely satisfied by the results of the
right plot of Fig. 1 while they are barely possible in the plot at the left.

The results discussed so far have been obtained in the case when
${\mathcal{F}}=\pm{\mathcal{H}}$. Since this analysis can be generalized to
the case ${\mathcal{F}}=\pm\gamma\,{\mathcal{H}}$ we now briefly examine, for
illustration, the case of increasing gauge coupling (i.e.
${\mathcal{F}}=-\gamma{\mathcal{H}}$); in this case the spectral energy
density during inflation is given by

$\Omega_{Y}(k,\tau)=\frac{1}{\rho_{crit}}\,\frac{d\langle\rho_{Y}\rangle}{d\ln{k}}=\frac{2}{3H^{2}M_{P}^{2}a^{4}}\biggl{[}P_{E}(k,\tau)+P_{B}(k,\tau)\biggr{]},$
(31)

where $\rho_{crit}$ denotes the spectral energy density and $H$ is the
inflationary Hubble rate. Equation (31) can be written, in explicit terms, as:

$\Omega_{Y}(k,\tau)=\frac{2}{3}\biggl{(}\frac{H}{M_{P}}\biggr{)}^{2}\biggl{[}{\mathcal{C}}(|\gamma-1/2|)|k\tau|^{5-|2\gamma-1|}+{\mathcal{C}}(\gamma+1/2)|k\tau|^{4-2\gamma}\biggr{]},$
(32)

where ${\mathcal{C}}(x)=2^{2x-3}\,\Gamma^{2}(x)/\pi^{3}$. From Eq. (32) it
follows that $\gamma\leq 2$ to avoid that the spectral energy density
eventually becomes overcritical. After the end of inflation the spectral
energy density can be expressed as

$\Omega_{Y}(k,\tau)=\frac{2\,H_{1}^{4}}{3\,H^{2}\,M_{P}^{2}}\,\,\biggl{(}\frac{a_{1}}{a}\biggr{)}^{4}{\mathcal{C}}(\gamma+1/2)\,\,\biggl{(}\frac{k}{a_{1}\,H_{1}}\biggr{)}^{4-2\gamma}.$
(33)

If we now evaluate Eq. (33) for $\tau\to\tau_{r}$ and require that
$\Omega_{Y}(k,\tau_{r})<10^{-6}$ we get a bound in the plane ($H_{r}/H_{1}$,
$w$) which is incidentally satisfied in the dashed area of Fig 1.

We can therefore summarize the main findings of this analysis as follows. The
gauge fields do not experience any unexpected growth in a conformally flat
cosmological background. When the Fourier mode gets of the order of the
(comoving) Hubble radius the peculiar phases of the gauge power spectra are of
order $1$ and do not entail an increase of the final magnetic field. The
present results show that the Sakharov phases must be evaluated when the
wavelengths relevant for magnetogenesis cross the Hubble radius and then
corrected for the effect of the conductivity when the corresponding scales are
inside the horizon. For the typical parameters of the concordance scenario the
crossing happens slightly before matter-radiation equality it would be
arbitrary to evaluate the gauge power spectra well before horizon crossing or
simply at the end of inflation. In spite of recent claims the post-
inflationary stages different from the radiation phase do not always improve
the magnetogenesis constraints. Conversely, as suggested long ago, only the
presence of a sufficiently long stiff epoch improves the situation since, in
this case, the expansion rate is slower than radiation. If the expansion rate
is instead faster than radiation the final values of the magnetic power
spectra are comparatively smaller than in the case of a radiation-dominated
evolution. The previous conclusions are also consistent with the scaling of
the gauge fields dictated by the covariant conservation of the corresponding
energy momentum tensor since the hypermagnetic power spectra remain
approximately constant in comparison with the background energy density when
radiation is the dominant component of the post-inflationary plasma.

I wish to thank T. Basaglia, A. Gentil-Beccot, S. Rohr and the whole CERN
Scientific Information Service for their help during the preparation of this
manuscript.

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