or $b=b_{0}.$ In this case, the claim is trivial since the right hand side of
(25) is more than $1$.

Now consider the case when $a<a_{0}$ and $b<b_{0}.$ Let us analyze the
behaviour of the algorithm $\mathcal{A}^{\prime}$ on an $(a,b)$-good input
$(S,T,(\rho^{\prime}_{j})_{j\not\in S}).$ Assume that the current iteration is
of Type I (the other case is similar). Claim 20 tells us that the algorithm
does not output FAILURE in this iteration. By Claim 21, the probability that
the algorithm produces an input (for the next iteration) that is _not_
$(a+1,b)$-good is at most $\exp(-\Omega(k))$; in this case, we give up and
assume that the algorithm subsequently will output FAILURE. Finally, by Claim
21, the probability that the algorithm does not output SUCCESS in this round
is at most $(1-\eta_{1})$. In particular, this implies that the probability
that the algorithm does not output SUCCESS _and_ produces an $(a+1,b)$-good
input for the next iteration is also at most $(1-\eta_{1})$. However,
conditioned on this event, we can use the inductive hypothesis to bound the
probability of FAILURE. By the union bound and the induction hypothesis, we
get

$\displaystyle\mathop{\mathrm{Pr}}[\mathcal{A}^{\prime}(S,T,(\rho^{\prime}_{j})_{j\not\in
S})\text{ outputs FAILURE}]$
$\displaystyle\leq\exp(-\Omega(k))+(1-\eta_{1})\cdot(((a_{0}-(a+1))+(b_{0}-b))\cdot\exp(-\Omega(k))+(1-\eta_{1})^{a_{0}-(a+1)}+(1-\eta_{2})^{b_{0}-b})$
$\displaystyle\leq\exp(-\Omega(k))+((a_{0}-(a+1))+(b_{0}-b))\cdot\exp(-\Omega(k))+(1-\eta_{1})\cdot(1-\eta_{1})^{a_{0}-(a+1)}+(1-\eta_{2})^{b_{0}-b}$
$\displaystyle\leq((a_{0}-a)+(b_{0}-b))\cdot\exp(-\Omega(k))+(1-\eta_{1})^{a_{0}-a}+(1-\eta_{2})^{b_{0}-b}$

which completes the induction.

In particular, since the initial input $(S_{0},T_{0})$ is $(0,0)$-good, we see
that

$\displaystyle\mathop{\mathrm{Pr}}[\mathcal{A}^{\prime}\text{ outputs FAILURE
on }(S_{0},T_{0})]$
$\displaystyle\leq(a_{0}+b_{0})\cdot\exp(-\Omega(k))+(1-\eta_{1})^{a_{0}}+(1-\eta_{2})^{b_{0}}$
$\displaystyle\leq(a_{0}+b_{0})\cdot\exp(-\Omega(k))+\exp(-\Omega(a_{0}))+\exp(-\Omega(b_{0}\varepsilon/k))$
$\displaystyle\leq
M\cdot\exp(-\Omega(k))+\exp(-\Omega(k))+\exp(-\Omega(M^{0.45}\varepsilon/k))$
$\displaystyle\leq\exp(-\Omega(k))$

where the second inequality follows from the definition of $\eta_{1}$ and
$\eta_{2}$, the third from the definition of $a_{0}$ and $b_{0}$, and the last
follows from the fact that $M\leq
m^{\Delta^{\prime}}\leq\exp(k^{0.02+o(1)}),$191919Note that there are at most
$m^{\Delta^{\prime}}$ many $X^{(\Delta)}$-segments in $G^{(\Delta^{\prime})}$.
$\varepsilon\geq 1/M^{0.25},$ and $M=\Omega(m)=\Omega(k^{20}).$ This implies
(13) and hence completes the proof of Case 1.

####  5.2.2 Case 2

By renaming our segments if necessary, we assume that $X_{1},\dots,X_{M/2}$
are the segments that are not $(1/4)$-shattered. That is, for each of these
segments $X_{j}$ ($j\in[M/2]$), there exists an $F_{i_{j}}^{\prime}$
($i_{j}\in[t]$) such that $\left|\mathrm{Vars}(F_{i_{j}}^{\prime})\cap
X_{j}\right|\geq(1/4)\left|X_{j}\right|$. By an averaging argument, we can say
that there exists $b_{j}\in\\{1,2\\}$ such that
$\left|\mathrm{Vars}(F_{i_{j}}^{\prime})\cap
X_{j,b_{j}}\right|\geq(1/4)\left|X_{j,b_{j}}\right|$.

For each $j\leq M/2$, the restriction $\rho_{j}$ is sampled according to the
algorithm $\mathcal{A}$ described in Section 3.2. We first condition on the
choice (among the options $E_{1},E_{2},E_{3}$) made by the algorithm
$\mathcal{A}$ in sampling each $\rho_{j}.$ We say that a $\rho_{j}$ is _good_
if the algorithm $\mathcal{A}$ decides on option
$E_{b_{j}}\in\\{E_{1},E_{2}\\}$ in sampling $\rho_{j}.$ For each $j$,
$\rho_{j}$ is good with probability $1/3$. Let $\mathcal{E}_{2}$ denote the
event that no $\rho_{j}$ is good for $j\in[M/2]$ and let
$\bar{\mathcal{E}_{2}}$ be the complement of $\mathcal{E}_{2}$. Clearly, we
have

$\mathop{\mathrm{Pr}}_{\rho}[\mathcal{E}_{2}]\leq(2/3)^{M/2}\leq\exp(-\Omega(M)).$
(26)

Assume that $\mathcal{E}_{2}$ does not occur. Then, there is a $j\in[M/2]$
such that $\rho_{j}$ is good. By renaming segments once more, we can assume
that $j=1$. Further, by renaming the $F_{i}^{\prime}$s, we assume that
$i_{1}=1$; in particular, we have $\left|\mathrm{Vars}(F_{1}^{\prime})\cap
X_{1}\right|\geq\left|X_{1}\right|/4$. Finally, we can similarly assume that
$b_{1}=1$, which implies that $\left|\mathrm{Vars}(F_{1}^{\prime})\cap
X_{1,1}\right|\geq\left|X_{1,1}\right|/4.$

We condition on any choice of the restrictions $\rho_{j}$ for $j>1$; this
fixes the sets $\tilde{Y}^{\prime}_{j}$ and $\tilde{Z}^{\prime}_{j}$ defined
above for $j>1$. Conditioned on all these events, we see that the random
restriction $\rho_{1}$ sets all the variables in $X_{1,2}$ to $0$
deterministically, and hence can now be considered an
$(X_{1,1},Y_{1,1},Z_{1,1})$-restriction, which we will call
$\rho_{1}^{\prime}$ for clarity. Note that $X_{1,1},Y_{1,1}$ and $Z_{1,1}$ are
$X^{(\Delta-1)}$-, $Y^{(\Delta-1)}$\- and $Z^{(\Delta-1)}$-clones
respectively.

Recall that the $X^{(\Delta-1)}$-clone $X_{1,1}$ is made up of $m$
$X^{(\Delta-1)}$-segments $X_{1,1,1},\ldots,X_{1,1,m}$. We similarly have
$Y_{1,1}=\bigcup_{j\in[m]}Y_{1,1,j}$ and $Z_{1,1}=\bigcup_{j\in[m]}Z_{1,1,j}$.

The following claim is proved by a standard averaging argument.

######  Claim 22.

There exist at least $m/8$ many $j\in[m]$ such that $F^{\prime}_{1}$ is
$1/8$-heavy in $X_{1,1,j}$ i.e., $|\mathrm{Vars}(F^{\prime}_{1})\cap
X_{1,1,j}|\geq(1/8)\cdot|X_{1,1,j}|.$

Assuming Claim 22, we will first show how we can invoke the induction
hypothesis to prove the desired result.

By renaming if needed, let the $m/8$ segments from Claim 22 be
$X_{1,1,1},\dots,X_{1,1,m/8}$. The restriction
$\rho_{1}^{\prime}=\rho_{1,1}\circ\cdots\circ\rho_{1,m}$ where $\rho_{1,\ell}$
for $\ell\in[m]$ is an $(X_{1,1,\ell},Y_{1,1,\ell},Z_{1,1,\ell})$-restriction
sampled using the algorithm $\mathcal{A}$ in Section 3.2. We further condition
on any choice of restrictions $\rho_{1,\ell}$ for all $\ell>m/8$. This fixes
the sets
$\tilde{Y}_{1,1,\ell}^{\prime}:=\rho_{1,\ell}(\mathrm{Vars}(F^{\prime}_{1})\cap
X_{1,1,\ell})\cap Y_{1,1,\ell}$ and $\tilde{Z}_{1,1,\ell}^{\prime}$ (defined
similarly) for all $\ell>m/8$. Let $F^{\prime\prime}$ denote the restricted
formula thus obtained and $\rho_{1}^{\prime\prime}$ the random restriction
after this conditioning (i.e.
$\rho_{1}^{\prime\prime}=\rho_{1,1}\circ\cdots\circ\rho_{1,m/8}).$

We proceed to analyze $\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup
Y^{\prime},\tilde{Z}^{\prime}\cup Z^{\prime})}(F^{\prime}|_{\rho})$. Since
$\tilde{Y}^{\prime}\cup Y^{\prime}$ (resp. $\tilde{Z}^{\prime}\cup
Z^{\prime}$) can be partitioned as $\bigcup_{i\in[t]}\hat{Y}_{i}$ (resp.
$\bigcup_{i\in[t]}\hat{Z}_{i}$), we know (using Proposition 2 Item 3) that for
any choice of $\rho$ consistent with our choices so far,

$\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup Y^{\prime},\tilde{Z}^{\prime}\cup
Z^{\prime})}(F^{\prime}|_{\rho})=\prod_{i\in[t]}\mathrm{relrk}_{(\hat{Y}_{i},\hat{Z}_{i})}(F^{\prime}_{i}|_{\rho})\leq\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho})=\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime\prime}|_{\rho_{1}^{\prime\prime}}).$
(27)

To bound the latter term, we invoke the induction hypothesis on
$F^{\prime\prime}$ with $M$ replaced by $M^{\prime}=m/8$ and $\varepsilon$
replaced by $\varepsilon^{\prime}=1/8$. Here $\rho^{\prime\prime}$ is a
$(X^{\prime\prime},Y^{\prime\prime},Z^{\prime\prime})$-restriction where

$\displaystyle X^{\prime\prime}$ $\displaystyle=\cup_{j\in[m/8]}X_{1,1,j}$
$\displaystyle Y^{\prime\prime}$ $\displaystyle=\cup_{j\in[m/8]}Y_{1,1,j}$
$\displaystyle Z^{\prime\prime}$ $\displaystyle=\cup_{j\in[m/8]}Z_{1,1,j}$

and $\mathrm{Vars}(F^{\prime\prime})=\bar{X}\cup\bar{Y}\cup\bar{Z}$ where

$\displaystyle\bar{X}$ $\displaystyle=\mathrm{Vars}(F_{1}^{\prime})\cap
X^{\prime\prime}$ $\displaystyle\bar{Y}$
$\displaystyle=\left(\cup_{j>1}\tilde{Y}_{j}^{\prime}\right)\cup\left(\cup_{\ell>m/8}\tilde{Y}^{\prime}_{1,1,j}\right)\cup\hat{Y}_{1,0}$
$\displaystyle\bar{Z}$
$\displaystyle=\left(\cup_{j>1}\tilde{Z}_{j}^{\prime}\right)\cup\left(\cup_{\ell>m/8}\tilde{Z}^{\prime}_{1,1,j}\right)\cup\hat{Z}_{1,0}.$

Note that by Claim 22, it follows that $|\mathrm{Vars}(F^{\prime\prime})\cap
X_{1,1,j}|\geq\varepsilon^{\prime}\cdot|X_{1,1,j}|$ for each
$j\in[M^{\prime}]$ and hence the induction hypothesis for product-depth
$(\Delta-1)$ is applicable.

From the induction hypothesis, we see that

$\displaystyle\mathop{\mathrm{Pr}}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime\prime}|_{\rho^{\prime\prime}})\geq\exp(-k^{0.1}/(\Delta-1))]$
$\displaystyle\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1))$

Using (27), we get

$\displaystyle\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup
Y^{\prime},\tilde{Z}^{\prime}\cup
Z^{\prime})}(F^{\prime}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))\mid\bar{\mathcal{E}}_{2}]$
$\displaystyle\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1)).$

By (26), we thus have

$\displaystyle\mathop{\mathrm{Pr}}[\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup
Y^{\prime},\tilde{Z}^{\prime}\cup
Z^{\prime})}(F^{\prime}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))]$
$\displaystyle\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1))+\exp(-\Omega(M))$
$\displaystyle\leq\Delta\exp(-k^{0.1}/(\Delta-1)).$

The last inequality holds since $M\geq m/10>k$. This finishes the proof of
Case 2 modulo the (standard) proof of Claim 22, which we give below for
completeness.

###### Proof of Claim 22.

For $j\in[m]$, let $p_{j}$ denote $\left|\mathrm{Vars}(F_{1}^{\prime})\cap
X_{1,1,j}\right|/|X_{1,1,j}|=(m\left|\mathrm{Vars}(F_{1}^{\prime})\cap
X_{1,1,j}\right|)/|X_{1,1}|\,$. Since $\left|\mathrm{Vars}(F_{1}^{\prime})\cap
X_{1}\right|\geq(1/4)\left|X_{1}\right|$, we get that $\sum_{j=1}^{m}p_{j}\geq
m/4$. Assume that the number of $j$ such that $p_{j}\geq 1/8$ is strictly less
than $m/8$. Then, we have

$\displaystyle\sum_{j\in[m]}p_{j}\leq\sum_{j:p_{j}\geq 1/8}1+\sum_{j:p_{j}\leq
1/8}(1/8)<(m/8)+1/8\sum_{j:p_{j}\leq 1/8}1\leq m/8+(1/8)\cdot m=m/4,$

yielding $\sum_{j\in[m]}p_{j}<m/4,$ contradicting our assumption on
$\sum_{j\in[m/8]}p_{j}.$ This proves the claim. ∎

####  5.2.3 Case 3

Recall that in Case 3, we have that there is some $F^{\prime}_{i}$ ($i\in[t]$)
that is $(\varepsilon/k)$-heavy in at least $M/k$ many segments $X_{j}.$ We
will prove (4) in this case using induction. W.l.o.g., assume that $i=1$.

Since $\tilde{Y}^{\prime}\cup Y^{\prime}$ (resp. $\tilde{Z}^{\prime}\cup
Z^{\prime}$) can be partitioned as $\bigcup_{i\in[t]}\hat{Y}_{i}$ (resp.
$\bigcup_{i\in[t]}\hat{Z}_{i}$), we know (Proposition 2 Item 3) that for any
choice of $\rho,$

$\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup Y^{\prime},\tilde{Z}^{\prime}\cup
Z^{\prime})}(F^{\prime}|_{\rho})=\prod_{i\in[t]}\mathrm{relrk}_{(\hat{Y}_{i},\hat{Z}_{i})}(F^{\prime}_{i}|_{\rho})\leq\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho}).$
(28)

It is therefore sufficient to bound
$\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho})$.

W.l.o.g., we assume that $F^{\prime}_{1}$ is $(\varepsilon/k)$-heavy in
segments $X_{1},\ldots,X_{M/k}$. Each $X_{j}$ ($j\in[M/k]$) consists of two
half-segments, i.e., $X_{j}=X_{j,1}\cup X_{j,2}$. By averaging, we know that
there is some $\gamma\in[2]$ such that $F^{\prime}_{1}$ is
$(\varepsilon/k)$-heavy in at least half of
$X_{1,\gamma},\ldots,X_{M/k,\gamma}$. W.l.o.g. we assume that $\gamma=1$. By
renaming the segments, let us assume that $F^{\prime}_{1}$ is
$(\varepsilon/k)$-heavy in $X_{1,1},\ldots,X_{M/2k,1}$.

For each $j\leq M/2k$, the restriction $\rho_{j}$ is sampled according to the
algorithm $\mathcal{A}$ described in Section 3.2. Let us say that $\rho_{j}$
is _good_ if the algorithm $\mathcal{A}$ decides on option $E_{1}$ in sampling
$\rho_{j}.$ For each $j$, $\rho_{j}$ is good with probability $1/3$. For each
$j\in[M/2k],$ we condition on whether or not $\rho_{j}$ is good. Let
$\mathcal{E}_{3}$ denote the event that the number of good $\rho_{j}$ is at
most $M/8k$. Since the different $\rho_{j}$ are sampled independently, a
Chernoff bound (Theorem 5 Item 1) tells us that

$\mathop{\mathrm{Pr}}_{\rho}[\mathcal{E}_{3}]\leq\exp(-\Omega(M/k)).$ (29)

Assume that the event $\mathcal{E}_{3}$ does not occur. By renaming segments
once more, we may assume that $\rho_{j}$ is good for each $j\in[M/8k]$. We
condition on any choice of the restrictions $\rho_{j}$ for $j>M/8k$; this
fixes the sets $\tilde{Y}^{\prime}_{j}$ and $\tilde{Z}^{\prime}_{j}$ for
$j>M/8k$. Also, note that since we have conditioned on choosing the option
$E_{1}$ for sampling restrictions $\rho_{1},\ldots,\rho_{M/8k}$, all variables
in $X_{j,2}$ ($j\in[M/8k]$) are set to $0$ with probability $1$. We will
therefore think of $\rho_{j}$ as an $(X_{j,1},Y_{j,1},Z_{j,1})$-restriction
for the rest of the proof.

We will now prove our result by considering two subcases. Recall that each
half-segment $X_{j,1}$ is an $X^{(\Delta-1)}$-clone and hence a union of $m$
$X^{(\Delta-1)}$-segments $X_{j,1,1},\ldots,X_{j,1,m}.$ We will refer to these
as _sub-segments_ of $X_{j,1}.$

##### Case (3a): There are at least $M/16k$ many $j\in[M/8k]$ such that
$F^{\prime}_{1}$ is at least $(\varepsilon/2k)$-heavy in at least $m^{0.5}$
many sub-segments of $X_{j,1}$.

Without loss of generality, let us assume that the hypothesis of Case (3a)
holds for the half-segments $X_{1,1},\ldots,X_{M/16k,1}.$ That is,
$F^{\prime}_{1}$ is $(\varepsilon/2k)$-heavy in at least $m^{0.5}$ many sub-
segments of $X_{j,1}$ for each $j\in[M/16k]$. Here, we will be able to apply
induction on $F^{\prime}_{1}$ (which is a $\Sigma(\Pi\Sigma)^{(\Delta-1)}$
formula) to obtain an upper bound on its relative rank under the random
restriction.

We will apply the induction hypothesis with $M$ replaced by
$M^{\prime}=M/16k\times m^{0.5}$ and $\varepsilon$ replaced by
$\varepsilon^{\prime}=\varepsilon/2k$. We claim that we have
$\varepsilon^{\prime}\geq(1/M^{\prime})^{0.25}$ and $M^{\prime}\geq m/10$
(which are needed to apply induction). The latter inequality is trivial as
$M^{\prime}\geq M\geq m/10$. For the former, note that we have

$\frac{1}{(\varepsilon^{\prime})^{4}}=\frac{(2k)^{4}}{\varepsilon^{4}}\leq(2k)^{4}M\leq\frac{M\sqrt{m}}{16k}=M^{\prime}$

where the first inequality uses the fact that $\varepsilon\geq 1/M^{0.25}$ and
the last inequality uses the fact that $k\leq m^{0.05}.$ This shows that
$\varepsilon^{\prime}\geq(1/M^{\prime})^{0.25}$ as required.

We apply induction after some further processing. By renaming if necessary,
let us assume that $F_{1}^{\prime}$ is heavy in the first $\sqrt{m}$ sub-
segments $X_{j,1,1},\cdots,X_{j,1,m^{0.5}}$ of $X_{j,1}$ for each
$j\in[M/16k]$. Conditioned on what is known about $\rho$ so far, for each
$j\in[M/8k]$, the restriction $\rho_{j}$ can be written as
$\rho_{j}=\rho_{j,1}\circ\cdots\circ\rho_{j,m}$ where each $\rho_{j,\ell}$ for
$\ell\in[m]$ is an $(X_{j,1,\ell},Y_{j,1,\ell},Z_{j,1,\ell})$-restriction
sampled as per the sampling algorithm $\mathcal{A}$. We further condition on
any choice of the restrictions $\rho_{j_{1}}$ for all $j_{1}>M/16k$ and
$\rho_{j_{2},\ell}$ for all $j_{2}\in[M/16k]$ and $\ell>m^{0.5}$. This fixes
the sets $\tilde{Y}_{j_{1}}^{\prime}$, $\tilde{Z}_{j_{1}}^{\prime}$
($M/8k<j_{1}\leq M/16k$) and
$\tilde{Y}^{\prime}_{j_{2},1,\ell}:=\rho(\mathrm{Vars}(F^{\prime}_{1})\cap
X_{j_{2},1,\ell})\cap Y_{j_{2},1,\ell}$ and
$\tilde{Z}^{\prime}_{j_{2},1,\ell}:=\rho(\mathrm{Vars}(F^{\prime}_{1})\cap
X_{j_{2},1,\ell})\cap Z_{j_{2},1,\ell}$ ($j_{2}\in[M/16k],\ell>m^{0.5}$).

Conditioned on our choices so far, the restriction $\rho$ can now be
identified an
$(X^{\prime\prime},Y^{\prime\prime},Z^{\prime\prime})$-restriction
$\rho^{\prime\prime}$ where

$\displaystyle X^{\prime\prime}$
$\displaystyle=\cup_{j\in[M/8k]}\cup_{\ell\in[m^{0.5}]}X_{j,1,\ell}$
$\displaystyle Y^{\prime\prime}$
$\displaystyle=\cup_{j\in[M/8k]}\cup_{\ell\in[m^{0.5}]}Y_{j,1,\ell}$
$\displaystyle Z^{\prime\prime}$
$\displaystyle=\cup_{j\in[M/8k]}\cup_{\ell\in[m^{0.5}]}Z_{j,1,\ell}.$

and the restricted formula, which we denote by $F_{1}^{\prime\prime}$,
satisfies $\mathrm{Vars}(F^{\prime\prime}_{1})=\bar{X}\cup\bar{Y}\cup\bar{Z}$
where

$\displaystyle\bar{X}$ $\displaystyle=\mathrm{Vars}(F_{1}^{\prime})\cap
X^{\prime\prime}$ $\displaystyle\bar{Y}$
$\displaystyle=\left(\cup_{j>M/16k}\tilde{Y}_{j}^{\prime}\right)\cup\left(\cup_{j\in[M/16k]}\cup_{\ell>m^{0.5}}\tilde{Y}^{\prime}_{j,1,\ell}\right)\cup\hat{Y}_{1,0}$
$\displaystyle\bar{Z}$
$\displaystyle=\left(\cup_{j>M/16k}\tilde{Z}_{j}^{\prime}\right)\cup\left(\cup_{j\in[M/16k]}\cup_{\ell>m^{0.5}}\tilde{Z}^{\prime}_{j,1,\ell}\right)\cup\hat{Z}_{1,0}.$

Based on the discussion above, we can now invoke the induction hypothesis with
$M^{\prime}=(M/16k)m^{0.5}$ and $\varepsilon^{\prime}=\varepsilon/2k$ on
$F^{\prime\prime}_{1}$ to obtain

$\mathop{\mathrm{Pr}}_{\rho^{\prime\prime}}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime\prime}_{1}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))]\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1)).$

Since the above holds for an arbitrary sequence of fixings after it was
determined that $\mathcal{E}_{3}$ did not hold, we have thus shown the
following.

$\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))\
|\ \bar{\mathcal{E}}_{3}]\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1)).$ (30)

We will use this below, after the proof of Case 3(b), to finish the proof in
this case.

##### Case (3b): There are at least $M/16k$ many $j\in[M/8k]$ such that
$F^{\prime}_{1}$ is at least $(\varepsilon/2k)$-heavy in less than $m^{0.5}$
many sub-segments of $X_{j,1}$.

W.l.o.g., assume that for each $j\in[M/16k]$, there is a $h_{j}<\sqrt{m}$ such
that $F_{1}^{\prime}$ is $(\varepsilon/2k)$-heavy only in the sub-segments
$X_{j,1,1},\ldots,X_{j,1,h_{j}}.$

For each $j\in[M/16k]$, let $W_{j}$ be
$X_{j,1}\cap\mathrm{Vars}(F_{1}^{\prime})$. By our assumption in Case 3, we
know that

$|W_{j}|\geq\frac{\varepsilon}{k}\cdot|X_{j,1}|~{}~{}~{}~{}\forall
j\in[M/16k]$ (31)

Also, we know that for any $\ell\in\\{h_{j}+1,\ldots,m\\}$, $|X_{j,1,\ell}\cap
W_{j}|<\frac{\varepsilon}{2k}\cdot|X_{j,1,\ell}|$ for each $j\in[M/16k]$.
Therefore, we get that

$\sum_{\ell>h_{j}}|X_{j,1,\ell}\cap
W_{j}|\leq\frac{\varepsilon}{2k}\cdot|X_{j,1}|~{}~{}~{}~{}\forall j\in[M/16k]$
(32)

Using Equations (31) and (32) we get that

$\displaystyle\sum_{\ell\in[h_{j}]}|X_{(j,1),\ell}\cap
W_{j}|\geq\frac{\varepsilon}{2k}\cdot|X_{j,1}|~{}\mbox{ for each
}j\in[M/16k].$

Thus by averaging and by using the fact that $h_{j}\leq m^{0.5}$, we get that
for every $j\in[M/16k]$ there exists an $\ell_{j}\in[h_{j}]$ such that
$|X_{j,1,\ell_{j}}\cap
W_{j}|\geq\frac{\varepsilon}{2k}\cdot\frac{|X_{j,1}|}{m^{0.5}}$. By renaming
if necessary, let $\ell_{j}=1$ for all $j\in[M/16k]$.

Notice that in fact $|X_{j,1}|=m\cdot|X_{j,1,\ell}|$ for any $\ell\in[m]$.
Therefore, we get

$\displaystyle|X_{j,1,1}\cap\mathrm{Vars}(F^{\prime}_{1})|=|X_{j,1,1}\cap
W_{j}|$ $\displaystyle\geq\frac{\varepsilon}{2k}\cdot\frac{|X_{j,1,1}|\cdot
m}{m^{0.5}}=\frac{\varepsilon}{2k}\cdot m^{0.5}\cdot|X_{j,1,1}|$
$\displaystyle\forall j\in[M/16k]$ (33)

We will apply induction on the formula $F^{\prime}_{1}$ and the sub-segments
$X_{j,1,1}$ ($j\in[M/16k]$). For the induction, the parameter $M$ will thus be
replaced by $M^{\prime}=M/16k$ and $\varepsilon$ by
$\varepsilon^{\prime}=\frac{\varepsilon}{2k}\cdot m^{0.5}$ (we can take this
$\varepsilon^{\prime}$ by (33) above). To check that induction is possible
with these parameters, we need to ensure that $\varepsilon^{\prime}\geq
1/M^{\prime 0.25}$ and $M^{\prime}\geq m/10$. The first inequality follows as
in Case (3a). Formally, we note that

$\frac{1}{(\varepsilon^{\prime})^{4}}=\frac{(2k)^{4}}{\varepsilon^{4}m^{2}}\leq\frac{M}{16k}=M^{\prime}$

where the inequality follows from the fact that $\varepsilon\geq 1/M^{0.25}$
and $k=m^{0.05}.$ This shows that
$\varepsilon^{\prime}\geq(1/M^{\prime})^{0.25}.$

Now we will show that $M^{\prime}\geq m/10$. We know that $M\geq m/10$, but in
this case we will be able to get a better lower bound on the value of $M$,
which will then give us the intended lower bound on $M^{\prime}$. For this,
first observe that $\varepsilon^{\prime}=(\varepsilon/2k)\cdot m^{0.5}$. Note
that by (33) and the fact that
$|X_{j,1,1}\cap\mathrm{Vars}(F^{\prime}_{1})|\leq|X_{j,1,1}|$, we get

$(\varepsilon/2k)\cdot m^{0.5}\leq
1\Leftrightarrow\frac{1}{\varepsilon}\geq\frac{\sqrt{m}}{2k}.$

We also know that $\varepsilon\geq 1/M^{0.25}$ and therefore, $M\geq
1/\varepsilon^{4}\geq m^{2}/(2k)^{4}$. As $M^{\prime}=M/16k$, we get that
$M^{\prime}\geq\frac{m^{2}}{8k\cdot(2k)^{4}}\geq m$. This gives us (a bound
that is slightly better than) the desired bound on $M^{\prime}$.

We now apply induction. As in Case (3a), some processing is needed. Note that
conditioned on what is known about $\rho$ so far, for each $j\in[M/8k]$, the
restriction $\rho_{j}$ sets all
$\rho_{j}=\rho_{j,1}\circ\cdots\circ\rho_{j,m}$ where each $\rho_{j,\ell}$ for
$\ell\in[m]$ is an $(X_{j,1,\ell},Y_{j,1,\ell},Z_{j,1,\ell})$-restriction
sampled as per the sampling algorithm $\mathcal{A}$. We further condition on
any choice the restrictions $\rho_{j_{1}}$ for all $j_{1}>M/16k$ and
$\rho_{j_{2},\ell}$ for all $j_{2}\in[M/16k]$ and $\ell>1$. This fixes the
sets $\tilde{Y}_{j_{1}}^{\prime}$, $\tilde{Z}_{j_{1}}^{\prime}$
($M/8k<j_{1}\leq M/16k$) and
$\tilde{Y}^{\prime}_{j_{2},1,\ell}:=\rho(\mathrm{Vars}(F^{\prime}_{1})\cap
X_{j_{2},1,\ell})\cap Y_{j_{2},1,\ell}$ and
$\tilde{Z}^{\prime}_{j_{2},1,\ell}:=\rho(\mathrm{Vars}(F^{\prime}_{1})\cap
X_{j_{2},1,\ell})\cap Z_{j_{2},1,\ell}$ ($j_{2}\in[M/16k],\ell>1$).

Conditioned on our choices so far, the restriction $\rho$ can be identified
with an $(X^{\prime\prime},Y^{\prime\prime},Z^{\prime\prime})$-restriction
$\rho^{\prime\prime}$ where

$\displaystyle X^{\prime\prime}$ $\displaystyle=\cup_{j\in[M/16k]}X_{j,1,1}$
$\displaystyle Y^{\prime\prime}$ $\displaystyle=\cup_{j\in[M/16k]}Y_{j,1,1}$
$\displaystyle Z^{\prime\prime}$ $\displaystyle=\cup_{j\in[M/16k]}Z_{j,1,1}.$

and the restricted formula $F_{1}^{\prime\prime}$ satisfies
$\mathrm{Vars}(F^{\prime\prime}_{1})=\bar{X}\cup\bar{Y}\cup\bar{Z}$ where

$\displaystyle\bar{X}$ $\displaystyle=\mathrm{Vars}(F_{1}^{\prime})\cap
X^{\prime\prime}$ $\displaystyle\bar{Y}$
$\displaystyle=\left(\cup_{j>M/16k}\tilde{Y}_{j}^{\prime}\right)\cup\left(\cup_{j\in[M/16k]}\cup_{\ell>1}\tilde{Y}^{\prime}_{j,1,\ell}\right)\cup\hat{Y}_{1,0}$
$\displaystyle\bar{Z}$
$\displaystyle=\left(\cup_{j>M/16k}\tilde{Z}_{j}^{\prime}\right)\cup\left(\cup_{j\in[M/16k]}\cup_{\ell>1}\tilde{Z}^{\prime}_{j,1,\ell}\right)\cup\hat{Z}_{1,0}.$

Based on the discussion above, we can now invoke the induction hypothesis with
$M^{\prime}$ and $\varepsilon^{\prime}$ (as defined above) on
$F^{\prime\prime}_{1}$ to obtain

$\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime\prime}_{1}|_{\rho^{\prime\prime}})\geq\exp(-k^{0.1}/(\Delta-1))]\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1)).$

Hence, we obtain as in Case (3a),

$\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))\
|\ \bar{\mathcal{E}}_{3}]\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1)).$ (34)

We now see how to finish the proof in both Cases (3a) and (3b). Using (30) and
(34), we see that in each of the cases (3a) and (3b),

$\displaystyle\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\hat{Y}_{1},\hat{Z}_{1})}(F^{\prime}_{1}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))]$
$\displaystyle\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1))+\mathop{\mathrm{Pr}}_{\rho}[\mathcal{E}_{3}]$
$\displaystyle\leq(\Delta-1)\exp(-k^{0.1}/(\Delta-1))+\exp(-\Omega(M/k))$
$\displaystyle\leq\Delta\exp(-k^{0.1}/(\Delta-1))$

where the second inequality follows from (29) and the last inequality from the
fact that $M=\Omega(m)=\Omega(k^{20}).$

By (28), we thus get

$\mathop{\mathrm{Pr}}_{\rho}[\mathrm{relrk}_{(\tilde{Y}^{\prime}\cup
Y^{\prime},\tilde{Z}^{\prime}\cup
Z^{\prime})}(F^{\prime}|_{\rho})\geq\exp(-k^{0.1}/(\Delta-1))]\leq\Delta\exp(-k^{0.1}/(\Delta-1))$

proving inequality (4) in this case. This completes the proof in Case 4.

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# Primordial black holes formation in a early matter dominated era from the
pre-big bang scenario

Pietro Conzinua<EMAIL_ADDRESS>Giovanni Marozzia
<EMAIL_ADDRESS>aDipartimento di Fisica, Università di Pisa, Largo
B. Pontecorvo 3, 56127 Pisa, Italy,
and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Italy

###### Abstract

We discuss the production of primordial black holes in an early matter
dominated era, which typically takes place in string inspired early universe
cosmological models. In particular, we consider a pre-big bang scenario
(extending previous results regarding formation in the radiation dominated
era) where the enhancement of curvature perturbations is induced by a
variation of the sound-speed parameter $c_{s}$ during the string phase of
high-curvature inflation. After imposing all relevant observational
constraints, we find that the considered class of models is compatible with
the production of a large amount of primordial black holes, in the mass range
relevant to dark matter, only for a small range of the parameters space. On
the other hand, we find that a huge production of light primordial black holes
may occur both in such matter dominated era and in the radiation dominated
one.

##  I Introduction

Primordial black holes (PBHs) have attracted considerable attention in recent
years and several different formation mechanisms were proposed (see e.g. Carr
and Kuhnel (2022); Escrivà  _et al._ (2022) for a review of the different
proposals). PBHs may form due to the collapse of large fluctuations in the
early universe. In particular, one can distinguish two particular cases in
which the collapse happens in a radiation era and in a matter era. In the
first case, the most studied in literature, due to the presence of radiation
pressure, only large enough fluctuations can collapse. In the second case,
instead, there is no pressure contribution and the effects of shape
deformations become crucial to define properly the collapse Harada  _et al._
(2016, 2017). While in standard scenarios we have only one matter dominated
era happening after the radiation dominated one, in many alternative scenarios
there can be an early matter dominated era before the radiation one,
increasing the interest in studying the formation of PBHs in such era. Here,
we consider the possibility that PBHs may be form from the gravitational
collapse of primordial density fluctuations in the early matter era that
follows a pre-big bang scenario Gasperini and Veneziano (2003), extending our
previous work Conzinu  _et al._ (2020), where we considered the formation for
density fluctuations that re-enter the horizon in the radiation era. The paper
is organized as follows. In Sec. II we describe the PBHs mass at formation for
both matter and radiation domination era. In Sec. III we introduce the PBHs
abundance parameter, its relation with dark matter and with the primordial
power spectrum in the above two eras. In Sec. IV we show the results obtained
for formation in the early matter era that takes place for the pre-big bang
scenario. Finally, we draw our conclusions in Sec. V.

##  II Primordial Black Holes Mass at Formation: Matter vs Radiation Era

If the density contrast $\delta\equiv\delta\rho/\rho$ is large enough
(exceeding a critical value $\delta_{c}$ Carr and Kuhnel (2022); Escrivà  _et
al._ (2022); Carr (1975)) then the fluctuations, when re-enter the horizon,
collapse directly in black holes. The mass of a primordial black hole
$M_{pbh}$, when it forms, is proportional to the mass contained in the Hubble
horizon at the formation time $M_{H}=\frac{4\pi}{3}\rho H^{-3}=4\pi
M^{2}_{p}H^{-1}\,$ Carr and Hawking (1974).

Defining the re-entry time of a fluctuation with wave-number $k$ by
$k=a(t_{k})H(t_{k})\equiv a_{k}H_{k}$, the PBH mass can be written as

$M_{pbh}=4\pi\frac{M^{2}_{p}}{H_{k}}=4\pi\frac{M^{2}_{p}}{k}a_{k}\,.$ (1)

In this way one can express directly the mass $M_{pbh}$ in terms of the wave
number $k$ .
For matter dominated era, defining $t_{d}\sim H^{-1}_{d}$ as the time when the
early matter era ends (and begin the radiation era), it holds 111Here $a_{eq}$
and $H_{eq}$ are the scale factor and the Hubble parameter at the radiation-
matter equilibrium era.

$\begin{split}\frac{a_{k}}{a_{0}}=&\left(\frac{a_{k}}{a_{d}}\right)\left(\frac{a_{d}}{a_{eq}}\right)\left(\frac{a_{eq}}{a_{0}}\right)=\\\
=&\left(\frac{H_{d}}{H_{k}}\right)^{2/3}\left(\frac{H_{eq}}{H_{d}}\right)^{1/2}\left(\frac{H_{0}}{H_{eq}}\right)^{2/3}\,,\end{split}$

using the relation $H_{k}=k/a_{k}$ and setting $a_{0}=1$, we can solve in
terms of $a_{k}$

$a_{k}=\left(\frac{H_{eq}}{H_{d}}\right)^{-{1\over
2}}\left(\frac{H_{0}}{k}\right)^{2}\,,$ (2)

while in the case of radiation era, since

$\begin{split}&\frac{a_{k}}{a_{0}}=\left(\frac{a_{k}}{a_{eq}}\right)\left(\frac{a_{eq}}{a_{0}}\right)=\left(\frac{H_{eq}}{H_{k}}\right)^{1/2}\left(\frac{H_{0}}{H_{eq}}\right)^{2/3}\,,\end{split}$
(3)

we obtain

$a_{k}=\frac{H_{eq}}{k}\left(\frac{H_{0}}{H_{eq}}\right)^{4/3}\,.$ (4)

Therefore, the PBHs mass can be written as

$\displaystyle M_{pbh}\simeq$ $\displaystyle
4\pi\frac{M^{2}_{p}H_{eq}}{k^{2}}\left(\frac{H_{0}}{H_{eq}}\right)^{4/3},\quad\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \,(\textbf{RD})\,,$ (5)
$\displaystyle M_{pbh}\simeq$ $\displaystyle
4\pi\frac{M^{2}_{p}}{k^{3}}\left(\frac{H_{eq}}{H_{d}}\right)^{-{1\over
2}}H_{0}^{2}\;,\quad\quad(\textbf{MD})\,.$ (6)

One can note that in the two abobe cases the dependence from $k$ is different
and in the case of MD the mass depends also on the duration of the matter
phase by the factor $H_{d}$.

##  III Abundance and Relation with Cold Dark Matter

In order to quantify the constraints on PBHs, their abundance and their
possible dark matter nature, one can consider two different parameters. The
so-called energy density fraction $\beta$, that quantifies the abundance of
PBHs at formation (see e.g. Carr and Kuhnel (2022, 2020); Carr  _et al._
(2021)),

$\beta\equiv\frac{\rho_{pbh}}{\rho_{tot}}\Bigg{|}_{atformation}\,,$ (7)

where $\rho_{pbh}$ and $\rho_{tot}$ are the density energy in form of PBHs and
the total energy respectively.
And the parameter $f_{pbh}$, defined as (see e.g. Sasaki  _et al._ (2018))

$f_{pbh}\equiv\frac{\rho_{pbh}}{\rho_{cdm}}\Bigg{|}_{t_{0}}\,.$ (8)

which gives us the relative amount of cold dark matter in form of PBHs today,
where $\rho_{cdm}$ is the dark matter contribution. Since the majority of
constraints for PBHs are usually given in terms of this parameter it is
important to relate $f_{pbh}$ and $\beta$ and then translate constraints from
$f_{pbh}$ directly on $\beta$ and, as we will explicitly see later, on
constraints for the primordial power spectrum.
By definition, at the radiation-matter equilibrium era, after all the PBHs
have been formed, we have that Carr  _et al._ (2021)

$f_{pbh}=\frac{\beta}{\Omega_{cdm}}\Bigg{|}_{eq}\,.$ (9)

On the other hand, in the case of formation in radiation era, since
$\rho_{pbh}\sim a^{-3}$ and $\rho_{rad}\sim a^{-4}$, then $\beta\sim a$; while
in matter era formation it holds $\beta=\beta_{0}\simeq const$. Thus we can
manipulate equations (8) and (9) and obtain the relation between $f_{pbh}$ and
$\beta$ in these two cases.

Radiation era: as mentioned before, PBHs might form if the density contrast
exceeds a critical value $\delta_{c}$. Assuming a probability distribution for
the density contrast $P(\delta)$ then $\beta$ can be expressed as Carr  _et
al._ (2021)

$\beta=\int_{\delta_{c}}^{\infty}P(\delta)d\delta\,,$ (10)

where $\delta_{c}$ is such critical density Carr and Kuhnel (2020). If the
probability distribution is Gaussian we have

$P(\delta)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{\delta^{2}}{2\sigma^{2}}}\,,$
(11)

where $\sigma=\sigma(R)$ is the mass variance given by

$\sigma=\int_{0}^{\infty}\tilde{W}(kR)\mathcal{P}_{\delta}(k)\frac{dk}{k}\,,$
(12)

where $\mathcal{P}_{\delta}(k)$ is the primordial power spectrum of $\delta$
at horizon entry and $\tilde{W}$ is a window function smoothing over the
comoving scale $R\sim(a_{k}H_{k})^{-1}$.
The PBHs mass fraction (10) assumes the simple form

$\beta=\frac{1}{\sqrt{2\pi\sigma^{2}}}\int^{\infty}_{\delta_{c}}\exp{-\frac{\delta^{2}}{2\sigma^{2}}}d\delta=\operatorname{Erfc}\left(\frac{\delta_{c}}{\sqrt{2}\sigma}\right)\,,$
(13)

where $\operatorname{Erfc}$ is the complementary error function.

Combining Eq. (9) with entropy conservation, we get the connection between
$\beta$ and $f_{pbh}$

$f_{pbh}=\beta\frac{\Omega_{\gamma}^{0}}{\Omega^{0}_{cdm}}\frac{g(T_{k})g_{s}(T_{0})}{g(T_{0})g_{s}(T_{k})}\frac{T_{k}}{T_{0}}\,.$
(14)

where $\Omega_{\gamma}^{0}$ is the radiation energy density today, and $g$,
$g_{s}$ are the number of effective temperature and entropy relativistic
degrees of freedom respectively.

Matter era: The situation is different if the formation happens in a matter
dominated era. This case has been of less interest during the past years and
only few studies have been done (see e.g. Harada and Carr (2005); Harada  _et
al._ (2017); Ballesteros  _et al._ (2020) ), essentially because in the
standard scenario we have only one matter era happening after the radiation
dominated era. As a consequence only PBHs of enormous mass could be formed
during such era. On the other hand, in many scenarios, mostly coming from
string theory, there can be a matter dominated era also before the radiation
one, as it is the case of pre-big bang scenario Gasperini and Veneziano
(2003). This gives new motivations to study the PBHs formation in a matter era
and motivated this work.

While during the radiation era, as a consequence of the presence of radiation
pressure, one obtain a critical density $\delta_{c}$ of order unity, beyond
which one obtains a collapse of the overdense region, in a matter contest this
not holds and the effects of shape deformation are crucial to define properly
the critical density. Following Harada  _et al._ (2016), the criterion for
formation in matter era is given in terms of the hoop conjecture 222The hoop
conjecture Misner  _et al._ (1973) states that a black hole forms when a mass
$M$ collapse into a region whose circumferences in every direction are smaller
than $\frac{4\pi GM}{c^{2}}$.

. By numerical analysis it was shown in Harada  _et al._ (2016) that holds

$\beta_{0}\sim 0.056\sigma^{5}\,.$ (15)

Furthermore, in Harada  _et al._ (2017) it was pointed out that if one takes
into account the angular momentum the expression given before holds only for
$\sigma>\sigma_{ang}=0.005$, while for smaller values was derived the semi-
analytic expression

$\beta_{0}\sim 1.321\times
10^{-7}f_{q}(q_{c})\mathcal{I}^{2}\sigma^{2}\exp{\left(-0.1474\frac{\mathcal{I}^{4/3}}{\sigma^{2/3}}\right)}\,,$
(16)

where $\mathcal{I}$ is related to the variance of the angular momentum and
$f_{q}(q_{c})$ is the fraction of mass with quadrupole asphericity $q\ll
q_{c}\sim 2.4(\mathcal{I}\sigma)^{1/3}$, both $\mathcal{I}$ and $f_{q}$ are
parameters of order unity.

In order to connect the two parameters $\beta$ and $f_{pbh}$, we assume an
instantaneously matter-radiation transition, such that

$3M^{2}_{P}H_{d}^{2}=\rho_{d}=\frac{\pi^{2}g(T_{d})}{30}T_{d}^{4}\,,$ (17)

where as before the subscript $"d"$ states for the end of matter era and
$T_{d}$ is the correspondent temperature, finally $g(T_{d})$ counts the
effective number of degrees of freedom. Then, using again entropy
conservation, we obtain

$f_{pbh}=\beta_{0}\frac{\Omega_{\gamma}^{0}}{\Omega^{0}_{cdm}}\frac{g(T_{d})g_{s}(T_{0})}{g(T_{0})g_{s}(T_{d})}\frac{T_{d}}{T_{0}}\,.$
(18)

For a rough estimation we can assume $g\simeq g_{s}$ for all the epochs and
obtain

$\displaystyle
f^{RD}_{pbh}=\beta\frac{\Omega_{\gamma}^{0}}{\Omega^{0}_{cdm}}\frac{T_{k}}{T_{0}}\,,$
(19) $\displaystyle
f^{MD}_{pbh}=\beta_{0}\frac{\Omega_{\gamma}^{0}}{\Omega^{0}_{cdm}}\frac{T_{d}}{T_{0}}\,.$
(20)

where we have $\Omega^{0}_{cdm}\simeq 0.26,\,\Omega_{\gamma}^{0}\simeq
10^{-4},T_{0}\simeq 2.7\leavevmode\nobreak\ \mathrm{K}$. In Eq.(19) we can
make explicit the PBH mass dependence by the relation $T_{k}=(4\pi
M_{p}/M_{pbh})^{1/2}(3/g(T_{k}))^{1/4}$.

As pointed out above, if formation happens in the radiation era $\beta\sim a$,
while in matter era we have that $\beta=\beta_{0}$ is nearly constant. Thus,
we can relate Eqs. (13) and (15) by the following simple relation

$\beta_{0}\simeq\left(\frac{H_{k}}{H_{d}}\right)^{2/3}\beta\,.$ (21)

Using this result, all relations between the parameter $\beta$ and $f$
obtained for the radiation dominated era can be then directly converted to the
case of formation in an early matter era.

###  III.1 Relation with Primordial Power Spectrum

Starting from the above results, we can now translate the constraints on PBHs
abundance in terms of constraints on the correspondent amplitude of the
comoving curvature perturbation $\mathcal{R}$.
The density contrast $\delta$ is related to $\mathcal{R}$ by Ballesteros  _et
al._ (2020)

$\delta=\frac{2(1+\omega)}{5+3\omega}\mathcal{R}\,,$ (22)

where $\omega$ is the equation of state parameter ($\omega=0$ for matter era
and $\omega=1/3$ for radiation era). Then, since the variance $\sigma^{2}\sim
P_{\delta}$, we have

$\displaystyle\sigma^{2}\sim\frac{16}{81}\mathcal{P_{R}}\qquad\text{RD}\,,$
$\displaystyle\sigma^{2}\sim\frac{4}{25}\mathcal{P_{R}}\qquad\text{MD}\,.$

We can now invert Eqs. (13) and (15) to obtain, for a radiation dominated era

$P_{\delta}\sim\sigma^{2}=\left(\frac{\delta_{c}}{\sqrt{2}\operatorname{Erfc}^{-1}(\beta)}\right)^{2}\,,\qquad$
(24)

while for a matter dominated era we have

$\displaystyle
P_{\delta}\sim\sigma^{2}=\left(\frac{\beta_{0}}{0.056}\right)^{2/5}\,,\qquad\sigma>0.005\,,$
(25a) $\displaystyle
P_{\delta}\sim\sigma^{2}=10^{-4}W\left(\frac{0.05}{\beta^{1/3}_{0}}\right)^{-3}\,,\qquad\sigma<0.005\,,$
(25b)

where $W$ is the Lambert $W$-function.

For the case of a pre-Big Bang scenario, the power spectra for the curvature
perturbation sourced by the dilaton $\phi$ and the axion $\chi$ are given by
Conzinu  _et al._ (2020)

Axion spectrum
$\displaystyle\mathcal{P}_{\mathcal{R}}^{\chi}(\omega)\leavevmode\nobreak\ $
$\displaystyle\simeq\leavevmode\nobreak\ {f^{2}\over 2\pi^{2}}\left(H_{1}\over
M_{\rm
P}\right)^{2}\left(\omega\over\omega_{1}\right)^{3-|3+2\alpha|}c_{\chi}^{-1-|3+2\alpha|}\leavevmode\nobreak\
,$ $\displaystyle\simeq\leavevmode\nobreak\ {f^{2}\over
2\pi^{2}}\left(H_{1}\over M_{\rm
P}\right)^{2}\left(\omega_{s}\over\omega_{1}\right)^{3-|3+2\alpha|}\left(\omega\over\omega_{s}\right)^{4}\,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ ,$
$\displaystyle\simeq\leavevmode\nobreak\ {f^{2}\over 2\pi^{2}}\left(H_{1}\over
M_{\rm
P}\right)^{2}\left(\omega_{s}\over\omega_{1}\right)^{3-|3+2\alpha|}\left(\omega\over\omega_{s}\right)^{n_{s}-1},$
Dilaton spectrum
$\displaystyle\mathcal{P}_{\mathcal{R}}^{\phi}(\omega)\leavevmode\nobreak\ $
$\displaystyle\simeq\leavevmode\nobreak\ {1\over 2\pi^{2}}\left(H_{1}\over
M_{\rm
P}\right)^{2}\left(\omega\over\omega_{1}\right)^{3-|3-2\alpha|}c_{\phi}^{-1-|3-2\alpha|}$
, $\displaystyle{\omega_{s}\over c_{\phi}}<\omega<{\omega_{1}\over c_{\phi}},$
$\displaystyle\leavevmode\nobreak\ \simeq\leavevmode\nobreak\ {1\over
2\pi^{2}}\left(H_{1}\over M_{\rm
P}\right)^{2}\left(\omega_{s}\over\omega_{1}\right)^{3-|3-2\alpha|}\left(\omega\over\omega_{s}\right)^{4}$
, $\displaystyle\omega_{s}<\omega<{\omega_{s}\over c_{\phi}},$
$\displaystyle\leavevmode\nobreak\ \simeq\leavevmode\nobreak\ {1\over
2\pi^{2}}\left(H_{1}\over M_{\rm
P}\right)^{2}\left(\omega_{s}\over\omega_{1}\right)^{3-|3-2\alpha|}\left(\omega\over\omega_{s}\right)^{3}$
, $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \omega<\omega_{s}.$

Where $f^{2}\simeq 0.137$ is the transfer function that relates the axion
modes to the curvature perturbations, $\omega_{s}$ and $\omega_{1}$ are the
frequency scales at the beginning and at the end of the string phase, $\alpha$
parameterize the evolution in the string phase 333Note that we have changed
for convenience the notation for the axion field $\chi$ and the parameter
$\alpha$ with respect to Conzinu  _et al._ (2020), the two notations are
mapped by $\chi\rightarrow\sigma$ and $\alpha\rightarrow\beta$., and $H_{1}$
is the Hubble parameter at the end of the string phase (for more details we
refer to Conzinu  _et al._ (2020); Gasperini (2007)).

Requiring to have the total or at least an important fraction of cold dark
matter in form of PBHs, one obtains important constraints on the needed value
of the primordial power spectrum. Comparing Eqs. (24) and (25) it is then
clear the main difference between formation in radiation dominated and matter
dominated eras: in the former the PBHs abundance is exponentially sensitive to
the primordial power spectrum $\mathcal{P_{R}}$, while in the second case the
dependence is only a power-law.

Furthermore, in the case of matter era, it should be noted that in order to
have collapse we require that $\delta\sim 1$ and thus only the perturbations
which have time to grow until such value can collapse. Since the linear
density contrast grows as the scale factor during a matter dominated era, only
fluctuations that reach non-linearity before the end of the early matter era
could collapse into a black holes, i.e. only fluctuations for which
$\sigma>\sigma_{nl}=(H_{d}/H_{k})^{2/3}$ could really collapse to form a PBH
Ballesteros  _et al._ (2020).

##  IV Formation in the early matter era

For the early matter era, using Eq.(20) and inserting the numerical values, we
obtain the following dark matter abundance

$f^{MD}_{pbh}\sim\left(\frac{\beta_{0}}{5.5\times
10^{-15}}\right)\left(\frac{T_{d}}{10^{5}GeV}\right)\,.$ (27)

Following Conzinu  _et al._ (2020) we have ${H_{d}\over M_{P}}=({H_{1}\over
M_{P}})^{3}$ and $T_{d}\sim(H_{d}M_{p})^{1/2}$, thus requiring $f_{pbh}\sim 1$
in Eq. (27) we obtain

$\beta_{0}\sim 5\times 10^{-29}\left({H_{1}\over M_{p}}\right)^{-3/2}\,,$ (28)

where we have used $\sigma^{2}={4\over 25}P_{\mathcal{R}}$.
Being interested to the implications that PBHs formation has for dark matter,
one can consider firstly PBHs with mass $M_{pbh}\sim
10^{18}-10^{22}\mathrm{g}$ Escrivà  _et al._ (2022); Carr  _et al._ (2021). On
the other hand, if one not assume instantaneous collapse, since the density
contrast grows as the scale factor during matter era, one has that (as
mentioned) only fluctuations with $\sigma>\sigma_{nl}=(H_{d}/H_{k})^{2/3}$
reach non-linearity during the matter dominated era, while fluctuations with
$\sigma<\sigma_{nl}$ will not collapse.

###  IV.1 Collapse in matter dominated era neglecting $\sigma_{nl}$

As described above, in the case of formation in matter era we have two
different relations for the abundance of PBHs. Firstly, neglecting the
evolution of fluctuations during the matter era (i.e. neglecting
$\sigma_{nl}$), we can consider the two cases:

  * •

Case A: $\sigma>\sigma_{ang}$ .

Requiring all dark matter in form of PBHs, we obtain

$P_{\mathcal{R}}>10^{-10}\left({H_{1}\over M_{p}}\right)^{-3/5}\,,$ (29)

  * •

Case B: $\sigma<\sigma_{ang}$.

Requiring all dark matter in form of PBHs, we obtain

$P_{\mathcal{R}}>\frac{7\times 10^{-4}}{W\left(6\times
10^{5}\left(\frac{H_{1}}{M_{p}}\right)^{1/2}\right)^{3}}\,,$ (30)

###  IV.2 Collapse in matter dominated era taking into account non-linearity

Now we take into account the evolution of fluctuations during the matter era.
When $\sigma>\sigma_{ang}$, the rotation of the collapsing region can be
neglected, and the reach of linearity is expressed by $\sigma>\sigma_{nl}$;
while when $\sigma<\sigma_{ang}$ the angular momentum becomes important and
the constraint is stronger Ballesteros  _et al._ (2020). As a consequence, we
obtain the following constraints, that should be added to those obtained from
$\beta$

$\displaystyle\sigma>\sigma_{nl}$ $\displaystyle=\left({H_{d}\over
H_{k}}\right)^{2/3}\,,\leavevmode\nobreak\ \qquad\sigma>\sigma_{ang}\,,$ (31a)
$\displaystyle\sigma>\sigma_{nl}$ $\displaystyle=\left(5{H_{d}\over
2\mathcal{I}H_{k}}\right)\,,\qquad\sigma<\sigma_{ang}\,.$ (31b)

We finally obtain the following cases:

  * •

Case A: $\sigma>\sigma_{ang}$.

With respect to the previous case A we obtain the new condition

$P_{\mathcal{R}}>{25\over 4}\left({M_{pbh}\over
M_{p}}\right)^{4/3}\left({H_{1}\over M_{p}}\right)^{4}\,.$ (32)

  * •

Case B: $\sigma<\sigma_{ang}$.

In this case, we have the new condition for the power spectrum

$P_{\mathcal{R}}>\left({25\over 4}\right)^{2}\left({M_{pbh}\over
M_{p}}\right)^{2}\left({H_{1}\over M_{p}}\right)^{6}\,.$ (33)

The results above are shown in Fig. 1 where we show the parameter space (in
blue) obtained by the typical constraints of the particular model of pre-big
bang chosen Conzinu  _et al._ (2020); Gasperini (2016) and the region of
parameter space (in orange) compatible with a production of PBHs that gives
all the dark matter ($f\sim 1$). We note that there is not a huge
superposition, in particular for the case b ($\sigma<0.005$.)

Production in matter era neglecting
non-linear evolution

Production in matter era with non-linear evolution

Figure 1:  We show the production of PBHs (in orange) in matter era with and
without the non-linear evolution (in the right and in the left panel
respectively) for the case A (top) and B (bottom) for a mass of
$10^{20}\mathrm{g}$ at varying axion sound speed (in orange). The parameter
space (in light blue) is given in terms of $z_{s}=\tau_{s}/\tau_{1}$ and
$g_{s}/g_{1}=z_{s}^{-\alpha}$.

###  IV.3 Issues with too-light-PBHs

One of the main constraints for PBHs comes from Hawking radiation Carr  _et
al._ (2021), in particular PBHs with $m<10^{15}\mathrm{g}$ should already be
evaporated, with correspondent observable signatures. As a consequence, those
light PBHs cannot be associated to dark matter. Moreover, strong constraints
are given by the effects that the evaporated particles could have on the big-
bang nucleosynthesis Escrivà  _et al._ (2022); Carr  _et al._ (2021). So, one
should check if in the context of the pre-big bang scenario the production of
light PBHs is too efficient. Actually this seem the case, at least in a first
analysis. Indeed, let us follow the same analysis above for light masses
$10^{10}<m<10^{19}\mathrm{g}$. We show both the case of formation in radiation
era Fig.2 and in matter era Fig.3. We can see that, in particular for the case
of matter era, there is a huge superposition in the parameters space that
grows for smaller masses. However, such issue can be addressed in many ways.
One for all, in a more realistic scenario, the sound speed depends on the
modes and it changes differently for each mode considered. We then have that
the formation of lighter PBHs is related to high frequency modes that exit
from the horizon closer to the end of the string phase. During the end of such
string phase other corrections (even non-perturbative ones) should be taken
into account to produce a smooth transition. Such corrections can change
drastically the sound speed, possible toward the standard unitary value,
stopping the productions of PBHs and therefore addressing the above issue.

Production of light PBHs in radiation epoch

Figure 2: We show the production of light PBHs at varying mass (mass expressed
in grams) in the radiation era. The parameters space (in light blue) is given
in terms of $z_{s}=\tau_{s}/\tau_{1}$ and $g_{s}/g_{1}=z_{s}^{-\alpha}$.

Production of light PBHs in matter epoch

Figure 3: We show the production of light PBHs at varying mass (mass expressed
in grams) in the matter era. The parameter space (in light blue) is given in
terms of $z_{s}=\tau_{s}/\tau_{1}$ and $g_{s}/g_{1}=z_{s}^{-\alpha}$.

##  V Conclusions

In this letter we show how the production of PBHs, in the pre-big bang
scenario, described previously in Conzinu  _et al._ (2020), can be extended
also to the case of formation in a early matter dominated era. This early
matter era it is needed in order to produce the right amount of scalar
perturbations by the curvaton mechanism. We show that in this case there is
not a huge production of PBHs in the proper range of mass needed to explain
dark matter (see Fig.1). However, it seems that a large production of light
PBHs ($m<10^{18}g$) can happens both in the matter and radiation dominated
eras (see Figs. 2 and 3). As already discussed, this last eventuality could be
avoided in several ways, for example adding new higher-order corrections (even
non-perturbative) or with a more realistic model of the sound speed variation,
in which such variation depends on the particular mode considered. We postpone
this analysis to future works.

## Acknowledgements

We are very thankful to Maurizio Gasperini for useful discussions and
comments, and for feedback on the manuscript. We are supported in part by INFN
under the program TAsP (Theoretical Astroparticle Physics).

## References

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  * Escrivà  _et al._ (2022) A. Escrivà, F. Kuhnel, and Y. Tada, (2022), arXiv:2211.05767 [astro-ph.CO] .
  * Harada  _et al._ (2016) T. Harada, C.-M. Yoo, K. Kohri, K.-i. Nakao, and S. Jhingan, Astrophys. J. 833, 61 (2016), arXiv:1609.01588 [astro-ph.CO] .
  * Harada  _et al._ (2017) T. Harada, C.-M. Yoo, K. Kohri, and K.-I. Nakao, Phys. Rev. D 96, 083517 (2017), [Erratum: Phys.Rev.D 99, 069904 (2019)], arXiv:1707.03595 [gr-qc] .
  * Gasperini and Veneziano (2003) M. Gasperini and G. Veneziano, Phys. Rept. 373, 1 (2003), arXiv:hep-th/0207130 .
  * Conzinu  _et al._ (2020) P. Conzinu, M. Gasperini, and G. Marozzi, JCAP 08, 031 (2020), arXiv:2004.08111 [gr-qc] .
  * Carr (1975) B. J. Carr, Astrophys. J. 201, 1 (1975).
  * Carr and Hawking (1974) B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168, 399 (1974).
  * Carr and Kuhnel (2020) B. Carr and F. Kuhnel, Ann. Rev. Nucl. Part. Sci. 70, 355 (2020), arXiv:2006.02838 [astro-ph.CO] .
  * Carr  _et al._ (2021) B. Carr, K. Kohri, Y. Sendouda, and J. Yokoyama, Rept. Prog. Phys. 84, 116902 (2021), arXiv:2002.12778 [astro-ph.CO] .
  * Sasaki  _et al._ (2018) M. Sasaki, T. Suyama, T. Tanaka, and S. Yokoyama, Class. Quant. Grav. 35, 063001 (2018), arXiv:1801.05235 [astro-ph.CO] .
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  * Misner  _et al._ (1973) C. W. Misner, K. S. Thorne, and J. A. Wheeler,  _Gravitation_ (W. H. Freeman, San Francisco, 1973).
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# Comment on ”Reversible superconducting-normal phase transition in a magnetic
field and the existence of topologically protected loop currents that appear
and disappear without Joule heating” by Hiroyasu Koizumi

A.V. Nikulov  Institute of Microelectronics Technology and High Purity
Materials, Russian Academy of Sciences, 142432 Chernogolovka, Moscow District,
RUSSIA.

##  I Introduction

The article Koizumi2020EPL  is based on a not quite correct understanding of
the difference between reversible and irreversible thermodynamic processes.
The author Koizumi2020EPL  rightly writes at the beginning of the article that
the superconducting-normal transition in the presence of a magnetic field was
considered as irreversible before 1933: ”At that time it was assumed that the
transition in a magnetic field is substantially irreversible, since the
superconductor was considered as a perfect conductor (in the sense which was
discussed in Chapter II), in which, when the superconductivity is destroyed,
the surface current associated with the field are damped with the generation
of Joule heat” Shoenberg1952 . But all physicists began to consider this
transition as a reversible thermodynamic process Shoenberg1952  after the
discovery of the Meissner effect in 1933 Meissner1933 .

##  II Reversible thermodynamic processes are described with the help of the
free energy

A transition is reversible if the superconducting state and the normal state
have the same free energy $F_{s}=F_{n}$ at the point of the transition
$T_{c}$. The free energy used for the description of this transition equals
$F=H-ST$ where $H$ is the enthalpy (the sum of the system’s internal energy)
of normal or superconducting state, $T$ is the absolute temperature, and $S$
is the entropy of normal or superconducting state. All physicists believe
after 1933 that the phase transition is observed at the temperature $T=T_{c}$
since $F_{s}>F_{n}$ at $T>T_{c}$ and $F_{s}<F_{n}$ at $T<T_{c}$. According to
the point of view of all physicists Shoenberg1952 ; Huebener ; Tinkham , the
free energy of the superconducting state increases in the magnetic field $H$
by the magnetic field energy equal $F_{m}=\mu_{0}H^{2}/2$ per the unit volume
of the bulk superconductor $F_{s}(T,H)=F_{s}(T)+\mu_{0}H^{2}/2$, whereas the
energy of the normal state does not change $F_{n}(T,H)=F_{n}(T)$. Therefore
the equality

$F_{s}(T)+F_{m}=F_{s}(T)+\frac{\mu_{0}H_{c}^{2}}{2}=F_{n}(T)$ $None$

was postulated. $H_{c}(T)$ is the critical field above which superconductivity
disappears Shoenberg1952 ; Huebener ; Tinkham . The temperature dependence
$H_{c}(T)$ is determined by the temperature dependence of the free energy
difference $F_{n}(T)-F_{s}(T)$ without the magnetic field energy. The entropy
$S=-dF/dT$ changes by jump

$S_{n}-S_{s}=-\mu_{0}H_{c}\frac{dH_{c}}{dT}$ $None$

at the first-order phase transition observed in magnetic field $H_{c}$
Shoenberg1952 .

The author Koizumi2020EPL  assumes that the superconducting-normal phase
transition can be reversible if the kinetic energy of the supercurrent is
transferable to the magnetic field energy without dissipation. This energy
transfer is impossible not only within the standard theory of
superconductivity, but also in principle, because of the two reason: 1) both
the kinetic energy
$F_{k}=V^{-1}\int_{V}dVn_{s}mv^{2}/2=V^{-1}\int_{V}dV\mu_{0}\lambda_{L}^{2}j^{2}/2$
and the magnetic field energy increase during the transition from the normal
to the superconducting state; 2) the kinetic energy is not considered in
textbooks Shoenberg1952 ; Huebener ; Tinkham  since its value

$F_{k}=\mu_{0}H_{c}^{2}\frac{\lambda_{L}}{R}$ $None$

is much smaller than the magnetic field energy $F_{m}=\mu_{0}H^{2}/2$ of a
bulk cylinder with a macroscopic radius $R\gg\lambda_{L}$, where
$\lambda_{L}=(m/\mu_{0}q^{2}n_{s})^{0.5}$ is the London penetration depth;
$q=2e$ is the charge of superconducting pair; $n_{s}$ is the density of
superconducting pairs. If the reversible process would imply $\Delta
F_{m}+\Delta F_{k}=0$, as the author Koizumi2020EPL  claims, then the
superconducting-normal transition could not be reversible in principle. This
transition is considered as reversible in all books Shoenberg1952 ; Huebener ;
Tinkham  because of the validity of the equality (1). The enthalpy difference
equal $H_{n}-H_{s}=\mu_{0}H_{c}^{2}/2+(S_{n}-S_{s})T$ at $H=H_{c}(T)$
according to (1) can be found from the experimental results. The entropy
difference is positive $S_{n}-S_{s}>0$ since $dH_{c}/dT<0$ Shoenberg1952 .
Thus, any theory of superconductivity should explain why the enthalpy
decreases on the value $H_{n}-H_{s}=\mu_{0}H_{c}^{2}/2-T\mu_{0}H_{c}dH_{c}/dT$
in order to describe the normal-superconducting transition as a reversible
thermodynamic process.

The enthalpy decreases due to electron pairing according to the BCS theory
BCS1957 . The author Koizumi2020EPL  does not consider the enthalpy change
because of his wrong assumption that the kinetic energy (3) is transferable to
the magnetic field energy (1) during reversible superconducting-normal phase
transition. He does not consider also the change of the latent heat
$T(S_{n}-S_{s})=-T\mu_{0}H_{c}dH_{c}/dT$ at this first-order phase transition
which is a reversible thermodynamic process as any phase transition.

The author Koizumi2020EPL  claims, following Hirsch Hirsch2020EPL ;
Hirsch2020ModPhys ; Hirsch2020Physica , that the conventional BCS-London
theory of superconductivity cannot explain the normal-superconducting
transition because of Joule heating. Hirsch states that ”the conventional
theory of superconductivity predicts that Joule heat is generated”
Hirsch2020EPL . But such direct prediction is absent in the BCS theory BCS1957
. The presence of such prediction in the theory created within the framework
of equilibrium thermodynamics would mean that its authors did not know that
Joule heating is an irreversible thermodynamic process. Each physicist must
know that the equality of the free energy (1) cannot be valid for an
irreversible thermodynamic process.

The author Koizumi2020EPL  did not quite understand Hirsch’s arguments
correctly. Hirsch argues that Joule heating occurs at the transition from
superconducting to normal state of a metal in a magnetic field
Hirsch2020Physica . ”Joule heating is a non-equilibrium dissipative process
that occurs in a normal metal when an electric current flows, in an amount
proportional to the metal’s resistance” Hirsch2020Physica . The electric
current flows during a relaxation time in the normal state with a non-zero
resistance after the transition from the superconducting state in the magnetic
field $H>H_{c}$. Thus, the superconducting-normal transition in the magnetic
field $H_{c}$ cannot be considered as a reversible thermodynamic process
according to Hirsch’s opinion Hirsch2020Physica  and in contrast to the
opinion of the author Koizumi2020EPL .

##  III Gauge theory of the vector potential

The author Koizumi2020EPL  makes a mistake when he uses the London equation
London1950  and the Ginzburg-Landau theory GL1950 . The distinction, used in
Koizumi2020EPL , between the vector potential $A$ in the London equation and
the ordinary electromagnetic vector potential $A^{em}$ has no sense because of
the mathematic equality

$\nabla\times\nabla\theta=(\frac{d^{2}\theta}{dydz}-\frac{d^{2}\theta}{dzdy})i_{x}+\cdot\cdot\cdot\equiv
0$ $None$

which is correct for any function $\theta$. The right-hand side of eq. (16) in
Koizumi2020EPL  is identical to zero because of the mathematical identity (4)
rather than because the term with $\nabla\times\nabla\theta$ is often omitted,
as the author Koizumi2020EPL  thinks.

The term $\nabla\theta$ makes sense in the Ginzburg-Landau equation

$j=\frac{q}{m}n_{s}(\hbar\nabla\theta-qA)$ $None$

only if the unambiguity condition of the wave function
$\Psi_{GL}=|\Psi_{GL}|\exp{i\theta}=|\Psi_{GL}|\exp{i(\theta+2\pi n)}$ is used
Huebener . According to this condition $\oint_{l}dl\nabla\theta=n2\pi$ and the
quantization of the density of superconducting current

$\mu_{0}\oint_{l}dl\lambda_{L}^{2}j+\Phi=n\Phi_{0}$ $None$

along a closed path $l$ is deduced. Here $\oint_{l}dlA=\Phi$ is the magnetic
flux inside $l$; $n$ is an integer number and $\Phi_{0}=2\pi\hbar/q$ is the
flux quantum. The quantization of the magnetic flux $\Phi=n\Phi_{0}$ is
observed fluxquan1961  when $\oint_{l}dlj=0$ along $l$. The Meissner effect
$\Phi=0$ is observed when the wave function has no singularity inside $l$ and
therefore $n=0$

###### Acknowledgements.

This work was made in the framework of State Task No 075-00355-21-00.

## References

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