56.9055pt-\left(\int_{\pi/4}^{3\pi/4}C^{2}\frac{(2i+1)^{\frac{\nu+1}{2}}}{(i+1)^{2}}|\cos(\phi)|^{\nu}\sin(\phi)d\phi\right)^{1/2}$
$\displaystyle\geq Ci^{\nu/4}\left(J(i)^{1/2}-i^{-3/4}\right),$

where $C^{\prime}>0$ is another generic constant and

$J(i)=\int_{\pi/4}^{3\pi/4}\sin^{2}\left(\frac{3\pi}{4}+(\frac{i}{2}+\frac{1}{4})(\sin(2\phi)-2\phi)\right)|\cos(\phi)|^{\nu}d\phi.$

We can write the latter integral as

$J(i)=\frac{1}{2}\int_{\pi/4}^{3\pi/4}|\cos(\phi)|^{\nu}d\phi-\frac{1}{2}\int_{\pi/4}^{3\pi/4}\cos\left(3\pi+2(i+\frac{1}{2})(\sin(2\phi)-2\phi)\right)|\cos(\phi)|^{\nu}d\phi.$

The change of variable $\theta=\sin(2\phi)-2\phi$ and an integration by part
ensures that the second term above goes to $0$ as $i\to\infty$. Therefore,
there exists $i_{0}=i_{0}(\nu)\geq 1$ such that for all $i\geq i_{0}$:

$\displaystyle\left(\int_{\mathbb{R}}|\psi_{i}(x)|^{2}|x|^{\nu}dx\right)^{1/2}$
$\displaystyle\geq
Ci^{\nu/4}\left(\left(\frac{1}{4}\int_{\pi/4}^{3\pi/4}|\cos(\phi)|^{\nu}d\phi\right)^{1/2}-i^{-3/4}\right)\geq
C^{\prime}i^{\nu/4}.$

Up to the modification of the constant, we may gather in the same inequality
the remaining terms associated to $i<i_{0}$, to obtain for all $i\geq 1$ and
for some $c_{\nu}>0$:

$\int_{\mathbb{R}}|\psi_{i}(x)|^{2}|x|^{\nu}dx\geq~{}c_{\nu}i^{\nu/2}.$

∎

######  Lemma 5.9.

Let $\alpha\geq 0$ and $(\psi_{i})_{i\in I}$ be a family of Hermite wavelets
given by (20). Then, there exist two constants $0<c\leq C<\infty$ such that:

$c\sum_{i\in I}|i|^{\alpha/2}\leq\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}|k|^{\alpha}dk\leq
C\sum_{i\in I}|i|^{\alpha/2},$ (43) $\sum_{i_{1},i_{2}\in
I}\left(\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)|k|^{\alpha}dk\right)^{2}\leq
C\sum_{i\in I}|i|^{\alpha}.$ (44)

###### Proof.

If $I$ does not contain $0$, then the inequalities in (43) are obtained by
summing over $i\in I\setminus\\{0\\}$ the inequalities (42) of Lemma 5.8,
since for all $k$, $|\mathcal{F}[\psi_{i}](k)|=|\psi_{i}(k)|$. If $I$ contains
$0$, we start from these inequalities and we simply add the term $i=0$:

$\displaystyle c_{0}+c\sum_{i\in
I\setminus\\{0\\}}|i|^{\alpha/2}\leq\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}|k|^{\alpha}dk\leq
C\sum_{i\in I\setminus\\{0\\}}|i|^{\alpha/2}+c_{0},$

where
$c_{0}=\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{0}](k)|^{2}|k|^{\alpha}dk$. The
latter upper-bound is less than $(C+c_{0})\sum_{i\in I}|i|^{\alpha/2}$ since
$1\leq\sum_{i\in I\setminus\\{0\\}}|i|^{\alpha/2}$, yielding the upper-bound
of (43). The lower-bound in (43) is in turn obviously deduced.

Concerning inequality (44), using the change of variable $k\leftrightarrow-k$,
we note that

$\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)|k|^{\alpha}dk\in\mathbb{R},$

so the statement makes sense, and we have:

$\displaystyle\sum_{i_{1},i_{2}\in
I}\left(\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)|k|^{\alpha}dk\right)^{2}$
$\displaystyle\leq\sum_{i_{1}\in
I}\sum_{i_{2}\in\mathbb{N}^{d}}\left\langle\mathcal{F}[\psi_{i_{1}}](\cdot)|\cdot|^{\alpha},\mathcal{F}[\psi_{i_{2}}](\cdot)\right\rangle^{2}.$

We view the latter angle brackets as the coefficients of the function
$\mathcal{F}[\psi_{i_{1}}](\cdot)|\cdot|^{\alpha}$ in the orthonormal basis
$(\mathcal{F}[\psi_{i_{2}}])_{i_{2}\in\mathbb{N}^{d}}$, so that the sum over
$i_{2}$ above is nothing else than
$\|\mathcal{F}[\psi_{i_{1}}](\cdot)|\cdot|^{\alpha}\|_{2}^{2}$. Thereby,

$\displaystyle\sum_{i_{1},i_{2}\in
I}\left(\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)|k|^{\alpha}dk\right)^{2}$
$\displaystyle\leq\sum_{i_{1}\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i_{1}}](k)|^{2}|k|^{2\alpha}dk.$

The conclusion follows from the fact that for all $k$,
$|\mathcal{F}[\psi_{i_{1}}](k)|=|\psi_{i_{1}}(k)|$ and Lemma 5.8. ∎

######  Lemma 5.10.

Let $r>0$ and $i\in\mathbb{N}^{d}$, then:

$\int_{\mathbb{R}^{d}\setminus[-r,r]^{d}}|\psi_{i}(x)|^{2}dx\leq\frac{2d}{\pi^{1/4}}e^{|i|_{\infty}}\int_{y\geq
r-\sqrt{2|i|_{\infty}}}e^{-y^{2}/2}dy.$

###### Proof.

With Equation (1.2) of [81], $\forall
y\in\mathbb{N},~{}|H_{n}(y)|\leq\pi^{-1/4}\exp(\sqrt{2n}|y|),$ we obtain that
for $i=(i_{1},\dots,i_{d})\in\mathbb{N}^{d}$,
$x=(x_{1},\dots,x_{d})\in\mathbb{R}^{d}$ and $1\leq l_{0}\leq d$:

$\displaystyle|\psi_{i}(x)|$
$\displaystyle=e^{-\frac{1}{2}|x|^{2}}|H_{i_{l_{0}}}(x_{l_{0}})|\prod_{l=1,l\neq
l_{0}}^{d}|H_{i_{l}}(x_{l})|$ $\displaystyle\leq
e^{-\frac{1}{2}x_{l_{0}}^{2}}\pi^{-1/4}\exp(\sqrt{2i_{l_{0}}}|x_{l_{0}}|)\prod_{l=1,l\neq
l_{0}}^{d}|H_{i_{l}}(x_{l})|e^{-\frac{1}{2}x_{l}^{2}}.$

Using the fact that $H_{i_{l}}(x_{l})e^{-\frac{1}{2}x_{l}^{2}}$ has a
$L^{2}(\mathbb{R})$ norm equal to $1$ and the inequality:

$1-\prod_{i=1}^{d}\mathbf{1}_{|x_{l}|\leq
r}\leq\sum_{l=1}^{d}\mathbf{1}_{|x_{l}|>r},$

we get:

$\displaystyle\int_{\mathbb{R}^{d}\setminus[-r,r]^{d}}|\psi_{i}(x)|^{2}dx$
$\displaystyle\leq\sum_{l_{0}=1}^{d}\int_{\mathbb{R}^{d}}\mathbf{1}_{|x_{l_{0}}|\geq
r}|\psi_{i}(x)|^{2}dx\leq\frac{1}{\pi^{1/4}}\sum_{l=1}^{d}\int_{|y|\geq
r}e^{-\frac{1}{2}y^{2}}e^{\sqrt{2i_{l}}|y|}dy.$

To conclude, we use the changes of variable $y\leftrightarrow
y-\sqrt{2i_{l}}$:

$\displaystyle\int_{\mathbb{R}^{d}\setminus[-r,r]^{d}}|\psi_{i}(x)|^{2}dx$
$\displaystyle\leq 2\pi^{-1/4}\sum_{l=1}^{d}e^{i_{l}}\int_{y\geq
r}\exp\left(-\frac{1}{2}\left(y-\sqrt{2i_{l}}\right)^{2}\right)dy$
$\displaystyle=2\pi^{-1/4}\sum_{l=1}^{d}e^{i_{l}}\int_{y\geq
r-\sqrt{2i_{l}}}e^{-y^{2}/2}dy$
$\displaystyle\leq\frac{2d}{\pi^{1/4}}e^{|i|_{\infty}}\int_{y\geq
r-\sqrt{2|i|_{\infty}}}e^{-y^{2}/2}dy.$

∎

We now turn to the proof of Proposition 3.17

###### Proof of Propositon 3.17..

Let $\epsilon>0$. We want to upper bound:

$p_{R}:=\mathbb{P}\left(\mathbf{1}_{|\Phi_{R}|\geq
1}\log(R)\left|\widehat{\alpha}(I,J,R)-\alpha\right|\geq\epsilon\right).$

According to Equations (11) and (12), we can normalize the sum inside the
logarithm with a quantity close to expectation of the numerator:

$\displaystyle\mathbf{1}_{|\Phi_{R}|\geq
1}\log(R)\left(\widehat{\alpha}(I,J,R)-\alpha\right)$
$\displaystyle=\mathbf{1}_{|\Phi_{R}|\geq 1}\sum_{j\in
J}w_{j}\log\left(\sum_{i\in I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}\right)$
$\displaystyle=\mathbf{1}_{|\Phi_{R}|\geq 1}\sum_{j\in
J}w_{j}\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right).$

We deduce:

$\displaystyle p_{R}$
$\displaystyle\leq\mathbb{P}\left(\mathbf{1}_{|\Phi_{R}|\geq 1}\sum_{j\in
J}|w_{j}|\left|\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right)\right|\geq\epsilon\right)$
$\displaystyle\leq\mathbb{P}\left(\mathbf{1}_{|\Phi_{R}|\geq
1}|w|_{\infty}\sum_{j\in J}\left|\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right)\right|\geq\epsilon\right)$
$\displaystyle\leq\sum_{j\in J}\mathbb{P}\left(\mathbf{1}_{|\Phi_{R}|\geq
1}\left|\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right)\right|\geq\frac{\epsilon}{|J||w|_{\infty}}\right).$

Denoting $\epsilon^{\prime}=(\epsilon/(|J||w|_{\infty}))\wedge 1$, we also
have

$p_{R}\leq\sum_{j\in J}\mathbb{P}\left(\mathbf{1}_{|\Phi_{R}|\geq
1}\left|\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right)\right|\geq\epsilon^{\prime}\right).$

Since $\epsilon^{\prime}>0$, we can rewrite the previous bound as:

$p_{R}\leq\sum_{j\in J}\mathbb{P}\left[\left|\log\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi,R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}\right)\right|\geq\epsilon^{\prime},|\Phi_{R}|\geq
1\right].$

According to the mean value inequality, for all $y\in\mathbb{R}$ such that
$|y-1|\leq\delta$, for some $\delta\leq 1/2$, then $|\log(y)|\leq 2\delta$.
The contraposition of this result for $\delta=\epsilon^{\prime}/2$ implies:

$\displaystyle p_{R}$ $\displaystyle\leq\sum_{j\in
J}\mathbb{P}\left[\left|\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi,R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}-1\right|\geq\frac{\epsilon^{\prime}}{2},|\Phi_{R}|\geq
1\right]$ $\displaystyle\leq\sum_{j\in
J}\mathbb{P}\left[\left|\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi,R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}-1\right|\geq\frac{\epsilon^{\prime}}{2}\right].$

Using Markov inequality, we obtain:

$p_{R}\leq\frac{4}{\epsilon^{\prime 2}}\sum_{j\in
J}\mathbb{E}\left[\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi,R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}-1\right)^{2}\right]=\frac{4}{\epsilon^{\prime
2}}\sum_{j\in J}\mathcal{R}(I,R),$ (45)

where:

$\mathcal{R}(I,R):=\mathbb{E}\left[\left(\frac{\sum_{i\in
I}R^{(\alpha-d)j}T_{j}(\psi,R)^{2}}{\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk}-1\right)^{2}\right].$

To bound $\mathcal{R}(I,R)$, we split it between a bias term and a variance
term:

$\displaystyle\mathcal{R}(I,R)$ $\displaystyle=\frac{1}{\left(\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk\right)^{2}}\left(B(I,R)^{2}+V(I,R)\right),$
(46)

with

$B(I,R):=\sum_{i\in
I}\mathbb{E}\left[R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}-\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}t|k|^{\alpha}dk\right],$
$V(i,R):=\operatorname{Var}\left[R^{(\alpha-d)j}\sum_{i\in
I}|T_{j}(\psi_{i},R)|^{2}\right].$

We first bound the bias term. We denote as in the proof of Lemma 5.1,
$\psi_{i}^{R^{1-j}}:=\psi_{i}\mathbf{1}_{[-R^{1-j},R^{1-j}]^{d}}$. By (4), and
using the computations leading to Equation (26), we have

$\displaystyle\mathbb{E}\left(R^{(\alpha-d)j}T_{j}(\psi_{i},R)^{2}\right)$
$\displaystyle=\operatorname{Var}\left(R^{\frac{\alpha-d}{2}j}T_{j}(\psi_{i},R)^{2}\right)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left|\mathcal{F}[\psi_{i}^{R^{1-j}}](k)\right|^{2}R^{\alpha
j}S\left(k/R^{j}\right)dk$
$\displaystyle=\int_{\mathbb{R}^{d}}\left|\mathcal{F}[\psi_{i}](k)\right|^{2}R^{\alpha
j}S\left(k/R^{j}\right)dk+\delta_{1}(i,R)+\delta_{2}(i,R),$ (47)

where

$\displaystyle\delta_{1}(i,R)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\mathcal{F}[\psi_{i}^{R^{1-j}}](k)-\mathcal{F}[\psi_{i}](k)\right)\overline{\mathcal{F}[\psi_{i}^{R^{1-j}}](k)}R^{\alpha
j}S\left(k/R^{j}\right)dk,$ $\displaystyle\delta_{2}(i,R)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\overline{\mathcal{F}[\psi_{i}^{R^{1-j}}]}(k)-\overline{\mathcal{F}[\psi_{i}]}(k)\right)\overline{\mathcal{F}[\psi_{i}](k)}R^{\alpha
j}S\left(k/R^{j}\right)dk.$

Using, as in (5.1), the Cauchy-Schwarz inequality and Plancherel Theorem, we
obtain:

$\displaystyle|\delta_{1}(i,R)|+|\delta_{2}(i,R)|$ $\displaystyle\leq
2\|S\|_{\infty}R^{\alpha
j}\|\psi_{i}\mathbf{1}_{\mathbb{R^{d}}\setminus[-R^{1-j},R^{1-j}]^{d}}\|_{2}\|\psi_{i}\|_{2}$
$\displaystyle=2\|S\|_{\infty}R^{\alpha
j}\|\psi_{i}\mathbf{1}_{\mathbb{R^{d}}\setminus[-R^{1-j},R^{1-j}]^{d}}\|_{2}.$
(48)

Accordingly, gathering (5.6) and (5.6), the bound on the bias term is
decomposed as

$\displaystyle B(I,R)$ $\displaystyle\leq
B_{1}(I,R)+2\|S\|_{\infty}B_{2}(I,R),$

with

$\displaystyle B_{1}(I,R)$ $\displaystyle:=\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}R^{\alpha
j}\left|S(k/R^{j})-t|k/R^{j}|^{\alpha}\right|dk,$ $\displaystyle
B_{2}(I,R):=\sum_{i\in I}R^{\alpha
j}\|\psi_{i}\mathbf{1}_{\mathbb{R^{d}}\setminus[-R^{1-j},R^{1-j}]^{d}}\|_{2}.$

Concerning the first term, using the assumption that $|S(k)-t|k|^{\alpha}|\leq
C_{S}|k|^{\beta}$ with $\beta>\alpha$ and $C_{S}>0$, and then Lemma 5.9, we
get, for some $C>0$,

$\displaystyle B_{1}(I,R)\leq\sum_{i\in
I}\int_{\mathbb{R}^{d}}|\mathcal{F}[\psi_{i}](k)|^{2}R^{(\alpha-\beta)j}C_{S}|k|^{\beta}dk\leq
CR^{(\alpha-\beta)j}\sum_{i\in I}|i|^{\beta/2}.$

Concerning the second term of the bias, according to Lemma 5.10 for
$r=R^{1-j}$:

$\displaystyle B_{2}(I,R)\leq R^{\alpha j}\sum_{i\in
I}\left(\frac{2d}{\pi^{1/4}}e^{|i|_{\infty}}\int_{y\geq
R^{1-j}-\sqrt{2|i|_{\infty}}}e^{-y^{2}/2}dy\right)^{1/2}.$

Note that, for all $(i,j)\in I\times J$, $R^{1-j}-\sqrt{2|i|_{\infty}}\geq
R^{1-j_{\text{max}}}-\sqrt{2|i|_{\infty}}\geq
R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}}>0$ by assumption, so using
standard bound on the complementary error function, see, e.g., [1], we get:

$\displaystyle B_{2}(I,R)$
$\displaystyle\leq\frac{\sqrt{2d}}{\pi^{1/8}}R^{\alpha j}\sum_{i\in
I}e^{|i|_{\infty}/2}\left(\frac{\exp\left(-(R^{1-j}-\sqrt{2|i|_{\infty}})^{2}/2\right)}{R^{1-j}-\sqrt{2|i|_{\infty}}}\right)^{1/2}$
$\displaystyle\leq\frac{\sqrt{2d}}{\pi^{1/8}}R^{\alpha
j_{\text{max}}}|I|e^{i_{\text{max}}/2}\frac{\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/4\right)}{(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{1/2}}.$

Concerning the variance term in (46), we demonstrated in the proof of Theorem
3.9 (see Section 5.1) that all moments of the statistics
$(R^{(\alpha-d)j/2}T_{j}(\psi_{i},R))_{i\in I}$ converge to the moments of
$(N(i))_{i\in I}$ as $R\to\infty$ where $(N(i))_{i\in I}$ is a Gaussian vector
with covariance matrix
$\left(\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)t|k|^{\alpha}dk\right)_{i_{1}\in
I,i_{2}\in I}$. So there exists $R_{0}>1$ such that for all $R\geq R_{0}$:

$\displaystyle V(i,R)$ $\displaystyle\leq 2\operatorname{Var}\left[\sum_{i\in
I}N(i)^{2}\right]=2\sum_{i_{1}\in I,i_{2}\in
I}\operatorname{Cov}[N(i_{1})^{2},N(i_{2})^{2}].$

Moreover, for Gaussian vectors
$\operatorname{Cov}[N(i_{1})^{2},N(i_{2})^{2}]=2(\operatorname{Cov}[N(i_{1}),N(i_{2})])^{2}$,
so we can apply Lemma 5.9 to get, for some $C>0$,

$\displaystyle V(i,R)$ $\displaystyle\leq 4\sum_{i_{1}\in I,i_{2}\in
I}\left(\int_{\mathbb{R}^{d}}\mathcal{F}[\psi_{i_{1}}](k)\overline{\mathcal{F}[\psi_{i_{2}}]}(k)t|k|^{\alpha}dk\right)^{2}\leq
C\sum_{i\in I}|i|^{\alpha}.$

Finally, using again Lemma 5.9 for the denominator of $\mathcal{R}(I,R)$ in
(46), we obtain that there exists $C>0$ such that,

$\displaystyle\mathcal{R}(I,R)$ $\displaystyle\leq
C\left(B_{1}(I,R)^{2}+V(I,R)+B_{2}(I,R)^{2}\right)$ $\displaystyle\leq
C\frac{1}{\left(\sum_{i\in
I}|i|^{\alpha/2}\right)^{2}}\Bigg{(}R^{2(\alpha-\beta)j}\left(\sum_{i\in
I}|i|^{\beta/2}\right)^{2}+\sum_{i\in I}|i|^{\alpha}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+R^{2\alpha
j_{\text{max}}}|I|^{2}e^{i_{\text{max}}}\frac{\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/2\right)}{R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}}}\Bigg{)}.$

Remember that by assumption, $I$ is of the form
$I=\\{i\in\mathbb{N}^{d}|~{}|i|_{\infty}<i_{\text{max}}\\}$ with
$i_{\text{max}}\geq 1$. We then use the facts that for
$\nu\in\\{\alpha/2,\alpha,\beta\\}$,

$\displaystyle\sum_{i\in I}|i|^{\nu}$ $\displaystyle\leq d^{\nu/2}\sum_{0\leq
i_{1},\dots,i_{d}<i_{\text{max}}}(i_{1}^{\nu}+\dots+i_{d}^{\nu})\leq
d^{\nu/2}i_{\text{max}}^{d}i_{\text{max}}^{\nu}=d^{\nu/2}i_{\text{max}}^{d+\nu}$

and

$\displaystyle\sum_{i\in I}|i|^{\nu}$ $\displaystyle\geq\sum_{0\leq
i_{1},\dots,i_{d}<i_{\text{max}}}i_{1}^{\nu}=i_{\text{max}}^{d-1}\sum_{0\leq
i_{1}<i_{\text{max}}}i_{1}^{\nu}\geq\frac{i_{\text{max}}^{d-1}}{\nu+1}i_{\text{max}}^{\nu+1}=\frac{1}{\nu+1}i_{\text{max}}^{d+\nu},$

to obtain for $R\geq R_{0}$, since $|I|=i_{\text{max}}^{d}$,

$\displaystyle\mathcal{R}(I,R)$ $\displaystyle\leq
C\left(R^{2(\alpha-\beta)j}i_{\text{max}}^{\beta-\alpha}+i_{\text{max}}^{-d}+\frac{R^{2\alpha
j_{\text{max}}}|I|^{2}}{i_{\text{max}}^{2d+\alpha}}e^{i_{\text{max}}}\frac{\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/2\right)}{R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}}}\right)$
$\displaystyle\leq
C\left(\left(\frac{|I|^{1/d}}{R^{2{j_{\text{min}}}}}\right)^{\beta-\alpha}+\frac{1}{|I|}+R^{2\alpha
j_{\text{max}}}\frac{e^{|I|^{1/d}}}{|I|^{\alpha/d}}\frac{\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/2\right)}{R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}}}\right).$

Because $j_{\text{max}}<1$, we may choose $R_{0}$ larger, if necessary, to
ensure that for $R\geq R_{0}$:

$R^{2\alpha
j_{\text{max}}}\frac{e^{|I|^{1/d}}}{|I|^{\alpha/d}}\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/2\right)\leq\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/4\right),$

and consequently:

$\displaystyle\mathcal{R}(I,R)\leq
C\left(\left(\frac{|I|^{1/d}}{R^{2{j_{\text{min}}}}}\right)^{\beta-\alpha}+\frac{1}{|I|}+\frac{\exp\left(-(R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}})^{2}/4\right)}{R^{1-j_{\text{max}}}-\sqrt{2i_{\text{max}}}}\right).$

Plugging in this inequality in (45) concludes the proof. ∎

###  5.7 Asymptotic covariance matrix for Hermite wavelets

As announced in Remark 3.15, $\Sigma_{R}$, defined in equation (17), can be
implemented numerically without costly numerical integrations. This section
focuses on the dimension $d=2$, but the method can be adapted in dimension
$d=1$ and $d\geq 3$.

Beyond the explicit formula provided in the next proposition, one may also use
properties of $\Sigma_{R}$ in order to avoid the computation of certain
coefficients. Specifically, denoting
$\Sigma_{R}=(\Sigma_{R}(i_{1},i_{2},j_{1},j_{2}))_{(i_{1},i_{2})\in
I^{2},(j_{1},j_{2})\in J^{2}}$, we have
$\Sigma_{R}(i_{1},i_{2},j_{1},j_{2})=\Sigma_{R}(i_{2},i_{1},j_{2},j_{1})$.
Moreover, denoting $i=(i^{1},i^{2})$ for $i\in\mathbb{N}^{2}$, we have by
parity that $\Sigma_{R}(i_{1},i_{2},j_{1},j_{2})=0$ whenever
$(i_{1}^{1}-i_{2}^{1})$ or $(i_{1}^{2}-i_{2}^{2})$ is odd.

In the next proposition, we denote

$\forall z>0,~{}\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}dt,\quad\forall
p,q\geq 0,~{}B(p,q)=\int_{0}^{2\pi}\cos(\theta)^{p}\sin(\theta)^{q}d\theta.$

Remark that $B(p,q)$ can be easily computed, since on the one hand $B(p,q)\neq
0$ if and only if $p$ and $q$ are even, and on the other hand, for all
$p,q\geq 2$,

$B(p,q)=\frac{(p-1)(q-1)}{(p+q)(p+q-2)}B(p-2,q-2),~{}B(0,p)=B(p,0)=\prod_{l=1}^{p/2}\left(1-\frac{1}{2l}\right).$

######  Proposition 5.11.

Assume $d=2$ and denote
$(\Sigma_{R}(i_{1},i_{2},j_{1},j_{2}))_{(i_{1},i_{2})\in
I^{2},(j_{1},j_{2})\in J^{2}}$ the covariance matrix defined in (17). Then
$\Sigma_{R}(i_{1},i_{2},j_{1},j_{2})$ is given by

$\frac{R^{\frac{\beta+2}{2}(j_{1}+j_{2})}}{2}\sum_{l_{1}^{1},l_{1}^{2},l_{2}^{1},l_{2}^{2}\geq
0}c_{l_{1}^{1}}c_{l_{1}^{2}}c_{l_{2}^{1}}c_{l_{2}^{2}}(-\bm{i})^{|l_{1}|_{1}}\bm{i}^{|l_{2}|_{1}}B(|l_{1}|_{1},|l_{2}|_{1})\Gamma\left(\frac{2+\beta+|l_{1}|_{1}+|l_{2}|_{1}}{2}\right)\\\
\times
R^{j_{1}|l_{1}|_{1}+j_{2}|l_{2}|_{1}}\left(\frac{R^{2j_{1}}+R^{2j_{2}}}{2}\right)^{-(2+\beta+|l_{1}|_{1}+|l_{2}|_{1})/2},$

where $(c_{p})_{p\geq 0}$ are the coefficients of the Hermite polynomial
$H_{p}$.

###### Proof of Proposition 5.11.

Using the fact that Hermite wavelets are eigenvectors of the Fourier
transform, i.e., for any $i\in\mathbb{N}^{2}$,
$\mathcal{F}[\psi_{i}]=~{}(-\bm{i})^{i^{1}+i^{2}}\psi_{i}$ where
$i=(i^{1},i^{2})$, we get:

$\displaystyle\Sigma_{R}(i_{1},i_{2},j_{1},j_{2})$
$\displaystyle=\bm{i}^{i_{2}^{1}+i_{2}^{2}}(-\bm{i})^{i_{1}^{1}+i_{1}^{2}}R^{\frac{\beta+d}{2}(j_{1}+j_{2})}\underbrace{\int_{\mathbb{R}^{2}}\psi_{i_{1}}(R^{j_{1}}k)\psi_{i_{2}}(R^{j_{2}}k)|k|^{\beta}dk}_{=:A}.$

To compute the latter integral, we use polar coordinates:

$\displaystyle A$
$\displaystyle=\int_{\mathbb{R}^{2}}H_{i_{1}^{1}}(R^{j_{1}}k_{1})H_{i_{1}^{2}}(R^{j_{1}}k_{2})e^{-\frac{1}{2}R^{2j_{1}}k_{1}^{2}-\frac{1}{2}R^{2j_{2}}k_{2}^{2}}$
$\displaystyle\qquad\qquad\times
H_{i_{2}^{1}}(R^{j_{2}}k_{1})H_{i_{2}^{2}}(R^{j_{2}}k_{2})e^{-\frac{1}{2}R^{2j_{2}}k_{2}^{2}-\frac{1}{2}R^{2j_{2}}k_{1}^{2}}|k|^{\beta}dk$
$\displaystyle=\displaystyle\int_{0}^{\infty}\int_{0}^{2\pi}H_{i_{1}^{1}}(R^{j_{1}}r\cos(\theta))~{}H_{i_{1}^{2}}(R^{j_{1}}r\sin(\theta))~{}H_{i_{2}^{1}}(R^{j_{2}}r\cos(\theta))~{}H_{i_{2}^{2}}(R^{j_{2}}r\sin(\theta))$
$\displaystyle\qquad\qquad\times\exp\left(-\frac{1}{2}(R^{2j_{1}}+R^{2j_{2}})r^{2}\right)r^{\beta+1}d\theta
dr.$

To get the announced formula, we expand the product of the four polynomials,
perform the change of variable $r\leftrightarrow
r\sqrt{(R^{2j_{1}}+R^{2j_{2}})/2}$ and finally use the fact that for $\nu>0$,
$\int_{0}^{\infty}r^{\nu}e^{-r^{2}}dr=\Gamma((\nu+1)/2)/2$. ∎

## Cumulant and Brillinger mixing for point processes

We recall in this appendix the definition of factorial cumulant moment
measures of a simple point process and their reduced counterpart in case of
stationarity; see also [15, 16]. We then present the property of Brillinger
mixing of a simple stationary point process, as assumed in our Theorem 3.9;
see for instance [34, 5, 7].

As a generalisation of the second order factorial moment measure (1), the
factorial moment measures $\rho^{(r)}$, for $r\geq 1$, of a simple point
process $\Phi$ is defined for any bounded measurable sets $A_{1},\dots,A_{r}$
in $\mathbb{R}^{d}$ by

$\displaystyle\rho^{(r)}(A_{1}\times\ldots\times
A_{r})=\mathbb{E}\left(\sum_{x_{1},\ldots,x_{r}\in\Phi}^{\neq}1\\{x_{1}\in
A_{1},\ldots,x_{m}\in A_{r}\\}\right).$

The $m$-th order factorial cumulant moment measure $\gamma^{(m)}$ are in turn
defined by the following expression:

$\displaystyle\gamma^{(m)}(A_{1}\times\dots\times
A_{m})=\sum_{\sigma\in\Pi[p]}(-1)^{|\sigma|-1}(|\sigma|-1)!\prod_{i=1}^{|\sigma|}\rho^{(|\sigma_{i}|)}\left(\prod_{k_{i}\in\sigma(i)}A_{k_{i}}\right),$

where, as in (31), $\Pi[m]$ denotes the set of all unordered partitions of
$\\{1,\ldots,m\\}$.

Furthermore, when the point process $\Phi$ is stationary, we may consider for
any $m\geq 2$ the reduced $m$-th order factorial cumulant moment measure
$\gamma_{\text{red}}^{(m)}$. This is a locally finite signed measure on
$(\mathbb{R}^{d})^{(m-1)}$ that satisfies

$\gamma^{(m)}\left(A_{1}\times\dots\times
A_{m}\right)=\int_{A_{m}}\gamma_{\text{red}}^{(m)}\left(\prod_{i=1}^{m-1}(A_{i}-x)\right)\,dx,$

where for $i=1,\ldots,m-1$, $(A_{i}-x)$ is the translation of the set $A_{i}$
by $x$.

Finally, a simple stationary point process $\Phi$ is said Brillinger-mixing
if, for $m\geq 2$, and any bounded measurable set $A$ of $\mathbb{R}^{d}$,
$\mathbb{E}\left[|\Phi\cap A|^{m}\right]<+\infty$ and if
$|\gamma_{\text{red}}^{(m)}|(\mathbb{R}^{d(m-1)})<+\infty,$ where the total
variation of $\gamma_{\text{red}}^{(m)}$ is defined by
$|\gamma_{\text{red}}^{(m)}|:={\gamma_{\text{red}}^{(m)}}^{+}+{\gamma_{\text{red}}^{(m)}}^{-}$
where
$\gamma_{\text{red}}^{(m)}={\gamma_{\text{red}}^{(m)}}^{+}-{\gamma_{\text{red}}^{(m)}}^{-}$
is the Jordan decomposition of the signed measure $\gamma_{\text{red}}^{(m)}$.

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Pleasant behavior of swampland conjectures in the face of specific
inflationary models
S. Noori Gashti⋆111Email<EMAIL_ADDRESS>J.
Sadeghi⋆,†222Email<EMAIL_ADDRESS>B. Pourhassan⋆⋆,†333Email:
<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
⋆⋆School of Physics, Damghan University, Damghan, 3671641167, Iran
†Canadian Quantum Research Center 204-3002 32 Avenue Vernon
British Columbia V1T 2L7 Canada

###### Abstract

Recently researchers have been studying various conditions as swampland
criteria in cosmological implications. They have studied many inflation models
with different swampland conditions. Occasionally these conjectures are
modified and lead to fascinating points. The swampland conjecture is in
contradiction with the slow-roll single-field inflation model. So in this
paper, we will briefly describe resolving this contradiction with the special
method; we consider a small parameter $\lambda$ with respect to a coupling
function in inflation and resolve this important antithesis. Then we will
discuss a new point of view in studying inflation models according to the
swampland criteria. Therefore we introduce some inflation models and challenge
the swampland criteria. We specify the allowable range of cosmological
parameters as the tensor-to-scalar ratio ($r$), the scalar spectral index
($n_{s}$) according to the latest observable data. Next, we study the new
constraints to determine the compatibility or incompatibility of the
corresponding model with the swampland conjectures. Finally, we express the
results and compare them with the latest observational data by plotting some
figures.

Keywords: Swampland criteria, Cosmological parameters, Single-field inflation

###### Contents

  1. 1 Introduction
  2. 2 The basic equations
  3. 3 The Inflation Models
    1. 3.1 Case I
    2. 3.2 Case II
  4. 4 Conclusions

##  1 Introduction

Due to problems in standard cosmology such as horizons, monopole, and
flatness, various solutions have been proposed to address these problems,
including inflation. With science development, new modifications are
accompanied for solutions, and also, there will exist new questions for
researchers. Recent studies in cosmology and other sciences such as particle
physics have led to remarkable developments. Cosmic microwave background (CMB)
measurements that show fluctuations in matter and energy are always unstable
on a large scale. The main cause of these fluctuations is still unknown.
Researchers have provided explanations for the reasons for these fluctuations,
such as the inflationary universe like all cosmological questions and
problems. So, cosmic inflation has been proposed as an amazing pattern for
these problems. To solve these problems, the universe has gone through an
early phase of accelerated expansion [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18]. Different types of these inflation models with
various conditions and their implications in cosmology have been studied
thoroughly. The simplest model is single-field inflation with the slow-roll
condition. It can be shown that inflation is practical for more than one
field. However, the study of single-field inflation models and their
consequences is of more interest to researchers [19, 20, 21, 22, 23, 24].
Recently, the concept of the swampland program has been introduced, which has
some conditions as swampland dS conjecture, distance conjecture, TCC, etc. One
of them is weak gravity conjecture that states gravity is the weakest force in
theories coupled with gravity. On the other hand, a set of theories are placed
in the landscape (an area compatible with quantum gravity), and other
effective theories are usually placed in an area called the swampland, which
is incompatible with quantum gravity. The swampland is wider than the
landscape, and it surrounds the landscape. So most effective theories are
located in the swampland. To adapt to quantum gravity at low energy levels,
they must satisfy the swampland conditions [25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62]. Single-field inflation with the
slow-roll conditions is usually based on effective low-energy theories derived
from Einstein equations. So to be consistent with quantum gravity theories,
they must meet several criteria known as swampland criteria. The swampland
distance conjecture, which normalizes the scalar field excursion with reduced
Planck mass, and the field space is limited from above. Also the refined
swampland dS conjecture, which provides a limit to the potential [46, 47, 48,
49, 50, 51, 63, 64, 65, 66, 67]. Single-field inflation models with slow-roll
and swampland conjectures have a small problem, i.e., there is a contradiction
between single-field slow-roll inflation with one of the swampland criteria,
which we will explain in the following. Researchers have developed a powerful
method for this problem by connecting a coupling as Gauss-Bonnet to a
potential that solves the problems associated with swampland by a special
mechanism using a parameter $(1-\lambda)^{2}$ [68]. Gauss-Bonnet is an
exciting theory and also may even solve problems of the universe’s
singularity. The speed of gravitational waves in such a model differs from the
speed of light. In the last decade, different Gauss-Bonnet structures, and
their implications have been studied in various cosmological topics [69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
90, 91, 92, 93, 94, 95, 96]. However, regarding the swampland conjecture, an
important question is relation of Gauss-Bonnet term with the string theory.
Indeed, Gauss-Bonnet term may reproduced from the first order
$\alpha^{\prime}$ correction of the string theory [97, 98]. Then, the black
hole solutions in the superstring effective field theory with the Gauss-Bonnet
term may considered [99]. Also charged black hole solutions in Einstein-Gauss-
Bonnet theory with the dilaton field as the low-energy effective theory of the
heterotic string has been studied [100].
Important cosmology parameters in inflation are calculated as the scalar
spectral index $n_{s}$, and the tensor-to-scalar ratio ($r$) under the slow-
roll conditions regarding $\epsilon\ll 1$, and $\eta\ll 1$. With the Gauss-
Bennet connection, the slow-roll condition is converted to
$(1-\lambda)(\frac{V^{\prime}}{V})^{2}\ll 1$. So the swampland criteria are
satisfied as long as $(1-\lambda)\ll 1$ and compatible with the slow-roll
condition. The related results are also well-satisfying. Also in various works
in literature determine that the modified theory resolved this like problem.
So in this paper, we examine some inflation models from this perspective, and
we will show this compatibility between the slow-roll condition and swampland
conjecture. Recently in the literature to solve this antithesis, some
solutions have been proposed that are related to the tensor to scalar ratio
and properties of ($\Delta\phi$), you can see the Refs. [101, 102, 103, 104,
105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119,
120, 121, 122, 123, 124, 125, 126, 127, 128]. So this paper is organized as
follows.
In section 2, we will briefly describe the base equation about inflation and
some points related to refined swampland dS conjectures completely. In section
3, we study a new perspective in studying inflation models according to the
swampland criteria. We introduce some inflation models, and we investigate the
important cosmological parameters as $N$, $r$, and $n_{s}$ with respect to the
concept mentioned in this paper. We specify the allowable area of cosmological
parameters such as ($r$), ($n_{s}$), and reconstructed new restrictions as
($C_{1}-r$), ($C_{1}-n_{s}$), ($C_{2}-r$), and ($C_{2}-n_{s}$). We determine
the allowable range for the swampland component corresponding to the models by
plotting some figures and advantaging with the latest observational data.
Finally, we state the result in section 4.

##  2 The basic equations

In this section, we study the basic equation according to the concepts
presented. Hence, we generally express the action in the following form [129,
130, 131, 132, 133, 134, 135, 136, 137],

$S=\frac{1}{2}\int\sqrt{-g}d^{4}x\bigg{(}R-g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-2V(\phi)-\xi(\phi)R^{2}_{GB}\bigg{)},$
(1)

where $\xi(\phi)$, $R^{2}_{GB}$ are Gauss-Bonnet coupling function and Gauss-
Bonnet term, respectively, which are as follows,

$R^{2}_{GB}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(2)

and the Gauss-Bonnet coupling function [68],

$\xi(\phi)=\frac{3\lambda}{4V(\phi)+\Lambda_{0}},$ (3)

where $\Lambda_{0}\ll(10^{16}Gev)^{4}$, so in this paper we ignore the
$\Lambda_{0}$. Also, the observational constraints [3] suggests $0<\lambda<1$
[68]. Indeed, the marginalized $1\sigma-3\sigma$ confidence level contours for
scalar spectral index and tensor-to-scalar ratio from Planck 2015 and
BICEP2/Keck data [3] suggest the constant $\lambda$ of order unity to match
mentioned observations.
The slow-roll parameter satisfy the conditions as
$\epsilon_{1}=-\frac{\dot{H}}{H^{2}}\ll 1$ and
$\epsilon_{2}=\frac{\dot{\epsilon_{1}}}{H\epsilon_{1}}\ll 1$. Also, the number
of e-folds ($N$) at the horizon, before the end of inflation and according to
the potential, is expressed in the following form,

$N\approx\int_{\phi_{e}}^{\phi}\frac{3V}{3V,\phi+4\xi,\phi
V^{2}}d\phi=\int_{\phi_{e}}^{\phi}\frac{d\phi}{Q},$ (4)

where $Q=\frac{V,\phi}{V}+\frac{4\xi,\phi V}{3}$. According to the equation
(4), we will have [68]

$N\approx\int_{\phi_{e}}^{\phi}\frac{1}{\sqrt{2\epsilon_{1}}}d\phi.$ (5)

The scalar spectral index expressed in the following form [68],

$n_{s}=1-2\epsilon_{1}-\epsilon_{2}$ (6)

where the $n_{s}$ is the scalar spectral tilt. Also, the tensor-to-scalar
ratio is as following,

$r=16(1-\lambda)\epsilon_{1}.$ (7)

The above slow-roll parameters can state with the potential term, which is
introduced as follows [138, 139, 140],

$\epsilon_{1}=\frac{1-\lambda}{2}\left(\frac{V^{\prime}}{V}\right)^{2},$ (8)

and the second parameter is as follows,

$\epsilon_{2}=-2(1-\lambda)\left[\frac{V^{\prime\prime}}{V}-(\frac{V^{\prime}}{V})^{2}\right].$
(9)

From the two equations (8) and (9) as well as the matching of potential and
swampland conditions, with respect to ($\lambda$), that be very close to 1,
the slow-roll conditions as $\epsilon_{1}$ and $\epsilon_{2}$ are satisfied.
Therefore, it can be concluded that equations (6) and (7) are correct and
valid. Assuming the number of e-folds is equal to 60. So, from equations (7)
and (8), we will have,

$r=8(1-\lambda)^{2}\left(\frac{V^{\prime}}{V}\right)^{2}.$ (10)

From the equation (10), the second swampland dS conjecture is satisfied, and
the contradiction is resolved. Also, if we consider the $(1-\lambda)$ is very
small, the tensor-to-scalar ratio $r$ can be consistent with observable data.
The first swampland criteria, according to the Lyth bound, is given by [68],

$\Delta\phi=(1-\lambda)\Delta N\frac{V^{\prime}}{V}.$ (11)

If we ignore the $\lambda$, the second swampland criteria are satisfied, but
it is not met for slow-roll single-field inflation. Therefore if the
$(1-\lambda)$ be very small, both of the swampland criteria are satisfied.
Here with respect to the second swampland conjecture, and $\Delta\phi\leq d$,
one can obtain,

$1-\lambda<\frac{d}{c\Delta N}.$ (12)

The remarkable point in the concepts presented in this section is the use of
the Gauss-Bonnet term. With the help of the Gauss-Bonnet term, it satisfies
both swampland criteria as long as the conditions expressed for the $\lambda$
are met. Besides, the tensor-to-scalar ratio ($r$) will be in line with
observable data[3]. Therefore, Given all the above concepts, we calculate each
of the mentioned cosmological quantities in Einstein’s framework for each
model separately by introducing several potentials. By calling swampland
conjectures in this framework, we will examine the results about each of these
variables and the parameter $\lambda$. We compare the results of different
models with each other as well as the latest observable data. We also plot
some figures to determine the range associated with each of these cosmological
parameters.

##  3 The Inflation Models

In this section, we introduce some inflation models as follows, and apply the
concepts expressed in the previous section to these inflation models [141,
142, 143],

$\bullet U(\psi)=V_{0}\bigg{(}1-\bigg{[}\psi/\mu\bigg{]}^{n}\bigg{)}$
$\bullet U(\psi)=V_{0}\tanh^{2n}\bigg{[}\psi/\sqrt{6\alpha}\bigg{]}$

We calculate the cosmological parameters as $r$, $n_{s}$, etc., and determine
the range of each of these parameters by plotting some figures. We also use
the small parameter $\lambda$ to solve the problems associated with single-
field inflation with slow-roll and swampland conditions. Hence, we first
introduce the swampland conjectures, then we describe the results associated
with any potential. As mentioned above, swampland criteria are expressed in
the following form,

$\frac{\Delta\phi}{M_{pl}}<\mathcal{O}(1),$ (13)

and

$\displaystyle M_{pl}\frac{|U^{\prime}|}{U}>C_{1},$
$\displaystyle\frac{min(|\nabla_{i}\nabla_{j}U|)}{U}=M_{pl}^{2}\frac{|U^{\prime\prime}|}{U}\leq-
C_{2}.$ (14)

###  3.1 Case I

As a first case, we consider a hilltop attractor potential, So we have

$U(\psi)=V_{0}\bigg{(}1-\bigg{[}\psi/\mu\bigg{]}^{n}\bigg{)}.$ (15)

With respect to the above equation and swampland conjectures in equation (3),
one can obtain,

$\frac{U^{\prime}}{U}=\frac{n(1+\frac{1}{-1+(\frac{\psi}{\mu})^{n}})}{\psi}>C_{1},$
(16)

and

$\frac{U^{\prime\prime}}{U}=\frac{(-1+n)n(\frac{\psi}{\mu})^{n}}{\psi^{2}(-1+(\frac{\psi}{\mu})^{n})}>-C_{2},$
(17)

where $U^{\prime}$, $U^{\prime\prime}$ are the first and second derivatives of
$\psi$. The tensor-to-scalar ratio and scalar spectral tilt for this model in
Einstein framework can obtain with respect to the above equations as

$r=\frac{8(1-\lambda)^{2}}{\mu^{2}(\mathcal{W}^{\frac{(1-\lambda)-1}{(1-\lambda)-2}}-\mathcal{W}^{-\frac{1}{(1-\lambda)-2}})^{2}}$
(18)

and

$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle
1-\frac{2(1-\lambda)-1}{((1-\lambda)-2)N+(1-\lambda)-1}$ (19) $\displaystyle-$
$\displaystyle\frac{(1-\lambda)\bigg{\\{}(5(1-\lambda)-2)\mathcal{W}^{\frac{1-\lambda}{((1-\lambda)-2)}}-2((1-\lambda)-1)\bigg{\\}}}{\mu^{2}\mathcal{W}\bigg{\\{}-2\mathcal{W}^{\frac{1-\lambda}{((1-\lambda)-2)}}+\mathcal{W}^{\frac{2(1-\lambda)}{((1-\lambda)-2)}}+1\bigg{\\}}},$

where
$\mathcal{W}=\frac{(1-\lambda)[((1-\lambda)-2)N+(1-\lambda)-1]}{\mu^{2}}$.
Here we invert equation (18) in terms of ($r$) and equation (19) in terms of
($n_{s}$) as $N-r$, and $N-n_{s}$. By considering the equations (4), and
(16)-(19) the new expressions will be obtained in terms of ($r$) and
($n_{s}$). So we will have

$\begin{split}&C_{1}-n_{s}=1/\mu 2^{1/n}n\bigg{(}(-1+\lambda^{2})(-2\hskip
5.69054pt_{2}F_{1}\bigg{[}1,-1/2+1/n,1/2+1/n,4\lambda/-1+\lambda\bigg{]}\\\
&+2\hskip 5.69054pt_{2}F_{1}\bigg{[}1,1/n,1+1/n,4\lambda/-1+\lambda\bigg{]}\\\
&-n\hskip
5.69054pt_{2}F_{1}\bigg{[}1,1/n,1+1/n,4\lambda/-1+\lambda\bigg{]}\bigg{/}(-2+n)n(1-4\lambda)\bigg{(}\lambda-\lambda^{2}+4^{-\frac{1+\lambda}{-1+\lambda}}\times\\\
&\bigg{[}\frac{-5+n_{s}+5\lambda+2\lambda
n_{s}+\sqrt{57-42n_{s}+9n_{s}^{2}-18\lambda-90\lambda n_{s}+36\lambda
n_{s}^{2}+25\lambda^{2}-12\lambda^{2}n_{s}+36\lambda^{2}n_{s}^{2}}}{-2+n_{s}+2\lambda
n_{s}}\bigg{]}^{\frac{1+\lambda}{-1+\lambda}}\\\
&\times\mu^{2}\bigg{)}\bigg{)}\end{split}$ (20)

and

$\begin{split}&\mathcal{A}=2\hskip
5.69054pt_{2}F_{1}\bigg{[}1,-\frac{1}{2}+\frac{1}{n},\frac{1}{2}+\frac{1}{n},1+\frac{1}{-1+4\lambda}\bigg{]}+(-2+n)\\\
&\times\hskip
5.69054pt_{2}F_{1}\bigg{[}1,\frac{1}{n},1+\frac{1}{n},1+\frac{1}{-1+4\lambda}\bigg{]}\\\
&\mathcal{B}=(4^{1+\lambda/-1+\lambda}(-1+\lambda)\lambda\\\
&-(\frac{-5+n_{s}+5\lambda+2\lambda
n_{s}+\sqrt{57-6n_{s}(7+\lambda)(1+2\lambda)+\lambda(-18+25\lambda)+9(n_{s}+2n_{s}\lambda)^{2}}}{-2+n_{s}+2\lambda
n_{s}})^{\frac{(1+\lambda)}{(-1+\lambda)}}\mu^{2})\\\
&C_{2}-n_{s}=\bigg{\\{}4^{1/n}(-1+n)n\bigg{(}\frac{2^{2+\frac{4}{-1+\lambda}}(-1+\lambda^{2})\mathcal{A}}{(-2+n)n(-1+4\lambda)\mathcal{B}}\bigg{)}^{-2/n}\times\bigg{(}\bigg{[}\frac{2^{(3+\lambda)/(-1+\lambda)}(-1+\lambda)\mathcal{A}}{(-2+n)n(-1+4\lambda)\mathcal{B}}\bigg{]}^{1/n}\bigg{)}^{n}\bigg{\\}}\\\
&\bigg{/}\bigg{(}\mu^{2}\bigg{[}-1+\bigg{(}\bigg{[}\frac{2^{(3+\lambda)/(-1+\lambda)}(-1+\lambda^{2})\mathcal{A}}{(-2+n)n(-1+4\lambda)\mathcal{B}}\bigg{]}^{1/n}\bigg{)}^{n}\bigg{]}\bigg{)}\end{split}$
(21)

and swampland conjectures in terms of $r$ as,

$\begin{split}&\mathcal{X}=2\hskip
5.69054pt_{2}F_{1}\bigg{[}1,-\frac{1}{2}+\frac{1}{n}+\frac{1}{2}+\frac{1}{n},1+\frac{1}{-1+4\lambda}\bigg{]}+(-2+n)\\\
&\times\hskip
5.69054pt_{2}F_{1}\bigg{[}1,\frac{1}{n}+1+\frac{1}{n},1+\frac{1}{-1+4\lambda}\bigg{]}\\\
&\mathcal{Y}=(-2+n)n(-1+4\lambda)\bigg{[}\sqrt{r}(-1+\lambda)\lambda+2^{3(1+\lambda)/2\lambda}\sqrt{(-1+\lambda)^{2}}(-\frac{\sqrt{(-1+\lambda)^{2}}}{\mu\sqrt{r}})^{1/\lambda}\mu\bigg{]}\\\
&C_{1}-r=1/\mu
2^{1/n}n\bigg{(}\frac{\sqrt{r}(-1+\lambda^{2})\mathcal{X}}{\mathcal{Y}}\bigg{)}^{-1/n}\times\bigg{[}1+\frac{1}{-1+\bigg{(}2^{-1/n}\bigg{[}\frac{\sqrt{r}(-1+\lambda^{2})\mathcal{X}}{\mathcal{Y}}\bigg{]}^{1/n}\bigg{)}^{n}}\bigg{]}\end{split}$
(22)

and

$\begin{split}&\mathcal{G}=-2\hskip
5.69054pt_{2}F_{1}\bigg{[}1,-\frac{1}{2}+\frac{1}{n}+\frac{1}{2}+\frac{1}{n},\frac{4\lambda}{-1+4\lambda}\bigg{]}\\\
&+2\hskip
5.69054pt_{2}F_{1}\bigg{[}1,\frac{1}{n}+1+\frac{1}{n},\frac{4\lambda}{-1+4\lambda}\bigg{]}\\\
&-n\hskip
5.69054pt_{2}F_{1}\bigg{[}1,\frac{1}{n}+1+\frac{1}{n},\frac{4\lambda}{-1+4\lambda}\bigg{]}\\\
&\mathcal{K}=\bigg{(}(-2+n)n(-1+4\lambda)\bigg{[}\lambda-\lambda^{2}+2^{3(1+\lambda)/2\lambda}\bigg{(}-\frac{\sqrt{1-2\lambda+\lambda^{2}}}{\sqrt{r}\mu}\bigg{)}^{(1+\lambda)/\lambda}\mu^{2}\bigg{]}\bigg{)}\\\
&C_{2}-r=\bigg{(}2^{2/n}(1+n)n\bigg{(}\bigg{[}(-1+\lambda^{2})\bigg{\\{}\mathcal{G}\bigg{\\}}\bigg{]}\bigg{/}\mathcal{K}\bigg{)}^{-2/n}\bigg{(}2^{-1/n}\bigg{[}\bigg{(}(-1+\lambda^{2})\bigg{\\{}\mathcal{G}\bigg{\\}}\bigg{)}\bigg{/}\mathcal{K}\bigg{]}^{1/n}\bigg{)}^{n}\bigg{)}\\\
&\bigg{/}\bigg{(}\mu^{2}\bigg{[}-1+\bigg{(}2^{-1/n}\bigg{[}\bigg{(}(-1+\lambda^{2})\bigg{\\{}\mathcal{G}\bigg{\\}}\bigg{)}\bigg{/}\mathcal{K}\bigg{]}^{1/n}\bigg{)}^{n}\bigg{]}\bigg{)}.\end{split}$
(23)

Now, we can determine these cosmological parameters range by plotting some
figures.

Figure 1: The plot of $n_{s}$ in term of $r$.

Figure 2: The plot of ($C_{1}$ and $C_{2}$) in term of ($n_{s}$), and (r) in
Fig. 2, 2 and Fig. 2, 2, respectively.  Figure 3: the constraint of
($C_{2}-C_{1}$).

As you can see in the figures of model I, we first plotted the two important
cosmological parameters; (tensor-to-scalar ratio) $r$ in terms of $n_{s}$ in
figure 1. As shown in figure 1, the allowable areas of these two parameters
are well determined for this model. But the important point that we should
mention is the study of swampland conditions according to important
cosmological parameters such as $r$, and $n_{s}$ with advantaging the
parameter $\lambda$. we will assume $\lambda=0.990$, $\mu<1,\simeq 0.33$,
$n=3$. Figure 2 shows the changes in the swampland conjectures component,i.e.,
($C_{1}$), and ($C_{2}$) in terms of two important cosmological parameters,
($n_{s}$), and ($r$). The changes of ($C_{1}$), and ($C_{2}$) in terms of
($n_{s}$) in figure 2, and 2 are specified. The allowable range of these
parameters is well determined. As shown in the figures, the swampland
conjecture components are within the allowable range. As stated in the
literature, ($C_{1}$) and ($C_{2}$) are positive and unit order. Also, the
values of the $C_{2}$ are smaller than the $C_{1}$. Figures 2 and 2 show the
variations of two components, ($C_{1}$), and ($C_{2}$) in terms of $r$. The
allowable range associated with each of these components is well represented.
Here, the important point is the role of parameter $\lambda$ to resolve the
contradiction between parameter $r$ and swampland conjectures, which has
eliminated this defect well. Using equations (4), (16), (17), and (19), we
will examine another constraint from the perspective of swampland conjectures
and obtain the allowable range or the extent to which the above-inflation
model can be valid in swampland conjectures. We compute the
$C_{1}^{2}C_{2}^{2}$ according to the condition $f>C_{1}^{2}C_{2}^{2}$ to
investigate this constraint. With respect to the latest observable data and
the mentioned condition, the allowable range for swampland conjectures of this
model is specified. In figure 3, we plot the plan ($C_{2}-C_{1}$). The
allowable area for swampland conjectures regarding the model I for the
mentioned constant parameters will be $c<0.000910209$. In the remainder of
this paper, we examine the changes of cosmological parameters with respect to
another potential. We determine the range of these parameters. It should be
noted that the exact values of each of these cosmological parameters, such as
the scalar spectral index ($n_{s}$), the tensor to scale ratio ($r$), etc.,
can be examined. Our main focus is to study the swampland conjecture behavior
for different values and coefficients as $\lambda$.

###  3.2 Case II

The second example of potential is expressed in the following form (T-model
attractor),

$U(\psi)=V_{0}\tanh^{2n}(\frac{\psi}{\sqrt{6\alpha}}).$ (24)

Like the previous part with respect the equations (3) and (24), one can
obtain,

$\frac{U^{\prime}}{U}=\frac{2\sqrt{\frac{2}{3}}nCsch[\frac{\sqrt{\frac{2}{3}}\psi}{\sqrt{\alpha}}]}{\sqrt{\alpha}}>C_{1},$
(25)

and

$\frac{U^{\prime\prime}}{U}=-\frac{4n\bigg{(}-2n+\cosh[\frac{\sqrt{\frac{2}{3}}\psi}{\sqrt{\alpha}}]\bigg{)}(Csch[\frac{\sqrt{\frac{2}{3}}\psi}{\sqrt{\alpha}}])^{2}}{3\alpha}>-C_{2},$
(26)

where $U^{\prime}$, $U^{\prime\prime}$ are the first and second derivatives of
$\psi$. Also, the tensor-to-scalar ratio and scalar spectral tilt, one can
obtain,

$r=\frac{12\alpha}{N^{2}+\frac{N}{2(1-\lambda)}\times\mathcal{M}+3\alpha/4},$
(27)

and

$n_{s}=\frac{1-\frac{2}{N}-\frac{3\alpha}{4N^{2}}+\frac{1}{2(1-\lambda)N}(1-\frac{1}{N})\mathcal{M}}{1+\frac{1}{2(1-\lambda)N}\mathcal{M}+\frac{3\alpha}{4N^{2}}},$
(28)

where $\mathcal{M}=\sqrt{3\alpha\bigg{(}4(1-\lambda)^{2}+3\alpha\bigg{)}}$. As
mentioned in the previous section, we will go through a straightforward
calculations process to challenge each of the components of the refined
swampland dS conjecture, namely $C_{1}$ and $C_{2}$ in terms of two of the
most important cosmological quantities $n_{s}$ and $r$. We will rewrite (27)
and (28) inversely and in terms of functions as $N-n_{s}$ and $N-r$, then like
the previous part with respect to equations (4), and (25-(28), the swampland
conjectures are rewritten according to two very important cosmological
quantities, scalar spectral index and tensor-to-scalar ratio as
$C_{1,2}-n_{s}$ and $C_{1,2}-r$. Now you can see the changes of these
parameters in the following figures, which we will analyze the results. After
the above calculations, we specify the appropriate range of these cosmological
parameters by plotting some figures.

Figure 4: The plot of $n_{s}$ in term of $r$.

Figure 5: The plot of ($C_{1}$ and $C_{2}$) in term of ($n_{s}$), and ($r$) in
Fig. 5, 5 and Fig. 5, 5, respectively.  Figure 6: the constraint of
($C_{2}-C_{1}$).

The second model follow the same process as the model (I). But different
results are evaluated for each of the parameters. We also describe the results
of each of these models in detail. We will discuss the differences and
similarities of the mentioned models and introduce the best model compatible
with swampland conjectures. We plotted the $r$ in terms of $n_{s}$ in figure
4. The changes of these two parameters about each other for the mentioned
constant parameters are well shown in this figure. The difference can be seen
well in comparison with the previous model. The important point in this model
is related to the parameter $\alpha=1-\lambda$, and $n=3$ which expresses good
results for the swampland conjectures. different values of $n$, and $\alpha$,
also valid the range of swampland conjectures, which is true of all figures.
Also, these values viz ($r$) and ($n_{s}$) are within the range of the latest
observable data [3]. Figure 5 shows the changes the swampland conjectures
component,i.e., $C_{1}$ and $C_{2}$ in terms of two important cosmological
parameters; $n_{s}$, and $r$. $C_{1}$ and $C_{2}$ in terms of $n_{s}$ in Fig.
5, 5 and in terms of $r$ in Fig. 5, 5 are plotted regarding
$\alpha=1-\lambda$. The allowable range of these parameters is well
determined. As shown in the figures, the swampland conjecture components are
within the allowable range. As can be clearly seen in the figures, the
component $C_{2}$ has smaller values than $C_{1}$, which is, in fact,
consistent with the results in the literature and the latest observational
data [3]. Like the previous part, we will examine another constraint from the
perspective of swampland conjectures and obtain the allowable range, i.e., we
investigate the $C_{1}^{2}C_{2}^{2}$ with respect to the latest observable
data and the mentioned condition. The allowable range for swampland
conjectures of this model is calculated regarding the $\alpha=1-\lambda$. In
figure 6, we plot the plan ($C_{2}-C_{1}$) with respect to $\alpha=1-\lambda$.
The allowable range of the swampland conjectures for $\alpha=1-\lambda$, will
be $c<0.000012645$. As mentioned above, single-field inflation models with the
slow-roll conditions and swampland conjecture have a series of inconsistencies
that methods were used to solve this problem in the literature. This paper has
resolved these differences and inconsistencies by a known method and small
parameter $\lambda$. By introducing several inflation models, we applied each
of these concepts to the inflation models. We then obtained each of the
cosmological parameters and determined each of these parameters’ range by
plotting some figures. However, according to the above mentioned, this method
can be studied with different theories of gravity and other conditions.

##  4 Conclusions

Recently many researchers have been studied various conditions as the
swampland program in cosmological implications. Sometimes these expressed
conjectures are modified and lead to fascinating points. Different inflation
models, dark energy, and other cosmological implications have also been
investigated with swampland conjectures. The swampland conjecture is in
contrast to the single-field inflation model with slow-roll conditions. So in
this paper, first, we described the special method to solve this problem with
the help of a small parameter with respect to a coupling function. So we
introduced some cosmological parameters as the number of e-folds $N$, tensor-
to-scalar ratio $r$, and scalar spectral index $n_{s}$ associated with this
method, and we explained this issue entirely. Also, we studied a new
perspective in inflation models according to the swampland criteria. We
introduced some inflation models and also examined the various swampland
conditions. We specified the allowable area of cosmological parameters such as
($r$), ($n_{s}$), and reconstructed new restrictions as ($C_{1}-r$),
($C_{1}-n_{s}$), ($C_{2}-r$), and ($C_{2}-n_{s}$). We determined the allowable
range for the swampland component corresponding to the models by plotting some
figures and advantaging with the latest observational data [3]. Finally, we
expressed the results of these different conditions and compared the result
with each other. In studying the models, we identified the permissible areas
of swampland conjectures. These studies found that the contradiction between
the swampland conditions and slow-roll single-field inflation can be resolved
with these special situations. The important point is that both models, in
addition to compatibility with all the constraints of the swampland dS
conjecture, have deeper consistency of these swampland conjecture components
with important cosmological parameters such as $r$ and $n_{s}$. The allowable
range of swampland conjectures was determined for each model. It was found
that swampland conjectures accommodate more volume from the model (I) and are
more compatible with this model. These constraints can also be examined by
finding a common area in combining different concepts such as inflation and
dark energy, and determining the commonalities of these concepts. Different
inflation models, and the implications of other cosmological studies such as
dark energy, etc., can also be evaluated with these new conditions.

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The search for strongly decaying exotic matter
R.S. Longacrea
aBrookhaven National Laboratory, Upton, NY 11973, USA

###### Abstract

In this paper we explore the possibility of detecting strongly decaying exotic
states. The dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$
is the example we use in this report.

##  1 Introduction

At the STAR experiment we can collect hundreds of million ultra-relativistic
heavy ion collisions. Light nuclei and anti-nuclei emerge from these
collisions during the last stage of the collision process. The quantum wave
functions of the constituent nucleons or anti-nucleons, if close enough in
momentum and coordinate space, will overlap to produce composite systems. The
production rate for the systems with baryon or anti-baryon B is proportional
to the baryon or anti-baryon density in momentum and coordinate space, raised
to the power $|$B$|$, and therefore exhibits exponential behavior as a
function of $|$B$|$. Figure 1 shows the exponential[1] invariant yields versus
baryon number in $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions.
Empirically, the production rate reduces by a factor of 1.1 x $10^{3}$(1.6 x
$10^{3}$) for each additional nucleon (anti-nucleon) added. The measurement of
hundreds of million of events make it possible to probe up to a scale of five
in baryon number. The baryon four data points come from a STAR measurement
published in Ref.[2]. It should be noted that there are no baryon five nuclear
fragments that live long enough or decay weakly such that they would have a
displaced vertex[2].

The paper is organized in the following manner:

Sec. 1 explores nuclear states that have been measured.

Sec. 2 explores the possibility of detecting strongly decaying exotic states.
The dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$ is
considered.

Figure 1: Differential invariant yields as a function of baryon number B,
evaluated at $p_{T}$/$|$B$|$ = 0.875 GeV/c, in central $\sqrt{s_{NN}}$=200 GeV
Au+Au collisions. Yields for tritons ${}^{3}H$ (anti-tritons
$\overline{{}^{3}H}$) lie close to the position for ${}^{3}He$ and
$\overline{{}^{3}He}$. The lines represent fits with the exponential formula
$\alpha$ $e^{-r|B|}$ for positive and negative particles separately, where $r$
is the production reduction factor.

##  2 Exotic States through strong decay.

In the above section the states decayed by the weak interaction. The
possibility of detecting strongly decaying exotic states is considered using
the dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$ as an
example. In the QGP(Quark Gluon Plasma) six quarks or anti-quarks could come
together to form a deuteron or anti-deuteron. However such states are loosely
bound and easily destroyed in the hadronic phase. The cross section for $d$
$\pi$ scattering is 240 mb. This implies that the deuteron can only be formed
in the final freeze-out of the hadronic system. At the time of freeze-out many
hadrons scatter and coalesce into compound or excited states(see Figure 2).

The dibaryon state interacts in three two-body scattering channels. Its mass
is 2.15 GeV and has a strong interaction resonance decay width of 100 MeV. It
interacts in the $N$ $N$ d-wave spin anti-aligned[3], $d$ $\pi$ p-wave spin
aligned[4], and $\Delta$ $N$ s-wave spin aligned[5]. The dibaryon system
mainly resonates in the s-wave $\Delta$ $N$ mode with a pion rotating in a
p-wave about a spin aligned $N$ $N$ system which forms a isospin singlet. The
pion moves back and forth forming $\Delta$ states with one nucleon and then
the other(see Figure 3). All three isospin states of the pion can be achieved
in this resonance. Thus we can have $\pi^{+}$ $d$, $\pi^{0}$ $d$, and
$\pi^{-}$ $d$ states. If the pion is absorbed by any of the nucleons it under
goes a spin flip producing a d-wave $N$ $N$ system. The resonance decays into
$N$ $N$, $\pi$ $d$, or $\pi$ $N$ $N$. In a meson system an analogous resonance
is formed where a pion is orbiting in a p-wave about a $K\overline{K}$ in a
s-wave[6](see Figure 4). Both systems have a similar lifetime or width of
$\sim$ .100 GeV.

Figure 2: Above are a few of the compound or excited states that will form
during the last stage of hadronic freeze-out.

In order to predict the rate for dibaryon production we turn to a Monte Carlo
heavy ion event generator[7]. This generator was a cradle to grave going from
initial partons to final state hadrons. Figure 5 shows the time line in the
center-of-mass frame for partons, then pre-hadrons and final hadrons. What is
happening in the early times of the collision is of no importance for dibaryon
formation, while the conditions of the hadrons at later times will determine
the dibaryon production. For $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions
the spectrum is well measured. Therefore we can start the Monte Carlo at the
intermediate times with a fireball of excited hadrons and let it evolve to the
final state.

We start with an expanding cylinder of radius 10.0 Fermi filled with excited
hadrons with density and $p_{t}$ distribution that reproduces the
$\sqrt{s_{NN}}$=200 GeV central Au+Au collisions. Figure 6 gives the measured
Au+Au spectrum which we will tune for. Mesons in the fireball cascade include
$\pi$, $K$, $\rho$, $\omega$, $a_{1}$, $\eta$, $\eta\prime$, $\phi$, $K^{*}$,
$K^{*}(1420)$, $f_{0}(975)$, $a_{0}(980)$, $f_{2}(1270)$, $a_{2}(1320)$,
$h_{1}(1170)$, $\rho(1700)$, $f_{2}(1800)$, $b_{1}(1235)$ and $f_{2}(1525)$.
The cross section for $\pi\pi$ and $\pi K$ was determined from S-matrix phase
shifts, while for $K\overline{K}$ we used the production of $\phi$,
$f_{0}(975)$, $a_{0}(980)$, $f_{2}(1270)$, $a_{2}(1320)$,and $f_{2}(1525)$(see
Figure 7). For $\rho\pi$ cross sections we used the production of
$h_{1}(1170)$, $a_{1}(1260)$, and $a_{2}(1320)$, while for $K^{*}\pi$ we used
the production of $K^{*}(1420)$. For $a_{1}\pi$ we used $\rho(1700)$ and
$f_{2}(1800)$(see Figure 8). Finally for $\omega\pi$ we used $b_{1}(1235)$ and
$\eta\pi$ we used $a_{0}(980)$ and $a_{2}(1320)$. For all other meson meson
cross section we used the additive quark model. The particles produced from
such scatterings were determined by a multi-pomeron chain model using a Field-
Feymann algorithm(see Figure 9).

Since we are detecting baryons and anti-baryons the $NN$, $N\pi$, $NK$, and
$\Delta N$ cross section and scattering ratios are obtained from data and
extracted S-matrix amplitudes(see Figure 10). All other cross sections for
baryon meson and baryon baryon systems we use the additive quark model(see
Figure 11). The particles produced from such scatterings are determined by a
multi-pomeron chain model using a Field-Feymann algorithm. For baryon($B$)
anti-baryon($\overline{B}$) scattering and cross section, data is used for
$N\overline{N}$ annihilation and elastic scattering. For annihilation yields,
we use a flavor consistent meson meson multi-pomeron chain model. For the rest
of the yield a $B\overline{B}$ multi-pomeron chain model is used. The elastic
scattering obtained by this method is close to the data for the
$N\overline{N}$ system.

Figure 3: The dibaryon system mainly resonates in the s-wave $\Delta$ $N$ mode
with a pion rotating in a p-wave about a spin aligned $N$ $N$ system which
forms a isospin singlet. The pion moves back and forth forming $\Delta$ states
with one nucleon and then the other.

Figure 4: The meson system mainly resonates in the s-wave $K^{*}$
$\overline{K}$ and $K$ $\overline{K^{*}}$ mode with a pion rotating in a
p-wave about a $K$ $\overline{K}$ system which forms a isospin triplet. The
pion moves back and forth forming $K^{*}$ and $\overline{K^{*}}$ states with
one $K$ or $\overline{K}$.

Figure 5: Time evolution of the total numbers of produced partons Np, pre-
hadronic clusters Nc, and hadrons Nh during Au + Au collisions. The time
refers to the center-of-mass frame of the colliding nuclei.

Figure 6: The measured Au+Au spectrum which we will tune for.

Figure 7: Cross sections for $\pi\pi$, $\pi K$, and $K\overline{K}$.

Figure 8: Cross sections for $\rho\pi$, $K^{*}\pi$, and $a_{1}\pi$.

Figure 9: Cross sections for $\omega\pi$, $\eta\pi$, and others.

Figure 10: Cross sections for $N\pi$, $NK$, $NN$ and others.

Figure 11: The additive quark model calculates the cross section for the
scattering of any two particles based on a product of geometric factors.

The annihilation threshold effect is scaled to other $B\overline{B}$
scatterings using the $N\overline{N}$ ratios obtained in the above algorithm.

We need to add the production of d’s into the Monte Carlo code. Let us assume
that the formation of the $J^{p}=2^{+}$ dibaryon state is the driving source
of d’s. We fit the d-wave $NN$ elastic scattering[3], p-wave $d\pi$ elastic
scattering[4], and p-wave $d\pi$ to d-wave $NN$[4]. A three channel K-matrix
was used to form a S-matrix, where the channels are d-wave $NN$, p-wave
$d\pi$, and s-wave $\Delta N$ data. We are able to fit the above data if we
use one K-matrix pole to generate the dibaryon 2.15 GeV state plus a far away
pole and a flat none factorable background. Figure 12 shows the fit to elastic
$NN$ scattering amplitude. Figure 13 shows the fit to $d\pi$ elastic
scattering, while Figure 14 is the connection between $NN$ going to $d\pi$.

The cross sections for $N$ $N$ $\rightarrow$ $\pi$ $d$, $\Delta$ $N$
$\rightarrow$ $\pi$ $d$, $\pi$ $d$ $\rightarrow$ $\Delta$ $N$ and $N$ $N$
where added to the hadron cascade part of the code. When we consider the known
cross sections for $NN$ and $\Delta N$, the yield for charge pairs of $d\pi$
can be calculated and is plotted in Figure 15. In our hadron cascade these
scatterings are the only source of d’s. The production of d’s and anti-d’s is
close to the values measured in Figure 1. The value of d’s in the cascade
would be much larger than the measured value except d’s are destroyed by
interacting with pions. Figure 16 show the large $d\pi$ cross section of
$\sim$ 250 mb. About 3/4 of these scattering remove the d’s from the cascade.

We achieve the yield and spectrum for Au+Au $\sqrt{s_{NN}}$=200 GeV central
collisions by adjusting the excited hadrons in our cylinder of radius 10.0
Fermi. We generate enough events at $\sqrt{s_{NN}}$=200 GeV central Au+Au
collisions in order to obtain 1 million $d$ or $\overline{d}$ events in the
STAR acceptance. Out of the 1 million events there were 230,000 pairs of
either $d$ $\pi$ or $\overline{d}$ $\pi$ which decayed in the STAR acceptance.
The effective mass distribution of these pairs are plotted as solid points in
Figure 17.

In order to obtain the mass spectrum from the data, We need to determined the
uncorrelated background of either $d$ or $\overline{d}$ paired with a charge
particle in a average event. For each of the 1 million events we can pair up
either the $d$ or $\overline{d}$ with all charge particles(which then is
assumed to be a pion) in that event and plot the total number of pairs as a
function of effective mass. From this pair spectrum we then subtract the
average uncorrelated spectrum times the number of events. We can determine
this average uncorrelated spectrum by mixed event methods taking the same $d$
and $\overline{d}$ paired with the charged particles from other events. The
subtracted effective mass spectrum is the open points of Figure 17. We see
that we have recovered the mass spectrum.

##  3 Summary and Discussion

In the first section of this manuscript we consider baryons and anti-baryons
up to a baryon number five. These states decayed by the weak interaction. The
exotic states that decay strongly is considered in the second section. In
order to develop methods for such research we consider a dibaryon(2.15)
$J^{P}$ = $2^{+}$ state which decays into d $\pi$.

Figure 12: The real and imaginary parts of the elastic scattering T-matrix
amplitude for $NN\rightarrow NN$ as a function of mass in GeV.

Figure 13: The real and imaginary parts of the elastic scattering T-matrix
amplitude for $d\pi\rightarrow d\pi$ as a function of mass in GeV.

Figure 14: The modulus of the inelastic scattering T-matrix amplitude for
$NN\rightarrow d\pi$ as a function of mass in GeV.

Figure 15: The percentage of $d\pi$ charge pairs produced in $NN$ and $\Delta
N$ scattering as a function of mass in GeV.

Figure 16: The $d\pi$ total and elastic cross section in millibarns(mb) as a
function of mass in GeV.

Figure 17: The number of $d$ or $\overline{d}$ paired with charged pions
coming from $10^{6}$ dibaryons decays within the STAR acceptance plotted as
solid points. The open points are form by all $d$ and $\overline{d}$ paired
with the charged particles in each event in the star acceptance minus the same
$d$ and $\overline{d}$ paired with the charged particles from other
events(mixed events).

We create a Monte Carlo simulation that should give realistic events structure
with realistic dibaryon production. With the ability to measure hundreds of
million ultra-relativistic heavy ion collisions, we predicted that a clear
dibaryon signal decaying into $d\pi$ should be measured.

##  4 Acknowledgments

This research was supported by the U.S. Department of Energy under Contract
No. DE-AC02-98CH10886.

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