the overlap of $D_{N}(r;\tau)$ with the Maass cusp forms vanishes, because the
latter have no zero mode:

$(D_{N}(r),\phi_{n})=0\leavevmode\nobreak\ \leavevmode\nobreak\
\forall\leavevmode\nobreak\ n.$ (E.3)

Proceeding with the Eisenstein overlap, we have

$\\{D_{N}(r),E_{1-s}\\}={2\over\Lambda(1-s)}\int_{0}^{\infty}{dy\over
y^{2}}\,d_{N}(r;y)\,y^{s}+(s\rightarrow 1-s)$ (E.4)

Now use

${\rm Li}_{2r}(x)=\sum_{k=1}^{\infty}{x^{k}\over k^{2r}}$ (E.5)

Swapping the sum and integral and performing them in succession is elementary.
The integral yields323232Convergence of the integral near the cusp requires
${\rm Re\,}(r+s)<1$.

$\\{D_{N}(r),E_{1-s}\\}={\pi^{s-1}4^{2r+2s-1}N^{r+s-1}\Gamma(2(1-r-s))\over\Lambda(1-s)}\sum_{k=1}^{\infty}k^{2(s-1)}+(s\rightarrow
1-s)$ (E.6)

Performing the sum leads to the final result,

${\\{D_{N}(r),E_{1-s}\\}={4^{1+2r-2s}N^{r-s}\Gamma(2(s-r))\over\Gamma(s)}+(s\rightarrow
1-s)}$ (E.7)

Note that while $D_{N}(r;\tau)$ parameterizes exponentially suppressed terms
at $p\gg 1$, the spectral overlap itself is not exponential per se. Rather,
the sign of such terms is the fact the overlap is factorially divergent at
$s\rightarrow\pm\infty$.

For generic $r$, the modular average of $D_{N}(r;\tau)$ is

$\begin{split}\langle D_{N}(r)\rangle&:=\int_{\cal F}{dxdy\over
y^{2}}D_{N}(r;\tau)\\\ &={1\over 2}\lim_{s\rightarrow
1}\\{D_{N}(r),E_{1-s}\\}\\\
&=2^{4r-3}N^{r-1}\Gamma(2-2r)\qquad(2r\notin\mathbb{Z}_{\geq 0})\end{split}$
(E.8)

For $2r\in\mathbb{Z}_{\geq 0}$, the result is finite but takes a different
form which may be easily extracted from (E.7). The full spectral decomposition
is therefore

$D_{N}(r;\tau)=\langle D_{N}(r)\rangle+{1\over 2\pi i}\int_{{\rm
Re\,}s={1\over
2}}ds\,{4^{1+2r-2s}N^{r-s}\Gamma(2(s-r))\over\Gamma(s)}E_{s}^{*}(\tau)$ (E.9)

#### ’t Hooft limit

Using the above result, we now show that the non-perturbative corrections in
the strong ’t Hooft coupling expansion (3.36) can be derived directly from the
non-perturbative corrections in the large charge expansion at finite coupling
given in (3.16). We begin by writing the leading term in ${\cal
F}^{(2)}_{\text{NP}}(p;\tau)$ in the spectral form

$\displaystyle\big{(}2\pi^{2}p\big{)}^{1/4}D_{p/2}\left(-\frac{1}{4};\tau\right)=$
$\displaystyle\leavevmode\nobreak\ \frac{3\pi}{16\sqrt{2}p}+{1\over 2\pi
i}\int_{{\rm Re\,}s={1\over
2}}ds\frac{\sqrt{\pi}2^{\frac{3}{2}-3s}p^{-s}\Gamma\left(2s+\frac{1}{2}\right)}{\Gamma(s)}\left(\Lambda(s)y^{s}+\Lambda(1-s)y^{1-s}\right)$
$\displaystyle+\text{(non-zero modes)}\,.$ (E.10)

Shifting the contour slightly to ${\rm Re\,}s={1\over 2}-\epsilon$ for an
infinitesimal $\epsilon>0$, taking the double-scaling limit of fixed
$\lambda_{p}$, and keeping only the leading order term $\sim p^{0}$, we get

$\displaystyle\big{(}2\pi^{2}p\big{)}^{1/4}D_{p/2}\left(-\frac{1}{4};\tau\right)\approx{1\over
2\pi i}\int_{{\rm Re\,}s={1\over
2}-\epsilon}ds\sqrt{\pi}2^{\frac{3}{2}-s}\lambda_{p}^{-s}\zeta(2s)\Gamma\left(2s+\frac{1}{2}\right),$
(E.11)

which comes from the first term in the parenthesis of (E). The integrand has
poles at $s=-\frac{1}{4}(2n+1)$ for all $n\in\mathbb{Z}_{\geq 0}$, and decays
exponentially at infinity for $\lambda_{p}<2\pi^{2}$. Deforming the contour to
the left and using the series representation of $\zeta(2s)$ we find that the
sum over the residues is given by

$\displaystyle{1\over 2\pi i}\int_{{\rm Re\,}s={1\over
2}-\epsilon}ds\sqrt{\pi}2^{\frac{3}{2}-s}\lambda_{p}^{-s}\zeta(2s)\Gamma\left(2s+\frac{1}{2}\right)$
$\displaystyle=\sqrt{2\pi}(2\lambda_{p})^{1/4}\sum_{k=1}^{\infty}\sqrt{k}e^{-\sqrt{2}k\sqrt{\lambda_{p}}}$
$\displaystyle=\sqrt{2\pi}(2\lambda_{p})^{1/4}\text{Li}_{-\frac{1}{2}}\left(e^{-\sqrt{2\lambda_{p}}}\right).$
(E.12)

This matches the result obtained in (3.36) for the leading-order non-
perturbative corrections in $\lambda_{p}\gg 1$.

##  Appendix F Resurgence of the $\lambda_{p}\gg 1$ expansion at $N=3$

In this appendix we apply resurgence to the strong coupling expansion (3.37)
to compute the exponentially small non-perturbative terms quoted in (3.39).
Following [34], we consider the modified Borel transform of the infinite sum
in (3.37),333333For convenience, here we use the Borel transform with rescaled
argument $\sqrt{2}w$, compared to $2w$ as in [34].

$\displaystyle\mathcal{B}:\quad\sum_{n=1}^{\infty}c_{3}(n)\,x_{p}^{-2n-1}\quad\mapsto\quad\phi(w):=\sum_{n=1}^{\infty}\frac{2\pi
c_{3}(n)}{\zeta(2n+1)\Gamma(2n+2)}(\sqrt{2}w)^{2n+1}\,,$ (F.1)

where we have introduced $x_{p}=\sqrt{\lambda_{p}}$ and the coefficients
$c_{3}(n)$ have been given previously in (3.38). For the Borel transform (F.1)
we then find

$\displaystyle\begin{split}\phi(w)=\frac{9}{35}\pi
w^{3}\Big{[}&280\,_{3}F_{2}\big{(}-\tfrac{3}{2},\tfrac{3}{2},\tfrac{3}{2}\big{|}1,\tfrac{5}{2}\big{|}w^{2}\big{)}-168w^{2}\,_{3}F_{2}\big{(}-\tfrac{1}{2},\tfrac{5}{2},\tfrac{5}{2}\big{|}2,\tfrac{7}{2}\big{|}w^{2}\big{)}\\\
&+25w^{4}\,_{3}F_{2}\big{(}\tfrac{1}{2},\tfrac{7}{2},\tfrac{7}{2}\big{|}3,\tfrac{9}{2}\big{|}w^{2}\big{)}\Big{]}\,,\end{split}$
(F.2)

which is input to the directional Borel resummation of (3.37) given by

$\displaystyle\mathcal{S}_{\theta}{\mathcal{G}}_{\mathfrak{g}=0}^{(3)}(x_{p})=\frac{3}{2}\log\Big{(}\frac{x_{p}^{2}}{32\pi^{2}}\Big{)}+\frac{7}{2}+3\gamma_{E}+\frac{x_{p}}{\sqrt{2}\pi}\int_{0}^{e^{i\theta}\infty}\frac{dw}{4\sinh^{2}(x_{p}w/\sqrt{2})}\,\phi(w)\,,$
(F.3)

for some $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$. However, due to the branch
cut along $[1,\infty)$ of the hypergeometric functions in $\phi(w)$, the
result of the Borel resummation is ambiguous, and in fact not real for
$x_{p}>0$. This suggests that the asymptotic expansion is not Borel summable
and hence requires exponentially small corrections. Such terms are encoded in
the so-called ‘Stokes automorphism’, defined as the difference between the two
lateral resummations just above and below the branch cut of $\phi(w)$,

$\displaystyle\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(x_{p}):=(\mathcal{S}_{+}-\mathcal{S}_{-})\mathcal{G}_{\mathfrak{g}=0}^{(3)}(x_{p})=\frac{x_{p}}{\sqrt{2}\pi}\int_{1}^{\infty}\frac{dw}{4\sinh^{2}(x_{p}w/\sqrt{2})}\,\text{disc}\big{[}\phi(w)\big{]},$
(F.4)

where
$\mathcal{S}_{\pm}=\lim_{\theta\to\theta^{\pm}}\mathcal{S}_{\theta}\mathcal{G}_{\mathfrak{g}=0}^{(3)}(x_{p})$
and we already anticipated that the $w$-integration actually starts from 1
(since $\text{disc}\big{[}\phi(w)\big{]}=0$ for $0\leq w\leq 1$).

To compute the discontinuity of $\phi(w)$ across the branch cut,

$\displaystyle\text{disc}\big{[}\phi(w)\big{]}:=\phi(w+i0)-\phi(w-i0)\,,$
(F.5)

we make use of Euler’s integral transform for the hypergeometric
${}_{3}F_{2}$’s appearing in (F.2):

$\displaystyle_{3}F_{2}\big{(}a_{1},a_{2},c\big{|}b_{1},d\big{|}z\big{)}=\frac{\Gamma(d)}{\Gamma(c)\Gamma(d-c)}\int_{0}^{1}dt\leavevmode\nobreak\
t^{c-1}(1-t)^{d-c-1}\,_{2}F_{1}\big{(}a_{1},a_{2};b_{1};tz\big{)}\,.$ (F.6)

The only piece of the integrand which has a branch cut is the ${}_{2}F_{1}$,
whose discontinuity is given by the formula

$\displaystyle\begin{split}\text{disc}\big{[}{{}_{2}F_{1}}\big{(}a,b;c;z\big{)}\big{]}=\frac{2\pi
i\,\Gamma(c)}{\Gamma(a)\Gamma(b)\Gamma(c-a-b+1)}&z^{1-c}(z-1)^{c-a-b}\\\
&\times{{}_{2}F_{1}}\big{(}1-a,1-b;c-a-b+1;1-z\big{)}\,.\end{split}$ (F.7)

Note that since the discontinuity of the hypergeometric function starts at
argument 1, when plugging (F.7) into (F.6) the range of the $t$-integration is
modified to $1/z\leq t\leq 1$.343434 In the computation of the discontinuity
of the last ${}_{3}F_{2}$ in (F.2), after using (F.6), one finds that formula
(F.7) diverges. To avoid this divergence, we first use the shift identity
$\displaystyle{{}_{3}F_{2}}\big{(}\tfrac{1}{2},\tfrac{7}{2},\tfrac{7}{2}\big{|}3,\tfrac{9}{2}\big{|}w^{2}\big{)}=8\,{{}_{3}F_{2}}\big{(}-\tfrac{1}{2},\tfrac{7}{2},\tfrac{7}{2}|3,\tfrac{9}{2}|w^{2}\big{)}-7\,{{}_{2}F_{1}}\big{(}-\tfrac{1}{2},\tfrac{7}{2};3;w^{2}\big{)}\,,$
(F.8) which then leads to a manifestly finite answer in the computation of the
discontinuity.

We find that the discontinuity of $\phi(w)$ can be expressed as

$\displaystyle\begin{split}\text{disc}\big{[}\phi(w)\big{]}&=\frac{48i}{\pi}\Bigg{[}-3wG_{2,2}^{2,2}\left(w^{2}\left|\begin{array}[]{c}\frac{1}{2},\frac{5}{2}\\\
1,1\\\
\end{array}\right.\right)-3wG_{3,3}^{2,3}\left(w^{2}\left|\begin{array}[]{c}\frac{1}{2},\frac{1}{2},\frac{7}{2}\\\
1,2,-\frac{1}{2}\\\ \end{array}\right.\right)\\\
&\qquad\quad\leavevmode\nobreak\
\,+3wG_{3,3}^{2,3}\left(w^{2}\left|\begin{array}[]{c}\frac{1}{2},\frac{1}{2},\frac{7}{2}\\\
2,2,-\frac{1}{2}\\\
\end{array}\right.\right)-4wG_{3,3}^{2,3}\left(w^{2}\left|\begin{array}[]{c}\frac{1}{2},\frac{1}{2},\frac{9}{2}\\\
1,3,-\frac{1}{2}\\\ \end{array}\right.\right)\\\
&\qquad\quad\leavevmode\nobreak\
\,+3G_{2,2}^{2,2}\left(1\left|\begin{array}[]{c}-\frac{1}{2},\frac{3}{2}\\\
0,0\\\
\end{array}\right.\right)+3G_{3,3}^{2,3}\left(1\left|\begin{array}[]{c}-\frac{1}{2},-\frac{1}{2},\frac{5}{2}\\\
0,1,-\frac{3}{2}\\\ \end{array}\right.\right)\\\
&\qquad\quad\leavevmode\nobreak\
\,+4G_{3,3}^{2,3}\left(1\left|\begin{array}[]{c}-\frac{1}{2},-\frac{1}{2},\frac{7}{2}\\\
0,2,-\frac{3}{2}\\\
\end{array}\right.\right)-3G_{3,3}^{2,3}\left(1\left|\begin{array}[]{c}-\frac{3}{2},-\frac{3}{2},\frac{3}{2}\\\
0,0,-\frac{5}{2}\\\ \end{array}\right.\right)\\\
&\qquad\quad\leavevmode\nobreak\
\,+\pi^{2}w^{3}\,_{2}F_{1}\left(-\frac{5}{2},\frac{3}{2};1;1-w^{2}\right)\Bigg{]},\end{split}$
(F.9)

where $w>1$ and $G_{p,q}^{m,n}\left(z\left|\begin{array}[]{c}\\{a_{i}\\}\\\
\\{b_{i}\\}\end{array}\right.\right)$ denotes the Meijer-G function.

This is then plugged into (F.4), which can be computed by first shifting the
integration variable $w\mapsto w+1$ and expanding the
$\sinh^{2}(x_{p}w/\sqrt{2})$ in the denominator:

$\displaystyle\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(x_{p})=\frac{\tilde{x}_{p}}{\pi}\sum_{n=1}^{\infty}n\,e^{-2n\tilde{x}_{p}}\int_{0}^{\infty}d{w}\leavevmode\nobreak\
e^{-2n\tilde{x}_{p}{w}}\,\text{disc}\big{[}\phi({w}+1)\big{]}\,,$ (F.10)

where for convenience we introduced
$\tilde{x}_{p}=x_{p}/\sqrt{2}=\sqrt{\lambda_{p}/2}$. The integration over
${w}$ can be done by expanding the discontinuity around $w=0$ and using the
integral identity $\int_{0}^{\infty}dw\leavevmode\nobreak\
e^{-2n\tilde{x}_{p}w}w^{m}=\frac{m!}{(2n\tilde{x}_{p})^{m+1}}$ term by term.
The series expansion of the discontinuity (F.9) is of the form

$\displaystyle\text{disc}\big{[}\phi(w+1)\big{]}=\pi
i\,\sum_{m=0}^{\infty}a_{m}w^{m}\,,$ (F.11)

where the coefficients $a_{m}$ turn out to be rational numbers. The first few
are given by

$\displaystyle
a_{m}=\Big{\\{}48,\,408,\,1839,\,\frac{6737}{2},\,\frac{192939}{64},\,\frac{834603}{640},\,\frac{222319}{1024},\,\ldots\Big{\\}},\qquad\text{for
}m=0,1,2,\ldots\,.$ (F.12)

Putting all together, we have

$\displaystyle\begin{split}\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(x_{p})&=i\tilde{x}_{p}\sum_{n=1}^{\infty}n\,e^{-2n\tilde{x}_{p}}\Big{[}\frac{48}{(2n\tilde{x}_{p})}+\frac{408}{(2n\tilde{x}_{p})^{2}}+\frac{3678}{(2n\tilde{x}_{p})^{3}}+\frac{20211}{(2n\tilde{x}_{p})^{4}}+\frac{578817}{8(2n\tilde{x}_{p})^{5}}+\ldots\Big{]}\\\
&=i\sum_{n=1}^{\infty}e^{-2n\tilde{x}_{p}}\Big{[}24+\frac{102}{n\tilde{x}_{p}}+\frac{1839}{4(n\tilde{x}_{p})^{2}}+\frac{20211}{16(n\tilde{x}_{p})^{3}}+\frac{578817}{256(n\tilde{x}_{p})^{4}}+\ldots\Big{]},\end{split}$
(F.13)

where each term in this sum corresponds to a perturbative contribution in
$1/\tilde{x}_{p}\sim 1/\sqrt{\lambda_{p}}$ to a non-perturbative correction of
order $e^{-2n\tilde{x}_{p}}=e^{-n\sqrt{2\lambda_{p}}}$. Note that this is in
exact agreement with the scale $\Lambda^{2}(\lambda_{p})$ predicted by the
weak coupling radius of convergence, c.f. (3.32).

Term by term in $1/\tilde{x}_{p}$, the sum over $n$ resums into polylogarithms
and we have

$\displaystyle\begin{split}\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(\lambda_{p})=i\,\bigg{[}&24\,\text{Li}_{0}\big{(}e^{-\sqrt{2\lambda_{p}}}\big{)}+\frac{102\sqrt{2}}{\lambda_{p}^{1/2}}\,\text{Li}_{1}\big{(}e^{-\sqrt{2\lambda_{p}}}\big{)}+\frac{1839}{2\lambda_{p}}\,\text{Li}_{2}\big{(}e^{-\sqrt{2\lambda_{p}}}\big{)}\\\
&+\frac{20211}{4\sqrt{2}\lambda_{p}^{3/2}}\,\text{Li}_{3}\big{(}e^{-\sqrt{2\lambda_{p}}}\big{)}+\frac{578817}{64\lambda_{p}^{2}}\,\text{Li}_{4}\big{(}e^{-\sqrt{2\lambda_{p}}}\big{)}+\ldots\bigg{]},\end{split}$
(F.14)

which is the result already quoted in the main text.

We can now assemble the median Borel resummation,
$\mathcal{S}_{\text{med}}:=\frac{1}{2}\big{(}\mathcal{S}_{+}-\mathcal{S}_{-}\big{)}$,
by adding the discontinuity (F.14) to one of the lateral resummations:

$\displaystyle\begin{split}\mathcal{S}_{\text{med}}\mathcal{G}_{\mathfrak{g}=0}^{(3)}(\lambda_{p})&=\mathcal{S}_{\pm}\mathcal{G}_{\mathfrak{g}=0}^{(3)}(\lambda_{p})\mp\frac{1}{2}\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(\lambda_{p})\\\
&=\frac{3}{2}\log\Big{(}\frac{\lambda_{p}}{32\pi^{2}}\Big{)}+\frac{7}{2}+3\gamma_{E}+\frac{\sqrt{\lambda_{p}}}{\sqrt{2}\pi}\int_{0}^{\infty}\frac{dw}{4\sinh^{2}(w\sqrt{\lambda_{p}}/\sqrt{2})}\leavevmode\nobreak\
\text{Re}\big{[}\phi(w)\big{]}\,,\end{split}$ (F.15)

with $\phi(w)$ given in (F.2). This is now a manifestly real expression for
any $\lambda_{p}>0$, where the non-perturbative corrections are captured by
the discontinuity $\delta\mathcal{G}_{\mathfrak{g}=0}^{(3)}(\lambda_{p})$
given in (F.14).

##  Appendix G The genus-one correlator
$\mathcal{G}_{\mathfrak{g}=1}^{(N)}(\lambda_{p})$

For the first sub-leading contribution at order $p^{-1}$ one finds

$\displaystyle\begin{split}\mathcal{G}_{\mathfrak{g}=1}^{(N)}(\lambda_{p})&=-\frac{3N(N^{2}-1)\zeta(3)}{16\pi^{2}}\,\lambda_{p}+\frac{N(N^{2}-1)(N-1)}{8}\\\\[5.0pt]
&\quad+{1\over 2\pi i}\int_{{\rm Re\,}s=\frac{1}{2}}ds\,{\pi\over\sin(\pi
s)}s(1-s)(2s-1)^{2}\,\Lambda(1-s)\left(\frac{\lambda_{p}}{4\pi}\right)^{s-1}g_{2}^{(N)}(s)\,F^{(1)}(N,s)\,.\end{split}$
(G.1)

Its large $\lambda_{p}$ expansions reads

$\displaystyle\begin{split}\frac{\text{(2nd line of
\eqref{eq:large_p_g1})}}{N^{2}-1}=\frac{\sqrt{\frac{\pi}{2}}\,\Gamma\big{(}\frac{N^{2}+1}{2}\big{)}}{\Gamma\big{(}\frac{N^{2}-4}{2}\big{)}}\bigg{[}-\frac{3\zeta(3)}{4\lambda_{p}^{\frac{3}{2}}}\,\widehat{f}_{2}^{(N)}(-\tfrac{1}{2})+\frac{45(N^{2}-6)\zeta(5)}{32\lambda_{p}^{\frac{5}{2}}}\,\widehat{f}_{2}^{(N)}(-\tfrac{3}{2})++O(\lambda_{p}^{-\frac{7}{2}})\bigg{]},\end{split}$
(G.2)

where the same comments about the truncation for $N$ even apply again. For the
perturbative expansion at small $\lambda_{p}$ we have

$\displaystyle\frac{{\mathcal{G}}_{\mathfrak{g}=1}^{(N)}(\lambda_{p})}{N^{2}-1}=-\frac{75N^{2}\zeta(5)}{128\pi^{4}(N^{2}+1)}\,\lambda_{p}^{2}+\frac{3675N^{3}\zeta(7)}{2048\pi^{6}(N^{2}+1)(N^{2}+3)}\,\lambda_{p}^{3}+O(\lambda_{p}^{4})\,,$
(G.3)

where the constant and linear terms in $\lambda$ have cancelled (i.e. the
entire first line of (3.40) gets cancelled by the $s=1$ and $s=2$ residues
from the second line).

##### Bessel integral representations:

Proceeding as in the $\mathfrak{g}=0$ case, for $N=2$ we find the expression

$\displaystyle{\mathcal{G}}_{\mathfrak{g}=1}^{(2)}(\lambda_{p})=-\int_{0}^{\infty}dw\leavevmode\nobreak\
\frac{3w^{2}}{4\pi^{2}\sinh^{2}(w)}\leavevmode\nobreak\
\bigg{(}w\lambda_{p}-\sqrt{2\lambda_{p}}\pi
J_{1}\Big{(}\tfrac{w\sqrt{2\lambda_{p}}}{\pi}\Big{)}\bigg{)}\,.$ (G.4)

The $N=3$ result takes again a more complicated form given by

$\displaystyle\begin{split}{\mathcal{G}}_{\mathfrak{g}=1}^{(3)}(\lambda_{p})&=-\int_{0}^{\infty}dw\leavevmode\nobreak\
\frac{3}{w^{3}\lambda_{p}^{2}\pi^{2}\sinh^{2}(w)}\leavevmode\nobreak\
\bigg{(}w^{6}\lambda_{p}^{3}+16\pi^{2}w^{2}\lambda_{p}(24\pi^{2}-w^{2}\lambda_{p})J_{0}^{2}\Big{(}\tfrac{w\sqrt{\lambda_{p}}}{\sqrt{2}\pi}\Big{)}\\\
&\qquad\qquad\qquad-4\sqrt{2\lambda_{p}}w\pi(384\pi^{4}-40\pi^{2}w^{2}\lambda_{p}+w^{4}\lambda_{p}^{2})J_{0}\Big{(}\tfrac{w\sqrt{\lambda_{p}}}{\sqrt{2}\pi}\Big{)}J_{1}\Big{(}\tfrac{w\sqrt{\lambda_{p}}}{\sqrt{2}\pi}\Big{)}\\\
&\qquad\qquad\qquad+4(768\pi^{6}-128\pi^{4}w^{2}\lambda_{p}+5\pi^{2}w^{4}\lambda_{p}^{2})J_{1}^{2}\Big{(}\tfrac{w\sqrt{\lambda_{p}}}{\sqrt{2}\pi}\Big{)}\bigg{)}\,.\end{split}$
(G.5)

##  Appendix H Derivation of Bessel integral representation in $p=\alpha
N^{2}$ regime

Here we resum the weak coupling expansion of
$\mathcal{G}_{\alpha}^{(0)}(\lambda)$, the planar integrated correlator in the
gravity regime $p=\alpha N^{2}\gg 1$ with fixed $\alpha$, to derive the Bessel
integral representation (4.16).

Using the zeta function integral identity (3.28), we can resum the weak
coupling expansion (4.10). As before, we split the computation again into two
parts. For the first sum over $\widetilde{h}_{0}(s)$, we find

$\displaystyle\widetilde{h}_{0}(s):\quad\int_{0}^{\infty}dw\leavevmode\nobreak\
\frac{w}{\sinh^{2}(w)}\bigg{(}-\frac{1}{2}+\frac{2\pi^{2}}{\lambda
w^{2}}J_{1}^{2}\Big{(}\tfrac{w\sqrt{\lambda}}{\pi}\Big{)}\bigg{)}\,.$ (H.1)

To perform the second sum over $\widetilde{h}_{\alpha}(s)$ we make use of the
integral representation
$P_{n}(z)=\frac{1}{\pi}\int_{0}^{\pi}dt\,(z-\sqrt{z^{2}-1}\cos(t))^{n}$ for
the Legendre polynomial. This yields

$\displaystyle\widetilde{h}_{\alpha}(s):\qquad\int_{0}^{\infty}dw\int_{0}^{\pi}dt\leavevmode\nobreak\
\frac{w}{\sinh^{2}(w)}\Bigg{(}\frac{1}{2\pi}-\frac{2\pi}{\lambda
w^{2}}\frac{J_{1}\left({w\sqrt{\lambda
u_{\alpha}}\over\pi}\right)^{2}}{u_{\alpha}}\Bigg{)}\,,$ (H.2)

where $u_{\alpha}:=1+2\alpha-2\sqrt{\alpha(\alpha+1)}\cos(t)>0$ for
$\alpha\geq 0$. Performing the $t$-integral for the first term in the bracket
gives $1/2$, which then cancels against the first term from (H.1) when adding
the two parts. Altogether,

$\displaystyle\mathcal{G}_{\alpha}^{(0)}(\lambda)=\frac{2\pi^{2}}{\lambda}\int_{0}^{\infty}dw\leavevmode\nobreak\
{J_{1}^{2}\left({w\sqrt{\lambda}\over\pi}\right)-K_{11}\left({w\sqrt{\lambda}\over\pi};\alpha\right)\over
w\sinh^{2}(w)}\,,$ (H.3)

where we have defined the localized353535More general Bessel kernels may be
defined as functions of two positions $(x,x^{\prime})$. integral Bessel kernel
$K_{11}(x;\alpha)$ in (4.17). Note that
$dt={du_{\alpha}\over\sqrt{u_{\alpha}(4\alpha+2-u_{\alpha})-1}}$. Since
$u_{0}=1$, the first term in (H.3) is simply the second term at $\alpha=0$,
which allows us to nicely combine them:

$\displaystyle\mathcal{G}_{\alpha}^{(0)}(\lambda)={2\pi^{2}\over\lambda}\int_{0}^{\infty}dw\leavevmode\nobreak\
{K_{11}\left({w\sqrt{\lambda}\over\pi};0\right)-K_{11}\left({w\sqrt{\lambda}\over\pi};\alpha\right)\over
w\sinh^{2}(w)}\,.$ (H.4)

This is the final result (4.16).

We point out that this vanishes linearly as $\alpha\rightarrow 0$, as is
manifest from the spectral representation (4.4). Near $\alpha=0$,

$\begin{split}\mathcal{G}_{\alpha}^{(0)}(\lambda)&=\alpha\,\int_{0}^{\infty}dw\,w\,{J_{1}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}^{2}-J_{2}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}^{2}\over\sinh^{2}(w)}\\\
&\leavevmode\nobreak\ -\,\alpha^{2}\int_{0}^{\infty}dw\,\frac{\lambda
w^{3}}{4\pi^{2}}\left({J_{1}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}^{2}-J_{2}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}^{2}+{\pi\over\sqrt{\lambda}w}J_{1}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}J_{2}\big{(}\frac{w\sqrt{\lambda}}{\pi}\big{)}\over\sinh^{2}(w)}\right)+O(\alpha^{3})\,.\end{split}$
(H.5)

The first term is precisely the $p=2$ integrated correlator in the planar ’t
Hooft limit [25]; this correspondence was anticipated from (4.4) expanded near
$\alpha=0$,

$h^{(0)}_{\alpha}(s)={\alpha\over 2}\,g^{(0)}_{2}(s)+O(\alpha^{2})\,.$ (H.6)

At $O(\alpha^{2})$ and beyond, one finds a three-term linear combination of
the same general form as the $O(\alpha^{2})$ result, with the same Bessel
functions appearing, dressed by increasing powers of $w\sqrt{\lambda}/\pi$.

## References

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# Worry No More, The Hubble Tension is Relieved:
A Truly Direct Measurement of the Hubble Constant from Mooniversal Expansion

Gagandeep S. Anand  Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218, USA  Zachary R. Claytor  Institute for Astronomy,
University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA  Ryan
Dungee  Institute for Astronomy, University of Hawai‘i, 640 North A‘ohōkū,
Hilo, HI 96720, USA

(1 April 2022)

###### Abstract

Using sedimentary and eclipse-based measurements of the lunar recession
velocity, we derive a new local-Universe measurement of the Hubble constant
($H_{0}$) from the recession rate of Earth’s Moon. Taking into account the
effects of tides, we find a value of $H_{0}=$ 63.01 $\pm$ 1.79 km s-1 Mpc-1,
which is in approximate agreement with the Planck space mission’s measurement
using the cosmic microwave background (CMB) and base $\Lambda$CDM. Our new
measurement represents the first ever model-independent, single-step
measurement of the Universe’s current expansion rate. This is also the first
major local Universe measurement of $H_{0}$ which is below the measurement
from Planck. Importantly, it is robust to the systematic errors that may be
present in other $H_{0}$ measurements using other cosmological probes such as
type Ia supernovae, baryon acoustic oscillations, or lensed quasars. Our work
provides key evidence towards the notion that the existing Hubble tension may
indeed be a result of systematic uncertainties in the local distance ladder.

††thanks: All authors contributed equally to this manuscript.††thanks: All
authors contributed equally to this manuscript.††thanks: All authors
contributed equally to this manuscript.

##  1 Motivation

Cosmology today is at an important juncture. There is a large amount of
discussion on the value of the Hubble Constant ($H_{0}$), which denotes the
present expansion rate of the Universe. Measurements of the Cosmic Microwave
Background (CMB) from the Planck Satellite provide an extremely precise value
of 67.4 $\pm$ 0.5 km/s/Mpc (Planck Collaboration et al., 2020). On the other
hand, the latest results from the most prominent “local” measure of the Hubble
constant is from the SH0ES team, who use observations of Cepheids in 42 nearby
galaxies to derive of value of $H_{0}$ = 73.0 $\pm$ 1.0 km/s/Mpc (Riess et
al., 2021). These two measurements are in 5$\sigma$ disagreement with each
other, a fact which has sparked an incredible amount of interest within the
community (Schöneberg et al., 2021) to come up with a possible solution to
this Hubble “Tension”.

Other local Universe measurements also support values of $H_{0}$ which are
higher than that obtained from Planck, with many different techniques being
explored to provide a cross-check on the Cepheid results (although some of
these measurements are somewhat consistent with Planck, such as those from the
Tip of the Red Giant Branch). These techniques include Mira variables (Huang
et al., 2020), Surface Brightness Fluctuations (Blakeslee et al., 2021), the
Tip of the Red Giant Branch (TRGB, Freedman et al. 2019; Anand et al. 2021),
and the Tully-Fisher relation (Kourkchi et al., 2020), among many others (see
Di Valentino et al. 2021 for a more complete list).

It has been previously pointed out that the Moon’s recessional velocity is on
the same order as the implied Hubble flow expansion in the Earth-Moon system
(Maeder & Gueorguiev, 2021), although the origin of this coincidence is
unclear. In this work, we combine this suggestion with improved measurements
on the Moon’s recessional velocity and make the assertion that the origin of
the Moon’s recessional velocity (after discounting the known effects of tides)
is from the Hubble expansion. In doing so, we provide a precise local
measurement of the Hubble constant from the first truly local measurement of
$H_{0}$.

Figure 1: A comparison of numerous recent $H_{0}$ results, including the one
we derive in this paper, which we stress is the only measurement which is both
model-independent and performed in a single step. Methods plotted include
Cepheid variables (Riess et al., 2021), a restriction on the peculiar velocity
monopole from Cosmicflows-3 (Tully et al., 2016), the Dark Energy Survey +
Baryon Acoustic Oscillations + Big Bang Nucelosynthesis (Abbott et al., 2018),
time delay cosmography of lensed quasars from TDCOSMO (Millon et al., 2020),
megamasers from the Megamaser Cosmology Project (Pesce et al., 2020), Mira
variables (Huang et al., 2020), Planck measurements of the CMB (Planck
Collaboration et al., 2020), surface brightness fluctuations (Blakeslee et
al., 2021), type II supernovae (de Jaeger et al., 2020), the Tully-Fisher
relation (Kourkchi et al., 2020), and the Tip of the Red Giant Branch
(Freedman, 2021; Anand et al., 2021).

##  2 Method and Data

We begin with the Hubble law and solve for the Hubble constant:

$H_{0}=\frac{v}{d},$

where $v$ is the recession velocity, and $d$ is the distance from Earth.
Scaling by some typical values, we obtain the scaling relation

$H_{0}=70~{}\mathrm{km/s/Mpc}\left(\frac{v}{2.755~{}\mathrm{cm/yr}}\right)\left(\frac{d}{385,000~{}\mathrm{km}}\right)^{-1}.$

The question, then, is which are the appropriate values of $v$ and $d$ for an
estimate of the Hubble constant?

###  2.1 Lunar Distance

Because of the Moon’s elliptical orbit, the Earth-Moon distance varies from
363,300 km at perigee to 405,500 km at apogee (NASA, 2022). Table 1 lists
important distance values from the Moon’s orbit. Calculating a true cosmic
expansion rate from the Moon’s recession would require a series of precise,
instantaneous measurements of both the Earth-Moon distance and the lunar
recession velocity, spanning many orbits. In the absence of that, we will use
our best guess. Since we are looking at the Moon’s recession over time, we
adopt the time-averaged distance of 385,000 km $\pm$ 1 mm as the nominal
Earth-Moon distance.

Distance | Value [km]
---|---
Semimajor Axis | 384,400
Perigee | 363,300
Apogee | 405,500
Time-Averaged | 385,000
Table 1: Important distances in the Moon’s orbit from NASA (2022). These
distances are precisely determined by laser measurements and have 1 mm
precision. Since we consider the Moon’s recession over time, we adopt the
time-averaged Earth-Moon distance as the appropriate distance.

###  2.2 Lunar Recession Velocity

The value of the lunar recession velocity is the subject of some debate
(Poliakow, 2005; Riofrio, 2012; Maeder & Gueorguiev, 2021). Laser measurements
suggest an instantaneous, present-day recession velocity of 3.8 cm/yr, but
this has been regarded by some as anomalously high (Riofrio, 2012). Sediment
data and eclipse data suggest slower recession speeds of $2.9\pm 0.6$ and
$2.82\pm 0.08$ cm/yr, respectively, and these are supported by detailed
numerical simulations (Poliakow, 2005). We favor the sediment and eclipse
values due to their robustness and agreement with simulations, and we adopt
the weighted average of the two values, $v=2.82\pm 0.08$ cm/yr.

###  2.3 Correction for Tides

Tides have long been known to influence the Earth-Moon system (Darwin, 1878),
and any measure of the lunar recession must account for this effect. To do
this we assume the two effects are entirely independent and each contribute to
the total recession linearly. Thus, the cosmological lunar recession
$v_{\mathrm{cosmo}}$ is given simply by:
$v_{\mathrm{total}}=v_{\mathrm{tides}}+v_{\mathrm{cosmo}}$.
$v_{\mathrm{tides}}$ varies over time due to a variety of effects and Poliakow
(2005) provide a table containing historical values dating back 600 Myr.
Taking the $v_{\mathrm{tides}}$ for the approximate age of the sedimentary
data gives $v_{\mathrm{tides}}=0.34$ cm/yr. This then yields a
$v_{\mathrm{cosmo}}$ value of $2.48$ cm/yr.

##  3 Results and Discussion

Evaluating the Hubble law at the distance and recession velocity of the Moon
(after taking into account the effect of tides), we obtain $H_{0}=$ 63.01
$\pm$ 1.79 km s-1 Mpc-1, a 2.8% determination of the Hubble constant. Figure 1
shows our measurement along with several others from the recent literature. We
see that previous results which are from the “late” Universe (e.g. Cepheids)
are consistently higher than those from the “early” Universe (e.g. Planck).
Our measurement is important not only because it is the only measurement of
the Hubble Constant which is both model-independent and performed in a single
step, but because it is the first local measurement of $H_{0}$ whose value is
*below* the value measured by Planck.

##  4 Summary and Future Outlook

In this letter, we examine the dynamics of the Earth-Moon system to provide
the first truly local measurement of the Hubble constant of $H_{0}$ = 63.01
$\pm$ 1.79 km s-1 Mpc-1. This is somewhat lower than the value measured by the
Planck satellite, though it is roughly consistent within the errors (see
Figure 1). We emphasize that our measurement of $H_{0}$ is the first which is
both model-independent and is done within a single step (i.e. does not rely on
a multi-step distance ladder). Our work is an important result, as it suggests
that other local Universe results whose measurements are higher than those of
Planck may be biased by systematic errors.

In the near future, we hope to improve the precision on our lunar-based
measurement of $H_{0}$ by both 1) collecting more geological samples (to
improve the sedimentary data), and 2) observing in detail more eclipses (to
improve the eclipse-based measurements). Both of these avenues will allow us
to decrease the uncertainty on our value of $H_{0}$ and solve once and for all
this cosmological crisis.

The idea for this paper came in 2019 out of late night desperation of three
grad students trying to procrastinate real work. Nearly 3 PhDs and some
unknown fraction of a pandemic later, we are enthralled to report this very
significant result, which is definitely not a joke.

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