$\begin{split}&\mathbb{P}(Q^{\lambda}_{0,N}\\{\tau<t^{2/3}N^{2/3}\\}\geq
1/10)=\mathbb{P}(Q^{\lambda}_{{\bf x},N}\\{\tau<t^{2/3}N^{2/3}\\}\geq 1/10)\\\
&\leq\mathbb{P}(Q^{\lambda}_{0,N\boldsymbol{\xi}[\lambda]}\\{\tau<-t^{2/3}N^{2/3}\\}\geq
1/10)\leq e^{-Ct}.\end{split}$ (7.16)

Define

$A=\\{Q^{\lambda}_{0,N}\\{\tau\leq t^{2/3}N^{2/3}\\}\leq 1/10\\}.$

Then (LABEL:def_Ac) shows $\mathbb{P}(A)\geq 1-e^{-Ct}$, and on $A$, (7.15)
holds. With this, we have completed the proof of this proposition. ∎

Next, we extend the range of $t$ from the previous proposition.

######  Proposition 7.2.

Suppose the boundary condition $W$ satisfies assumption (1.2). There exist
positive constants $C_{1},N_{0},t_{0}$ such that for each $N\geq N_{0}$,
$t\geq t_{0}$, and each positive integer
$a\leq\min\\{\tfrac{1}{2}t^{2/3}N^{2/3},N\\}$, we have

$\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{a}_{N}}-\log Z^{W}_{N}\geq
C_{1}t\sqrt{a}\Big{)}\leq e^{-t^{1/10}}.$

###### Proof.

First, note that when $t\leq c_{0}N^{1/2}$ the result holds from the previous
proposition. Now, suppose $t\geq c_{0}N^{1/2}$, let $t=zN^{1/2}$ for some real
positive number $z\geq c_{0}$. Then, if $a\leq N^{2/3}$,we have

$\displaystyle\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{a}_{N}}-\log
Z^{W}_{N}\geq C_{1}t\sqrt{a}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}^{a}_{N}}-\log Z^{W}_{N}\geq C_{1}t\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{N^{2/3}}_{N}}-\log
Z^{W}_{N}\geq C_{1}(zN^{1/6})N^{1/3}\Big{)}$ (by Proposition 2.14 and
Proposition 2.15) $\displaystyle\leq e^{-CzN^{1/6}}\leq e^{-t^{1/10}}.$

Otherwise, if $N^{2/3}\leq a\leq\min\\{\tfrac{1}{2}t^{2/3}N^{2/3},N\\}$, we
have

$\displaystyle\mathbb{P}\Big{(}\log Z^{W}_{\mathcal{L}^{a}_{N}}-\log
Z^{W}_{N}\geq C_{1}t\sqrt{a}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\log
Z^{W}_{\mathcal{L}^{a}_{N}}-\log Z^{W}_{N}\geq C_{1}tN^{1/3}\Big{)}$ by
Proposition 2.14 and Proposition 2.15 $\displaystyle\leq e^{-t^{1/10}}.$

∎

The next proposition captures the transversal fluctuation of the favorite
point in the polymer. Fix $N/2\leq r\leq N$. Let ${\bf p}_{*}$ denote the
random maximizer in

$\max_{{\bf p}\in\mathcal{L}_{r}}\big{\\{}\log Z^{W}_{{\bf p}}+\log\mathaccent
869{Z}_{{\bf p},N}\big{\\}}=\log Z^{W}_{{\bf p}_{*}}+\log\mathaccent
869{Z}_{{\bf p}_{*},N}.$

The proposition below captures the KPZ transversal fluctuation which says that
the maximizer ${\bf p}_{*}$ cannot be too far from the diagonal on the local
scale $(N-r)^{2/3}$.

######  Proposition 7.3.

Suppose the boundary condition $W$ satisfies assumption (1.2). There exist
positive constants $C_{1},c_{0},t_{0},N_{0}$ such that for each $N\geq N_{0}$,
$N/2\leq r\leq N-c_{0}$ and $t\geq t_{0}$, we have

$\mathbb{P}(|{\bf p}_{*}-(r,r)|_{\infty}>t(N-r)^{2/3})\leq
e^{-C_{1}{t}^{3/2}}.$

###### Proof.

Abbreviate $J^{h}=\mathcal{L}^{(N-r)^{2/3}}_{r+\underline{{2h(n-r)^{2/3}}}}$.
We bound the probability as follows.

$\displaystyle\mathbb{P}(|{\bf p}_{*}-(r,r)|_{\infty}>t(N-r)^{2/3})$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}\setminus\mathcal{L}_{r}^{t(N-r)^{2/3}}}\Big{\\{}\log
Z^{W}_{{\bf p}}+\log\mathaccent 869{Z}_{{\bf p},N}\Big{\\}}>\log
Z^{W}_{r}+\log\mathaccent 869{Z}_{r,N}\Big{)}$
$\displaystyle\leq\sum_{|h|=\lfloor
t/2\rfloor}^{(N-r)^{1/3}}\mathbb{P}\Big{(}\max_{{\bf x}\in J^{h}}\log
Z^{W}_{{\bf x}}+\max_{{\bf x}\in J^{h}}\log\mathaccent 869{Z}_{{\bf x},N}>\log
Z^{W}_{r}+\log\mathaccent 869{Z}_{r,N}\Big{)}$
$\displaystyle=\sum_{|h|=\lfloor
t/2\rfloor}^{(N-r)^{1/3}}\mathbb{P}\Big{(}\Big{[}\max_{{\bf x}\in J^{h}}\log
Z^{W}_{{\bf x}}-\log Z^{W}_{r}\Big{]}+\Big{[}\max_{{\bf x}\in
J^{h}}\log\mathaccent 869{Z}_{{\bf x},N}-\log\mathaccent
869{Z}_{r,N}\Big{]}>0\Big{)}$ $\displaystyle=\sum_{|h|=\lfloor
t/2\rfloor}^{(N-r)^{1/3}}\mathbb{P}\Big{(}\max_{{\bf x}\in J^{h}}\log
Z^{W}_{{\bf x}}-\log Z^{W}_{r}\geq Dh^{2}(N-r)^{1/3}\Big{)}$ (7.17)
$\displaystyle\qquad\qquad\qquad\qquad+\mathbb{P}\Big{(}\max_{{\bf x}\in
J^{h}}\log\mathaccent 869{Z}_{{\bf x},N}-\log\mathaccent 869{Z}_{r,N}\geq-
Dh^{2}(N-r)^{1/3}\Big{)}.$ (7.18)

where $D$ is a small positive constant that we will fix later.

For (7.18), provided $t_{0}$ is fixed sufficiently large, we may upper bound
it using Proposition 2.6 and Proposition 2.4 as below

$\displaystyle\eqref{close_max}$ $\displaystyle=\mathbb{P}\Big{(}[\max_{{\bf
x}\in J^{h}}\log\mathaccent 869{Z}_{{\bf x},N}-\Lambda_{N-r}]-[\log\mathaccent
869{Z}_{r,N}-\Lambda_{r}]\geq-Dh^{2}(N-r)^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf x}\in J^{h}}\log\mathaccent
869{Z}_{{\bf x},N}-\Lambda_{N-r}\geq-2Dh^{2}(N-r)^{1/3}\Big{)}$
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\log\mathaccent
869{Z}_{r,N}-\Lambda_{N-r}\leq-Dh^{2}(N-r)^{1/3}\Big{)}$ $\displaystyle\leq
e^{-C|h|^{3}}$

provided $D\leq\tfrac{1}{10}C^{*}$ where $C^{*}=C_{1}$ from Proposition 2.6.

For (7.17), we will split the estimate into two cases depending on the value
of $N-r$, whether $N-r\leq\epsilon_{0}r$ or $N-r\geq\epsilon_{0}r$, for
$\epsilon_{0}$ which we will fix below between the math displays (7.20) and
(7.21).

When $N-r\leq\epsilon_{0}r$, we upper bound (7.17) by

$\eqref{far_max}\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{r}^{4|h|(N-r)^{2/3}}}\log Z^{W}_{{\bf x}}-\log Z^{W}_{r}\geq
Dh^{2}(N-r)^{1/3}\Big{)}.$ (7.19)

To apply Proposition 7.1, let us set $a=4|h|(N-r)^{2/3}$ and
$t=8|h|^{3/2}\epsilon_{0}^{3/2}$, and we have

$\displaystyle\eqref{bound_small_r}=\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{r}^{a}}\log Z_{{\bf x},N}-\log
Z_{r,N}\geq\tfrac{D}{16\epsilon_{0}^{3/2}}t\sqrt{a}\Big{)}.$ (7.20)

Next, we fix $\epsilon_{0}$ sufficiently small so that $t\leq\mathaccent
869{c}_{0}r^{1/2}$, where $\mathaccent 869{c}_{0}=c_{0}$ from Proposition 7.1.
Next, we lower the value of $D$ to get

$\tfrac{\frac{1}{10}D}{8\epsilon_{0}^{3/2}}\leq C^{*}$ (7.21)

where $C^{*}=C_{1}$ appearing in Proposition 7.1. Finally, by Proposition 7.1,
the above probability in (7.20) will always be bounded by
$e^{-Ct}=e^{-C|h|^{3/2}}$.

On the other hand, let us look at the case when $N-r\geq\epsilon_{0}r$, and
note that now $\epsilon_{0}$ has already been fixed. Since the value of $N-r$
will always be less than $r$ when $r\geq N/2$, let us set $N-r=zr$ for some
positive constant $z$ with $\epsilon_{0}\leq z\leq 1$. We can obtain the
following upper bound as

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf x}\in J^{h}}\log Z^{W}_{{\bf
x}}-\log Z^{W}_{r}\geq Dh^{2}(N-r)^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{r}^{4|h|(N-r)^{2/3}}}\log Z^{W}_{{\bf x}}-\log Z^{W}_{r}\geq
Dh^{2}(N-r)^{1/3}\Big{)}$ $\displaystyle=\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{r}^{4|h|(N-r)^{2/3}}}\log Z^{W}_{{\bf x}}-\log Z^{W}_{r}\geq
Dh^{2}z^{1/3}r^{1/3}\Big{)}$ $\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
x}\in\mathcal{L}_{r}^{4|h|(N-r)^{2/3}}}\log Z^{W}_{{\bf
x}}-\Lambda_{r}\geq\tfrac{1}{2}Dh^{2}z^{1/3}r^{1/3}\Big{)}+\mathbb{P}\Big{(}\log
Z^{W}_{r}-\Lambda_{r}\leq-\tfrac{1}{2}Dh^{2}z^{1/3}r^{1/3}\Big{)}$
$\displaystyle\leq e^{-C|h|^{3/2}}\qquad\text{by Proposition \ref{b_up_tail}
and Proposition \ref{b_low_tail}.}$

To summarize, the arguments above show that

$\sum_{|h|=\lfloor
t/2\rfloor}^{(N-r)^{1/3}}\eqref{close_max}+\eqref{far_max}\leq\sum_{|h|=\lfloor
t/2\rfloor}^{\infty}e^{-C|h|^{3/2}}\leq e^{-Ct^{3/2}},$

with this, we have finished the proof of this proposition. ∎

Finally, we prove Theorem 2.18.

###### Proof of Theorem 2.18.

Note because of

$\log Z^{W}_{N}\leq\max_{{\bf p}\in\mathcal{L}_{r}}\\{\log Z^{W}_{{\bf
p}}+\log\mathaccent 869{Z}_{{\bf p},N}\\}+2\log(N-r),$

it suffices to show that

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf p}\in\mathcal{L}_{r}}\Big{\\{}\log
Z^{W}_{{\bf p}}+\log\mathaccent 869{Z}_{{\bf p},N}\Big{\\}}-\Big{[}\log
Z^{W}_{r}+\log\mathaccent 869{Z}_{r,N}\Big{]}\geq t(N-r)^{1/3}\Big{)}\leq
e^{-t^{1/10}}.$

By a union bound, we split the above maximum over ${\bf p}\in\mathcal{L}_{r}$
to ${\bf p}\not\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$ and ${\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$.

In the case when ${\bf p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$, we have

$\displaystyle\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}}\Big{\\{}\log Z^{W}_{{\bf
p}}+\log\mathaccent 869{Z}_{{\bf p},N}\Big{\\}}-\Big{[}\log
Z^{W}_{r}+\log\mathaccent 869{Z}_{r,N}\Big{]}\geq t(N-r)^{1/3}\Big{)}$
$\displaystyle\leq\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}}\log Z^{W}_{{\bf p}}-\log\mathaccent
869{Z}_{0,r}\geq\tfrac{1}{2}t(N-r)^{1/3}\Big{)}$ (7.22)
$\displaystyle\qquad\qquad+\mathbb{P}\Big{(}\max_{{\bf
p}\in\mathcal{L}_{r}^{t(N-r)^{2/3}}}\log\mathaccent 869{Z}_{{\bf
p},N}-\log\mathaccent 869{Z}_{r,N}\geq\tfrac{1}{2}t(N-r)^{1/3}\Big{)}.$ (7.23)

Both probabilities (7.22) and (7.23) can be upper bounded by $e^{-Ct^{1/10}}$
using Proposition 7.2. In the case when ${\bf
p}\not\in\mathcal{L}_{r}^{t(N-r)^{2/3}}$, this follows from Proposition 7.3. ∎

##  8 Parallel results in exponential LPP

As already mentioned in the introduction, the results of this paper all have
analogues in the setting of exponential LPP on $\mathbb{Z}^{2}$. Our proofs
carry over almost verbatim to this set-up upon replacing the free energy by
the last passage time, the restricted free energy over a class of paths by the
maximum passage time among the paths in that class, and the quenched polymer
measure by the measure that puts mass one on the geodesic in the LPP setting.
We shall not be repeating the arguments, but will formulate proper statements
to this effect.

Let $\\{Y_{v}\\}_{v\in\mathcal{L}_{0}^{>}}$ denote a collection of i.i.d. rate
one exponential random variables. Let $H_{k}$ be a sequence of random
variables with $H_{0}=0$ such that $\\{\mathaccent
869{X}_{i}:=H_{i}-H_{i-1}\\}$ is a collection of independent mean $0$ random
variables satisfying the following conditions:

  1. (i)

There exists $\lambda_{1},K_{1}>0$ such that for all $i$,
$\log\mathbb{E}(\exp(\lambda|\mathaccent 869{X}_{i}|))\leq K_{1}\lambda^{2}$
for $\lambda\in[-\lambda_{1},\lambda_{1}]$.

  2. (ii)

$\inf_{i}\textup{$\mathbb{V}$ar}\mathaccent 869{X}_{i}>0$.

The last passage time between ${\bf u}$ and ${\bf v}$ (without any boundary
condition), denoted $\mathaccent 869{T}_{{\bf u},{\bf v}}$ is defined by

$\mathaccent 869{T}_{{\bf u},{\bf v}}:=\max_{{\bf
x}_{{{\scaleobj{0.6}{\bullet}}}}\in\mathbb{X}_{{\bf a},{\bf
b}}}\sum_{i=1}^{|{\bf v}-{\bf u}|_{1}}Y_{{\bf x}_{i}}.$

For $\mathbf{v}\in\mathcal{L}^{>}_{0}$, let us define the last passage time to
$v$ with initial condition $H$ by

$T^{H}_{\bf v}=\max_{k\in\mathbb{Z}}\left(H_{k}+T_{(k,-k),{\bf v}}\right).$

We then have the following results corresponding to Theorems 1.1 and 1.2.

######  Theorem 8.1.

Consider exponential LPP with initial condition $H$ on $\mathcal{L}_{\bf 0}$
satisfying the above hypotheses. There exist positive constants
$C_{1},C_{2},c_{0},N_{0}$ such that, whenever $N\geq N_{0}$ and $N/2\leq r\leq
N-c_{0}$, we have

$1-C_{1}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}\leq\textup{$\mathbb{C}$orr}\Big{(}T^{H}_{(r,r)},T^{H}_{(N,N)}\Big{)}\leq
1-C_{2}\Big{(}\frac{N-r}{N}\Big{)}^{2/3}.$

######  Theorem 8.2.

Consider exponential LPP with initial condition $H$ on $\mathcal{L}_{\bf 0}$
satisfying the above hypotheses. There exist positive constants
$C_{3},C_{4},c_{0},N_{0}$ such that, whenever $N\geq N_{0}$ and $c_{0}\leq
r\leq N/2$, we have

$C_{3}\Big{(}\frac{r}{N}\Big{)}^{1/3}\leq\textup{$\mathbb{C}$orr}\Big{(}T^{H}_{(r,r)},T^{H}_{(N,N)}\Big{)}\leq
C_{4}\Big{(}\frac{r}{N}\Big{)}^{1/3}.$

As the reader might have noticed, our arguments for the inverse-gamma polymer
essentially used only the curvature of the limit shape (Proposition 2.1), and
the moderate deviation estimates for the free energy, see Section 3.3.1 of [9]
for a discussion about this. All the other estimates were then developed using
these ingredients together with certain random walk comparison results. The
curvature of limit shape for exponential LPP is known [24], as are the
moderate deviation estimates [28]. Many of the required auxiliary estimates
have already been developed in the exponential LPP setting (see [7, 8]).
Random walk comparisons in the zero temperature setting first appeared in the
seminal work [13], since then it has been employed in many places [4, 5, 33].
By quoting these auxiliary estimates as necessary, the proofs of Theorems 8.1
and 8.2 can be completed following the proofs of Theorems 1.1 and 1.2 almost
verbatim.

Before finishing this section, we point out that a variant of Theorem 8.1 (in
the $n\to\infty$ limit with $\tau=r/n$ close to $1$) was established in [18]
in the special case where $\mathaccent 869{X}_{i}$ are i.i.d. having the same
distribution as $\sigma(U_{i}-V_{i})$ where $U_{i},V_{i}$ are two independent
sequences of i.i.d. random variables distributed as $\exp(1/2)$ and
$\sigma\in[0,\infty)$. Theorem 8.1 extends this to more general initial
conditions and to the pre-limiting set-up. Theorem 8.2 appears to be new
except in the stationary case with density $1/2$ (i.e., when $H_{k}$ is a two
sided random walk with i.i.d. increments with distribution $W-W^{\prime}$
where $W,W^{\prime}$ are independent copies of $\mbox{Exp}(1/2)$ random
variables) which was also dealt with in [18], again only in the $n\to\infty$
limit.

We also emphasize that several extensions of Theorems 1.1 and 1.2 remain valid
in this set-up as well. We can extend the hypothesis on the initial condition
to an analogue of Assumption B1, Assumption B2, Assumption B3 or relax even
further as discussed in Remark 1.7. It should also be possible cover
stationary initial conditions with different densities for time correlations
if the endpoint varies along the corresponding characteristic direction, but
we shall not get into the details here.

##  Appendix A Appendix

###  A.1 Sub-exponential random variables.

First, we state a general result for the running maximum of sub-exponential
random variables. Recall that a random variable $X_{1}$ is sub-exponential if
there exist two positive constants $K_{0}$ and $\lambda_{0}$ such that

$\log(\mathbb{E}[e^{\lambda(X_{1}-\mathbb{E}[X_{1}])}])\leq
K_{0}\lambda^{2}\quad\textup{ for $\lambda\in[0,\lambda_{0}]$}.$ (A.1)

Let $\\{X_{i}\\}$ be a sequence of i.i.d. sub-exponential random variables
with the parameters $K_{0}$ and $\lambda_{0}$. Define $S_{0}=0$ and
$S_{k}=X_{1}+\dots+X_{k}-k\mathbb{E}[X_{1}]$ for $k\geq 1$. The following
theorem gives an upper bound for the right tail of the running maximum.

######  Theorem A.1 ([9, Theorem D.1]).

Let the random walk $S_{k}$ be defined as above. Then,

$\mathbb{P}\Big{(}\max_{0\leq k\leq n}S_{k}\geq
t\sqrt{n}\Big{)}\leq\begin{cases}e^{-t^{2}/(4K_{0})}\quad&\textup{if $t\leq
2\lambda_{0}K_{0}\sqrt{n}$}\\\
e^{-\frac{1}{2}\lambda_{0}t\sqrt{n}}\quad&\textup{if $t\geq
2\lambda_{0}K_{0}\sqrt{n}$}\end{cases}.$

Next, we will state a lower bound for the right tail probability which was
obtained from [37] using Paley–Zygmund inequality. Let $\\{Z_{i}\\}$ be a
sequence of i.i.d. random variables such that there exist parameters $K_{1}$
and $\lambda_{1}$ with

$\tfrac{1}{K_{1}}\lambda^{2}\leq\log(\mathbb{E}[e^{\lambda(Z_{1}-\mathbb{E}[Z_{1}])}])\leq
K_{1}\lambda^{2}\quad\textup{ for $\lambda\in[0,\lambda_{1}]$}.$ (A.2)

Similar to before, define $H_{0}=0$ and
$H_{k}=Z_{1}+\dots+Z_{k}-k\mathbb{E}[Z_{1}]$ for $k\geq 1$, then the following
holds.

######  Theorem A.2 ([37, Theorem 4]).

Let the random walk $H_{n}$ be defined as above. Then, there exists positive
constants $C_{1},C_{2},C_{3}$ depending on $K_{1}$ and $\lambda_{1}$ such that

$e^{-C_{1}t^{2}}\leq\mathbb{P}(H_{n}\geq t\sqrt{n})\leq
e^{-C_{2}t^{2}}\qquad\text{ for each $0\leq t\leq C_{3}\sqrt{n}$.}$

####  A.1.1 Verifying (A.2) for stationary model weights

Since assumption (A.2) is strictly stronger than (A.1), we will just verify
(A.2) for our applications in this paper. First, we have a proposition which
shows that both $\log(\textup{Ga})$ and
$\log(\textup{Ga}^{-1})=-\log(\textup{Ga})$ satisfies (A.2).

######  Proposition A.3.

Fix $\epsilon\in(0,\mu/2)$. There exists positive constants
$K_{1},\lambda_{1}$ depending on $\epsilon$ such that for each
$\alpha\in[\epsilon,\mu-\epsilon]$, let $X\sim\textup{Ga}(\alpha)$ and we have

$\tfrac{1}{K_{1}}\lambda^{2}\leq\log(\mathbb{E}[e^{\pm\lambda(\log(X)-\Psi_{1}(\alpha))}])\leq
K_{1}\lambda^{2}\qquad\textup{ for $\lambda\in[0,\lambda_{1}]$}.$

###### Proof.

First, note that
$\mathbb{E}[X^{\pm\lambda}]=\frac{\Gamma(\alpha\pm\lambda)}{\Gamma(\alpha)}$,
provided that $\alpha\pm\lambda>0$. Then, the proof essentially follows from
Taylor’s theorem,

$\displaystyle\log(\mathbb{E}[e^{\pm\lambda(\log(X)-\Psi_{1}(\alpha))}])$
$\displaystyle=\log(\mathbb{E}[X^{\pm\lambda}]e^{\mp\lambda\Psi_{1}(\alpha)})$
$\displaystyle=\log(\Gamma(\alpha\pm\lambda))-[\log(\Gamma(\alpha))\pm\lambda\Psi_{1}(\alpha)]$
(recall $\log(\Gamma(\alpha))^{\prime}=\Psi_{1}(\alpha)$)
$\displaystyle=\frac{\Psi_{1}^{\prime}(\alpha)}{2}\lambda^{2}+o(\lambda^{2}).$

The last line is bounded between $\tfrac{1}{K_{1}}\lambda^{2}$ and
$K_{1}\lambda^{2}$, provided $\lambda_{1}$ is fixed sufficiently small. And
the constant $K_{1}$ can be chosen uniformly for all $\alpha$ from the compact
interval $[\epsilon,\mu-\epsilon]$ because $\Phi_{1}$ is a smooth function on
$\mathbb{R}_{\geq 0}$. ∎

###  A.2 Monotonicity results for the polymer model

Recall the definition of a polymer with initial condition $W$ from Section 1.1
and the exit time $\tau$ defined at the beginning of Section 2.4. We state two
monotonicity results for the ratios of partition functions. In this section,
the weights can take any arbitrary positive values.

The first proposition gives certain monotonicity between the exit times and
ratios of the partition function. The proof follows from a “corner flipping”
induction. For a similar result in the setting of two different southwest
boundaries (instead of for different exit times), see Lemma A.1 from [12].

######  Proposition A.4.

Fix a positive initial condition $W$ with $W_{0}=1$. Let for each ${\bf
a}\in\mathbb{Z}^{2}_{>0}$ and $k\geq 1$,

$\frac{Z^{W}_{{\bf a}}(|\tau|\leq k)}{Z^{W}_{{\bf a}-{\bf e_{1}}}(|\tau|\leq
k)}\leq\frac{Z^{W}_{{\bf a}}(\tau\geq 0)}{Z^{W}_{{\bf a}-{\bf e_{1}}}(\tau\geq
0)}\qquad\text{ and }\qquad\frac{Z^{W}_{{\bf a}}(|\tau|\leq k)}{Z^{W}_{{\bf
a}-{\bf e_{2}}}(|\tau|\leq k)}\geq\frac{Z^{W}_{{\bf a}}(\tau\geq
0)}{Z^{W}_{{\bf a}-{\bf e_{2}}}(\tau\geq 0)}.$

###### Proof.

Fix $W$, let us define two more boundary weights $W^{(1)}$ and $W^{(2)}$,
given by

$W^{(1)}_{i}=\begin{cases}W_{i}\quad&\text{for }|i|\leq k\\\
0\quad&\text{otherwise }\end{cases}\qquad\text{ and }\qquad
W^{(2)}_{i}=\begin{cases}W_{i}\quad&\text{for }i\geq 0\\\ 0\quad&\text{for
}i\leq-1\end{cases}.$

Then, $Z^{W}_{{\bf x}}(|\tau|\leq k)=Z_{{\bf x}}^{W^{(1)}}$ and $Z_{{\bf
x}}^{W}(\tau\geq 0)=Z_{{\bf x}}^{W^{(2)}}$. In the ratio calculation below,
let us follow the convention $0/0=0$, then for each $i\in\mathbb{Z}_{>0}$, it
holds that

$\displaystyle\frac{\log Z^{W^{(1)}}_{(i,-i)}}{\log
Z^{W^{(1)}}_{(i-1,-i+1)}}=\frac{W^{(1)}_{i}}{W^{(1)}_{i-1}}$
$\displaystyle\leq\frac{W^{(2)}_{i}}{W^{(2)}_{i-1}}=\frac{\log
Z^{W^{(2)}}_{(i,-i)}}{\log Z^{W^{(2)}}_{(i-1,-i+1)}}$ (A.3)
$\displaystyle\frac{\log Z^{W^{(1)}}_{(-i,i)}}{\log
Z^{W^{(1)}}_{(-i+1,i-1)}}=\frac{W^{(1)}_{-i}}{W^{(1)}_{-i+1}}$
$\displaystyle\geq\frac{W^{(2)}_{-i}}{W^{(2)}_{-i+1}}=\frac{\log
Z^{W^{(2)}}_{(-i,i)}}{\log Z^{W^{(2)}}_{(-i+1,i-1)}}.$ (A.4)

These will be the base case of our induction.

Next, using the inductive relation $Z^{W}_{{\bf x}}=(Z^{W}_{{\bf x}-{\bf
e}_{1}}+Z^{W}_{{\bf x}-{\bf e}_{2}})Y_{\bf x},$ and the base case (A.3),
(A.4), we have

$\displaystyle\frac{Z^{W^{(1)}}_{{\bf x}}}{Z^{W^{(1)}}_{{\bf x}-{\bf
e_{1}}}}=\Big{(}1+\frac{Z^{W^{(1)}}_{{\bf x}-{\bf e}_{2}}}{Z^{W^{(1)}}_{{\bf
x}-{\bf e_{1}}}}\Big{)}Y_{\bf x}\leq\Big{(}1+\frac{Z^{W^{(2)}}_{{\bf x}-{\bf
e}_{2}}}{Z^{W^{(2)}}_{{\bf x}-{\bf e_{1}}}}\Big{)}Y_{\bf
x}=\frac{Z^{W^{(2)}}_{{\bf x}}}{Z^{W^{(2)}}_{{\bf x}-{\bf e_{1}}}}$
$\displaystyle\frac{Z^{W^{(1)}}_{{\bf x}}}{Z^{W^{(1)}}_{{\bf x}-{\bf
e_{2}}}}=\Big{(}\frac{Z^{W^{(1)}}_{{\bf x}-{\bf e}_{1}}}{Z^{W^{(1)}}_{{\bf
x}-{\bf e_{2}}}}+1\Big{)}Y_{\bf x}\geq\Big{(}\frac{Z^{W^{(2)}}_{{\bf x}-{\bf
e}_{1}}}{Z^{W^{(2)}}_{{\bf x}-{\bf e_{2}}}}+1\Big{)}Y_{\bf
x}=\frac{Z^{W^{(2)}}_{{\bf x}}}{Z^{W^{(2)}}_{{\bf x}-{\bf e_{2}}}}.$

∎

The second proposition is similar to the first one, and it has appeared as
Lemma A.4 from [31] for the polymer model with south-west boundary. The same
“corner flipping” induction there also applies to the anti-diagonal boundary,
so we omit the proof.

######  Proposition A.5.

Fux a positive initial condition $W$ with $W_{0}=1$. Let for each ${\bf
a}\in\mathbb{Z}^{2}_{>}0$ and $k\geq 1$,

$\frac{Z^{W}_{{\bf a}}(\tau\geq 0)}{Z^{W}_{{\bf a}-{\bf e_{1}}}(\tau\geq
0)}\leq\frac{Z^{W}_{{\bf a}}(\tau\geq k)}{Z^{W}_{{\bf a}-{\bf e_{1}}}(\tau\geq
k)}\qquad\text{ and }\qquad\frac{Z^{W}_{{\bf a}}(\tau\geq 0)}{Z^{W}_{{\bf
a}-{\bf e_{2}}}(\tau\geq 0)}\geq\frac{Z^{W}_{{\bf a}}(\tau\geq k)}{Z^{W}_{{\bf
a}-{\bf e_{2}}}(\tau\geq k)}.$

The last result is a specific instance of Lemma A.7 from [31]. While this
result applies to positive weights of any value, we will present it within the
context of the stationary inverse-gamma polymer. Because its statement is
better suited to the definition of the partition function as outlined in
Section 5 (instead of Section 1.1).

######  Proposition A.6.

Fix ${\bf a}\in\mathbb{Z}^{2}$. Then, for each $\rho\in(0,\mu)$,
$k\in\mathbb{Z}$, ${\bf z}\in\mathcal{L}_{\bf a}^{\geq}$, ${\bf
w}\in\mathcal{L}_{\bf a}^{>}$ and $i=1,2$

$Q^{\rho}_{{\bf a},{\bf w}}\\{\text{paths go through $[\\![{\bf z},{\bf
z}+e_{i}]\\!]$}\\}=Q^{\rho}_{{\bf a}+\underline{{k}},{\bf w}}\\{\text{paths go
through $[\\![{\bf z},{\bf z}+e_{i}]\\!]$}\\}.$

###  A.3 Assumption A implies Assumption B

######  Proposition A.7.

Let $X_{i}$ be a sequence of independent mean $0$ random variables satisfying
the hypotheses of Assumption A. Then $\\{W_{k}\\}_{k\in\mathbb{Z}}$ defined by
$W_{0}=1$ and $X_{i}=\log W_{i}-\log W_{i-1}$ satisfy Assumption B1,
Assumption B2 and Assumption B3.

###### Proof.

Since $X_{i}$ are independent, many of the hypotheses in Assumption B2 and
Assumption B3 are redundant. Indeed, (1.5) and (1.6) follow from (1.2) and
(1.4) respectively. The mixing hypothesis (1.3) is also a trivial consequence
of independence. The FKG inequality hypothesis also follows from the standard
FKG inequality for product measures. In (1.7) the conditioning is redundant
and $\textup{$\mathbb{V}$ar}(\log W_{N})\geq C_{1}N$ follows from the
hypothesis that $\inf_{i}\textup{$\mathbb{V}$ar}X_{i}>0$. It therefore only
remains to verify (1.2), (1.4) and (1.8).

For (1.2) and (1.4), notice that since $\mathbb{E}X_{i}=0$, the first
hypothesis of Assumption A implies that $|X_{i}|$ are uniformly sub-
exponential. Using Theorem A.1, which is a consequence of Doob maximal
inequality, we get (1.2) and (1.4). For (1.8), we appeal to a Berry-Essen
theorem for sums of non i.i.d. variables (see e.g. [10]). Since
$\mathbb{E}X_{i}=0$ and $\sup_{i}\mathbb{E}|X_{i}|^{3}<\infty$ (by the
subexponentiality hypothesis), it follows that for $Z\sim N(0,1)$,

$\sup_{t\in\mathbb{R}}\mathbb{P}(\log W_{n}\geq
t\sqrt{\textup{$\mathbb{V}$ar}(\log W_{n})})-\mathbb{P}(Z\geq t)=O(n^{-1/2}).$

Since $\textup{$\mathbb{V}$ar}\log W_{n}\in[C_{1}N,C_{2}N]$ for some $C_{1}>0$
(by the second hypothesis in Assumption A) and $C_{2}<\infty$ (by the first
hypothesis in Assumption A), (1.8) follows. ∎

## References

  * [1] Ian Alevy and Arjun Krishnan, _Negative correlation of adjacent Busemann increments_ , Ann. Inst. Henri Poincaré Probab. Stat. 58 (2022), no. 4, 1942–1958. MR 4492966
  * [2] Jinho Baik, Percy Deift, Ken T.-R. McLaughlin, Peter Miller, and Xin Zhou, _Optimal tail estimates for directed last passage site percolation with geometric random variables_ , Adv. Theor. Math. Phys. (2001).
  * [3] Jinho Baik, Patrik Lino Ferrari, and Sandrine Péché, _Convergence of the two-point function of the stationary tasep_ , pp. 91–110, Springer International Publishing, 2014.
  * [4] Márton Balázs, Ofer Busani, and Timo Seppäläinen, _Non-existence of bi-infinite geodesics in the exponential corner growth model_ , Forum Math. Sigma 8 (2020), Paper No. e46, 34. MR 4176750
  * [5] Márton Balázs, Eric Cator, and Timo Seppäläinen, _Cube root fluctuations for the corner growth model associated to the exclusion process_ , Electron. J. Probab. 11 (2006), no. 42, 1094–1132 (electronic). MR 2268539
  * [6] Riddhipratim Basu, Ofer Busani, and Patrik L. Ferrari, _On the exponent governing the correlation decay of the $\rm Airy_{1}$ process_, Comm. Math. Phys. 398 (2023), no. 3, 1171–1211. MR 4561801
  * [7] Riddhipratim Basu and Shirshendu Ganguly, _Time correlation exponents in last passage percolation_ , In and out of equilibrium 3. Celebrating Vladas Sidoravicius, Progr. Probab., vol. 77, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 101–123. MR 4237265
  * [8] Riddhipratim Basu, Shirshendu Ganguly, and Lingfu Zhang, _Temporal correlation in last passage percolation with flat initial condition via Brownian comparison_ , Comm. Math. Phys. 383 (2021), no. 3, 1805–1888. MR 4244262
  * [9] Riddhipratim Basu, Timo Seppäläinen, and Xiao Shen, _Temporal correlation in the inverse-gamma polymer_ , 2023, arXiv:2305.11086.
  * [10] Kh. Batirov, D. V. Manevich, and S. V. Nagaev, _The esseen inequality for sums of a random number of differently distributed random variables_ , Mathematical notes of the Academy of Sciences of the USSR 22 (1977), no. 1, 569–571.
  * [11] Richard C. Bradley, _Basic properties of strong mixing conditions. A survey and some open questions_ , Probab. Surv. 2 (2005), 107–144, Update of, and a supplement to, the 1986 original. MR 2178042
  * [12] Ofer Busani and Timo Seppäläinen, _Non-existence of bi-infinite polymers_ , Electron. J. Probab. 27 (2022), Paper No. 14, 40. MR 4372098
  * [13] Eric Cator and Piet Groeneboom, _Hammersley’s process with sources and sinks_ , Ann. Probab. 33 (2005), no. 3, 879–903. MR 2135307 (2006b:60216)
  * [14] Ivan Corwin and Promit Ghosal, _KPZ equation tails for general initial data_ , Electronic Journal of Probability 25 (2020), no. none, 1 – 38\.
  * [15] Elnur Emrah, Nicos Georgiou, and Janosch Ortmann, _Coupling derivation of optimal-order central moment bounds in exponential last-passage percolation_ , 2022, arXiv:2204.06613.
  * [16] Elnur Emrah, Chris Janjigian, and Timo Seppäläinen, _Right-tail moderate deviations in the exponential last-passage percolation_ , 2020, arXiv:2004.04285.
  * [17] Elnur Emrah, Christopher Janjigian, and Yongjia Xie, _Moderate deviation and exit point estimates for solvable directed polymer models_ , (2023+), Forthcoming.
  * [18] P. L. Ferrari and A. Occelli, _Time-time covariance for last passage percolation with generic initial profile_ , Math. Phys. Anal. Geom. 22 (2019), no. 1, Paper No. 1, 33. MR 3895778
  * [19] Patrik L. Ferrari and Alessandra Occelli, _Time-time covariance for last passage percolation in half-space_ , 2022, arXiv:2204.06782.
  * [20] Patrik L. Ferrari and Herbert Spohn, _On time correlations for KPZ growth in one dimension_ , SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 074, 23. MR 3529743
  * [21] Nicos Georgiou, Firas Rassoul-Agha, Timo Seppäläinen, and Atilla Yilmaz, _Ratios of partition functions for the log-gamma polymer_ , Ann. Probab. 43 (2015), no. 5, 2282–2331. MR 3395462
  * [22] Takashi Imamura and Tomohiro Sasamoto, _Free energy distribution of the stationary O’Connell-Yor directed random polymer model_ , J. Phys. A 50 (2017), no. 28, 285203, 35. MR 3673307
  * [23] Christopher Janjigian and Firas Rassoul-Agha, _Busemann functions and Gibbs measures in directed polymer models on $\mathbb{Z}^{2}$_, Ann. Probab. 48 (2020), no. 2, 778–816. MR 4089495
  * [24] Kurt Johansson, _Shape fluctuations and random matrices_ , Comm. Math. Phys. 209 (2000), no. 2, 437–476. MR 1737991 (2001h:60177)
  * [25] Johannes HB Kemperman, _On the FKG-inequality for measures on a partially ordered space_ , Indagationes Mathematicae (Proceedings), vol. 80, North-Holland, 1977, pp. 313–331.
  * [26] Benjamin Landon and Philippe Sosoe, _Tail bounds for the O’Connell-Yor polymer_ , 2022, arXiv:2209.12704.
  * [27] Benjamin Landon and Philippe Sosoe, _Upper tail bounds for stationary KPZ models_ , (2022), arXiv:2208.01507.
  * [28] Michel Ledoux and Brian Rider, _Small deviations for beta ensembles_ , Electron. J. Probab. 15 (2010), no. 41, 1319–1343. MR 2678393
  * [29] Matthias Löwe, Franz Merkl, and Silke Rolles, _Moderate deviations for longest increasing subsequences: the lower tail_ , J. Theoret. Probab. 15 (2002), no. 4, 1031–1047. MR 1937784
  * [30] , _Moderate deviations for longest increasing subsequences: the lower tail_ , J. Theoret. Probab. 15 (2002), no. 4, 1031–1047. MR 1937784
  * [31] Firas Rassoul-Agha, Timo Seppäläinen, and Xiao Shen, _Coalescence and total-variation distance of semi-infinite inverse-gamma polymers_ , 2023, arXiv:2305.10265.
  * [32] Timo Seppäläinen, _Scaling for a one-dimensional directed polymer with boundary conditions_ , Ann. Probab. 40 (2012), no. 1, 19–73, Corrected version available at arXiv:0911.2446. MR 2917766
  * [33] Timo Seppäläinen and Xiao Shen, _Coalescence estimates for the corner growth model with exponential weights_ , Electron. J. Probab. 25 (2020), 31 pp., Corrected version available at arXiv:1911.03792.
  * [34] Subhendu B Singha, _Persistence of surface fluctuations in radially growing surfaces_ , Journal of Statistical Mechanics: Theory and Experiment 2005 (2005), no. 08, P08006.
  * [35] Kazumasa A Takeuchi, _Statistics of circular interface fluctuations in an off-lattice eden model_ , Journal of Statistical Mechanics: Theory and Experiment 2012 (2012), no. 05, P05007.
  * [36] Kazumasa A. Takeuchi and Masaki Sano, _Evidence for geometry-dependent universal fluctuations of the kardar-parisi-zhang interfaces in liquid-crystal turbulence_ , Journal of Statistical Physics 147 (2012), no. 5, 853–890.
  * [37] Anru R. Zhang and Yuchen Zhou, _On the non-asymptotic and sharp lower tail bounds of random variables_ , Stat 9 (2020), e314, 11. MR 4193419


# Non-local $R^{2}$-like inflation, Gravitational Waves and Non-Gaussianities

K. Sravan Kumar∗ Department of Physics, Tokyo Institute of Technology
1-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
∗E-mail<EMAIL_ADDRESS>
www.titech.ac.jp/english

###### Abstract

The emergence of $R^{2}$ (Starobinsky) inflation from the semi-classical
modification of gravity due to matter quantum fields (trace anomaly) clearly
points out the importance of fundamental physics and the first principles in
the construction of successful cosmological models. Along with the
observational success, $R^{2}$ gravity is also an important step beyond
general relativity (GR) towards quantum gravity. Furthermore, several
approaches of quantum gravity to date are strongly indicating the presence of
non-locality at small time and length scales. In this regard, ultraviolet (UV)
completion of $R^{2}$ inflation has been recently studied in a string theory-
inspired ghost-free analytic non-local gravity. We discuss the promising
theoretical predictions of non-local $R^{2}$-like inflation with respect to
the key observables such as tensor-to-scalar ratio, tensor tilt which tell us
about the spectrum of primordial gravitational waves, and scalar Non-
Gaussianities which tell us about the three-point correlations in the CMB
fluctuations. Any signature of non-local physics in the early Universe will
significantly improve our understanding of fundamental physics at UV energy
scales and quantum gravity.

###### keywords:

Quantum gravity, Inflation, CMB, Gravitational Waves

##  1 Status of inflation and the quests for UV-completion

Inflationary cosmology has become the most important field of study in
theoretical higher energy physics (THEP) in the recent years not only because
of its success with respect to the data of Cosmic Microwave Background (CMB)
[1, 2, 3] but also it gives a best chance to test our approaches to build
physics at the fundamental scales such as Planck energy or length scales.
Recent detection of Gravitational Waves from binary black hole merger events
further corroborated to build new hopes and strategies to probe the physics of
early Universe [4, 5].

Inflationary cosmology is so far developed in two equivalent ways. One is by
geometrical modification i.e., by extension of General Relativity (GR) by
higher curvature terms (for example ”$R+R^{2}$ gravity which is also known as
Starobinsky theory [6, 7, 8, 9]). Second way is by the addition of
hypothetical (scalar) matter fields which result in modification of right hand
side of Einstein equations [10, 11]. This second way motivated in the view of
particle physics beyond the standard model and several ultraviolet (UV)
complete approaches such as string theory and supergravity (SUGRA) [12]. In
this regard, inflationary cosmology is so far studied from the point of view
of beyond the standard model of particle physics and as a low-energy limit of
UV-complete approaches such as string theory/M-theory (see right panel of Fig.
1). In the last decades plethora of inflationary models have been constructed
with different scalar fields emerging from simplest set ups to all the way up
to the most general scalar tensor theories [13, 14] (see left panel of Fig.
1). However the latest data from Planck satellite [1] has found out
Starobinsky and Higgs inflation be the most compatible models with the
following predictions of spectral index and the tensor to scalar ratio

$n_{s}=1-\frac{2}{N}\,,\qquad r=\frac{12}{N^{2}}\,,$ (1)

where $N=50-60$ number of e-folds before the end of inflation. The recent
Planck data [1, 2] has constrained $n_{s}=0.9649\pm
0.0042\,\,\textrm{at}\,\,68\%\textrm{CL},r<0.036\,\,\textrm{at}\,\,95\%\,\textrm{CL}$.
The success of Starobinsky and Higgs inflation has lead to the emergence of
equivalent frameworks of inflation in string theory and SUGRA [12].

Figure 1: In the left panel, arrow of complexity shows how one can start with
a simplest scalar field Lagrangian and generalized all the way up to most
general scalar-tensor theories whose equations of motion are second order in
field derivative. In the right panel we can see that inflationary cosmology
can be understood well either from beyond the standard model of particle
physics or from low-energy limit of a UV-complete approaches such as string
theory (This Figure taken from [13]).

The challenging problem of the current status of inflationary cosmology is to
break several degeneracies in the frameworks of inflation and find definitive
clues towards achieving quantum gravity and UV-completion. For this purpose,
from observational point of view, one must go beyond the two observables
$(n_{s},\,r)$. In this regard there are two more important observational
parameters known as tensor tilt ($n_{t}$) which is the tilt of the tensor
power-spectrum and non-Gaussianity parameter ($f_{NL}$) which is related to
the 3-point scalar correlations. If inflation is driven by single (canonical)
scalar field with standard slow-roll then the tensor consistency relation [13]
and the Maldacena consistency relation [15] given by

$r=-8n_{t}\quad f^{\rm sq}_{NL}=\frac{5}{12}(1-n_{s})$ (2)

serves as a crucial test. Here $f^{\rm sq}_{NL}$ is so-called squeezed shape
of non-Gaussianity. If inflation is driven by multiple fields then the above
consistency relations are understood to be violated [16]. If inflation is
driven by non-canonical scalar fields and if non-standard quantum initial
conditions are assumed we will find a strong signal in the equilateral
($f^{\rm eq}_{NL}$) and orthogonal ($f^{\rm ortho}_{NL}$) shapes of non-
Gaussianities. These shapes are constrained by the latest Planck data as

$f_{\rm NL}^{\rm loc}=0.8\pm 5.0\,,\quad f_{\rm NL}^{\rm equi}=-4\pm
43\,,\quad f_{\rm NL}^{\rm ortho}=-26\pm 21\,,\quad{\rm at}\,\,\,68\%{\rm
CL}.$ (3)

Violation of consistency relations (2) and any detection of $f_{NL}\sim O(1)$
is definitely a new physics in the context of early Universe cosmology which
will be crucial for further advancements in THEP. Moreover, in a quest to find
signatures of new physics, recently a new program of research has gained
significant attention known as ”cosmological collider physics” [17] whose aim
is to find signatures of higher energy particles interacting with primordial
fields during inflation.

Despite numerous phenomenological approaches in understanding inflation and
find new observables to probe physics of inflation, it is very vital to focus
on consistency of our theoretical endeavors. The success of Starobinsky and
Higgs inflationaris (which are almost identical during inflation) covey us a
very crucial property that at the scale of inflation approximate ”scale
invariance” symmetry of the Lagrangian is important. The fact that various
setups of SUGRA leading to Starobinsky-like or $R^{2}$-like inflation [12]
indicate that $R+R^{2}$ is an important modification of GR. In the next
section, we will discuss in detail about the foundations of Starobinsky
inflation and how a very straightforward procedure from geoemertric
modification of gravity lead us towards a UV-complete framework [18, 19, 20,
21].

##  2 From Starobinsky inflation to Stelle gravity and the emergence of ghost-
free UV-non-local gravity

$R+R^{2}$ gravity should not be viewed as a toy model in the class of f(R)
gravity. The theory is well-motivated from generic considerations and
important questions about UV-completion which dates back the discovery of
trace anomaly [22]. The trace anomaly comes from 1-loop quantum corrections to
the graviton propagator due to matter conformal fields leading to an anomalous
non-zero trace of energy momentum tensor which is expressed as

$M_{p}^{2}R=g^{\mu\nu}\langle T_{\mu\nu}\rangle=aF+bG+c\square R\,,$ (4)

where $F=W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}$ is the the Weyl tensor
square and
$G=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$ is
the so-called Gauss-Bonnet term and $R_{\mu\nu\rho\sigma}$, $R_{\mu\nu}$, $R$
Riemann tensor, Ricci tensor and Ricci scalar respectively. Here $a,b,c$ are
dimensionless coefficients quantified by the number of conformal fields. Trace
anomaly has been widely studied and applied in various fundamental theories
including anticipated UV complete theories such as string theory and
supergravity (SUGRA) [22]. Neglecting the coefficients $a,b$ the trace anomaly
equation reduces to the trace equation of the $R+R^{2}$ given by the action
[6, 23]

$S_{R^{2}}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R+\frac{M_{p}^{2}}{12M^{2}}R^{2}\right]\,.$
(5)

where $M\ll M_{p}$ becomes the mass of a propagating scalar in this model,
named scalaron. Independent of what is the right framework of UV-completion
$R^{2}$ term is very natural term that appears as a classical addition to the
GR and it naturally leads to a quasi-de Sitter evolution (which is ”cosmic
inflation”) in the high curvature regime $R\gg M^{2}$. Therefore it is a
matter of fact that $R+R^{2}$ gravity indeed an unavoidable and very important
extension of GR. Furthermore, if we add Weyl square term to the action of
$R+R^{2}$ gravity (5) the resultant gravity theory becomes renormalizable
(which is an important property to achieve quantum gravity) as it was proven
by K. S. Stelle long ago [24]

$S_{\rm Stelle}=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R+\frac{M_{p}^{2}}{12M^{2}}R^{2}+\frac{M_{p}^{2}}{M_{g}^{2}}W_{\mu\nu\rho\sigma}W^{\mu\nu\rho\sigma}\right]$
(6)

However, Stelle gravity unfortunately falls short in becoming a consistent
theory of quantum gravity because of the presence of a massive tensor ghost in
the spectrum which spoils Unitarity. A natural question one can ask is that
whether this ghost can be removed by any higher derivative extension of Stelle
gravity. Due to Ostrogradsky theorem any finite derivative extension of Stelle
gravity results in further introduction of ghosts in the spectrum and it
becomes not a viable option. However, Krasnikov has shown that analytic
infinite derivative extension of Stelle gravity (i.e., analytic non-local
gravity) can give a ghost-free theory [25].

$S=\int
d^{4}x\sqrt{-g}\left(\frac{M_{p}^{2}}{2}R+\frac{1}{2}\bigg{[}R\mathcal{F}_{R}\left(\square_{s}\right)R+W_{\mu\nu\rho\sigma}\mathcal{F}_{W}\left(\square_{s}\right)W^{\mu\nu\rho\sigma}\bigg{]}\right)\,.$
(7)

where $\square_{s}=\frac{\square}{\mathcal{M}_{s}^{2}}$ with $\square$ is
being the d’Alembertian and $\mathcal{F}_{R},\mathcal{F}_{W}$ are analytic
infinite derivative operators or non-local form factors. This theory (7) has
been studied over the years and understood to have good properties of UV-
completion with respect to renormalizability and Unitarity (See [21] and the
references therein). In a nutshell, the action (7) is a very logical extension
of Stelle gravity and we very much straightforwardly end up with introducing
non-locality in the gravity sector. Note that ”Non-locality” is a very
important common concept that appears in several approaches to quantum gravity
such as string theory, loop quantum gravity, causal sets and non-commutative
gravity [21, 26]. Therefore, (7) is a consistent culmination of various
approaches which allow us to give a consistent UV-completion for Starobinsky
inflation111Worth to metion that Starobinsky inflation has also been embedded
in SUGRA which gives an interesting route for UV-completion [27]. Here we
focus on non-supersymmetric aspects of UV-completion.. Further note that (7)
should not be understood as an effective theory but rather a consistent part
of a UV-complete or a fundamental theory. We do not have any cut off scales in
the theory and the scale $\mathcal{M}_{s}$ represents the scale at which non-
local nature of gravity is prominent. In the next section, we will discuss the
cosmological application of (7) and identify the specific features of non-
local gravity which can be probed in the future observations.

##  3 Non-local $R^{2}$-like inflation and future targets for CMB and
Gravitatinal wave observations

In this section, we apply UV non-local quadratic curvature gravity in the
context of inflationary cosmology. The first and foremost difficulty is
associated with solving equations of motion of an infinite derivative theory
which perhaps appears to be an impossible task. However, this was consistency
achieved [18] and it was also quite rigorously shown in [19] that the theory
contains an inflationary attractor solution exactly same as the local
$R+R^{2}$ or Starobinsky theory. To see this explicitly let us write down the
background equations of motion for (7) Friendmann-Lemaître-Robertson-Walker
background for which the Weyl tensor is zero

$\displaystyle\bar{E}_{\nu}^{\mu}\equiv$
$\displaystyle-\left[M_{p}^{2}+2\lambda\mathcal{F}\left(\frac{\square}{\mathcal{M}_{s}^{2}}\right)\bar{R}\right]\bar{G}_{\nu}^{\mu}\delta^{\mu}_{\nu}-\frac{1}{2}\bar{R}\mathcal{F}\left(\frac{\square}{\mathcal{M}_{s}^{2}}\right)\bar{R}\delta_{\nu}^{\mu}$
$\displaystyle+2\left(\nabla^{\mu}\partial_{\nu}-\delta_{\nu}^{\mu}\square\right)\mathcal{F}\left(\frac{\square}{\mathcal{M}_{s}^{2}}\right)R+\mathcal{\bar{K}}_{\nu}^{\mu}-\frac{\lambda}{2}\delta_{\nu}^{\mu}\left(\mathcal{\bar{\mathcal{K}}}_{\sigma}^{\sigma}+\bar{\mathcal{K}}\right)=0\,,$

where

$\displaystyle\mathcal{\bar{\mathcal{K}}}_{\nu}^{\mu}=\frac{1}{\mathcal{M}_{s}^{2}}\sum_{n=1}^{\infty}f_{n}\sum_{l=0}^{n-1}\partial^{\mu}\frac{\square^{l}}{\mathcal{M}_{s}^{2l}}\bar{R}\partial_{\nu}\left(\frac{\square}{\mathcal{M}_{s}^{2}}\right)^{n-l-1}\bar{R}\,,\bar{\mathcal{K}}=\sum_{n=1}^{\infty}f_{n}\sum_{l=0}^{n-1}\frac{\square^{l}}{\mathcal{M}_{s}^{2l}}\bar{R}\left(\frac{\square}{\mathcal{M}_{s}^{2}}\right)^{n-l}\bar{R}\,.$

The trace equation is

$\bar{E}=M_{p}^{2}\bar{R}-6\bar{\square}\mathcal{F}\left(\frac{\bar{\square}}{\mathcal{M}_{s}^{2}}\right)\bar{R}-\bar{\mathcal{K}}_{\mu}^{\mu}-2\bar{\mathcal{K}}=0\,.$
(8)

The overbars in (3) and (8) is used to indicate background quantities. It was
showed in [19] that the only solution of the above equations is
$\bar{\square}\bar{R}=M^{2}\bar{R}$ given the following Unique conditions on
the form factor are satisfied.

$\mathcal{F}^{\prime}_{R}\left(\frac{M^{2}}{M_{s}^{2}}\right)=0,~{}\frac{M_{P}^{2}}{2\lambda}=3M^{2}\mathcal{F}_{1},~{}\text{
where
}\mathcal{F}_{1}\equiv\mathcal{F}_{R}\left(\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)$
(9)

where ′ denotes derivative with respect to argument. Note that
$\bar{\square}\bar{R}=M^{2}\bar{R}$ is the trace equation of local $R+R^{2}$
which gives Starobinsky inflationary solution. Here given the conditions (9)
are met we can solve exactly the full non-local gravity equations of motion
and also we have good inflationary solution. Now it is important to see how
non-local effects of gravity appear through inflationary perturbations. Before
we go this step we can draw a heuristic bound on the scale of non-locality
shown in Fig. 2 based on the fact that we require near scale invariance at the
scale of inflation.

Figure 2: Hierarchy of energy scales in non-local $R^{2}$-like inflation.

This can be seen heuristically by expanding the quadratic Ricci scalar part of
the action as

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R+\frac{M_{p}^{2}}{12M^{2}}R^{2}+\mathcal{O}\left(\frac{M_{p}^{2}R\square
R}{M^{2}\mathcal{M}_{s}^{2}}\right)\right]\,.$

In the high curvature regime $R\gg M^{2}$ local quadratic curvature term is
naturally dominant and it is known to be scale invariance.

###  3.1 Scalar and tensor power spectrum

To study perturbations around quasi-de Sitter we take the perturbed metric of
the form

$ds^{2}=a^{2}\left(\eta\right)\left[-\left(1+2\Phi\right)d\eta^{2}+\left(\left(1-2\Psi\right)\delta_{ij}+2h_{ij}\right)dx^{i}dx^{j}\right]\,.$
(10)

where $\Phi,\,\Psi$ are Bardeen potentials and $h_{ij}$ is tensor fluctuation.
The study of non-local perturbed equations of motion [19] has revealed that
$\Phi+\Psi\approx 0$ during inflation. Since $\delta
W_{\mu\nu\rho\sigma}\propto\left(\Phi+\Psi\right)$ the Weyl tensor part in (7)
does not contribute to the scalar perturbations during inflation. Following
the computations in [19, 20] the second order action of scalar degree of
freedom in de Sitter approximation becomes

$\delta^{2}S_{(s)}=\frac{1}{2\mathcal{F}_{1}\bar{R}}\int d\tau
d^{3}x\sqrt{-\bar{g}}\Upsilon\frac{\mathcal{W}\left(\frac{\bar{\square}}{\mathcal{M}^{2}}\right)}{\mathcal{F}_{R}\left(\frac{\bar{\square}}{\mathcal{M}^{2}}\right)}(\bar{\square}-M^{2})\Upsilon\,.$
(11)

where $\Upsilon\approx 2\bar{R}\Psi$. In order not to have any more ghost
degrees of freedom we choose the structure of form factor as

$\mathcal{F}_{R}\left(\square_{s}\right)=\mathcal{F}_{1}\frac{3e^{\gamma_{S}\left(\square_{s}\right)}\left(\square_{s}-\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)+\left(\frac{\bar{R}}{\mathcal{M}_{s}^{2}}+3\frac{M^{2}}{\mathcal{M}_{s}^{2}}\right)}{3\square_{s}+\frac{\bar{R}}{\mathcal{M}_{s}^{2}}}\,.$
(12)

where $\bar{R}\approx{\rm constant}$ and $\gamma_{S}$ is an entire function.
Therefore, the theory now has only one scalar(on) degree of freedom similar to
local $R+R^{2}$ theory but however, the the scalar field here is non-local in
nature.

Computation of the power spectrum and the scalar spectral index of curvature
perturbation $\mathcal{R}=\Psi+H\frac{\delta
R}{\dot{\bar{R}}}=\Psi+\frac{24H^{3}}{24H\dot{H}}\Phi\approx-\frac{1}{\epsilon}\Psi$
we get

$\left.\mathcal{P}_{\mathcal{R}}=\frac{H^{2}}{16\pi^{2}\epsilon^{2}}\frac{1}{3\lambda\mathcal{F}_{1}\bar{R}}\right|_{k=aH}\,,\quad
n_{s}\equiv\left.\frac{d\ln\mathcal{P}_{\mathcal{R}}}{d\ln
k}\right|_{k=aH}\approx 1-\frac{2}{N}\,,$ (13)

Similarly the computation of the second order action of tensor perturbations
gives (with addition of the non-trivial contributions from the Weyl tensor
part in (7))

$\displaystyle\delta^{2}S_{(T)}=$ $\displaystyle\frac{1}{2}\int
d^{4}x\sqrt{-\bar{g}}h_{ij}^{\perp}e^{2\gamma_{T}\left(\square_{s}-\frac{2\bar{R}}{3\mathcal{M}_{s}^{2}}\right)}\left(\bar{\square}-\frac{\bar{R}}{6}\right)h^{\perp{ij}}\,,$
(14)

where

$\mathcal{F}_{W}\left(\square_{s}\right)=\frac{\mathcal{F}_{1}\bar{R}}{\mathcal{M}_{s}^{2}}\frac{e^{\gamma_{T}\left(\square_{s}-\frac{2}{3}\frac{\bar{R}}{\mathcal{M}_{s}^{2}}\right)}-1}{\square_{s}-\frac{2\bar{R}}{3\mathcal{M}_{s}^{2}}}\,.$
(15)

which fixes the tensor degrees of freedom to be just one massless graviton.

Computing the tensor power spectrum in the leading order in slow-roll we
obtain

$\left.\mathcal{P}_{\mathcal{T}}\right|_{k=aH}=\frac{H^{2}}{\pi^{2}\mathcal{F}_{1}\bar{R}}e^{-2\gamma_{T}\left(\frac{\bar{R}}{6\mathcal{M}_{s}^{2}}\right)}\,.$
(16)

As a result the key inflationary parameter called tensor-to-scalar ratio
becomes

$r=\frac{12}{N^{2}}e^{-2\gamma_{T}\left(\frac{-\bar{R}}{2\mathcal{M}_{s}^{2}}\right)}\Bigg{|}_{k=aH}$
(17)

The tilt of the tensor power spectrum can be obtained as

$\displaystyle n_{t}\equiv\left.\frac{d\ln\mathcal{P}_{\mathcal{T}}}{d\ln
k}\right|_{k=aH}$
$\displaystyle\approx\left.-\frac{d\ln\mathcal{P}_{\mathcal{T}}}{dN}\left(1+\frac{1}{2N}\right)\right|_{k=aH}$
(18)
$\displaystyle\approx\left.-\frac{3}{2N^{2}}-\left(\frac{2}{N}+\frac{3}{2N^{2}}\right)\frac{\bar{R}}{2\mathcal{M}_{s}^{2}}\gamma_{T}^{(\prime)}\left(-\frac{\bar{R}}{2\mathcal{M}_{s}^{2}}\right)\right|_{k=aH}\,,$

where ′ denotes derivative with respect to the argument. The above predictions
of non-local $R^{2}$-like inflation are depicted in Fig. 3 and Fig. 4.

Figure 3: We depict the predictions of non-local $R^{2}$-like inflation in the
$(n_{s},r)$ plane of the latest CMB S-4 science paper about future forecast of
detecting $B$-modes [28, 19, 21].  Figure 4: We note that the predictions of
non-local $R^{2}$-like inflation can be anywhere within the likelihood
projected $(n_{t},r)$ plane of latest Planck 2018 [29, 19, 21].

##  4 Primordial (scalar) non-Gaussianities

Beyond two point correlations it is interesting to see if there will be non-
Gaussianities in non-local $R^{2}$-like inflation. As we discussed before the
scalar(on) in the non-local $R^{2}$-like inflation is non-local in nature.
Therefore, we generally expect the non-local interactions to play a role in
predicting inflationary 3-point correlations different from the local
theories. The primordial (scalar) non-Gaussianities in the non-local
$R^{2}$-like inflation are computed in [20] and here we briefly review the
main results. The 3-point correlations and the parameter $f_{NL}$ are defined
by

$\displaystyle\langle\mathcal{R}\left(\mathbf{k_{1}}\right)\mathcal{R}\left(\mathbf{k_{2}}\right)\mathcal{R}\left(\mathbf{k_{3}}\right)\rangle=\left(2\pi\right)^{3}\delta^{3}\left(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}}\right)\mathcal{B}_{\mathcal{R}}\left(k_{1},k_{2},k_{3}\right)\,,$

and

$B_{\mathcal{R}}\left(k_{1},k_{2},k_{3}\right)=(2\pi)^{4}\frac{1}{\prod_{i}k_{i}^{3}}\mathcal{P}_{\mathcal{R}}^{\ast
2}f_{NL}\left(k_{1},\,k_{2},\,k_{3}\right)\,,$

where

$f_{NL}=-\frac{5}{6}\frac{A_{\mathcal{R}}\left(k_{1},\,k_{2},\,k_{3}\right)}{\sum_{i}k_{i}^{3}}\,.$
(19)

The interactions of the curvature perturbations in the non-local $R^{2}$-like
inflation can be schematically explained as in Fig. 5.

Figure 5: In the above plot
$\mathbb{R}=\\{\mathcal{R},\,\partial\mathcal{R},\,\mathcal{D}^{\mu}\partial\mathcal{R}\\}$
imply various tree level interactions of different modes of the curvature
perturbation in the local $R^{2}$ and the non-local $R^{2}$-like inflation.
$O\left(\frac{\bar{\square}}{\mathcal{M}_{s}^{2}}\right)$ is some analytic
non-local operator.

To calculate bi-spectrum, first we expand our action to cubic order to the
leading order in slow-roll parameter. Then we compute the correlations using
the following definition

$\langle\mathcal{R}\left(\mathbf{k_{1}}\right)\mathcal{R}\left(\mathbf{k_{2}}\right)\mathcal{R}\left(\mathbf{k_{3}}\right)\rangle=-i\int_{-\infty}^{\tau_{e}}ad\tau\langle
0|[\mathcal{R}(\tau_{e},\,\mathbf{k_{1}})\mathcal{R}(\tau_{e},\,\mathbf{k_{2}})\mathcal{R}(\tau_{e},\,\mathbf{k_{3}}),\,H_{int}]|0\rangle\,,$
(20)

where $\boldsymbol{k}_{i}$ are wave vectors and $H_{int}$ is the interaction
Hamiltonian. It is related to a perturbation of the Lagrangian (7) expanded up
to the 3rd order in curvature perturbations ($\mathcal{L}_{3}$) as
$H_{int}\approx-\mathcal{L}_{3}$. We consider the mode functions as

$\mathcal{R}\approx-\Psi/\epsilon,\quad\bar{\square}\mathcal{R}=M^{2}\mathcal{R}$
(21)

New Interactions arise via the commutation relation in de Sitter approximation

$\square\nabla_{\mu}\mathcal{R}=\nabla_{\mu}\square\mathcal{R}+\frac{\bar{R}}{4}\nabla_{\mu}\mathcal{R}.$
(22)

After numerous steps of computation the cubic order action in $\mathcal{R}$ of
(7) in the leading order slow-roll approximation is obtained as [20]

$\displaystyle\delta^{(3)}S_{(S)}=$ $\displaystyle 4\epsilon M_{p}^{2}\int
d\tau
d^{3}x\Bigg{\\{}T_{1}^{\ast}\mathcal{R}\nabla\mathcal{R}\cdot\nabla\mathcal{R}+T_{2}^{\ast}\mathcal{R}\mathcal{R}^{\prime
2}+T_{3}^{\ast}\mathcal{H}^{2}\mathcal{R}^{3}$ $\displaystyle+$ $\displaystyle
T_{4}^{\ast}\mathcal{H}\mathcal{R}\mathcal{R}\mathcal{R}^{\prime}+T_{5}^{\ast}\mathcal{H}^{-1}\nabla\mathcal{R}\cdot\nabla\mathcal{R}\mathcal{R}^{\prime}+T_{6}^{\ast}\mathcal{H}^{-1}\mathcal{R}^{\prime
3}$
$\displaystyle+T_{7}^{\ast}\mathcal{H}^{-2}\mathcal{R}^{\prime}\nabla\mathcal{R}\cdot\nabla\mathcal{R}^{\prime}\Bigg{\\}}\,,$

where $T_{i}$’s are dimensionless constants whose can be found in [20]

Computing different shapes of $f_{NL}$ parameter when $k=aH$ we get [20]

$\displaystyle f_{NL}^{sq}\Bigg{|}_{k=aH}$
$\displaystyle\approx\frac{5}{12}\left(1-n_{s}\right)-23\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}-1\right]-\frac{4\bar{R}}{\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{\prime}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)\,$
(23) $\displaystyle f_{NL}^{eq}\Bigg{|}_{k=aH}$
$\displaystyle\approx\frac{5}{12}\left(1-n_{s}\right)-49\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{S}^{2}}\right)}-1\right]-\frac{9\bar{R}}{\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{\prime}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)$
$\displaystyle f_{NL}^{ortho}\Bigg{|}_{k=aH}$
$\displaystyle\approx\frac{5}{12}\left(1-n_{s}\right)-43\epsilon^{2}\left[e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}-1\right]-\frac{22\bar{R}_{\ast}}{3\mathcal{M}_{s}^{2}}\epsilon^{3}e^{\gamma_{S}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)}\gamma_{S}^{\prime}\left(\frac{\bar{R}}{4\mathcal{M}_{s}^{2}}\right)\,.$

These characterize the non-local interactions of curvature perturbation.
Clearly from the above expressions single field Maldacena consistency relation
is violated in non-local theories and one can obtain detectable level of non-
Gaussianities due to non-local nature of interactions (See Fig. 6). This is
totally non-trivial effect in comparison with all the known mechanisms of
producing large non-Gaussianities in the local theories.

Figure 6: In the above plots, $f_{NL}$ versus the scale of non-locality
$\mathcal{M}_{s}$ (in the units of $M_{p}$) is depicted for squeezed (blue),
equilateral (red), and orthogonal (green) configurations for the polynomial
entire functions $\gamma_{T}$ given by eqs. (4.19) and (4.20) of
arXiv:2003.00629 and represented by solid and dashed lines respectively. Here
$N=55$ of $e$-foldings is assumed. In the limit $\mathcal{M}_{s}\to M_{p}$ the
predictions of the local $R^{2}$ model are recovered.

##  5 Concluding remarks

We conclude here with highlighting the notable results of early Universe
cosmology in the context of UV-non-local ghost-free theories of gravity. First
of all we note that the UV non-local gravity action (7) is a consistent and
necessary extension of Stelle gravity in order to achieve UV-completion based
on the first principles. Another point to note here is ”non-locality” is an
important concept that is needed for building a consistent theory of quantum
gravity. We have discussed how one can have a successful realization of
inflation with the action action (7) and how non-local nature of gravity can
be seen through inflationary observables. These are the important conclusions
we can draw here to probe non-local nature of gravity in the scope of future
CMB and gravitational wave observations [29, 28, 30, 16, 5].

  * •

In the non-local $R^{2}$-like inflation the primordial scalar power spectrum
remains same as in the case of local $R+R^{2}$ inflation. However the tensor
power spectrum is modified and one can have any value of tensor-to-scalar
ratio[2, 20, 21] $r<0.036$. As a result of modification of tensor power
spectrum the tensor tilt in this theory is modified and ultimately tensor
consistency relation $r=-8n_{t}$ is violated. Note that the violation of
consistency relation is purely a non-local effect, the sound speed of
primordial fluctuations are unity in this theory.

  * •

The blue tensor tilt $n_{t}>0$ is possible with non-local $R^{2}$-like
inflation. Since not many frameworks of inflation can predict blue tilt any
detection of it automatically favours non-local theory. Also $n_{t}<0$ can
also equally possible too.

  * •

Non-local nature of gravity leads to detectable range of non-Gaussianities
especial one can have squeezed limit of $f^{\rm eq}_{NL}\sim O(1)$ despite
there is only one scalar degree of freedom and nature of inflation is
standard-slow-roll. Further more, other shapes of $f_{NL}$ can also be large
despite the sound speed of primordial scalar field is unity.

In summary, non-local nature of gravity brings very unique predictions which
are indeed very interesting in the view of future observations as well as
further theoretical investigations.

##  6 Acknowledgments

KSK acknowledges financial support from JSPS and KAKENHI Grant-in-Aid for
Scientific Research No. JP20F20320.

## References

  * [1] Planck, Y. Akrami et al.,  (2018), 1807.06211.
  * [2] BICEP, Keck, P. A. R. Ade et al.,  Phys. Rev. Lett. 127, 151301 (2021), 2110.00483.
  * [3] Planck, Y. Akrami et al.,  (2019), 1905.05697.
  * [4] LIGO Scientific, Virgo, B. P. Abbott et al.,  Phys. Rev. Lett. 116, 061102 (2016), 1602.03837.
  * [5] A. Ricciardone,  J. Phys. Conf. Ser. 840, 012030 (2017), 1612.06799.
  * [6] A. A. Starobinsky,  Phys. Lett. B91, 99 (1980).
  * [7] A. A. Starobinsky,  Nonsingular Model of the Universe with the Quantum Gravitational De Sitter Stage and its Observational Consequences,  in *Moscow 1981, Proceedings of the second seminar “Quantum Theory of Gravity”*, 58-72, 1981.
  * [8] A. A. Starobinsky,  JETP Lett. 34, 438 (1981).
  * [9] A. A. Starobinsky,  Cosmological models with the intermediate de Sitter stage: theory and observational consequences,  in Fundamental Interactions, MGPI Press, Moscow, 1984, p. 55-79, 1984.
  * [10] A. H. Guth,  Phys. Rev. D23, 347 (1981).
  * [11] A. D. Linde,  Phys. Lett. B129, 177 (1983).
  * [12] A. Linde,  Inflationary Cosmology after Planck 2013,  in Proceedings, 100th Les Houches Summer School: Post-Planck Cosmology: Les Houches, France, July 8 - August 2, 2013, pp. 231–316, 2015, 1402.0526.
  * [13] S. K. Korumilli,  Inflaton candidates: from string theory to particle physics,  PhD thesis, UBI, Covilha, 2017, 1808.03701.
  * [14] J. Martin, C. Ringeval, and V. Vennin,  Phys. Dark Univ. 5-6, 75 (2014), 1303.3787.
  * [15] J. M. Maldacena,  JHEP 05, 013 (2003), astro-ph/0210603.
  * [16] P. D. Meerburg et al.,  (2019), 1903.04409.
  * [17] N. Arkani-Hamed and J. Maldacena,  (2015), 1503.08043.
  * [18] A. S. Koshelev, L. Modesto, L. Rachwal, and A. A. Starobinsky,  JHEP 11, 067 (2016), 1604.03127.
  * [19] A. S. Koshelev, K. Sravan Kumar, and A. A. Starobinsky,  JHEP 03, 071 (2018), 1711.08864.
  * [20] A. S. Koshelev, K. Sravan Kumar, A. Mazumdar, and A. A. Starobinsky,  JHEP 06, 152 (2020), 2003.00629.
  * [21] A. S. Koshelev, K. S. Kumar, and A. A. Starobinsky,  Int. J. Mod. Phys. D 29, 2043018 (2020), 2005.09550.
  * [22] M. J. Duff,  Class. Quant. Grav. 11, 1387 (1994), hep-th/9308075.
  * [23] S. W. Hawking, T. Hertog, and H. S. Reall,  Phys. Rev. D63, 083504 (2001), hep-th/0010232.
  * [24] K. S. Stelle,  Phys. Rev. D16, 953 (1977).
  * [25] N. V. Krasnikov,  Theor. Math. Phys. 73, 1184 (1987),  [Teor. Mat. Fiz.73,235(1987)].
  * [26] L. Buoninfante,  (2021), 2105.08167.
  * [27] S. V. Ketov and A. A. Starobinsky,  Phys. Rev. D83, 063512 (2011), 1011.0240.
  * [28] K. Abazajian et al.,  (2019), 1907.04473.
  * [29] CMB-S4, K. N. Abazajian et al.,  (2016), 1610.02743.
  * [30] P. D. Meerburg, M. Munchmeyer, J. B. Muñoz, and X. Chen,  JCAP 1703, 050 (2017), 1610.06559.