###### Abstract

We study the statistical mechanics of binary systems under gravitational
interaction of the Modified Newtonian Dynamics (MOND) in three-dimensional
space. Considering the binary systems, in the microcanonical and canonical
ensembles, we show that in the microcanonical systems, unlike the Newtonian
gravity, there is a sharp phase transition, with a high-temperature
homogeneous phase and a low temperature clumped binary one. Defining an order
parameter in the canonical systems, we find a smoother phase transition and
identify the corresponding critical temperature in terms of physical
parameters of the binary system.

###### keywords:

Modified Newtonian Dynamics (MOND), phase transition, structure formation,
binary systems.

xx 1 5 Received: date; Accepted: date; Published: date Phase transition in
Modified Newtonian Dynamics (MONDian) self-gravitating systems Mohammad H.
Zhoolideh Haghighi 1,2*, S. Rahvar 3 and M Reza Rahimi Tabar 3,4, Firstname
Lastname, Firstname Lastname and Firstname Lastname Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

The growth of structures from the initial condition in the early Universe to
the galaxies and clusters of galaxies is addressed by the standard model of
cosmology. In this scenario, the origin of the structures is quantum
fluctuations of a scalar field, the so-called inflaton field. The amplitude of
the structures grows after the end of inflation. The standard paradigm for
structure formation, $\Lambda$CDM Blumenthal , has a very good agreement in
the early Universe from the CMB observations Planck  because the free
parameters are tuned to fit the CMB. However, $\Lambda$CDM runs into severe
difficulties with local and large scale structures 2021MNRAS.500.5249A ;
2020MNRAS.499.2845H ; MNRAS.491.3042P ; 2021ApJ…908L..51S .

There is another approach to structure formation theory from the statistical
mechanics point of view, where the structures form when a phase transition
occurs in gravitating systems Emden1 ; deVega1 ; Chavanis ; Rahimi . One
classical example in statistical mechanics is the 2D self-gravitating system
with a logarithmic gravitational potential Rahimi . In this approach, one
takes an ensemble of $N$-body particles as a thermodynamical system, where
from the partition function one can derive the thermodynamical quantities. We
note that in this system all the particles are under their mutual
gravitational interaction; this has to be taken into account in calculating
the partition function. It has been shown that for such a system there exist
two phases of (a) high-temperature gaseous phase, and (b) clumped low-
temperature phase. This approach has an analytical solution only in 2D
(logarithmic) gravity. To extend it to 3D, one has to deal with a simpler
thermodynamical approach such as studying two-body systems Emden1 .

In the standard theory of structure formation, to have a compatible
theoretical result with observation, we need to have a dark matter component
in addition to the baryonic component of the cosmic fluid. The effect of dark
matter in structure formation is important when the universe was dominated by
radiation whose pressure repelled the baryonic matter and prevented the
formation of baryonic structures. The perturbation in the dark matter fluid of
the Universe (unlike the baryonic and radiation components) would grow through
the gravitational instability to form over-dense regions. The consequence of
dark matter structures is that after recombination, the gravitational
potential of dark matter could accumulate the baryonic matter to form the
baryonic part of galaxies within dark matter halos Peebles ; 1993Padmanabhan .
There is another approach to deal with the dynamics of large scale structures
by replacing the dark matter with a modification to the gravity law. In some
of these models, the extra degrees of freedom such as the scalar or the vector
sectors are considered for the gravitational field. A generic formalism of
this theory is the Scalar-Vector-Tensor theory which is the so-called Modified
gravity model (MOG) moffat . Although this theory can explain the dynamics of
galaxies and clusters of galaxies without a need for dark matter rahvar1 ;
rahvar2 , it predicts unexpected mass-to-light ratio 2019ApJ…884L..25H . There
are also non-local gravity models, where in one of these theories, the
Einstein gravity has been supplemented by the non-local terms, in analogy to
the non-local electrodynamics mashhoon . This theory also educes to the
standard Poisson equation in the weak field approximation with an extra term
that plays the role of dark matter. The theory provides compatible dynamics
for galaxies and clusters of galaxies without the need for a dark matter
component rahvar3 . There is also another popular model which is called
Modified Newtonian Dynamics (MOND) milgrom . In this theory, Newton’s second
law or usually the Poisson equation is modified in systems with accelerations
smaller than a universal acceleration, $a_{0}$. This theory also provides
compatible dynamics to the spiral and elliptical galaxies McGaugh . It is
worth mentioning that the main problem of modified gravity models is that they
can not explain the observational data on different scales, though see
2020MNRAS.499.2845H  for a possible hybrid solution using both MOND and hot
dark matter in the form of sterile neutrinos.

In this work, we study the phase transition for a binary interacting via
MONDian gravity, in non-expanding and expanding spaces. For simplicity, we
start with an ensemble of two-body objects instead of the $N$-body system and
investigate the phase transition by decreasing the temperature of the system.

The rest of the paper is organized as follows: In section II, we review the
statistical mechanics of binary interacting systems under a Newtonian
potential in the microcanonical ensemble. In section III, we present the
statistical mechanics of a binary system in MOND. Section IV is devoted to
studying the statistical mechanics of MONDian systems in the canonical
ensemble. We then study and discuss the influence of expanding Universe with
scale factor $a(\tau)$ on the critical temperature of the phase transition in
section V. Section VI summarizes the paper and discusses possible further
research.

##  2 Statistical mechanics of a self-gravitating binary under Newtonian
potential: Microcanonical ensemble

The statistical mechanics of self-gravitating systems has been the subject of
attention for many years Emden1 ; deVega1 ; Levin1 . Such systems have
distinguished physical properties due to the long-range nature of the
gravitational force. At thermal equilibrium, these systems are not spatially
homogeneous, and intrinsic inhomogeneity character suggests that fractal
structures can emerge in a system of gravitationally interacting particles
deVega1 . Here we review the statistical mechanics of a self-gravitating
binary system in Newtonian gravity in the microcanonical ensemble Emden1 . We
start with the Hamiltonian of a two-body system,

$H({\bf P},{\bf Q};{\bf p},{\bf r})={{\bf P}^{2}\over 2M}+{{\bf p}^{2}\over
2\mu}+V(r)\quad.$ (1)

where in 3D the potential is $V(r)=-Gm^{2}/r$, $({\bf Q},{\bf P})$ are
coordinates and momentum of the center of mass, and $({\bf r},{\bf p})$ are
the relative coordinates and momentum with the reduced mass. In what follows,
for the sake of simplicity we take the mass of these two bodies to be
identical, (i.e. $M=2m$ and $\mu=m/2$). We assume a spherical shape for the
two objects with radius of ${b}/{2}$ and that the two-body system is confined
in a spherical box of radius $R$, where $r$ in equation (1) varies within the
interval of $(b,R)$. The volume associated with a constant energy of this
system (i.e. $H=E$) in the phase-space 2008arXiv0812.2610P ; 2002LNP…602…..D
is given by the density of states

$g(E)=\int\delta(E-H(r,p,Q,P))d^{3}Pd^{3}pd^{3}Qd^{3}r\quad.$ (2)

First we integrate over $Q$, which leads to $4{\pi}r^{3}/3$, and inserting the
explicit form of the Hamiltonian leads to the following relation:

$g(E)=\frac{4\pi R^{3}}{3}\int d^{3}Pd^{3}r\int_{0}^{\infty}\delta(E-{{\bf
p}^{2}\over 2\mu}-{{\bf P}^{2}\over 2M}-V(r))4\pi p^{2}dp\quad.$ (3)

Now we perform integration over $p$-space. Using the new variable $x\equiv
p^{2}/\mu$, the square-root term appears from the integration as a result of a
property of the Dirac $\delta$-function.

$g(E)=\frac{8\pi^{2}m^{3/2}R^{3}}{3}\int
d^{3}Pd^{3}r\sqrt{E-\frac{P^{2}}{2m}-V(r)}\quad.$ (4)

This time we integrate over $P$-space similar to the previous step in equation
(3). The result is

$g(E)=AR^{3}\int_{b}^{r_{\rm max}}r^{2}dr\left(E-V(r)\right)^{2}\quad,$ (5)

where $A=64\pi^{5}m^{3}/3$. In the case of Newtonian gravity, the potential
energy is $V(r)=-Gm^{2}/r$ and since the kinetic energy is always positive
definite (i.e $E-V(r)>0$), the upper limit of the integral ($r_{max}$) should
be taken in such a way that guarantees the positive sign of the kinetic
energy. The upper bound of integration for the following ranges of energy is
given by,

$\Bigg{\\{}\begin{tabular}[]{c}$(-Gm^{2}/b)<E<(-Gm^{2}/R)\hskip
28.45274ptr_{max}=Gm^{2}/E$\\\ $(-Gm^{2}/R)<E<+\infty\hskip
42.67912ptr_{max}=R$\end{tabular}$ (6)

The second condition has no specific meaning in astrophysics, though it has an
analogy with the standard thermodynamics where the size of the box is $R$. We
assume that the container has a fixed volume while we can increase the
velocity of the particles, and they are constrained to stay in this volume.
Integrating (5) results in

$\frac{g(E)}{A(Gm^{2})^{3}}=\left\\{\begin{array}[]{l}{R^{3}\over
3}(-E)^{-1}\left(1+{bE\over Gm^{2}}\right)^{3},\hskip
28.45274pt(-Gm^{2}/b)<E<(-Gm^{2}/R)\\\ {}\\\ {R^{3}\over
3}(-E)^{-1}\left[\left(1+{RE\over Gm^{2}}\right)^{3}-\left(1+{bE\over
Gm^{2}}\right)^{3}\right],\quad(-Gm^{2}/R)<E<\infty.\end{array}\right.$ (7)

From the density of states function $g(E)$ which represents the phase space
volume covered by this system with energy $E$, we can calculate the entropy
and the temperature of the system according to the following relations (with
Boltzmann constant $K_{B}=1$):

$S(E)=\ln g(E);\quad T^{-1}(E)=\beta(E)={\partial S(E)\over\partial E}\quad.$
(8)

From equation (7), we obtain the dimensionless temperature for the interval
$(-Gm^{2}/b)<E<(-Gm^{2}/R)$ as

$t(\epsilon)=\left[{3\over 1+\epsilon}-{1\over\epsilon}\right]^{-1}\quad.$ (9)

where the dimensionless temperature $t$ and energy $\epsilon$ are defined as

$t\equiv(bT/Gm^{2}),\hskip 28.45274pt\epsilon\equiv(bE/Gm^{2})\quad.$ (10)

Also for the second interval of $-Gm^{2}/R<E<\infty$ in (7), $t(\epsilon)$ is
given by

$t(\epsilon)=\left[{3\left[(1+\epsilon)^{2}-{R\over b}(1+{R\over
b}\epsilon)^{2}\right]\over(1+\epsilon)^{3}-(1+{R\over
b}\epsilon)^{3}}-{1\over\epsilon}\right]^{-1}\quad.$ (11)

In Figure (1), we depict $t(\epsilon)$ in terms of $\epsilon$
2008arXiv0812.2610P . It can be seen that the specific heat is positive along
AB and CD while it is negative along BC. For a system with energy in the range
AB, the two solid spherical objects are in contact, and increasing the energy
of the system increases the kinetic energy, or in other words, the temperature
of the system. Over the range BC, the two objects detach from each other and
start circular motion according to the mutual gravitational force between
them. For the CD path in Figure (1) the total energy is larger than zero (i.e.
$E>0$) and the two objects decouple from each other and behave as free
particles.

Figure 1: Dimensionless temperature versus dimensionless energy (Eq. 9) for a
binary system interacting under Newtonian gravity (dashed orange curve) and
MONDian gravity (solid blue curve). As expected, there is a flat part for MOND
due to entering the DML. Since for constant temperature the energy has a
finite change there must be a sharp phase transition.

##  3 Statistical mechanics of a self-gravitating binary in MOND:
Microcanonical ensemble

In the Modified Newtonian Dynamics (MOND), proposed to solve the dark matter
problem on galaxy scalesmilgrom , the second law of Newtonian mechanics is
modified for small accelerations. The characteristic acceleration for this
modification is $a_{0}=1.2\times 10^{-10}ms^{-2}$ 1991MNRAS.249..523B  where
for $a\leq a_{0}$, the definition of force is modified to $F=ma\mu(a/a_{0})$.
Here $\mu(x)$ is larger than unity for smaller accelerations and is unity for
the large accelerations (i.e. $a\gg a_{0}$). This model can explain the
rotation curve of spiral galaxies and predicts the Tully-Fisher relation Tully
. However, in this theory, energy and momentum are not well-defined Felten .
One of the simple solutions to this problem is that one may interpret MOND as
modified gravity rather than modified dynamics. The gravitational acceleration
in the deep MOND limit where $g\ll a_{0}$ will be milgrom

$g_{DML}=-\sqrt{g_{N}.a_{0}}=-\frac{\sqrt{Gma_{0}}}{r}\quad.$ (12)

The subscript DML stands for the deep-MOND limit, in which the potential from
the gravitational acceleration is given by

$\phi_{DML}=\sqrt{Gma_{0}}~{}\ln(r/R)\quad.$ (13)

For an isolated binary system in the DML, the gravitational force between the
two particles scales as $F\sim 1/r$ which for objects in circular motion
results in a constant velocity, compatible with the flat rotational curves of
galaxies. We note that for $g\gg a_{0}$ the Newtonian gravity is recovered
(i.e. $g=g_{N}$) and the potential will behave as $\phi_{N}=-Gm/r$.

Since in MOND the acceleration of a test particle in a gravitating system is
stronger than in Newtonian gravity, in addition to studying the dynamics of a
system, we can investigate the formation of structures in these two scenarios.
For a self-gravitating system, the free-fall time scale represents the
strength of clustering of a structure. The ratio of characteristic timescales
in Newtonian and MOND gravity is 1987gady.book…..B ; Golovnev ,

$\frac{T_{MOND}}{T_{N}}=\Big{[}\frac{16}{\pi^{2}}\cdot\frac{g_{N}}{a_{0}}\Big{]}^{\frac{1}{4}}\quad.$

For $g_{N}<a_{0}$ we would expect a shorter time-scale for the clustering in
MOND compare to the Newtonian gravity. We refer interested readers to
2021MNRAS.506.5468Z  for a thorough investigation into the free-fall
timescales in MOND.

In order to have a continuous transition from the DML to the Newtonian
gravity, one needs a transition function $\mu(a)$ Beyken and milgrom . There
are various transition functions. One of the simple models is $\mu(x)=x/(1+x)$
or a sigmoid function, other transition function can be found in Famaey ;
Banik . The sigmoid function better fits the observational data
2018MNRAS.480.2660B .

Now let us study the statistical mechanics of two-body objects in MOND in the
microcanonical ensemble. The enumeration of the total number of possible
states (phase volume) for a binary system with energy $E$ is given by

$g(E)=\int\delta(E-H(r,p,Q,P))d^{3}Pd^{3}pd^{3}Qd^{3}r\quad.$ (14)

By integrating over $P$, $p$ and $Q$, ( similar to the calculations in section
(2) for Newtonian gravity, the function $g(E)$ is given by

$g(E)=AR^{3}\int_{b}^{r_{\rm max}}r^{2}dr\left(E-V(r)\right)^{2}\quad,$ (15)

where $A=64\pi^{4}m^{3}/3$. Taking into account the Newtonian and DML phases
for the gravitational potential, the total potential is given in the following
two domains,

$V(r)=\Bigg{\\{}\begin{tabular}[]{c}$-Gm^{2}/r,\hskip 71.13188ptr<r_{M}$\\\
\\\ $m\sqrt{Gma_{0}}ln(r/R),\hskip 28.45274ptr>r_{M}$\end{tabular}$ (16)

Where the MOND radius $r_{M}=\sqrt{Gm/a_{0}}$ is the scale of transition
between the two domains. In order to calculate integral (15), one needs
precisely determine $r_{max}$ in terms of energy ranges, i.e.,

$\left\\{\begin{array}[]{l}\frac{-Gm^{2}}{b}<E<\frac{-Gm^{2}}{r_{M}},\quad\qquad
r_{max}=\frac{-Gm^{2}}{E}\\\ {}\\\ \frac{-Gm^{2}}{r_{M}}<E<0,\quad\qquad\qquad
r_{max}=R\exp(\frac{E}{m\sqrt{Gma_{0}}})\\\ {}\\\
0<E<\infty,\quad\qquad\qquad\qquad r_{max}=R\end{array}\right.$ (17)

Here the bound of $r_{max}$ results from the positive sign of the kinetic
energy. We define the temperature of system from equation (8) and plot
$t=t(\epsilon)$ for three energy ranges in Fig (1). Here we adopt
${R}/{b}=10^{10}$ and ${e_{2}}/{e_{1}}=100$ where $e_{1}=m\sqrt{Gma_{0}}$ and
$e_{2}=Gm^{2}/b$. In this figure, the $(t,\epsilon)$ diagram for MONDian
gravity is almost similar to that for Newtonian, except for $\epsilon$ just
slightly below 0, where for the case of Newtonian gravity $t(0)=0$ but for
MONDian gravity $t(0)>0$. In fact, for MONDian gravity (the solid curve),
there is a flat part, in which for constant temperature, the energy has a
finite change, indicating a sharp phase transition where the heat capacity
diverges.

We can interpret this area as when the binary system enters the deep-MOND
where the rotation velocity of the binary objects around their center of mass
is $v_{rot}=\sqrt{Gma_{0}/2}$. In this case, with increasing energy of the
system, the orbital size of the binary increases; however, the kinetic energy,
which represents the temperature of the system, remains constant. In the next
section, we study statistical mechanics and phase transition of a binary
system in the MONDian gravity for the canonical ensemble.

##  4 Statistical mechanics of a self-gravitating binary in MOND: Canonical
ensemble

Let us at first assume an ensemble of binary objects under Newtonian gravity.
The system is composed of an ensemble of thermalized binaries with an
associated temperature. The partition function associated with a binary system
in this ensemble is given by

$Z(\beta)=\int d^{3}Pd^{3}pd^{3}Qd^{3}r\exp(-\beta H)\quad,$ (18)

where the parameters and the Hamiltonian are defined in Sec. (2). Integrating
over momenta $P$, $p$ and position $Q$, equation (18) simplifies to
1987gady.book…..B ; 2008arXiv0812.2610P :

$Z(\beta)=R^{3}\beta^{-3}\int_{b}^{R}dr\,r^{2}\exp\left(\beta\frac{GM^{2}}{r}\right)\quad.$
(19)

In dimensionless form, using the definition of $t$ as introduced in Sec. (2),
the partition can be written as

$Z(t)=\left(\frac{R}{b}\right)^{3}t^{3}\int_{1}^{{R}/{b}}x^{2}\exp\left(\frac{1}{tx}\right)\,dx\quad\\\
=\frac{R^{3}}{12b^{6}}\Bigg{[}b^{3}(-2\text{Ei}\left(\frac{b}{Rt}\right)-\log\left(\frac{Rt}{b}\right)+2\log\left(-\frac{Rt}{b}\right)+\log\left(\frac{b}{Rt}\right)+2\text{Ei}\left(\frac{1}{t}\right)-2e^{1/t}t\left(2t^{2}+t+1\right)\\\
-\log\left(\frac{1}{t}\right)-2\log(-t)+\log(t))+2Rte^{\frac{b}{Rt}}\left(b^{2}+bRt+2R^{2}t^{2}\right)\Bigg{]}$
(20)

Equation (20) is expressed in term of exponential integral function
$\left(Ei(x)=-\int_{-x}^{\infty}\frac{\exp\left(-t\right)}{t}\,dt\right)$ and
has no known analytical solution, so here we solve it numerically and
calculate the mean energy of the system by using:

$E(\beta)=-\partial\ln Z/\partial\beta\quad.$ (21)

In order to identify a phase transition, we calculate numerically the
derivative of energy with respect to the thermodynamical variables. If this
quantity diverges or becomes discontinuous, the system undergoes a phase
transition 2011ConPh..52…98S . We calculate the derivative of energy with
respect to the temperature, which is defined as the specific heat and is shown
in Fig. (2).

Figure 2: Normalized specific heat of the system made of a binary under
Newtonian (dashed orange curve) and MONDian (solid blue curve) gravity. The
phase transition temperature is at the peak of these curves.

From the numerical calculation of specific heat $c_{v}$ in terms of the
temperature, we find that it has a peak at $t=t_{critical}$. To understand the
nature of the detected phase transition, we define an order parameter and
study its behavior near the critical temperature. Here we take the mean
distance between the companions of a binary system as the order parameter and
define it as Rahimi :

$<r^{2}>=\frac{\int_{b}^{R}drr^{4}\exp(\beta\frac{Gm^{2}}{r})}{\int_{b}^{R}drr^{2}\exp(\beta\frac{Gm^{2}}{r})}\quad.$
(22)

For the first-order phase transition, we expect to have an abrupt change in
the order parameter. For instance, if we apply it for matter in the liquid and
gaseous phases, this parameter sharply changes when a liquid changes to the
gaseous state. For the second order phase transition, the order parameter will
be a continuous function at the critical temperature. For simplicity, we
rewrite Eq.(22) in dimensionless form as:

$<x^{2}>=\frac{\int_{1}^{{R}/{b}}x^{4}\exp\left(\frac{1}{tx}\right)\,dx}{\int_{1}^{{R}/{b}}x^{2}\exp\left(\frac{1}{tx}\right)\,dx}\quad$
(23)

and calculate $<x^{2}>$ numerically allowing us to plot the order parameter as
a function of temperature in Fig. (3). We notice that there is the phase
transition for the order parameter at exactly the same temperature that the
specific heat has a peak, whereas the order parameter is a differentiable
function at the critical temperature. The order parameter shows that a high-
temperature system has a homogeneous phase, while at low temperature $<x^{2}>$
vanishes.

Figure 3: Order parameter as a criterion to detect the phase transition. The
orange dashed curve represents the Newtonian system, and the blue solid one is
for MOND. This plot shows that the mean value of the distance between the
particles changes very fast but remains differentiable near the phase
transition.

Now we perform a similar calculation for the temperature dependence of the
order parameter in the combination of deep MOND and Newtonian gravity, where
for small and large accelerations (i.e $a<a_{0}$ and $a>a_{0}$), DML and
Newtonian gravity will have dominant contributions, respectively. Since the
potential is a function of distance, we need to make an approximation before
calculating the partition function. The potential is defined as $\phi(r)=-\int
F.dr$, we can break this integration into separate three parts with different
gravitational potentials. For $r\ll r_{M}$ we have Newtonian gravity, and
$r\gg r_{M}$ potential is in DML, and between we are in a regime that is a
combination of Newtonian and MONDian, i.e.,

$\displaystyle\phi(r)$ $\displaystyle=$ $\displaystyle-\int
F.dr=-\int_{0<r<r_{M}-\delta}(-GM/r^{2})dr$ (24) $\displaystyle-$
$\displaystyle\int_{r>r_{M}+\delta}(-\sqrt{(GMa_{0}/r^{2})})dr-\int_{r_{M}-\delta<r<r_{M}+\delta}F.dr\quad.$
(25)

Three terms in the integration belong to the Newtonian, DML, and mixing of
Newtonian and DML, respectively. Here $\delta$ is a small constant. The result
of integration is,

$\phi(r)=-GM/r+\sqrt{GMa_{0}}\ln(r)-\int_{r_{M}-\delta}^{r_{M}+\delta}F.dr+C\quad.$
(26)

where C is the constant of integration, and we chose it to be $\sim-\ln(R)$.
We note that the third term in r.h.s of Eq.(26), that is
$\int_{r_{M}-\delta}^{r_{M}+\delta}F.dr$, depends on the interpolating
function, ($F=m\mu(a/a_{0})$). A simple approach is by choosing a proper $\mu$
with a very fast transition from Newtonian regime to MONDian regime. Then we
can ignore this term and the potential simplifies to

$\phi(r)\simeq-GM/r|_{(r<r_{M})}-\sqrt{(GMa_{0})}\ln(r/R)|_{(r>r_{M})}\quad.$
(27)

An alternative approach is to use a simple interpolating function for a single
point mass using a hyperbolic substitution Famaey ; Banik ,

$\displaystyle\phi\left(r\right)$ $\displaystyle=$
$\displaystyle\sqrt{Gma_{{}_{0}}}\left[\ln\left(1+\sqrt{1+\tilde{r}^{2}}\right)-\frac{1}{\tilde{r}}-\sqrt{\frac{1}{\tilde{r}^{2}}+1}\right]\quad.$
$\displaystyle~{}~{}{where}\quad\tilde{r}$ $\displaystyle\equiv$
$\displaystyle\frac{2r}{r_{{}_{M}}}\quad and\quad
r_{{}_{M}}\equiv\sqrt{\frac{Gm}{a_{{}_{0}}}}\quad.$ (28)

We provide the influence of the interpolating function on the critical
temperature, see below.

For simplification, we continue with Eq. 27. As a result, the partition
function of the ensemble of the binary system in the MONDian gravity (using
the MONDian potential in Eq. 18) by integrating over variables $P,p$ and $Q$
is

$Z(\beta)=R^{3}\beta^{-3}\Big{[}\int_{b}^{r_{M}}drr^{2}\exp(\beta\frac{Gm^{2}}{r})+\int_{r_{M}}^{R}drr^{2}\exp(-e_{1}\beta
ln({r}/{R}))\Big{]}\quad,$ (29)

Which in dimensionless representation it simplifies to:

$Z(t)=(\frac{R}{b})^{3}t^{3}\Big{[}\int_{1}^{{r_{M}}/{b}}dxx^{2}\exp(\frac{1}{tx})+\int_{{r_{M}}/{b}}^{{R}/{b}}dxx^{2}\exp(-\frac{e_{1}\ln(bx/R)}{e_{2}t})\Big{]}\,\quad.$
(30)

Here $e_{1}=m\sqrt{Gma_{0}}$ and $e_{2}=Gm^{2}/b$. Following the same
procedure as the Newtonian case, we calculate the specific heat and order
parameter and plot them in Figs. 2 & 3, where the specific heat and the order
parameter are given by:

$C_{v}=\partial E/\partial T=\frac{\partial(t^{2}\partial\ln Z/\partial
t)}{\partial t}\quad,$ (31)
$<r^{2}>=\frac{\int_{b}^{r_{M}}r^{4}\exp(\beta{Gm^{2}}/{r})dr+\int_{r_{M}}^{R}r^{4}\exp(-\beta
e_{1}ln(r/R))dr}{\int_{b}^{r_{M}}r^{2}\exp(\beta{Gm^{2}}/{r})dr+\int_{r_{M}}^{R}r^{2}\exp(-\beta
e_{1}ln(r/R))dr}\quad.$ (32)

Eq. 32 can also be rewritten in the dimensionless form as:

$<x^{2}>=\frac{\int_{1}^{\frac{{r_{M}}}{b}}x^{4}\exp\left(\frac{1}{tx}\right)\,dx+\int_{\frac{{r_{M}}}{b}}^{\frac{R}{b}}x^{4}\exp\left(-\frac{{e_{1}}\ln(bx/R)}{{e_{2}}t}\right)\,dx}{\int_{1}^{\frac{{r_{M}}}{b}}x^{2}\exp\left(\frac{1}{tx}\right)\,dx+\int_{\frac{{r_{M}}}{b}}^{\frac{R}{b}}x^{2}\exp\left(-\frac{{e_{1}}\ln(bx/R)}{{e_{2}}t}\right)\,dx}\quad.$
(33)

Comparing Figs. 2 and 3, the phase transition temperature is identical whether
we obtain it from the divergence of $C_{v}$ or from a fast change of
$<x^{2}>$. Also, we note that $C_{v}$ is always positive for the canonical
case, unlike the micro-canonical case.

If we use the specific interpolating function that is introduced in Eq. (28),
we observe a similar behavior, but at a different critical temperature (see
Fig 4). We see that qualitatively the critical behavior of systems is the same
but the interpolating function affects the details of the phase transition.

Figure 4: Order parameter as a criterion to detect the phase transition. This
plot shows that the variance of the distance between the particles changes
fast but remains differentiable near the phase transition.

##  5 Statistical mechanics of a self-gravitating binary in MOND: Comoving
coordinates

In an expanding universe with characteristic scale factor $a(\tau)$, the
physical coordinates of $r$ is related to the comoving coordinate $q$ as
deVega2 ,

$r\equiv a(\tau)q\quad.$ (34)

The Hamiltonian of a binary system with potential given in Eq. (27) can be
written in the comoving coordinate as

$H=\frac{p^{2}}{2m{a(\tau)}^{2}}-\frac{Gm^{2}}{a(\tau)q}\left|{}_{(q<q_{M}{a(\tau)})}+{m}\sqrt{Gma_{0}}\ln(qa(\tau)/R)\right|_{(q>q_{M}{a(\tau)})}\quad.$
(35)

This Hamiltonian in an expanding universe can also be obtained from the
Minkowski-Hamiltonian with the following replacements:

$m\rightarrow ma(\tau)^{2},\quad\quad G\rightarrow G{a(\tau)}^{-5},\quad\quad
a_{0}\rightarrow a_{0}{a(\tau)}^{-1},\quad\quad R\rightarrow
R{a(\tau)}^{-1}\quad.$ (36)

We can interpret the renormalization process by assigning a dynamics to these
parameters. For instance the $m\rightarrow ma(\tau)^{2}$ renormalization is
related to the kinetic energy of particles that decrease as $1/a^{2}$ with the
expansion of the Universe. Here we have the effective gravitational constant
$(G\rightarrow G{a(\tau)}^{-5})$ as well as the acceleration parameter of MOND
($a_{0}\rightarrow a_{0}{a(\tau)}^{-1}$) changing with the scale factor.
To consider a self-gravitating binary at any time in approximate thermal
equilibrium, we assume that the characteristic time of the particle motions
under their mutual gravitation is shorter than the time variation of the scale
factor. This hypothesis is valid for structures that are almost decoupled from
the expansion and become virialized. Given this, the Eqs. 29 & 32 in the
comoving frame will be as follow:

$Z(\beta)=R^{3}a(\tau)^{3}\beta^{-3}\Bigg{[}\int_{b}^{q_{M}}q^{2}\exp(\beta{Gm^{2}}/{a(\tau)q})dq+\int_{q_{M}}^{R}q^{2}\exp(-\beta
m\sqrt{Gma_{0}}ln(a(\tau)q/R))dq\Bigg{]}\quad.$ (37)
$<q^{2}>=a(\tau)^{2}\frac{\int_{b}^{q_{M}}q^{4}\exp(\beta{Gm^{2}}/{a(\tau)q})dq+\int_{q_{M}}^{R}q^{4}\exp(-\beta
m\sqrt{Gma_{0}}ln(a(\tau)q/R))dq}{\int_{b}^{q_{M}}q^{2}\exp(\beta{Gm^{2}}/{a(\tau)q})dq+\int_{q_{M}}^{R}q^{2}\exp(-\beta
m\sqrt{Gma_{0}}ln(a(\tau)q/R))dq}\quad.$ (38)

Eqs. 37 & 38 can be rewritten in dimensionless form as:

$Z(t)=(\frac{R}{b})^{3}a(\tau)^{3}t^{3}\Bigg{[}\int_{1}^{\frac{{q_{M}}}{b}}x^{2}\exp\left(\frac{1}{txa(\tau)}\right)\,dx+\int_{\frac{{q_{M}}}{b}}^{\frac{R}{b}}x^{2}\exp\left(-\frac{{e_{1}}\ln(ba(\tau)x/R}{{e_{2}}t}\right)\,dx\Bigg{]}\quad,$
(39)
$<x^{2}>=a(\tau)^{2}\frac{\int_{1}^{\frac{{q_{M}}}{b}}x^{4}\exp\left(\frac{1}{txa(\tau)}\right)\,dx+\int_{\frac{{q_{M}}}{b}}^{\frac{R}{b}}x^{4}\exp\left(-\frac{{e_{1}}\ln(ba(\tau)x/R)}{{e_{2}}t}\right)\,dx}{\int_{1}^{\frac{{q_{M}}}{b}}x^{2}\exp\left(\frac{1}{txa(\tau)}\right)\,dx+\int_{\frac{{q_{M}}}{b}}^{\frac{R}{b}}x^{2}\exp\left(-\frac{{e_{1}}\ln(ba(\tau)x/R}{{e_{2}}t}\right)\,dx}\quad,$
(40)

Figure 5: Order parameter (left) and critical temperature (right) for
different values of the scale factor. When the scale factor increases, the
critical temperature decreases.

where $x\equiv\frac{q}{b}$. In Fig. (5) , the order parameter $<x^{2}>$ for
various scale factors $a(\tau)=1,0.1,0.05$ is plotted in the left panel, while
the right panel demonstrates critical temperature versus scale factor, showing
that the critical temperature decreases as the scale factor increases.

##  6 Conclusions

In this work, we have studied thermodynamical phase transition under MONDian
gravity. In addition influence of the cosmic expansion on the critical
temperature of the detected phase transition has been studied. We have shown
that in the microcanonical ensemble of binary systems under MONDian gravity, a
sharp phase transition is not present in the Newtonian gravity. Furthermore,
we find a smoother phase transition with finite critical temperature by
studying the specific heat $C_{v}$ and an order parameter of a binary system
in a canonical ensemble. One interesting result is that although both
Newtonian and MONDian systems experience a phase transition in the canonical
ensemble, they have different critical temperatures. The next steps in our
research will be considering all interactions in the $N$-body system with the
MONDian gravity and its connection with cosmological MOND 2019ApJ…884L..25H .
In other directions, it would also be interesting to study the fractal
structure of clusters in the crumpled phase of the system, as well as the
equation of state (using the partition function in the grand canonical
ensemble) and complexity in the view of 2018PhRvD..97d4010H . Investigation of
the critical temperature of rotating MONDian self-gravitating systems will be
another interesting problem.

##  Appendix A

Here we calculate $g(E)$ for the binary system in MOND for the microcanonical
ensemble. For the case with energy $-\frac{Gm^{2}}{b}<E<-\frac{Gm^{2}}{r_{M}}$
from Eq. (15), we have:

$g(E)=\int_{b}^{-\frac{b{e_{2}}}{E}}r^{2}(E+\frac{b{e_{2}}}{r})^{2}\
dr=-\frac{b^{3}(E+{e_{2}})^{3}}{3E}\quad.$ (41)

The temperature of this system with energy $E$ is given by
$T^{-1}=\frac{\partial\ln(g(E))}{\partial E}$,

$t=\frac{\epsilon(\epsilon+1)}{2\epsilon-1},$ (42)

where dimensionless temperature and energy defined as $t=(bT/Gm^{2})$ and
$\epsilon=(bE/Gm^{2})$ in Eq. (10). If we increase the energy, the particles’
accessible energy would be $\frac{-Gm^{2}}{r_{M}}<E<0$ making the integral
(15)

$\displaystyle g(E)$
$\displaystyle=\int_{b}^{\frac{b{e_{2}}}{{e_{1}}}}r^{2}\left(E+\frac{b{e_{2}}}{r}\right)^{2}dr+\int_{\frac{b{e_{2}}}{{e_{1}}}}^{Re^{\left(\frac{E}{{e_{1}}}\right)}}r^{2}\left[E-{e_{1}}\log\left(\frac{r}{R}\right)\right]^{2}dr$
$\displaystyle=A_{1}+A_{2}+A_{3},$

where $e_{1}=m\sqrt{Gma_{0}}$, $e_{2}=Gm^{2}/b$ and

$\displaystyle A_{1}$
$\displaystyle=\frac{\left(\frac{bE{e_{2}}}{{e_{1}}}+b{e_{2}}\right)^{3}-b^{3}(E+{e_{2}})^{3}}{3E}-\frac{b^{3}{e_{2}}^{3}}{27{e_{1}}^{3}}\left(6E{e_{1}}+2{e_{1}}^{2}\right),$
$\displaystyle A_{2}$
$\displaystyle=\frac{1}{27}R^{3}e^{\frac{3E}{{e_{1}}}}[9E^{2}+9{e_{1}}^{2}\log^{2}\left(e^{E/{e_{1}}}\right)+6E{e_{1}}-6{e_{1}}(3E+{e_{1}})\log\left(e^{E/{e_{1}}}\right)+2{e_{1}}^{2}],$
$\displaystyle A_{3}$
$\displaystyle=-\frac{b^{3}{e_{2}}^{3}}{27{e_{1}}^{3}}\left[-6{e_{1}}(3E+{e_{1}})\log\left(\frac{b{e_{2}}}{{e_{1}}R}\right)+9{e_{1}}^{2}\log^{2}\left(\frac{b{e_{2}}}{{e_{1}}R}\right)+9E^{2}\right].$

Similarly to the first part, we define the temperature of system in this range
of energy as

$t=B_{1}+B_{2}+B_{3},$

where

$\displaystyle B_{1}$
$\displaystyle=b^{3}\big{[}3\log(\frac{b{e_{2}}}{{e_{1}}R})(-3\frac{e_{1}}{e_{2}}\log(\frac{b{e_{2}}}{{e_{1}}R})+6\epsilon+2\frac{e_{1}}{e_{2}})-9\epsilon^{2}(\frac{e_{1}}{e_{2}})^{2}+3\epsilon(7-9\frac{e_{1}}{e_{2}}^{2})+(\frac{e_{1}}{e_{2}})(25-27\frac{e_{1}}{e_{2}})\big{]},$
$\displaystyle B_{2}$
$\displaystyle=(\frac{e_{1}}{e_{2}})^{2}R^{3}e^{\frac{3\epsilon{e_{2}}}{{e_{1}}}}(9\epsilon^{2}+6\epsilon\frac{e_{1}}{e_{2}}+3\frac{e_{1}}{e_{2}}\log(e^{\epsilon{e_{2}}/{e_{1}}})(3\frac{e_{1}}{e_{2}}\log(e^{\epsilon{e_{2}}/{e_{1}}})-2(3\epsilon+\frac{e_{1}}{e_{2}})),$
$\displaystyle B_{3}$
$\displaystyle=2(\frac{e_{1}}{e_{2}})^{2}\big{[}3\frac{e_{1}}{e_{2}}R^{3}e^{\frac{3\epsilon}{{e_{1}}}}(9e^{2}+6\epsilon\frac{e_{1}}{e_{2}}+3\frac{e_{1}}{e_{2}}\log(e^{\epsilon{e_{2}}/{e_{1}}})(3\frac{e_{1}}{e_{2}}\log(e^{\epsilon{e_{2}}/{e_{1}}})$
$\displaystyle-2(3\epsilon+\frac{e_{1}}{e_{2}}))+2(\frac{e_{1}}{e_{2}})^{2}-3b^{3}(-6\log(\frac{b{e_{2}}}{{e_{1}}R})+6\epsilon(\frac{e_{1}}{e_{2}})^{2}+9(\frac{e_{1}}{e_{2}})^{2}-7))^{-1}\big{]}.$

Finally for the energy range $0<E<\infty$, we have:

$g(E)=\int_{b}^{\frac{b{e_{2}}}{{e_{1}}}}r^{2}\big{[}E+\frac{b{e_{2}}}{r}\big{]}^{2}dr+\int_{\frac{be_{2}}{e_{1}}}^{R}r^{2}\big{[}E-{e_{1}}\log\left(\frac{r}{R}\right)\big{]}^{2}dr=C_{1}+C_{2},$

where

$\displaystyle C_{1}$
$\displaystyle=\frac{\left(9E^{2}+6Ee_{1}+2e_{1}^{2}\right)\left(e_{1}^{3}R^{3}-b^{3}e_{2}^{3}\right)}{27e_{1}^{3}}+\frac{3b^{3}e_{1}e_{2}^{3}\log\left(\frac{be_{2}}{e_{1}R}\right)\left[-3e_{1}\log\left(\frac{be_{2}}{e_{1}R}\right)\right]}{27e_{1}^{3}},$
$\displaystyle C_{2}$
$\displaystyle=\frac{\left(6E+2e_{1}\right)}{27e_{1}^{3}}+\frac{\left(\frac{bEe_{2}}{e_{1}}+ae_{2}\right)^{3}-b^{3}(E+e_{2})^{3}}{3E}.$

The temperature of the system is similar to parts one and two above, and is
given by:

$t=D_{1}+D_{2}+D_{3},$

where

$D_{1}=\frac{b^{3}\left[9\epsilon^{2}(\frac{e_{1}}{e_{2}})^{2}+3\epsilon\left(9(\frac{e_{1}}{e_{2}})^{2}e_{2}-7\right)+\frac{e_{1}}{e_{2}}(27\frac{e_{1}}{e_{2}}-25)\right]}{3\left[b^{3}\left(6\epsilon(\frac{e_{1}}{e_{2}})^{2}+9(\frac{e_{1}}{e_{2}})^{2}-7\right)-6b^{3}\log\left(\frac{be_{2}}{e_{1}R}\right)-2(\frac{e_{1}}{e_{2}})^{2}R^{3}(3\epsilon+e_{1})\right]},$

$D_{2}=\frac{3b^{3}e_{2}^{3}\log\left(\frac{be_{2}}{e_{1}R}\right)\left[3\frac{e_{1}}{e_{2}}\log\left(\frac{be_{2}}{e_{1}R}\right)-2(3\epsilon+e_{1})\right]}{3\left[b^{3}\left(6\epsilon(\frac{e_{1}}{e_{2}})^{2}+9(\frac{e_{1}}{e_{2}})^{2}-7\right)-6b^{3}\log\left(\frac{be_{2}}{e_{1}R}\right)-2(\frac{e_{1}}{e_{2}})^{2}R^{3}(3\epsilon+e_{1})\right]},$

$D_{3}=-\frac{e_{1}^{2}R^{3}\left[9\epsilon^{2}+6\epsilon\frac{e_{1}}{e_{2}}+2(\frac{e_{1}}{e_{2}})^{2}\right]}{3\left[b^{3}\left(6\epsilon(\frac{e_{1}}{e_{2}})^{2}+9(\frac{e_{1}}{e_{2}})^{2}-7\right)-6b^{3}\log\left(\frac{be_{2}}{e_{1}R}\right)-2(\frac{e_{1}}{e_{2}})^{2}R^{3}(3\epsilon+e_{1})\right]}.$

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KSW criterion in large field models

Oliver Janssen ††e-mail<EMAIL_ADDRESS>
Laboratory for Theoretical Fundamental Physics, EPFL, 1015 Lausanne,
Switzerland

We extend the analytic description of complex no-boundary solutions in the
context of inflation to large field models. We discuss the Kontsevich-Segal-
Witten (KSW) criterion and find it is satisfied in small field models, while
in large field models it depends on an integral involving $V^{\prime}(\phi)$
over the range of inflation. It follows that the criterion does not truly
constrain inflationary phenomenology since one can complete any inflaton
potential beyond observable scales so as to satisfy KSW.

###### Contents

  1. 1 Introduction and conclusions
  2. 2 Review of recent literature
  3. 3 Regime of validity of (6)
  4. 4 KSW in small field models
  5. 5 Approximate solution in large field models
  6. 6 KSW in the extended region
  7. 7 Numerical checks
    1. 7.1 Monomial inflation
    2. 7.2 $\alpha$-attractors
    3. 7.3 Natural inflation
  8. A Slow roll corrections to the on-shell action
    1. A.1 Small field regime
    2. A.2 Large field regime

##  1 Introduction and conclusions

The no-boundary proposal [1] is a mathematically compelling candidate for the
wave function of a closed universe. It has proven difficult however to
reconcile its predictions with observations, as, without further input, it
favors histories of the universe with small amounts of inflation with
exponential pressure. It appears to predict a value for the spatial curvature
parameter which is as large as possible, or, in extremis, an empty de Sitter
universe. For reviews about the proposal and possible ways around this
substantial unresolved issue see [2, 3, 4].

In these notes we assume this hurdle may one day be overcome, and that after
the dust has settled the no-boundary geometry is still a potentially relevant
contribution to the wave function of the universe in the semiclassical limit.
In this context an interesting suggestion was made by Witten [5], building on
work in axiomatic quantum field theory by Kontsevich and Segal [6]: only
complex metrics belonging to the class of “allowable geometries” can serve as
contributing saddles in the semiclassical limit. Colloquially, the allowable
geometries are those on which correlation functions of probe matter can be
computed by a Euclidean path integral. For diagonal complex metrics
$g_{\mu\nu}=\lambda_{\mu}\delta_{\mu\nu}$ in some basis, the Kontsevich-Segal-
Witten (KSW) criterion of allowability can be simply recast into the
inequality

$\sum_{\mu=0}^{D-1}\left|\arg\lambda_{\mu}\right|<\pi\,.$ (1)

Euclidean metrics lie in the interior of the class of allowable geometries
while Lorentzian ones lie on the boundary.

In [7], complex no-boundary solutions that arise in the context of single-
field inflation were numerically subjected to the KSW criterion in a
collection of models. Intriguingly, the corresponding slow roll histories that
passed the criterion were found to have a small tensor-to-scalar ratio $r$ in
accordance with observations. This is exciting because $r$ may be our most
direct probe of quantum gravity through the CMB, and constraining it on
theoretical grounds could explain why we do not observe a large signal. An
analytic understanding of this unexpected feature was lacking, however, and in
[2] significant progress was made in this direction. The main point of this
paper is to complete the calculation of [2] to the large field regime (if
there is one), where it turns out the criterion (1) has a chance of failing.
To leading order in slow roll we find the KSW criterion translates to

$\frac{V^{\prime}_{*}}{V_{*}}\int_{\phi_{*}}^{\chi}|\mathrm{d}\phi|\frac{V_{*}^{\prime}}{V^{\prime}(\phi)}<1\,,$
(2)

where $\phi_{*}$ is the start of inflation and $\chi$ is the end.

The left-hand side of (2) decreases with an increasing amount of efolds if the
potential has a hilltop-like shape ($|V^{\prime}(\phi)|$ decreases with
increasing $N_{e}$). This suggests that any potential $V(\phi)$ could be
modified beyond the point $\phi_{60}$, corresponding to the regime where
observable modes exited the horizon during inflation, so as to satisfy the KSW
criterion when the amount of efolds is increased beyond 60. In other words one
can engineer the factor $V^{\prime}_{*}/V_{*}$ to decrease dramatically
without changing the value of the integral by much. We demonstrate this point
in §7.2 and §7.3 with examples.

Unfortunately this nullifies an ambitious prospect expressed in [7] and
prohibits KSW from having universal phenomenological implications for
inflation (e.g. small $r$) because we do not probe the inflationary potential
beyond $\phi_{60}$, and there is no reason to expect inflation lasted
precisely long enough to solve the standard problems in cosmology. One can
still contemplate using KSW to rule out any particular slow roll history of
the universe, although it is clear that further work on the foundations of the
criterion is overdue.111Examples of complex saddles that do not satisfy KSW
but give physically sensible results have been noted in the recent literature,
e.g. [8].

##  2 Review of recent literature

For metric and scalar field

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a(t)^{2}\mathrm{d}\Omega_{3}^{2}\,,~{}~{}~{}\phi(t)\,,$
(3)

the EOM are ($1/8\pi G_{N}=1$)

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{1}{a^{2}}-\frac{1}{3}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right)$
$\displaystyle=0\,,$ (4)
$\displaystyle\ddot{\phi}+3\frac{\dot{a}}{a}\dot{\phi}+V^{\prime}(\phi)$
$\displaystyle=0\,.$ (5)

In [2], an analytic solution to these equations was obtained to first order in
slow roll and with no-boundary initial conditions. This reads, in slightly
different notation
($\tau=H_{*}t,H_{*}=\sqrt{V_{*}/3},\varepsilon_{*}=(V^{\prime}_{*}/V_{*})^{2}/2$):

$\begin{aligned}
\phi(\tau)&=\phi_{*}+\sqrt{2\varepsilon_{*}}\varphi(\tau)\,,\\\
\varphi(\tau)&=f_{\phi}(\tau)+c_{\phi}\,,\\\
f_{\phi}(\tau)&=\frac{1+i\sinh\tau}{\cosh^{2}\tau}-\log\left(1-i\sinh\tau\right)\,,\end{aligned}\qquad\qquad\begin{aligned}
a(\tau)&=\frac{1}{H_{*}}\cosh(\tau)\left[1+\varepsilon_{*}\gamma(\tau)\right]\,,\\\
\gamma(\tau)&=\tanh\tau\left(f_{a}(\tau)+c_{a}\right)\,,\\\
f_{a}(\tau)&=\text{(long explicit expression)}\,.\end{aligned}$ (6)

$c_{\phi,a}$ are integration constants which we have not yet fixed, and
$f_{a}$ depends on $c_{\phi}$. The center of the ball, or “south pole” of the
geometry, is located near $\tau=i\pi/2$ where

$f_{\phi}(\tau)=\frac{1}{2}-\log
2+\mathcal{O}\left(\left(\tau-\frac{i\pi}{2}\right)^{2}\right)\,,\quad
f_{a}(\tau)=\mathcal{O}\left(\left(\tau-\frac{i\pi}{2}\right)^{3}\right)\,.$
(7)

For $\tau\gg 1$ on the real line, where the $S^{3}$ boundary lies,

$\displaystyle\varphi(\tau)$ $\displaystyle=-\tau+\log
2+\frac{i\pi}{2}+c_{\phi}+3e^{-2\tau}-\frac{16i}{3}e^{-3\tau}+\cdots\,,$ (8)
$\displaystyle\gamma(\tau)$
$\displaystyle=-\frac{\tau^{2}}{2}+\frac{1}{6}\left(1+6c_{\phi}+6\log 2+3\pi
i\right)\tau+\frac{1}{72}\left(136-72c_{\phi}-6\pi i-36\pi
ic_{\phi}-3\pi^{2}-72\log 2-6\pi i\log 64\right)$
$\displaystyle+c_{a}+(\cdots)e^{-2\tau}+\frac{32i}{9}e^{-3\tau}+\cdots\,.$ (9)

To get an approximately real solution on the real line we must set
$\text{Im}\,c_{\phi}=-\pi/2$.222In any exact no-boundary solution, $a,\phi$
must become exactly real at the same point on the real line. To make this
happen, $c_{\phi,a}$ will be corrected by terms exponentially small in $\tau$
and higher order in slow roll. The real part of $c_{\phi}$ is redundant with
$\phi_{*}$. As explained in [2], setting it to zero causes $\phi_{*}$ to be
identified with the start of inflation in the familiar slow roll solution in
flat slicing, viz. $\phi\sim\phi_{*}-\sqrt{2\varepsilon_{*}}N_{e}$ at large
$N_{e}\gg 1$ where $N_{e}=\log a/a_{*}\sim\tau-\log 2$ is the amount of efolds
of inflation and $a_{*}=1/H_{*}$. Likewise, the imaginary part of $c_{a}$ is a
fixed constant dictated by the expansion (9).333It depends on
$\text{Re}\,c_{\phi}$. Picking these imaginary parts for $c_{\phi,a}$ causes
the leading imaginary terms in (8)-(9) to be the exponentially small ones
$\propto e^{-3\tau}$ that are written. The real part is redundant with a shift
in the time coordinate. Again one could fix it by matching onto the solution
in flat slicing. In any case, the south pole is located at

$\tau_{\textsf{SP}}=\frac{i\pi}{2}-\varepsilon_{*}c_{a}$ (10)

including the first slow roll correction. Near this point
$a(t)=i\left(t-t_{\textsf{SP}}\right)+\cdots$ as required by regularity;
moving downwards from here, $t=t_{\textsf{SP}}-i\theta$,
$\mathrm{d}s^{2}\sim\mathrm{d}\theta^{2}+\theta^{2}\mathrm{d}\Omega_{3}^{2}$.
Following the matching condition of [2], $c_{a}$ has both real and imaginary
parts. Not following this matching, we could choose $c_{a}=0$, causing no
shift in the location of the south pole, provided we set $c_{\phi}=5/6-\log
2-i\pi/2$. A third possibility is to set $\text{Re }c_{a}=0$, as was done in
[9]. Further, instead of identifying $\phi_{*}$ with the start of inflation in
the flat slicing solution, one could identify $\phi_{*}=\text{Re
}\phi_{\textsf{SP}}$. We have to first non-trivial order

$\phi_{\textsf{SP}}=\phi_{*}+\frac{V^{\prime}_{*}}{V_{*}}\left(\frac{1}{2}-\log
2+c_{\phi}\right)\,,$ (11)

so setting $\text{Re }c_{\phi}=\log 2-1/2$ achieves $\text{Re
}\phi_{\textsf{SP}}=\phi_{*}$. In turn this sets $\text{Im }c_{a}=\pi(\log
2-2/3)$. In this case the south pole gets shifted by

$\tau_{\textsf{SP}}=\frac{i\pi}{2}\left(1+\alpha\right)$ (12)

where

$\alpha=2\left(\frac{2}{3}-\log
2\right)\varepsilon_{*}=(-0.052961\dots)\varepsilon_{*}\,.$ (13)

This explains analytically what was found approximately in [9], where it was
estimated that $\alpha\approx-0.05\varepsilon_{*}$. The other numerical
estimate of [9] was $|\text{Im }\phi_{\textsf{SP}}|\approx
2.21\sqrt{\varepsilon_{*}}$. Here we see $2.21$ is really
$\pi/\sqrt{2}=(2.2214\dots)$. Finally notice that (6) has a singularity
($a,\phi\to\infty)$ at $\tau=i\pi/2+i(2n+1)\pi$ for integer $n$. In particular
there is a singularity at $\tau=-i\pi/2$ at a distance $\pi$ from
$\tau_{\textsf{SP}}$. This was anticipated in [9] where the first few terms of
the series expansion of the solution (6) around the south pole were found, and
from them it was estimated that the radius of convergence of the series should
be about $\pi$.

##  3 Regime of validity of (6)

The approximation (6) is accurate as long as

$\tau\ll\frac{1}{\sqrt{\varepsilon_{*}}}\,.$ (14)

Beyond this regime, from (9), (6) would predict a negative scale factor. More
formally the breakdown can be seen from the equation for $\varphi$,

$\ddot{\varphi}+H_{*}\tanh\tau\,\dot{\varphi}+V^{\prime}_{*}=0\,,$ (15)

where it was assumed that

$\frac{\dot{a}}{a}\approx H_{*}\tanh\tau\,.$ (16)

Instead, from (6), keeping only the leading correction term
$\propto\varepsilon_{*}\tau^{2}$ for $\tau\gg 1$,

$\frac{\dot{a}}{a}=H_{*}\left(\tanh\tau-\frac{2\varepsilon_{*}\tau}{2-\varepsilon_{*}\tau^{2}}\right)\,.$
(17)

The approximation (16) is not good when $\varepsilon_{*}\tau^{2}$ is not
small. In other words (6) is no longer valid when
$\varepsilon_{*}\gamma=\mathcal{O}(1)$. Another way to put it is that (6) is
accurate as long as $\Delta\phi\sim\sqrt{\varepsilon_{*}}\tau\ll 1$. In large
field models of inflation this can be violated. Beyond $\tau\gtrsim
1/\sqrt{\varepsilon_{*}}$ higher order corrections in slow roll
$\propto\varepsilon_{*}^{n}f(\tau)$ become important.

This last point can be seen explicitly from the usual slow roll solution (in
closed or flat slicing), where
$a\sim\exp\left(\tau-\varepsilon_{*}\tau^{2}/2\right)$ in the regime
$1\ll\tau\ll 1/\varepsilon_{*}$. This follows from the relation
$H=\dot{a}/a\approx H_{*}(1-\varepsilon_{*}\tau)$ in this regime. In large
field models such as quadratic inflation $\varepsilon_{*}\tau^{2}$ may become
large, so in these cases we cannot expand the exponential in the expression
for $a$ and keep the first term.

The other approximation that was made to arrive at (6) is $V^{\prime}\approx
V^{\prime}_{*}$. This is valid as long as

$\tau\ll\frac{1}{\eta_{*}}\,.$ (18)

##  4 KSW in small field models

We now turn to the KSW criterion in the small field regime where (6) is
accurate. This has also been discussed in [2]. The equation for the extremal
curve $t=\gamma_{\textsf{e}}(\ell)$, saturating the KSW bound (1), reads

$\arg\gamma^{\prime}_{\textsf{e}}+\arg a(\gamma_{\textsf{e}})^{3}=0\,.$ (19)

This is solved e.g. by

$\gamma^{\prime}_{\textsf{e}}=\frac{1}{a^{3}}\,,$ (20)

which in turn is integrated to yield

$\int_{\gamma_{\textsf{SP}}}^{\gamma_{\textsf{e}}}\mathrm{d}\gamma\,a(\gamma)^{3}=\ell\,,$
(21)

or, taking the imaginary part444A parametrization of $\gamma_{\textsf{e}}$ has
implicitly been chosen by the solution (20), but this choice has been removed
by taking the imaginary part in (22).,

$\text{Im}\left(\int_{\gamma_{\textsf{SP}}}^{\gamma_{\textsf{e}}}\mathrm{d}\gamma\,a(\gamma)^{3}\right)=0\,.$
(22)

We can compute the integral by splitting the contour into 3 pieces: calling
$I=I_{\gamma_{\textsf{SP}}\to\gamma_{\textsf{e}}}\equiv\int_{\gamma_{\textsf{SP}}}^{\gamma_{\textsf{e}}}\mathrm{d}\gamma\,a(\gamma)^{3}$
we write

$I=I_{\gamma_{\textsf{SP}}\to 0}+I_{0\to u}+I_{u\to u+i\sigma}\,,$ (23)

where $u+i\sigma=\gamma_{\textsf{e}}$. The dominant contribution to the
imaginary part of the first piece comes from the
non-$\varepsilon_{*}$-suppressed part of $a$, that is, the de Sitter solution
$a(\tau)=(1/H_{*})\cosh\tau$. We have

$\text{Im}\left(I_{\gamma_{\textsf{SP}}\to
0}\right)=-\frac{2}{3H_{*}^{4}}\left[1+\mathcal{O}\left(\varepsilon_{*}\right)\right]\,.$
(24)

Then,

$I_{0\to
u}=\text{(real)}+\frac{3\varepsilon_{*}}{H_{*}^{3}}\int_{0}^{u}\mathrm{d}t\cosh^{3}(H_{*}t)\gamma(t)+\cdots\,.$
(25)

We have $\text{Im }\gamma(t)\approx(32/9)e^{-3H_{*}t}$ after a time $t\gtrsim
1/H_{*}$ when slow roll kicks in. Therefore we can split the integral into a
piece $0\to\mathcal{O}(1/H_{*})$ giving a small contribution
$(\varepsilon_{*}/H_{*}^{4})\times\mathcal{O}(1)$ plus the contribution

$\text{Im}(I_{0\to
u})=\mathcal{O}\left(\frac{\varepsilon_{*}}{H_{*}^{4}}\right)+\frac{3\varepsilon_{*}}{H_{*}^{3}}\int_{\mathcal{O}(1)}^{u}\mathrm{d}t\,\frac{1}{8}e^{3H_{*}t}\frac{32}{9}e^{-3H_{*}t}=\frac{4\varepsilon_{*}}{3H_{*}^{4}}\times\left[H_{*}u+\mathcal{O}(1)\right]$
(26)

for $H_{*}u\gg 1$. For the last piece

$\text{Im}\left(I_{u\to
u+i\sigma}\right)\approx\frac{e^{3H_{*}u}}{8H_{*}^{4}}H_{*}\sigma\,.$ (27)

Putting it all together in (22), solving for $\sigma$ and calling
$H_{*}u=\tau$,

$H_{*}\sigma=\frac{16}{3}e^{-3\tau}\left[1+\mathcal{O}(\varepsilon_{*})-2\varepsilon_{*}\tau\right]\,,$
(28)

which was obtained in [2].

We argued in §3 that the approximation (6), which was used in this
calculation, is valid as long as $\sqrt{\varepsilon_{*}}\tau\ll 1$. So surely
in this case $\varepsilon_{*}\tau\ll 1$ and the extremal curve remains above
the real axis ($\sigma>0$ in (28)), where the boundary lies. This means we can
construct a KSW-allowable curve that reaches the boundary. We conclude that
small field models satisfy the KSW criterion.555Perhaps this is not quite
rigorous. We have argued that if $\sqrt{\varepsilon_{*}}\tau\ll 1$, then there
is a small (sub-Planckian) field variation and KSW is satisfied. For the
converse, standard manipulations yield
$\Delta\phi=\int\mathrm{d}t\,H\sqrt{2\varepsilon_{H}}\gtrsim\sqrt{2\varepsilon_{*}}\int\mathrm{d}t\,H\geq\sqrt{2\varepsilon_{*}}\tau(H_{\textsf{end}}/H_{*})$,
where we have used $\varepsilon_{H}=\varepsilon_{V}$ to first order in slow
roll (for a mini review of slow roll see §5). So if $\Delta\phi\ll
H_{\textsf{end}}/H_{*}$, which is perhaps slightly stronger than small field
($\ll 1$), KSW would be satisfied. Our final result is really Eq. (59) below
which makes no independent reference to the distance traversed by $\phi$.

Formula (28) suggests that KSW could be violated when $\varepsilon_{*}\tau$
becomes of order one, as it does in some large field models of inflation e.g.
monomial inflation where $V=\phi^{p}$ and $\varepsilon_{*}\tau\sim p/4$ for
large $\phi_{*}\gg\chi$, where $\chi$ is the end of inflation. This is not
sharp, however, because the approximation used to arrive at (28) is not valid
in this regime. In the next section we extend the approximate solution (6)
beyond $\sqrt{\varepsilon_{*}}\tau\ll 1$ to the whole slow roll patch and then
discuss the KSW criterion in the extended regime.

##  5 Approximate solution in large field models

To extend the solution (6) beyond $\tau\ll 1/\sqrt{\varepsilon_{*}}$, we
reconsider the EOM (4)-(5) with the scale factor and scalar field written as a
dominant real part plus a small imaginary part:

$a=a^{\textsf{Re}}+ia^{\textsf{Im}}\,,\quad\phi=\phi^{\textsf{Re}}+i\phi^{\textsf{Im}}\,.$
(29)

Linearizing in the imaginary parts, the zeroth order equations are simply
(4)-(5) with $(a,\phi)\to(a^{\textsf{Re}},\phi^{\textsf{Re}})$. For $\tau\gg
1$ these become the usual slow roll equations [10, 11], about which we recall
some useful formulas. Neglecting the curvature term altogether in what
follows666This is reasonable because this term is volume-suppressed compared
to others, which may only be slow roll-suppressed., the EOM for the real parts
can be put in the otherwise exact form

$\displaystyle
3H^{2}=\frac{(\dot{\phi}^{\textsf{Re}})^{2}}{2}+V(\phi^{\textsf{Re}})\,,\quad(3-\eta_{H})H\dot{\phi}^{\textsf{Re}}=-V^{\prime}(\phi^{\textsf{Re}})\,,$
(30)

where

$H\equiv\frac{\dot{a}^{\textsf{Re}}}{a^{\textsf{Re}}}\,,\quad\eta_{H}\equiv-\frac{\ddot{\phi}^{\textsf{Re}}}{H\dot{\phi}^{\textsf{Re}}}\,.$
(31)

Is it also useful to define

$\varepsilon_{H}\equiv-\frac{\dot{H}}{H^{2}}\,,\quad\zeta_{H}^{2}\equiv\frac{1}{H^{2}}\partial_{t}(\ddot{\phi}/\dot{\phi})\,.$
(32)

One can derive the (exact) relations

$\displaystyle V$ $\displaystyle=(3-\varepsilon_{H})H^{2}\,,$ $\displaystyle
V^{\prime}$ $\displaystyle=\sqrt{2\varepsilon_{H}}(3-\eta_{H})H^{2}\,,$
$\displaystyle V^{\prime\prime}$
$\displaystyle=(3\varepsilon_{H}+3\eta_{H}-\eta_{H}^{2}-\zeta_{H}^{2})H^{2}\,,$
(33) $\displaystyle\varepsilon_{H}$
$\displaystyle=\frac{(\dot{\phi}^{\textsf{Re}})^{2}}{2H^{2}}\,,$
$\displaystyle\ddot{a}^{\textsf{Re}}$
$\displaystyle=a^{\textsf{Re}}(1-\varepsilon_{H})H^{2}\,.$ (34)

Although we will not use them, for completeness we recall relations between
the Hubble slow roll parameters $\varepsilon_{H},\eta_{H},\zeta_{H}$ and the
potential slow roll parameters $\varepsilon_{V},\eta_{V},\zeta_{V}$, defined
as

$\varepsilon_{V}=\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\,,\quad\eta_{V}=\frac{V^{\prime\prime}}{V}\,,\quad\zeta_{V}^{2}=\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}\,,$
(35)

To second order [10],

$\varepsilon_{H}=\varepsilon_{V}-\frac{4}{3}\varepsilon_{V}^{2}+\frac{2}{3}\varepsilon_{V}\eta_{V}+\cdots\,,\quad\eta_{H}=\eta_{V}-\varepsilon_{V}+\frac{8}{3}\varepsilon_{V}^{2}-\frac{8}{3}\varepsilon_{V}\eta_{V}+\frac{1}{3}\eta_{V}^{2}+\frac{1}{3}\zeta_{V}^{2}+\cdots\,.$
(36)

Returning to the EOM, the first order equations (or, leading imaginary parts)
are [9]

$\displaystyle
H\dot{a}^{\textsf{Im}}-\frac{a^{\textsf{Re}}}{6}\left(V^{\prime}(\phi^{\textsf{Re}})\phi^{\textsf{Im}}+\dot{\phi}^{\textsf{Re}}\dot{\phi}^{\textsf{Im}}\right)-H^{2}a^{\textsf{Im}}=\frac{a^{\textsf{Im}}}{(a^{\textsf{Re}})^{2}}$
$\displaystyle\approx 0\,,$ (37)
$\displaystyle\ddot{\phi}^{\textsf{Im}}+3H\dot{\phi}^{\textsf{Im}}+3\frac{\dot{\phi}^{\textsf{Re}}}{a^{\textsf{Re}}}\left(\dot{a}^{\textsf{Im}}-Ha^{\textsf{Im}}\right)+V^{\prime\prime}(\phi^{\textsf{Re}})\phi^{\textsf{Im}}$
$\displaystyle=0\,.$ (38)

In (37) we have again neglected a term which is volume-suppressed compared to
the others. Inspired by (6), which we must reproduce when $1\ll\tau\ll
1/\sqrt{\varepsilon_{*}}$, we try the Ansatz

$\phi^{\textsf{Im}}=\frac{g_{\phi}}{(a^{\textsf{Re}})^{3}}\,,\quad
a^{\textsf{Im}}=\frac{g_{a}}{(a^{\textsf{Re}})^{2}}\,,$ (39)

where $g_{\phi,a}$ are slowly-varying functions, i.e. the dimensionless
$\dot{g}/(Hg)$ are slow roll-suppressed. In (6) these functions are just
constants. In this way

$\dot{\phi}^{\textsf{Im}}=-3H\phi^{\textsf{Im}}\left(1-\frac{1}{3}\frac{\dot{g_{\phi}}}{Hg_{\phi}}\right)\,,\quad\dot{a}^{\textsf{Im}}=-2Ha^{\textsf{Im}}\left(1-\frac{1}{2}\frac{\dot{g_{a}}}{Hg_{a}}\right)\,,$
(40)

where the correction terms are small. More precisely we will set up a
perturbative expansion

$\displaystyle g_{\phi}=g_{\phi}^{(0)}+g_{\phi}^{(1)}+\cdots\,,\quad
g_{a}=g_{a}^{(0)}+g_{a}^{(1)}+\cdots\,,$ (41)

treating $g^{(i+1)}\ll g^{(i)}$ but
$g^{(i+1)}=\mathcal{O}\left(\dot{g}^{(i)}/Hg^{(i)}\right)$.

Plugging (39) into (37), using (30)-(34), gives

$\displaystyle{\color[rgb]{0.295,0.295,0.295}g_{a}^{(0)}+\frac{\sqrt{2\varepsilon_{H}}}{3}g_{\phi}^{(0)}}{\color[rgb]{1,.5,0}+g_{a}^{(1)}+\frac{\sqrt{2\varepsilon_{H}}}{3}g_{\phi}^{(1)}-\frac{\sqrt{\varepsilon_{H}}}{9\sqrt{2}}\left(\eta_{H}+\frac{\dot{g}_{\phi}^{(0)}}{Hg_{\phi}^{(0)}}\right)g_{\phi}^{(0)}-\frac{\dot{g}_{a}^{(0)}}{3H}}{\color[rgb]{1,0,0}+\cdots}=0\,,$
(42)

where leading terms are indicated in green, subleading ones in orange, and we
have neglected even further subleading terms in red. Solving the leading piece
gives

${\color[rgb]{0.295,0.295,0.295}g_{a}^{(0)}=-\frac{\sqrt{2\varepsilon_{H}}}{3}g_{\phi}^{(0)}}\,.$
(43)

Plugging (39) into (38) yields instead

$\displaystyle{\color[rgb]{0.295,0.295,0.295}3\sqrt{2\varepsilon_{H}}\,g_{a}^{(0)}+(2\varepsilon_{H}+\eta_{H})g_{\phi}^{(0)}-\frac{\dot{g}_{\phi}^{(0)}}{H}}$
(44)
$\displaystyle{\color[rgb]{1,.5,0}+3\sqrt{2\varepsilon_{H}}g_{a}^{(1)}-\frac{1}{3}\left(\eta_{H}^{2}+\zeta_{H}^{2}\right)g_{\phi}^{(0)}+(2\varepsilon_{H}+\eta_{H})g_{\phi}^{(1)}-\sqrt{2\varepsilon_{H}}\frac{\dot{g}_{a}^{(0)}}{H}-\frac{\dot{g}_{\phi}^{(1)}}{H}+\frac{\ddot{g}_{\phi}^{(0)}}{3H^{2}}}{\color[rgb]{1,0,0}+\cdots}=0\,.$
(45)

Considering the leading equation, using (43), we get

${\color[rgb]{0.295,0.295,0.295}\frac{\dot{g}_{\phi}^{(0)}}{Hg_{\phi}^{(0)}}=\eta_{H}}\,,$
(46)

which, as promised, is slow roll-suppressed. From the definition of $\eta_{H}$
in (31) we see the solution is

$g_{\phi}^{(0)}=\frac{C}{\dot{\phi}^{\textsf{Re}}}=-\frac{C}{\sqrt{2\varepsilon_{H}}H}\,,$
(47)

for a constant $C$ (we assume $V^{\prime}(\phi^{\textsf{Re}})>0$.) It follows
that

$g_{a}^{(0)}=\frac{C}{3H}\,.$ (48)

$C$ is determined by matching onto the intermediate $\tau$ solution of [2] in
(6) where $H,\varepsilon_{H}\approx H_{*},\varepsilon_{*}$:

$C\approx\frac{4\varepsilon_{*}}{3H_{*}^{2}}\,.$ (49)

We summarize the leading behavior:

$\phi^{\textsf{Im}}=\frac{g_{\phi}^{(0)}+\cdots}{(a^{\textsf{Re}})^{3}}=-\frac{C}{\sqrt{2\varepsilon_{H}}H(a^{\textsf{Re}})^{3}}\left(1+\cdots\right)\,,\quad
a^{\textsf{Im}}=\frac{g_{a}^{(0)}+\cdots}{(a^{\textsf{Re}})^{2}}=\frac{C}{3H(a^{\textsf{Re}})^{2}}\left(1+\cdots\right)\,.$
(50)

We stress that $H$ and $\varepsilon_{H}$ are slowly-varying functions (their
relative change per efold is small), but varying nonetheless. Their variation
over the entirety of inflation may be large.

(50) should be a good approximation as long as slow roll persists. So if slow
roll lasts longer than $\tau\sim 1/\sqrt{\varepsilon_{*}}$, it extends the
approximation (6) which was valid only for $\tau\ll 1/\sqrt{\varepsilon_{*}}$.
In terms of $\varepsilon_{*}$, in a sense (50) involves all powers
$\varepsilon_{*}^{n}$, as it depends on the functions $H,\varepsilon_{H}$
which depend on the entire function $V(\phi^{\textsf{Re}})$. We are familiar
with this feature in slow roll, e.g. the amount of efolds $N_{e}$ depends on
an integral of the function $1/\sqrt{2\varepsilon}$, which is slowly-varying,
but its total variation may be large.

Finally we solve the subleading orange equations, which are unimportant for
the main point of this paper (the KSW criterion in §6), but important to
analyze potential subleading corrections to the on-shell action coming from
the large field regime. These are discussed in Appendix A.

The subleading equation of (42) sets

${\color[rgb]{1,.5,0}g_{a}^{(1)}=\frac{C(\varepsilon_{H}-\eta_{H})}{9H}-\frac{\sqrt{2\varepsilon_{H}}}{3}g_{\phi}^{(1)}}\,.$
(51)

We use this in the subleading equation (45). A useful relation is

$\dot{\eta}_{H}=H(\varepsilon_{H}\eta_{H}-\zeta_{H}^{2})\,,\quad\text{so
}\ddot{g}_{\phi}^{(0)}=\left(\zeta_{H}^{2}-\eta_{H}^{2}\right)\frac{CH}{\sqrt{2\varepsilon_{H}}}\,.$
(52)

We obtain

${\color[rgb]{1,.5,0}\frac{\dot{g}_{\phi}^{(1)}}{H}=\eta_{H}g_{\phi}^{(1)}+\frac{2C}{3\sqrt{2\varepsilon_{H}}H}\left(\zeta_{H}^{2}-\varepsilon_{H}\eta_{H}\right)}\,.$
(53)

The solution is

$g_{\phi}^{(1)}=\frac{2\eta_{H}}{3}g_{\phi}^{(0)}\,,$ (54)

and so

$g_{a}^{(1)}=\frac{1}{3}\left(\varepsilon_{H}+\eta_{H}\right)g_{a}^{(0)}\,.$
(55)

Summarizing the subleading behavior:

$\phi^{\textsf{Im}}=-\frac{C}{\sqrt{2\varepsilon_{H}}H(a^{\textsf{Re}})^{3}}\left(1+\frac{2}{3}\eta_{H}+\cdots\right)\,,\quad
a^{\textsf{Im}}=\frac{C}{3H(a^{\textsf{Re}})^{2}}\left(1+\frac{1}{3}\left(\varepsilon_{H}+\eta_{H}\right)+\cdots\right)\,,$
(56)

with $C$ given in (49).

Before turning to KSW, we note that (56) corrects two formulas that appeared
in [9], where it was claimed that

$\phi^{\textsf{Im}}\approx-\frac{\tilde{C}}{\sqrt{2}H(a^{\textsf{Re}})^{3}}\,,\quad
a^{\textsf{Im}}\approx\frac{\tilde{C}\sqrt{\varepsilon}}{3H(a^{\textsf{Re}})^{2}}\quad\quad\quad\text{[incorrect
for $\tau\gtrsim 1/\sqrt{\varepsilon_{*}}$]}$ (57)

at large $\tau\gg 1$, for a constant $\tilde{C}$. From (56) one sees that the
leading slowly-varying factors are inaccurate. As long as $\tau\ll
1/\sqrt{\varepsilon_{*}}$, we have $H,\varepsilon_{H}\approx
H_{*},\varepsilon_{*}$ and (57) agrees with the solution (6) of [2] which is
valid in this regime. Outside of this regime (57) is inaccurate.

##  6 KSW in the extended region

Following the steps of §4 we now use (50) to calculate the extremal curve.
Instead of (28), after a straightforward calculation involving the standard
change of variables $t\to\phi$ in slow roll, we obtain

$H_{*}\sigma=\frac{2}{3(bH_{*})^{3}}\left(1+\mathcal{O}(\varepsilon_{*})-\sqrt{2\varepsilon_{*}}\int_{\phi_{*}}^{\chi}|\mathrm{d}\phi|\frac{V_{*}^{\prime}}{V^{\prime}(\phi)}\right)\,,$
(58)

where we have indicated the final (real) values of the scale factor and scalar
field by $b$ and $\chi$ respectively, and we have also used that
$\dot{\phi}=-V^{\prime}/3H$ to leading order in slow roll. This reduces to
(28) for $\tau\ll 1/\sqrt{\varepsilon_{*}},1/\eta_{*}$ since there
$V^{\prime}(\phi)\approx V^{\prime}_{*}$ and
$\Delta\phi\approx\sqrt{2\varepsilon_{*}}\tau$. It is also valid beyond this
regime, however, as long as we are in slow roll ($\varepsilon$ and $\eta$ are
small in the region between $\phi_{*}$ and $\chi$). We conclude KSW will be
satisfied as long as

$\mathcal{A}(\phi_{*},\chi)\equiv\frac{V^{\prime}_{*}}{V_{*}}\int_{\phi_{*}}^{\chi}|\mathrm{d}\phi|\frac{V_{*}^{\prime}}{V^{\prime}(\phi)}<1+\mathcal{O}(\varepsilon_{*})\,.$
(59)

By $|\mathrm{d}\phi|$ we mean the overall sign should be chosen so that
$\mathcal{A}$ is positive. We have neglected an $\mathcal{O(\varepsilon_{*}})$
correction to the right-hand side of the bound here. This corresponds to a
fuzziness of order $\sqrt{\varepsilon_{*}}$ in what we mean by $\phi_{*}$, or
equivalently a change by an $\mathcal{O}(1)$ amount of efolds of inflation.

For large field models in which $|V^{\prime}_{*}|$ is decreasing with
increasing $|\phi_{*}-\chi|$ (or, increasing amount of efolds) at fixed
endpoint $\chi$, i.e. plateaux or hilltop potentials, this bound is
qualitatively different from the naive extension to
$\tau=\mathcal{O}(1/\varepsilon_{*})$ of the small field bound
$2\varepsilon_{*}\tau<1$ extrapolated from (28). This is because $\mathcal{A}$
could decrease with an increasing amount of efolds, whereas
$2\varepsilon_{*}\tau\sim 2\varepsilon_{*}N_{e}$ always increases. We will see
explicit instances of this in §7.2 and §7.3 below.

##  7 Numerical checks

###  7.1 Monomial inflation

For monomial inflation,

$V=\phi^{p}$ (60)

with $\phi,p>0$, we find

$\displaystyle\frac{V^{\prime}_{*}}{V_{*}}\int_{\phi_{*}}^{\chi}|\mathrm{d}\phi|\frac{V_{*}^{\prime}}{V^{\prime}(\phi)}=\begin{dcases}2\,\log\left(\frac{\phi_{*}}{\chi}\right)&\text{if
$p=2$,}\\\
\frac{p}{2-p}\left[1-\left(\frac{\chi}{\phi_{*}}\right)^{2-p}\right]&\text{otherwise.}\end{dcases}$
(61)

Setting this to one and solving for $\phi_{*}$ gives us the value
$\phi_{*,\textsf{KSW}}(\chi)$ beyond which KSW will be violated:

$\phi_{*,\textsf{KSW}}(\chi)=\begin{dcases}\sqrt{e}\,\chi&\text{if $p=2$,}\\\
\frac{\chi}{\left(2-\frac{2}{p}\right)^{\frac{1}{2-p}}}&\text{if $p>1$,}\\\
\infty&\text{if $p\leq 1$.}\end{dcases}$ (62)

That is, if we start rolling from $\phi_{*}>\phi_{*,\textsf{KSW}}(\chi)$ and
end at $\chi$, the trajectory will not be allowable, while rolling from
$\phi_{*}<\phi_{*,\textsf{KSW}}(\chi)$ and ending at $\chi$ is allowable. The
structure is this way – from non-allowable at small $\phi_{*}\approx\chi$ to
allowable at large $\phi_{*}\gg\chi$ – because $\mathcal{A}$ is monotonically
increasing with $\phi_{*}$ at fixed $\chi$.

For $p=2$ we have
$\chi=e^{-1/2}\phi_{*,\textsf{KSW}}(\chi)\approx(0.606\dots)\phi_{*,\textsf{KSW}}(\chi)$.
This value is consistent with numerics, see Figure 1. Notice that our
approximations should be trustworthy in this case since both $\chi$ and
$\phi_{*}$ lie parametrically far in the slow roll region when $\phi_{*}\gg
1$.

Figure 1: Results in $m^{2}\phi^{2}$ inflation ($m$ drops out in slow roll):
for a given endpoint of inflation $\chi$, $\phi_{*,\textsf{KSW}}$ is the
maximal starting point of inflation so that the slow roll history from
$\phi_{*,\textsf{KSW}}$ to $\chi$ is KSW-allowable. Dots are numerical data,
the line is the analytic prediction $\phi_{*,\textsf{KSW}}=\sqrt{e}\chi$.
Figure courtesy of J. Karlsson.

This result can be contrasted with a naive extension of the small field KSW
bound $2\varepsilon_{*}\tau<1$ inferred from (28), which would not rule out
any trajectory since

$2\varepsilon_{*}\tau\sim
2\frac{p^{2}}{2\phi_{*}^{2}}\times\frac{1}{2p}\left(\phi_{*}^{2}-\chi^{2}\right)=\frac{p}{2}\left[1-\left(\frac{\chi}{\phi_{*}}\right)^{2}\right]\,,$
(63)

which is smaller than $1$ (and so the trajectory would be allowable) for any
$p\leq 2$ and $\chi<\phi_{*}$.

For $p\leq 1$, $\phi_{*,\textsf{KSW}}(\chi)=\infty$ for all $\chi$ and so all
trajectories are allowable according to (59). For $p>1$ there is an upper
bound on the amount of efolds of inflation that can reach a given endpoint
$\chi$. This is

$N_{e,\textsf{max}}(\chi)=\frac{1}{2p}\left[\frac{1}{\left(2-\frac{2}{p}\right)^{\frac{2}{2-p}}}-1\right]\chi^{2}\,.$
(64)

As $p\to 1^{+}$ this diverges.

Strictly speaking, as we have stressed before, these formulas are valid only
in slow roll. In particular $\chi$ must lie in the slow roll regime. In [7],
trajectories were examined which end at the exit of slow roll where the slow
roll parameters become order one. Therefore we should not expect the above
analysis to be exactly correct in that case, but we can proceed with the idea
that most of the slow roll history should be described as above. In
particular, it was found numerically in [7] that
$N_{e,\textsf{max}}(\chi=\phi_{\textsf{end}})=60$ for $p\approx 1.05.$
According to the analytic estimate (64), setting
$\chi=\phi_{\textsf{end}}=p/\sqrt{2}$, $N_{e,\textsf{max}}=60$ and solving for
$p$ yields

$p_{\textsf{KSW},60}=(1.03773\dots)\,,$ (65)

which is close.

Finally we have gathered one more isolated – rather arbitrary – data point for
$p=3/2$. In this case we set $\phi_{*}=8$, for which (62) predicts inflation
should be allowable until $\chi=(3.55\dots)$, or for $N_{e}\approx 17.12$
efolds. We have $\varepsilon_{\chi}\approx 0.09$ so we are approximately in
slow roll throughout the trajectory from $\phi_{*}$ to $\chi$. Numerically we
found $3.530<\chi_{\textsf{KSW}}<3.534$. Some discrepancy is to be expected
here since in (59) we have neglected an $\mathcal{O}(\varepsilon_{*})$
correction to the right-hand side, corresponding to a change of $\phi_{*}$ by
an amount $\mathcal{O}(\sqrt{\varepsilon_{*}}\sim 0.13)$.

###  7.2 $\alpha$-attractors

Here

$V(\phi)=\left[1-\exp\left(-\sqrt{2}\phi/\sqrt{3\alpha}\right)\right]^{2}\,.$
(66)

As for the monomials we can estimate analytically what the value
$\alpha_{\textsf{KSW},60}$ should be, which in [7] was found to be $\approx
93.9$. In this case the structure is opposite to that of monomial inflation:
at fixed $\chi$, $\mathcal{A}$ initially increases with $\phi_{*}$ for
$\phi_{*}\gtrsim\chi$, but then when $\phi_{*}\gg\chi$ it decreases again,
tending to zero as $\phi_{*}\to\infty$. This is because here $V^{\prime}_{*}$
decreases with increasing $\phi_{*}$, in contrast to monomial inflation where
$V^{\prime}_{*}$ grows with $\phi_{*}$. So here if
$\phi_{*}>\phi_{*,\textsf{KSW}}(\chi)$ (more inflation) the trajectory is
allowable while if $\phi_{*}<\phi_{*,\textsf{KSW}}(\chi)$ (less inflation) it
is not allowable, unless $\phi_{*}\approx\chi$ in which case it becomes
allowable again. We leave out the explicit expressions and just state the
result $\alpha_{\textsf{KSW},60}=(83.89\dots)$, which is not too far off. We
summarize these findings in Figure 2.

Figure 2: The integral $\mathcal{A}$ of Eq. (59) for $\alpha$-attractor
inflation with $\alpha=83.88$, which must be smaller than one for KSW to be
satisfied. Here we have chosen
$\chi=\phi_{\textsf{end}}=\sqrt{3\alpha/2}\,\log\left(1+2/\sqrt{3\alpha}\right)$
at the end of inflation where $\varepsilon_{V}=1$. We see that starting
inflation beyond $\phi_{*}\approx 12.65$, which produces 60 efolds of
inflation, is predicted to be allowable, while starting at smaller values is
not allowable, unless the amount of efolds is very small (in this case
$\approx$ 1.14 efolds; intuitively this regime is KSW-allowable because the
geometry is approximately Euclidean). This analytic result is not in poor
agreement with numerics (which tell us $\alpha_{\textsf{KSW},60}\approx 93.9$
instead of 83.88), and as we have stressed some disagreement is to be expected
since our analytic approximation is not valid towards the end of inflation.

It is also curious to note that $\alpha\approx 17$ is the critical value below
which all slow roll trajectories are allowable. For larger $\alpha$, like in
Figure 2, there is some exclusion. We show this in Figure 3.

Figure 3: $\alpha\approx 17$ is the critical value beyond which KSW begins to
rule out some inflationary histories. For smaller $\alpha$ all trajectories
(with any amount of inflation) are allowable.

###  7.3 Natural inflation

Here

$V=1+\cos(\phi/f)\,.$ (67)

The allowability-structure is similar to that of $\alpha$-attractors because
we have a hilltop: there is a small region close to the end of inflation,
$\phi_{*}\lesssim\chi\sim\pi f$, with an $\mathcal{O}(1)$ amount of efolds
which are allowable, after which there is a non-allowable region, and then
close to the hilltop near $\phi=0$ trajectories become allowable once more. We
find $f_{\textsf{KSW},60}=(5.866\dots)$ while [7] found numerically
$f_{\textsf{KSW},60}\approx 6.09$. We illustrate this in Figure 4.

Figure 4: Same setup as in Figure 2, but now for natural inflation where
$V=1+\cos(\phi/f)$. Inflation starts near the hilltop at $\phi\approx 0$ and
ends near the minimum at $\phi\approx\pi f$. More precisely
$\phi_{\textsf{end}}=2f\tan^{-1}(\sqrt{2}f)$ is where $\varepsilon_{V}=1$.
Shown here again is the KSW integral (59) which must be smaller than one for
an allowable history. We see in this case that if $\phi_{*}\lesssim 5.02$,
which would have produced 60 efolds of inflation, the history will be
allowable. The corresponding value of $f$, $(5.866\dots)$, is close to the
numerical value $\approx 6.09$ that was obtained in [7].

## Acknowledgements

We thank Victor Gorbenko, Thomas Hertog, Joel Karlsson and Juan Maldacena for
inspiring conversations.

##  Appendix A Slow roll corrections to the on-shell action

The action is

$\displaystyle S$
$\displaystyle=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{R}{2}-\frac{1}{2}\left(\partial\phi\right)^{2}-V\right)+\text{(boundary)}$
(68)
$\displaystyle=2\pi^{2}\int\mathrm{d}t~{}3a(1-\dot{a}^{2})+a^{3}\left(\frac{\dot{\phi}^{2}}{2}-V\right)\,.$
(69)

Using the Friedmann equation (4), the on-shell action can be expressed as

$\bar{S}=4\pi^{2}\int\mathrm{d}t~{}a(3-a^{2}V)\,.$ (70)

For de Sitter, $a(t)=(1/H_{*})\cosh(H_{*}t)$, $V_{*}=3H_{*}^{2}$, and
calculating the integral along the no-boundary contour from $t=i\pi/2H_{*}$ to
$t=T$ on the real line where $a=b$, we find the classic result

$\frac{\bar{S}_{\textsf{dS}}}{4\pi^{2}}=-\frac{3i}{V_{*}}-\frac{3}{V_{*}}\left[(H_{*}b)^{2}-1\right]^{3/2}\,.$
(71)

If as in [2] we call $\alpha_{r}\equiv H_{*}b$ and expand at large
$\alpha_{r}$:

$\frac{i\bar{S}_{\textsf{dS}}}{4\pi^{2}}\times
H_{*}^{2}=1-i\left(\alpha_{r}^{3}-\frac{3}{2}\alpha_{r}+\mathcal{O}(1/\alpha_{r})\right)\quad\text{as
}\alpha_{r}\to\infty\,.$ (72)

This agrees with Eq. (48) in [2]. Now we would like to compute the leading
slow roll correction to this.

###  A.1 Small field regime

In the regime $\tau\ll 1/\sqrt{\varepsilon_{*}}$ we can use the analytic
solution (6) of [2]. We will fix the integration constants $c_{\phi,a}$ as [2]
does. In this way, with $N_{e}\equiv\tau-\log 2$,

$\varphi(\tau)\sim-
N_{e}\,,\quad\gamma(\tau)\sim-\frac{N_{e}^{2}}{2}+\frac{N_{e}}{6}\quad\text{as
}\tau\to\infty\,.$ (73)

We will calculate the integral in (70) from the south pole at
$\tau_{\textsf{SP}}=i\pi/2-\varepsilon_{*}c_{a}$ to a point
$\tau_{\textsf{int}}$ on the real line which satisfies
$1\ll\tau_{\textsf{int}}\ll 1/\sqrt{\varepsilon_{*}}$, keeping track of the
leading corrections in $\varepsilon_{*}$ to the de Sitter answer. The point
$\tau_{\textsf{int}}$ must be chosen so that the same value $H_{*}b$ is
attained at it as in the de Sitter solution. The relevant relation is

$H_{*}b=\alpha_{r}=\cosh\tau_{\textsf{int}}\left(1+\varepsilon_{*}\gamma(\tau_{\textsf{int}})\right)\,.$
(74)

Inverting to first order in $\varepsilon_{*}$ we find777Notice the small
imaginary contribution. This arises because we did not completely kill the
imaginary part of $a(\tau)$ (or $\phi(\tau)$) on the real line with our choice
of integration constants $c_{\phi,a}$ in (6).

$\tau_{\textsf{int}}=\cosh^{-1}\alpha_{r}+\varepsilon_{*}\left[\frac{1}{2}\log\alpha_{r}\left(\log\alpha_{r}-\frac{1}{3}\right)-\frac{3\log\alpha_{r}-2}{4\alpha_{r}^{2}}-\frac{4i}{9\alpha_{r}^{3}}+\mathcal{O}\left(\frac{\log\alpha_{r}}{\alpha_{r}^{4}}\right)\right]\quad\text{as
}\alpha_{r}\to\infty\,.$ (75)

To first order in $\varepsilon_{*}$ we find888Because the integrand vanishes
at the center of the ball, there is no contribution from the shift of the
south pole at this order.

$\displaystyle\frac{\bar{S}_{1}}{4\pi^{2}}=\frac{1}{H_{*}}\int_{\tau_{\textsf{SP}}}^{\tau_{\textsf{int}}}\mathrm{d}\tau\,a(3-a^{2}V)$
$\displaystyle=\frac{\bar{S}_{\textsf{dS}}}{4\pi^{2}}+\frac{3\varepsilon_{*}}{V_{*}}\int_{i\pi/2}^{\cosh^{-1}\alpha_{r}}\mathrm{d}\tau\left(3\gamma\cosh\tau-9\gamma\cosh^{3}\tau-6\varphi\cosh^{3}\tau\right)$
(76)
$\displaystyle+\frac{3\varepsilon_{*}}{V_{*}}\left(\frac{1}{2}\log\alpha_{r}\left(\log\alpha_{r}-\frac{1}{3}\right)-\frac{3\log\alpha_{r}-2}{4\alpha_{r}^{2}}-\frac{4i}{9\alpha_{r}^{3}}+\cdots\right)3\alpha_{r}(1-\alpha_{r}^{2})\,.$
(77)

The integral can be evaluated analytically; we do so and finally obtain

$\frac{i(\bar{S}_{1}-\bar{S}_{\textsf{dS}})}{4\pi^{2}}\times\frac{H_{*}^{2}}{\varepsilon_{*}}=i\alpha_{r}^{3}\left(\log\alpha_{r}-\frac{1}{6}\right)+\frac{i\alpha_{r}}{4}\left(6\log\alpha_{r}-11\right)+i\pi+\left(\log
4-\frac{7}{2}\right)+\mathcal{O}(1/\alpha_{r})\,.$ (78)

This answer was obtained in [2] using different coordinates.

###  A.2 Large field regime

A priori (78) is valid in the regime $1\ll\tau_{\textsf{int}}\ll
1/\sqrt{\varepsilon_{*}}$, or
$1\ll\alpha_{r}\ll\exp\left(1/\sqrt{\varepsilon_{*}}\right)$. Beyond this the
approximation (6) we used breaks down as discussed in §3. In §5 we discussed
how the imaginary parts of the fields decay beyond this regime – as long as
inflation lasts – and we can use this knowledge to estimate the contribution
to the real part of $i\bar{S}$ coming from these late times. From (70) and
(56) we have

$\displaystyle\frac{1}{4\pi^{2}}\text{Im }\bar{S}\big{|}_{\textsf{late}}$
$\displaystyle=-\int_{t_{\textsf{int}}}^{T}\mathrm{d}t~{}3V(\phi^{\textsf{Re}})(a^{\textsf{Re}})^{2}a^{\textsf{Im}}+V^{\prime}(\phi^{\textsf{Re}})(a^{\textsf{Re}})^{3}\phi^{\textsf{Im}}+\text{(volume-
suppressed)}$ (79)
$\displaystyle=-C\int_{t_{\textsf{int}}}^{T}\mathrm{d}t~{}\frac{V}{H}\left(1+\frac{1}{3}\left(\varepsilon_{H}+\eta_{H}\right)+\mathcal{O}(\varepsilon^{2})\right)-\frac{V^{\prime}}{\sqrt{2\varepsilon_{H}}H}\left(1+\frac{2\eta_{H}}{3}++\mathcal{O}(\varepsilon^{2})\right)$
(80)
$\displaystyle=-C\int_{t_{\textsf{int}}}^{T}\mathrm{d}t~{}H\,\mathcal{O}(\varepsilon^{2})\,,$
(81)

where we have used (34). Notice the cancellation that has occurred at
subleading order in slow roll. Without it, since
$C=\mathcal{O}(\varepsilon_{*}/V_{*})$, one might have expected a contribution
$\mathcal{O}(\varepsilon_{*}/V_{*})$ from this part of the evolution, since
$\Delta\tau\sim 1/\varepsilon_{*}$ in large field models. This calculation
shows the contribution is instead $\mathcal{O}(\varepsilon^{2}/V_{*})$, where
$``\varepsilon^{2}$” stands for second order in slow roll. So both the leading
and first subleading contribution in slow roll to $\log|\Psi|^{2}$ originate
from the approximately Euclidean (small field) part of the solution.

We have focussed here on potential subleading corrections to the real part of
$i\bar{S}$ from the large field regime, which turn out to be absent. For a
discussion of the leading behavior of the imaginary part of $i\bar{S}$ – the
phase of the wave function – from the large field regime we refer the reader
to [2, 9].

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