# Fractional Growth Portfolio Investment

A.E. Brockwell<EMAIL_ADDRESS>
###### Abstract

We review some fundamental concepts of investment from a mathematical
perspective, concentrating specifically on fractional-Kelly portfolios, which
allocate a fraction of wealth to a growth-optimal portfolio while the
remainder collects (or pays) interest at a risk-free rate. We elucidate a
coherent continuous-parameter time-series framework for analysis of these
portfolios, explaining relationships between Sharpe ratios, growth rates, and
leverage. We see how Kelly’s criterion prescribes the same leverage as
Markowitz mean-variance optimization. Furthermore, for fractional Kelly
portfolios, we state a simple distributional relationship between portfolio
Sharpe ratio, the fractional coefficient, and portfolio log-returns. These
results provide critical insight into realistic expectations of growth for
different classes of investors, from individuals to quantitative trading
operations.

We then illustrate application of the results by analyzing performance of
various bond and equity mixes for an investor. We also demonstrate how the
relationships can be exploited by a simple method-of-moments calculation to
estimate portfolio Sharpe ratios and levels of risk deployment, given a fund’s
reported returns.

Key words: stochastic differential equations, geometric Brownian motion, Kelly
portfolio, leverage, Sharpe ratio, fractional Kelly, growth-optimal portfolio

###### Contents

  1. 1 Introduction
  2. 2 Problem Formulation
    1. 2.1 Objective
    2. 2.2 Mathematical Analysis
      1. 2.2.1 Univariate Case
      2. 2.2.2 Multivariate Case
    3. 2.3 Sharpe Ratios and Fractional Kelly Investment
  3. 3 Examples and Applications
    1. 3.1 Equity and Bonds Mix
      1. 3.1.1 Parameter Estimation
      2. 3.1.2 Leverage Value Configurations
      3. 3.1.3 Simulation Results
    2. 3.2 Application to Fund Evaluation
  4. 4 Concluding Remarks
  5. 5 Acknowledgements
  6. A Supporting Results
    1. A.1 Constrained Maximization of Expected Log-Return per Unit Time

##  1 Introduction

This document describes quantitative relationships which are important in
investment, including the concept of leverage. In doing so, it provides a
range of insights that could be useful to home investors, quantitative
traders, or anyone who is managing liquid investments.

We are interested in managing a block of capital which can be invested into
one or more risky _instruments_ , or held in cash/debt at a “risk-free”
interest rate. An instrument is simply a particular type of asset that can be
bought and sold, and exhibits some growth (or decay) in value over time. It is
typical to aim for good long-term growth rate in total capital by investing in
a mix of different instruments. Below we study the question of determining the
“optimal” mix. By nature of the construction of the portfolio, we ensure
theoretically, that if gains/losses on capital are re-invested continuously in
time, then the pool of capital will retain non-negative value. In other words,
we cannot lose more than our initial block of capital. 111However, it is worth
noting that this is a theoretical result, since reinvestment of profits/losses
continuously in time is a practical impossibility in most cases.

Leverage is a critical tool in optimizing capital growth. Roughly speaking,
leverage is a mechanism by which we can amplify the returns of an investment.
While it might seem natural that we should amplify the returns of a “good”
investment as much as possible, we will see that this is not a good idea. To
give a concrete example, imagine that on two successive days we observe a 10%
loss followed by a 10% gain. This leaves us with a cumulative 1% loss
($0.9\times 1.1=0.99$). If we were to amplify the two returns by a factor of
two, however, a 20% loss followed by a 20% gain leaves us with a 4% loss
($0.8\times 1.2=0.96$). Thus, somewhat counter-intuitively, doubling the size
of our returns causes our cumulative loss to be be _worse_ than double the
original loss. This asymmetry suggests to us a potential connection between
volatility, expected growth, and optimal leverage.

In the remainder of this paper, we consider this problem from a continuous-
parameter time series perspective, studying the impact of leverage
analytically using stochastic calculus. It is beyond the scope of this
document to provide an introduction to stochastic calculus, but for a useful
starting point, see, e.g. Øksendal (2003), or Karatzas and Shreve (1991) for a
more comprehensive treatment. To the author’s knowledge, the results stated in
theorem form in this paper do not appear in existing financial literature,
although components of the framework itself, along with associated topics of
discussion, can be found in the literature. In particular, Breiman (1961);
Hakansson (1971); Thorpe (1975) give much detailed discussion of Kelly
portfolios and their advantages and disadvantages. Useful related information
can also be found in MacLean et al. (2012). As in Browne (1997), we will adopt
a continuous-time framework based on the use of stochastic differential
equations, to analyze a portfolio of instruments whose prices follow a
multivariate geometric Brownian motion. MacLean et al. (2004) (and references
therein) point out the importance of avoiding “bad” outcomes in such
portfolios, pointing out that fractional Kelly schemes (among other solutions)
are useful in this context. In this paper, we also address this problem,
obtaining a specific characterization of the trade-off between growth and
safety.

##  2 Problem Formulation

Suppose that, at time $t=0$, an investor has a block of capital $A_{0}$ to
invest, and assume that it can be allocated to cash, which earns interest
continuously at the _risk-free rate_ $r$ per unit time, or to one or more
different risky instruments which generate returns over time. Over time, we
will denote the total capital owned by the investor as

$\\{A_{t},~{}t\in{\mathbb{R}}\\}.$ (1)

This quantity should be thought of as the total value of all assets, including
investments in particular instruments, and cash/debt. We will assume the
existence of a set of $m$ different investment instruments (not including
cash), which have price time series

$\\{P_{t,j},~{}t\geq 0\\},~{}j=1,2,\ldots,m.$ (2)

In what follows, we will often refer to the vector of prices by

${\bf P}_{t}=(P_{t,1},\ldots,P_{t,m})^{T}.$ (3)

The collection of investments in the $m$ instruments as well as cash will be
referred to as the _portfolio_. As mentioned above, the value of the portfolio
at time $t$ is denoted by $A_{t}.$ Any cash component of the portfolio
receives continuously-compounded interest at the risk-free rate $r$. It will
be useful to define a vector version of the risk-free rate

${\bf r}=(r,r,\ldots,r)^{T}\in{\mathbb{R}}^{m}.$ (4)

We define the _leverage_ vector to be

${\bf k}=(k_{1},\ldots,k_{m})^{T}.$ (5)

and we define _total-leverage_

$\kappa=\sum_{j=1}^{m}k_{j}.$ (6)

We will say that our portfolio is _non-leveraged_ if $\kappa\leq 1.$

At any given time $t$, the investor invests fractions $k_{j}$ of his/her
capital $A_{t}$ into the instruments with price $P_{t,j},$ holding the
remainder in cash if that remainder is positive, or maintaining the required
debt otherwise. Cash is assumed to earn the interest-free rate $r$, while debt
pays interest at the same rate. By construction, the amount of cash held at
time $t$ is clearly

$A_{t}(1-\kappa).$ (7)

The leverage vector determines the manner in which returns on the $m$
investment instruments are related to returns in the total capital. To state
this property formally, let us define the infinitesimal return of an
investment instrument by

$dP_{t,j}/P_{t,j}.$ (8)

Then define the infinitesimal return of our total capital by

$dA_{t}/A_{t}.$ (9)

The infinitesimal returns are related by

$dA_{t}/A_{t}=(1-\kappa)rdt+\sum_{j=1}^{m}k_{j}dP_{t,j}/P_{t,j}$ (10)

Equation (10) can be viewed as a mathematical definition of leverage. The
first term on the right represents the return from risk-free rate interest
accrual/payment on the cash/debt portion of the portfolio, while the summation
represents return contributed by the leveraged investments.

The components of our leverage vector do not necessarily have to be less than
one, or add up to one, or even be non-negative.

For example, suppose that we have $m=2$ possible investments: an S&P500 mutual
fund and a long-term bond fund. (We study a case like this in more detail
later in the paper.) A leverage vector of $(0.2,0.2)$ would indicate that at
any given point in time, we keep one fifth of our total capital in the S&P500
fund, one fifth in the bond fund, and the remaining three fifths held in cash,
accruing interest payment at the risk-free rate. Alternately, a leverage
vector of $(0.2,1.8)$ would indicate that we would invest one fifth of initial
capital in the S&P500 fund, borrow an amount equal to the initial capital, and
invest the four fifths remaining initial capital, along with the borrowed sum,
in the bond fund.

###  2.1 Objective

There are many possible investment objectives. For example, one could attempt
to minimize the probability of ultimately losing all their capital, attempt to
maximize the expected return relative to some benchmark, maximize some utility
function of wealth, etc. In this paper we aim for good long-term growth
profiles for our capital $\\{A_{t}\\}$. To state this more precisely, we will
be particularly interested in the _expected log-return per unit time_

$L={{\bf E}\left[\log(A_{t+\delta}/A_{t})\right]}/\delta$ (11)

and _log-return variance per unit time_

$V=\mbox{\bf Var}(\log(A_{t+\delta}/A_{t}))/\delta.$ (12)

As we will see below, this approach leads us to a convenient stochastic
calculus-based derivation of Kelly’s formula, and yields additional useful
insights.

Direct maximization of $L$ (with no regard for $V$) yields the so-called
_growth-optimal portfolio_ , also referred to as the _Kelly portfolio_. In the
long-run, the growth-optimal portfolio almost-surely leads to more wealth than
any competing portfolio. 222More formally, if $\\{A_{t}\\}$ represents the
capital associated with the growth-optimal portfolio, and
$\\{A^{\prime}_{t}\\}$ represents the capital from another portfolio in the
same instruments, chosen in a different manner, then under a modest set of
regularity conditions, we can show that
$(A^{\prime}_{t}/A_{t})\stackrel{{\scriptstyle a.s.}}{{\rightarrow}}0$ as
$t\rightarrow\infty.$ The price to pay for this growth dominance however, is
relatively large up and down-swings in $\\{A_{t}\\}$. _Fractional Kelly
schemes_ , which simply apply a fractional multiplier to the growth-optimal
portfolio leverage vector, provide one way to address this problem.

###  2.2 Mathematical Analysis

We are now in a position to analyze the impact of leverage choice. For the
sake of exposition, we address the univariate case before developing the
multivariate case.

####  2.2.1 Univariate Case

In the univariate ($m=1$) case, we can write down a formal stochastic
differential equation to describe the price $\\{P_{t}\\}$ of an instrument
over time $t\in{\mathbb{R}},t\geq 0,$ as

$\displaystyle dP_{t}$ $\displaystyle=$ $\displaystyle\mu P_{t}dt+\sigma
P_{t}dW_{t},$ (13) $\displaystyle P_{0}$ $\displaystyle=$ $\displaystyle 1,$
(14)

where $\\{W_{t}\\}$ is a standard Brownian motion.

The solution to (13,14) is a geometric Brownian motion, for which

$d\log(P_{t})=(\mu-\sigma^{2}/2)dt+\sigma dW_{t}.$ (15)

(This follows directly from an application of Itô’s formula to (13,14).) The
process has several important properties, which we will simply state here
without proof.

  1. 1.

It provides a realistic description of many real-life instruments that can be
bought as investments. The parameters $\mu$ and $\sigma$ vary, however, over
different investments and arguably also over time for a particular investment.

  2. 2.

$P_{t}>0.$

  3. 3.

The log-return of the price satisfies

$\log(P_{t+\delta}/P_{t})\sim\mbox{N}(\mu\delta-\sigma^{2}\delta/2,\sigma^{2}\delta).$
(16)

Now let us consider the behavior of total capital $\\{A_{t},~{}t\geq 0\\}$
over time when we invest a fraction $k$ (leverage) in the instrument whose
price $P_{t}$ is given by (13,14). It follows directly from (10) that

$\displaystyle dA_{t}$ $\displaystyle=$
$\displaystyle(1-k)rA_{t}dt+kA_{t}dP_{t}/P_{t}$ (17) $\displaystyle=$
$\displaystyle[(1-k)r+k\mu]A_{t}dt+kA_{t}\sigma dW_{t}.$ (18)

In other words, like the underlying instrument price, our amount of capital
$A_{t}$ also follows a geometric Brownian motion process, but with different
parameters. Consequently,

$d\log(A_{t})=[(1-k)r+k\mu-k^{2}\sigma^{2}/2]dt+k\sigma dW_{t}.$ (19)

It is very important to note333It is a common mistake to presume, for example,
that a triple-leverage investment yields triple the log-returns over time.
This is not the case. that

$d\log(A_{t})\neq k\cdot d\log(P_{t}).$ (20)

It follows directly from (19) that

  1. 1.

$A_{t}>0.$ This is our guarantee that we do not lose more than our initial
capital. However, note that it relies on the unrealistic assumption that we
are able to carry out continuous reinvestment of profits/losses.

  2. 2.

The log of total amount of capital (including reinvested profits/losses)
satisfies

$\log(A_{t+\delta}/A_{t})\sim\mbox{N}(r\delta+k(\mu-r)\delta-k^{2}\sigma^{2}\delta/2,k^{2}\sigma^{2}\delta).$
(21)

At any time $t$, equation (21) provides a predictive distribution for our
amount of capital at a time point $(t+\delta)$ in the future. Specifically, it
is log-normal.

Following on from (21), we see that the _expected log-return per unit time_ of
our capital is

$L(k)={{\bf
E}\left[\log(A_{t+\delta}/A_{t})\right]}/\delta=r+k(\mu-r)-k^{2}\sigma^{2}/2.$
(22)

Differentiating this with respect to $k$ and setting to zero gives us a
formula for the value of leverage $k$ that maximizes expected log-return per
unit time. This is simply

$k^{*}=(\mu-r)/\sigma^{2},$ (23)

and the corresponding expected log-return per unit time is

$L(k^{*})=r+\frac{1}{2}(\mu-r)^{2}/\sigma^{2}.$ (24)

The expression (23) is well-known, and is typically referred to as _Kelly’s
formula_ , for Kelly’s analysis of a closely-related problem (see Kelly
(1956)).

####  2.2.2 Multivariate Case

It is straightforward to extend both (22) and (23) to the multivariate case
when we have multiple investments ($m>1$), and we allocate proportions ${\bf
k}=(k_{1},k_{2},\ldots,k_{m})^{T}$ of capital to the respective investments.
In this case, we define ${\bf\mu}=(\mu_{1},\ldots,\mu_{m})^{T},$
${\bf\sigma}=(\sigma_{1},\ldots,\sigma_{m})^{T},$ as well as a correlation
matrix $R\in{\mathbb{R}}^{m\times m}.$ The multivariate analog of (13) becomes

$d{\bf P}_{t}=\mbox{diag}({\bf\mu}){\bf
P}_{t}dt+\mbox{diag}({\bf\sigma})\mbox{diag}({\bf P}_{t})d{\bf U}_{t},$ (25)

where $\\{{\bf U}_{t}\\}$ is a multivariate Brownian motion with correlation
matrix $R$, so that

${{\bf E}\left[{\bf U}_{t}\right]}=0,\quad\mbox{\bf Var}({\bf U_{t}})=Rt,$
(26)

and $\mbox{diag}(\cdot)$ represents a square matrix with diagonal elements
given by the vector argument, and zeros in all off-diagonal positions. It will
be convenient to define

$\Sigma=[\mbox{diag}({\bf\sigma})]R[\mbox{diag}({\bf\sigma})].$ (27)

Also, recall the definition $\kappa=\sum k_{j}={\bf k}\cdot{\bf e}_{m},$ where
${\bf e}_{m}=(1,1,\ldots,1)^{T}\in{\mathbb{R}}^{m}.$

Applying (10), it is straightforward to show that there is another standard
Brownian motion $\\{W_{t}\\}$ such that the total capital $\\{A_{t}\\}$
satisfies

$\displaystyle dA_{t}$ $\displaystyle=$ $\displaystyle((1-{\bf k}\cdot{\bf
e}_{m})r+{\bf k}\cdot{\bf\mu})A_{t}dt+({\bf k}^{T}\Sigma{\bf
k})^{1/2}A_{t}dW_{t}$ (28) $\displaystyle=$ $\displaystyle(r-{\bf k}\cdot{\bf
r}+{\bf k}\cdot{\bf\mu})A_{t}dt+({\bf k}^{T}\Sigma{\bf k})^{1/2}A_{t}dW_{t},$
(29)

and it follows directly from application of Itô’s formula that

$d\log(A_{t})=(r+{\bf k}\cdot({\bf\mu}-{\bf r})-{\bf k}^{T}\Sigma{\bf
k}/2)dt+({\bf k}^{T}\Sigma{\bf k})^{1/2}dW_{t},$ (30)

so the expected log-return per unit time is given by

$L({\bf k})={{\bf E}\left[\log(A_{t+\delta}/A_{t})\right]}/\delta=r+{\bf
k}\cdot({\bf\mu}-{\bf r})-{\bf k}^{T}\Sigma{\bf k}/2,$ (31)

Consequently Kelly’s formula becomes

${\bf k}^{*}=\Sigma^{-1}({\bf\mu}-{\bf r}),$ (32)

with maximum expected log-return per unit time

$L({\bf k}^{*})=r+\frac{1}{2}({\bf\mu}-{\bf r})^{T}\Sigma^{-1}({\bf\mu}-{\bf
r}).$ (33)

We will also be interested in the variance of the log-return per unit time.
From (30), this is easily seen to be

$V({\bf k}^{*})=\mbox{\bf Var}(\log(A_{t+\delta}/A_{t}))/\delta={\bf
k}^{T}\Sigma{\bf k}.$ (34)

Readers may note that (32) is recognizable as the solution of a standard
Markowitz mean-variance portfolio weight selection problem (see Markowitz,
1952).

###  2.3 Sharpe Ratios and Fractional Kelly Investment

In the finance industry, in the univariate case, it is common practice to
refer to the quantity $(\mu-r)/\sigma$, as the _Sharpe ratio_ of the
investment. 444It is important to differentiate between this quantity, and
estimators thereof. Unfortunately, people often use the same term to refer to
both the quantity and its estimator(s). More generally, we will define the
_Sharpe ratio_ of a single investment or a set of investments as

$S=[({\bf\mu}-{\bf r})^{T}\Sigma^{-1}({\bf\mu}-{\bf r})]^{1/2},$ (35)

where ${\bf\mu}$ and $\Sigma$ are the parameters defining the growth and
volatility of the component investments, and ${\bf r}$ is the vector whose
elements are all equal to the risk-free rate. (Note that the one-dimensional
multivariate expression is consistent with the univariate definition.)

Sharpe ratios are clearly important because the maximum growth rates in (24,
33) increase quadratically with respect to the Sharpe ratio.

We have seen that Kelly’s formula prescribes the leverage that maximizes
expected log-return per unit time $L$ of an investment. However, to limit
up/down-swings in capital, it is desirable to operate with a lower total level
of risk. To achieve this, we can apply a multiplier to (32),

${\bf k}_{\alpha}=\alpha\Sigma^{-1}({\bf\mu}-{\bf r}),\quad\alpha\in[0,1].$
(36)

This quantity is often referred to as _fractional Kelly_ leverage. In this
light, fractional Kelly investment can be viewed as a particular means of
leverage selection, with $\alpha$ specifying the level of risk relative to
that required for the growth-optimal portfolio.

The following result makes a formal connection between leverage, log-returns
and Sharpe ratios.

######  Theorem 2.1 (Sharpe-Leverage Performance Profile)

Suppose that we apply fractional Kelly leverage (36) with multiplier
$\alpha\in[0,1]$ to the portfolio with prices governed by (25), earning/paying
the risk-free interest rate $r$ on the cash/debt component. Then the resulting
capital process $\\{A_{t}\\}$ is a geometric Brownian motion satisfying

$d\log(A_{t})=[r+(\alpha-\alpha^{2}/2)S^{2}]dt+\alpha SdW_{t},$ (37)

where $S=[({\bf\mu}-{\bf r})^{T}\Sigma^{-1}({\bf\mu}-{\bf r})]^{1/2}$ denotes
the Sharpe ratio of the portfolio. Consequently, $\\{A_{t}\\}$ has expected
log-return per unit time

$L({\bf k}_{\alpha})={{\bf
E}\left[\log(A_{t+\delta}/A_{t})\right]}/\delta=r+(\alpha-\alpha^{2}/2)S^{2}$
(38)

and log-return variance per unit time

$V({\bf k}_{\alpha})=\mbox{\bf
Var}(\log(A_{t+\delta}/A_{t}))/\delta=\alpha^{2}S^{2}.$ (39)

Proof: The derivation of (30) above establishes that $\\{A_{t}\\}$ is a
geometric Brownian motion. Substituting (36) into (30) and making use of the
definition (35) yields (37). The remaining results then follow directly.
$\Box$

When the conditions of Theorem 2.1 are met and $\alpha=1,$ we obtain the
growth-optimal portfolio. More generally, equations (38, 39) show the trade-
off we obtain with different values of $\alpha.$ Values closer to one yield
higher expected log-return per unit time on capital, but values closer to zero
give better (reduced) variance of the log-return around the mean.

##  3 Examples and Applications

We now consider some practical implications of the theory outlined above,
beginning with analysis that would be of interest to a typical individual with
some savings to invest.

###  3.1 Equity and Bonds Mix

Let us consider a portfolio consisting of only two instruments:
$\\{P_{t,1}\\}$ will represent the price of Vanguard 500 Mutual Fund Index
shares (VFIAX), and $\\{P_{t,2}\\}$ will represent the price of Vanguard Long-
Term Treasury Fund shares (VUSUX).

Figure 1: Equity (VFIAX) and Bond (VUSUX) performance from Feb. 12th, 2001 to
Aug. 24th, 2021. Data source: Yahoo Finance.

Figure 1 shows differences of logarithms of adjusted daily closing prices for
these two instruments from 2001 to 2021. This time period covers the dot-com
bubble of 2002 and the sub-prime crisis of 2008, as well as more recent market
turbulence at the onset of the coronavirus pandemic in early 2020.

In what follows, we will assume that the risk-free rate $r$ is equal to zero.
This is a fairly reasonable approximation for the post-sub-prime period from
2008 to 2021, although results here can be easily adapted if $r$ is assumed to
be non-zero.

####  3.1.1 Parameter Estimation

From the raw data shown in the plots in Figure 1, we can easily compute
estimators of ${\bf\mu}=(\mu_{1},\mu_{2})$,
${\bf\sigma}=(\sigma_{1},\sigma_{2})$ and $R$. Let the unit of time be one
year, assume there are 260 trading days each year, and define the daily log-
returns (shown in the left part of Figure 1) as

$D_{t,j}=\log(P_{(t+1)/260,j})-\log(P_{t/260,j}),\quad t=1,2,\ldots,n-1,$ (40)

where $n$ is the total number of trading days of data. We know from (16) that

$D_{t,j}\sim\mbox{N}((\mu_{j}-\sigma^{2}_{j}/2)/260,\sigma^{2}/260).$ (41)

Furthermore,

$\mbox{Corr}(D_{t,1},D_{t,2})=R_{12}.$ (42)

We can therefore take sample mean and variance from our two daily price
series, compute sample correlation, and use them to construct method-of-
moments estimators

$\displaystyle\hat{\sigma}^{2}_{j}$ $\displaystyle=$ $\displaystyle{260\over
n-2}\sum_{t=1}^{n-1}(D_{t,j}-\bar{D}_{j})^{2},\quad\bar{D}_{j}={1\over
n-1}\sum_{t=1}^{n-1}D_{t,j},$ (43) $\displaystyle\hat{\sigma}_{j}$
$\displaystyle=$ $\displaystyle\sqrt{\hat{\sigma}^{2}_{j}},$ (44)
$\displaystyle\hat{\mu}_{j}$ $\displaystyle=$ $\displaystyle
260\bar{D}_{j}+\hat{\sigma}^{2}_{j}/2,$ (45) $\displaystyle\hat{R}_{jk}$
$\displaystyle=$
$\displaystyle{260\over\hat{\sigma}_{j}\hat{\sigma}_{k}(n-2)}\sum_{t=1}^{n-1}(D_{t,j}-\bar{D}_{j})(D_{t,k}-\bar{D}_{k})$
(46)

With the raw data described above, we obtain

$\hat{{\bf\mu}}=(0.099,0.051)^{T},\quad\hat{{\bf\sigma}}=(0.199,0.123)^{T},\quad\hat{R}_{12}=-0.377.$
(47)

From a practical perspective, we also want to consider the different impact of
taxation on these two instruments. In the United States, tax on a mutual fund
that tracks the S&P500 is typically close to the long-term capital gain rate
of 20%, while tax on bond funds, although more complex to compute, is
typically closer to standard income tax, which is often around 40%. To
approximate the impact of taxation, we will therefore adjust the estimates
(47) above by multiplying components of $\mu$ by one minus the corresponding
tax rate, giving us final estimates

$\hat{{\bf\mu}}=(0.079,0.031),\quad\hat{\Sigma}=\left[\begin{array}[]{cc}\hphantom{-}0.0396&-0.0093\\\
-0.0093&\hphantom{-}0.0152\end{array}\right].$ (48)

From these estimates, in turn, we can determine the Kelly leverage for the
growth-optimal portfolio by applying (32), obtaining

${\bf k}^{*}=\hat{\Sigma}^{-1}\hat{{\bf\mu}}\simeq(2.89,3.78)^{T}.$ (49)

If ${\bf\mu}$ and $\Sigma$ were indeed the same as these estimates, and if we
could borrow money to provide leverage, this says we could maximize growth
over time of capital by borrowing $5.67$ times our initial capital (assuming a
no-interest loan) to obtain total leverage of $2.89+3.78=6.67$. We would then
invest the original capital along with the borrowed capital into the two
components of the portfolio, rebalancing to hold the fraction of capital in
the two components constant. The resulting expected annual log-return on
initial (non-borrowed) capital would be

${\bf k}^{*}\cdot\hat{{\bf\mu}}-{{\bf k}^{*}}^{T}\hat{\Sigma}{\bf
k}^{*}/2\simeq 0.172$ (50)

In this argument we have pretended that there is no estimation error. In fact,
that source of noise can have a significant impact, but that analysis is
beyond the scope of this paper.

####  3.1.2 Leverage Value Configurations

We now consider a range of different investment possibilities. Some of these
will be applied to only one of our two instruments, and for these cases we
will use $m=1$ along with the corresponding (marginal) components of
$\hat{{\bf\mu}}$, $\hat{\Sigma}$ and the corresponding estimate $\hat{S}$ of
Sharpe ratio. In the more interesting cases we will examine portfolios
consisting of both instruments. We consider the following cases.

| Leverage | Drift/Diffusion | Kelly-Fraction
---|---|---|---
Name | $k$ | $\hat{\mu}$ | $\hat{\Sigma}$ | $\hat{S}$ | $\hat{\alpha}$
Single-instrument ($m=1$) portfolios
Non-Leveraged Equity | 1.00 | 0.079 | $0.199^{2}$ | 0.398 | 0.502
Double-Equity | 2.00 | 0.079 | $0.199^{2}$ | 0.398 | 1.003
Triple-Equity | 3.00 | 0.079 | $0.199^{2}$ | 0.398 | 1.505
Non-Leveraged Bonds | 1.00 | 0.031 | $0.123^{2}$ | 0.252 | 0.488
Double-Bonds | 2.00 | 0.031 | $0.123^{2}$ | 0.252 | 0.976
Two-instrument ($m=2$) portfolios
Fractional Kelly(0.30) | (0.87,1.13) | (0.079,0.031) |  $\hat{\Sigma}$ (eqn 48) | 0.588 | 0.30
Constr. Kelly(2.0) | (1.33,0.67) | (0.079,0.031) |  $\hat{\Sigma}$ (eqn 48) | 0.588 | NA
Full-Kelly | (2.89,3.78) | (0.079,0.031) |  $\hat{\Sigma}$ (eqn 48) | 0.588 | 1.0

Table 1: Example portfolios.

Cases are constructed as follows.

  * •

_Non-leveraged/Double/Triple Equities/Bonds:_ The non-leveraged approach
invests all capital directly into the equities/bonds instrument. Double and
triple versions invest all capital in the same instrument but with leverage
$2$ or $3$, respectively.

  * •

_Fractional Kelly (0.30):_ This applies leverage $\alpha=0.30$ times the full-
Kelly leverage. In this example, it results in total leverage $\kappa$
approximately equal to $2.0.$

  * •

_Constrained Kelly (2.0):_ Here we choose ${\bf k}$ so as to maximize expected
growth $L$ subject to the constraint that $\kappa=2.$ This can be done using
Lagrange multipliers, as described in Appendix A.1.

  * •

_Full-Kelly:_ This choice, given by (49) leads to maximum growth rate over
time, at the cost of high variance and draw-downs.

Table 1 also shows a term

$\hat{\alpha}={\bf k}/(\hat{\Sigma}^{-1}\hat{{\bf\mu}}),$ (51)

which is only defined when the leverage vector ${\bf k}$ is a multiple of
$\hat{\Sigma}^{-1}\hat{{\bf\mu}}$, and in that case represents an estimate of
the ratio of portfolio leverage to growth-optimal leverage. Values less than
$1.0$ are desirable in this context, since it is obvious from (38) and (39)
that $\alpha>1$ implies we are taking unnecessary extra risk for a return that
can also be achieved with $\alpha<1.$

####  3.1.3 Simulation Results

It is straightforward to simulate hypothetical capital invested in the
portfolios corresponding to the test cases. Using data ranging from Feb 13th,
2001 to March 18th, 2021, we simply compute the daily portfolio returns using
the recorded data. In order to approximate the tax-effect on the two
components, we also apply a negative drift by subtracting the tax adjustment
used on ${\bf\mu}$ in (48) on a daily basis from the log-returns of the data.

Figure 2: Simulated post-tax capital of hypothetical portfolios built of VFIAX
and VUSUX, assuming starting capital of $100000, for a range of different
leverage vectors.

Figure 2 shows cumulative capital $\\{A_{t}\\}$ over time for a range of
different leverage settings. The full-Kelly leverage case is omitted from the
plot since it distorts the scale, but the full-Kelly case is included in
Figure 3, which shows the log of cumulative capital.

Figure 3: Simulated post-tax log-return of hypothetical portfolios built of
VFIAX and VUSUX, for a range of different leverage vectors. The full-Kelly
case clearly dominates the others in terms of growth, but it also loses 87.7%
of its capital value from Dec. 5th, 2007 to Mar. 6th, 2009 during the sub-
prime crisis. The simple triple-leverage equity investment also stands out
with its low return and 96.3% loss from Feb 14, 2001 to Mar 6, 2009.

| Growth |  | Draw-down |
---|---|---|---|---
Name | $\hat{L}$ | ${\hat{V}}^{1/2}$ | Max. | Start | End | Final Value
Single-instrument ($m=1$) portfolios
Non-Leveraged Equity | 0.060 | 0.200 | 56.4% | Oct 8 2007 | Mar 6 2009 | $328,143
Double-Equity | 0.081 | 0.400 | 84.6% | Oct 8 2007 | Mar 6 2009 | $486,418
Triple-Equity | 0.060 | 0.603 | 96.4% | Feb 14 2001 | Mar 6 2009 | $319,574
Non-Leveraged Bonds | 0.023 | 0.123 | 28.3% | Mar 6 2020 | Mar 30 2021 | $158,109
Double-Bonds | 0.031 | 0.247 | 50.7% | Mar 6 2020 | Mar 30 2021 | $184,841
Two-instrument ($m=2$) portfolios
Fractional Kelly(0.30) | 0.089 | 0.176 | 41.9% | Dec 5 2007 | Mar 6 2009 | $576,464
Constr. Kelly(2.0) | 0.097 | 0.240 | 62.7% | Oct 8 2007 | Mar 6 2009 | $675,775
Full-Kelly | 0.172 | 0.594 | 89.8% | Dec 5 2007 | Mar 6 2009 | $3,002,829

Table 2: Summary of simulated performance of example portfolios derived from
the VFIAX and VUSUX instruments, simulated from Feb. 12th, 2001 to Aug. 24th,
2021, with initial capital $\$100,000.$ Risk-free interest rate is assumed to
be zero. $\hat{L}$ denotes the sample average of the annualized log-return,
${\hat{V}}^{1/2}$ denotes the annualized sample standard deviation of those
log-returns, and the maximum _draw-down_ is defined as the maximum drop from
peak to trough with trough occurring after the peak.

Table 2 shows corresponding summary statistics, including the sample
(annualized) growth rate $\hat{L}$, the corresponding sample variance
$\hat{V}$, as well as the final value and maximum draw-down over the time
period, which is defined as the maximum peak-to-trough drop such that the
trough occurs after the peak. From a psychological perspective, investors pay
significant attention to difference between the historical maximum and the
current value of an investment. Large values of this “draw-down” typically
induce great anxiety. Several important observations can be made from the
results. First, the full-Kelly case and the triple-equity case both exhibit
very large maximum draw-downs over this time period - 89.8% and 96.4%,
respectively. Few investors would hold onto an investment after losses that
large. Both the fractional Kelly and the constrained Kelly options appear to
give a fairly good compromise between greater growth and higher variance/draw-
downs. Of course different investors may different preferences, but it is
notable that of the choices in Table 2, apart from the non-leveraged bond
case, the fractional Kelly option has the smallest value of $\hat{V}$ and
still yields a respectable growth rate $\hat{L}.$

###  3.2 Application to Fund Evaluation

Hedge fund managers generally release very little information about the nature
of their trading operations. However, they often release yearly or quarterly
statments of actual returns on their hedge funds. We have already seen that
Theorem 2.1 assists in choosing a desirable leverage vector. Here we see how
it leads to a simple method-of-moments approach to estimate portfolio Sharpe
ratio and risk deployment of a fund, using only such publicly available return
data. The approach relies on an implicit assumption that the fund is applying
a fractional Kelly approach, or something similar. In light of the previous
observation at the end of Subsection 2.2.2 that fractional Kelly leverage is
equivalent to industry-standard Markowitz mean-variance optimization, this is
not an unrealistic assumption. This “reverse-engineering” allows us to
evaluate funds in a more nuanced way than simple inspection of past returns.
For example, we can easily differentiate between funds that obtain high
returns by taking excessive risk, and those that obtain high returns by
deployment of high-Sharpe ratio portfolios.

Consider, for example, the annual returns of the well-known (and very strongly
performing) Renaissance Medallion fund. These returns are publicly available
in Appendix 1 of Zuckerman (2019), and can easily be translated into log-
returns and viewed in light of equations (38) and (39). Without re-printing
the full table itself, we give summary statistics of their resulting (before-
fee) log-returns in Table 3 below.

Date Range | 1988-2018 inclusive
---|---
# Data Points | 31
Average Log-Return | 0.490
Std. Dev. (Log-Return) | 0.187

Table 3: Summary statistics of Renaissance Medallion Fund returns as given in
Zuckerman (2019).

Re-arranging the pair of equations (38,39), we see that for a fractional Kelly
investor,

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle 2V(2L+V)^{-1}$ (52)
$\displaystyle S^{2}$ $\displaystyle=$ $\displaystyle\alpha^{-1}(L+V/2).$ (53)

By simply matching $L=0.490$ and $V=0.187^{2}$ from the table above, we see
that the Medallion returns are consistent with deployment of a portfolio with
Sharpe ratio $S\simeq 2.72$ at approximately $\alpha=0.068$ times the Kelly-
optimal leverage. The (estimated) value of $0.068$ places them comfortably
below the obvious danger point. A value equal to $1.0$ would indicate
deployment of risk at the growth-optimal level, with its extreme volatility.
Such a value is not palatable to a typical investor. Any value larger than
$1.0$ would indicate a sub-optimal deployment of risk. A value larger than
$2.0$ would indicate that the fund will likely eventually collapse in the
sense that its value will converge in probability to zero.

##  4 Concluding Remarks

Given a portfolio of instruments with risk, some capital $A_{0},$ and a risk-
free interest rate $r$, we have seen that the Kelly portfolio maximizes long-
term growth of capital, but with distressingly large draw-downs along the way.
This undesirable quality can be mitigated by the use of fractional Kelly
portfolios, at the cost of a reduction in long-term growth rate. In Theorem
2.1, we have directly quantified this distributional relationship, and given
explicit expressions that help us to find a desirable balance between growth
and variance of returns. We have also seen how, under a set of reasonable
assumptions, fractional Kelly investment is the same as Markowitz mean-
variance optimization.

The stochastic differential equation methodology used in this paper has been
used by others in the same context, notably Browne (1997), but appears not to
be widely appreciated. Within this framework, one could potentially generalize
the geometric Brownian motion price models (13, 25) to jump-diffusion models,
thereby allowing for the more realistic case of skewed and heavy-tailed log-
returns. Using multivariate stochastic calculus, it would also possible to
carry out analysis of situations where drift and/or diffusion coefficients
${\bf\mu}$ and $\Sigma$ are not known and must be estimated.

On the practical side, the relationships given in Theorem 2.1 have broad
applicability. They apply equally well to simple investment of an individual’s
personal funds as to the more sophisticated operations of a quantitative
trading organization or other fund. For example, individual investors can use
the implied return distributions to assess the quality of leveraged mutual
funds under various assumptions. At the more sophisticated end of the
spectrum, a quantitative trading operation with a porfolio of known Sharpe
ratio could use the result to determine how much capital/leverage to deploy,
while remaining within distributional constraints imposed by their investors.
More generally, the stochastic differential equation framework provides a
solid foundation with which to address important yet-unsolved practical
investment problems.

##  5 Acknowledgements

The author is grateful to Peter Brockwell, Jonathan Baxter, and Peter Dodds
for their valuable comments and suggestions. This work has benefited
substantially from their feedback.

##  Appendix A Supporting Results

###  A.1 Constrained Maximization of Expected Log-Return per Unit Time

Suppose we wish to choose the leverage vector $k\in{\mathbb{R}}^{m}$ to
maximize expected log-return per unit time (restating equation (31))

$L({\bf k})={\bf k}\cdot{\bf\mu}-\frac{1}{2}{\bf k}^{T}\Sigma{\bf k},$ (54)

subject to the constraint that total-leverage is

$\kappa=\sum_{j=1}^{m}k_{j}={\bf k}\cdot{\bf e}_{m}=\kappa_{0},$ (55)

where ${\bf e}_{m}=(1,\ldots,1)^{T}\in{\mathbb{R}}^{m}.$ We can express the
constraint as

$({\bf k}\cdot{\bf e}_{m}-\kappa_{0})=0.$ (56)

Thus the Lagrangian is

${\cal L}({\bf k})={\bf k}\cdot{\bf\mu}-\frac{1}{2}{\bf k}^{T}\Sigma{\bf
k}-\lambda({\bf k}\cdot{\bf e}_{m}-\kappa_{0}),$ (57)

Differentiating with respect to the vector ${\bf k}$, and setting to zero, we
find that the value maximizing (54) takes the form

${\bf k}=\Sigma^{-1}({\bf\mu}-\lambda{\bf e}_{m}).$ (58)

Substituting (58) into the constraint (56), we can solve for $\lambda$,
obtaining

$\lambda=({\bf e}_{m}^{T}\Sigma^{-1}{\bf\mu}-\kappa_{0})({\bf
e}_{m}^{T}\Sigma^{-1}{\bf e}_{m})^{-1}.$ (59)

Equations (58) and (59) together specify the leverage vector maximizing $L$
subject to the required constraint.

## References

  * Breiman (1961) Leo Breiman.  Optimal gambling systems for favorable games.  _Fourth Berkeley Symposium on Probability and Statistics_ , 1:65–78, 1961.
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  * MacLean et al. (2012) L.C. MacLean, E.O. Thorp, and W.T. Ziemba, editors.  _The Kelly Capital Growth Investment Criterion_ , volume 3 of _World Scientific Handbook in Financial Economic Series_.  World Scientific, 2012.
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  * Øksendal (2003) B. Øksendal.  _Stochastic Differential Equations_.  Springer, 6th edition, 2003.
  * Thorpe (1975) E.O. Thorpe.  Portfolio choice and the Kelly criterion.  In W.T. Ziemba and T.G. Vickson, editors, _Stochastic Optimization Models in Finance_ , pages 599–619. Academic Press, 1975.
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aainstitutetext: Department of Physics and IPAP, Yonsei University, Seoul
03722, Republic of Koreabbinstitutetext: Korea Institute for Advanced Study,
Seoul 02455, Republic of Korea

# Reheating in Models with Non-minimal Coupling
in metric and Palatini formalisms

Dhong Yeon Cheong a Sung Mook Lee a, b and Seong Chan Park
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study reheating of inflationary models with general non-minimal coupling
$K(\phi)R$ with $K(\phi)\sim\sqrt{V(\phi)}$ where $R$ is the Ricci scalar and
$V$ is the inflaton potential. In particular, when we take the monomial
potential $K(\phi)\propto\phi^{m}$ with $m\in\mathbb{Z}_{+}$, we provide
general analytic expressions for cosmological observables. We consider a wide
range of non-minimal coupling $\xi\in[0,\infty)$ in metric and Palatini
formalisms and derive the predictions for cosmological observables and the
reheating temperature taking a general equation of state parameter
$w_{\text{reh}}$.

##  1 Introduction

Cosmological inflation Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi , now
widely accepted as a ‘standard procedure’ in early universe, has been heavily
tested with increasing precision WMAP:2010qai ; Planck:2018jri ;
BICEPKeck:2021gln . However, the particle physics details governing reheating
after inflation, which ultimately provides the initial conditions for the hot
big bang, is largely unknown. Reheating involves the physics in the
perturbative and non-perturbative decay, resonances and nonlinear dynamics of
inflaton Dolgov:1989us ; Traschen:1990sw ; Kofman:1994rk ; Shtanov:1994ce ;
Kofman:1997yn .

Phenomenologically, the reheating stage is effectively parameterized by the
phenomenological parameters $(T_{\text{reh}},N_{\text{reh}},w_{\text{reh}})$
where $T_{\rm reh}$ is the reheating temperature, $N_{\text{reh}}$ is the
e-folding number, and $w_{\text{reh}}$ is the (constant) effective equation of
state during reheating, respectively. In general, direct cosmological
observables are hardly traceable up to the period of reheating, but some
indirect bounds are available especially in the single field inflation models:
taking the CMB data into account, consistency relations among the parameters
$(T_{\text{reh}},N_{\text{reh}},w_{\text{reh}})$ can be derived Cook:2015vqa ;
Drewes:2015coa . Numerous studies have been conducted along this line
Cai:2015soa ; Rehagen:2015zma ; Ellis:2015pla ; deFreitas:2015xxa ;
Ueno:2016dim ; Dalianis:2016wpu ; Drewes:2017fmn ; Krajewski:2018moi ;
Rashidi:2018ois ; DiMarco:2018bnw ; Sakhi:2019vcx ; Gialamas:2019nly ;
Asadi:2019iod ; German:2020iwg ; Das:2020kff ; Haro:2020qos ; Kanfon:2020yxx .

Among the various models compatible with observations that one may consider,
models with a non-minimal coupling between the Ricci scalar $R$ and the
inflaton field $\phi$ are highly motivated Futamase:1987ua ; Fakir:1990eg ;
Komatsu:1999mt ; Park:2008hz ; Hertzberg:2010dc ; Kim:2013ehu ; Park:2018kst .
It is known that a generic class of cases exist where the non-minimal coupling
($K(\phi)R$) induces the asymptotic flatness
($\frac{V(\phi)}{K(\phi)^{2}}\sim\text{const.}$), hence guarantees approximate
shift symmetry of the inflationary potential and suppresses the tensor-to-
scalar ratio $r$ Park:2008hz ; Park:2008tj . This class is also known as the
$\alpha$-attractor Kallosh:2013yoa . and it is being constrained by precision
CMB measurements (i.e. Planck, WMAP, COBE and BICEP/Keck), and the detection
of $r$ is one of the main targets of several future experiments including
CMB-S4 and LiteBIRD Abazajian:2019eic ; LiteBIRD:2020khw . One notable example
in the inflationary model class with a non-minimal coupling is the Higgs
inflation, where the Standard Model (SM) Higgs $h$ has the role of the
inflaton with a non-minimal coupling $K(h)=\xi h^{2}$ and a quartic potential
$V(h)=\frac{\lambda}{4}h^{4}$ Bezrukov:2007ep , especially in the vicinity of
a critical point Hamada:2014wna ; Hamada:2014iga . (See Ref. Cheong:2021vdb
for a recent review.)

Although various theoretical and phenomenological issues of non-minimally
coupled inflation models have been explored Futamase:1987ua ; Fakir:1990eg ;
Makino:1991sg ; Komatsu:1999mt ; Park:2008hz ; Hertzberg:2010dc ; Linde:2011nh
; Jinno:2019und ; Lee:2020yaj , in this work, we further refine the
cosmological predictions of these models implementing effects of reheating.
Reheating is often implicitly assumed to be nearly instantaneous, and
conventional wisdom for $\alpha$-attractor class of models (including
$R^{2}$-driven Starobinsky model) says the cosmological observables are
degenerate among different realizations, but its specific reheating dynamics
may depend on microscopic details and alter the observational predictions
Chung:1998rq ; Giudice:2000ex ; Davidson:2000er ; Harigaya:2013vwa .
Considering various theoretical possibilities, we want to cover directions
worth pursuing as follows:

  * •

Metric and Palatini formalism: In the Palatini formalism of gravity, the
affine connection $\Gamma^{\rho}_{\mu\nu}$ and the metric $g_{\mu\nu}$ are
introduced in an independent manner. Although the Palatini formalism is
equivalent to the metric formalism at the level of pure Einstein-Hilbert
action, it is known that they provide different predictions when non-minimal
couplings are present Bauer:2008zj ; Tamanini:2010uq ; Borowiec:2011wd ;
Tenkanen:2017jih ; Racioppi:2017spw ; Jarv:2017azx ; Racioppi:2018zoy ;
Jinno:2019und ; Tenkanen:2020dge ; Jarv:2020qqm .

  * •

General monomial potentials: A generic monomial potential $V(\phi)\propto
K(\phi)^{2}\propto\phi^{2m}$ guarantees the asymptotic flat potential in the
Einstein frame Park:2008hz :

$\displaystyle\lim_{\phi\rightarrow\infty}\left(\frac{V}{K^{2}}\right)=\text{Const.}>0.$
(1)

The condition essentially describes the $\alpha$-attractor behavior
Kallosh:2013yoa . This criterion is applied in the metric formalism as well as
Palatini cases.

  * •

Wide range of non-minimal coupling $\xi\in[0,\infty)$: Even though large non-
minimal couplings are usually considered, a wide range of $\xi$ (including
ones much smaller than unity) consistent to observations still remains valid.
We consider general cases including $\xi=0$ (minimal, monomial) and
$\xi=\infty$ (maximally non-minimal) Cook:2015vqa .

This paper is organized as follows. In Section. 2, we set our model in both
metric and Palatini formalism of gravity and provide the inflationary
predictions. In Section. 3, we relate the cosmological observables $(n_{s},r)$
and reheating parameters. We also discuss current/future bound on reheating
temperatures from these observables. We conclude in Section. 4.

##  2 Model

In this section, we set the inflationary model with non-minimal coupling term
in metric and Palatini formalism.

###  2.1 Non-minimal Coupling in metric and Palatini formalisms

As the Einstein-Hilbert action already includes the $(Mass)^{2}$ dimensional
coupling to the Ricci scalar $R$, we are enforced to include the operators of
the form $\hat{\cal O}R$ with $[\hat{\cal O}]=(Mass)^{2}$ following the
effective field theory approach unless a symmetry forbids it. Therefore when
we introduce a scalar field $\phi$ to our theory, we include an arbitrary
(gauge invariant) function $K(\phi)$ directly coupling to gravity:

$\displaystyle S_{J}=\int
d^{4}x\sqrt{-g_{J}}\left[-\frac{M^{2}+K(\phi)}{2}R+\frac{1}{2}(\partial\phi)^{2}-V(\phi)\right],$
(2)

with $[K]=(Mass)^{2}$ and an arbitrary mass parameter $M$. The potential
$V(\phi)$ is a function of $[V]=(Mass)^{4}$. In metric formalism, the Ricci
scalar $R$ is solely determined by the metric $g_{\mu\nu}$, but in the
Palatini formalism it is also determined by the connection
$\Gamma^{\mu}_{\nu\lambda}$ which is taken to be independent of the metric. In
the present of non-minimal coupling $K(\phi)R$, the two formalisms are not
equivalent and predicts different Universe Bauer:2008zj .

By performing the Weyl transformation,

$\displaystyle g_{E,\mu\nu}=\Omega^{2}g_{J,\mu\nu},$
$\displaystyle\Omega^{2}\equiv\frac{M^{2}+K(\phi)}{M_{P}^{2}}$ (3)

the action is conveniently transformed into the canonical form of gravity in
Einstein frame

$\displaystyle S=\int
d^{4}x\sqrt{-g_{E}}\left[-\frac{M_{P}^{2}}{2}R_{E}+\frac{1}{2}\Pi(\phi)(\partial\phi)^{2}-\frac{V(\phi)}{\Omega^{4}}\right],$
(4)

where the non-trivial kinetic term is given by

$\displaystyle\Pi(\phi)\equiv\frac{1}{\Omega^{2}}+\frac{3\zeta}{2M_{P}^{2}}\frac{K^{\prime}(\phi)^{2}}{\Omega^{4}},$
$\displaystyle\zeta=\begin{dcases}1&\text{(Metric) }\\\ 0&\text{(Palatini)
}\end{dcases}.$ (5)

in metric and Palatini formalism, respectively. The second term ($\propto
K^{\prime 2}$) originates from the transformation of the Ricci scalar
$R(\Gamma)$, making it absent in the Palatini formalism. The kinetic term is
easily canonicalized by $\sqrt{\Pi(\phi)}\partial h=\partial\phi$ or

$\displaystyle\frac{dh}{d\phi}=\sqrt{\Pi(\phi)}.$ (6)

When the explicit function $K(\phi)$ is given, we obtain the canonical scalar
$h(\phi)$ by integration.

An asymptotically flat, positive potential in the Einstein frame
$V_{E}(h(\phi))$ should satisfy a general condition at large fields
Park:2008hz :

$\displaystyle\lim_{\phi\to\infty}\frac{V(\phi)}{K(\phi)^{2}}=\text{Const.}>0.$
(7)

We request this condition $V\sim K^{2}$ or equivalently $K\sim\sqrt{V}$ to
realize successful slow-roll inflation. We emphasize again that the condition
essentially describes the $\alpha$-attractor behavior Kallosh:2013yoa .

###  2.2 Monomial functions: $K(\phi)\sim\phi^{m}\sim\sqrt{V(\phi)}$

To explicitly see the inflationary behavior, we choose $K$ and $V$ to be
monomial functions with $m\geq 1$ 111For models with a global $U(1)$ symmetry,
a $K(\phi)$ function with an odd power of $\phi$ is usually forbidden.

$\displaystyle K(\phi)=\xi M_{P}^{2}\left(\frac{\phi}{M_{P}}\right)^{m},$
$\displaystyle V=\frac{\lambda
M_{P}^{4}}{2m}\left(\frac{\phi}{M_{P}}\right)^{2m},$ (8)

where $\xi$ and $\lambda$ are arbitrary dimensionless parameters. In addition,
without a (or with a sufficiently small) vacuum expectation value of the field
$\phi$, we have $M\simeq M_{P}$ to guarantee the canonicalized Einstein frame
field (later we define as $h$) and potential $U(\phi(h))$ coincide with the
Jordan frame field $\phi$ and potential $V(\phi)$ near the origin. Therefore,
we set $M=M_{P}$ without losing precision in our predictions. Then the
potential in Einstein frame becomes

$\displaystyle
U(\phi)=\frac{M_{P}^{4}}{\left(M_{P}^{2}+K\right)^{2}}V=\frac{\lambda
M_{P}^{4}\left(\frac{\phi}{M_{P}}\right)^{2m}}{2m\left(1+\xi\left(\frac{\phi}{M_{P}}\right)^{m}\right)^{2}}.$
(9)

The slow-roll parameters are defined in the Einstein frame:

$\displaystyle\epsilon(\phi)$
$\displaystyle\equiv\frac{M_{P}^{2}}{2}\left(\frac{\partial
U(\phi(h))/\partial
h}{U}\right)^{2}=\frac{M_{P}^{2}}{2\Pi_{\zeta}(\phi)}\left(\frac{\partial
U/\partial\phi}{U}\right)^{2},$ (10) $\displaystyle\eta(\phi)$
$\displaystyle\equiv M_{P}^{2}\frac{\partial^{2}U/\partial
h^{2}}{U}=\frac{M_{P}^{2}}{\sqrt{\Pi_{\zeta}(\phi)}U}\frac{\partial}{\partial\phi}\left(\frac{1}{\sqrt{\Pi_{\zeta}(\phi)}}\frac{\partial
U}{\partial\phi}\right),$ (11)

where $\Pi_{\zeta}$ is defined in Eq. (5) for metric ($\zeta=1$) and Palatini
($\zeta=0$) cases with $K(\phi)$ in Eq. (8) and $x=\phi/M_{P}$:

$\displaystyle\Pi_{\zeta}(\phi=M_{P}x)=\frac{1+\xi
x^{m}+\zeta\frac{3m^{2}\xi^{2}}{2}x^{2(m-1)}}{(1+\xi x^{m})^{2}},$
$\displaystyle\zeta=\begin{dcases}1&\text{(Metric) }\\\ 0&\text{(Palatini)
}\end{dcases}.$ (12)

The explicit form of the slow-roll parameters are obtained

$\displaystyle\epsilon(\phi=M_{P}x)$
$\displaystyle=\frac{4m^{2}}{2x^{2}\left(\xi x^{m}+1\right)+3\zeta
m^{2}\xi^{2}x^{2m}},$ (13) $\displaystyle\eta(\phi=M_{P}x)$
$\displaystyle=\frac{4m\left[x^{2}\left(\xi x^{m}+1\right)\left\\{(m+2)\xi
x^{m}-4m+2\right\\}-3\zeta m^{3}\xi^{2}x^{2m}\left(\xi
x^{m}-1\right)\right]}{\left[2x^{2}\left(\xi x^{m}+1\right)+3\zeta
m^{2}\xi^{2}x^{2m}\right]^{2}}.$ (14)

It is noticed that the slow-roll parameters are independent of $\lambda$.

The cosmological time (in Einstein frame) $t_{k}$ when the mode corresponding
to the pivot scale $k$ leaves the horizon is determined by
$k=a(t_{k})H(t_{k})$ with the scale factor $a$ and $H$ being the Hubble
parameter. In this work, the pivot scale is chosen to be
$k=0.05\text{Mpc}^{-1}$. On the other hand, the time $t_{e}$ at the end of the
inflation is set by $\epsilon(\phi(t_{e}))=1$.

Then, the number of e-foldings during expansion from a pivot scale with
$a_{k}=a(t_{k})$ to the end of inflation at $a_{e}=a(t_{e})$ is given by

$\displaystyle N(\phi_{k})\equiv N_{k}$
$\displaystyle=\log\frac{a_{e}}{a_{k}}=\frac{1}{M_{P}^{2}}\int_{h(\phi_{e})}^{h(\phi_{k})}dh\frac{U}{\partial
U/\partial h}$
$\displaystyle=\int_{\phi_{e}}^{\phi_{k}}\frac{d\phi}{M_{P}}\sqrt{\frac{\Pi_{\zeta}(\phi)}{2\epsilon(\phi)}}=\left[\frac{x^{2}}{4m}+\frac{3}{4}\zeta\left(\xi
x^{m}-\ln(M_{P}^{m}+\zeta
M_{P}^{m}x^{m})\right)\right]_{\phi_{e}/M_{P}}^{\phi_{k}/M_{P}}$ (15)

with $\phi_{k}\equiv\phi(t_{k})$, $\phi_{e}\equiv\phi(t_{e})$, and
$H_{k}\equiv H(t_{k})$ where we used $dh=d\phi\sqrt{\Pi_{\zeta}}$ in the 2nd
line.

###  2.3 $\xi\ll 1$ (Metric $\approx$ Palatini)

We first analyze the $\xi\ll 1$ limit close to the minimal case. The exact
slow-roll parameters Eq. (13) and Eq. (14) in this limit then take the
approximate expression

$\displaystyle\epsilon(\phi=M_{P}x)$
$\displaystyle\simeq\frac{2m^{2}}{x^{2}}-2m^{2}\xi
x^{m-2}+\left(2x^{2}-3m^{2}\zeta\right)m^{2}\xi^{2}x^{2m-4}+\mathcal{O}(\xi^{3}),$
(16) $\displaystyle\eta(\phi=M_{P}x)$
$\displaystyle\simeq\frac{2m(2m-1)}{x^{2}}-5m^{2}\xi
x^{m-2}+\left[5x^{2}+3(2-3m)m\zeta\right]m^{2}\xi^{2}x^{2m-4}+\mathcal{O}(\xi^{3})$
(17)

where $\zeta=0,1$ each correspond to the Palatini and metric cases,
respectively. Note that the $\zeta$ parameter dependence appears in the
$\mathcal{O}(\xi^{2})$ order, i.e. the predictions of both cases will deviate
in $\mathcal{O}(\xi^{2})$, and the observables approximately coincide for
$\xi\ll 1$.

The field value where inflation ends for both cases is determined by
$\epsilon=1$, hence

$\displaystyle\phi_{e}\simeq\sqrt{2}mM_{P}-2^{\frac{m-1}{2}}m^{1+m}M_{P}\xi+2^{m-\frac{5}{2}}m^{2m+1}(2m+3-3\zeta)M_{P}\xi^{2}+\mathcal{O}(\xi^{3}).$
(18)

Consequently, the e-folding number $N_{k}$ from the pivot scale $a_{k}$ to the
end of inflation $a_{e}$ then becomes, following Eq. (2.2)

$\displaystyle
N_{k}\simeq\frac{\phi_{k}^{2}-\phi_{e}^{2}}{4mM_{P}^{2}}+\frac{3}{8}\zeta\left[\frac{\phi_{k}^{2m}-\phi_{e}^{2m}}{M_{P}^{2m}}\right]\xi^{2}.$
(19)

As $\xi\ll 1$, the field value $\phi_{k}$ at pivot scale is represented in
terms of $N_{k}$ as

$\displaystyle\phi_{k}\simeq\sqrt{2m(m+2N_{k})}M_{P}-\frac{2^{\frac{m}{2}-\frac{1}{2}}m^{m+2}}{\sqrt{m(m+2N_{k})}}M_{P}\xi+\mathcal{O}(\xi^{2})$
(20)

leading to the slow-roll parameters

$\displaystyle\epsilon\left(\phi_{k}\right)$
$\displaystyle\simeq\frac{m}{m+2N_{k}}+\frac{2^{\frac{m}{2}}\left[m^{m+2}-m^{\frac{m}{2}+1}(m+2N_{k})^{\frac{m}{2}+1}\right]\xi}{(m+2N_{k})^{2}}+\mathcal{O}(\xi^{2}),$
(21) $\displaystyle\eta\left(\phi_{k}\right)$
$\displaystyle\simeq\frac{2m-1}{2N_{k}+m}+\frac{2^{\frac{m}{2}-1}\left[2m^{m+1}(2m-1)-5m^{\frac{m}{2}+1}(m+2N_{k})^{\frac{m}{2}+1}\right]\xi}{(m+2N_{k})^{2}}+\mathcal{O}(\xi^{2}).$
(22)

These expressions yield the spectral index $n_{s}$ and the tensor-to-scalar
ratio $r$ in this limit as

$\displaystyle n_{s}$ $\displaystyle\simeq 1-6\epsilon+2\eta|_{\phi=\phi_{k}}$
(23) $\displaystyle\simeq
1-\frac{2(m+1)}{m+2N_{k}}+\frac{2^{\frac{m}{2}}\left[m^{\frac{m}{2}+1}(m+2N_{k})^{\frac{m}{2}+1}-2m^{m+1}(m+1)\right]\xi}{(m+2N_{k})^{2}}+\mathcal{O}(\xi^{2})$
(24) $\displaystyle r$ $\displaystyle\simeq
16\epsilon|_{\phi=\phi_{k}}\simeq\frac{16m}{m+2N_{k}}+\frac{2^{\frac{m}{2}+4}\left[m^{m+2}-m^{\frac{m}{2}+1}(m+2N_{k})^{\frac{m}{2}+1}\right]\xi}{(m+2N_{k})^{2}}+\mathcal{O}(\xi^{2})$
(25)

along with the curvature power spectrum amplitude being

$\displaystyle A_{s}$
$\displaystyle\simeq\left.\frac{1}{24\pi^{2}\epsilon}\frac{U}{M_{P}^{4}}\right|_{\phi=\phi_{k}}$
(26)
$\displaystyle\simeq\frac{2^{m-4}m^{m-2}(m+2N_{k})^{m+1}\lambda}{3\pi^{2}}$
$\displaystyle\qquad-\frac{2^{\frac{3m}{2}-4}m^{\frac{3m}{2}-2}\lambda\left[(m+1)m^{\frac{m}{2}+1}(m+2N_{k})^{m}+(m+2N_{k})^{\frac{3m}{2}+1}\right]\xi}{3\pi^{2}}$
(27)

where $A_{s}\simeq 2.1\times 10^{-9}$ is given by Planck and BICEP/Keck 2018
results at the pivot scale $k=0.05\text{Mpc}^{-1}$ Planck:2018jri . The
$\zeta$ dependent terms are suppressed, with its leading order terms emerging
at $\mathcal{O}(\xi^{2})$ order.

Note that, the specific $\xi$ range that resembles an approximate minimal
coupling case differs for each $m$ value. In order for a noticeable deviation
of $(n_{s},r)$ from minimal cases, the required $\delta\xi$ involves

$\displaystyle\left|\frac{\delta r}{r}\right|\sim\left|\frac{\delta
n_{s}}{n_{s}}\right|\sim(4m{N_{k}})^{\frac{m}{2}}|\delta\xi|$ (28)

which is $m$ dependent. A larger power monomial potential will lead to a
smaller $\delta\xi$ that shows a reasonable deviation from $\xi=0$
predictions.

In the minimal coupling case $(\xi=0)$, the relations of the inflationary
observables simplify to

$\displaystyle n_{s}\simeq 1-\frac{m+1}{N_{k}},$ $\displaystyle
r\simeq\frac{8m}{N_{k}},$ (29)

and these two parameters are correlated as

$\displaystyle N_{k}\simeq\frac{m+1}{1-n_{s}}\simeq\frac{8m}{r}$
$\displaystyle\Leftrightarrow$ $\displaystyle
r\simeq\frac{8m}{m+1}(1-n_{s})\approx\left.\frac{0.32m}{m+1}\right|_{n_{s}\approx
0.96}.$ (30)

The resulting value of $r$ is a bit too large compared to the observed limit
$r\lesssim 0.1$ WMAP:2010qai ; Planck:2018jri ; BICEPKeck:2021gln .

###  2.4 $\xi\gg 1$ (Metric $\neq$ Palatini)

We now turn to the opposite limit and examine the case with a large non-
vanishing non-minimal coupling. Now the predictions from metric and Palatini
formalisms drastically differ. Therefore we discuss the metric case and
Palatini case below separately.

####  2.4.1 Metric Formalism

In the large $\xi$ limit, slow roll parameters are approximated as

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{4}{3\xi^{2}}\left(\frac{M_{P}}{\phi}\right)^{2m}+\mathcal{O}(\xi^{-3}),$
(31) $\displaystyle\eta$
$\displaystyle\simeq-\frac{4}{3\xi}\left(\frac{M_{P}}{\phi}\right)^{m}+\frac{4}{3\xi^{2}}\left[\left(\frac{M_{P}}{\phi}\right)^{2m}+\frac{3m-2}{3m^{3}}\left(\frac{M_{P}}{\phi}\right)^{2m-2}\right]+\mathcal{O}(\xi^{-3}).$
(32)

Therefore, $\epsilon(\phi_{e})=1$ gives

$\displaystyle\phi_{e}\simeq\left(\frac{4}{3\xi^{2}}\right)^{\frac{1}{2m}}M_{P}.$
(33)

E-folding number and the field value at the pivot scale are obtained

$\displaystyle
N_{k}\approx\frac{3\xi}{4}\left(\frac{\phi_{k}}{M_{P}}\right)^{m}$
$\displaystyle\Rightarrow$
$\displaystyle\phi_{k}\simeq\left(\frac{4N_{k}}{3\xi}\right)^{\frac{1}{m}}M_{P},$
$\displaystyle(\xi\gg 1,~{}\text{Metric}).$ (34)

Then, the slow roll parameters and cosmological observables are represented by
$N_{k}$ as

$\displaystyle\epsilon(\phi_{k})\simeq\frac{3}{4N_{k}^{2}},$
$\displaystyle\eta(\phi_{k})\simeq-\frac{1}{N_{k}}+\frac{3}{4N_{k}^{2}}+\frac{3m-2}{4N_{k}^{2}m^{3}}\left(\frac{4N_{k}}{3\xi}\right)^{\frac{2}{m}}$
(35)

and the spectral index and tensor-to-scalar-ratio are

$\displaystyle n_{s}\simeq 1-6\epsilon+2\eta|_{\phi=\phi_{k}}\simeq
1-\frac{2}{N_{k}}-\frac{3}{N_{k}^{2}},$ $\displaystyle r_{s}\simeq
16\epsilon|_{\phi=\phi_{k}}\simeq\frac{12}{N_{k}^{2}},$ (36)

which enjoys the sweet spot of the Planck, WMAP and BICEP/Keck bounds on
$(n_{s},r)$ for $N_{k}\simeq 50-60$. Prospects from future measurements are
discussed in Section. 3.

From the fact that the potential during inflation approaches a constant value
at the large field regime with a large non-minimal coupling,

$\displaystyle U(\phi)\xrightarrow[]{\xi\gg 1,~{}\phi\gg
M_{P}/\xi^{1/m}}U_{\text{inf}}\simeq\frac{\lambda M_{P}^{4}}{2m\xi^{2}}$ (37)

we have the normalization of $\lambda/\xi^{2}$ as

$\displaystyle
A_{s}\simeq\left.\frac{1}{24\pi^{2}\epsilon}\frac{U}{M_{P}^{4}}\right|_{\phi=\phi_{k}\gg
M_{P}/\xi^{1/m}}\simeq\frac{N_{k}^{2}\lambda}{36m\pi^{2}\xi^{2}}\simeq
2.1\times 10^{-9}.$ (38)

####  2.4.2 Palatini Formalism

The slow-roll parameters in Palatini formalism are approximated as

$\displaystyle\epsilon\simeq\dfrac{2m^{2}M_{P}^{m+2}}{\xi\phi^{m+2}},$
$\displaystyle\eta\simeq-\frac{m(m+2)M_{P}^{2}}{\phi^{2}}.$ (39)

By requiring $\epsilon(\phi_{e})=1$,

$\displaystyle\phi_{e}=2^{\frac{1}{2+m}}M_{P}\left(\frac{\xi}{m^{2}}\right)^{-\frac{1}{2+m}}.$
(40)

The e-folding number and the field value at pivot scale are

$\displaystyle
N_{k}\simeq\frac{1}{4mM^{2}}\left(\phi_{k}^{2}-\phi_{e}^{2}\right)$
$\displaystyle\Rightarrow$ $\displaystyle\phi_{k}\simeq 2\sqrt{mN_{k}}M_{P},$
$\displaystyle(\xi\gg 1,~{}\text{Palatini})$ (41)

where we neglect $\phi_{e}\ll\phi_{k}$. Now slow roll parameters and
cosmological observables are represented as

$\displaystyle\epsilon(\phi_{k})\simeq\frac{m^{1-\frac{m}{2}}}{2^{1+m}N_{k}^{1+\frac{m}{2}}\xi},$
$\displaystyle\eta(\phi_{k})\simeq-\frac{2+m}{4N_{k}}+\frac{5m^{1-\frac{m}{2}}}{2^{2+m}N_{k}^{1+\frac{m}{2}}\xi}$
(42)

and the spectral index and the tensor-to-scalar ratio are

$\displaystyle n_{s}\simeq
1-\frac{2+m}{2N_{k}}-\frac{m^{1-\frac{m}{2}}}{2^{1+m}N_{k}^{1+\frac{m}{2}}\xi},$
$\displaystyle
r\simeq\frac{m^{1-\frac{m}{2}}}{2^{m-3}N_{k}^{1+\frac{m}{2}}\xi}.$ (43)

One should note that $r\propto 1/\xi$ is highly suppressed by $\xi\gg 1$. In
Palatini case, CMB power spectrum normalization determines $\lambda/\xi$ as

$\displaystyle
A_{s}\simeq\left.\frac{1}{24\pi^{2}\epsilon}\frac{U}{M_{P}^{4}}\right|_{\phi\simeq\phi_{k}}\simeq\frac{2^{m-3}m^{\frac{m}{2}-2}N_{k}^{\frac{m}{2}+1}\lambda}{3\pi^{2}\xi}.$
(44)

The fact that $\lambda/\xi$ is normalized requires a larger order of magnitude
of $\xi$ for the same $\lambda$ compared to metric cases.

##  3 Reheating and Cosmological Predictions

###  3.1 Reheating

In this section we enlist the calculations of $N_{\text{reh}}$ and
$T_{\text{reh}}$ by modeling the reheating epoch to be described with a
constant equation of state $w_{\text{reh}}$. Note that the equation of state
$w(t)\equiv p/\rho$, in general, is a time dependent parameter determined by
the particle physics details of the reheating process Podolsky:2005bw ;
Lozanov:2016hid ; Saha:2020bis ; DiMarco:2021xzk . In our approach following
Ref. Cook:2015vqa , we rather take $w_{\text{reh}}$ as an average value over
the reheating

$\displaystyle
w_{\text{reh}}\equiv\frac{1}{N_{\text{reh}}}\int_{N_{k}}^{N_{k}+N_{\text{reh}}}w(N)~{}dN.$
(45)

In this work, we allows a wide range of $w_{\text{reh}}\in[0,1/3]$. Recall
that $w=0$ for matter domination and $w=1/3$ for radiation domination.222In
principle, one might take $w_{\text{reh}}\in[-1,1]$ depending on the detail of
the model.

Now we take the standard assumptions for the late time cosmology after the
reheating and get the relations for $T_{\rm reh}$ and $N_{\rm reh}$. First,
the e-folds for the epoch is expressed as

$\displaystyle
N_{\text{reh}}\equiv\ln\left(\frac{a_{\text{reh}}}{a_{e}}\right)=\frac{1}{3(1+w_{\text{reh}})}\ln\left(\frac{\rho_{e}}{\rho_{\text{reh}}}\right),$
(46)

where $N_{\text{reh}}$ represents the duration of reheating from the end of
inflation, $a_{\text{reh}}$ and $a_{e}$ is the scale factor at the end of
reheating and the end of inflation, respectively, while $\rho_{\text{reh}}$
and $\rho_{e}$ is the corresponding energy density at the end of reheating and
the end of inflation. From $\epsilon_{H}=-\frac{\dot{H}}{H^{2}}=1$ at the end
of inflation, we have

$\displaystyle\rho_{e}=\frac{3}{2}U_{e},$ (47)

hence $w(t_{e})=-1/3$. Also, the radiation energy with the relativistic
degrees of freedom $g_{\text{reh}}$ and the temperature $T_{\text{reh}}$ at
the end of reheating is given by

$\displaystyle\rho_{\text{reh}}=\frac{\pi^{2}}{30}g_{\text{reh}}T_{\text{reh}}^{4}.$
(48)

Assuming no additional entropy production after reheating, we can link $T_{\rm
reh}$ and the temperature of our current universe $T_{0}$ as

$\displaystyle
T_{\text{reh}}=T_{0}\left(\frac{a_{0}}{a_{\text{reh}}}\right)\left(\frac{g_{0}}{g_{\text{reh}}}\right)^{1/3}=\left(\frac{43}{11g_{\text{reh}}}\right)^{1/3}\left(\frac{a_{0}T_{0}}{k}\right)H_{k}e^{-N_{k}}e^{-N_{\text{reh}}}$
(49)

where $g_{0}=2+\frac{7}{8}\times N_{\text{eff}}\times\frac{4}{11}$ with
$N_{\text{eff}}=3.046$ in the SM is the relativistic degree of freedom at
current universe and $g_{\text{reh}}$ is that at the end of reheating,
respectively. In the second line, we take the length of e-folds of radiation
dominance, $N_{\rm RD}=\ln(a_{\rm eq}/a_{\rm reh})$ with the scale factor at
the matter-radiation equality $a_{\rm eq}$, and the ratio $a_{0}/a_{\rm
eq}=a_{0}(H_{k}/k)e^{-N_{k}}e^{-N_{\rm reh}}e^{-N_{\rm RD}}$ from the pivot
scale $k=a_{k}H_{k}$.

Combining the expressions, we obtain $N_{\text{reh}}$ as

$\displaystyle
N_{\text{reh}}=\frac{4}{3(1+w_{\text{reh}})}\left[\frac{1}{4}\ln\left(\frac{45}{\pi^{2}g_{\text{reh}}}\right)+\ln\left(\frac{U_{e}^{1/4}}{H_{k}}\right)+\frac{1}{3}\ln\left(\frac{11g_{\text{reh}}}{43}\right)\right.\qquad\qquad$
$\displaystyle\ss\left.+\ln\left(\frac{k}{a_{0}T_{0}}\right)+N_{k}+N_{\text{reh}}\right].$
(50)

Taking the standard values $g_{\text{reh}}=106.75$,
$k=0.05~{}\text{Mpc}^{-1}$, $T_{0}=2.725~{}\text{K}$ for
$w_{\text{reh}}\neq-1/3$ we arrive at

$\displaystyle
N_{\text{reh}}=\frac{4}{(1-3w_{\text{reh}})}\left[61.6-\ln\left(\frac{U_{e}^{1/4}}{H_{k}}\right)-N_{k}\right],$
(51)

and accordingly, inserting this expression to $T_{\text{reh}}$ gives

$\displaystyle
T_{\text{reh}}=\left[\left(\frac{43}{11g_{\text{reh}}}\right)^{1/3}\frac{a_{0}T_{0}}{k}H_{k}e^{-N_{k}}\left(\frac{45U_{e}}{\pi^{2}g_{\text{reh}}}\right)^{-\frac{1}{3(1+w_{\text{reh}})}}\right]^{\frac{3(1+w_{\text{reh}})}{3w_{\text{reh}}-1}}.$
(52)

###  3.2 Results : Observables & Reheating Temperature

We present the exact results in Figure. 1 for metric cases and Figure. 2,
Figure. 3 for Palatini cases. For all figures, we include
Planck18+BICEP/Keck(BK)18+BAO $1\sigma$ (yellow) and $2\sigma$ (green) results
on $(n_{s},r)$ as well as prospective results from future CMB-S4 observations
with a fiducial detection of $r=3\times 10^{-3}$ (purple) and null results
(blue), both assuming a similar $n_{s}$ center value with Planck18+BK18+BAO
BICEPKeck:2021gln ; Abazajian:2019eic .

To begin with, we discuss how inflationary predictions differ in common in
both metric and Palatini formalisms. First, the predictions of $(n_{s},r)$
change depending on $(w_{\text{reh}},T_{\text{reh}})$. For illustration, we
denote predictions on $(n_{s},r)$ for $w_{\text{reh}}=0$ (black) and
$w_{\text{reh}}=1/5$ (brown) depending on
$T_{\text{reh}}\in[10^{-2}~{}\text{GeV}-T_{\text{max}}]$. $T_{\text{max}}$ is
determined by imposing instantaneous reheating

$\displaystyle\rho_{e}=\frac{3}{2}U_{e}=\frac{\pi^{2}}{30}g_{\text{reh}}T_{\text{max}}^{4}$
(53)

at the end of the inflation. Lines corresponding to $T_{\text{max}}$ are
denoted by solid lines. For metric cases, $T_{\text{max}}\simeq
10^{16}~{}\text{GeV}$ and for Palatini cases, $T_{\text{max}}$ decreases for
larger $\xi$. (See Figure. 6.) On the other hand, the lower bound
$T_{\text{reh}}\sim 10^{-2}~{}\text{GeV}$ corresponds to the BBN scale
temperature $T_{\text{BBN}}$ as a conservative assumption and is depicted in
dotted lines. Dot-dashed and dashed lines correspond to
$T_{\text{reh}}=10^{5}~{}\text{GeV}$ and
$T_{\text{reh}}=10^{10}~{}\text{GeV}$, respectively.333
$T_{\text{reh}}=10^{5}~{}\text{GeV}$ can be understood as an energy scale of
future collider experiments for new physics searches.
$T_{\text{reh}}=10^{10}~{}\text{GeV}$ is usually regarded as a typical upper
bound of the reheating temperature of the ‘gravitino overproduction problem’
if one takes this literally. Lower reheating temperature gives smaller $n_{s}$
predictions while the $r$ dependence on reheating temperature is weak.
Constraints on the reheating temperature from current/future $n_{s}$ and $r$
observations are also discussed below.

Figure 1: $(n_{s},r)$ plot with $m=(1,2,3,4)$ in metric formalism. Red solid
lines shows the dependence of inflationary predictions on various $\xi\geq 0$
values for $w_{\text{reh}}\in\left[0,\frac{1}{3}\right]$ and
$T_{\text{reh}}\in\left[T_{\text{BBN}},T_{\text{max}}\right]$. Dotted, dash-
dotted, dashed, solid lines denote inflationary predictions with
$T_{\text{reh}}=(10^{-2},10^{5},10^{10},T_{\text{max}})~{}\text{GeV}$
respectively, along with $w_{\text{reh}}=0$ for black and
$w_{\text{reh}}=\frac{1}{5}$ for brown lines. Current Planck18+BK18+BAO and
expected future CMB-S4 measurements are denoted each in yellow/green and
blue/purple.

Second, the regions of prediction become narrower and converge to the solid
line on the right as $w_{\text{reh}}\rightarrow 1/3$. In this limit, the end
of the reheating is ambiguous, because there is no distinct change in the
equation of state compared to the radiation dominant phase. Instead, the
predictions on $(n_{s},r)$ does not depend on reheating processes and is
uniquely determined solely by inflationary model parameters.

Figure 2: $(n_{s},r)$ plot for $m=(1,2,3,4)$ in Palatini formalism. Solid
lines are for instantaneous reheating, and dotted lines are for
$T_{\text{reh}}=T_{\text{BBN}}$. Except for $m=1$ and $m=2$, other power
potentials are inconsistent to current observations, regardless of the
reheating temperature for equation of state
$w_{\text{reh}}\in\left[0,\frac{1}{3}\right]$.

Figure 3: $(n_{s},r)$ plot for $m=1$ and $m=2$ in Palatini formalism. Plotting
conventions are the same as Figure. 1. Depending on the reheating temperature,
predictions on $n_{s}$ significantly changes as well.

We note that, depending on $m$, even $\xi\lesssim\mathcal{O}(1)$ still
provides predictions compatible to current constraints. For example, in the
metric formalism, intermediate $\xi$ values $\xi\sim\mathcal{O}(1)$ for $m=1$,
and $\xi\sim\mathcal{O}(10^{-1})$ for $m=2$ and each of these show different
predictions for $(n_{s},r)$, but still fully consistent to observations.

On the other hand, there are several notable differences between the two
formalisms of gravity. For the Palatini case, each $m$ give different
predictions even in the $\xi\rightarrow\infty$ limit. Potentials with the form
$m\geq 3$ for $w_{\text{reh}}\in\left[0,\frac{1}{3}\right]$ are ruled out
regardless of the reheating history. This is shown in Figure. 2. (However, if
one considers $w_{\text{reh}}>1/3$, the predictions shift to larger $n_{s}$
values so that these can be inside current observational bounds.)

Also, in the Palatini case, from the fact that
$r|_{\xi\rightarrow\infty}\propto 1/\xi$, the tensor-to-scalar ratio is highly
suppressed, while for metric cases, predictions converge to specific values
depending on $T_{\text{reh}}$ as long as $\xi\gg 1$. This was precisely why,
Palatini Higgs inflation has hardly been regarded to be proven by observation
in general. However, in this work, we show that the predictions of $n_{s}$ for
non-minimal inflation in the Palatini formalism actually possesses a
dependence on $\xi$, when we take the reheating processes more carefully.

In the $m=1$ case, the predictions are still highly compatible with the CMB
observations, regardless of the $w_{\text{reh}}$ value. Interestingly, for
$m=2$ in the Palatini case, we observe that quartic potential inflation with a
large non-minimal coupling $\xi\gtrsim\mathcal{O}(10)$ is already outside of
current Planck18+BK18+BAO $1\sigma$ bounds, even for instantaneous reheating.
(See the right of Figure. 3.) This also includes conventionally considered
non-minimal coupling values for Palatini-Higgs inflation with
$\xi\gtrsim\mathcal{O}(10^{8})$ (See Eq. (44) with
$\lambda\gtrsim\mathcal{O}(10^{-2})$). Also, $w_{\text{reh}}$ values deviating
from $\frac{1}{3}$ tend to worsen the compatibility with observations for
lower $T_{\text{reh}}$. For example, for $\xi=10^{8}$, current CMB bounds
require $T_{\text{reh}}\gtrsim\mathcal{O}(10^{10})~{}\text{GeV}$ for
$w_{\text{reh}}=0$ and $T_{\text{reh}}\gtrsim\mathcal{O}(10^{5})~{}\text{GeV}$
for $w_{\text{reh}}=\frac{1}{5}$. These constraints will be more stringent for
future observations such as CMB-S4 Abazajian:2019eic  and LiteBIRD
LiteBIRD:2020khw , lying beyond the $2\sigma$ expected bounds. This fact also
can be used to distinguish metric Higgs inflation models from Palatini cases
Bauer:2010jg ; Rasanen:2017ivk ; Markkanen:2017tun ; Jinno:2019und ;
Shaposhnikov:2020fdv ; Gialamas:2020vto ; Enckell:2020lvn . Note that the
early stage of the reheating (namely ‘preheating’) of Palatini Higgs inflation
possesses $w\simeq-1$ Rubio:2019ypq , which makes the compatibility to the
observations worse if the reheating processes has an averaged equation of
state $w_{\text{reh}}<1/3$.444As the reheating stage of the Palatini Higgs
inflation is tachyonic, we expect the reheating to be very efficient so that
$w_{\text{reh}}$ is very close to $1/3$. However, detailed analysis on the
reheating of this model and determination of the precise $w_{\text{reh}}$ is
out of the scope of this work.

Figure 4: Constraints on the reheating temperature $T_{\text{reh}}$ depending
on $n_{s}$ for the metric $m=(1,2,3,4)$ cases. The spectrum of $r$ is also
depicted. The solid, point-dotted, square-dotted lines each correspond to
$w_{\text{reh}}=0$, $w_{\text{reh}}=1/5$, $w_{\text{reh}}=1/3$, respectively.
The Planck18+BK18+BAO $1\sigma$ and $2\sigma$ constraints are depicted in
yellow and green, respectively. Future CMB-S4 null and fiducial constraints
are also expressed in blue and purple, where the dashed lines and solid
boundaries represent $1\sigma$ and $2\sigma$ bounds.

Following the preceding results, bounds on $(n_{s},r)$ also impose constraints
on $T_{\text{reh}}$ depending on $w_{\text{reh}}$. Relations between
$(n_{s},T_{\text{reh}})$ for metric cases are depicted in Figure. 4. The $\xi$
values range from $\xi=0$ to $\xi=\infty$ for all $m=(1,2,3,4)$ potentials.
The allowed range of $T_{\text{reh}}$ varies depending on the potential form
along with the specific non-minimal coupling value. We also note that, for
$m=1$, there exists some range of $\xi$ which gives upper bound on
$T_{\text{reh}}$ as well depending on $w_{\text{reh}}$. The
$(r,T_{\text{reh}})$ relation for metric cases is also shown in Figure. 5. As
the prospected sensitivity at CMB-S4 improves significantly compared to
Planck18+BK18+BAO up to $r\lesssim 10^{-3}$, these observations may be used to
effectively constrain $T_{\text{reh}}$.

Figure 5: Constraints on the reheating temperature $T_{\text{reh}}$ depending
on $r$ for the metric $m=(1,2,3,4)$ cases. The spectrum of $n_{s}$ is also
depicted. The solid, point-dotted, square-dotted lines each correspond to
$w_{\text{reh}}=0$, $w_{\text{reh}}=1/5$, $w_{\text{reh}}=1/3$, respectively.
The Planck18+BK18+BAO $1\sigma$ and $2\sigma$ constraints are depicted in
yellow and green, respectively. Future CMB-S4 null and fiducial constraints
are also expressed in blue and purple, where the dashed lines and solid
boundaries represent $1\sigma$ and $2\sigma$ bounds.  Figure 6: Constraints on
the reheating temperature $T_{\text{reh}}$ depending on $n_{s}$ (top) and $r$
(bottom) for the Palatini $m=1$ and $m=2$ cases. The spectrum of $r$ and
$n_{s}$ is also depicted accordingly. The solid, point-dotted, square-dotted
lines each correspond to $w_{\text{reh}}=0$, $w_{\text{reh}}=1/5$,
$w_{\text{reh}}=1/3$, respectively. The Planck18+BK18+BAO $1\sigma$ and
$2\sigma$ constraints are depicted in yellow and green, respectively. Future
CMB-S4 null and fiducial constraints are also expressed in blue and purple,
where the dashed lines and solid boundaries represent $1\sigma$ and $2\sigma$
bounds. For cases $m=3,4$, the predictions contradict with current CMB
observations in terms of $n_{s}$, regardless of the reheating dynamics.

The $T_{\text{reh}}$ implications for the Palatini case with $m=(1,2)$, which
is depicted in Figure. 6, differ from the metric scenario. The suppression of
$r\propto 1/\xi$ in the Palatini formalism makes it nearly impossible for
future observations to provide reasonable constraints on the allowed
$T_{\text{reh}}$ range for large $\xi$ parameters. Instead, the $\xi$
dependence on $n_{s}$ may lead to providing a more stringent (upper/lower)
bound on $T_{\text{reh}}$.

##  4 Conclusion and Discussions

In this work, we clarified the effects of reheating to predictions of
inflation models with non-minimal coupling to gravity, covering (i) both
metric and Palatini formalisms, (ii) arbitrary monomial potential with
asymptotic flatness in the Einstein frame, and (iii) a wide range of non-
minimal coupling $\xi$ depending on the power of the potential, from the
consistency relations between reheating parameters and inflationary
observables $n_{s}$ and $r$. We also obtained a range of compatible reheating
temperatures for a given equation of state parameter $w_{\text{reh}}$ during
reheating. This in turn can be used complementarily to further decipher the
microscopic particle physics governing the reheating process for each model.

For models in both formalisms, we note that the CMB compatible $\xi$ values
also allow $\xi\leq 1$, with its predictions deviating from those for large
$\xi\gg 1$ limits. The specific dependence on $(n_{s},r)$ in this particular
$\xi$ range differ with a definite dependence on the details of the model’s
potential.

Especially, for $m=2$ in the Palatini formalism, while
$\xi\gtrsim\mathcal{O}(10^{-2})$ are allowed for current CMB $2\sigma$ bounds,
the dependence on $\xi$ indicates that typically considered large
$\xi\sim\mathcal{O}(10^{8})$ lead to $N_{k}$ smaller than 50, which is not
preferred within the $1\sigma$ limit of current CMB bounds. Current and future
CMB observations are able to constrain the particular parameter range of
$\xi$s, giving definite lower and upper bounds on the non-minimal coupling,
with the constraints being stronger when taking the effects of reheating into
account.

We emphasize that our results are applicable to a wider class of models, which
exhibit $\alpha$-attractor behavior. $R^{2}$-driven Starobinsky inflation is
also transformable to this class, giving the same inflationary dynamics.
Application to multi-field inflationary models that are reducible into an
effective single-field $\alpha$-attractor case is also straightforward.
However, the reheating details of these equivalent inflationary classes differ
depending specifically on the model that describes its microscopic nature, and
predicts different values of $(T_{\text{reh}},N_{\text{reh}},w_{\text{reh}})$.
Therefore, our result could be used as a generic template for the inflationary
predictions of wide classes of models, breaking the degeneracy of the various
models not caught solely from considering the inflation dynamics. This
approach will provide valuable information on deciphering the contents of the
post-inflationary universe, which may feature - but most definitely not
restricted to - multiple fields encompassing spectator fields, new
interactions including higher order gravitational couplings.555Single field
inflation models including higher order $R^{3}$ have been analyzed in Ref.
Cheong:2020rao . We will further investigate this direction in future works.

###### Acknowledgements.

We thank Fedor Bezrukov, Kin-ya Oda, and Minxi He for discussions throughout
various stages of the research. This work is supported in part by the National
Research Foundation of Korea (NRF) grant funded by the Korea government
(NRF-2019R1A2C1089334), (NRF-2021R1A4A2001897) (SCP), and
(NRF-2020R1A6A3A13076216) (SML). SML is supported by the Hyundai Motor Chung
Mong-Koo Foundation Scholarship.

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