CytoSaddleSum: a functional enrichment analysis plugin for Cytoscape based on
sum-of-weights scores

Aleksandar Stojmirović , Alexander Bliskovsky and Yi-Kuo Yu***to whom
correspondence should be addressed

National Center for Biotechnology Information
National Library of Medicine
National Institutes of Health
Bethesda, MD 20894
United States

### Summary:

CytoSaddleSum provides Cytoscape users with access to the functionality of
SaddleSum, a functional enrichment tool based on sum-of-weight scores. It
operates by querying SaddleSum locally (using the standalone version) or
remotely (through an HTTP request to a web server). The functional enrichment
results are shown as a term relationship network, where nodes represent terms
and edges show term relationships. Furthermore, query results are written as
Cytoscape attributes allowing easy saving, retrieval and integration into
network-based data analysis workflows.

### Availability:

www.ncbi.nlm.nih.gov/CBBresearch/Yu/downloads
The source code is placed in Public Domain.

### Contact:

<EMAIL_ADDRESS>
##  1 Introduction

CytoSaddleSum is a Cytoscape (Smoot et al., 2011) plugin to access the
functionality of SaddleSum, an enrichment analysis tool based on sum-of-
weights-score (Stojmirović and Yu, 2010). Unlike most other enrichment tools,
SaddleSum does not require users to directly select significant genes or
perform extensive simulations to compute statistics. Instead, it uses weights
derived from measurements, such as log-expression ratios, to produce a score
for each database term. It then estimates, depending on the number of genes
involved, the P-value for that score by using the saddlepoint approximation
(Lugannani and Rice, 1980) to the empirical distribution function derived from
all weights. This approach was shown (Stojmirović and Yu, 2010) to yield
accurate P-values and internally consistent retrievals.

As a popular and flexible platform for visualization, integration and analysis
of network data, Cytoscape allows gene expression data import and hosts
numerous plugins for functional enrichment analysis. However, none of these
plugins are based on the ‘gene set analysis approach’ that takes into account
gene weights. Therefore, to fill this gap, we have developed CytoSaddleSum, a
Cytoscape interface to SaddleSum. To enable several desirable features of
CytoSaddleSum, however, we had to significantly extend the original SaddleSum
code (see descriptions below).

##  2 Implementation

While CytoSaddleSum is implemented in Java using Cytoscape API, it functions
by running either locally or remotely a separate instance of SaddleSum,
written in C. In either mode, CytoSaddleSum takes the user input through a
graphical user interface, validates it, and passes a query to SaddleSum. Upon
receiving the entire query results, CytoSaddleSum stores them as the node and
network attributes of the newly-created term relationship graph. Consequently,
the query output can be edited or manipulated within Cytoscape. Furthermore,
saving term graph through Cytoscape also preserves the results for later use.

The most important extension to SaddleSum involved construction of extended
term databases (ETDs). Each ETD contains the mappings of genes to Gene
Ontology (Gene Ontology Consortium, 2010) terms and KEGG (Kanehisa et al.,
2008) pathways, as well as an abbreviated version of the NCBI Gene (Maglott et
al., 2011) database for all genes mapped to terms. Thanks to the latter, when
using an ETD, SaddleSum is able to interpret the provided gene labels as NCBI
Gene IDs, as gene symbols and as gene aliases. Each ETD also contains
relations among terms that are used by SaddleSum for term graph construction.

Figure 1: CytoSaddleSum user interface consists of the query form (left), the
results panel (right) and the term relationship network (center), which here
partially covers the original network. The results stored as attributes of the
term network can be edited through Cytoscape Data Panel.

##  3 Usage

CytoSaddleSum operates on the currently selected Cytoscape network whose nodes
represent genes or gene products. The queries are submitted through the query
form embedded as a tab into the Cytoscape Control Panel, on the left of the
screen. The selected network must contain at least one node mapped to a
floating-point Cytoscape attribute, which would provide node weights.
CytoSaddleSum considers only the selected nodes within the network. The user
can select the weight attribute through a dropdown box on the query form. Any
selected node without specified weight is assumed to have weight 0. The user-
settable cannonicalName attribute, automatically created by Cytoscape for each
network node, serves as the gene label.

After selecting the network and the nodes within it, the user needs to select
a term database and set the statistical and weight processing parameters. The
latter enable users to transform the supplied weights within SaddleSum. This
includes changing the sign of the weights, as well as applying a cutoff, by
weight or by rank. All weights below the cutoff are set to 0. The statistical
parameters are E-value cutoff, minimum term size, effective database size and
statistical method. We define the effective database size as the number of
terms in the term database that map to at least $k$ genes among the selected
nodes, where $k$ is the minimum term size. Apart from the default ‘Lugannani-
Rice’ statistics, it is also possible to select ‘One-sided Fisher’s Exact
test’ statistics, which are based on the hypergeometric distribution. In that
case, the user must select a cutoff under the weight processing parameters.

To run local queries, a user needs the command-line version of SaddleSum and
the term databases, both available for download from our website, and install
them on the same machine that runs Cytoscape. The advantages of running local
queries include speed, independence of Internet connection, and support of
queries to custom databases in the GMT file format used by the GSEA tool
(Subramanian et al., 2005). Furthermore, the standalone program can be used
outside of Cytoscape for large sets of queries. On the other hand, running
remote queries require no installation of additional software, since queries
are passed to the SaddleSum server over an HTTP connection. The disadvantage
of running remote queries is that it can take much longer to run and that the
choice of term databases is restricted to ETDs available only for some model
organisms.

CytoSaddleSum also displays warning or error messages reported by SaddleSum.
For example, when a provided gene label is ambiguous, depending on whether the
ambiguity could be resolved, CytoSaddleSum will relay a warning or an error
message reported by SaddleSum. CytoSaddleSum presents query results as a term
relationship network (Fig. 1), consisting of significant terms or their
ancestors linked by hierarchical relations available in the term database. The
statistical significance of each term is indicated by the color of its
corresponding node. To facilitate browsing of the results, CytoSaddleSum
generates a set of summary tables, which contain the lists of significant
terms and various details about the query. These summary tables are embedded
into Cytoscape Results Panel, on the right of the screen. Clicking on a
significant term in a summary table will select that term in the term
relationship network and select all nodes mapping to it in the original
network. The results can be exported as text or tab-delimited files and can be
restored from tab-delimited files through the Export and Import menus of
Cytoscape.Detailed instructions and explanations can be found in SaddleSum
manual available from our website.

## Acknowledgments

This work was supported by the Intramural Research Program of the National
Library of Medicine at National Institutes of Health.

## References

  * Gene Ontology Consortium (2010) Gene Ontology Consortium (2010).  The gene ontology in 2010: extensions and refinements.  Nucleic Acids Res, 38(Database issue), D331–5.
  * Kanehisa et al. (2008) Kanehisa, M. et al. (2008).  KEGG for linking genomes to life and the environment.  Nucleic Acids Res, 36(Database issue), D480–4.
  * Lugannani and Rice (1980) Lugannani, R. and Rice, S. (1980).  Saddle point approximation for the distribution of the sum of independent random variables.  Adv. in Appl. Probab., 12(2), 475–490.
  * Maglott et al. (2011) Maglott, D. et al. (2011).  Entrez Gene: gene-centered information at NCBI.  Nucleic Acids Res, 39(Database issue), D52–7.
  * Smoot et al. (2011) Smoot, M. E. et al. (2011).  Cytoscape 2.8: new features for data integration and network visualization.  Bioinformatics, 27(3), 431–2.
  * Stojmirović and Yu (2010) Stojmirović, A. and Yu, Y.-K. (2010).  Robust and accurate data enrichment statistics via distribution function of sum of weights.  Bioinformatics, 26(21), 2752–9.
  * Subramanian et al. (2005) Subramanian, A. et al. (2005).  Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles.  Proc Natl Acad Sci USA, 102(43), 15545–15550.


Draft whatever

Scaling Characteristics of Hilltop and Hilltop-Squared Inflation

Monika<EMAIL_ADDRESS>and Rolf<EMAIL_ADDRESS><EMAIL_ADDRESS>

Dept. of Physics

Indiana University South Bend

1700 Mishawaka Ave., South Bend, IN 46615

Abstract

One of the longstanding goals in the framework of inflation is the
construction of tools that can be used to classify models in theory space. An
idea that has been put forward in this context is to consider the energy
dependent scaling behavior of observables to characterize different models. We
implement this approach in the framework of hilltop and hilltop-squared
inflation by analyzing their observables when the small-field approximation is
not imposed and the energy scale $\mu$ of these models is varied as a free
parameter, subject to observational constraints. We show that the scalar
spectral tilt and the tensor ratio $r$ exhibit $\mu$-dependent scaling
behavior and that the scaling exponents as functions of $\mu$ in turn lead to
functional forms that are model dependent. Scaling relations of the type
discussed here are of interest as characteristics of the inflationary theory
space as well as in the context of the post-inflationary reheating process. We
further observe a bifurcation behavior in the behavior of $p$-families in the
spectral-tensor plane for a critical value of $\mu$.

###### Contents

  1. 1 Introduction
  2. 2 Observables in hilltop inflation
    1. 2.1 Slow-roll approximation
    2. 2.2 Small-field approximations of hilltop inflation
  3. 3 Reheating constraint
  4. 4 Scaling behavior of hilltop inflation
  5. 5 Scaling distributions in hilltop inflation via $N_{*}$
  6. 6 Scaling exponents in hilltop inflation
    1. 6.1 Scaling exponents for the quartic hilltop model
    2. 6.2 Behavior of hilltop inflation at $p=10$
    3. 6.3 Implications
  7. 7 Bifurcation among $p$-families in hilltop inflation
    1. 7.1 The bifurcation phenomenon for $p$-families
    2. 7.2 Small-field structure of $p$-families
  8. 8 Hilltop-squared inflation
    1. 8.1 Observables in hilltop-squared inflation
    2. 8.2 Distribution of $(r,n_{{\cal R}{\cal R}})$ in hilltop-squared inflation
    3. 8.3 Scaling distributions for hilltop-squared observables via $N$
    4. 8.4 $r(N)$ scaling in hilltop-squared inflation
    5. 8.5 Scaling exponents in hilltop-squared inflation
  9. 9 Scaling exponents as model characteristics
    1. 9.1 Characteristic intervals
    2. 9.2 Characteristic functions of scaling exponents
  10. 10 Conclusions

##  1 Introduction

Hilltop type potentials have remained among the canonical models of inflation.
While the general class of hilltop theories encompasses a wide variety of
potentials that have been discussed in the context of a number of embedding
theories, see e.g. the reviews [1, 2] for many references, our focus here is
on a more specific set of models, defined as

$V_{p,n}~{}=~{}\Lambda^{4}\left(1~{}-~{}\left(\frac{\phi}{\mu}\right)^{p}\right)^{n}.$
(1)

where $n=1,2$. Some of the models in this class have become benchmark models
for the CMB collaborations [3, 6, 4, 5, 7, 8]. While hilltop models in
general, and the $V_{p,n}$ above in particular, have traditionally been
considered in the context of singlefield inflation (see e.g. [2, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35]), this class of potentials is also of interest in the
framework of multifield inflation models that describe saddle point inflation.
In this context hilltop potentials can be used as singlefield approximations
of inflaton trajectories that roughly evolve along a cross-section in the
concave direction of the multifield potentials. Examples of this latter type
include the models of $j$-inflation and $h$-inflation in the framework of
modular inflation [36, 37, 38, 39, 40, 41].

The potentials $V_{p,n}$ describe two-parameter singlefield inflationary
models and as such are amenable to the scaling type analysis that was
introduced in ref. [42] in the context of multifield inflation. In was
observed there that theories with two parameters may lend themselves to a
characterization in terms of a family of scaling relations with associated
scaling exponents $\beta_{p}$ that are specific to the models and therefore
could be used to distinguish regions in the inflationary theory space.
Furthermore, if the exponents $\beta_{p}$ are functions $\beta_{p}(\mu)$ of
the energy scale $\mu$ one can ask whether they in turn show scaling behavior
with an associated exponent $\gamma_{p}$ that would then provide a single
number associated to the parameter space of the model. Such scaling relations
were established in [42] in the context of modular inflation, which is a class
of twofield models. The same question can be raised in the context of other
models with two parameters and in the present analysis we focus on hilltop
type potentials. We vary the exponent $p$ in $V_{p,n}$ systematically and
consider the dependence of the observables on the energy scale $\mu$. We will
focus on pure slow-roll inflation, without making any further approximations
that were adopted in early papers on these models. Making such additional
approximations eliminates the dependence of the observables on the energy and
thereby trivializes the scaling behavior. Imposing such approximations
implies, for example, that the critical exponent of the tensor ratio is energy
independent, hence there is no functional dependence of the scaling exponents
on $\mu$.

Depending on the exponent $n$ in (1) the models exhibit a quite different
structure close to the minimum at $\phi=\mu$. While hilltop inflation ($n=1$)
intersects the zero line with a non-zero slope, hilltop-squared models have a
smooth valley around the minimum. The fact that there is no valley for the
hilltop models has sometimes been considered a weakness of these models
because strictly speaking this means that these model do not have a domain
that allows the inflaton to oscillate around the minimum, thereby releasing
its energy to other particles whose couplings to the inflaton are assumed to
be small and negligible during inflation but become relevant during the post-
inflation process of reheating. As a practical matter this issue can be
resolved by simply gluing an appropriate potential to $V(\phi_{e})$ for
$\phi>\phi_{e}$ to the hilltop potential. The fact that this issue does not
arise for hilltop-squared theories has been a motivation to consider these
models, see e.g. [7, 32, 34, 35].

We have organized this paper as follows. Section 2 describes the observables
we will focus on in both the pure slow-roll and in their different small-field
approximations. In section 3 we briefly consider the reheating constraint so
as to be able to gauge its effect when imposed in addition to the CMB
constraints. In section 4 we present the impact of the CMB and reheating
constraints for the class of hilltop models and in section 5 we discuss the
scaling distributions that will be made quantitative for hilltop models in
section 6. In section 7 we analyze $p$-families of hilltop models in
dependence of the energy scale $\mu$ and describe a bifurcation phenomenon
that appears at a critical value of this scale. In section 8 we extend our
analysis to the hilltop-squared models and in section 9 we discuss the
implications of our scaling analysis as a tool that provides characteristic
quantities defined on the inflationary landscape. We conclude in section 10.

##  2 Observables in hilltop inflation

Our focus in this work is on hilltop inflation in the pure slow-roll
approximation but it is of interest to compare our results with the behavior
of this class of models in the small-field approximation. We discuss these
approximations it turn.

###  2.1 Slow-roll approximation

The analysis in the following is concerned with the observables at horizon
crossing as functions of the inflaton $\phi_{*}$ and the number of e-folds
$N_{*}$ between horizon crossing time $t_{*}$ and the end of inflation
$t_{e}$. For the Lukash-Bardeen perturbation [43, 44, 45]

${\cal R}~{}=~{}H\delta u~{}-~{}\psi,$ (2)

where $H={\dot{a}}/a$ is the Hubble-Slipher parameter, $\psi$ is the metric
potential, and $\delta u$ is obtained from $\delta T_{0i}$, the spectral index
and the tensor-to-scalar ratio take the form

$\displaystyle n_{{\cal R}{\cal R}}$ $\displaystyle=$ $\displaystyle
1-p\frac{M_{\rm
Pl}^{2}}{\mu^{2}}\frac{\left(\frac{\phi}{\mu}\right)^{p-2}}{\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)^{2}}\left[2(p-1)+(p+2)\left(\frac{\phi}{\mu}\right)^{p}\right]$
$\displaystyle r$ $\displaystyle=$ $\displaystyle 8p^{2}\frac{M_{\rm
Pl}^{2}}{\mu^{2}}\frac{\left(\frac{\phi}{\mu}\right)^{2(p-1)}}{\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)^{2}}.$
(3)

while the number of e-folds is given by

$N_{*}~{}=~{}\frac{\mu^{2}}{pM_{\rm
Pl}^{2}}\left[\frac{1}{p-2}\left(\left(\frac{\mu}{\phi_{*}}\right)^{p-2}~{}-~{}\left(\frac{\mu}{\phi_{e}}\right)^{p-2}\right)-\frac{1}{2}\left(\left(\frac{\phi_{e}}{\mu}\right)^{2}-\left(\frac{\phi_{*}}{\mu}\right)^{2}\right)\right]$
(4)

The end-of-inflation value $\phi_{e}$ will be determined by the constraint
$\epsilon_{V}(\phi_{e})=1$, which is obtained from the constraint

$\left(\frac{\phi_{e}}{\mu}\right)^{p}~{}+~{}\frac{p}{\sqrt{2}}\frac{M_{\rm
Pl}}{\mu}~{}\left(\frac{\phi_{e}}{\mu}\right)^{p-1}~{}-~{}1~{}=~{}0.$ (5)

We will use two different ways to analyze these models phenomenologically. The
first is the “scan method”, in which the strategy is to systematically scan
the parameter space $(\Lambda,\mu,\phi_{*})$ for viable models within the
available constraints, in particular the CMB results for $n_{{\cal R}{\cal
R}},r$ from Planck as well as the adopted range for the number of e-folds.
This is in the spirit of earlier analyses in a different context, such as
[41]. The second is the “$p$-family method”, where the idea is to choose a
fixed number of e-folds $N$, say $N=60$, and determine $\phi_{*}$ in terms of
$N_{*}$, where $\phi_{e}$ has been solved via $\epsilon_{V}(\phi_{e})=1$, as
usual. Having determined $\phi_{*}(N)$ by inverting the $N_{\star}$ equation
in (4) leads to

$\left(\frac{\phi_{*}}{\mu}\right)^{p}~{}-~{}A_{p}\left(\frac{\phi_{*}}{\mu}\right)^{p-2}~{}+~{}\frac{2}{p-2}~{}=~{}0$
(6)

with

$A_{p}~{}:=~{}2p\frac{M_{\rm
Pl}^{2}}{\mu^{2}}N_{*}~{}+~{}\left(\frac{\phi_{e}}{\mu}\right)^{2}~{}+~{}\frac{2}{p-2}\left(\frac{\mu}{\phi_{e}}\right)^{p-2}.$
(7)

In this way one can compute the observables for a given $\mu$. This method was
considered for example in the early hilltop papers [9, 10] (see also [21]).

###  2.2 Small-field approximations of hilltop inflation

The small-field approximation can be implemented in different forms. The
weakest form requires only that the initial value $\phi_{*}$ is much smaller
than the energy scale $\mu$. More interesting is the stronger approximation
that $\phi$ is much smaller than $\mu$ throughout inflation, because this
allows to obtain analytical expressions for the observables. While consistency
with the model does place restrictions on $\mu$, these are model dependent and
are weaker than the sub-Planckian constraint often encountered as part of the
small-field definition. In order to distinguish these different approximations
we use the designations SF.I for the weakest version, SF.II for the version
that covers all of inflation, and SF.III for the version that furthermore
imposes an a priori constraint on the energy scale $\mu$ that is model
independent.

The motivation for SF.II comes from the fact that in this case one can use the
resulting e-fold formula

$N_{*}~{}=~{}\frac{\mu^{2}}{p(p-2)M_{\rm
Pl}^{2}}\left(\left(\frac{\mu}{\phi_{*}}\right)^{p-2}~{}-~{}\left(\frac{\mu}{\phi_{e}}\right)^{p-2}\right)$
(8)

to solve analytically for $\phi_{*}$ as a function of $N_{*}$. Using this
solution we can write the e-fold dependence of the spectral index as

$n_{{\cal R}{\cal
R}}(p,\mu,N_{*})~{}=~{}1-\frac{2(p-1)}{(p-2)N_{*}~{}+~{}\frac{1}{p}\left(\frac{p}{\sqrt{2}}\right)^{(p-2)/(p-1)}\left(\frac{\mu}{M_{\rm
Pl}}\right)^{p/(p-1)}}.$ (9)

and the tensor-to-scalar ratio as

$r(N_{*},\mu,p)~{}=~{}\frac{8p^{2}\left(\frac{\mu}{M_{\rm
Pl}}\right)^{2p/(p-2)}}{\left(p(p-2)N_{*}+\left(\frac{\mu}{M_{\rm
Pl}}\right)^{p/(p-1)}\left(\frac{p}{\sqrt{2}}\right)^{(p-2)/(p-1)}\right)^{2(p-1)/(p-2)}},$
(10)

This $\mu$-dependent form of the small-field approximation does not lead to an
analytical $N_{*}$-scaling relation for either of the observables, which is
presumably why it has not been considered in the literature. However, we can
think of these expressions as functions that indicate a kind of “effective”
scaling relation with energy dependent exponents and amplitudes. This point
will turn out to be useful later in this paper to gain a qualitative
understanding of the $\mu$-dependence of hilltop inflation.

While neither $r$ nor $n_{{\cal R}{\cal R}}$ scale with $N_{*}$ directly in
the SF.II approximation, the tensor ratio does scale analytically with the
spectral tilt $\delta_{n}=1-n_{{\cal R}{\cal R}}$ as

$r(n_{{\cal R}{\cal R}},\mu)~{}=~{}\frac{8p^{2}\left(\frac{\mu}{M_{\rm
Pl}}\right)^{2p/(p-2)}}{(2p(p-1))^{2(p-1)/(p-2)}}~{}\delta_{n}^{2(p-1)/(p-2)}.$
(11)

This shows that even in the SF.II approximation the tensor ratio varies for
the class of hilltop models and that typicality statements made in the
literature about the behavior of the tensor ratio as a function of $n_{{\cal
R}{\cal R}}$ do not apply in general. The slow-roll form of this relation will
be discussed in section 4.

Our results below will show that while the SF.II approximation does capture
qualitatively certain features of hilltop inflation it is only a good
approximation in a limited parameter range that is not enforced by
observations. Furthermore, while in some parameter regions the small-field
approximation might be reasonable at the beginning of hilltop inflation it is
in general less so toward the end of inflation. If the $\mu$-terms in the
denominators are neglected the spectral index reduces in the resulting SF.III
approximation to the $\mu$-independent spectral index

$n_{{\cal R}{\cal R}}(p,N_{*})~{}=~{}1-\frac{2(p-1)}{(p-2)}\frac{1}{N_{*}}$
(12)

considered in the early hilltop papers [9, 10]. Similarly, in this limit the
tensor ratio reduces to the form discussed in [11]

$r(N_{*})~{}=~{}8p^{2}\left(\frac{\mu}{M_{\rm
Pl}}\right)^{2p/(p-2)}\frac{1}{\left(p(p-2)N_{*}\right)^{2(p-1)/(p-2)}}.$ (13)

In both relations the exponents are independent of the energy, which is not
the case in the slow-roll or the small field approximation SF.II.

##  3 Reheating constraint

While our main focus is in the energy dependence of the scaling behavior in
hilltop and hilltop-squared inflation, it is of interest to see how the
reheating constraints affect the distributions associated with the initial
conditions of the inflaton field. For this we consider the usual insertion of
phases between horizon crossing and today, in combination with the evolution
of the energy density during reheating and the conservation of entropy between
the end of reheating and today. The simplest analyses assume that the equation
of state parameter $w_{\rm rh}$ is constant, so that

$\rho_{\rm rh}~{}=~{}\rho_{e}\left(\frac{a_{e}}{a}\right)^{3(1+w_{\rm rh})},$
(14)

an assumption that is made in much of the literature. The detailed form of the
phase evolution analysis of the scale factor depends on how the expansion of
the $k_{*}/k_{0}$ is done but the end-result is essentially the same, see for
example [46, 47, 48, 49, 50]. For our purpose it is useful to evaluate the
resulting formula for the e-folds $N_{\rm rh}$ during reheating for potentials
of the type $V=\Lambda^{4}f(\phi/\mu)$, where $f$ is a dimensionless function.
The overall energy scale $\Lambda$ is determined by the CMB amplitude ${\cal
A}_{\cal R}$, leading to

$N_{\rm rh}~{}=~{}\frac{4}{1-3w_{\rm
rh}}\left(-N_{*}~{}+~{}\frac{1}{4}\ln\frac{(f^{\prime}_{*})^{2}}{f_{*}f_{e}}~{}+~{}\frac{1}{2}\ln\frac{M_{\rm
Pl}}{\mu}~{}+~{}\sigma_{\rm piv}\right),$ (15)

where $f_{*}$ is the dimensionless potential evaluated at $\phi_{*}$, $f_{e}$
is its value at the end-of-inflation, and $f^{\prime}$ is the dimensionless
derivative of $f$. The parameter $\sigma_{\rm piv}$ is a model independent
constant

$\sigma_{\rm
piv}~{}=~{}\ln\frac{a_{0}T_{0}}{k_{*}}~{}-~{}\frac{1}{3}\ln\frac{11g_{\rm
rh}}{43}-\frac{1}{4}\ln\frac{405}{\pi^{2}g_{\rm
rh}}+\frac{1}{4}\ln(12\pi^{2}{\cal A}_{\cal R}),$ (16)

determined by the CMB amplitude ${\cal A}_{\cal R}$, the pivot scale $k_{*}$,
as well as $g_{\rm rh}$ and $T_{0}$. For the reheating temperature we obtain
in terms of the dimensionless potential $f$

$T_{\rm rh}~{}=~{}M_{\rm Pl}\left(\frac{540}{g_{\rm rh}}\frac{M_{\rm
Pl}^{2}}{\mu^{2}}{\cal A}_{\cal
R}\frac{(f^{\prime}_{*})^{2}f_{e}}{f_{*}^{3}}\right)^{1/4}~{}e^{-\frac{3}{4}(1+w_{\rm
rh})N_{\rm rh}}.$ (17)

This compact form is convenient for computations even though it is written in
a slightly redundant form. The energy dependent scaling relations obtained
below for the observables lead to scaling behavior of $N_{\rm rh}$ and $T_{\rm
rh}$ in dependence of these variables.

##  4 Scaling behavior of hilltop inflation

Hilltop inflation has mostly been analyzed at low $p$ and low $\mu$. The model
at $p=4$ in particular has been discussed often in the literature, see e.g.
[9, 10, 11, 12, 2, 24, 25, 26, 29, 30, 32, 33], and has been adopted as a
bench-mark model by the CMB collaborations (see e.g. [3, 6, 4, 7, 8]). In this
section we investigate the impact of the CMB and reheating constraints more
generally for these models and in particular analyze the behavior of the
observables as the energy parameter $\mu$ is varied, similar to the analysis
in [42] in the context of multifield inflation. The resulting data will be
used in later sections for the scaling analyses.

We vary $p$ through a wide set of models, ranging from 3 to 100, and for each
$p$ consider a range of $\mu$ values that is compatible with the CMB
constraints on the spectral index $n_{{\cal R}{\cal R}}$ and the tensor-to-
scalar ratio $r$, and also lead to sufficient inflation $N_{*}\in[50,70]$.
This $\mu$-range increases as $p$ gets larger, both in the sub-Planckian and
the super-Planckian regime. For low $p$ in particular the models are not
consistent with the CMB and e-folds constraints for sub-Planckian values for
$\mu$, but these become viable as $p$ increases. We scan the complete
$\mu$-range for super-Planckian values in an integral parametrization
$\mu=mM_{\rm Pl}$, $m$ an integer, and impose for larger $p$ the generic
$p-$independent lower bound $\mu=10^{-2}M_{\rm Pl}$ for the sub-Planckian
range. This lower bound is sufficient to indicate the behavior of these
models, in particular toward the low end of the tensor ratio that are probed
by such $\mu$-values. The parametrization of these sub-Planckian ranges is via
rational values $\mu=M_{\rm Pl}/m$. The upper bound on $\mu$ grows linearly
with $p$ with a slope of slightly less than eight, while the full sub-
Planckian regime reaches for large $p$ far below the Planck scale. While there
is a weak correlation between the number of inflationary e-folds and $\mu$,
there is quite some degeneracy even for low $p$ and this degeneracy increases
for larger $p$. For the purpose of our discussion a full scan of $\mu$-range
is not necessary.

In Fig. 1 we illustrate the distributions mapped out by viable initial
conditions in the spectral-tensor plane considered by the experimental
collaborations. We adopt the values $n_{{\cal R}{\cal R}}\in[0.96,0.97]$ and
$r_{*}\leq 0.035$, as suggested by the CMB experiments [8]. The color coding
is blue for super-Planckian $\mu$-range and pure inflationary constraints,
while red indicates the results after taking into account the reheating
constraint $N_{\rm rh}$, which is implemented here for illustration with
$w_{\rm rh}=0$. Black is for $\mu$ in the sub-Planckian range with
inflationary constraints and yellow indicates the inclusion of the reheating
constraint. The first panel in Fig. 1 shows that the quartic hilltop model has
no viable initial values for sub-Planckian $\mu$-values, which also holds for
$p=3$. The ranges of the inflaton field in both models are such that these
models are no longer small-field inflation, given the current observational
constraints. As $p$ increases sub-Planckian energy values become possible and
the viable $\mu$-bounds extend both in the sub- and the super-Planckian
regime, leading to the regions in the parameter space in which these models
can describe small-field inflation.

> Fig. 1 The $(n_{{\cal R}{\cal R}},r)$ distributions for $p=4,7,10,20$ in the
> panels from the upper left to the lower right respectively. Here blue (red)
> indicates the super-Planckian $\mu$ without (with) the reheating constraint,
> while black (yellow) shows sub-Planckian domains without (with) the
> reheating constraint.

The overall shapes of the viable regions in the spectral-tensor plane of Fig.
1 are determined by a combination of constraints. The upper and lower boundary
curves of the distributions come from the lower and upper bound of the number
of inflationary e-folds. A smaller interval of allowed values $N_{*}$ would
lead to regions that are more narrow. The flat plateaus of the upper regions
of the distributions in Fig. 1 show that the Planck bound on $r$ constrains
these models for all $p$. Future experiments like the Simons Observatory [5],
CMBS-4 [6, 7] and LiteBird [51] aim to reach $r\cong 0.001$ and will put more
pressure on some of these models if gravitational waves are not found. In
particular the often considered quartic model could be excluded and the same
applies even more to the cubic hilltop model because its lower bound of $r$ is
higher than for the quartic model. For larger $p$ none of the upcoming CMB
experiments will be able to exclude these models based on their target values
of $r$ in the range of a milli, even if the energy scale $\mu$ is restricted
to super-Planckian values.

The results in Fig. 1 also show that for low $p$ hilltop models improved
constraints for the spectral index alone have the potential to significantly
reduce the viable region in the spectral-tensor plane, while for higher $p$
such constraints will have less impact since for larger $p$ the region in
these graphs covers the whole range of the spectral index, at least in the
tensor ratio range considered here for sub-Planckian energy scales $\mu$. This
pattern is stable when $p$ is increased to $p>20$ as far as the CMB
constraints above are concerned. Including the reheating constraint does not
change the viable range of $n_{{\cal R}{\cal R}}$ but does decrease the
regions dramatically by its effect on the tensor ratio. An improved accuracy
of the spectral index $n_{{\cal R}{\cal R}}$ could again limit the viability
of this class of models with a matter dominated reheating phase. The
combination of the results of the Planck probe with the upcoming large scale
survey data, such as Euclid and SKA, will therefore have a significant impact
on the hilltop models. A discussion of the uncertainties of different
combinations of these three experiments is for example given in ref. [52].

The $\mu$-stratifications of Fig. 1 show that in hilltop inflation the tensor-
to-scalar ratio $r$ follows scaling relations in dependence of $n_{{\cal
R}{\cal R}}$ and that the scaling parameters vary as the energy scale $\mu$ is
varied. This will be analyzed quantitatively in later sections by extending
and generalizing the strategy adopted in the context of multifield inflation
in ref [42]. Here we point to the fact that this energy dependence generalizes
and refines assumptions made in the literature that neglect a possible
variation of the amplitude and the exponents in the scaling relation [53, 54].
These results make explicit the difference between the strong small-field
approximation discussed earlier in this paper and the slow-roll approximation.
In the former the scaling exponents are energy independent, while Fig. 1
illustrates the significant variation of these parameters over the viable
parameter space. As a result of this energy dependence the parameter spaces of
the different hilltop models are constrained to a different degree, depending
on $p$. The $(r,n_{{\cal R}{\cal R}})$ scaling relations of Fig. 1 can be
viewed as a consequence of the energy dependent scaling relations of the
spectral tilt $\delta_{n}(N)$ and the tensor-ratio $r(N)$ as functions of the
number of inflationary e-folds, which will be discussed in the next section.

The $\mu$-stratification of the spectral-tensor plane also establishes
correlations between some observables and the energy parameter that defines
the deformation of the model. These are of different strength in that the
correlation, for example, between the energy $\mu$ and the tensor ratio is
quite strong, while the correlation between $\mu$ and the spectral index is
much weaker. Other correlations, such as the anti-correlation between the
number of inflationary e-folds $N_{*}$ and the reheating e-folds $N_{\rm rh}$
depend on the model, becoming weaker as $p$ increases. The bounds on these
e-folds are however stable and lead to upper bounds of about forty and about
sixty for the reheating e-folds and the inflationary e-folds, respectively.

##  5 Scaling distributions in hilltop inflation via $N_{*}$

Scaling functions in dependence of the model parameters can be used as
characteristics of the inflationary theory space, as discussed in the context
of multifield theories in [42]. While this is a particularly interesting tool
in multifield models with nontrivial potentials, even singlefield inflationary
theories are usually not solvable analytically. Making sufficiently strong
approximations to yield analytical methods leads to a loss of significant
features. To gain a broader perspective of scaling in inflation we analyze the
scaling behavior of hilltop inflation for both the spectral tilt $\delta_{n}$
and the tensor ratio $r$ as functions of the number of e-folds. We extend this
analysis for a wide range of models and a large range of $\mu$-values that
have been shown in the previous section to be viable (see e.g. the
distributions of the viable $\mu$-data in the $(n_{{\cal R}{\cal R}},r)$-plane
for different $p$ shown in Fig. 1).

In Fig. 2 we collect the distributions for $\delta_{n}(N,\mu)$ as $p$
increases and $\mu$ is varied over its viable range. The results show that
$\delta_{n}$ scales with $N$ in dependence of the energy scale $\mu$

$\delta_{n}(N,\mu)~{}=~{}\frac{\alpha_{\delta_{n}}(\mu)}{N^{\beta_{\delta_{n}}(\mu)}},$
(18)

where the existence of families of the energy dependent exponents extends the
limit defined by the strong small-field approximation SF.III with its energy
independent scaling exponent. Fig. 2 shows again that the range of viable
e-folds is dramatically affected by the reheating constraint.

> Fig. 2 Scaling distributions for $(\delta_{n},N_{*})$ in hilltop inflation
> models with $p=4,7,10,20$. The color scheme here is the same as in Fig. 1.

It was noted in the previous section that there is a weak correlation between
the energy scale $\mu$ and the number of inflationary e-folds $N_{*}$. In Fig.
2 this degeneracy can be seen explicitly for the different models and that the
distribution does become more narrow as $p$ increases.

In Fig. 3 we show the scaling distributions of the tensor-to-scalar ratio
$r(N,\mu)$ as $p$ and $\mu$ are varied on a logarithmic scale. The resulting
behavior

$r(N,\mu)~{}=~{}\frac{\alpha_{r}(\mu)}{N^{\beta_{r}(\mu)}}$ (19)

again shows an energy dependence of the exponents that goes beyond the small-
field approximation.

Fig. 3 The tensor ratio distribution $(r,N)$ for $p=4,7,10,20$.

The gap at low $r$ and low $N$ in the graphs of Fig.3 is quite pronounced for
small $p$ but becomes smaller as $p$ increases. This pattern continues for
$p>20$. Furthermore the viable regions including the reheating constraint
increase in particular for the sub-Planckian range of $\mu$.

The two scaling relations (18) and (19) of Figs. 2 and 3 explain the scaling
relations in Fig. 1, leading to a non-linear relation between the spectral
tilt and the tensor ratio. They also explain the scaling behavior of the
number of reheating e-folds and the reheating temperature as functions of the
spectral tilt $\delta_{n}$ or the tensor ratio $r$. This will be analyzed
quantitatively in the next section.

##  6 Scaling exponents in hilltop inflation

In this section we make the scaling behavior of the spectral tilt and the
tensor ratio of the hilltop potentials more precise. In Figs. 1, 2 and 3 the
scaling behavior of $r(\delta_{n})$, $\delta_{n}(N)$ and $r(N)$ is shown for
several hilltop models in a systematic way for the full super-Planckian range
of $\mu$-values as well as a partial range of sub-Planckian values. The
resulting stratifications in these distributions suggest a scaling behavior of
the form $\delta_{n}(N)=\alpha_{\delta}N^{-\beta_{\delta}}$ and
$r(N)=\alpha_{r}N^{-\beta_{r}}$, with exponents $\beta$ and amplitudes
$\alpha$ that depend on the energy scale $\mu$. Hence we obtain for a given
model a family of exponents for each of these observables. This is
qualitatively similar to the multifield behavior considered in [42]. In the
present discussion we will determine the resulting exponents $\beta$
explicitly in order to be able to compare these characteristic numbers of
hilltop inflation with those of other models and place this class in the
inflationary landscape. The results below can be compared to the small
$\mu$-limit of the small-field approximation prediction where both exponents
are constants for a fixed model. In the slow-roll approximation these families
of exponents instead lead to a non-trivial functional behavior.

###  6.1 Scaling exponents for the quartic hilltop model

As noted above, the hilltop model at $p=4$ has been discussed extensively in
the literature, see e.g. [9, 10, 11, 12, 16, 2, 24, 25, 26, 29, 30, 32, 33],
and has been adopted as a bench-mark model by the CMB collaborations (see e.g.
[3, 4, 7]). We therefore begin our analysis with this model for the most
general range of $\mu$. In the previous section we have seen that this model
leads to viable initial values compatible with the CMB results only for super-
Planckian $\mu$ values in a fairly narrow range. As a consequence parts of the
analyses of some of the earlier references are no longer applicable.

The distribution in the spectral-tensor plane is shown in the previous section
in Fig. 1, both with and without the reheating constraint. In the present
discussion we focus on the scaling behavior of the observables as functions of
$N$. Fig. 2 illustrates again that the spectral tilt $\delta_{n}=1-n_{{\cal
R}{\cal R}}$ of the spectral index away from the Harrison-Zeldovich spectrum
$n_{{\cal R}{\cal R}}=1$ shows for the quartic hilltop model scaling behavior
in dependence of the inflationary e-fold number $N_{*}$ as indicated above in
eq. (18). In Fig. 4 we make this scaling relation more explicit for several
curves describing $\delta_{n}(N)$ for the hilltop model at $p=4$.

> Fig. 4 Scaling behavior of the spectral tilt $\delta_{n}(N,\mu)$ for the
> selection of $\mu$ of Table 1. Here the red sections along the curves cover
> the range that is compatible with the CMB and the $e-$fold constraints.

The curves in Fig. 4 lead to the exponents collected in Table 1.

$\mu$ | 5 | 10 | 15 | 20 | 25 | 30
---|---|---|---|---|---|---
$\beta_{\delta_{n}}(\mu)$ | 0.924 | 0.838 | 0.833 | 0.86 | 0.886 | 0.906 A

> Table 1. Characteristic exponents $\beta_{\delta_{n}}(\mu)$ for the spectral
> tilt $\delta_{n}(N)$ in the quartic hilltop model.

The results of Fig. 4 and Table 1 show an energy dependent scaling exponent
$\beta_{\delta_{n}}(\mu)$, which for small $\mu$ approaches the limiting value
of the SF.III approximation, given by $\beta_{\delta_{n}}=1$, while the
amplitude is in this limit given by $2(p-1)/(p-2)=3$. Roughly speaking, the
slow-roll approximation leads to a scaling behavior for $p=4$ hilltop of the
form $\delta_{n}(N)\sim N^{-0.8*}$ in the $\mu$-range indicated in the above
Table 1.

For the tensor ratio $r$ the first panel in Fig. 3 shows a similar power law
scaling behavior $r(N)~{}=~{}\alpha_{r}N^{-\beta_{r}}$ for the quartic hilltop
model and a similar analysis leads to scaling functions that for a selection
of $\mu$-scales are illustrated in Fig. 5.

> Fig. 5. Scaling behavior of the tensor ratio $r(N)$ curves for quartic
> hilltop inflation.

The scaling exponents $\beta_{r}$ associated to the curves of $r(N)$ in Fig. 5
lead to the results in Table 2.

$\mu/M_{\rm Pl}$ | 5 | 10 | 15 | 20 | 25 | 30
---|---|---|---|---|---|---
$\beta_{r}(\mu)$ | 2.748 | 2.268 | 1.897 | 1.667 | 1.532 | 1.446 A

Table 2. The scaling exponents of the tensor ratio $r(N)$ at $p=4$.

This can be compared to the analytical tensor ratio scaling behavior obtained
in the SF.III approximation. In this approximation the scaling exponent
$\beta_{r}$ is independent of $\mu$ and its value is given by $\beta_{r}^{{\rm
III}}=2(p-1)/(p-2)=3$. The above results show the $\mu$-dependence of the
scaling exponents $\beta_{r}(\mu)$ and that for small $\mu$ they approach the
value of the strong small field approximation SF.III.

With the scaling relations $\delta_{n}(N)$ and $r(N)$ determined above we can
also eliminate $N$ in the second by using the tilt scaling to obtain a scaling
relation of the form $r=r(\delta_{n})$. The resulting exponents $\beta_{r}$
and $\beta_{\delta_{n}}$ can be compared with the analytical scaling relation
determined in the SF.II approximation in eq. (11). Alternatively, one can
analyze the distributions of Fig. 1 directly in the same way as done above for
the $N$-scaling functions. This places $r(\delta_{n})$ directly in spectral-
tensor plane and with a measurement of $r$ would allow tests not only of the
model but also of the parameter space spanned by $\mu$.

The post-inflationary evolution constraint is illustrated in this paper with a
matter dominated reheating phase. The implications of the reheating constraint
for the different models are shown in the first three figures for both the
super-Planckian and the sub-Planckian energy scales $\mu$. For hilltop
inflation with $p=4$ the reheating e-fold range is with our run-parameters
roughly $\Delta N_{\rm rh}=40$ while the reheating temperature $T_{\rm rh}$ is
roughly between $10^{3}{\rm GeV}$ and $10^{15}{\rm GeV}$.

###  6.2 Behavior of hilltop inflation at $p=10$

For $p\geq 5$ the range of viable energies $\mu$ includes the sub-Planckian
regime. In the present section we briefly summarize here the scaling results
at $p=10$, which are obtained in the same way as described in detail in the
previous subsection for $p=4$. For this model the sub-Planckian values for the
energy scale $\mu$ lead to a much deeper range for the tensor ratio.

The distribution $(\delta_{n},N)$ for all viable initial values at $p=10$
takes the form shown in Fig. 2. Similar to the case $p=4$, the distribution at
$p=10$ is again formed by a set of curves (that are described by a scaling
relation in dependence of $\mu$ of the type in eq. (19). The scaling curves
for a selection of energy scales $\mu$ are collected in Fig. 6.

Fig. 6 Scaling behavior of $\delta_{n}(N)$ for $p=10$ hilltop inflation.

The scaling exponents for the curves in Fig. 6 can be found in Table 3. The
values obtained can again be compared to those of the small-field
approximation, where the amplitude evaluates for $p=10$ to $\alpha_{10}=9/4$.

$\mu/M_{\rm Pl}$ | 1 | 20 | 40 | 60 | 80
---|---|---|---|---|---
$\beta_{\delta_{n}}(\mu)$ | 0.998 | 0.922 | 0.916 | 0.934 | 0.947 A

Table 3. Scaling exponents $\beta_{p}$ for $\delta_{n}$ in hilltop inflation
at $p=10$.

We consider again the behavior of the tensor-to-scalar ratio $r$, for which
Fig. 7 shows the dependence on the inflationary number of e-folds for
$r(N_{*})$ for several energy values $\mu$ extracted from the distribution of
$\mu$-family curves in Fig. 3.

Fig. 7 Scaling behavior of $r(N)$ at $p=10$.

The scaling curves in Fig. 7 lead to the scaling exponents compiled in Table
4. These results can again be compared to the analytical tensor ratio scaling
behavior that can be obtained in the SF.III approximation. The latter predicts
that $\beta_{r}^{\rm III}=2.25$.

$\mu/M_{\rm Pl}$ | 1 | 20 | 40 | 60 | 80
---|---|---|---|---|---
$\beta_{r}(\mu)$ | 2.24 | 1.92 | 1.62 | 1.46 | 1.37 A

Table 4. The scaling exponents of $r(N)$ for $p=10$ hilltop inflation.

The reheating parameters $N_{\rm rh}$ and $T_{\rm rh}$ for the $p=10$ hilltop
model are comparable to the values for the quartic hilltop model.

As the exponent $p$ of the hilltop models increases the pattern of the
distributions in Figure 1 is stable in that the spectral-tensor plane region
that is viable under CMB + e-fold constraints remains saturated as for the
$p=20$ model. Furthermore, the range of the scaling exponents decreases and
remains within the viable region for larger ranges for the energy scale $\mu$.

The above results for the hilltop models at $p=4$ and $p=10$ already indicate
that these two inflationary scenarios can be characterized by the ranges they
define for the scaling exponents of the spectral tilt and the tensor ratio.
This will be discussed in more detail in section 9, after the analysis of the
hilltop-squared models.

###  6.3 Implications

The energy dependence of the $N-$scaling behavior of any observable

${\cal O}(N)~{}=~{}\alpha_{\cal O}N^{-\beta_{\cal O}},$ (20)

can be used to determine how the uncertainties of parameters describing the
inflationary and reheating phases are affected by varying $\mu$. Denoting
these parameters by ${\cal P}$, for example the e-folds of the inflationary
and reheating phases, and the reheating temperature, their uncertainty is
determined by the uncertainties $\Delta{\cal O}$ associated with either the
measured values of ${\cal O}$, or the uncertainties of fiducial values
predicted for future experiments. By inverting (20) the induced uncertainty of
the inflationary number of e-folds is determined by

$\Delta N~{}=~{}-\frac{\alpha_{\cal O}^{1/\beta_{\cal O}}}{\beta_{\cal
O}}\frac{\Delta{\cal O}}{{\cal O}^{1+\frac{1}{\beta_{\cal O}}}}$ (21)

which depends on $\mu$ via the energy dependence of the amplitude
$\alpha_{\cal O}(\mu)$ and the exponent $\beta_{\cal O}(\mu)$. This in turn
can be used to trace the implications of the uncertainties of ${\cal O}$ for
$\Delta T_{\rm rh}$. To do so it is useful to make the dependence of $T_{\rm
rh}$ on the number of inflationary e-folds explicit by writing eq. (17) for
the reheating temperature via eq. (15) for the number of reheating e-folds as

$T_{\rm rh}~{}=~{}\kappa F(\phi_{*})\exp\left(\frac{3(1+w_{\rm
rh})}{(1-3w_{\rm rh}}N_{*}\right),$ (22)

where

$F(\phi_{*})~{}=~{}\left(\frac{f_{*}^{3w_{\rm
rh}}}{(f^{\prime}_{*})^{1+3w_{\rm rh}}}\right)^{1/(1-3w_{\rm rh})}$ (23)

and $\kappa$ collects all the remaining terms that are independent of
$\phi_{*}$ and $N_{*}$. With these ingredients we obtain

$\frac{\Delta T_{\rm rh}}{T_{\rm rh}}~{}=~{}\left(\frac{3(1+w_{\rm
rh})}{1-3w_{\rm rh}}~{}+~{}\frac{F^{\prime}_{*}}{F_{*}}\right)\Delta N_{*}.$
(24)

where here the dimensionless derivative is relative to the number of e-folds
$F^{\prime}=dF/dN$. The energy dependence of the scaling amplitude
$\alpha_{\cal O}(\mu)$ and the exponent $\beta_{\cal O}(\mu)$ therefore leads
to a $\mu$-dependence of these uncertainties. Using for example the estimates
of the CMB-S4 and the Simons collaborations [6, 7, 5] this leads to
uncertainties induced by the spectral index and the tensor ratio.

##  7 Bifurcation among $p$-families in hilltop inflation

In the previous sections we have analyzed the behavior of the spectral tilt
$\delta_{n}$ and the tensor-to-scalar ratio $r$ of hilltop models in
dependence of model parameters $p$ and $\mu$, as well as the number of e-folds
$N$, leading to functions on the theory space. Our strategy has been to
consider a wide range of models and determine the energy ranges that are
compatible with the constraints provided by the CMB data and the e-fold
constraints. This leads to energy dependent scaling exponents that we interpet
as diagnostic tools to characterize inflationary models.

An alternative view of the class hilltop models can be obtained by varying the
models systematically by letting $p$ run while keeping $\mu$ and $N_{*}$
fixed, leading to a $p$-family. By subsequently varying $\mu$ we obtain a set
of $p$-families in the spectral-tensor plane, as shown in Fig. 8. The strategy
here is to determine $\phi_{e}$ in the slow-roll approximation as the solution
of the degree $p$-constraint of eq. (5) and to determine $\phi_{*}(N,p)$ via
eq. (6). This type of analysis gives further insight into the differences
between the hilltop models and it also illustrates the energy dependence of
the observables described earlier in this paper in a different way.

###  7.1 The bifurcation phenomenon for $p$-families

Naively, one might have expected that as $\mu$ increases the $p$-family curve
becomes ever steeper in the spectral-tensor plane, perhaps approaching the
vertical line. This turns out not to be the case. Instead, when changing the
$\mu$-parameter we find that at some “critical value” $\mu_{c}$ a bifurcation
arises in the plane spanned by the spectral index and the tensor ratio. For
low $\mu$ the trajectories begin at low $n_{{\cal R}{\cal R}}$ for low
$p$-values, with increasing $n_{{\cal R}{\cal R}}$ as $p$ increases. Beyond
the critical value of $\mu$ the $p$-families begin for small $p$ at large
$n_{{\cal R}{\cal R}}$, with decreasing $n_{{\cal R}{\cal R}}$ as $p$
increases. This is illustrated in Fig. 8, which also contains an overlay of
the forecasts of the Simons Observatory to illustrate how different models are
affected by experimental constraints. The precise location of the bifurcation
in the plane depends on the choice of the e-folds. In a first approximation
this location is clear from the approximation of the spectral index in the
form of eq. (12), which ignores the $\phi_{e}$ dependence of the e-folds. In
this approximation the large $p$-limit approaches the spectral index $n_{{\cal
R}{\cal R}}=1-2/N_{*}$.

> Fig. 8 A selection of $p$-families in the $(n_{{\cal R}{\cal R}},r)$-plane
> for different energy scales $\mu/M_{\rm Pl}$ values between one (purple, low
> left corner) and one-hundred (in black, upper right corner). The forecasts
> here are by the Simons Observatory collaboration [5] for different
> scenarios.

The structure of the $p-$families in Fig. 8 illustrates in a different way the
results of the scan analysis earlier in this paper, in particular the fact
that for any super-Planckian energy scale $\mu$ there are hilltop models that
are compatible with the CMB and lead to sufficient inflation. It also
illustrates that for low $p$-models there is a cut-off in the super-Planckian
regime below which no viable models can be found. This includes in particular
the quartic model for which sub-Planckian energy is ruled out.

###  7.2 Small-field structure of $p$-families

It is clear that the bifurcation of the $p$-families in the spectral-tensor
plane cannot be understood in the approximation of hilltop inflation in which
the dependence of the number of e-folds on the field $\phi_{e}$ at the end of
inflation is neglected. This approximation leads to the spectral index (12)
and the tensor-ratio (13) considered in the early literature [9, 10, 11]. The
reason why the $\phi_{e}$-terms have often been neglected is because the focus
was on the low range of the energy scale $\mu$, in which case the
$N$-independent terms in the denominators of the small-field approximation are
small and it is justified to neglect them. An unusual feature of this
procedure is that the spectral index does not depend on the energy scale.
While it is usually the case that $n_{{\cal R}{\cal R}}$ does not depend on
the overall energy scale $\Lambda$, it does in general depend on the remaining
parameters of the theory, and we see here that in the less severe SF.II
approximation this is also the case for hilltop inflation.

To illustrate that the SF.II approximation does capture the bifurcation we
consider in Fig. 9 several $p$-family curves in this approximation for several
$\mu$-values between 1 and 200 $M_{\rm Pl}$. This graph is truncated compared
to the numerical slow-roll graph in Fig. 8 because of the $p$-dependent
consistency constraints on the energy scale $\mu$ in this approximation.
Nevertheless, it does show that the bifurcation phenomenon can be captured by
the SF.II approximation even though the detailed numerics of the trajectories
is of course different in pure slow-roll as compared to the SF.II
approximation.

> Fig. 9 An illustration of the SF.II approximation of the bifurcation
> phenomenon in hilltop inflation for energy scales between 5$M_{\rm Pl}$ and
> 100 $M_{\rm Pl}$ with $p$ running for $p>p_{\mu}$, where $p_{\mu}$ is the
> consistency bound.

##  8 Hilltop-squared inflation

We have noted already that hilltop inflation $V_{p,n}$ as defined in eq. (1)
with $n=1$ does not contain a minimum valley in which the inflaton can
oscillate during reheating. While this is in principle easy to fix with a
small deformation of the potential close to its zero, this feature has
motivated the consideration of more global changes of the hilltop models, such
as hilltop-squared inflation (HSI), with potentials of the type considered in
eq. (1) with $n=2$ [23]. Higher exponents $n$ have been considered in [26, 32,
34, 35].

In the present section we extend our analysis of the scaling behavior of the
spectral tilt and the tensor ratio considered above for hilltop inflation to
potentials of the type

$V_{p,2}~{}=~{}\Lambda^{4}\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)^{2}.$
(25)

We focus in particular on the phenomenological aspects and the scaling
behavior of these models. We find that the scaling results for the hilltop-
squared models differ from those of the hilltop models, but that they follow a
similar overall pattern.

###  8.1 Observables in hilltop-squared inflation

The spectral index of the scalar perturbations for the hilltop-squared
inflation models (25) is given by

$n_{{\cal R}{\cal R}}^{\rm HSI}~{}=~{}1-\frac{4p\frac{M_{\rm
Pl}^{2}}{\mu^{2}}}{\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)^{2}}\left(\frac{\phi}{\mu}\right)^{p-2}\left[(p-1)+(p+1)\left(\frac{\phi}{\mu}\right)^{p}\right]$
(26)

and the tensor ratio $r^{\rm HSI}$ relates to the hilltop tensor ratio as

$r^{\rm HSI}(\phi)~{}=~{}32p^{2}\frac{M_{\rm
Pl}^{2}}{\mu^{2}}\frac{\left(\frac{\phi}{\mu}\right)^{2(p-1)}}{\left(1-\left(\frac{\phi}{\mu}\right)^{p}\right)^{2}}~{}=~{}4r.$
(27)

The inflaton field at the end of inflation $\phi_{e}$ is determined in HSIp by

$x_{e}^{p}~{}+~{}\sqrt{2}p\frac{M_{\rm Pl}}{\mu}x_{e}^{p-1}~{}-~{}1~{}=~{}0,$
(28)

where $x_{e}=\phi_{e}/\mu$. The number of e-folds $N_{*}^{\rm HSI}$ is related
to that of hilltop inflation as

$N_{*}^{\rm HSI}~{}=~{}\frac{1}{2}N_{*}^{\rm HI}.$ (29)

The reheating e-fold number $N_{\rm rh}$ and the reheating temperature $T_{\rm
rh}$ can be obtained from the general formulae given above in (15) and (17)
via the dimensionless hilltop-squared potential.

###  8.2 Distribution of $(r,n_{{\cal R}{\cal R}})$ in hilltop-squared
inflation

In the present subsection we consider a sample of HSIp models for the same
range of $p$-values considered above for hilltop inflation and with the same
CMB and reheating constraints so as to be consistent with the Planck results
for the observables. The number of e-folds is again in the canonical interval
$[50,70]$. Fig. 10 shows the distribution in the spectral-tensor plane with
the same color code that was used in Fig. 1 for the hilltop models.

> Fig. 10 The spectral-tensor distributions for hilltop-squared models with
> $p=4,7,10,20$ from top left to bottom right. For $p=4,7$ the reheating
> constraint rules out sub-Planckian $\mu$.

These results imply that hilltop-squared inflation is not small-field
inflation in general in the sense that for some $p$-models sub-Planckian field
values are not compatible with the CMB constraints. This is similar to the
behavior of the hilltop models discussed above. The panel in Fig. 10 for
hilltop-squared at $p=4$ shows in particular that the allowed range in the
spectral-tensor plane is quite small and that there are in particular no sub-
Planckian energies $\mu$ that lead to viable initial values. One can check
that this implies that the inflaton values $\phi_{*}$ for this model are all
super-Planckian and hence hilltop-squared inflation at $p=4$ is not small-
field inflation. The same holds for the $p=3$ model.

###  8.3 Scaling distributions for hilltop-squared observables via $N$

The behavior of the spectral tilt $\delta_{n}$ is of interest both from a
theoretical and a practical perspective, as discussed above in the context of
the hilltop models. Fig. 11 shows the scaling distributions for the spectral
tilt $\delta_{n}$ for different $p$-models in hilltop-squared inflation. Like
in hilltop inflation these distributions show a weak correlation between the
energy scales $\mu$ and the number of inflationary e-folds.

> Fig. 11 $\delta_{n}(N)$ distribution for $p=4,7,10,20$. The color coding is
> the same as in the hilltop inflation distributions. For $p=4$ there are no
> contributions from sub-Planckian $\mu$, hence there are no black and yellow
> regions in the corresponding panel.

###  8.4 $r(N)$ scaling in hilltop-squared inflation

The behavior of the tensor-to-scalar ratio $r$ as a function of the number of
inflationary e-folds $N_{*}$ in hilltop-squared inflation is similar to that
of hilltop inflation. Fig. 12 illustrates the $(r,N)$ scaling relations for
several hilltop models. As in the case of the hilltop models the viable region
for $p=4$ is much smaller than that of the higher $p$ models.

Fig. 12 Distributions of $(r,N)$ for $p=4,7,10,20$.

###  8.5 Scaling exponents in hilltop-squared inflation

The analysis of the scaling behavior in hilltop-squared inflation follows the
same strategy as in hilltop inflation. The structure of the two classes of
models is similar although the numerical results differ. We have analyzed a
wide range of models between $p=3$ and $p=100$, based on the distributions
discussed in the previous sections, and extensions thereof, but here we focus
on the $p=4$ hilltop-squared model to illustrate the results. The distribution
in the spectral-tensor plane shown in Fig. 12 establishes the regions that are
viable given the CMB inflationary constraints with or without the reheating
constraint. In the following we focus on the scaling exponents of the tensor
ratio in dependence of the number of e-folds, shown for several models in Fig.
12 as a function of the energy scale $\mu$. Some aspects of this model other
than scaling have also been discussed in [32]. In Fig. 13 we select several
curves from the distribution shown in the $p=4$ panel of Fig. 12.

> Fig. 13 A selection of $r(N)$ curves for hilltop-squared inflation at $p=4$.

The scaling exponents for the curves in Fig. 13 lead to the results listed in
Table 5.

$\mu$ | 5 | 10 | 20 | 25 | 30
---|---|---|---|---|---
$\beta_{r}(\mu)$ | 2.859 | 2.546 | 1.958 | 1.798 | 1.687 A

Table 5. Scaling exponents $\beta_{r}(\mu)$ for at $p=4$ HSI.

For hilltop-squared models the range of the scaling exponents becomes more
narrow as $p$ increases because the upper boundary decreases and the lower
boundary changes only slightly.

##  9 Scaling exponents as model characteristics

In the previous sections we have determined the scaling exponents of the two
main observables $n_{{\cal R}{\cal R}},r$ as functions of the number of
inflationary e-folds in both hilltop and hilltop-squared inflation models.
These power law scaling exponents depend on the energy parameter $\mu$ of the
models, leading for a given model to functions $\beta_{\delta}(\mu)$ and
$\beta_{r}(\mu)$. More generally, we can think of observables ${\cal O}(N)$
and their associated exponents $\beta_{\cal O}(\mu)$ as functions on the model
parameter space. These energy scales are constrained, depending on the
conditions imposed on the observables. If one is interested in what present
and future experiments can observe then these constraints include those
obtained from CMB and LSS data, assumptions about the number of inflationary
e-folds $N_{*}$ and, in principle, assumptions about the reheating period and
other constraints. Given such a set of $\mu$-dependent scaling exponents for
an observable ${\cal O}$ it is natural to ask how they characterize different
models. The discussion above suggests that this can be achieved in different
ways. An immediate comparison of models can be obtained by comparing the
ranges of the exponents for different models, leading to what we will call
“characteristic intervals”. A second way is to focus on the functional form of
the scaling exponents, leading to “characteristic functions”.

###  9.1 Characteristic intervals

One possibility to distinguish different inflationary models in an observable-
based way is by comparing the characteristic intervals obtained by imposing
the CMB constraints and the number of inflationary e-folds. The results above
of the analyses of hilltop and hilltop-squared inflation show that for both
types of models the strong small-field approximation SF.III leads to an upper
bound of the corresponding characteristic intervals. For small $p$ this upper
limit is quite different from the actual upper limit, but for larger $p$ the
energy independent SF.III exponent $\beta_{r}^{\rm III}=2(p-1)/(p-2)$ is quite
close to the slow-roll upper bound of the tensor ratio scaling $r(N)$.

In order to gain perspective it is useful to compare the hilltop and hilltop-
squared results above with results obtained in other models. In ref. [42] a
scaling analysis of the type considered in the present paper was constructed
for two modular inflation models, theories that are based on softly broken
groups ${\rm SL}(2,{\rm I\kern-1.79993ptR})$, associated to a hyperbolic
target space. In these models the potentials are invariant under discrete
subgroups of the Möbius group and the theories have two energy scales
$(\Lambda,\mu)$, much like the hilltop and hilltop-squared families. The
observables $n_{{\cal R}{\cal R}}$ and $r$ can be viewed as functions of the
number of e-folds and show scaling behavior that is also energy dependent. The
resulting exponents again sweep out characteristic intervals and hence can be
compared with the results obtained in the present paper. The fact that the
$\beta_{r}^{\rm III}$ define the upper limit for the energy dependent
exponents immediately implies that the hilltop and hilltop-squared models
define scaling classes that are distinct from the scaling classes defined by
modular inflation models considered in [42] because the characteristic
intervals of hilltop and hilltop-squared inflation have no overlap with the
models considered there. This shows that characteristic intervals provide a
useful diagnostic for both singlefield and twofield inflation.

The comparison of the characteristic intervals of the two hilltop classes on
the other hand shows that they are similar, with a somewhat higher lower bound
of $\beta_{r}(\mu)$ for hilltop-squared models as compared to hilltop models.
Roughly speaking, the distribution of these values in the hilltop-squared
class is more narrow than in hilltop inflation.

###  9.2 Characteristic functions of scaling exponents

A second defining feature that can be considered, once the distribution of the
scaling exponents $\beta_{\cal O}(\mu)$ of an observable ${\cal O}$ is known,
is their functional form. Here we focus on the tensor-to-scalar ratio for
hilltop and hilltop-squared inflation, similar to the analysis considered for
modular inflation in ref. [42]. In both HI and HSI inflation it is tempting to
use as a guide for the functional fit the fact that for large $p$ the
exponents $\beta(\mu)$ are approximately flat because for large $p$ these
models behave similar to the strong small-field approximation SF.III and in
this approximation there is no $\mu$-dependence. This suggests a linear ansatz
for the $\beta(\mu)$ as a comparison to the horizontal line determined by
SF.III. It turns out however that this is not a good approximation in general,
as seen already earlier in the analyses above.

We make the functional $\mu$-dependence of the tensor scaling exponents
explicit in Fig. 14 for the classes of hilltop and hilltop-squared inflation
for six different models between $p=3$ and $p=100$ and an energy range
$\mu\leq 150M_{\rm Pl}$. The grey bands indicate the characteristic intervals
of the approximate range of exponents for which the models are compatible with
the CMB and e-fold constraints. For small $p$ these $\mu$ have a lower bound
that is super-Planckian, hence for these models the exponent functions leave
this CMB region at smaller values of $\mu$ and again for larger $\mu$,
depending on $p$. For larger $p$ the $\mu$ range reaches down quite far into
the sub-Planckian regime and the functional form of scaling exponents
$\beta_{r}(\mu)$ becomes flatter as the viable $\mu$-range expands. It is only
for the largest $p$ models considered here that the exponents remain
completely within the characteristic band in the whole range of $\mu$
considered in the graphs.

The behavior of the $r(N)$-exponents in Fig. 14 for hilltop and hilltop-
squared models shows that for the complete range of $\mu$-values, irrespective
of the precise observational constraints, the functions $\beta_{r}(\mu)$ can
be described by higher degree polynomials for low $p$, with the degrees
decreasing as the model exponents $p$ increase. These polynomial fits for the
$\beta_{r}(\mu)$ thus provide intrinsic characteristic functions of the
models. In the parameter ranges that are viable relative to the CMB and e-fold
constraints the functions $\beta_{r}(\mu)$ become simpler in the sense that
fewer parameters suffice for a good description.

> Fig. 14 The functional dependence of the scaling exponents $\beta_{r}$ of
> the tensor ratio $r(N)$ on the energy scale $\mu$ at $p=3,4,7,10,20,100$ for
> hilltop (top panel) and hilltop-squared inflation (lower panel). The grey
> region indicates the values for which the different models are compatible
> with the CMB and e-fold based constraints.

The results obtained here for hilltop and hilltop-squared inflation can now be
compared with the results obtained in [42] for modular inflation models. For
those models the $\mu$ range was restricted to the super-Planckian range
$\mu>10M_{\rm Pl}$ and the models showed a simple scaling-of-scaling behavior
with exponents $\delta_{r}$ that were very similar, both close to
$\delta_{r}\cong 2$. In order to compare the hilltop models to these modular
inflation models it is useful to restrict them to the same energy range. This
leads again to an approximate simple scaling behavior with exponents
$\delta_{r}$ in dependence of $p$. These hilltop scaling exponents are much
smaller than 2, establishing that the resulting scaling-of-scaling behavior
can be used as a classification tool that distinguishes between these two
different classes of models. The advantage of this analysis is that a single
numerical exponent allows to characterize a model and its parameter space.

##  10 Conclusions

In this paper we have shown how the energy dependent scaling exponents
$\beta_{\cal O}(\mu)$ associated to observables ${\cal O}(N)$ as functions of
the number $N$ of inflationary e-folds can be used to characterize the classes
of hilltop and hilltop-squared inflation. Our focus has been on the spectral
tilt $\delta_{n}$ and the tensor-to-scalar ratio $r$, leading to an analysis
of the behavior of the scaling exponents $\beta_{\delta_{n}}(\mu)$ and
$\beta_{r}(\mu)$ of $\delta_{n}(N)$ and $r(N)$. Here the energy scale $\mu$ is
constrained by the conditions imposed, which include the CMB and the e-fold
constraint, but may also include other restrictions, for example the reheating
constraint. In the classes of hilltop and hilltop-squared inflation these
constraints lead to $\mu$-ranges that increase as the exponent $p$ is
increased, leading to increasing domains of functions $\beta_{\cal O}$. The
resulting scaling exponents $\beta_{r}(\mu)$ then lead to two different modes
in which models can be classified, first via their characteristic intervals,
and second via the functional types of the scaling exponents.

The scaling results for the hilltop and hilltop-squared for $\beta_{\cal
O}(\mu)$ are quite similar, as expected, indicating that the differences seen
by the inflaton toward the end of inflation do not lead to significant changes
in the present context. The behavior of these two classes is however quite
different from that of the modular inflation models considered in [42]. Using
the characteristic interval as a diagnostic tool shows that the modular
inflation models are quite distinct from hilltop inflation and hilltop-squared
inflation since the tensor exponents of the former have no overlap with the
tensor exponents of the latter. Hence these models belong to different regions
of the inflationary landscape. The functional behavior of hilltop and hilltop-
squared models is also quite similar, but again quite different from that of
the modular models. Comparing modular inflation with hilltop/hilltop-squared
inflation shows that the CMB + e-fold compatible range of $\mu$ for hilltop
and hilltop-squared inflation is much larger than for the modular inflation
models. As discussed in section 9, for the full viable range the functional
behavior of the exponents depends quite dramatically on the model parameter
$p$. In order to compare HI and HSI models with modular inflation it is
necessary to restrict the energy scale to about $\mu\geq 10M_{\rm Pl}$. In
this range the functional behavior of HI and HSI simplifies and can be
approximated as a power law scaling relation, leading different inflationary
scaling regimes.

An immediate implication of the energy dependence of the scaling exponents of
both $\delta_{n}(N)$ and $r(N)$ is that this leads to an energy dependent
scaling behavior for $r(\delta_{n})$, leading to a family of curves in the
spectral-tensor plane. This will have consequences for the type of reheating
analysis considered for example in [55].

The fact that hilltop and hilltop-squared models lead to scaling regimes that
are different from that of the modular inflation models considered in [42]
raises the question whether the scaling diagnostics considered here generalize
to other models. It would be interesting to place other inflationary models in
the scaling framework considered here, either for singlefield inflationary
models or for theories with more than one inflaton field. Analyzing models
such as those considered in the singlefield encyclopedia [2], or multifield
models, including for example the models considered in the references [56, 57,
58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68], will provide further insight into
the scaling structure of the inflationary landscape.

Acknowledgement.
This work was supported in part by a sabbatical leave of absence. It is a
pleasure to thank Ali Ishaq and Vipul Periwal for discussions and
correspondence.

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