# A generalized Bayesian framework for the analysis of subscription based
businesses

Rahul Madhavan Ankit Baraskar

Atidiv
Pune, IN-411014
<EMAIL_ADDRESS>

###### Abstract

We have created a framework for analyzing subscription based businesses in
terms of a unified metric which we call SCV (single customer value). The major
advance in this paper is to model customer churn as an exponential decay
variable, which directly follows from experimental data relating to
subscription based businesses. This Bayesian probabilistic model was used to
compute an expected value for the revenue contribution of a single user. We
obtain an exact closed-form solution for the constant churn model, and an
approximate closed-form solution for the exponential decay model. In addition,
we define a general methodology for decision making processes using
sensitivity analysis of the model equation, which we illustrate with a real-
life case study for a food based subscription business.

##  1 Introduction

Subscription based businesses (referred to as SBS henceforth) have been around
for quite some time - newspapers being one of the very first of these, where a
subscription entitles you to the daily news. Post-paid mobile service was one
of the early instances of an SBS delivering a service rather than a product on
a subscription basis.

Some of the largest businesses today are exploring the subscription based
model - Amazon is attempting to make grocery subscription commonplace. Netflix
is fast becoming the alternative television, with Amazon prime video, Hulu and
other services vying for a share of this new segment.

One of the reasons for the popularity of an SBS is its guaranteed revenue
stream per customer - much like rent in real estate, the company is guaranteed
its returns till the customer quits, also referred to as a customer churning.
Thus, for an existing SBS, there are two primary ways to increase its revenue
- customer acquisition and customer retention. In this paper, we model both of
these processes and explore how the interplay between them can be optimized
for overall revenue by addressing a single metric - the Single Customer Value
(SCV).

##  2 Motivation

###  2.1 Need for a single customer-based metric

KPI alignment of individual functions towards overall company metrics is a
largely un-addressed area for many companies. Individual functions within
companies tend to optimize their own goals, often to the detriment of overall
company progress. The need for alignment of divisions and functions towards an
overall framework drives our idea of a single customer score.

Here, we outline an equation that models the business process for an SBS in
terms of both internal as well as any external factors known about customers.
This in turn creates a framework where we then have a single metric that we
are optimizing for. An optimization differential equation within this
framework leads us to the most important action points for the process as a
whole, and therefore the business.

###  2.2 Comparisons to existing models

Most major attempts to attack complete business modeling have come from the
finance side. However, these suffer from four main problems. Firstly,
financial models apply a revenue and profit based approach versus our process
based approach. Secondly, the low detailing into the process parameters in a
financial model limits visibility of outcomes on process changes. Thirdly,
financial models tend to focus on internal parameters like rate of return for
judging a process change whereas the below model allows both internal and
external variations.

Models for CLV (Customer Lifetime Value) have been developed in the context of
non-subscription and subscription based companies[1],[2]. However, these
models are hard to interpret owing their inherently stochastic nature - here
we provide a highly accessible and flexible deterministic model in an SBS
context.

##  3 Assumptions

We list the assumptions that go into this model.

### General Assumptions

  1. 1.

The model is for a single product.

  2. 2.

Customer differences can be modeled as categorical variables.

  3. 3.

The model does not deal with nonlinear interactions between different
features.

### Process Unit Assumptions

  1. 1.

The customer is a single, well-defined entity in the process flow.

  2. 2.

The model assumes that a user who has not churned generates fixed revenue per
unit time.

  3. 3.

The model does not explicitly address cross-customer correlation effects.

### Process Assumptions

  1. 1.

The process is self-contained.

  2. 2.

The process parameters are constant across process units.

  3. 3.

The process parameters are constant across time.

### Customer Flow Assumptions

  1. 1.

The flow is uni-directional and conserves process units.

  2. 2.

Each exit point is differentiated by a unit of subscription time.

  3. 3.

Each exit point has a certain ab-initio probability of exit (churn) associated
with it.

### Path Assumptions

  1. 1.

Each path is uniquely defined by a point of entry a the point of exit.

  2. 2.

The ab-initio probability associated with an exit is the conditional
probability of exit given the process unit reaches the previous point in the
flow.

  3. 3.

The expected value of the overall process is a weighted sum of the expected
values of all possible paths

Parameter | Definition
---|---
SCV | Single Customer Value
$AC_{\\_}i$ | Acquisition Path through Channel $i$
Tr | Path to Trial Post Acquisition
$C_{\\_}{Tr}$ | Path to failure of Trial (month 0)
$C_{\\_}i$ | Path to failure at month $i$, post success of month $i-1$
$S_{\\_}{Tr}$ | Path to success of Trial
$S_{\\_}i$ | Path to success at month $i$, post success of month $i-1$
$P_{\\_}{i}$ | Overall probability of path $i$
$p_{\\_}{i}$ | Conditional probability of path $i$ given success at $i-1$
$P_{\\_}{Tr}$ | Probability of entering trial after acquisition
$P_{\\_}{C_{\\_}{Tr}}$ | Probability of churn during trial
$P_{\\_}{S_{\\_}{Tr}}$ | Probability of continuation from Trial
$P_{\\_}{C_{\\_}t}$ | Overall probability of churn after $t$ time periods
$P_{\\_}{S_{\\_}t}$ | Overall probability of continuing after $t$ time periods
$CAC_{\\_}j$ | Cost of Acquiring Customer through channel $j$
$CAC_{\\_}{mean}$ | Average Cost of Acquiring Customer through all channels
$CC$ | Cost of retaining a customer beyond the trial period
$R_{\\_}{i}$ | Overall contribution to revenue from path $i$
$r_{\\_}{i}$ | Conditional revenue given path $i$
$r_{\\_}{pt}$ | Revenue per unit time net of costs
$R_{\\_}{C_{\\_}{Tr}}$ | Contribution to overall net revenue given churn at Trial
$R_{\\_}{C_{\\_}t}$ | Contribution to overall net Revenue given churn at time $t$
$r_{\\_}{C_{\\_}{Tr}}$ | Net Revenue on the path churning out during trial
$r_{\\_}{C_{\\_}t}$ | Net Revenue on the path churning at time $t$
$F$ | Factor variable external the process
Table 1: Variable definitions used in this work

##  4 Setup of Equations

We set up the equations for our model described in figure 1 using the
assumptions defined in the previous section. Refer to Table 1 for the variable
definitions.

###  4.1 Probability Equations

Figure 1: The subscription business as a Bayesian churn model

$\sum_{\\_}{i=1}^{n}P_{\\_}{AC_{\\_}i}=1$ (1)
$P_{\\_}{C_{\\_}{Tr}}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t}=1$ (2)

$P_{\\_}{S_{\\_}{Tr}}=1-p_{\\_}{C_{\\_}{Tr}}$ (3) $\displaystyle
P_{\\_}{S_{\\_}i}$
$\displaystyle=(1-p_{\\_}{C_{\\_}{Tr}})\times\prod_{\\_}{t=1}^{i}(1-p_{\\_}{C_{\\_}t})$
(4)

Since a customer cannot churn before the end of the trial stage,

$P_{\\_}{C_{\\_}{Tr}}=p_{\\_}{C_{\\_}{Tr}}$ (5) $\displaystyle
P_{\\_}{C_{\\_}t}$ $\displaystyle=P_{\\_}{S_{\\_}{t-1}}\times
p_{\\_}{C_{\\_}t}$ (6) $\displaystyle\implies P_{\\_}{C_{\\_}t}$
$\displaystyle=(1-p_{\\_}{C_{\\_}{Tr}})\times\prod_{\\_}{i=1}^{t-1}(1-p_{\\_}{C_{\\_}i})\times
p_{\\_}{C_{\\_}t}$ (7)

###  4.2 Revenue Equations

$\displaystyle r_{\\_}{CAC_{\\_}i}$
$\displaystyle=P_{\\_}{AC_{\\_}i}*CAC_{\\_}i$ (8) $\displaystyle
CAC_{\\_}{mean}$ $\displaystyle=\sum_{\\_}{i=1}^{n}r_{\\_}{CAC_{\\_}i}$ (9)
$\displaystyle=\sum_{\\_}{i=1}^{n}P_{\\_}{AC_{\\_}i}*CAC_{\\_}i$

$r_{\\_}{Tr}=CAC_{\\_}{mean}+CC\\\ $ (10) $\displaystyle R_{\\_}{Tr}$
$\displaystyle=P_{\\_}{C_{\\_}{Tr}}\times r_{\\_}{Tr}$ (11)
$\displaystyle=P_{\\_}{C_{\\_}{Tr}}\times(CAC_{\\_}{mean}+CC)$
$\displaystyle=p_{\\_}{C_{\\_}{Tr}}\times(CAC_{\\_}{mean}+CC)$

$\displaystyle r_{\\_}{C_{\\_}t}$ $\displaystyle=CAC_{\\_}{mean}+CC+t\times
r_{\\_}{pt}$ (12)

$\displaystyle R_{\\_}{C_{\\_}t}$ $\displaystyle=r_{\\_}{C_{\\_}t}\times
P_{\\_}{C_{\\_}t}$ (13)

This expands to

$\displaystyle R_{\\_}{C_{\\_}t}$ $\displaystyle=(CAC_{\\_}{mean}+CC+t\times
r_{\\_}{pt})\times P_{\\_}{C_{\\_}t}$ (14)

$SCV=\sum_{\\_}{t=1}^{\infty}{R_{\\_}{C_{\\_}t}}+R_{\\_}{Tr}\\\ $ (15)

##  5 Constant churn probability model

Here we assume that each point of exit in our process flow has a constant
conditional probability of churn post the trial period. This has been found to
be a decent approximation for businesses with a good product and high customer
retention.

###  5.1 Probability Equations

Let

$p_{\\_}{C_{\\_}t}=1-\theta$ (16) $p_{\\_}{C_{\\_}{Tr}}=1-\gamma$ (17)

Consequently, we can write the overall probabilities using Eq (8) as:

$\displaystyle P_{\\_}{C_{\\_}t}$
$\displaystyle=(1-p_{\\_}{C_{\\_}{Tr}})\times\prod_{\\_}{i=1}^{t-1}(1-p_{\\_}{C_{\\_}i})\times
p_{\\_}{C_{\\_}t}$ (18)
$\displaystyle=(1-\theta)(1-p_{\\_}{C_{\\_}{Tr}})\theta^{t-1}$
$\displaystyle=\gamma(1-\theta)\theta^{t-1}$

###  5.2 Value Equations

$R_{\\_}{Tr}=P_{\\_}{Tr}\times(CAC_{\\_}{mean}+CC)$ (19)
$R_{\\_}{C_{\\_}t}=P_{\\_}{C_{\\_}t}\times(CAC_{\\_}{mean}+CC+t\times
R_{\\_}{Pt})$ (20)

###  5.3 Overall Value

From equations 15, 19 and 20

$\displaystyle SCV$
$\displaystyle=P_{\\_}{Tr}\times(CAC_{\\_}{mean}+CC)+\sum_{\\_}{t=1}^{\infty}{R_{\\_}{C_{\\_}t}}$
(21)
$\displaystyle=(CAC_{\\_}{mean}+CC)\times(P_{\\_}{Tr}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t})+\frac{R_{\\_}{Pt}\gamma}{(1-\theta)}$

Here, we used the fact that $p_{\\_}{C_{\\_}t}$ is constant to arrive at an
infinite geometric series, which was condensed into the above equation. From
equation 2

$P_{\\_}{Tr}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t}=1$ $\displaystyle SCV$
$\displaystyle=(CAC_{\\_}{mean}+CC)\times(P_{\\_}{Tr}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t})+\frac{R_{\\_}{Pt}\gamma}{(1-\theta)}$
(22) $\displaystyle=(CAC_{\\_}{mean}+CC)+\frac{R_{\\_}{Pt}\gamma}{(1-\theta)}$

Re-substituting for $\theta$, we obtain the final SCV equation for the
constant churn model as follows:

$SCV=(CAC_{\\_}{mean}+CC)+\frac{R_{\\_}{Pt}\times\
(1-p_{\\_}{C_{\\_}{Tr}})}{p_{\\_}{C_{\\_}t}}\\\ $ (23)

##  6 Exponential decay churn probability model

Although the constant churn model is a fairly good approximation for cohort
analysis post trial, we observed from experimental data such as that depicted
in Figure 2 that the overall churn rate for an actual SBS is closer to an
exponentially decaying quantity. This motivated us to formulate churn in terms
of variables defined in Table 2 as follows.

Figure 2: Churn is modelled through an exponential decay from the initial rate
to the natural rate

###  6.1 Model Formulation

Variable | Description
---|---
$CR_{\\_}{init}$ | Churn Rate at time period 0
$CR_{\\_}{nat}$ | Natural Rate of Churn to which churn rate decays
$CR_{\\_}{t}$ | Churn Rate at time period t
$k$ | decay constant describing rate of decay of churn rate
Table 2: Additional variable definitions for the exponential model

$\displaystyle\alpha=CR_{\\_}{nat}$ (24)
$\displaystyle\beta=(CR_{\\_}{init}-CR_{\\_}{nat})$ (25) $\displaystyle
CR_{\\_}{t}=CR_{\\_}{nat}+(CR_{\\_}{init}-CR_{\\_}{nat})\times e^{-kt}$ (26)
$\displaystyle p_{\\_}{C_{\\_}t}=\alpha+\beta\times e^{-kt}$ (27)

###  6.2 Extending to Overall Probability Equations

From Eq 17, $\gamma=1-p_{\\_}{C_{\\_}{Tr}}$

$\displaystyle P_{\\_}{C_{\\_}t}$
$\displaystyle=(1-p_{\\_}{C_{\\_}{Tr}})\times
p_{\\_}{C_{\\_}t}\times\prod_{\\_}{i=0}^{t-1}(1-p_{\\_}{C_{\\_}i})$ (28)
$\displaystyle=\gamma\times(\alpha+\beta\times
e^{-kt})\times\prod_{\\_}{i=0}^{t-1}(1-(\alpha+\beta\times e^{-i\times k}))$

###  6.3 Analysis of churn rate approximations

The exponential decay model in its original state proves to be analytically
intractable. Consequently, we considered 6 different simplifying assumptions,
where we compared the numerical values for the exact solution series and the
approximate closed-form solutions. This helped us determine ranges for process
parameters wherein these approximations would hold good. Based on the
analysis, we then chose the following single simplifying assumption that holds
true within reasonable error for an adequate fraction of the valid parameter
space.

$\displaystyle\gamma\times\prod_{\\_}{i=0}^{t-1}(1-(\alpha+\beta\times
e^{-i\times k}))\times(\alpha+\beta\times
e^{-kt})=\gamma\times\prod_{\\_}{i=0}^{t-1}(1-(\alpha+\beta))\times(\alpha+\beta\times
e^{-kt})$ (29)

where $\alpha\in(0.001,1),$ $\beta\in(0.001,1-\alpha),$ $\gamma\in(0.001,1),$
$k\in(0.001,1)$

The following measures were used for validation:

  * •

Average:
$\overline{\sum_{\\_}{t=1}^{1000}|P_{\\_}{C_{\\_}t}^{model}-P_{\\_}{C_{\\_}t}^{truth}|}=0.0488$

  * •

Min:
$\min{(\sum_{\\_}{t=1}^{1000}|P_{\\_}{C_{\\_}t}^{model}-P_{\\_}{C_{\\_}t}^{truth}|)}=1.19*10^{-7}$

  * •

Max:
$\max{(\sum_{\\_}{t=1}^{1000}|P_{\\_}{C_{\\_}t}^{model}-P_{\\_}{C_{\\_}t}^{truth}|)}=0.5849$

  * •

Stdev:
$\sigma{(\sum_{\\_}{t=1}^{1000}|P_{\\_}{C_{\\_}t}^{model}-P_{\\_}{C_{\\_}t}^{truth}|)}=0.0765$

A further analysis of input parameters that gave the highest deviation for
this suggests that the maximum absolute deviation for any 1 time period is
less than 0.01%. Large deviations are observed when $k$ is large and $\alpha$,
$\beta$ and $\gamma$ are all small. Hence we recommend this model for all
parameter values where $k<1$.

###  6.4 Overall Probability

Let

$1-\alpha=\alpha^{\prime}$ (30)

From Eq: 28 and Eq: 29, we get

$\displaystyle P_{\\_}{C_{\\_}t}$
$\displaystyle=\gamma\times(1-\alpha-\beta)^{t-1}\times(\alpha+\beta\times
e^{-kt})$ (31)
$\displaystyle=\gamma\times(\alpha^{\prime}-\beta)^{t-1}\times(\alpha+\beta\times
e^{-kt})$ $\displaystyle=\alpha\ \gamma\ (\alpha^{\prime}-\beta)^{t-1}+\beta\
\gamma\ (\alpha^{\prime}-\beta)^{t-1}\ e^{-kt}$

###  6.5 Overall Value

Since the underlying process is the same, we can write the $SCV$ expression
using equation 15 as

$\displaystyle SCV$
$\displaystyle=R_{\\_}{Tr}+\sum_{\\_}{t=1}^{\infty}{R_{\\_}{C_{\\_}t}}$ (32)
$\displaystyle=P_{\\_}{Tr}\times(CAC_{\\_}{mean}+CC)+\sum_{\\_}{t=1}^{\infty}{R_{\\_}{C_{\\_}t}}$
$\displaystyle=(CAC_{\\_}{mean}+CC)\times(P_{\\_}{Tr}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t})+R_{\\_}{Pt}\times\sum_{\\_}{t=1}^{\infty}{P_{\\_}{C_{\\_}t}\times
t}$

Substituting from equation 2,

$\displaystyle SCV$
$\displaystyle=(CAC_{\\_}{mean}+CC)+R_{\\_}{Pt}\times\sum_{\\_}{t=1}^{\infty}{P_{\\_}{C_{\\_}t}\times
t}$ (33)

From equation 31:

$P_{\\_}{C_{\\_}t}=\alpha\ \gamma\ (\alpha^{\prime}-\beta)^{t-1}+\beta\
\gamma\ (\alpha^{\prime}-\beta)^{t-1}\ e^{-kt}$

Therefore,

$\displaystyle SCV$
$\displaystyle=(CAC_{\\_}{mean}+CC)+R_{\\_}{Pt}\times\sum_{\\_}{t=1}^{\infty}{(\alpha\
\gamma\ (\alpha^{\prime}-\beta)^{t-1}+\beta\ \gamma\
(\alpha^{\prime}-\beta)^{t-1}\ e^{-kt})\times t}$ (34)
$\displaystyle=(CAC_{\\_}{mean}+CC)+\gamma\ R_{\\_}{Pt}\
\bigg{\\{}\frac{\alpha}{{(\alpha+\beta)}^{2}}+\frac{\beta\ e^{-k}}{((1-\
e^{-k})+(\alpha+\beta)\ e^{-k})^{2}}\bigg{\\}}$

Substituting back $CR_{\\_}{nat}$ and $CR_{\\_}{init}$ we get

$\displaystyle SCV$ $\displaystyle=(CAC_{\\_}{mean}+CC)+\gamma\ R_{\\_}{Pt}\
\bigg{\\{}\frac{CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})\
e^{-k}}{((1-\ e^{-k})+(CR_{\\_}{init})\ e^{-k})^{2}}\bigg{\\}}$ (35)

##  7 Partial Differentials of the SCV Equation

###  7.1 Overview

In the first part of our paper, we have arrived at a single equation
describing the entire process as one key metric - the Single Customer Value.
We now proceed to determine the sensitivity of the SCV to various factors
using equation 35. Knowing the sensitivity of the model to its various
parameters, we can further compute its response to changes in external factor
variables by studying their impact on model parameters, thus giving us a
powerful decision making tool that does not rely on vague human intuition but
on simple numerical computations.

###  7.2 Partial Differentials

From Eq: 35

$\displaystyle SCV$ $\displaystyle=(CAC_{\\_}{mean}+CC)+\gamma\ R_{\\_}{Pt}\
\bigg{\\{}\frac{CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})\
e^{-k}}{((1-\ e^{-k})+(CR_{\\_}{init})\ e^{-k})^{2}}\bigg{\\}}$
$\displaystyle=f(CAC_{\\_}{mean},CC,\gamma,R_{\\_}{pt},CR_{\\_}{nat},CR_{\\_}{init},k)$

Writing the vector of variables as
$\textbf{x}=(CAC_{\\_}{mean},CC,\gamma,R_{\\_}{pt},CR_{\\_}{nat},CR_{\\_}{init},k)$,
we can compute the sensitivity of $SCV$ to features $F$ using the chain rule:

$\displaystyle\frac{\partial SCV}{\partial
F}=\sum_{\\_}{i=1}^{7}\frac{\partial SCV}{\partial
x_{\\_}i}\times\frac{\partial x_{\\_}i}{\partial F}$ (36)

From Eq: 35, we can show the partial differentials with respect to each of the
sub-variables to the SCV equation.

$\frac{\partial SCV}{\partial CAC_{\\_}{mean}}=1$ (37)

$\frac{\partial SCV}{\partial CC}=1$ (38)

$\frac{\partial SCV}{\partial
P_{\\_}{Tr}}=-R_{\\_}{Pt}\bigg{(}\frac{CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})\
e^{-k}}{((1-\ e^{-k})+(CR_{\\_}{init})\ e^{-k})^{2}}\bigg{)}$ (39)

$\frac{\partial SCV}{\partial
R_{\\_}{Pt}}=\gamma\bigg{(}\frac{CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})\
e^{-k}}{((1-\ e^{-k})+(CR_{\\_}{init})\ e^{-k})^{2}}\bigg{)}$ (40)

$\frac{\partial SCV}{\partial CR_{\\_}{nat}}=\quad\frac{\gamma\
R_{\\_}{Pt}}{{(CR_{\\_}{init})}^{2}}+\frac{-\gamma R_{\\_}{Pt}e^{-k}}{((1-\
e^{-k})+(CR_{\\_}{init})\ e^{-k})^{2}}$ (41)

$\frac{\partial SCV}{\partial CR_{\\_}{init}}=\gamma\
R_{\\_}{Pt}\Big{(}-\frac{2CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{3}}+\frac{e^{-k}(1-\
e^{-k})+(2CR_{\\_}{nat}-\ CR_{\\_}{init})\ e^{-2k}}{((1-\
e^{-k})+(CR_{\\_}{init})\ e^{-k})^{3}}\Big{)}$ (42)

$\frac{\partial SCV}{\partial k}=-\gamma\ R_{\\_}{Pt}\
((CR_{\\_}{init}-CR_{\\_}{nat})\
e^{-k})\bigg{(}\frac{(1+(1-CR_{\\_}{init})e^{-k})}{((1-\
e^{-k})+(CR_{\\_}{init})\ e^{-k})^{3}}\bigg{)}$ (43)

###  7.3 Computing the mean time to churn ($\tau_{\\_}{mean}$)

The mean time for a customer to be churned can be computed by a simple
weighted time average, and can be expressed in terms of known parameters using
equations (3) and (37), the latter being where we compute the quantity

$\sum_{\\_}{t=1}^{\infty}tP_{\\_}{C_{\\_}t}=\gamma\Big{\\{}\frac{CR_{\\_}{nat}}{CR_{\\_}{init}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})e^{-k}}{(1-e^{-k}+CR_{\\_}{init}e^{-k})^{2}}\Big{\\}}$

The mean time to churn for a user can be written as

$\sum_{\\_}{t=1}^{\infty}(t+1)P_{\\_}{C_{\\_}t}+1\times
P_{\\_}{C_{\\_}Tr}=\sum_{\\_}{t=1}^{\infty}tP_{\\_}{C_{\\_}t}+\sum_{\\_}{t=1}^{\infty}P_{\\_}{C_{\\_}t}+P_{\\_}{C_{\\_}{Tr}}$
(44)

This implies

$\tau_{\\_}{mean}=1+\gamma\Big{\\{}\frac{CR_{\\_}{nat}}{CR_{\\_}{init}^{2}}+\frac{(CR_{\\_}{init}-CR_{\\_}{nat})e^{-k}}{(1-e^{-k}+CR_{\\_}{init}e^{-k})^{2}}\Big{\\}}$
(45)

##  8 Case Study : Food Based Subscription Business

###  8.1 Overview

The company in question is a small-sized silicon valley based subscription
food company offering an externally sourced single food type operating on a
monthly subscription basis. They have three kinds of subscription offers, viz.
1 6 and 12 month subscriptions. Most of their uptake is on a monthly payment
basis. They also offer prepaid options which customers seldom opt for (<1%
usage). Every month customers get a shipment containing three packets of
similar food items. The customer gets an option to choose, but sometimes the
company may send over the shipment based on a pre-decided menu.

The company only acquires customers through online channels. Most of their
acquisition spend is on channels like Facebook and Google based advertising.
Their customers are heavily discounted during the first month, which we refer
to as a trial period where the company may end up losing or making a small
amount of money net of COGS and shipping costs. Further we note that their
initial churn rates are very high, though their longer term customers are much
less likely to churn than a new customer into the system. While many of their
customers initially opt for a 6m or 12m subscription period, we find that
these too cancel their subscription quite often or simply default on their
payment.

On this note, we can proceed to analyse the company through the SCV equation
lens.

###  8.2 Churn Timeline

An essential part of the pipeline to estimating the variables is to plot the
percentage active customers by time period. This enables us to estimate the
variables. The nature of the curve found, which we expect to be true for most
subscription based businesses, was also part of the motivation for the
exponential decay churn model

Figure 3: Cohortwise decay of percentage active customers

###  8.3 SCV computation and local sensitivity analysis

From equation 35, we can compute the $SCV$ as -$31.48. This indicates that the
company is losing money, and that the revenue generated per user is much
lesser than the cost spent on acquiring them. This leads us to believe that
the company should focus on reducing their $CAC$ and improving customer
retention. Although it is trying to address the former, the latter issue in
our opinion is equally important and can potentially have a much greater
impact on the company.

The values for our model variables were perturbed in order to study the
effects on SCV. We computed the gradients as described by our approximate
exponential model. We see that the SCV value is most sensitive to the initial
and natural (or final) churn rates. Even a 1 % increase in the initial monthly
churn rate can potentially push the SCV down by $ 1.11, whereas a similar
change in the natural churn rate bumps it up by $ 1.67. Thus, we expect the
companies performance to improve significantly if they focus more on customer
retention.

Variable | Value | Sensitivity
---|---|---
$CAC_{\\_}{mean}$ | -$35 | 1
$CC$ |  $\approx$ $0 | 1
$R_{\\_}{P_{\\_}t}$ | $6 | 1.58
$P_{\\_}{Tr}$ | 0.36 | -14.78
$CR_{\\_}{nat}$ | 0.05 | 167.17
$CR_{\\_}{init}$ | 0.30 | -111.81
$k$ | 0.6 | -2.01
Table 3: Sensitivities computed across different model parameters

###  8.4 Checking for variability with single feature

As noted previously, we can use the SCV equation to determine the impact of an
external variable on the customer value. As an illustration, we choose a
categorical variable that takes three distinct values. This variable does not
pertain to any particular path in the process and therefore can be considered
an external variable. We show the effect of this variable on the mean months
to cancel in the process. This leads us to the difference in SCV for every
differential increase in the percentage of customers in one particular bucket.
We split this categorical variable into multiple features, each representing
one particular category.

Figure 4: Change in Mean months to cancel by subscription period

We choose the categorical variable subscription period, and within that, focus
our calculation on the 6MM feature bucket and check the effects of having
customers move into that bucket. As seen in Figure 4, the mean months to
cancel varies across these features. We note that this is the same as the
previously defined $\tau_{\\_}{mean}$ for this process, which for the overall
process can be calculated using equation 45, which gives us a value of 1.6884
($\approx 2$) months. We assume that the natural churn rate and the
exponential decay coefficient ($CR_{\\_}{nat}$ and $k$) are unaffected by a
change in the categorical variable. Thus, we can relate change in
$\tau_{\\_}{mean}$ across buckets to change in $CR_{\\_}{init}$, computed
explicitly in the following section.

###  8.5 Checking for change in SCV by changing Subscription Period

$\displaystyle\Delta(SCV)$ $\displaystyle=\frac{\partial SCV}{\partial
CR_{\\_}{init}}\times\frac{\partial CR_{\\_}{init}}{\partial
F}\times\Delta(F)$ (46)

From Equation 42,

$\displaystyle\frac{\partial SCV}{\partial CR_{\\_}{init}}=\gamma\
R_{\\_}{Pt}\Big{(}-\frac{2CR_{\\_}{nat}}{{(CR_{\\_}{init})}^{3}}+\frac{e^{-k}(1-\
e^{-k})+(2CR_{\\_}{nat}-\ CR_{\\_}{init})\ e^{-2k}}{((1-\
e^{-k})+(CR_{\\_}{init})\ e^{-k})^{3}}\Big{)}$

We note that $\tau_{\\_}{mean}$ changes from 1.6884 to 2.653. This changes the
$CR_{\\_}{init}$ values from 30% to 14.83%
In other words

$\displaystyle\lim_{\\_}{\tau\to 1.6884}CR_{\\_}{init}$ $\displaystyle=0.3$
$\displaystyle\lim_{\\_}{\tau\to 2.653}CR_{\\_}{init}$ $\displaystyle=0.1483$
$\displaystyle\Delta(CR)$ $\displaystyle=\frac{\partial
CR_{\\_}{init}}{\partial F}\times\Delta(F)$ $\displaystyle\Delta(CR)$
$\displaystyle=0.1483-0.3=-0.1517$ $\implies\frac{\partial
CR_{\\_}{init}}{\partial F}\times\Delta(F)=-0.1517\\\ $

We also note the change in partial differentials through these two limits

$\displaystyle\lim_{\\_}{\tau\to 1.6884}\frac{\partial SCV}{\partial
CR_{\\_}{init}}$ $\displaystyle=-11.14$ $\displaystyle\lim_{\\_}{\tau\to
2.653}\frac{\partial SCV}{\partial CR_{\\_}{init}}$ $\displaystyle=-111.81$
$\implies\bigg{(}\frac{\partial SCV}{\partial
CR_{\\_}{init}}\bigg{)}_{\\_}{mean}=-(111.81+11.14)/2=-61.47\\\ $

Substituting the values,

$\displaystyle\Delta(SCV)$ $\displaystyle=\frac{\partial SCV}{\partial
CR_{\\_}{init}}\times\frac{\partial CR_{\\_}{init}}{\partial
F}\times\Delta(F)$ $\displaystyle=\bigg{(}\frac{\partial SCV}{\partial
CR_{\\_}{init}}\bigg{)}_{\\_}{mean}\times\Delta(CR)$
$\displaystyle=-61.47*-0.1517$ $\displaystyle=\$9.325$

This implies a SCV of -$22, or a total change of 30% on the SCV through a
complete change to this feature. Note that this is a linear extrapolation
based on the derivative and does not account for non-linear changes.

##  9 Conclusions

The churn rates were expressed using an exponential decay model so as to embed
the data in our SCV framework. Using an approximation for certain exponential
terms lead to a surprisingly clean expression for SCV as a function of
parameters such as $R_{\\_}{pt}$, $P_{\\_}{C_{\\_}{Tr}}$ and most importantly
$CR_{\\_}{init}$ \- let us call this the AEM (Approximate Exponential Model)
as opposed to EM (the "exact" Exponential Model).

A local sensitivity analysis of SCV as computed by the AEM with respect to the
feature $CR_{\\_}{init}$ indicates a high negative value for sensitivity,
which is in agreement with our hypothesis as well as the EM. The AEM also
allows us to use real data values for the mean time to churn
$\tau_{\\_}{mean}$ in order to exactly back-compute the initial churn rate
$CR_{\\_}{init}$. This relation allows us to tune our initial churn rate in
accordance with the mean time to churn.

It should be noted, however, that the AEM works best within the specified
parameter bounds. The AEM also shows a lower local sensitivity to
$CR_{\\_}{nat}$, especially in the partial derivative than what we expect,
though it is likely due to related parameters $k$ (the decay rate) and
$CR_{\\_}{init}$ not being varied in a more realistic fashion. Further work
would involve improving upon the AEM to ensure it performs equally well across
greater parameter ranges. More data from similar businesses would also help to
verify the model and its underlying assumptions, as well as allow us to
develop better and more generalized churn models with predictive value. We
hope to integrate this model with a more general approach that allows us to
use this single metric along with the number of customers to compute overall
company value, with additional factors such as cash flow discounting taken
into account in a follow up paper.

## References

[1] Romero J. ,van der Lans R. ,Wierenga B. (2013) A partially hidden markov
model of customer dynamics for CLV measurement, Journal of Interactive
Marketing, vol. 27

[2] McCarthy D. ,Fader P. ,Hardie B. (2015) Valuing Subscription-Based
Businesses Using Publicly Disclosed Customer Data SSRN Electronic Journal,
vol. 81, pp. 17-35


# Propagation measurements and channel models in Indoor Environment at 6.75
GHz FR1(C) and 16.95 GHz FR3 Upper-mid band Spectrum for 5G and 6G

Dipankar Shakya†1, Mingjun Ying†2, Theodore S. Rappaport†3,
Hitesh Poddar*, Peijie Ma†, Yanbo Wang†, and Idris Al-Wazani† This research is
supported by the New York University (NYU) WIRELESS Industrial Affiliates
Program. *Sharp Laboratories of America (SLA), Vancouver, Washington, USA †NYU
WIRELESS, Tandon School of Engineering, New York University, USA
{1dshakya, 2yingmingjun<EMAIL_ADDRESS>
###### Abstract

New spectrum allocations in the 4–8 GHz FR1(C) and 7–24 GHz FR3 mid-band
frequency spectrum are being considered for 5G/6G cellular deployments. This
paper presents results from the world’s first comprehensive indoor hotspot
(InH) propagation measurement campaign at 6.75 GHz and 16.95 GHz in the NYU
WIRELESS Research Center using a 1 GHz wideband channel sounder system over
distances from 11 to 97 m in line-of-sight (LOS) and non-LOS (NLOS). Analysis
of directional and omnidirectional path loss (PL) using the close-in free
space 1 m reference distance model shows a familiar waveguiding effect in LOS
with an omnidirectional path loss exponent (PLE) of 1.40 at 6.75 GHz and 1.32
at 16.95 GHz. Compared to mmWave frequencies, the directional NLOS PLEs are
lower at FR3 and FR1(C), while omnidirectional NLOS PLEs are similar,
suggesting better propagation distances at lower frequencies for links with
omnidirectional antennas at both ends of the links, but also, importantly,
showing that higher gain antennas will offer better coverage at higher
frequencies when antenna apertures are kept same over all frequencies.
Comparison of the omnidirectional and directional RMS delay spread (DS) at
FR1(C) and FR3 with mmWave frequencies indicates a clear decrease with
increasing frequency. The mean spatial lobe and omnidirectional RMS angular
spread (AS) is found to be wider at 6.75 GHz compared to 16.95 GHz indicating
more multipath components are found in the azimuthal spatial domain at lower
frequencies.

###### Index Terms:

FR3, FR1(C), 6G, InH, indoor, upper mid-band, delay spread, angular spread,
path loss, propagation, channel model

##  I Introduction

The continued increase in mobile data traffic requires wide bandwidths to
deliver gigabits-per-second data throughput for next-generation wireless
communication systems. As the spectrum below 6 GHz is congested, the search
for higher bandwidth has driven the telecommunications industry towards the
large swaths of bandwidth available at higher frequencies for future 5G and 6G
deployments. With new spectrum allocations in the 4–8 GHz FR1(C) and 7–24 GHz
FR3 frequencies being considered by major government agencies and the
International Telecommunication Union (ITU), there is an urgent need for
characterizing the radio propagation in the upper mid-band spectrum. The new
allocations offer promise of balancing wide area coverage of sub-6 GHz
frequencies with higher data rates and capacity of mmWave frequency bands.
Telecommunications agencies such as the ITU and the National
Telecommunications and Information Administration (NTIA) are exploring
effective usage of the FR3 frequencies with other applications such as
satellite communications and radio astronomy[1, 2]. Particularly, ITU and NTIA
have focused on frequency bands including 7.125-8.4 GHz, 4.40-4.80 GHz, and
14.8-15.35 GHz, as identified by the ITU World Radio Conference 2023 (WRC-23),
Fig. 1 [1]. There are major efforts from the 3rd Generation Partnership
Project (3GPP) and the Federal Communications Commission (FCC) for exploring
and standardizing deployments at these frequencies. Industry bodies including
the Alliance for Telecommunications Industry Solutions (ATIS) and several
mobile operators have identified frequency bands up to 15 GHz for future
deployments[3, 4]. There is, therefore, a strong interest and necessity to
understand the radio propagation characteristics in the FR3 and FR1(C)
frequency bands, including indoor office environments.

Figure 1: Bands of interest identified in the FR1(C) and FR3 by ITU, NTIA,
FCC, and ATIS [1, 3, 4, 5].

Few empirical channel measurements have been reported in the literature in the
FR3 and FR1C spectrum. Authors in [6] conducted propagation measurements along
an office corridor at 11 and 14 GHz using a vector network analyzer
(VNA)-based channel sounder with biconical antennas and observed PLEs of 1.52
and 1.59 in LOS and 3.06 and 2.76 in NLOS. Mean RMS DS were reported as 19.5
ns and 17.9 ns in LOS at 11 GHz and 14 GHz, and 23.43 ns and 22.03 ns at 11
GHz and 14 GHz, respectively. Wei et al. conducted propagation measurements in
university corridors in [7] with a MIMO array at 6 GHz with 100 MHz bandwidth
using a sliding correlation channel sounder and recorded $\sim$21 ns RMS DS in
LOS and $\sim$39 ns in NLOS. Continuous wave measurements conducted along a
university corridor in [8] using a signal generator and analyzer pair at 14
and 22 GHz using horn antennas with 19.5 and 22 dBi gain at the respective
frequencies resulted in a directional LOS PLE of 1.6 and 1.7, respectively.
Wideband VNA measurements in [9] conducted at 2.4, 4.75, and 11.5 GHz in an
indoor office/laboratory environment with biconical antennas yielded LOS PLEs
of 1.86, 1.98 and 1.94 and NLOS PLEs of 3.33, 3.75, and 4.46 at the three
respective frequencies. Work in [10] measured diffraction loss around drywall,
plastic board, and wooden corners, and found good agreement with the Knife
Edge Diffraction (KED) model.

The Indoor Hotspot (InH) measurement campaign at the NYU WIRELESS Research
Center is the world’s first comprehensive radio propagation study at 6.75 GHz
in FR1(C) and 16.95 GHz in FR3 bands with a 1 GHz bandwidth channel sounder.
The results presented in this paper are analyzed from over 30,000 power delay
profiles (PDP) with measurements being conducted at 31 dBm EIRP transmit power
employing 15 and 20 dBi gain rotatable horn antennas at 6.75 GHz and 16.95
GHz, respectively, in both co-polarized and cross-polarized antenna
configurations. The system and calibration procedures are described in Section
II and III. The measurement procedure employed is detailed in Section IV with
measured path loss models presented in Section V. Section VI presents spatio-
temporal statistics from the measurements before concluding in Section VII.

##  II Wideband FR3/FR1C Sliding Correlation Channel Sounder at NYU

The working principle of the sliding-correlation channel sounder using a
pseudo-random noise (PN) sequence is detailed in [11, 12]. A 500 Mcps PN
sequence, generated at baseband, is upconverted to a 1 GHz wideband signal
centered around 6.75 GHz. The FR1(C) channel sounder directly transmits the
upconverted signal, whereas the FR3 sounder upconverts the 6.75 GHz IF to a
16.95 GHz RF signal in a heterodyne architecture. At the RX, the received
signal is correlated with an identical PN sequence running at a slightly
slower rate of 499.9375 Mcps, causing them to slide past one another and
resulting in a time-dilated PDP. The work in [13] highlights the FR3/FR1(C)
channel sounder with co-located specialized RF frequency converters developed
by Mini-Circuits for operation at 6.75 GHz in FR1(C) and 16.95 GHz in FR3. The
unique dual-band co-located design allows easy switching of operational
frequency bands by changing only the IF signal cable to the suitable RF module
input. The detailed specifications of the channel sounder system are provided
in Table I in [14].

##  III Calibration of the Channel Sounder System

The channel sounder calibration for InH measurements ensures accurate capture
of the power, delay, and direction of multipath components (MPC). Calibration
is conducted at least twice a day–once at the beginning of the measurement day
and once at the end–to assure daily and long-term accuracy, and is completed
in three distinct phases:

$\bullet$ Linearity and Power Calibration: The linear range of operation,
power limits, and system gain of the RX are defined and confirmed by
measurement twice daily, once at the beginning and once at the end of each
day. The transmit power (before the TX horn antenna) is measured with a
Keysight N1913A power meter via the Keysight N8487A power sensor. The TX
attenuator and LO powers are set to keep the TX power within the permissible
limit of 35 dBm and ensure linearity. A free space PL measurement at a
distance beyond 5$\times$far-field ($D_{f}$) of the horn antennas helps
determine the system gain and the correct attenuator settings. A 4 m
separation is selected with a 1.5 m TX and RX height to capture a single LOS
path avoiding any reflection paths, adhering to the calibration criteria
described in [15].

$\bullet$ Time Calibration: Accurate measurement of the absolute multipath
delay requires correct implementation of time calibration. Before the
measurement day, the two Rubidium (Rb) clocks used on the TX and RX are
synchronized overnight (12-13 hrs) using a physical cable to ensure accurate
time synchronization. During system startup each day, two Raspberry-Pi
computers connected to each Rb clock are initiated to run a patent-pending
Precision Time Protocol (PTP) synchronization for absolute timing over a Wi-Fi
link, as detailed in [16]. Following the linearity and power calibration at
the start of the day, the PDP which captures the single LOS peak is circular
shifted to match the free-space propagation delay of 13.3 ns for the four-
meter calibration distance. Then, the TX and RX are moved to the desired
location for measurements.

Upon completing measurements at a TX-RX location, the TX and RX are returned
to a 4 m separation in the laboratory to remeasure the propagation delay for
the LOS multipath and capture any drift in the propagation delay. The drift is
caused due to frequency and phase offsets between the untethered Rb clocks at
the TX and RX in the duration of the TX and RX measurement, which typically
takes several hours due to the intensive sweeping of the channel (described
subsequently in Section IV). The drift is found to be unaffected by the
movement of TX and RX carts to respective locations[16]. The observed drift is
recorded to correct the measured data to obtain the absolute propagation delay
of MPCs. The time calibration is repeated for every TX-RX location measured
throughout the day. Unlike past measurement campaigns at NYU WIRELESS, the
present campaign uses an active synchronization method that continuously
synchronizes the Rb clocks and the PDP propagation delays against a reference
MPC during the several hours of measurements at each TX-RX location, using
patent-pending methods in [16]. This new approach removes reliance on ray-
tracing or post-processing methods to assure absolute timing of measured PDPs.

$\bullet$ Spatial Calibration: At every TX-RX location, the TX and RX antennas
are raised to 2.4 m and 1.5 m above the ground, respectively. The geographical
North is used as the spatial 0∘ pointing reference at each TX-RX location pair
for consistent AOA/AOD recording across all locations and spatial coordination
between TX and RX during measurements.

Once the calibration (linearity and power, time, and space) process is
complete, the channel sounder is ready for multipath propagation measurements
at each TX-RX location pair.

##  IV Measurement environment and procedure for InH measurements

The InH measurement campaign is conducted in the NYU WIRELESS Research Center
at 370 Jay Street, 9th Floor, Brooklyn, NY. The environment is a typical open
office space with cubicles, office rooms, labs, and conference rooms. The TX
locations are marked as different color stars and RX locations as circles of
the corresponding color on the floorplan presented in Fig. 2. Identical T-R
locations are measured at both frequency bands. Table I lists the T-R
locations in LOS and NLOS for the 6.75 GHz FR1(C) and 16.95 GHz FR3
measurements, encompassing 7 LOS and 13 NLOS T-R location pairs. The T-R
separation for the locations range from 11 m to 97 m. No outages are observed
at any of the T-R locations in both bands with a transmit EIRP of 31 dBm and
link margin (at 5 dB SNR) of 156 dB at 6.75 GHz and 159 dB at 16.95 GHz.

Figure 2: The NYU WIRELESS research Center at 370 Jay Street, 9th Floor,
Brooklyn, NY is a typical open office space with low height cubicles. The four
TX locations are indicated as different color stars with corresponding RX
locations as circles of the same color. TABLE I: TX-RX location pairs for
indoor measurements at 16.95 GHz in FR3 and 6.75 GHz in FR1(C) 16.95 GHz
---
TX | LOS RX | NLOS RX
TX1 | RX1, RX4 | RX2, RX3
TX2 | RX1, RX2, RX6 | RX3, RX4, RX5
TX3 | RX5 | RX1, RX2, RX3, RX4, RX6
TX4 | RX1 | RX2, RX3, RX4
6.75 GHz
TX | LOS RX | NLOS RX
TX1 | RX1, RX4 | RX2
TX2 | RX1, RX2 | –
TX3 | RX5 | RX1 RX3, RX4, RX6
TX4 | RX1 | RX2, RX3, RX4

###  IV-A Measurement Procedure

The InH multipath statistics are obtained by capturing the propagation delay,
power, AOD, and AOA of all MPCs at each TX-RX location in a two-step process:
rapid RX scans, followed by RX azimuthal sweeps in HPBW steps.

First, the RX completes rapid continuous azimuthal scans (rapid scan) for each
TX boresight AOD to rapidly determine the TX AODs with significant energy
received at the RX. To initiate the rapid scans, as taught in [17], the TX and
RX are pointed at the direction with strongest received power; boresight in
LOS and a strong reflection direction in NLOS. At each TX AOD, the RX performs
a continuous 360∘ scan across the azimuth plane. The TX is rotated in
increments of HPBW until rapid scans are performed for each AOD. TX AODs are
selected using a threshold based on the minimum of two values; either 30 dB
below the maximum peak received power among all TX AODs, or 10 dB above the
noise floor. All TX AODs with peak power exceeding this threshold are selected
for the stepped RX azimuth sweeps.

Once key TX AODs are identified through rapid scan, the TX is pointed to the
strongest TX AOD and the RX AOA with the strongest received power is
identified. The PDP captured for this AOD-AOA pair is stored and the highest
peak in the PDP is marked as the ‘reference MPC’ for implementing the patent-
pending PTP synchronization [16]. After RX completes any azimuthal sweep, the
TX and RX return to the AOD-AOA pair of the reference MPC and recapture the
PDP to correct any drift in the propagation delay accumulated during an
azimuthal sweep to implement successive drift correction (Algorithm I in
[16]).

TABLE II: TX/RX Elevation angles for different RX azimuthal sweeps for a fixed TX azimuth angle at each TX-RX location Sweep# |  TX elevation |  RX elevation
---|---|---
1 |  TX is kept at boresight elevation |  RX is kept at boresight elevation. RX is then swept 360∘ in the azimuth plane in HPBW steps.
2 |  TX is kept at boresight elevation |  RX is tilted down by one HPBW (30∘ at 6.75 GHz/15∘ at 16.95 GHz). RX is then swept 360∘ in the azimuth plane in HPBW steps.
3 |  TX is kept at boresight elevation |  RX is tilted up by one HPBW (30∘ at 6.75 GHz/15∘ at 16.95 GHz). RX is then swept 360∘ in the azimuth plane in HPBW steps.
4 |  TX is tilted down by one HPBW (30∘ at 6.75 GHz/15∘ at 16.95 GHz). |  RX is kept at boresight elevation. RX is then swept 360∘ in the azimuth plane in HPBW steps.
5 |  TX is tilted down by one HPBW (30∘ at 6.75 GHz/15∘ at 16.95 GHz). |  RX is tilted down by HPBW (30∘ at 6.75 GHz/15∘ at 16.95 GHz). RX is then swept 360∘ in the azimuth plane in HPBW steps.

With key TX AODs and the reference MPC identified, the RX azimuthal sweeps,
where the RX is rotated 360 ∘ in the azimuth in HPBW increments, are initiated
to capture multipath in the channel. Five different RX azimuthal sweeps, as
shown in Table II are completed for each TX AOD. With the TX and RX at
boresight elevation, the RX steps through the azimuth in HPBW steps, referred
to as Sweep#1 in Table II. At each AOA, during the azimuthal scans, the RX
averages 20 instantaneous PDPs and logs the system and spatial information
corresponding to the captured directional PDP. In the 16.95 GHz FR3 campaign,
an RX azimuthal sweep includes 24 AOAs corresponding to the 15∘ HPBW
(15∘$\times$24 = 360∘), while the 6.75 GHz FR1(C) campaign includes 12 AOAs
using a 30∘ HPBW (30∘$\times$12 = 360∘). Omnidirectional PDPs are synthesized
from directional PDPs using the method described in [18] despite different
antenna beamwidths to obtain omnidirectional channel statistics.

Following Sweep#1, for the same particular TX AOD, the RX elevation is down-
and up- tilted by HPBW for sequential azimuthal sweeps (Sweep#2 and Sweep#3 in
Table II). For Sweep#4, the TX elevation is downtilted by HPBW and the stepped
sweep is performed with RX at boresight elevation. Then, Sweep#5 is conducted
with both TX and RX downtilted by HPBW. The TX then moves to the next
identified TX AOD and the five RX azimuthal sweeps are repeated until all
identified TX AODs are measured. After the TX steps through all identified
AODs, the RX azimuthal sweeps are repeated for a cross-polarized (V-H) antenna
configuration, whereby the TX horn antenna is oriented in V-polarization and
the RX horn antenna in H-polarization. Using the measured cross-polarization
discrimination of 35.7 and 38.4 dB at 6.75 GHz and 16.95 GHz respectively
[14], only the TX AODs that exhibited a peak received power in co-polarized
(V-V) configuration of 30 dB above the noise floor during rapid scans are
measured. The attenuation at RX must be lowered to capture the much weaker
multipath in the V-H configuration due to increased cross-polarization
propagation loss[19].

After collecting directional PDPs from RX azimuthal sweeps over all key TX
AODs at a T-R location pair, post processing on measured PDPs is conducted by
applying a power threshold to each directional PDP. The higher received power
of either 5 dB above the noise floor or 25 dB below the PDP peak is selected
as the threshold and powers below it are ignored in the processing of channel
statistics (Section IV B in [20]). The power azimuth spectrum (PAS) generated
from the directional PDPs at each T-R location produced wide spatial lobes
with power received at several contiguous HPBW pointing directions. A spatial
lobe is defined as contiguous pointing directions with received power
exceeding a spatial lobe threshold (SLT) in the PAS, below which powers in the
PAS are ignored for calculating angular statistics such as RMS AS[20]. An SLT
of 10 dB below the peak received power in the PAS is used for defining the
spatial lobes in the InH campaigns, shown in Fig. 3 as the orange dotted
circle. An example of a PAS recorded for different TX AOD and RX AOA in the
16.95 GHz campaign for the TX1-RX2 pair (Fig. 2) is depicted in Fig. 3.

(a)

(b)

Figure 3: Power Azimuth Spectrum for TX1-RX2 in the 16.95 GHz campaign at: (a)
RX AOAs (b) TX AODs. [T-R separation: 27 m]

##  V Large Scale Path Loss Modeling

The close-in 1 m free space reference distance (CI) model, (1), for describing
the path loss is employed on the directional and omnidirectional path loss
measurements. The CI model efficiently describes PL with a single PLE (n), as
demonstrated in Appendix A [21].

$\begin{split}PL^{CI}(f_{c},d_{\text{3D}})\;\text{[dB]}&=\text{FSPL}(f_{c},1m)+\\\
&10n\log_{10}\left(\dfrac{d_{3D}}{d_{0}}\right)+\chi_{\sigma},\\\
\text{FSPL}(f_{c},1m)&=32.4+20\log_{10}\left(\dfrac{f_{c}}{1\;\text{GHz}}\right),\end{split}$
(1)

where FSPL$(f_{c},1\;\text{m})$ is obtained for carrier frequency $f_{c}$ GHz
at 1 m, $n$ is the PLE, and $\chi_{\sigma}$ is large-scale shadow fading
(zero-mean Gaussian r.v. with s.d. $\sigma^{CI}$ in dB) [22].

TABLE III: Directional and Omnidirectional CI PL model parameters for 6.75 and 16.95 GHz FR1(C) and FR3 InH campaigns, with comparison at 28, 73 and 142 GHz FR2 campaigns Campaign |  Distance (m) |  Antenna HPBW (TX/RX) | Directional path loss | Omni path loss
---|---|---|---|---
LOS | NLOS Best | NLOS | LOS | NLOS
n |  $\sigma$ (dB) |  n |  $\sigma$ (dB) |  n |  $\sigma$ (dB) |  n |  $\sigma$ (dB) |  n |  $\sigma$ (dB)
6.75 GHz (This work) |  11-97 |  (30∘/30∘) |  1.55 |  2.52 |  2.74 |  7.14 |  3.05 |  9.71 |  1.40 |  3.41 |  2.42 |  7.87
16.95 GHz (This work) |  11-97 |  (15∘/15∘) |  1.45 |  1.87 |  3.52 |  9.28 |  3.93 |  14.90 |  1.32 |  2.66 |  3.07 |  9.03
28 GHz [23] |  4-46 |  (30∘/30∘) |  1.70 |  2.90 |  3.30 |  10.80 |  4.4 |  12.10 |  1.2 |  1.80 |  2.7 |  9.70
73 GHz [24] |  4-46 |  (15∘/15∘) |  1.63 |  3.06 |  3.30 |  8.76 |  5.51 |  8.94 |  1.36 |  2.30 |  2.81 |  8.71
142 GHz [24] |  4-39 |  (8∘/8∘) |  2.05 |  2.89 |  3.21 |  6.03 |  4.60 |  13.80 |  1.74 |  3.62 |  2.83 |  6.07

Directional PL modeling results are based on the definitions for LOS,
NLOS${}_{\text{Best}}$, NLOS presented in Table IV in [22]. The scatter plot
of recorded PL in V-V configuration and the CI PL models are presented in Fig.
4

(a)

(b)

Figure 4: InH directional CI PL models and scatter plot for V-V polarization
at: (a) 6.75 GHz FR1(C); (b) 16.95 GHz FR3. [T-R separation: 11–97 m]

Omnidirectional PL modeling is carried out by superposition of the directional
received powers at each unique non-overlapping azimuth and elevation pointing
angles removing the antenna gain, following the method given in [18].

The CI PL model results for directional and omnidirectional PL, removing the
antenna gains, are tabulated in Table III along with results from past
measurement campaigns in indoor offices at mmWave (FR2) frequencies.

(a)

(b)

Figure 5: InH omnidirectional CI PL models and scatter plot for V-V
polarization at: (a) 6.75 GHz FR1(C); (b) 16.95 GHz FR3. [T-R separation:
11–97 m]

The LOS directional PL exponent (PLE) is obtained as 1.55 at 6.75 GHz and 1.45
at 16.95 GHz, which are slightly lower (i.e. the channel is less lossy) than
the directional LOS PLEs at 28, 73, and 142 GHz (1.7, 1.63, and 2.05,
respectively). Further, the NLOS directional PLE at 6.75 and 16.95 GHz are
3.05 and 3.93, which are significantly lower than the PLEs at 28, 73, and 142
GHz (4.4, 5.51, and 4.6, respectively). The NLOS${}_{\text{Best}}$ PLEs are
found to be 2.74 and 3.52 at 6.75 and 16.95 GHz, respectively, which are
comparable to higher mmW channels. The omnidirectional PL modeling resulted in
LOS PLEs of 1.4 at 6.75 GHz and 1.32 at 16.95 GHz. Similarly, NLOS PLEs of
2.42 and 3.07 are obtained at 6.75 GHz and 16.95 GHz, respectively, which are
similar to the PLE values at the mmwave frequencies.

##  VI Spatio-Temporal InH Statistics

The RMS DS is a measure of the temporal dispersion of multipath in a wideband
channel. The power threshold of 25 dB below PDP maxima or 5 dB above noise
floor, discussed in Section IV, is used to calculate RMS DS. The mean of the
RMS DS observed in the directional PDPs captured in the 6.75 GHz campaign is
obtained as 19.3 ns and 21.7 ns in LOS and NLOS, whereas, the directional RMS
DS from the 16.95 GHz measurements are obtained as 19.5 ns and 17.01 ns. After
synthesizing the omnidirectional PDP for each TX-RX location [18], the RMS DS
in the omnidirectional PDP is observed as 33.7 ns and 43.5 ns in LOS and NLOS
at 6.75 GHz, and 22.1 ns and 40.7 ns in LOS and NLOS at 16.95 GHz. It is clear
that the 6.75 GHz band has slightly greater RMS DS compared to the 16.95 GHz
band.

Further, Table IV compares the RMS DS with results at higher frequencies. The
RMS DS experiences a clear decreasing trend with increasing frequency, which
is evident across both directional and omnidirectional RMS DS values in LOS
and NLOS conditions.

The RMS AS captures the azimuthal angular dispersion in the PAS, as shown in
Fig. 3. Fig. 7 shows the CDF of the lobe and omni AS for the AOA PAS at 6.75
and 16.95 GHz for both LOS and NLOS scenarios. The mean AS is found to be
higher in NLOS than LOS for both lobe and omni AS, which has also been
reported at higher frequencies[23, 24]. Omni AOA AS (ASA) are observed at 34∘
in LOS and 58∘ in NLOS at 6.75 GHz, whereas they are found lower at 16.95 GHz
at 18∘ in LOS and 43∘ in NLOS. This further confirms less multipath dispersion
in both time and space at higher mmWave frequencies, as seen in [24, 23].

(a)

(b)

(c)

(d)

Figure 6: InH LOS and NLOS RMS DSs at different frequencies: (a) Directional
at 6.75 GHz; (b) Directional at 16.95 GHz; (c) Omnidirectional at 6.75 GHz;
(d) Omnidirectional at 16.95 GHz.

(a)

(b)

(c)

(d)

Figure 7: InH LOS and NLOS RMS AS for AOA PASs at different frequencies: (a) Lobe ASA at 6.75 GHz; (b) Lobe ASA at 16.95 GHz; (c) Omni ASA at 6.75 GHz; (d) Omni ASA at 16.95 GHz.  TABLE IV: RMS Delay Spread Characteristics from 6.75 GHz to 142 GHz for LOS and NLOS Conditions. Frequency (GHz) | 6.75 | 16.95 | 28[23] | 73[24] | 142[24]
---|---|---|---|---|---
Dir RMS DS
LOS [$\mu$ (ns)] | 19.3 | 19.5 | 3.9 | 3.5 | 2.7
NLOS [$\mu$ (ns)] | 21.7 | 14.9 | 14.5 | 10.0 | 7.2
Omni RMS DS
LOS [$\mu$ (ns)] | 33.7 | 22.1 | 10.8 | 6.2 | 3.0
NLOS [$\mu$ (ns)] | 43.5 | 40.7 | 17.1 | 12.3 | 9.2

##  VII Conclusion

This paper reported comprehensive indoor propagation measurement campaigns for
the FR1(C) and FR3 frequency bands in the indoor office/labs at the NYU
WIRELESS Research Center. The results from the omnidirectional and directional
CI PL modeling with a 1 m free space reference distance showed a waveguiding
effect in the indoor hallways with omni PLEs of 1.40 and 1.32 in LOS at 6.75
and 16.95 GHz, respectively. A decrease in omnidirectional and directional RMS
DS and azimuthal angular spread with increasing frequency was observed when
compared with mmWave frequencies, indicating more multipath-rich propagation
at the FR3 and FR1(C) frequencies. PAS showed widespread multipath arrival and
departure directions, and spatial lobe and omnidirectional RMS AS was found
narrower at 16.95 GHz compared to 6.75 GHz. The empirical data and models
generated provide insights for future indoor propagation modeling in the 3GPP
frequency bands for 5G and 6G and support the analysis and design of wireless
systems and networks.

## Acknowledgment

Authors thank Profs. Sundeep Rangan and Marco Mezzavilla for support with
licensing and Tomoki Yoshimura from Sharp Labs of America for support in the
measurements.

## References

  * [1] A. Davidson, “National spectrum strategy implementation plan,” National Telecommunications and Information Administration, Tech. Rep., Mar. 2024.
  * [2] S. Kang _et al._ , “Cellular wireless networks in the upper mid-band,” _IEEE Open Journal of the Communications Society_ , pp. 1–18, 2024.
  * [3] Nokia, “Spectrum for 6G explained,” https://www.nokia.com/about-us/newsroom/articles/spectrum-for-6G-explained/, 2024, accessed: 2024-04-02.
  * [4] W. Tong and P. Zhu, _New Spectrum_. Cambridge University Press, 2021, p. 146–157.
  * [5] 3GPP, “Study on the 7 to 24 GHz frequency range for NR,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 38.820, 2021, https://www.3gpp.org/DynaReport/38820.htm.
  * [6] X. Zhou _et al._ , “Indoor wideband channel measurements and analysis at 11 and 14 GHz,” _IET Microwaves, Antennas & Propagation_, vol. 11, no. 10, pp. 1393–1400, 2017.
  * [7] Q. Wei et al., “Measurement-based Analysis of XL-MIMO Channel Characteristics in a Corridor Scenario,” in _(Accepted) IEEE VTC Spring 2024_ , 2024, pp. 1–6.
  * [8] N. O. Oyie and T. J. O. Afullo, “Measurements and analysis of large-scale path loss model at 14 and 22 GHz in indoor corridor,” _IEEE Access_ , vol. 6, pp. 17 205–17 214, 2018.
  * [9] G. J. Janssen, P. A. Stigter, and R. Prasad, “Wideband indoor channel measurements and ber analysis of frequency selective multipath channels at 2.4, 4.75, and 11.5 GHz,” _IEEE Transactions on Communications_ , vol. 44, no. 10, pp. 1272–1288, 1996.
  * [10] S. Deng, G. R. MacCartney, and T. S. Rappaport, “Indoor and outdoor 5G diffraction measurements and models at 10, 20, and 26 GHz,” in _2016 IEEE Global Communications Conference (GLOBECOM)_ , 2016, pp. 1–7.
  * [11] D. Shakya _et al._ , “A Wideband Sliding Correlation Channel Sounder in 65 nm CMOS: Evaluation Board Performance,” _IEEE Transactions on Circuits and Systems II: Express Briefs_ , vol. 68, no. 9, pp. 3043–3047, 2021\.
  * [12] G. R. MacCartney and T. S. Rappaport, “A flexible millimeter-wave channel sounder with absolute timing,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 6, pp. 1402–1418, 2017.
  * [13] D.Shakya et al., “Comprehensive FR3 and FR1C upper-mid band propagation measurements and models in Indoor Office environment for 5G and 6G communications,” _(Submitted) IEEE Open Journal of Comm. Soc._ , pp. 1–13, 2024.
  * [14] D. Shakya et al., “Wideband Penetration Loss through Building Materials and Partitions at 6.75 GHz in FR1(C) and 16.95 GHz in the FR3 Upper Mid-band spectrum,” in _(Submitted) IEEE GLOBECOM 2024_ , 2024, pp. 1–6.
  * [15] Y. Xing _et al._ , “Verification and calibration of antenna cross-polarization discrimination and penetration loss for millimeter wave communications,” in _2018 IEEE 88th Vehicular Technology Conference (VTC-Fall)_ , 2018, pp. 1–6.
  * [16] D. Shakya, H. Poddar, and T. S. Rappaport, “A sub-terahertz sliding correlator channel sounder with absolute timing using precision time protocol over wi-fi,” in _IEEE GLOBECOM 2023_ , 2023, pp. 5793–5798.
  * [17] S. Ju et al., “142 GHz sub-terahertz radio propagation measurements and channel characterization in factory buildings,” _IEEE Transactions on Wireless Communications_ , pp. 1–17, 2023.
  * [18] S. Sun _et al._ , “Synthesizing omnidirectional antenna patterns, received power and path loss from directional antennas for 5G millimeter-wave communications,” in _2015 IEEE GLOBECOM_ , 2015, pp. 1–7.
  * [19] Y. Xing _et al._ , “Indoor wireless channel properties at millimeter wave and sub-terahertz frequencies,” in _2019 IEEE GLOBECOM_ , 2019, pp. 1–6.
  * [20] D. Shakya _et al._ , “Radio Propagation Measurements and Statistical Channel Models for Outdoor Urban Microcells in Open Squares and Streets at 142, 73, and 28 GHz,” _IEEE Transactions on Antennas and Propagation_ , pp. 1–15, Feb 2024.
  * [21] G. R. Maccartney _et al._ , “Indoor office wideband millimeter-wave propagation measurements and channel models at 28 and 73 GHz for ultra-dense 5G wireless networks,” _IEEE access_ , pp. 2388–2424, 2015\.
  * [22] T. S. Rappaport _et al._ , “Wideband millimeter-wave propagation measurements and channel models for future wireless communication system design,” _IEEE Transactions on Communications_ , vol. 63, no. 9, pp. 3029–3056, 2015.
  * [23] S. Ju _et al._ , “Millimeter wave and sub-terahertz spatial statistical channel model for an indoor office building,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 6, pp. 1561–1575, 2021.
  * [24] Y. Xing, T. S. Rappaport, and A. Ghosh, “Millimeter wave and sub-thz indoor radio propagation channel measurements, models, and comparisons in an office environment,” _IEEE Communications Letters_ , vol. 25, no. 10, pp. 3151–3155, 2021.


# Braneworlds in Horndeski gravity

F.A. Brito,a,b and F.F. dos Santos,a aDepartamento de Física, Universidade
Federal da Paraíba, Caixa Postal 5008, 58051-970 João Pessoa, Paraíba, Brazil
bDepartamento de Física, Universidade Federal de Campina Grande, Caixa Postal
10071, 58109-970 Campina Grande, Paraíba, Brazil

###### Abstract

In this paper we address the issue of finding braneworld solutions in a five-
dimensional Horndeski gravity and the mechanism of gravity localization into
the brane via “almost massless modes” for suitable values of the Horndeski
parameters. We compute the corrections to the Newtonian potential and discuss
the limit where four-dimensional gravity is recovered.

###### pacs:

xxx,yyy

##  I Introduction

In this paper we will focus on the study of braneworld solutions and
localization of four-dimensional gravity in a Horndeski theory of gravity in
five dimensions. This theory follows the same class of coupled scalar-tensor
theories such as the Brans-Dicke theory whose scalar field is non-minimally
coupled to the Einstein-Hilbert term. The main difference here is that the
Einstein tensor couples non-minimally to the derivative squared of a dynamical
scalar field. In order to find such a theory of gravity Horndeski horndeski74
firstly assumed the most general second-order Euler-Lagrange tensors derivable
from a Lagrangian that is concomitant of a pseudo-Riemannian metric-tensor, a
scalar field and their derivatives of arbitrary order in four-dimensional
space — although higher dimensional gravity was also discussed in horndeski74
. The conclusion is that these Euler-Lagrange tensors may be obtained from a
Lagrangian which is at most of second order in the derivatives of the field
functions. Several recent studies in black holes and gravitational waves P ; 5
; Cisterna:2014nua ; Bettoni:2016mij  have been considered in this context
which imposes bounds on the parameters of the theory. Since braneworlds can be
understood as an effective low energy theory of superstrings which in turn can
enclose the most general theories of gravity, it seems mandatory to
investigate the phenomenon of localization of four-dimensional gravity in the
Horndeski theory. The localization of gravity on a 3-brane Randall  arises in
the sense of being an alternative for compactification involving infinite
extra dimensions. In the Randall-Sundrum scenarios Randall ; Sundrum ,
3-branes are embedded into $AdS_{5}$ $bulk$ space which is an ambient space
developing 5-dimensional gravity with a negative cosmological constant
($\Lambda_{5}<0$) and infinitely thin sources of 3-branes composed of delta
functions. In this setup there is a perfect fine-tuning between the brane
tension and the cosmological constant $\Lambda_{5}$. Thus, the fine adjustment
leads to a 4d brane in Minkowski $(M_{4})$ with a cosmological constant of
four dimensions $\Lambda_{4d}=0$, in such a way that only the space $AdS_{5}$
is curved. In the limit of thick branes their profile are described by scalar
fields, which is the main proposal here. The graviton zero-mode localized into
the 3-brane is responsible for a localized 4d gravity, so that the correction
for Newtonian potential due to Kaluza-Klein gravitons is highly suppressed in
the case of low energies. Even in the case of graviton ‘quasi zero mode’ 2 ,
i.e, the case of massive gravity, the localization of gravity can still be
achieved as metastable localization. In this work we will take an approach
with respect to the DeWolfe ; 1 ; Chico , in order to evaluate how the
graviton will be localized in the brane with the use of Horndeski gravity.

The paper is organized as follows. In Sec. II we present the Horndeski gravity
and in Sec. III we develop the first order formalism in five dimensions. In
Sec. IV we present two explicit examples and in Sec. V we compute numerical
solutions. In Sec. VI we address the issue of the graviton fluctuations and in
Sec. VII we compute the corrections to the Newtonian potential. Finally in
Sec. VIII we present our final comments.

##  II The Horndeski gravity with a scalar potential

In our present investigation we shall address the study of braneworlds in the
framework of the Horndeski gravity which action with a scalar potential reads

$I[g_{MN},\phi]=\int{\sqrt{-g}d^{D}x\left[k(R-2\Lambda)-\frac{1}{2}(\alpha
g_{MN}-\eta G_{MN})\nabla^{M}\phi\nabla^{N}\phi-V(\phi)\right]}$ (1)

The Einstein field equations can still be given in the form

$G_{MN}+\Lambda g_{MN}=\frac{1}{2k}T_{MN}$ (2)

where $T_{MN}=\alpha T^{(1)}_{MN}-g_{MN}V(\phi)+\eta T^{(2)}_{MN}$ with

$\begin{array}[]{rclrcl}T^{(1)}_{MN}&=&\nabla_{M}\phi\nabla_{N}\phi-\frac{1}{2}g_{MN}\nabla_{P}\phi\nabla^{P}\phi\\\
T^{(2)}_{MN}&=&\frac{1}{2}\nabla_{M}\phi\nabla_{N}\phi
R-2\nabla_{P}\phi\nabla_{(M}\phi R^{P}_{N)}-\nabla^{P}\phi\nabla^{K}\phi
R_{MPNK}\\\
&-&(\nabla_{M}\nabla^{P}\phi)(\nabla_{N}\nabla_{P}\phi)+(\nabla_{M}\nabla_{N}\phi)\square\phi+\frac{1}{2}G_{MN}(\nabla\phi)^{2}\\\
&-&g_{MN}\left[-\frac{1}{2}(\nabla^{P}\nabla^{K}\phi)(\nabla_{P}\nabla_{K}\phi)+\frac{1}{2}(\square\phi)^{2}-(\nabla_{P}\phi\nabla_{K}\phi)R^{PK}\right]\end{array}$
(3)

and the scalar field equation is

$\nabla_{M}[(\alpha g^{MN}-\eta G^{MN})\nabla_{N}\phi]=V_{\phi}$ (4)

The metric Ansatz to be studied in five dimensions is of the form

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{MN}dx^{M}dx^{N}$ (5)
$\displaystyle=$ $\displaystyle e^{2A(r)}g_{\mu\nu}dx^{\mu}dx^{\nu}-dr^{2}$

where the latin indices $M,N=$ $0$, $1$, $2$, $3$ and $5$ run on the bulk and
the greek indices $\mu,\nu=$ $0$, $1$, $2$ and $3$ run along the braneworld.

##  III Equations of motion

By considering the metric (5) we have the following Einstein equation
components involving the scalar potential. First, from equation (2) the
$rr$-component is given by

$2V(\phi)+4k(\Lambda+6A^{{}^{\prime}2})-\Psi^{2}(r)[\alpha-18\eta
A^{{}^{\prime}2}]=0,$ (6)

where $\Psi(r)=\phi^{{}^{\prime}}(r)$. This equation can be rewritten as
follows

$\Psi(r)=\pm\sqrt{\frac{2V(\phi)+4k\Lambda+24kA^{{}^{\prime}2}(r)}{\alpha-18\eta
A^{{}^{\prime}2}(r)}}$ (7)

Now taking the relationship between $A(r)$ and the superpotential $W(\phi)$
through the first-order differential equation

$A^{{}^{\prime}}(r)=-\frac{1}{3}W(\phi)$ (8)

and redefining $24k\equiv b$ we can rewrite the equation (7) in the form

$\Psi(r)=\pm\sqrt{\frac{2V(\phi)+\frac{b\Lambda}{6}+\frac{b}{9}W^{2}(\phi)}{(\alpha-2\eta
W^{2}(\phi))}}$ (9)

For a potential of the form

$V(\phi)=-\frac{b}{18}W^{2}(\phi)+\frac{c^{2}W^{2}_{\phi}(\phi)}{2}(\alpha-2\eta
W^{2}(\phi))$ (10)

and taking $\Lambda=0$, for simplicity, we find

$\Psi(r)=\pm{cW_{\phi}(\phi)}$ (11)

For $\alpha=1,c=1/2,\eta=0$ and $b=6$ (i.e., $4\pi G=1$) we find the familiar
scalar potential and scalar field equation of braneworlds in five-dimensional
Einstein gravity 1 ; DeWolfe .

Another equation from the Einstein equations can be found by combining the
$tt$-component with $xx$, $yy$ or $zz$-components to find

$2V(\phi)+6\eta\Psi(r)\Psi^{{}^{\prime}}(r)A^{{}^{\prime}}(r)+4k[\Lambda+6A^{{}^{\prime}2}(r)+3A^{{}^{\prime\prime}}(r)]+\Psi^{2}(r)[\alpha+6\eta
A^{{}^{\prime}2}(r)+3\eta A^{{}^{\prime\prime}}(r)]=0$ (12)

A third equation describing the scalar field dynamics comes from the equation
(4)

$V_{\phi}(\phi)+4A^{{}^{\prime}}\phi^{{}^{\prime}}(r)[-\alpha+6\eta
A^{{}^{\prime}2}(r)+3\eta A^{{}^{\prime\prime}}(r)]-(\alpha-6\eta
A^{{}^{\prime}2}(r))\phi^{{}^{\prime\prime}}=0$ (13)

Using the following relationships $\phi^{{}^{\prime}}=cW_{\phi}$,
$A^{{}^{\prime\prime}}=-cW^{2}_{\phi}/3$ we can write these equations as

$2cWW_{\phi\phi}+cW^{2}_{\phi}+\frac{4}{3}W^{2}-\frac{(2\alpha-b/6c)}{\eta}=0$
(14)

This implies that the equations (6), (12) and (13) are consistently satisfied
by equation (8) and (11) under the nontrivial condition (14) on the
superpotential $W(\phi)$.

##  IV examples

i) For $\alpha=b/12c$ with $c=1/2\eta$ 111Note this choice is dimensionally
correct since $[\eta]=1/mass^{2}$ and $[b]=[6/4\pi G]=mass^{2}$., Eq. (14)
becomes a homogeneous differential equation that gives the solution

$W(\phi)=\left(\frac{1}{2}\right)^{2/3}\left(-C_{1}\sin\left(\frac{\phi}{\sqrt{c}}\right)-C_{2}\cos\left(\frac{\phi}{\sqrt{c}}\right)\right)^{2/3}$
(15)

With this superpotential we can easily solve the first-order differential
equations by implementing a numerical Runge-Kutta method. We come back to this
approach shortly. However, by expanding this superpotential for $\eta\ll 1$ we
have

$W(\phi)=\gamma+\sigma\phi$ (16)

where the $\gamma=(-C_{2}/2)^{2/3}$ and
$\sigma=2C_{1}(-C_{2}/2)^{2/3}/3\sqrt{c}C_{2}$ are constants. Note that for
$C_{2}$ very small we can suppress $\gamma$. Using
$\phi^{{}^{\prime}}=cW_{\phi}$ we can write

$W(r)=c\sigma^{2}(r-r_{0})$ (17)

where $\phi=c\sigma r$. Thus, we find

$A(r)=-\frac{\sigma^{2}}{12\eta}(r-r_{0})^{2}$ (18)

or for $\sigma^{2}/6\eta=q$ and $r_{0}=0$ we can simply write

$A(r)=-\frac{q}{2}r^{2}$ (19)

ii) For $\beta=2(\alpha-1)/\eta$ with $b=6,c=1/2$, Eq. (14) is not a
homogeneous differential equation but can still be approximately solved by the
linear solution

$W(\phi)=\sigma\phi,\qquad\sigma=\sqrt{2\beta}$ (20)

where $\sigma\ll 1$ for very large $\eta$. Thus by using Eqs. (8) and (11) we
have

$\phi(r)=\frac{1}{2}\sigma r,\qquad A(r)=-\frac{q}{2}r^{2}$ (21)

where $q=\sigma^{2}/6$. $A(r)$ is depicted in Fig. 1 for distinct values of
$\eta$ — similar type of geometry has been already considered in confining
AdS/QCD 6 . The scalar solutions discussed in the two examples above can be
understood as very thick ‘kinks’ which have very small slope. We shall show
numerically an evidence that the slope of the kink type solution of the exact
differential equations indeed tends to decrease as $\eta$ increases.

Figure 1: The figure shows $A(r)$ for $q=\sigma^{2}/6$ and $\alpha=1.5$,
$\sigma=\sqrt{4(\alpha-1)/\eta}$, $\eta=10$-blue, $\eta=20$-red,
$\eta=40$-green and $\eta=60$-black.

The energy density is given by

$\displaystyle\rho(r)=-e^{2A}\left(\frac{3\alpha
A^{{}^{\prime\prime}}}{2}+3A^{{}^{\prime}2}\right)-\frac{9\eta
e^{2A}}{4}(A^{{}^{\prime}}A^{{}^{\prime\prime\prime}}-4A^{{}^{\prime}2}A^{{}^{\prime\prime}}+A^{{}^{\prime\prime}2})$
(22)

and for such solutions the energy density is depicted in Fig. 2 for several
large values of $\eta$.

Figure 2: The energy density $\rho(r)$ for small values of $q=\sigma^{2}/6$,
$\alpha=1.5$, $\sigma=\sqrt{4(\alpha-1)/\eta}$, $\eta=10$-blue, $\eta=20$-red,
$\eta=40$-green and $\eta=60$-black.

##  V Numerical solutions

In the examples above we were able to find explicit solutions in a restrict
regime of value of $\alpha$ and $\eta$. However, the pair of first-order
equations (8) and (11) can be solved numerically for a broader range of values
as long as we assume appropriate boundary conditions. In Fig. 3 we show the
behavior of the kink profile and geometry associated with the braneworld
solutions for $\beta=2$ and $5$ for $c=1/2$, which means $2(\alpha-1)/\eta$=2
and 5, respectively, that for $\alpha=3/2$ we have $\eta=0.5$ (red curve) and
$\eta=0.2$ (blue curve). This is the regime of small $\eta$, which was not
addressed previously in the non-homogeneous limit of Eq. (14). The warp factor
signalizes the existence of an asymmetric brane, which in general is not able
to localize graviton zero mode, but at least one may find metastable gravity.
Lower values of $\eta<1$ seems to force brane asymmetry. The boundary
conditions used was the following: $W(0)=1$, $W^{\prime}(0)=1$, $\phi(0)=0$
and $A(0)=0$.

Figure 3: The figure shows the kinks $\phi(r)$ (top-left) and $A(r)$ (top-
right) for $\beta=2$ (red) and $\beta=5$ (blue) for $c=1/2$. In the bottom we
show the warp factor $\exp{(2A(r))}$ signalizing the existence of an
asymmetric brane.

##  VI Equation for fluctuations

Let us now compute the linearization of the Einstein equations by considering
the following perturbations $\bar{g}_{MN}=g_{MN}+h_{MN}$ and
$\bar{\phi}=\phi+\tilde{\phi}$ in the axial gauge $h_{M5}=0$, where $M$ is a
bulk index. Thus, the fluctuation in the metric (5) can be written in the form

$\displaystyle ds^{2}=e^{2A(r)}(g_{\mu\nu}+h_{\mu\nu})dx^{\mu}dx^{\nu}-dr^{2}$
(23)

Now, performing the following first-order perturbations

$\displaystyle\delta^{(1)}g_{\mu\nu}=h_{\mu\nu},\quad\delta^{(1)}\phi=\tilde{\phi},$
(24)

into the Einstein equations in the Ricci form DeWolfe ; MG  the linearized
equations for the graviton sector become

$\displaystyle\delta^{(1)}R_{\mu\nu}=\frac{\delta^{(1)}\bar{T}_{\mu\nu}}{k}$
(25)

where

$\displaystyle\delta^{(1)}R_{\mu\nu}$ $\displaystyle=$ $\displaystyle
e^{2A}\left[\frac{1}{2}(\partial^{2}_{r}+4A^{{}^{\prime}}\partial_{r}-e^{-2A}\Box)h_{\mu\nu}+(4A^{{}^{\prime}2}+A^{{}^{\prime\prime}})h_{\mu\nu}\right]$
(26) $\displaystyle-$
$\displaystyle\frac{g^{\gamma\beta}}{2}(\partial_{\mu}\partial_{\nu}h_{\gamma\beta}-\partial_{\mu}\partial_{\gamma}h_{\nu\beta}-\partial_{\nu}\partial_{\gamma}h_{\mu\beta})+\frac{g_{\mu\nu}e^{2A}A^{{}^{\prime}}\partial_{r}(g^{\beta\gamma}h_{\gamma\beta})}{2},$
(27)

and

$\displaystyle\delta^{(1)}\bar{T}_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{{e^{2A}}}{3}\left(\frac{\partial
V(\phi)}{\partial\phi}\tilde{\phi}\
g_{\mu\nu}+V(\phi)h_{\mu\nu}\right)+\frac{3\eta
e^{2A}}{4}(8A^{{}^{\prime\prime}}A^{{}^{\prime}2}+A^{{}^{\prime\prime}2}+A^{{}^{\prime}}A^{{}^{\prime\prime\prime}})h_{\mu\nu}$
(28)

Then the $h_{\mu\nu}$ components in both sides of equation (25) can be
combined together by using the definition of the scalar potential (10) with
the relations between $\phi^{{}^{\prime}}(r)$ and $A(r)$ through Eqs. (8) and
(11). This enables to write the equation

$\displaystyle-\frac{4}{3}V(\phi)$ $\displaystyle=$ $\displaystyle
4A^{{}^{\prime}2}+\alpha A^{{}^{\prime\prime}}+18\eta
A^{{}^{\prime\prime}}A^{{}^{\prime}2}$ (29)

Finally we use the transverse traceless components of $h_{\mu\nu}$, defined by
the non-local projection:

$\bar{h}_{\mu\nu}=\left(\frac{1}{2}(\pi_{\mu\gamma}\pi_{\nu\beta}+\pi_{\mu\beta}\pi_{\nu\gamma})-\frac{\pi_{\mu\nu}\pi_{\gamma\beta}}{3}\right)h^{\beta\gamma}=h_{\mu\nu}+...$
(30)

where $\pi_{\mu\nu}\equiv(g_{\mu\nu}-\partial_{\mu}\partial_{\nu}/\Box)$. The
$\cdots$ indicate nonlocal terms. For $\bar{h}_{\mu\nu}$ satisfying
$\partial^{\nu}\bar{h}_{\mu\nu}=g^{\mu\nu}\bar{h}_{\mu\nu}=0$, we can write
the following equation for the graviton fluctuations

$\bar{h}^{{}^{\prime\prime}}_{\mu\nu}+4A^{\prime}\bar{h}^{{}^{\prime}}_{\mu\nu}-e^{-2A}\Box_{4d}\bar{h}_{\mu\nu}+2A^{{}^{\prime\prime}}(1-\alpha)\bar{h}_{\mu\nu}-6\eta(14A^{{}^{\prime\prime}}A^{{}^{\prime}2}+A^{{}^{\prime\prime}2}+A^{{}^{\prime}}A^{{}^{\prime\prime\prime}})\bar{h}_{\mu\nu}=0$
(31)

We now considering the use of the Ansatz
$\bar{h}_{\mu\nu}=h(r)M_{\mu\nu}(x^{\mu})$, we separate Eq. (31) into two
equations that describe the graviton. i.e.,
$\square_{4d}M_{\mu\nu}=m^{2}M_{\mu\nu}$ and another one whose masses and
eigenfunctions are governed by a Schroedinger type equation that appears
through the following change of variables: $h(r)=e^{3A(z)/2}\psi(z)$,
$z(r)=\int{e^{-A(r)}}dr$ (where we have taken the relationship
$dz=e^{-A(r)}dr$). The Schroedinger type equation is then given by

$-\partial^{2}_{z}\psi(z)+V(z)\psi(z)=m^{2}\psi(z)$ (32)

with a potential of the form

$V(z)=\frac{9}{4}A^{{}^{\prime}2}(z)+\frac{3}{2}A^{{}^{\prime\prime}}(z)+2A^{{}^{\prime\prime}}(1-\alpha)-6\eta(14A^{{}^{\prime\prime}}A^{{}^{\prime}2}+A^{{}^{\prime\prime}2}+A^{{}^{\prime}}A^{{}^{\prime\prime\prime}})$
(33)

This is an unusual potential as compared with those in the literature DeWolfe
; 1 ; Chico ; 2 . However, one can easily recover the usual case as $\alpha=1$
and $\eta=0$, otherwise we have the potential derived from Horndeski gravity.
To solve the equation (32) let us first recall the above transformation of
variables between $r$ and $z$ by using $z(r)=\int{e^{-A(r)}}dr$ such that we
have

$z(r)=\sqrt{\frac{\pi}{2q}}{\rm
erf}\left(\sqrt{\frac{q}{2}}r\right)=\frac{2}{\sqrt{2q}}\sum^{\infty}_{k=0}\frac{(-1)^{k}\left(\sqrt{\frac{q}{2}}r\right)^{2k+1}}{k!(2k+1)}$
(34)

whose first terms are

$z(r)=\frac{2}{\sqrt{2q}}\left(\sqrt{\frac{q}{2}}r-\frac{(\sqrt{\frac{q}{2}}r)^{3}}{3}+\frac{(\sqrt{\frac{q}{2}}r)r^{5}}{10}+...\right)$
(35)

Now recalling that for $\eta\gg 1$ implies $q\ll 1$, for the second example
above, then taking the leading term for this approximation we find $z=r$,
which allows us to write

$A(z)=-\frac{q}{2}z^{2}$ (36)

Using this geometry the potential (33) can be written as follows

$V(z)=84q^{3}\eta z^{2}+\frac{9}{4}q^{2}z^{2}+q\left(2\alpha-6\eta
q-\frac{7}{2}\right)$ (37)

Figure 4: The potential $V(z)$ in equation (38) for small values of
$q=\sigma^{2}/6$ due to the geometry given with the values of
$\sigma=\sqrt{4(\alpha-1)/\eta}$, $\alpha=1/4$, $\eta=10$-blue, $\eta=20$-red,
$\eta=40$-green and $\eta=60$-black.

As $\eta$ is sufficiently large, $q$ is tiny enough and assuming $\alpha=1/4$,
thus only the squared term in $q^{2}z^{2}$ becomes dominant and we can
approximate the potential (37) as follows

$\displaystyle V(z)=M^{2}z^{2},\qquad M\equiv\ \frac{9q^{2}}{4}$ (38)

such a way we can now rewrite the equation (32) in the form

$-\partial^{2}_{z}\psi(z)+M^{2}z^{2}\psi(z)=E\psi(z)$ (39)

Note that this equation is a Schroedinger-like problem for a quantum harmonic
oscillator whose solution is well-known in the quantum mechanics literature.
The equation (39) can be written as an eigenvalue equation, i.e.,
$H\psi=E\psi$ with the Hamiltonian

$H=-\partial^{2}_{z}+M^{2}z^{2}$ (40)

We can define the following “annihilation” and “creation” operators written as
follows

$Q=\frac{1}{\sqrt{M}}(-\partial_{z}-Mz)\quad\mbox{and}\quad
Q^{+}=\frac{1}{\sqrt{M}}(\partial_{z}-Mz)$ (41)

which implies in a Hamiltonian written in terms of such operators in the form

$H=M(N+1)$ (42)

where $N=Q^{+}Q$. Note that the energies for this “oscillator” are of the form
$E=M(n+1)$ which implies $m^{2}_{n}=M(n+1)$. Thus, the graviton spectrum reads

$m^{2}_{n}=M(n+1)$ (43)

For AdS/QCD theories that follows a similar linear glueball spectrum see, e.g.
G ; 3 . For $n=0$ we have that $m^{2}_{0}=M$. In the limit of large $\eta$ one
achieves $m^{2}_{0}\rightarrow 0$ that plays the role of a “quasi-zero mode”
which is responsible for the $4d$ gravity on the brane. Let us now evaluate
the wave-function of the quasi-zero mode through the equation $Q\psi_{0}(z)=0$
which implies

$\psi_{0}(z)=Ce^{\frac{-Mz^{2}}{2}}$ (44)

Figure 5: The wave-function $\psi_{0}(z)$ for values of $q=\sigma^{2}/6$,
$\alpha=1/4$, $\sigma=\sqrt{4(\alpha-1)/\eta}$, $\eta=10$-blue, $\eta=20$-red,
$\eta=40$-green and $\eta=60$-black.

The eigenfunctions of the tower of higher massive modes are given by the
solutions of equation (39) which read

$\psi_{n}(z)=\frac{C}{\sqrt{2^{n}n!}}\bar{H}_{n}(Mz)e^{\frac{-Mz^{2}}{2}}$
(45)

Figure 6: The wavefunctions $\psi_{n}(z)$ for values of $q=\sigma^{2}/6$ due
to the geometry given with the values of $\sigma=\sqrt{4(\alpha-1)/\eta}$,
$\alpha=1/4$, $n=0$/$\eta=10$-blue, $n=1$/$\eta=20$-red, $n=2$/$\eta=40$-green
and $n=3$/$\eta=60$-black.

where the functions $\bar{H}_{n}(Mz)$ are the well-known Hermite polynomials
and $C=\left(\frac{M}{\pi}\right)^{1/4}$.

##  VII Correction to Newtonian potential

The effective Newtonian potential in braneworlds scenarios can be found as in
the form 1

$\displaystyle
V(r)=\psi^{2}_{0}(0)\frac{e^{-m_{0}r}}{r}+\sum_{n}\psi^{2}_{n}(0)\frac{e^{-m_{n}r}}{r}$
(46)

Notice that the Hermite polynomials at $z=0$ contribute to the potential only
for even modes, that means

$\displaystyle\psi^{2}_{0}(0)$ $\displaystyle=$ $\displaystyle C^{2},$ (47)
$\displaystyle\psi^{2}_{n}(0)$ $\displaystyle=$
$\displaystyle\frac{C^{2}\bar{H}^{2}_{n}(0)}{2^{n}n!}=\frac{\pi
2^{n}C^{2}}{\Gamma\left(\frac{1-n}{2}\right)^{2}\Gamma(n+1)}$ (48)

As $\eta$ is very large $q\rightarrow 0$ the graviton mass spectrum approaches
a continuum and we can substitute the sum for an integral. Thus, we can write
7

$\displaystyle V(r)$ $\displaystyle\approx$ $\displaystyle
C^{2}\frac{e^{-\sqrt{M}r}}{r}+\frac{C^{2}}{r}\int{\frac{\pi
2^{n}}{\Gamma\left(\frac{1-n}{2}\right)^{2}\Gamma(n+1)}e^{-\sqrt{M(n+1)}r}dn}$
(49) $\displaystyle\approx$
$\displaystyle\frac{1}{r}\sqrt{\frac{M}{\pi}}\left(e^{-\sqrt{M}r}+\pi\int{\frac{2^{n}}{\Gamma\left(\frac{1-n}{2}\right)^{2}\Gamma(n+1)}e^{-\sqrt{M(n+1)}r}dn}\right)$
(50)

Let us now make a change of variables as in the form $x=M(n+1)$ to which we
have

$\displaystyle
V(r)\approx\frac{1}{r}\sqrt{\frac{M}{\pi}}\left(e^{-\sqrt{M}r}+\frac{\pi}{2M}\int{\frac{2^{\frac{x}{M}}}{\Gamma\left(1-\frac{x}{2M}\right)^{2}\Gamma\left(\frac{x}{M}\right)}e^{-x^{1/2}r}dx}\right)$
(51)

In the limit of highly excited modes, that is $x/M\rightarrow\infty$, the
arguments of the $\Gamma$ functions become large and we can use the following
approximation

$\displaystyle\frac{2^{\frac{x}{M}}}{\Gamma\left(1-\frac{x}{2M}\right)^{2}\Gamma\left(\frac{x}{M}\right)}\approx{\frac{2\sqrt{2}}{\pi^{3/2}}}\frac{\sqrt{M}}{\sqrt{x}\left(1-\cos^{2}\left(\frac{\pi}{2}(n+1)\right)\right)^{-1}}$
(52)

As we have previously mentioned (and also easily checked from Fig. 6) the
Hermite polynomials at $z=0$ do not contribute to the potential for odd modes
since $\bar{H}_{2n+1}(0)=0$, thus the cosine vanishes because its argument is
odd multiples of $\pi/2$ for any even number $n$. As such, we can write

$\displaystyle
V(r)\approx\frac{1}{r}\sqrt{\frac{M}{\pi}}\left(e^{-\sqrt{M}r}+\frac{\sqrt{2}}{\sqrt{\pi
M}}\int^{\infty}_{1/r_{c}^{2}}{\frac{1}{\sqrt{x}}e^{-x^{1/2}r}dx}\right)$ (53)

where $r_{c}$ is a crossover scale at large energies. We can now perform the
integral in the second term to find

$\displaystyle
V(r)\approx\frac{1}{r}\sqrt{\frac{M}{\pi}}\left(e^{-\sqrt{M}r}+\frac{2\sqrt{2}}{\sqrt{\pi
M}}\frac{e^{-\frac{r}{r_{c}}}}{r}\right)$ (54)

The result of the integral when calculated is real for the condition
$\Re[r\sqrt{M}>0]$ or $\Re(r)>0$. Recalling that $M$ is very small, the
exponential can still be approximated to unity. Thus, as $r\ll r_{c}$ we
obtain the final form of the corrected Newtonian potential as follows

$\displaystyle
V(r)\approx\sqrt{\frac{M}{\pi}}\left(\frac{1}{r}+\frac{2\sqrt{2}}{r^{2}\sqrt{\pi
M}}\right)$ (55)

However, for $r\gg r_{c}$ one can recover the four-dimensional gravity. In
addition, the computation of the weakly excited modes has shown to be able to
correct the potential with an extra term that goes with $1/r^{5}$, which is
easily suppressed at large distances in comparison with the $1/r$ term, but
strongly dominant at small distances.

##  VIII Conclusions

In this paper we have addressed the problem of localizing gravity in
braneworlds solutions found in Horndeski gravity. We found numerical solutions
and approximated explicit solutions. The parameter that deviates such theory
of gravity in relation to Einstein gravity controls the localization of four-
dimensional gravity in a non-trivial way as can be easily checked directly in
the induced Newtonian potential. Interestingly for sufficiently large $\eta$
the four-dimensional gravity is safely localized on the brane. This is also
the regime where one can easily find explicit braneworlds solutions. It was in
this regime that we restricted ourselves to the analysis of the Newtonian
potential. Further studies for arbitrary values of $\eta$ should be addressed
elsewhere.

###### Acknowledgements.

We would like to thank CNPq and CAPES for partial financial support. We also
thank Cristián Erices for useful discussions in the early stages of this work.

## References

  * (1) HORNDESKI, G.W. Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10, 363 (1974). doi:10.1007/BF01807638.
  * (2) FENG, X. H.; LIU, H. S.; LÜ, H.; POPE, C. N. Black Hole Entropy and Viscosity Bound in Horndeski Gravity. JHEP., v.83, p. 4690, 2015; [arXiv:hep-th/1509.07142].
  * (3) ANABALON, A.; CISTERNA, A.; OLIVA, J. Asymptotically locally AdS and flat black holes in Horndeski theory. (2014); [arXiv:hep-th/13123597].
  * (4) CISTERNA, A.; ERICES, C. “Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario,” Phys. Rev. D 89, 084038 (2014) doi:10.1103/PhysRevD.89.084038 [arXiv:1401.4479 [gr-qc]].
  * (5) BETTONI, D.; EZQUIAGA, J.M.; HINTERBICHLER, K.; ZUMALAC RREGUI, M. Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity, Phys. Rev. D 95, no. 8, 084029 (2017) doi:10.1103/PhysRevD.95.084029 [arXiv:1608.01982 [gr-qc]].
  * (6) RANDALL, L.; SUNDRUM, R. An Alternative to Compactification. Phys. Rev. Lett., v.83, p. 4690, 1999; [arXiv:hep-th/9906064].
  * (7) RANDALL, L.; SUNDRUM, R. A Large Mass Hierarchy from a Small Extra Dimension. Phys. Rev. Lett., v.83, p. 3370, (1999); [arXiv:hep-ph/9905221].
  * (8) KARCH, A.; RANDALL, L. Locally Localized Gravity. JHEP, v.8, n.5, p. 008, (2001); [arXiv:hep-th/0011156].
  * (9) BAZEIA, D.; BRITO, F. A.; GOMES, A. R. Locally localized gravity and geometric transitions. JHEP, v.4, n.11, p. 070, (2004); [arXiv:hep-th/0411088.
  * (10) CSAKI, C.; ERLICH, J.; HOLLOWOOD, T. J.; SHIRMAN, Y. Universal aspects of gravity localized on thick branes. Nucl. Phys. B, v.581, p. 309-338, (2000); [arXiv:hep-th/0001033].
  * (11) DEWOLFE, O.; FREEDMAN, D. Z.; GUBSER, S. S.; KARCH, A. Modeling the fifth dimension with scalars and gravity Phys. Rev. D 62, v.62, n.4, p. 046008, (2000); [arXiv:hep-th/9909134].
  * (12) ANDREEV, O.; ZAKHAROV V. I. Heavy-quark potentials and AdS/QCD. Phys.Rev. D, v.74, p. 025023, (2012); [arXiv:hep-ph/0604204].
  * (13) GASPERINI, M. ELEMENTS OF STRING COSMOLOGY. (Cambridge University Press, 2007).
  * (14) BROWER, R. C; MARTHUR, D. S.; TAN, C. I. Glueball Spectrum for QCD from AdS Supergravity Duality, Nucl. Phys. B, v.587, p. 249-276. (2000); [arXiv:hep-th/0003115].
  * (15) KARCH, A.; KATZ, E.; SON, D. T.; STEPHANOV, M. A. Linear Confinement and AdS/QCD. Phys.Rev.D, v.74, p. 015005. (2006) ; [arXiv:hep-ph/0602229].
  * (16) SCHWARTZ, M. D. The Emergence of localized gravity. Phys. Lett. B, v.502, p. 223-228, (2001); [arXiv:hep-th/0011177].


###### Lemma 30

For each $t\leq T$, we have

$\displaystyle{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\left[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\right]\hat{H}_{t-1}(f)$
$\displaystyle=$
$\displaystyle\eta{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\;\mathcal{O}\left(({\hat{Z}}_{t}^{h_{t}}(f)+{\mathcal{E}}^{h_{t}}(f,f,x_{t}^{h_{t}},a_{t}^{h_{t}})^{2})(\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+\ln
T)\right)+\mathcal{O}(\eta(\kappa(\epsilon)+1)/T).$

Proof  Let the indicator $I(A_{t})$ denotes that the event of Lemma 25 holds.
We have

$\displaystyle{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\left[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\right]\hat{H}_{t-1}(f)$
$\displaystyle=$
$\displaystyle{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\left[{\mathbb{E}}_{x_{t},a_{t},r_{t}}\big{[}e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\big{]}\right]\hat{H}_{t-1}(f)$
(Since $x_{t},a_{t},r_{y}$ are independent of $S_{t-1}$) $\displaystyle=$
$\displaystyle\eta{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\mathcal{O}\left({\mathbb{E}}_{x_{t},a_{t},r_{t}}{\hat{Z}}_{t}^{h_{t}}(f)+{\mathbb{E}}_{x_{t},a_{t},r_{t}}{\mathbb{E}}_{f^{\prime}|S_{t-1}}{\mathcal{E}}^{h_{t}}(f^{\prime},f^{\prime},x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right)\hat{H}_{t-1}(f)$
(Lemma 29) $\displaystyle=$
$\displaystyle\eta{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\mathcal{O}\left({\mathbb{E}}_{x_{t},a_{t},r_{t}}{\hat{Z}}_{t}^{h_{t}}(f)+{\mathbb{E}}_{x_{t},a_{t},r_{t}}{\mathbb{E}}_{f^{\prime}|S_{t-1}}{\mathcal{E}}^{h_{t}}(f^{\prime},f^{\prime},x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right)\hat{H}_{t-1}(f)I(A_{t})$
$\displaystyle+\eta{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\mathcal{O}\left({\mathbb{E}}_{x_{t},a_{t},r_{t}}{\hat{Z}}_{t}^{h_{t}}(f)+{\mathbb{E}}_{x_{t},a_{t},r_{t}}{\mathbb{E}}_{f^{\prime}|S_{t-1}}{\mathcal{E}}^{h_{t}}(f^{\prime},f^{\prime},x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right)\hat{H}_{t-1}(f)(1-I(A_{t}))$
$\displaystyle=$
$\displaystyle\eta{\mathbb{E}}\;\mathcal{O}\left({\mathbb{E}}_{f|S_{t-1}}{\hat{Z}}_{t}^{h_{t}}(f)+{\mathbb{E}}_{f^{\prime}|S_{t-1}}{\mathcal{E}}^{h_{t}}(f^{\prime},f^{\prime},x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right)\mathcal{O}(\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+\ln
T)$ (Definition of $A_{t}$ in Lemma 25 and
${\mathbb{E}}_{x_{t},a_{t},r_{t}}[{\hat{Z}}_{t}^{h_{t}}(f)]\geq 0$)
$\displaystyle+\eta\mathcal{O}(\kappa(\epsilon)+T){\mathbb{E}}(1-I({A}_{t}))$
(Lemma 26) $\displaystyle=$
$\displaystyle\eta{\mathbb{E}}\;\mathcal{O}\left({\mathbb{E}}_{f|S_{t-1}}{\hat{Z}}_{t}^{h_{t}}(f)+{\mathbb{E}}_{f^{\prime}|S_{t-1}}{\mathcal{E}}^{h_{t}}(f^{\prime},f^{\prime},x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right)\mathcal{O}(\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+\ln
T)$ $\displaystyle+\eta\mathcal{O}((\kappa(\epsilon)+1)/T).$ (Probability of
$A_{t}$ in Lemma 25)

This implies the desired bound.

###### Lemma 31

We have

$\displaystyle{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}\left|\left[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\right]{\mathbb{E}}_{g\sim{\tilde{p}}(\cdot|f)}[-{\hat{\delta}}_{t}^{h_{t}}(g,f)+{\hat{Z}}_{t}^{h_{t}}(f)]\right|$
$\displaystyle=$
$\displaystyle\eta{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}\mathcal{O}\left(\epsilon+|{\mathcal{E}}^{h_{t}}(f,f;x_{t}^{h},a_{t}^{h})|^{2}+{\hat{Z}}_{t}^{h}(f)\right).$

Proof  We have

$|{\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}|=\mathcal{O}(1),$

and $\forall g\in\mathcal{F}(\epsilon,f)$, we have
${\hat{\delta}}_{t}^{h_{t}}(g,f)=\mathcal{O}(\epsilon)$. Therefore by the
definition of ${\tilde{p}}(\cdot|f)$ in Lemma 23, we have

$\displaystyle\bigg{|}[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1]{\mathbb{E}}_{g\sim{\tilde{p}}(\cdot|f)}{\hat{\delta}}_{t}^{h_{t}}(g,f)\bigg{|}=$
$\displaystyle\eta\mathcal{O}(\epsilon).$

Moreover,

$\displaystyle{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}\bigg{|}[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1]{\hat{Z}}_{t}^{h_{t}}(f)\bigg{|}$
$\displaystyle=$
$\displaystyle\eta{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}O\left([|{\hat{\delta}}_{t}^{h_{t}}(f,f)|+|{\hat{Z}}_{t}^{h_{t}}(f))|+|{\hat{Z}}_{t}^{h_{t}}|]|{\hat{Z}}_{t}^{h_{t}}(f)|\right)$
$\displaystyle=$
$\displaystyle\eta{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}O\left(|{\hat{\delta}}_{t}^{h_{t}}(f,f)|^{2}+|{\hat{Z}}_{t}^{h_{t}}(f))|^{2}+|{\hat{Z}}_{t}^{h_{t}}|^{2}\right)$
(Cauchy-Schwarz) $\displaystyle=$
$\displaystyle\eta{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}O\left(|{\mathcal{E}}_{t}^{h}(f,f;x_{t}^{h},a_{t}^{h})|^{2}+|{\hat{Z}}_{t}^{h}(f))|^{2}+|{\hat{Z}}_{t}^{h}|^{2}\right)$
(Lemma 17) $\displaystyle=$
$\displaystyle\eta{\mathbb{E}}_{x_{t},a_{t},r_{t}|h_{t}=h}{\mathbb{E}}_{f|S_{t-1}}O\left(|{\mathcal{E}}_{t}^{h}(f,f;x_{t}^{h},a_{t}^{h})|^{2}+{\hat{Z}}_{t}^{h}(f))\right)$

where the second to the last equation was obtained by taking conditional
expectation conditioned on $(x_{t}^{h},a_{t}^{h})$ with respect to the
transition and reward, followed by Lemma 17. The last equation used Lemma 27
and Lemma 28.

We are now ready to prove Proposition 13.

#### Proof of Proposition 13

Proof  From the proof of Lemma 24, we obtain

$\displaystyle{\mathbb{E}}[{\mathbb{E}}_{f\sim
p_{t}}\hat{H}_{t}(f)-{\mathbb{E}}_{f\sim p_{t-1}}\hat{H}_{t-1}(f)]$
$\displaystyle=$ $\displaystyle{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}\left[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\right]\hat{H}_{t-1}(f)$
$\displaystyle-\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}\left[e^{-\eta[({\hat{\delta}}_{t}^{h_{t}}(f,f)-{\hat{Z}}_{t}^{h_{t}}(f))-{\hat{Z}}_{t}^{h_{t}}]}-1\right]{\mathbb{E}}_{g\sim{\tilde{p}}(\cdot|f)}[-{\hat{\delta}}_{t}^{h_{t}}(g,f)+{\hat{Z}}_{t}^{h_{t}}(f)]$
$\displaystyle-\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}{\mathbb{E}}_{g\sim{\tilde{p}}(\cdot|f)}[-{\hat{\delta}}_{t}^{h_{t}}(g,f)+{\hat{Z}}_{t}^{h_{t}}(f)]$
$\displaystyle\leq$ $\displaystyle\eta c_{0}{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}\left(|{\mathcal{E}}^{h_{t}}(f,f;x_{t}^{h_{t}},a_{t}^{h_{t}})|^{2}+{\hat{Z}}_{t}^{h_{t}}(f)\right)(\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+\ln
T)+\eta c_{0}(\kappa(\epsilon)+1)/T$ (Lemma 30) $\displaystyle+\gamma\eta
c_{0}{\mathbb{E}}{\mathbb{E}}_{f\sim
p_{t-1}}\left(|{\mathcal{E}}^{h_{t}}(f,f;x_{t}^{h_{t}},a_{t}^{h_{t}})|^{2}+{\hat{Z}}_{t}^{h_{t}}(f)\right)$
(Lemma 31) $\displaystyle-\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}{\hat{Z}}_{t}^{h_{t}}(f)+\mathcal{O}(\epsilon)$ ($\forall
g\in\mathcal{F}(\epsilon,f)$, we have
${\hat{\delta}}_{t}^{h_{t}}(g,f)=\mathcal{O}(\epsilon)$) $\displaystyle\leq$
$\displaystyle c_{1}\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}|{\mathcal{E}}^{h_{t}}(f,f;x_{t}^{h_{t}},a_{t}^{h_{t}})|^{2}-0.5\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}{\hat{Z}}_{t}^{h_{t}}(f)+\mathcal{O}(\epsilon+\eta(\kappa(\epsilon)+1)/T).$

The last inequality used the condition of the Lemma concerning the choice of
$\eta$. Now by summing over $t=1$ to $t=T$, and apply Lemma 23 to bound
${\mathbb{E}}\hat{H}_{0}(f)$, we obtain

$\displaystyle\gamma{\mathbb{E}}\;{\mathbb{E}}_{f\sim
p_{t-1}}{\hat{Z}}_{t}^{h_{t}}(f)\leq$ $\displaystyle(\epsilon
T+\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+1)+2c_{1}\gamma\sum_{t=1}^{T}{\mathbb{E}}\;|{\mathcal{E}}^{h_{t}}(f,f;x_{t}^{h_{t}},a_{t}^{h_{t}})|^{2}.$

The proof is now completed by recalling Lemma 18.

##  Appendix I Proof of Theorem 9

In this section, we provide a proof for Theorem 9. We focus on the differences
from the proof of Theorem 5, which are limited to our decoupling analysis. In
particular, in the setting of Theorem 9, we have the following improved analog
for Lemma 15.

######  Lemma 32 (Design based decoupling)

We have for all $x^{h}\in\mathcal{X}^{h}$ and $f\in\mathcal{F}$:

$\displaystyle{\mathcal{E}}^{h}(f,f;x^{h},\pi_{f}(x^{h}))^{2}\leq{\mathcal{E}}^{h}(f,f,x^{h},\rho^{h}(x^{h}))^{2},$

where $\rho^{h}(x^{h})$ is the $G$-optimal design (6) at the state $x^{h}$.

Proof  Let $\Sigma_{\star}(x^{h})$ be the covariance
${\mathbb{E}}_{a^{h}\sim\rho(x^{h})}[\phi^{h}(x^{h},a^{h})\phi^{h}(x^{h},a^{h})^{\top}]$.
The definition of $G$-optimal design implies that

$\sup_{a^{h}\in{\mathcal{A}}}\|\phi^{h}(x^{h},a^{h})\|_{\Sigma_{\star}(x^{h})^{-1}}^{2}\leq
d_{2}.$ (10)

By Assumption 4, we have

$\displaystyle{\mathcal{E}}^{h}(f,f;x^{h},\pi_{f}(x^{h}))^{2}$
$\displaystyle=\left\langle w^{h}(f,x^{h}),\phi^{h}(x^{h},a)\right\rangle^{2}$
$\displaystyle\leq\|w^{h}(f,x^{h})\|_{\Sigma_{\star}(x^{h})}^{2}\|\phi^{h}(x^{h},a^{h})\|_{\Sigma_{\star}(x^{h})^{-1}}^{2}$
$\displaystyle\leq d_{2}{\mathbb{E}}_{a^{h}\sim\rho^{h}(x^{h})}\left\langle
w^{h}(f,x^{h}),\phi^{h}(x^{h},a^{h})\right\rangle^{2}$
$\displaystyle=d_{2}{\mathcal{E}}^{h}(f,f,x^{h},\rho^{h}(x^{h}))^{2}.$

The first inequality used Cauchy-Schwarz. The second inquality used (10).

Combining this with the Bellman rank decoupling in Lemma 14 and Proposition
10, we get

$\displaystyle\lambda{\mathbb{E}}\;{\mathrm{Regret}}(f_{t},x_{t}^{1})+\lambda{\mathbb{E}}{\mathbb{E}}_{f|S_{t-1}}\;\Delta
f^{1}(x_{t}^{1})=\lambda
H{\mathbb{E}}\;{\mathcal{E}}^{h_{t}}(f_{t},f_{t},x_{t}^{h_{t}},a_{t}^{h_{t}})$
$\displaystyle\leq$
$\displaystyle\lambda\left[\epsilon_{1}HB_{1}+\mu_{1}^{-1}{\mathrm{br}}(\epsilon_{1})+H\mu_{1}{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}{\mathcal{E}}^{h_{t}}(f,f,x_{t}^{h_{t}},\pi_{f}(x_{t}^{h_{t}}))^{2}\right]$
$\displaystyle\leq$
$\displaystyle\lambda\left[\epsilon_{1}HB_{1}+\mu_{1}^{-1}{\mathrm{br}}(\epsilon_{1})+H\mu_{1}d_{2}{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}{\mathcal{E}}^{h_{t}}(f,f,x_{t}^{h_{t}},\rho^{h_{t}}(x^{h_{t}}_{t}))^{2}\right]$
$\displaystyle=$
$\displaystyle\lambda\left[\epsilon_{1}HB_{1}+\mu_{1}^{-1}{\mathrm{br}}(\epsilon_{1})+H\mu_{1}d_{2}{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}{\mathcal{E}}^{h_{t}}(f,f,x_{t}^{h_{t}},a^{h_{t}}_{t})^{2}\right]$

Further choosing $\lambda H\mu_{1}d_{2}=0.5\eta$, we get the following analog
of Proposition 11.

$\displaystyle\lambda{\mathbb{E}}\;{\mathrm{Regret}}(f_{t},x_{t}^{1})\leq$
$\displaystyle{\mathbb{E}}\;{\mathbb{E}}_{f|S_{t-1}}\left[-\lambda\Delta
f(x_{t}^{1})+0.5\eta{\mathcal{E}}^{h_{t}}(f,f,x_{t}^{h_{t}},a_{t}^{h_{t}})^{2}\right]$
$\displaystyle+\lambda\left(\epsilon_{1}HB_{1}+\frac{2\lambda{\mathrm{br}}(\epsilon_{1})H_{1}d_{2}}{\eta}\right).$

Now we combine this result with Proposition 12 and Proposition 13 (both
results still hold without modification, because the proofs did not rely on
how $a_{t}^{h}$ is generated given $x_{t}^{h}$), leading to

$\displaystyle\sum_{t=1}^{T}\lambda{\mathbb{E}}\;{\mathrm{Regret}}(f_{t},x_{t}^{1})\leq$
$\displaystyle\mathcal{O}\left(\lambda\epsilon T+4\eta
T\epsilon^{2}+\kappa(\epsilon)+1.5\lambda^{2}T\right)$
$\displaystyle+\mathcal{O}(\epsilon
T+\kappa(\epsilon)+\kappa^{\prime}(\epsilon)+1)$ $\displaystyle+\lambda
T\left(\epsilon_{1}HB_{1}+\frac{2\lambda{\mathrm{br}}(\epsilon_{1})H_{1}d_{2}}{\eta}\right).$

This implies the result of Theorem 9.