# SonicPACT: An Ultrasonic Ranging Method for the Private Automated Contact
Tracing (PACT) Protocol
††thanks: DISTRIBUTION STATEMENT A. Approved for public release. Distribution
is unlimited.

This material is based upon work supported by the Defense Advanced Research
Projects Agency under Air Force Contract No. FA8702-15-D-0001 and the MIT–IBM
Watson Artificial Intelligence Laboratory. Any opinions, findings, conclusions
or recommendations expressed in this material are those of the author(s) and
do not necessarily reflect the views of the Defense Advanced Research Projects
Agency.

R. L. Rivest received support from the Center for Science of Information
(CSoI), an NSF Science and Technology Center, under grant agreement
CCF-0939370.

LC Ivers received funding from the Sullivan Family Foundation.

Principal authors Meklenburg, Specter and Wentz each made equal contributions
to research and analysis reported in this paper.

John Meklenburg12, Michael Specter13, Michael Wentz12,
Hari Balakrishnan3, Anantha Chandrakasan3, John Cohn4, Gary Hatke2, Louise
Ivers5,
Ronald Rivest3, Gerald Jay Sussman3, Daniel Weitzner3,
1Corresponding Authors, 2MIT Lincoln Laboratory, 3Massachusetts Institute of
Technology,
4MIT-IBM Watson AI Lab, 5Massachusetts General Hospital

###### Abstract

Throughout the course of the COVID-19 pandemic, several countries have
developed and released contact tracing and exposure notification smartphone
applications (apps) to help slow the spread of the disease. To support such
apps, Apple and Google have released Exposure Notification Application
Programming Interfaces (APIs) to infer device (user) proximity using Bluetooth
Low Energy (BLE) beacons. The Private Automated Contact Tracing (PACT) team
has shown that accurately estimating the distance between devices using only
BLE radio signals is challenging [1]. This paper describes the design and
implementation of the SonicPACT protocol to use near-ultrasonic signals on
commodity iOS and Android smartphones to estimate distances using time-of-
flight measurements. The protocol allows Android and iOS devices to inter-
operate, augmenting and improving the current exposure notification APIs. Our
initial experimental results are promising, suggesting that SonicPACT should
be considered for implementation by Apple and Google.

###### Index Terms:

COVID-19, contact tracing, proximity detection, smartphone, ultrasonic,
ultrasound, acoustic, ranging, PACT, Bluetooth

##  I Introduction

Since the onset of the COVID-19 pandemic there has been significant interest
in the use of smartphone applications (apps) to facilitate contact discovery
and tracing. Researchers have been pursuing methods using Bluetooth Low Energy
(BLE) signals to estimate the duration and proximity of smartphones (and those
carrying them) to one another. One such effort is the Private Automated
Contact Tracing (PACT) Protocol [2], which is a decentralized approach modeled
after Apple’s “Find My” protocol [3]. A smartphone using a PACT-based app
periodically emits BLE chirps derived from a secret seed known only to that
phone. All smartphones log their seeds, the times the seeds were active, and
any BLE chirps received with timestamps. If a user is diagnosed as COVID-19
positive, a record of all BLE chirps sent by the user’s phone can be
voluntarily uploaded to a database through a trusted public health authority.
Other smartphones frequently check against this database to see if they have
received any BLE chirps designated from a COVID-19 positive individual, and if
so, the phone’s owner can be notified to take the appropriate action (for
instance, self-isolate or seek COVID-19 testing).

In order for PACT and related efforts to be successful, a Too Close For Too
Long (TCFTL) detector must be developed based on suitable signals (e.g., BLE)
and associated measurements, such as the transmit power level, Received Signal
Strength Indicator (RSSI), and received timestamp. There are three desired
features of the TCFTL detector:

  1. 1.

a high probability of detection when the two users (devices) are within a
threshold distance (i.e., high recall),

  2. 2.

low probability of false alarm (high precision), and

  3. 3.

ability to be tuned based on guidelines from local health authorities (for
example, based on an exposure of 15 or more minutes within 6 feet of an
infected individual [4]).

Apple and Google ($A|G$) have released BLE-based exposure notification
protocols and Application Programming Interfaces (APIs) for iOS and Android
devices, a significant step forward for automated contact tracing efforts [5].
Unfortunately, the initial implementations of these protocols only log
received BLE chirps when the device is awake, or no more often than every 5
minutes when the device is not awake. This limits the data available to a
TCFTL detector, and it has been shown to have relatively poor performance (up
to 50% false alarms and less than 25% of true contact events detected), with
an effectiveness that varies across smartphones [1, 6, 7, 8, 9]. Increasing
the BLE sample rate and estimating the carriage state of each smartphone has
been shown to help [1], although may not be possible due to battery life and
privacy constraints.

In this paper, we show that ultrasonic (US)111Ultrasonic frequencies are
typically defined as those exceeding 20 kHz. In the context of commodity
smartphones and this paper, we use “ultrasonic” for frequencies around 20 kHz,
but not necessarily exceeding that threshold. range measurement using the
speakers and microphones built-in to all smartphones is a promising concept
that could improve the accuracy of BLE-based estimates significantly. By using
a BLE protocol augmented with US ranging (BLE+US), smartphones exchange time-
tagged inaudible acoustic pulses in the ultrasonic frequency range to measure
the time-of-flight between them. Because the speed of sound is known, the
devices can then jointly solve for the range between them. By improving the
ranging accuracy, this technique can be used together with BLE-based protocols
to reduce the false alarm and miss rates of exposure notification services.
For example, a BLE-based approach may flag neighbors in an apartment building
as positive contacts, when they are actually separated by physical barriers
(ceilings or walls) because the 2.4 GHz BLE signals will propagate through
such barriers. A BLE+US approach would identify these BLE contacts as false
alarms and eliminate them, since US frequencies are more significantly
attenuated by the same barriers.

This paper introduces SonicPACT, an ultrasonic ranging protocol developed by
the PACT team to augment a BLE-based TCFTL detector for contact tracing.
SonicPACT has the following key features:

  1. 1.

It can be implemented on both Android and iOS and allow interoperability
between them.

  2. 2.

It operates at inaudible near-ultrasonic frequencies to avoid disturbing users
while measurements are made.

  3. 3.

It uses BLE advertisements (rather than Wi-Fi, although it could also use Wi-
Fi) to coordinate measurements and exchange information. This feature
simplifies integration with BLE-based exposure notification protocols.

  4. 4.

It uses pseudorandom noise (PN) waveforms generated by a random number
generator with a seed based on the device’s unique Bluetooth Universally
Unique Identifier (UUID), allowing both devices to generate matched filters
for the other’s waveform without prior coordination.

We have implemented SonicPACT as user-level apps on Android and iOS and have
conducted several experiments with the initial implementation. Our results are
promising:

  1. 1.

In experiments done placing phones at distances between 2 feet (60 cm) and 12
feet (3,6 meters) of each other in multiples of 2 feet, and considering a
threshold of $\leq$ 6 feet as the critical distance at which to flag a
“contact”, we find that the missed detection rate indoors is between 0% and
5.1% with no false alarms.

  2. 2.

Indoors, the estimated distances are within 1 foot (30 cm) of the true
distance between 56.7% and 70.9% of the time; outdoors, due to the absence of
significant multipath, estimates are within 1 foot between 82.9% and 100% of
the time.

  3. 3.

These are proof-of-concept implementations with significant optimizations that
can be done, but the low false alarm and miss rates compared to a BLE-only
method when estimating if two devices are “too close” (within 6 feet or 1.8
meters) is an encouraging result for exposure notification and is therefore
worth studying for implementation by $A|G$.

Our goal in developing SonicPACT was to demonstrate that ultrasonic signaling,
coupled with BLE technology widely available on $A|G$ APIs, can provide a more
robust implementation of a TCFTL detector. Ultimately, a power-efficient and
ubiquitous implementation of an ultrasonic ranging protocol may (will?) need
to be integrated within the iOS and Android operating systems, and may
therefore need to be developed by $A|G$ themselves. As such, SonicPACT as
proposed here is not a turnkey app solution to contact discovery, but we view
it as a promising technology and protocol to incorporate into Android and iOS.
In addition to discussing our initial promising experimental results, we
discuss at length the limitations of both our SonicPACT implementation and of
ultrasonic ranging in general, along with directions for future work.

###  I-A Related Work

Several research groups have sought to use the acoustic channel on smartphones
or mobile devices for communications [10, 11, 12, 13], and indoor and outdoor
localization [14, 15, 16, 17, 18, 19] with some success. Most prior acoustic
indoor localization techniques rely on external hardware to serve as beacons
or receivers. Ranging between smartphones without any external hardware is
more challenging, since depending on the method employed, accurate transmit
and receive timestamps may be required to compute time-of-flight of the US
signal between devices. Obtaining the timestamps with low or predictable
latency from a user-space application is not easy.

Peng et al.’s BeepBeep method [18] provides a ranging capability using only
commercial off-the-shelf (COTS) hardware available on smartphones. BeepBeep
ranging uses linear frequency modulated (LFM) chirp waveforms in the 1-6 kHz
frequency range and a sample-counting technique to eliminate the latency and
uncertainty introduced by time-stamping the transmit and receive pulses. The
protocol uses Wi-Fi to exchange measurements between devices, enabling each
device to compute the range to the other device. The BeepBeep paper reports
that sub-centimeter ranging accuracy was achievable in many cases, and that
performance could be achieved even in challenging indoor environments with
multipath mitigation techniques. Fotouhi [20] extended this work by
implementing the BeepBeep protocol on Android devices using spread-spectrum
waveforms to differentiate between devices and maintain user privacy.

In April 2020, Loh released the NOVID app, which was the first-available
COVID-19 contact tracing app to use both Bluetooth and ultrasound for contact
detection. Loh’s laboratory tests of the app shows excellent ranging
performance in a controlled setting [21]. At this time, the technical details
of the NOVID implementation have not been made available to the public.

###  I-B Security and Privacy Concerns

Here we review the security and privacy trade-offs inherent in the use of
SonicPACT, or any exposure notification app that uses acoustic measurement.

Potential Privacy Risks: Any acoustic measurement implementation will require
the ability to process audio, which poses a risk to user privacy. At the
outset, we suggest it is appropriate to assess the privacy risks of this
protocol relative to the design and implementation of the current ($A|G$)
service. This protocol ought to satisfy the core privacy and security
commitments of the ($A|G$) BLE protocol, including

  * •

voluntary adoption, implemented by requiring affirmative opt-in to the service
for all users

  * •

protection of user location privacy, including no collection of GPS or other
location information

  * •

confidentiality protection for all users of the system, as between potential
contacts, between those contacts and the public health authority, and as may
be observed by third party adversaries

Requiring constant audio access also allows an adversary with control over the
operating system or ranging app itself to covertly record potentially
sensitive conversations, and normalizing the use of such systems may
potentially lead to overreach by the developers involved. We believe, however,
that such risks can be mitigated through carefully designed technical and
policy mechanisms to avoid leakage or overreach.

First, we advocate that the protocol be implemented by the mobile operating
system vendor (e.g., Apple or Google) at the OS level via a privileged app, or
as a part of the audio processing path itself. This approach can limit the
exposure of such data to trusted applications with OS-level privileges,
222This could be the SEPolicy equivalent of priv_app or system_app for
Android. which, beyond preventing leakage to apps that would already be
capable of such spying, would also represent the same risk profile as
currently-existing wake-word activation (“Hey Siri” and “OK Google”).

Furthermore, ultrasonic sensing does not require the audible range of the
spectrum. Any frequency below, say, 18 kHz may (should) be filtered and
removed by the operating system (or, if possible, at the hardware level),
ensuring that the app is only capable of listening at near-US frequencies.

Finally, we would expect that there would be little need to store any
information about prior interactions after a range as been estimated. Indeed,
our proof of concept implementations discard audio samples after processing,
and such information is never intentionally stored on disk. We recommend the
same no-storage policy for a production implementation.

Resolving Weaknesses in the ($A|G$) Protocol: A weakness of the ($A|G$)
protocol is that Bluetooth packets may be heard, recorded, and rebroadcast
later (a _replay attack_) or somewhere else (a _relay attack_). An interactive
protocol, such as the SonicPACT protocol presented in this paper, can be
augmented to avoid this weakness. Such protection may be obtained by blending
SonicPACT with a standard Diffie-Hellman key-agreement protocol.

Here, each device’s unique identifier key can serve a dual purpose as a
freshly-generated Diffie-Hellman public key (as is done in the Apple Find My
protocol [3]). A pair of devices can therefore create an agreed upon shared
secret key $K$, which can then be used to seed the generation of the waveforms
used in the protocol, and to generate message authentication codes (MACs) for
additional confirmation messages after the fact. If the parties use distinct
keys for each session bounded by some fixed time period, then they also have
“freshness” guarantees since the cryptographic key $K$ will inherently depend
on the time period.

##  II Laptop Experiment

Before starting on Android and iOS devices, we conducted experiments using
laptops as smartphone surrogates to demonstrate that accurate US ranging is
possible without specialized hardware, and to understand what the performance
of smartphone ranging performance might be. We used the built-in laptop
speakers and microphones to exchange US pulses and processed the signals on
both laptops using MATLAB software to compute the range between the devices.

We used LFM chirp waveforms ranging with the parameters listed in Table I.
Laptop A (a Dell Precision 5540) transmitted up-chirps (frequency swept from
low to high) while Laptop B (a MacBook Pro, 2018 model, 13-inch display)
transmitted down-chirps (frequency swept from high to low) to allow the
processing code to easily differentiate between the two signals using a
matched filter. We staggered the signal transmissions by 1 second to avoid
interference. After data collection, we matched the signals recorded on both
laptops against both transmit waveforms, yielding four signal comparisons
between laptops A and B:

  1. 1.

A’s recording against A’s transmit waveform

  2. 2.

A’s recording against B’s transmit waveform

  3. 3.

B’s recording against A’s transmit waveform

  4. 4.

B’s recording against B’s transmit waveform

Then, we computed the range between laptops using the sample-counting method
described in [18].

We conducted an outdoor experiment by setting the laptops on lawn chairs as
shown in Figure 1. We varied the distance between laptops and measured the
ground truth distance between them with a tape measure. We took US range
measurements at each position by averaging around 10 pulse exchanges.

Figure 1: Laptop experiment, outdoor setup.

Figure 2: Laptop experiment results: range measurement vs truth. 1 ft = 30 cm.

The results (Figure 2) show that the measured range values were consistently
within 1 foot (30 cm) of ground truth. Likely explanations for the apparent
systematic error that manifests as a bias in Figure 2 include multipath and
non-colocated speakers and microphones on the laptops.

TABLE I: Laptop Experiment Waveform Parameters Parameter | Value |
---|---|---
Pulse Width | 200 ms |
Pulse Repetition Interval | 2 s |
Center Frequency | 20 kHz |
Sampling Rate | 48 kHz |
Bandwidth | 2 kHz |
Rise/Fall Time (to reduce audible clicks) | 10 ms |

##  III System Design

Though BeepBeep served as a starting point for our work, the protocol has a
few weaknesses that make it less functional for practical, widespread
adoption. SonicPACT attempts to improve on this in the following ways:

  * •

The audio signal processing uses LFM chirps that are not easily differentiable
between devices and will be more difficult to scale to tens or hundreds of
devices.

  * •

BeepBeep’s broadcasts are in audible frequency ranges

###  III-A Protocol

The intent of SonicPACT is to act in concert with BLE measurement estimates
that indicate that the devices may be in range for a sufficient amount of time
(as defined by public health authorities), following the ($A|G$) Exposure
Notification framework. Each phone maintains a table of BLE UUIDs (or other
identifiers) and the length of time for which they have been observed over a
sliding window, for instance 15 minutes. The table is updated during each BLE
scan period, and if packets are received from a device for greater than or
equal to some threshold (which we anticipate would be lower than the threshold
for contact duration), then SonicPACT ultrasonic waveforms are exchanged to
more accurately estimate the proximity between phones.

SonicPACT must then perform the following tasks after BLE has initiated the
ranging process:

  1. 1.

Generate the ranging waveforms, ensuring that the random samples are
consistent between devices (drawn from the same random number generator using
the same seed).

  2. 2.

Transmit and receive audio, accounting for any delays by using a transmit-
receive loopback measurement.

  3. 3.

All devices must be able to detect waveforms from both themselves and nearby
counterparts.

  4. 4.

Finally, each pair of nearby devices exchanges timing measurements over BLE to
compute the range to the other device.

The SonicPACT protocol is shown in Figure 3. Each phone generates waveform
samples for its own UUID and prepares to transmit that signal. To process a
received waveform, a phone must know the UUID of the transmitter, which has
been previously broadcast on the BLE channel. We describe here how a pair of
phones can estimate their range, but this method can be extended to multiple
devices in the same neighborhood as well.

The two phones select a leader and follower, so they know roughly when to
expect the other’s waveform. In practice this could be chosen a number of
ways. One idea is to compute an integer hash from the UUIDs of both phones and
always choose the lower integer to be the leader. The leader then starts the
process by sending a start command via BLE. In our implementation, the leader
sends its ultrasonic signal after a delay of 50 ms, and the follower sends its
ultrasonic signal 250 ms after receiving the BLE start command. Both phones
receive samples for a total of 400 ms, after which they perform matched
filtering and detection for their own waveform and the other phone’s waveform.
These are unoptimized and over-engineered parameters to demonstrate the
feasibility of the underlying concepts.

Figure 3: Prototype BLE/Ultrasonic Protocol.

Figure 4: Prototype BLE/Ultrasonic Protocol Message Timeline.

For the initial implementation we allocated up to 600 ms for receiver
processing and 500 ms for the BLE data exchange. A single measurement between
two phones would therefore take a total of 1.5 seconds, but this included a
generous margin. Based on the discussion in the next section, this time could
be reduced substantially.

###  III-B Alternative Protocols

Before settling on the above protocol, we considered several other approaches
and discarded then. We discuss why we didn’t pursue two of these promising
alternatives here.

Receive and echo: An initial design involved the first phone emitting a signal
and the second phone receiving, amplifying, and “echoing” the signal back to
the first phone.

The benefit of the design is its simplicity; in particular, the task of the
second phone is simple and all timing measurements are made with respect to
time on the first phone, and the time difference measured at the first phone
will include twice the time-of-flight and any loopback delays.

Unfortunately, while this approach might work in clear outdoor environments,
multipath reflections will also return to the first phone from objects in the
environment. Distinguishing the desired return from the multipath reflection
is challenging without significant processing.

Time difference of arrival between BLE and audio: If a device were capable of
sending a BLE packet and audio at the exact same time, any receiver would be
able to learn the sender’s distance by marking the time difference between
when it received the start of the BLE packet and initially received the audio
waveform, as in the Cricket system [14]. Unfortunately, the actual action of
sending a packet (and, indeed, all of the RF capabilities of a smartphone),
are controlled by the device’s baseband, which runs an unmodifiable
proprietary OS that is outside the purview of phone OS vendors. For example,
the Qualcomm baseband on the Pixel 2 phone provides no explicit guarantees
about the timing of packets, nor any feedback to the kernel software about
when a packet was actually sent.

###  III-C SonicPACT Ranging Protocol

A key challenge in estimating the range using time-of-flight between phones is
the lack of accurate synchronization of time between the devices. At the speed
of sound (346 meters/second at sea level), a timing offset of 10 ms is
equivalent to a range error of 3.46 meters (11.4 feet). It is likely that
independent phones will have varying offsets in practice, depending on
variables such as the processing load, OS version, manufacturer Stock Keeping
Unit (SKU), or other details, making it necessary to use an approach that can
account for variability between phones.

To account for these timing uncertainties, the protocol uses a _loopback_
measurement—each device estimates the time it takes to hear its own broadcasts
in order to estimate its own processing delay. As long as this process takes
place before the range between the phones has changed, and the timing offsets
are the same between hearing its own signal and the signal from the other
phone, the offsets will cancel and allow each phone to estimate the range.
This is a well known technique for two-way time transfer, and is summarized by
the steps below:

  1. 1.

Leader sends a waveform $x_{1}$ at $t_{1}=\Delta_{1}+T_{1}+\delta_{T_{1}}$

     * •

$\Delta_{1}$ is the Leader’s offset from global time

     * •

$T_{1}$ is the time the Leader thinks it is sending $x_{1}$

     * •

$\delta_{T_{1}}$ is the delay before the Leader actually sends $x_{1}$

  2. 2.

The Follower receives $x_{1}$ at $t_{2}=\Delta_{2}+T_{2}-\delta_{R_{2}}$

     * •

$\Delta_{2}$ is the Follower’s offset from global time

     * •

$T_{2}$ is the local time when the Follower detects $x_{1}$

     * •

$\delta_{R_{2}}$ is the delay between $x_{1}$ being received and timestamped
by software

  3. 3.

The Follower sends a waveform $x_{2}$ at
$t_{3}=\Delta_{2}+T_{3}+\delta_{T_{2}}$

     * •

$T_{3}$ is the time the Follower thinks it is sending $x_{2}$

     * •

$\delta_{T_{2}}$ is the delay before the Follower actually sends $x_{2}$

  4. 4.

The Leader receives $x_{2}$ at $t_{4}=\Delta_{1}+T_{4}-\delta_{R_{1}}$

     * •

$T_{4}$ is the local time when the Leader detects $x_{2}$

     * •

$\delta_{R_{1}}$ is the delay between $x_{2}$ being received and timestamped
by software

We now have two equations for the range between the phones:

$R=c(t_{2}-t_{1})=c(t_{4}-t_{3})$

where $c$ is the speed of propagation of sound. By adding the two equations
(twice the range), substituting, and rearranging, the range is

$R=\frac{c}{2}[(T_{4}-T_{1})-(\delta_{T_{1}}+\delta_{R_{1}})]-[(T_{3}-T_{2})+(\delta_{T_{2}}+\delta_{R_{2}})],$
(1)

where the quantity within the first set of brackets is computed by the Leader
and the quantity within the second set of brackets is computed by the
Follower. This equation can also be obtained more directly from Figure 4.
Thus, each phone needs to know the time when it sends its own waveform, the
time when it detects the waveform from the other phone, and its
transmit/receive loopback delay $(\delta_{T}+\delta_{R})$. If each phone also
receives its own waveform to measure the loopback delay, say at $T_{5}$ on the
Leader for the signal sent at $T_{1}$ and at $T_{6}$ on the Follower for the
signal sent at $T_{3}$, then

$(\delta_{T_{1}}+\delta_{R_{1}})=(T_{5}-T_{1})$
$(\delta_{T_{2}}+\delta_{R_{2}})=(T_{6}-T_{3})$

and substituting into the equation for range shows that the transmit times
$T_{1}$ and $T_{3}$ also cancel. Therefore, the range can also be written as

$R=\frac{c}{2}([T_{4}-T_{5}]+[T_{2}-T_{6}])$ (2)

and each phone only needs to compute the difference in arrival times of the
other phone’s waveform and their own waveform. In this case the variability in
loopback delay is less of a concern. However, Eq. (1) may still be useful for
situations where reduced processing is necessary, since potentially the
loopback delay is a constant calibration factor. Either way, after each phone
has computed the quantity within the brackets, it exchanges its measurement
with the other phone over BLE and both phones can estimate the range between
them.

###  III-D Waveform Generation

In order for the ultrasonic ranging concept to be implemented, the
smartphone’s speaker and microphone hardware must be capable of passing
signals at ultrasonic frequencies (ideally 20 kHz and above) without
significant attenuation. Public documentation suggests that most newer iPhones
support an audio sample rate of 48 KSPS, and so should have the digital
bandwidth to represent signals up to 24 kHz. However, the analog hardware is
likely not optimized for this range. As a first step, some simple experiments
were done to evaluate the frequency response of the iPhones available to the
group to get some idea of how well ultrasonic ranging would work.

We wrote an application to simultaneously play and record audio waveforms
using Apple’s AVFoundation framework. The first test was to loopback White
Gaussian Noise (WGN) from the speaker to microphone in order to characterize
the frequency response. While this doesn’t inform us of the independent
frequency responses of the speaker and microphone, it gives us a reasonable
sense of the hardware capabilities. We used Welch’s method to estimate the
Power Spectral Density (PSD) from the received samples. The results (Figure 5)
for an iPhone 7 and 8 show that both responses drop sharply above 20 kHz, but
may be able to pass frequencies in the 17-19 kHz range. The attenuation
profile is not ideal, but by using signals with a large time-bandwidth product
we can expect a large processing gain and high-resolution time delay estimate.

Figure 5: iPhone 7 and iPhone 8 Frequency Responses.

Ranging systems, such as those used in RADAR, SONAR, and radio communications,
typically use signals with a large time-bandwidth product [22]. Signals with a
long duration can be integrated to boost signal-to-noise ratio (SNR) for
detection, and signals with a wide bandwidth have fine time resolution (and
therefore fine range resolution). Some popular signals include LFM chirps and
Binary Phase Shift Keying (BPSK) modulated pulses. For the ultrasonic use
case, we desired a signal that is:

  * •

Uniquely identifiable for each phone

  * •

Simple to generate and detect

  * •

Capable of high-accuracy ranging

The Cramér-Rao Lower Bound (CRLB) is one starting point for understanding the
minimum variance of an unbiased range estimator. For a signal in Additive
White Gaussian Noise (AWGN), the CRLB for range is [23]:

$\sigma^{2}_{r}\leq\frac{c^{2}}{\frac{E}{N_{0}/2}F^{2}},$ (3)

where $\sigma^{2}_{r}$ is the variance of the range, $c$ is the speed of
propagation, $\frac{E}{N_{0}/2}$ is a SNR, and $F^{2}$ is the mean squared
bandwidth of the signal. Note that the CRLB does not tell us how to estimate
range, but it is informative to see that the variance is reduced linearly with
the SNR and quadratically with the bandwidth of the signal.

The CRLB for range is plotted in Figure 6 as a function of integrated SNR for
various rectangular (uniform) bandwidths. Also shown is an error floor if the
device were to have variability in its reported timestamps with a standard
deviation of 0.1 ms. It is worth noting that practical estimators require an
integrated SNR of 13-16 dB for an unambiguous solution [24]. Based on this
plot, we could choose a waveform with 200 Hz bandwidth and target an
integrated SNR of 28 dB, or 500 Hz bandwidth and target an integrated SNR of
20 dB, and have a similar performance bound.

The simple model used to derive the CRLB does not consider correlated noise or
interference. We built a large margin into the signal design to help boost the
SNR and overcome these effects while keeping the receiver processing
relatively simple. As a starting point we used a waveform with 500 Hz
bandwidth and 100 ms duration, which has a time-bandwidth product of 50 and
can yield a processing gain of about 17 dB. Based on Figure 5, the iPhone 8’s
frequency response starts to significantly attenuate signals above 19 kHz, so
we used a carrier frequency of 18.5 kHz. While this is not quite in the
ultrasonic range, it should be inaudible to the majority of the population
(particularly for adults).

Figure 6: CRLB for the standard deviation of range error.

We choose a modulation scheme such that each phone generates a waveform
orthogonal to any other phone. Initially, we considered LFM chirps and BPSK-
modulated signals, but pursued instead a noise-like waveform due to its ease
of generation. The noise waveform has a high peak-to-average power ratio, and
will therefore produce less output power than other modulations. Additional
schemes should be considered for scenarios where phones may be in pockets,
purses, or other sound-deadening material. We considered orthogonal chirps and
chirp spread spectrum signals, but did not implement them.

We generate WGN samples using a Gaussian Random Number Generator (RNG) seeded
by a hash of a UUID. The WGN is then passed through a bandpass filter with 500
Hz of bandwidth centered at 18.5 kHz, as shown in Figure 7. Finally, a tone at
18.5 kHz is embedded at 10 dB below full scale. The reason for the tone is to
aid in any frequency recovery that must be done by the receiver, since clock
offsets or the Doppler effect will cause the received waveform to decorrelate
with the reference copy. A Fast Fourier Transform (FFT) can be used to
estimate the frequency of the received tone and enable the receiver to correct
for any offset, avoiding the need to compute the full time/frequency ambiguity
function.

An example signal is shown in the time and frequency domains in Figure 8 and
Figure 9, respectively. The signal has a high correlation with itself when
time aligned, low correlation with itself when not time aligned, and low
correlation with signals generated using other UUIDs. These are favorable
properties for multiple signals to coexist simultaneously on a channel with
minimal interference to one another. An example of the correlation is shown in
Figure 10 over the full span of the signal and in Figure 11 expanded to +/- 3
ms around the peak. The oscillations are due to the fact that we are directly
processing a real bandpass signal (without first shifting to baseband and
filtering), however, with a large excess SNR we should be able to correctly
detect the peak at the receiver.

Finally, to prevent audible clicks due to the discontinuities when the signal
begins and ends, a linear ramp-up and ramp-down of 2.5 ms is applied to the
signal before it is transmitted for iOS, and a 1.5 ms ramp is used for
Android. These timings were determined empirically for the iPhone 7, iPhone 8,
and Pixel 2 hardware specifically – it is likely that other devices will
require different ramps for the transmission to be inaudible.

A summary of the waveform design is shown in Table II.

TABLE II: Smartphone Waveform Parameters Parameter | Value |  |  |
---|---|---|---|---
Pulse Width | 100 ms |  |  |
Center Frequency | 18.5 kHz |  |  |
Sampling Rate | 48 kHz |  |  |
Bandwidth | 500 Hz |  |  |
Modulation 333Processing gain and orthogonality for robustness against co-channel interference | Bandpass WGN generated from BLE UUID |  |  |
Rise/Fall Time 444To reduce audible clicks | 2.5 ms |  |  |

Figure 7: Bandpass filter centered at 18.5 kHz.

Figure 8: Ultrasonic signal in the time domain.

Figure 9: Ultrasonic signal in the frequency domain (note tone at 18.5 kHz).

Figure 10: Correlation of two bandpass noise signals $x_{1}$ and $x_{2}$.

Figure 11: Expanded peak from 10 showing oscillations that result from
directly processing a real bandpass signal.

###  III-E Waveform Detector

The detection processing is based on the concept of matched filtering, wherein
we design a digital filter with its coefficients (taps) matched to a set of
known waveform samples. As covered by the previous section, these samples are
generated by a Gaussian RNG, filtered, and loaded into a buffer to be
transmitted. This process is done twice: once based on a phone’s own UUID, and
again for the UUID of the remote phone. A block diagram of the transmitter is
shown in Figure 12, where the contents of the buffer are transmitted at time
$t_{0}$.

The receiver was designed to perform batch processing based on a single buffer
of samples returned with timestamp $t_{1}$. Based on experimenting with the
iPhone 7 and 8, it was determined that a 400 ms buffer (19200 samples at 48
KSPS) could be reliably supported. Once a buffer was returned the first step
was to apply a matched filter (correlate) with the phone’s own transmitted
signal samples and perform peak detection to estimate the time of arrival as
$t_{2}$. The time of arrival is then used for time domain excision (zeroing)
of the phone’s own signal in the buffer, so that it will not interfere with
the detection for the much weaker signal from the remote phone. This
processing is shown by the top portion of Figure 13.

After time domain excision the remaining samples contain the remote waveform
along with any noise and interference. As mentioned in the previous section,
any carrier frequency offset between the two phones (potentially due to motion
and the Doppler effect) will cause the output of the matched filter to be
reduced. Therefore, the next step is to perform frequency recovery by
estimating an offset $f_{0}$ from the ideal carrier frequency of $f_{c}$. This
is done by performing an FFT on the buffer (including samples that were zeroed
out), finding the bin with the peak magnitude, and identifying the difference
between that bin’s frequency and the nominal carrier frequency. The offset
between the two is used to generate a sinusoid to be mixed with the buffer
samples and correct for the offset. Note that the reason for including the
zero samples in the FFT is to interpolate the frequency domain data (which
helps in identifying the true peak).

After frequency correction the samples are processed by a matched filter
designed for the remote phone’s transmitted signal. We used a basic detector
with an adaptive threshold to attempt to find the first received copy of this
signal by comparing the filter output power to an estimate of the background
noise. An adaptive threshold is important for the processing to work when
delayed copies of the signal arrive via multipath propagation, or when the
background noise changes significantly due to interference. Some example
matched filter outputs are shown in Figure 14. The top plot shows a case
without multipath and the two phones within inches of each other. The bottom
plot shows multiple peaks from multipath off of the walls and furniture in a
room and the phones 12 feet apart. Note the significant difference in
magnitudes and potentially challenging problem detecting the first true peak
in the bottom plot.

Figure 12: Transmitter processing block diagram.

Figure 13: Receiver processing block diagram.

Figure 14: Matched filter output with and without multipath. The red circle in
each figure is the true peak. The times on the $x$ axis of the two graphs are
not comparable (i.e., time 0 is not when the signal was sent or received).

Ideally a large amount of data and truth would be collected to design a robust
detector. However, the short time span of this project did not permit that, so
a simple scheme was devised based on a limited amount of data. The adaptive
threshold is set to 15 times the mean power output from the matched filter.
The first sample out of the filter to cross this threshold is identified and
used to return the next 50 samples (1 ms, about 1 foot in range). Within these
samples we search for the two maxima and sort them by their times of arrival.
If the maxima are separated by more than 15 samples (0.3 ms, about 3 inches in
range), we choose the first maxima for the time of arrival estimate $t_{3}$.
Otherwise we choose the largest of the two maxima for $t_{3}$. These
heuristics seemed to work reasonably well on the initial data sets, but other
schemes, such as a Constant False Alarm Rate (CFAR) detector [22], may perform
better.

Once the phone has estimated the times of arrival, it can compute the terms in
Eq. (2) for use by the ranging algorithm. As described in III-C, the transmit
time cancels and we are left with the difference in the receive times to
exchange with the other phone.

##  IV Implementation

In this section, we describe system design aspects that are common to both iOS
and Android implementations. We then go on to discuss OS-specific
implementation details for iOS and Android. We discuss our experiments and
results in the section that follows. All source code is available under MIT
license at https://github.com/mit-ll/BluetoothProximity for iOS and
https://github.com/mspecter/SonicPACT for Android.

The iOS app was written entirely in Swift, while the Android version is a
combination of C++ and Java. For convenience and processing speed, both the
iOS and Android implementations use libraries that take advantage of system
hardware (e.g., ARM’s NEON extensions). For iOS, we used GameplayKit [25] and
vDSP [26]; for Android we used OpenCV [27].

###  IV-A iOS Implementation Details

The ranging algorithm and BLE/ultrasonic protocol were implemented on an
iPhone 7 and 8 for initial evaluation. The implementation follows Figures 12,
13, and 4, apart from the FFT processing not being included in the real-time
version due to time constraints (this was separately evaluated using offline
processing and shown to perform as expected). This real-time limitation meant
that tests could only be done between stationary phones, and that any motion
or frequency offsets between the devices would result in degraded performance.
Additionally, the implementation was only focused on the ultrasonic ranging
measurements, so did not include the full PACT BLE protocol or contact
duration estimator. Below are a few notes from working through the
implementation:

  * •

Sinusoid generation, such as that for the embedded tone, required double
precision arithmetic for the phase argument. Using single precision resulted
in audible tones during the transmission period (hypothesized to be harmonic
distortion).

  * •

The Accelerate framework and vDSP provide highly efficient signal processing
functions. As one data point, the convolution and correlation functions were
over 2000 times faster using vDSP instead of nested for loops in Swift.

  * •

The total receiver processing time for 400 ms worth of data was benchmarked to
take 240 ms on the iPhone 7. This model is nearly 4 years old, and new ones
are likely to require even less processing time.

  * •

The iOS BLE stack does not provide a means for inserting custom data bytes
into advertising packets, but it can encode an advertised name (string). Since
we did not want to establish a full BLE connection to transfer data, we
encoded our commands and data into the advertised name.

  * •

The delay between a BLE command being sent from one phone and received by the
other appeared to nearly always be within the 50 ms window, but varied
significantly. Even if we lost 10 ms of the waveform (due to the follower
starting late), it is unlikely to be an issue because of the excess processing
gain.

###  IV-B Android Implementation Details

We implemented and tested SonicPACT on a Google Pixel2. As with the iOS
implementation, we left FFT processing and BLE contact duration estimation for
future work.

Below are a few notes from working through the implementation:

  * •

Audio processing on Android is more complicated than on iOS. For example,
initial implementations that attempted to record and play audio directly from
Java resulted in unacceptable variations in timing and loss of specificity in
time. We instead used Java only for the high-level protocol implementation,
relying on C/C++ for most processing. We used Oboe [28], a real-time audio
processing library, for high-speed audio access.

  * •

There are no OS-specific libraries for fast DSP processing (similar to vDSP on
iOS). We instead had to rely on third party libraries for general matrix
operations – specifically OpenCV [27] – to provide equivalent processing
speeds.

  * •

Android’s BLE stack is far more customizable than iOS, allowing for use of
Bluetooth 5’s extended advertisements to share timing and UUID information.
Unfortunately, due to the privacy concerns associated with Bluetooth, this
access also requires that the user accept that the app have access to “fine-
grained location information.”

  * •

Unlike iOS, Android does not allow us to pre-buffer audio for a particular
time delay. Instead, all audio is sent to the device through a high-priority
process called MediaFlinger. MediaFlinger also performs OS-level mixing and
filtering, further complicating matters.

Beyond the differences provided in the above list, Android is a fragmented
ecosystem with a diverse set of hardware-dependent quirks. It is likely that
optimizing SonicPACT for each device type will require some device-specific
engineering.

##  V Evaluation

Figure 15: Indoor test setup for iOS devices (one phone on stool at the end of
the tape measure). Figure 16: Indoor test setup for Android devices.

(a) Indoor range errors for iOS devices.

(b) Indoor range errors for Android devices.

Figure 17: Indoor Results.

(a) Outdoor range errors for iOS devices.

(b) Outdoor range errors Android devices.

Figure 18: Outdoor Results.

To evaluate our protocol, we took indoor and outdoor measurements to
characterize the performance with various ranges between the phones in both
Line-of-Sight (LOS) conditions and with one of the phones in a pocket. The
test conditions were as follows:

  * •

Outdoor, no obstructions, some multipath, ranges 2 to 20 feet (60 cm to 6 m)
in 2-foot increments

  * •

Indoor, no obstructions, high multipath, ranges 2 to 12 feet (60 cm to 3.6 m)
in 2-foot increments

In each case, we conducted two sets of tests:

  * •

LOS: One phone in hand, second phone on a chair

  * •

Pocket: One phone in hand, second phone in a pants pocket

The indoor test setups for iOS and Android are shown in Figures 15 and 16,
respectively. In each case we used a tape measure to record the true distance
between the midpoints of each phone. We took at least ten measurements at each
true distance. Both phones were set to one step below their maximum volume in
the case of iOS, and at maximum volume in the case of Android.

###  V-A Metrics

We studied two sets of metrics. The first set is the miss rate and the false
alarm rate, where we took the ranging data from each individual test and
calculated the miss rate assuming a certain “too close” threshold. We show
results for two such thresholds, 6 feet (1.8 m) and 8 feet (2.4 m). The miss
rate measures how often the true distance is under the threshold but the
measured distance was higher. The false alarm rate measures how often the
measured distance was under the threshold but the true distance was higher
than the threshold. Because the measurements have some noise, we permit a
1-foot slack; if the true distance is 6 feet and that is also the threshold,
then we treat any measured distance $<$ 7 feet to be a correct detection
(after all, if the measured distance were 6.3 feet, for example, any practical
exposure notification system would consider that to be close enough to 6
feet). We use the same logic to assess false alarms.

We also report the fraction of measurements in each test where the measured
distance was within 1 foot (30 cm) of the truth.

Because the thresholds of interest for COVID-19 are in the 6-foot range, we
restrict our analysis in the outdoor case to true distances $\leq$ 12 feet;
larger distances artificially lower miss rates.

The second set of metrics evaluates the raw ranging performance showing the
measured distance as a function of the true distance along with the standard
deviations. It is worth noting that in many of these results single outliers
cause the standard deviation to be large.

###  V-B Results

Indoor: The miss rates and false alarm rates for the indoor tests are shown in
Table III.

Test | 6-ft | 6-ft | 8-ft | 8-ft | Within
---|---|---|---|---|---
| miss % | false % | miss % | false % | 1 foot %
LOS Android | 0.0 | 0.0 | 0.0 | 0.0 | 80.3
LOS iOS | 5.1 | 0.0 | 11.9 | 0.0 | 57,6
Pocket Android | 0.0 | 0.0 | 1.3 | 0.0 | 70.9
Pocket iOS | 3.3 | 0.0 | 13.3 | 0.0 | 56.7
TABLE III: Miss rates and false alarm rates for the indoor experiments. The
“Within 1 foot” column shows the percentage of measured distances in the tests
between 2 feet and 12 feet that were within 1 foot of the truth. These results
show low miss rates and false alarm rates with a 1-foot slack across both
platforms. We caution against drawing conclusions that Android is somehow
better than iOS because the tests were done in different indoor conditions. It
is likely that the iOS tests had higher multipath and Android tests had more
sound absorbers.

The results of measured versus true distance in the indoor experiments are
shown in Figure 17(a) for iOS and Figure 17(b) for Android. For both charts,
the points show the mean error and the vertical lines represent the standard
deviation of the error. The errors are generally larger at shorter distances,
especially at ranges beyond 6 feet. Based on a preliminary inspection, in our
iOS experiments, strong multipath components are causing the detector to
select a peak after the true peak, resulting in an estimated range a little
greater than the true range. The impact of phone placement (in hand or in a
pocket) is small, possibly due to additional indoor multipath corrupting the
SNR estimates. It is important to note that the iOS and Android experiments
were done in different locations, so we caution against comparing them
directly because the multipath effects are different.

Outdoor: The miss rates and false alarm rates for the outdoor tests are shown
in Table IV.

Test | 6-ft | 6-ft | 8-ft | 8-ft | Within
---|---|---|---|---|---
| miss % | false % | miss % | false % | 1 foot %
LOS Android | 0.0 | 0.0 | 0.0 | 0.0 | 82.9
LOS iOS | 5.0 | 0.0 | 0.0 | 0.0 | 93.3
Pocket Android | 0.0 | 0.0 | 1.3 | 0.0 | 100.0
Pocket iOS | 1.7 | 0.0 | 1.7 | 0.0 | 96.7
TABLE IV: Miss rates and false alarm rates for the indoor experiments. The
“Within 1 foot” column shows the percentage of measured distances in the tests
between 2 feet and 12 feet that were within 1 foot of the truth. Outdoor
performance is strong across both platforms both for LOS and when one device
is in a pocket.

The outdoor results are shown in Figure 18(a) for iOS and Figure 18(b) for
Android. Here we see that the mean error is within 1 foot of truth, and the
standard deviation is contained within 1 foot for several ranges, particularly
before the device is outside of the 10 ft radius.

For iOS, the large spread at ranges such as 6, 14, and 16 feet are from single
outliers that can likely be eliminated with a more robust detector. We also
see that the performance with one phone in and out of a pocket is similar.

For Android, we note that the range errors worsen dramatically outside of 10
ft. We found experimentally that the Pixel2’s speakers are forward-firing and
highly directional, which likely explains some of the error rate, though there
is likely some systematic error causing the negative bias. The TCFTL
constraint is currently set (by public health authorities) at 6 ft, so this
limitation is not a major problem.

The results of all these experiments show that accurate ultrasonic ranging is
possible on both iOS and Android smartphone platforms, and that SonicPACT is a
viable technique for COVID-19 contact discovery and exposure notification.

##  VI Open Questions

Though these early results are promising, several open questions remain and
there are several opportunities for future work on this subject. It is also
worth noting that the path to this demonstration was not a direct one; several
naive approaches were considered and ultimately discarded.

Our implementation serves as an existence proof that US ranging can work, but
there are hurdles that must be overcome for practical use. Many of these can
be considered for future work, but it is worth noting that some can only be
resolved by trusted computing within the operating system.

####  VI-1 Operation at Scale

Though the use of a matched filter should minimize the impact of hidden
terminals and other interference, careful protocol design will still be
required to enable US ranging technology to operate effectively at scale. A
crowded subway car with 50 devices and potential sources of interference is a
significantly more challenging scenario than two devices in a quiet home or
office. Additionally, US phenomenology is poorly understood and it is unclear
how much interference at US frequencies will be present in real-world
scenarios. The impact of multipath on measurement accuracy is unclear, as is
the extent to which signal attenuation from fabric or body blockage will
reduce the ability to make US measurements.

####  VI-2 Hardware Compatibility

The speakers and microphones used by various smartphone models have different
frequency responses. The percentage of devices currently in-use by consumers
that support reliable operation in the US frequency bands is unknown. However,
anecdotal evidence observed by the PACT (and NOVID) teams suggests that this
will not be a significant issue. Additionally, Lee et al. [10] have measured
the speaker and microphone frequency responses of several devices, and they
all appear to be quite similar.

####  VI-3 Bluetooth Restrictions

iOS does not allow inserting a custom payload into a BLE advertising packet.
However, it does allow setting the advertised name to a custom string. Because
of this limitation, and the short time span of this project, the advertised
name was encoded with commands and measurements for initial demonstration of
the US ranging protocol. This approach was taken since it was preferred to
avoid establishing a full BLE connection and only use advertising packets,
similar to the Exposure Notification API. Android _does_ allow for this sort
of extended advertising, but at the cost of requiring further location
permissions from the operating system. We expect this can be solved on both
platforms with greater OS integration.

####  VI-4 Audio Routing and Volume Control

On the iOS version of the application, we found that applications cannot
control the precise route the audio will take. For instance, an attached pair
of headphones (either analog or Bluetooth) will claim the audio interface and
prevent the built-in speaker from being used by an app. On Android, the app is
allowed to select the appropriate input/output devices, though there is
significant variation on which devices exist, where they are, and which would
be the most advantageous. Further, an app which overrides the user’s volume
settings may be problematic and interfere with normal smartphone usage unless
properly designed.

####  VI-5 Efficiency & Power

Outside of cursory analysis, we currently do not consider, nor do we measure,
the impact on battery life. There is likely further engineering effort to
ensure that power and efficiency are correct, but, given the complexity of
this scheme as compared to normal activation commands, we believe that this
protocol can be made to be efficient and robust, particularly if the existing
voice-activation detection co-processors are used.

####  VI-6 Impact of US Broadcasts on Adults, Children, and Pets

Though adults typically cannot hear ultrasonic frequencies, they are often
audible to children and pets. It is unclear to what extent the US measurement
signals will be audible to children and/or pets and whether or not they will
be bothersome or harmful. The PACT team has not, at this point, observed any
such harmful effects. It would be useful consulting with an audiologist
regarding the use of US frequencies for this purpose.

####  VI-7 Variability of the Speed of Sound

The speed of sound depends on the medium in which it is propagating, to
include the temperature, pressure, and humidity. For a highly accurate
implementation, the speed of sound may need to configured based on a coarse
location (for example, the zip code) and/or a weather report.

####  VI-8 Variability of Devices

Android is a fragmented ecosystem with many devices running many different
versions of the operating system, baseband, hardware profiles and SKUs. Timing
is likely significantly different in these devices and, while this protocol
does attempt to account for the majority of them, differences in mixing,
hardware timing, and other variables leads us to believe that there is a
strong possibility that there is a not-insignificant task of testing this
protocol on a larger diversity of devices for particular quirks.

##  VII Conclusion

This paper provides a proof-of-concept that ultrasonic ranging is viable on
commodity smartphones for the purposes of COVID-19 contact discovery and
exposure notification, helping to reduce the false alarm and miss rates. A
lower false positive rate may boost public confidence in the technique and
encourage more widespread adoption. As noted in our discussion section, many
unanswered questions remain, but these are surmountable by Apple and Google,
as well as the research community. We urge Apple and Google to strongly
consider ultrasonic ranging in general, and SonicPACT in particular, as a way
to improve performance of the Exposure Notification APIs they have developed
recently.

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# Refined Swampland conjecture in deformed Starobinsky gravity

Phongpichit Channuie<EMAIL_ADDRESS>College of Graduate Studies, Walailak
University, Thasala, Nakhon Si Thammarat, 80160, Thailand  School of Science,
Walailak University, Thasala,
Nakhon Si Thammarat, 80160, Thailand  Thailand Center of Excellence in
Physics, Ministry of Education, Bangkok 10400, Thailand

###### Abstract

In order to validate or invalidate a large class of low energy effective
theories, the Swampland conjecture has attracted significant attention,
recently. It can be stated as inequalities on the potential of a scalar field
which is conjectured to satisfy certain constraints. In this work, we discuss
the theoretical viability of deformed Starobinky gravity in light of the
recently refined Swampland conjectures. We consider the deformation of the
form $f(R)\sim R^{2(1-\alpha)}$ with $\alpha$ being a constant. Our analysis
shows that the deformed gravity considered is consistent with the refined
swampland conjecture. The upper bound of the parameter $\alpha$ can be also
constrained to be $\alpha<0.041$.

##  I Introduction

In string theory, it has been suggested that a landscape of vacua is vast and
consistent quantum theory of gravity is believed to be formulated with
consistent low-energy effective field theories (EFTs). Even more recently, the
authors of Refs.Obied:2018sgi ; Agrawal:2018own ; Ooguri:2018wrx  suggested
that the landscape is possibly surrounded by an even more vast neighborhood
called “swampland ”where consistent EFTs, which are coupled to gravity, are
inconsistent with quantum theory of gravity. To be more concrete, the
swampland can be formulated by the set of consistent effective field theories
which cannot be completed into any quantum gravity in the high energy regime.
Hence, it is desirable for consistent EFTs not to lie in the swamplands. See a
comprehensive review on the Swampland Palti:2019pca .

In light of recently proposed swampland conjectures which can be translated to
inequalities on the potential for the scalar field driving inflation, we
consider

$\displaystyle\frac{|\Delta\chi|}{M_{P}}$ $\displaystyle\lesssim$
$\displaystyle\Delta\sim 1,$ (1) $\displaystyle\gtrsim$ $\displaystyle c\sim
1,$ (2)

where $M_{P}$ is the reduced Planck mass, and $\Delta,c$ are positive
constants of order unity, $U(\chi)$ is a potential of an canonical normalized
field $\chi$. Eq.(1) describes effective field theory description to be valid
at the different points in the field space. The conjectures require the above
inequalities to be satisfied by any EFT which has a self-consistent
ultraviolet (UV) completion. Notice that single-field inflation is apparently
satisfied by the first condition, and the field variation is in excellent
agreement with the well-known Lyth Bound for single-field inflation
Lyth:1996im . However, the second condition poses difficulties for single-
field paradigm Agrawal:2018own ; Garg:2018reu ; Dimopoulos:2018upl ;
Matsui:2018bsy . This is so since it is in tension with the slow roll
condition. This tension has been explored in a number of recent works, see for
example Refs.Kehagias:2018uem ; Achucarro:2018vey ; Kinney:2018nny ;
Das:2018hqy ; Ashoorioon:2018sqb ; Brahma:2018hrd ; Lin:2018rnx ;
Motaharfar:2018zyb ; Lin:2018kjm ; Holman:2018inr . Interestingly, however,
more complex models like, among many others, multifield inflation
Achucarro:2018vey  and warm inflation Motaharfar:2018zyb  are still allowed.
In light of alternative theories of gravity, Swampland criteria was also
investigated, see for instance Yi:2018dhl ; Heisenberg:2019qxz ;
Brahma:2019kch . In Artymowski:2019vfy , the viability of $f(R)$ and Brans-
Dicke theories of gravity was also investigated.

Very recently, the “refined” version of the swampland conjecture has been
suggested Garg:2018reu ; Ooguri:2018wrx . It was found that the refined
conjecture imposes a slightly weaker criterion on the scalar field potential
in inflation, and is consistent with the existence of a tachyonic instability.
In light of the refined swampland conjecture, a scalar field potential
associated with a self-consistent UV-complete effective field theory must
satisfy one of the two conditions:

$\displaystyle\Bigg{(}M_{P}\frac{|U^{\prime}|}{{๊}U$≳c) or (M^2_P—U”—–๊U≲-c’)

where $c$ and $c^{\prime}$ are constants of order unity. In the present work,
we consider the implications of this slightly weaker constraint on a
deformation of Starobinsky gravity. It is worth noting that a different model
of deformed Starobinsky inflation was so far studied in Ref.Sebastiani:2013eqa
, while log corrections to $R^{2}$ gravity were investigated in
Ref.Myrzakulov:2014hca ; Bamba:2014jia . Regarding the refined version, there
exist relevant discussions of general constraints on inflation and other
cosmological/astrophysical models, see Refs.Kinney:2018kew ; Haque:2019prw ;
Andriot:2018mav ; Fukuda:2018haz ; Garg:2018zdg ; Park:2018fuj ;
Schimmrigk:2018gch ; Cheong:2018udx ; Chiang:2018lqx .

This paper is organized as follows. In Sec.II, we present a summary of
deformed Starobinsky gravity by following the work proposed by
Ref.Codello:2014sua . Here we linearize the action by introducing an auxiliary
field method. We use the conformal transformations in order to transform the
theory in the Jordan frame to the Einstein frame. The above transformation
connects both theories and allows us to rewrite the action in terms of a
propagating scalar field minimally coupled to gravity. Sec.III is devoted to
discuss the theoretical viability of deformed Starobinky gravity in light of
the recent refined Swampland conjectures. Here we will constrain the model
parameters using the conjecture. Our conclusions are reported in the last
section.

##  II Deformed Starobinsky gravity: a short recap

Among many others, an intriguing possibility is that the gravity itself can be
directly responsible for the inflationary period of the universe. The
examination requires us to go beyond time-honored Einstein-Hilbert (EH)
action. This can be achieved by adding a $R^{2}$ term to the original EH
theory as in the Starobinsky model Starobinsky:1980te . The model is highly
natural since gravity itself drives cosmic inflation without the need of the
scalar fields. It is worth noting that the model predicts a nearly vanishing
ratio of tensor to scalar modes which is in excellent agreement with the
observations, e.g. PLANCK data Ade:2015lrj ; Akrami:2018odb .

In light of the observations, a discovery of primordial tensor modes can be
used to constrain the cosmological parameters at the inflationary scale, which
turn out to be close or at the grand unification energy scale. In general, the
effective action for gravity can be in principle derived by considering the
Taylor expansion in the Ricci scalar, $R$. Here without assuming a concrete
form for the function $f(R)$, we consider

$\displaystyle{\cal S}=\int d^{4}x\sqrt{-g}f(R)\equiv\int
d^{4}x\sqrt{-g}\Big{[}a_{0}+a_{1}R+a_{2}R^{2}+...\Big{]}\,.$ (4)

The first term $a_{0}$ is like the cosmological constant and must be small.
The next coefficient $a_{1}$ can be set to one as in general relativity.
Regarding the Starobinsky gravity, we have $a_{2}=1/(6M^{2})$ where a constant
$M$ has the dimensions of mass, see cosmological implications of the model
Chatrabhuti:2015mws . Here the ellipses may include the Weyl tensor $C^{2}$
and the Euler topological terms $E$. As mentioned in Ref.Codello:2014sua , the
$E$ terms can be ignored since it is just a total derivative. Moreover, the
Weyl terms are subleading when gravity is quantized around a flat background.
Higher powers of $R,\,C^{2}$ and $E$ are naturally suppressed by the Planck
mass. Interestingly, the authors of Ref.Codello:2014sua  also take into
account marginal deformations of the action (4) by including logarithmic
corrections. The authors consider a simple form of the gravitational action
formulated in the Jordan frame:

$\displaystyle{\cal S}_{J}=\int
d^{4}x\sqrt{-g}\Big{[}-\frac{M^{2}_{P}}{2}R+hM^{4\alpha}_{P}R^{2(1-\alpha)}\Big{]}\,,$
(5)

where $h$ is a dimensionless parameter and $\alpha$ is a real parameter which
is assumed as $2|\alpha|<1$. Note that the condition of the parameter $\alpha$
is further examined in the context of gravity’s rainbow Channuie:2019kus . One
can linearize the above action by introducing an auxiliary field $y$ such that
${\cal S}_{J}=\int d^{4}x\sqrt{-g}\Big{[}f(y)+f^{\prime}(y)(R-y)\Big{]}$ with
$f(R)=-M^{2}_{P}R/2+hM^{4\alpha}_{P}R^{2(1-\alpha)}$ where
$f^{\prime}(y)=df(y)/dy$. Here the equation of motion for $y$ implies $R=y$
provided $f^{\prime\prime}(y)$ does not vanish. The explicit relation between
(4) and the effective quantum-corrected nonminimally coupled scalar field
theory used in Ref.Joergensen:2014rya  can be done by introducing the
conformal mode $\psi=-f^{\prime}(y)$ with $V(\psi)=-y(\psi)\psi-f(y(\psi))$
and having introduced the mass-dimension one real scalar field $\varphi$ via
$2\psi-M^{2}_{P}=\xi\varphi^{2}$ Codello:2014sua  to obtain:

$\displaystyle{\cal S}_{J}=\int
d^{4}x\sqrt{-g}\Big{[}-\frac{M^{2}_{P}+\xi\varphi^{2}}{2}R+V(\varphi)\Big{]}\,,$
(6)

where

$\displaystyle
V(\varphi)=\lambda\varphi^{4}\Big{(}\frac{\varphi}{M_{P}}\Big{)}^{4\gamma}\,\,\,{\rm
with}\,\,\,\alpha=\frac{\gamma}{1+2\gamma},$ (7)

and

$\displaystyle
h^{1+2\gamma}=\Big{(}\frac{\xi(1+2\gamma)}{4(1+\gamma)}\Big{)}^{2(1+\gamma)}\frac{1}{\lambda(1+2\gamma)}.$
(8)

Notice from Eq.(6) that the kinetic term for the field $\varphi$ is absent in
the Jordan frame. However introducing the following conformal transformation
of the metric, the kinetic term of the field can be simply generated via:

$\displaystyle{\tilde{g}}_{\mu\nu}=\Omega(\varphi)^{2}g_{\mu\nu}\,,\,\,{\rm
with}\,\,\Omega(\varphi)=1+\frac{\xi\varphi^{2}}{M^{2}_{P}}.$ (9)

The above transformation connects both theories and allows us to rewrite the
action in terms of a propagating scalar field minimally coupled to gravity.
The resulting action is written in the Einstein frame and takes the form:

$\displaystyle{\cal S}_{E}=\int
d^{4}x\sqrt{-g}\Big{[}-\frac{M^{2}_{P}}{2}R+\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-U(\chi)\Big{]}\,,\,\,\,U(\chi)=\Omega^{-4}V(\varphi(\chi)).$
(10)

Notice that the action is written in terms of the canonically normalized field
$\chi$ which is related to $\varphi$ via Codello:2014sua

$\displaystyle\frac{1}{2}\Big{(}\frac{d\chi}{d\varphi}\Big{)}^{2}=\frac{M^{2}_{P}(\sigma
M^{2}_{P}+(\sigma+3\xi)\xi\varphi^{2})}{(M^{2}_{P}+\xi\varphi^{2})^{2}}.$ (11)

It is worth noting that when setting $\sigma=0$ a map from the Jordan frame of
$f(R)$ gravity to the Einstein frame with a canonically normalized field is
obtained. Throughout this work, we will set $\sigma=0$. An explicit relation
between $\chi$ and $\varphi$ can be obtained by assuming that inflation occurs
at large values of the scalar field, i.e. $\varphi\gg M_{P}/\sqrt{\xi}$ and we
obtain

$\displaystyle\chi\simeq\sqrt{6}M_{P}\log\Big{[}\frac{\sqrt{\xi}\varphi}{M_{P}}\Big{]}.$
(12)

Substituting the above canonical normalized field into Eq.(9), therefore the
Einstein frame potential takes the form

$\displaystyle U(\chi)=\frac{\lambda
M^{4}_{P}}{\xi^{2}}\Bigg{(}1+\exp\Bigg{(}\frac{-2\chi}{\sqrt{6}M_{P}}\Bigg{)}\Bigg{)}^{-2}\xi^{-2\gamma}\exp\Bigg{(}\frac{4\gamma\chi}{\sqrt{6}M_{P}}\Bigg{)}\,,\,\,\gamma=\frac{\alpha}{1-2\alpha}.$
(13)

It is worth noting that in the limit of $\gamma=0$ (or equivalently
$\alpha=0$) one recovers the Starobinsky model. It’s pointed out in
Ref.Codello:2014sua  that for $0<\alpha<0.5$ the potential grows
exponentially, and then an inflationary model with nonzero primordial tensor
modes can be successfully obtained. Note that the exact Einstein-frame
potential of $f(R)=R+\lambda R^{p}$ with $p$ being not necessarily an integer
in general was also derived in Ref.Motohashi:2014tra .

##  III The refined Swampland criteria in deformed Starobinsky gravity

We note here that in the previous section we adopted a conformal
transformation and transformed the theory from the Jordan to the Einstein
frame. Deriving this potential with respect to the field $\chi$, we obtain

$\displaystyle\frac{\partial U(\chi)/\partial\chi}{U}=\frac{4\gamma+4\gamma
e^{\frac{2\chi}{\sqrt{6}M_{P}}}+4}{\sqrt{6}M_{P}e^{\frac{2\chi}{\sqrt{6}M_{P}}}+\sqrt{6}M_{P}}.$
(14)

Hence the first condition of the refined swampland conjecture (2) suggests the
following inequality:

$\displaystyle
c\lesssim\frac{4\gamma+4\gamma\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+4}{\sqrt{6}\left(1+\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}\right)}.$
(15)

In establishing the connection among the swampland conditions and the
parameters of the model considered, we consider the well-known inflationary
parameters, i.e., the scalar spectral index $n_{s}$, and tensor to scalar
ratio $r$, and from the standard formulation we write

$\displaystyle n_{s}=1-6\epsilon_{U}+2\eta_{U},\,\,r=16\epsilon_{U},$ (16)

where in terms of the potential the slow-roll parameters can be defined as

$\displaystyle\epsilon_{U}=\frac{M^{2}_{P}}{2}\left(\frac{U^{\prime}}{U}\right)^{2}\,,\,\,\eta_{U}=M^{2}_{P}\left(\frac{U^{\prime\prime}}{U}\right),$
(17)

where primes denote derivatives with respect to the field $\chi$. Using the
above expressions, we can recast Eq.(16) in terms of the inflaton field $\chi$
as

$\displaystyle 1-n_{s}$ $\displaystyle=$
$\displaystyle\frac{16\left(2\gamma+\gamma^{2}\left(\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+1\right)+1\right)}{\sqrt{6}^{2}\left(\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+1\right)},$
$\displaystyle r$ $\displaystyle\lesssim$ $\displaystyle 8c^{2}\approx
8\Bigg{(}\frac{4\gamma+4\gamma\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+4}{\sqrt{6}\left(1+\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}\right)}\Bigg{)}^{2},$
(18)

where we have used the potential from Eq.(13). The first expression of (18)
allows us to rewrite the inflaton field in terms of the parameters $\gamma$
and $n_{s}$ to obtain

$\displaystyle\chi:=\frac{1}{2}\sqrt{6}M_{P}\log\left(\left|\frac{16\gamma^{2}+32\gamma-\sqrt{6}^{2}(1-n_{s})+16}{\sqrt{6}^{2}(1-n_{s})-16\gamma^{2}}\right|\right).$
(19)

After combining Eqs.(19) and (15), we will see that particular values of $c$
yield the values of scalar spectral index $n_{s}$ to be fitted in a region at
the $2\sigma$ level. Here we find

$\displaystyle
c\lesssim\frac{4\alpha^{2}-4\alpha-3(1-2\alpha)^{2}n_{s}+3}{2\sqrt{6}(1-2\alpha)},$
(20)

where the constant $c$ is rewritten in terms of the parameters $\alpha$ and
$n_{s}$. Together with the swampland condition, this also allows us to use the
observational data, e.g., PLANCK satellite, to constrain the model. The
behavior of $c$ in Eq.(20) is illustarted in Fig.(1).

Figure 1: We plot allowed values of $c$ as a function of $n_{s}$ parametrized
by Eq.(20) by varying the values of $\alpha$. Here the gray shaded region
represents the allowed values of $n_{s}$ at $2\sigma$ level observed by PLANCK
2018 Akrami:2018odb .

Similarly from the second condition of the conjecture, we find the following
further constraint on the inflaton field value

$\displaystyle\frac{8\left(2(\gamma+1)^{2}+2\gamma^{2}\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+\left(4\gamma^{2}+4\gamma-1\right)\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}\right)}{\sqrt{6}^{2}\left(\exp\Big{(}\frac{2\chi}{\sqrt{6}M_{P}}\Big{)}+1\right)^{2}}\lesssim-c^{\prime}.$
(21)

Combing the Eqs.(19), (20) and (21), we are able to find constraints on
$c,\,c^{\prime}$. The combined equation can be written as

$\displaystyle c^{2}\,c^{\prime 2}$ $\displaystyle\lesssim$ $\displaystyle
f(\alpha)\equiv\frac{(1-2\alpha)^{6}}{16384\sqrt{6}^{6}}\Bigg{(}\frac{12288(\alpha-1)^{3}\alpha^{3}}{(1-2\alpha)^{6}}-3\sqrt{6}^{6}n_{s}^{3}+\frac{16(9(\alpha-1)\alpha+1)\sqrt{6}^{4}n_{s}^{2}}{(1-2\alpha)^{2}}$
(22)
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad-\frac{256(\alpha-1)\alpha(9(\alpha-1)\alpha+1)\sqrt{6}^{2}n_{s}}{(1-2\alpha)^{4}}\Bigg{)}^{2}.$

By using the above constraint relation along with any one of the swampland
conjectures, we have the allowed region of the $(c,\,c^{\prime})$ space as
shown in Fig.2.

Figure 2: We show the resulting constraints for $c$ and $c^{\prime}$ in
Eq.(22) by using the precise value of $n_{s}=0.964$ Akrami:2018odb . Note that
the allowed values of $c$ and $c^{\prime}$ are displayed with color-shaded
regions controlled by the refined swampland conditions.

Further using the swampland distance conjecture (1), we find in a unit of the
Planck mass that the condition $|\Delta\chi|\lesssim{\cal O}(1)$ is satisfied
for the model we are considering here. Hence using Eq.(14) and the relation
$\Delta U=\Big{(}U_{\chi}/U\Big{)}\Delta\chi$ with $U_{\chi}\equiv\partial
U/\partial\chi$, we can write

$\displaystyle\frac{\Delta\chi}{M_{P}}=\frac{4\sqrt{6}(1+2\gamma)}{16\gamma^{2}+16\gamma+\sqrt{6}^{2}(1-n_{s})}\frac{\Delta
U}{U}\lesssim\Delta\sim 1.$ (23)

Therefore, we obtain in terms of $\alpha$:

$\displaystyle\Delta
U\lesssim\frac{4\alpha^{2}-4\alpha-3(1-2\alpha)^{2}n_{s}+3}{2\sqrt{6}(1-2\alpha)}U.$
(24)

Figure 3: We display the relation between $\Delta U/U$ and $n_{s}$ given in
Eq.(24). Here we use $\alpha=0.32$ and $\alpha=0.25$ for comparison. The gray
shaded region represents the values of $n_{s}$ at $2\sigma$ deviation.

Assuming the value $|\Delta\chi|\sim 1$ as the upper limit in the equation
above, we can see that for the value $\alpha\sim 0.32$ and $n_{s}=0.964$ the
first swampland condition is satisfied whenever

$\displaystyle\Delta U\sim U\,\,\longrightarrow\,\,\frac{\Delta U}{U}\sim{\cal
O}(1).$ (25)

However, we can come up with stronger condition if we take $\alpha\ll 0.32$.
For example, using $\alpha=0.032$ we find $\frac{\Delta U}{U}\sim 0.07\ll{\cal
O}(1)$. The inequality of (24) is displayed in Fig.(3). We can further
consider the relation of Eq.(16). Hence we are able to write

$\displaystyle\alpha\lesssim\frac{1}{4(3n_{s}-1)}\Big{(}6n_{s}+\sqrt{3r}-2-\sqrt{24n_{s}+3r-8}\Big{)}.$
(26)

Figure 4: We plot an allowed region parametrized by Eq.(26). The values of the
model parameters below the region are satisfied by the conjecture.

We display an allowed region satisfied an inequality given in Eq.(26) as shown
in Fig.(4). We find that using the upper bound on the tensor to scalar ratio
$r<0.064$ and the scalar spectral index $n_{s}=0.964$ Akrami:2018odb  the
upper bound of the $\alpha$ parameter as

$\displaystyle\alpha<4.1\times 10^{-2}.$ (27)

In this case, we find that the model we are considering may pose difficulty of
the first swampland conjecture. This is so since using $\alpha=0.04$ we
discover from Eq.(19) that $c\simeq 0.1$. As mentioned in Ref.Kehagias:2018uem
, however, the requirement that no critical de Sitter vacua exist allows any
value of $c$ as long as $c$ is positive, even tiny values. Given that no
quantitive argument for having $c\simeq 1$ is currently available, $c\simeq
0.1$ seems as good as $c\simeq 1$ to us.

Figure 5: We show a 2D plot of an allowed region satisfied an inequality given
in Eq.(26) by using $\alpha=0.04$ (left panel) and $\alpha=0.045$ (right
panel).

We display the relation between $r$ and $n_{s}$ given by Eq.(26). In Fig.(5),
we compare the parameters by using $\alpha=0.04$ and $\alpha=0.045$. We find
that when setting $\alpha=0.04$ both $r$ and $n_{s}$ are simultaneously
satisfied the bound reported by PLANCK Akrami:2018odb ; while using
$\alpha=0.045$ both $r$ and $n_{s}$ can not be satisfied the bound. Therefore
the upper bound of $\alpha$ given by Eq.(27) can be satisfactorily verified.

##  IV Conclusion

Recently, the Swampland conjecture has attracted significant attention. The
conjecture allows us to validate or invalidate a large class of low energy
effective theories. It can be formulated as inequalities on the potential of a
scalar field which is satisfied by certain constraints. In this work, We first
considered the deformation of the form $f(R)\sim R^{2(1-\alpha)}$ with
$\alpha$ being a constant. Here we linearized the original action by
introducing an auxiliary field method and used the conformal transformations
in order to transform the theory in the Jordan frame to the Einstein frame. We
rewrote the action in terms of the cannonical normalized scalar field
minimally coupled to gravity.

We discussed the theoretical viability of deformed Starobinky gravity in light
of the recent refined Swampland conjectures. Our analysis showed that the
deformed gravity considered is consistent with the refined swampland
conjecture. We showed the resulting constraints on $c$ and $c^{\prime}$ for
the precise value of the scalar spectral index $n_{s}$. For the model under
consideration, the value of $c$ turned out to be bounded within $(0,0.1)$.
Interestingly, we discovered the upper bound of the parameter $\alpha$ to be
$\alpha<0.041$.

###### Acknowledgements.

This research was partially supported by the New Strategic Research (P2P)
project, Walailak University, Thailand.

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