# Fast-field-cycling, ultralow-field nuclear magnetic relaxation dispersion

Sven Bodenstedt  ICFO – Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
Morgan W. Mitchell  ICFO – Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
ICREA – Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona,
Spain  Michael C. D. Tayler<EMAIL_ADDRESS>ICFO – Institut de
Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain

(May 31st 2021)

## KEYWORDS

Molecular motion; Ultralow-field nuclear magnetic resonance (ULF NMR);
Optically pumped magnetometers (OPMs); Relaxation; Porous materials.

## Abstract

Optically pumped magnetometers (OPMs) based on alkali-atom vapors are ultra-
sensitive devices for dc and low-frequency ac magnetic measurements. Here, in
combination with fast-field-cycling hardware and high-resolution spectroscopic
detection, we demonstrate applicability of OPMs in quantifying nuclear
magnetic relaxation phenomena. Relaxation rate dispersion across the
$\mathrm{nT}$ to $\mathrm{mT}$ field range enables quantitative investigation
of extremely slow molecular motion correlations in the liquid state, with time
constants $>$$1\text{\,}\mathrm{ms}$, and insight into the corresponding
relaxation mechanisms. The 10-20
$\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$ sensitivity of an OPM
between 10 Hz and 5.5 $\mathrm{kHz}$ 1H Larmor frequency suffices to detect
magnetic resonance signals from $\sim$0.1 mL liquid volumes imbibed in simple
mesoporous materials, or inside metal tubing, following nuclear spin
prepolarization adjacent to the OPM. High-resolution spectroscopic detection
can resolve inter-nucleus spin-spin couplings, further widening the scope of
application to chemical systems. Expected limits of the technique regarding
measurement of relaxation rates above $100\text{\,}{\mathrm{s}}^{-1}$ are
discussed.

## Introduction

Nano-scale dynamic processes that occur on $\mathrm{ms}$ to
$\mathrm{\SIUnitSymbolMicro s}$ time scales, such as protein folding, aqueous
complexation and surface adsorption phenomena, are often probed using nuclear
magnetic relaxation dispersion (NMRD) techniques[1, 2, 3], in which field-
dependent relaxation rates of nuclear spins are used to infer correlation
times for molecular reorientation[4, 5] and diffusive transport. Beyond
fundamental interests, insights from NMRD such as surface fractal dimension
and roughness provide models for industrial catalysis and petrology, where
liquids are confined inside porous solids and molecular diffusion is
restricted by surface geometry[6] as well as adsorption[7], and in medicine
assist the design of molecular agents for relaxation-contrast magnetic
resonance imaging (MRI)[8]. Furthermore, if coupled with spectroscopic
dispersion via chemical shifts or spin-spin couplings, the dynamics can be
related to specific molecular functional groups, facilitating analyses of
chemical mixtures and biological specimens[9].

Accurate correlation times $\tau_{\rm c}$ can be obtained by measuring nuclear
spin relaxation across a range of Larmor frequencies $B\gamma_{I}\ll\tau_{\rm
c}^{-1}$ to $B\gamma_{I}\gg\tau_{\rm c}^{-1}$, where $B$ is the field strength
and $\gamma_{I}$ is the nuclear gyromagnetic ratio. Extremely slow
correlations thus require measurements at ultralow magnetic fields within
shielded enclosures such as a MuMetal chamber. The main existing NMRD
technique uses fast-field-cycling (FFC) electromagnets[10, 11, 12, 13] of
around $1\text{\,}\mathrm{T}$ for efficient inductive NMR signal detection,
but these must be used unshielded with active cancelation of ambient fields to
access below the geomagnetic field range[14, 15]. Alternatively, NMRD is
performed by transporting samples between persistent high- and ultralow-field
locations[16, 17, 19, 18, 20], but relatively slow transport times limit the
observable $\tau_{\rm c}$ at the high end. The limits of these existing
techniques are illustrated by the magenta- and blue-shaded regions,
respectively, of Figure 1.

Figure 1: Limits of existing NMRD techniques. Boundaries of the colored zones
in the horizontal dimension indicate upper limits of longitudinal relaxation
rate, ($T_{1,I}^{-1}$ ${\mathrm{s}}^{-1}$), imposed by signal receiver dead
time and/or speed of field-switching events. Boundaries in the vertical
dimension indicate limits to the range of magnetic fields achievable by
electromagnetic sources. Cross-hatching indicates Larmor frequencies where NMR
signals are detected and corresponds to measurable transverse relaxation rates
$T_{2,I}^{-1}$ (${\mathrm{s}}^{-1}$). The diagonal of slope 1 corresponds to
the spectroscopic resolution limit where rates of Larmor precession and
transverse relaxation are equal: $B\gamma_{I}/(2\pi)=T_{2,I}^{-1}\sim
T_{1,I}^{-1}$.

In this work, we introduce a third scenario to address the the top-left
portion of Figure 1 that lies outside the reach of inductive NMR pickup. The
speed of the FFC approach is combined with the low-frequency sensitivity of a
spin-exchange-relaxation-free (SERF)[25, 27, 26, 21, 22, 28, 23, 24] optically
pumped magnetometer (OPM) to perform NMRD at 1H Larmor frequencies from
$1\text{\,}\mathrm{Hz}$ to $10\text{\,}\mathrm{kHz}$, corresponding to the
region of Figure 1 shaded in green. The high sensitivity of SERF OPMs of order
$1\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$[29, 30] at signal
frequencies down to a few Hz, rivals the best Superconducting Quantum
Interference Device (SQUID)[31, 32] and high-Q inductive-pickup magnetometers
below Earth’s field[33, 34], with the advantage of cryogen-free operation and
simple tuning based on Hartmann-Hahn matching of the OPM and NMR spin
ensembles. In the ultralow-field NMRD context, the OPM is compatible both with
MuMetal shielding and relatively weak prepolarizing fields of order
$10\text{\,}\mathrm{mT}$. Magnetic fields for relaxation and detection are
supplied accurately and precisely (within $1\text{\,}\mathrm{nT}$) following a
one-off calibration procedure and can be cycled in less than
$1\text{\,}\mathrm{ms}$, resulting in atomic-response-limited dead times.
Based on this configuration, we are able to study spin relaxation phenomena
that cannot be probed using conventional inductive field-cycling NMR
procedures: (1) relaxation of liquids encased in metal tubing, demonstrated
using aqueous solutions of paramagnetic impurities; (2) the full frequency
dependence for motional correlations in a system of n-octane (n-C8$H$18) and
n-decane (n-C10H22) absorbed in nanoscale confinement upon porous alumina and
titania, as an example relevant to research in catalysis. Results
unambiguously support dynamics models involving molecular diffusion among
paramagnetic sites on the pore surface; (3) chemical species resolution via
spin-spin J couplings.

### Diffusion dynamics from NMRD

When it covers the appropriate range of fields and time scales, the relaxation
measured in NMRD can relate model parameters of interest to molecular motion,
surface structure and molecule-surface interactions [35, 36, 37]. A central
quantity of interest is the time correlation function $g(\tau)=\langle
x(t)x(t+\tau)\rangle/\langle x(t)x(t)\rangle$ of the molecular motion. This is
related to observable relaxation by the quantity
$j(\omega)=\int_{0}^{\infty}{g(\tau)\cos(\omega\tau)\mathrm{d}\tau}$, i.e. the
cosine transform of $g(\tau)$. Here we assume a simple but useful model where
the local field is inhomogeneous, with a randomly oriented component of root-
mean-square amplitude $B_{\mathrm{rms}}$. The longitudinal relaxation rate is
$[T_{1,I}(\omega_{I})]^{-1}=\gamma_{I}^{2}B_{\mathrm{rms}}^{2}j(\omega_{I})$
under standard perturbation (i.e. Redfield[35]) assumptions. In the ideal case
of unrestricted diffusion, a single correlation time is found, where
$g(\tau)\propto\exp[-(\tau/\tau_{\rm c})]$ and the spectral density is a
Lorentzian: $j(\omega)=\tau_{\rm c}/(1+\omega^{2}\tau_{\rm c}^{2})$, where
$\tau_{\rm c}$ is the characteristic diffusion time. Inverse-square power law
behavior is thus expected for $T_{1,I}^{-1}$ vs. $\omega_{I}$, for
$\omega_{I}\gg\tau_{\rm c}^{-1}$.

Scenarios of constrained Brownian motion[38] such as diffusion in pores may
yield several concurrent dynamics modes. Fitting to a distribution of
correlation times $p(\tau^{\prime})$ may be more appropriate:
$[T_{1,I}(\omega_{I})]^{-1}=\gamma_{I}^{2}\int_{0}^{\infty}B_{\mathrm{rms}}^{2}p(\tau_{\rm
c})\tau_{\rm c}/(1+\omega_{I}^{2}\tau_{\rm c}^{2})\,\mathrm{d}\tau_{\rm c}$
[39, 40], where $p(\tau_{\rm c})$ represents a probability distribution
normalized to $\int_{0}^{\infty}p(\tau_{\rm c})\,\mathrm{d}\tau_{\rm c}=1$.
Kimmich and co-workers examine this approach to explain power-law relaxation
behavior in porous glasses: $T_{1,I}\propto\omega_{I}^{\xi}$, where
$0<\xi<2$[41, 42]. Surface-induced relaxation is attributed to “molecular
reorientation mediated by translational displacement” (RMTD), where diffusion
across a rugged pore surface modulates intra-molecular spin-spin dipolar
couplings and $p(\tau_{\rm c})$ is linked to the surface fractal dimension. A
breakdown of the power law at low frequencies $\omega_{I}$ indicates a maximum
$\tau_{\rm c}$, which is connected to the longest distance a molecule can
diffuse before leaving the surface phase or experiencing a different surface
structure. The value should depend on the molecule, due to different diffusion
coefficients, as well as the porous medium.

Moreover towards zero Larmor frequency, $T_{1,I}$ tends to a plateau defined
by
$[T_{1,I}(0)]^{-1}=\gamma_{I}^{2}\int_{0}^{\infty}B_{\mathrm{rms}}^{2}p(\tau_{\rm
c})\tau_{\rm c}=\gamma_{I}^{2}B_{\mathrm{rms}}^{2}\langle\tau_{c}\rangle$,
where $\langle\tau_{c}\rangle$ is the mean correlation time[37]. This and the
above measures all require $T_{1,I}$ to be known for frequencies below
$\tau_{\rm c,max}^{-1}$, motivating ultra-low field measurement capability.

## Results

### Tunable NMR detection using optical magnetometry

Figure 2a shows an experimental setup for FFC NMR with OPM detection. A
$2\text{\,}\mathrm{mL}$ vial containing the NMR sample sits within four
coaxial solenoids (S1 – S4), which provide $z$-oriented fields. Liquid coolant
flows around the sample chamber, with S1 moreover immersed in the flowing
coolant to maintain a sample temperature around
$30\text{\,}\mathrm{\SIUnitSymbolCelsius}$. The short S1 solenoid is used to
produce fields up to $20\text{\,}\mathrm{mT}$ to polarize the nuclear spins in
the sample, while S2 and S3 provide weaker fields for Larmor precession.
Adjacent to this chamber is a heated glass cell containing 87Rb vapor, with
optical access along $x$ and $z$ directions for probing and pumping of the
alkali spin angular momentum $\bm{\mathrm{S}}$, respectively.

A magnetic field $\bm{\mathrm{B}}=(B_{1}\cos{\omega t},B_{1}\sin{\omega
t},B_{z,S})$ in Cartesian coordinates is assumed at the atoms, as the sum of a
constant bias field $B_{z,S}$ along the $z$ axis and an effective rotating
field $B_{1}$ induced by precession of nuclear magnetization in the NMR sample
in the $xy$ plane. Assuming the nuclei experience a field $B_{z,I}$ along the
$z$ axis, then $\omega=\omega_{I}=\gamma_{I}B_{z,I}$, and $B_{1}$ is
proportional to the amplitude of the nuclear magnetization.

Figure 2: Tunable NMR detection via optically pumped magnetometer: (a)
Schematic view of the apparatus through a vertical cross section. Standoff
distance between sample container atomic vapor cell $d_{1}=3.5\,\mathrm{mm}$.
Center-to-center distance between atomic cell and NMR sample
$d_{2}=10\,\mathrm{mm}$; (b) Frequency dependence of magnetometer noise in
field-equivalent units at bias fields $B_{z,I}$ from 0.1 to 100 $\mu$T. Plot
markers indicate sensitivity at the 1H Larmor frequency under the conditions
of ($\blacksquare$) tuned and ($\blacktriangle$) untuned 87Rb Larmor
frequencies; error bars correspond to the sensitivity’s root-mean-square
deviation within a $1\text{\,}\mathrm{H}\mathrm{z}$ window around the Larmor
frequency. (c) Representative NMR signal and noise amplitudes in the tuned
case for a sample of 1.8 mL milli-Q water, measured after pre-polarization at
20 mT and a transverse $\pi/2$ pulse.

Dynamics of $\bm{\mathrm{S}}$ are adequately described in the SERF regime by a
polarization vector model where the $x$-axis component of $\bm{\mathrm{S}}$
under steady-state pump-probe and a transverse rf field of angular frequency
$\omega$ is given by[30]

$S_{x}=\frac{g_{S}R_{\mathrm{op}}T_{2,S}^{2}}{2q^{2}}\Bigl{[}\frac{\cos\omega
t+(\omega-\omega_{S})T_{2,S}\sin\omega
t}{1+(\omega-\omega_{S})^{2}T_{2,S}^{2}}\Bigr{]}B_{1}.$ (1)

Here, $\omega_{S}=g_{S}B_{z,S}/q$ is the Larmor frequency, $g_{S}$ is the
gyromagnetic ratio, $q$ is the nuclear slowing down factor, $R_{\mathrm{op}}$
is the optical pumping rate and $T_{2,S}^{-1}$ is the transverse relaxation
rate of the alkali atom ensemble. According to the above Equation 1, the
atomic response to $B_{1}$ is strongest for matched precession frequencies of
the spin species: $\omega_{I}=\omega_{S}$. Thus the OPM is tunable to a given
NMR frequency $\omega_{I}$ by setting the magnetic field at the atoms to
$B_{z,S}=\pm(q/g_{s})\omega_{I}=\pm\gamma_{I}(q/g_{s})B_{z,I}$. This
adjustment is permitted since $B_{z,I}$ is the superposition of the fields in
the interior of coils S2 + S3 + S4, while $B_{z,S}$ is the superposition of
fields from S4 and the much weaker exterior field of S2 + S3.

For Larmor frequencies $\omega_{I}/(2\pi)$ between 10 and 200 Hz the
magnetometer noise is below $10\,\mathrm{fT}/\sqrt{\mathrm{Hz}}$ (Figure 2b),
limited by noise in the lasers and to a lesser extent the Johnson noise of the
coils S1 + S2 + S3. The spin projection noise estimated from the atom density
$n_{S}\approx 10^{20}$ m-3, temperature
$150\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and coherence time $T_{2,S}\approx
3$ ms is $\sqrt{n_{S}g_{S}^{2}T_{2,S}/q}\sim$
$1.1\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$. Above fields
$B_{z,S}\approx 100$ $\mathrm{nT}$, $\omega_{S}$ starts to become comparable
to $1/T_{2,S}$, marking the limit of the SERF regime, and the magnetometer
noise rises above
$20\text{\,}\mathrm{f}\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$. Overall, as
Figure 2c illustrates, NMR signals are obtainable at fields where Larmor
frequencies are around 100 times higher than the atomic bandwidth. In
contrast, without tuning, the combined atomic and nuclear spin system yields a
relatively narrow operating range for NMR, quantified by the half-width at
half-height of Equation 1: $\Delta\omega_{I}/(2\pi)\approx g_{S}T_{2,S}/(2\pi
q)\approx$ $80\text{\,}\mathrm{Hz}$.

### Dissolved paramagnetic species in liquids

Figure 3: Ultralow-field NMR relaxation of $I$ = 1H in aqueous TEMPOL
solutions at $30$ ∘C: (a) Sequences A and B measure longitudinal
($T_{1,I}^{-1}$; diamond plot markers) and transverse ($T_{2,I}^{-1}$ and
$(T_{2,I}^{\ast})^{-1}$; triangle and circle plot markers) relaxation rates.
Vertical axis (not to scale) shows field strength with the polarization field
along z and the $\pi/2$ and $\pi$ pulses along y; (b Field dependence of rates
$T_{1,I}^{-1}$ across $25\text{\,}\mathrm{nT}$ $<B_{z,I}<$
$130\text{\,}\mathrm{\SIUnitSymbolMicro T}$, 1.8 mL sample volume. Rates are
linearly proportional to TEMPOL concentration; (c) Field dependence of
$(T_{2,I}^{\ast})^{-1}$ and $(T_{2,I})^{-1}$. $(T_{2,I}^{\ast})^{-1}$ depends
weakly on field due to instrument-specific gradients in $B_{z,I}$ and
$B_{z,S}$, while $T_{2,I}^{-1}\approx T_{1,I}^{-1}$; (d) Relaxation rates for
TEMPOL solutions encased in 0.5 mm thick titanium tube, 0.1 mL sample volume.
All vertical error bars represent estimated standard deviation errors obtained
from least-squares regression analysis.

Many single-component liquids and simple solutions are characterized by an
exponential correlation function for molecular tumbling, with a time constant
$\tau_{\rm c}$ in the low $\mathrm{ps}$ range. Unless much slower additional
motion processes exist the NMR relaxation times $T_{1,I}$ and $T_{2,I}$ are
independent of magnetic field below $B_{z,I}\ll(\gamma_{I}\tau_{\rm c})^{-1}$)
$\sim$ $0.1\text{\,}\mathrm{T}$, all the way down to ultralow field.

Here we observe the dependence of relaxation in aqueous solutions of the
paramagnetic compound 4-hydroxy-2,2,6,6-tetramethylpiperidin-1-oxyl (TEMPOL).
TEMPOL is a chemical oxidant under study elsewhere for potential therapeutic
properties[43], as well as a source of nuclear spin hyperpolarization[44, 45]
that can achieve enhanced sensitivity in NMR. Sequences A and B are used,
respectively, to measure 1H $T_{1,I}$ and $T_{2,I}$ (Figure 3a).

In sequence A, nuclear spin prepolarization at $20\text{\,}\mathrm{mT}$ is
followed by switching to a lower magnetic field for a time $\tau_{1}$, before
a dc $\pi/2$ pulse induces NMR free nuclear precession about the $z$ axis. The
amplitudes of the NMR signal, $s_{A}$, are fit well by the function
$s_{A}\propto\exp(-\tau_{1}/T_{1,I})$ and the observed relaxation rates scale
linearly with concentration of the paramagnetic dopant as
$T_{1,I}^{-1}=(T_{1,I}^{(0)})^{-1}+k_{1}[\mathrm{TEMPOL}]$, where
$T_{1,I}^{(0)}$ is the relaxation time at zero solute (Figure 3b). The
relaxivity parameter $k_{1}=$
$0.453(5)\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mmol}}^{-1}\text{\,}{\mathrm{dm}}^{3}$
is in good agreement with literature values at the high-field end[44, 45],
which gives confidence in the method. In sequence B, the initial $\pi/2$ pulse
is followed by a Hahn-echo to refocus transverse magnetization after time
$\tau_{2}$. Signal amplitudes are fit well by the expected function
$\exp(-\tau_{1}/T_{1,I}-\tau_{2}/T_{2,I})$ and provide a transverse relaxivity
parameter $k_{2}=0.455(12)$
${\mathrm{s}}^{-1}\text{\,}{\mathrm{mmol}}^{-1}\text{\,}{\mathrm{dm}}^{-3}$
defined by
$T_{2,I}^{-1}(B_{z,I},[\mathrm{TEMPOL}])=T_{2,I}^{-1}(B_{z,I},0)+k_{2}(B_{z,I})[\mathrm{TEMPOL}]$
(Figure 3c, triangle plot markers). The result $k_{2}=k_{1}$ holds down to nT
fields, which confirms isotropic molecular tumbling in the fast motion limit
and absence of slow motional correlations.

The transverse decay rates are also well approximated by
$(T_{2,I}^{\ast})^{-1}=$ FWHM/$(2\pi)$ obtained from line widths in the
Fourier-transform NMR spectra of sequence A, due to relatively low
inhomogeneity in $B_{z,I}$. Here, S2+S3 produce field gradients
$\mathrm{d}B_{z,I}/\mathrm{d}z$ and smaller components along $x$ and $y$ due
to tilt imperfections in the coil windings, resulting in a linear dependence
$(\mathrm{d}/\mathrm{d}B_{z,I})(T_{2,I}^{\ast})^{-1}=$
$0.01\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{\SIUnitSymbolMicro T}}^{-1}$
(or 4 ppk of $B_{z,I}$) observed above 500 Hz 1H frequency. We also note that
S4 is centered on the 87Rb cell and not on the NMR sample, therefore gradients
may cancel out at some Larmor frequencies; this effect is attributed to the
line narrowing at around 200 Hz. Overall the results show that the NMR line
width stays below 1 Hz even up to geomagnetic fields, and that TEMPOL causes
no further detriment to spectroscopic resolution in the zero/ultralow-field
range.

To further demonstrate the application potential of the technique we highlight
that ultralow-field cycling and NMR detection is compatible with metal sample
enclosures. NMR signals can be detected without amplitude loss up to
$\text{\,}\mathrm{kHz}$ Larmor frequencies when $0.1\text{\,}\mathrm{mL}$
aliquots of the TEMPOL solutions are contained inside a titanium alloy tube
(outer diameter $8\text{\,}\mathrm{mm}$, inner diameter
$7\text{\,}\mathrm{mm}$, pressure rating $13\text{\,}\mathrm{MPa}$).
Relaxation rates $1/T_{1,I}$ for samples with and without the metal tube,
shown in Figure 3d, are identical within measurement error to those of Figure
3b. Larger error bars are due to the smaller sample volume giving lower snr.
This measurement is impossible via conventional fast-field cycling NMR
techniques, where eddy currents in metal strongly attenuate the amplitude of
high-frequency NMR signal and also limit the rate of field switching.
Transverse relaxation rates are also unaffected by the presence of the metal
tube, from which we may conclude that eddy currents are negligible over the
relatively small (mT) range of field switching. The approach may therefore
open the way to study relaxation in new contexts, for instance high-pressure
fluids (e.g. supercritical fluids), flow in pipes, foil-sealed products (e.g.
foods, pharmaceuticals) and (e.g. lead-, tungsten-) sealed radioactive
samples.

### Liquids confined in porous materials

To demonstrate new insight into molecular motion near pore surfaces we study
the 1H spin relaxation of n-alkane hydrocarbons confined within matrices of
alumina ($\gamma$ polymorph, $9\text{\,}\mathrm{nm}$ mean pore diameter) and
titania (anatase polymorph, 7–10 $\mathrm{nm}$ mean pore diameter). These
simple inorganic oxides in their mesoporous form possess catalytic features
due to their high specific surface area, Lewis acidic sites and option of
chemical treatments including metalization to activate the pore surface. Yet,
owing to the frequency range of conventional NMRD techniques, there is limited
understanding of how molecular dynamics and surface site properties relate to
long-$\tau_{\rm c}$ relaxation processes, even without surface
functionalization[11].

Figure 4: Ultralow-field FFC NMR of pore-confined fluids: (a) Magnetization
decay rates for n-octane and n-decane in porous $\gamma$-alumina and anatase
titania. For n-decane in $\gamma$-alumina, colors indicate the NMR detection
field, as discussed in the main text and represented in the rest of the
figure. Data above 100 kHz correspond to magnetization buildup rates in the S1
field; vertical error bars represent estimated standard deviation errors
obtained from least-squares regression analysis; (b) FFC sequence used to
measure $T_{1,I}$ decay at ultralow field; (c) frequency- and (d) time-domain
NMR signal of n-decane in $\gamma$-alumina after relaxation at
$\omega_{I}/(2\pi)=85$ $\mathrm{Hz}$. In (c), the NMR signal-to-noise ratios
(snr) illustrate the FFC requirement below 100 Hz: poor snr at
$\omega_{I}/(2\pi)=85$ $\mathrm{Hz}$ is due to $1/f$ and 50 Hz noise of the
OPM; FFC switching to $\omega_{I}/(2\pi)=$ $275\text{\,}\mathrm{Hz}$ between
relaxation and detection events is optimal for high snr and short dead time.
The blue and red color coding also serves to indicate the method used to
measure $T_{1,I}$ for n-decane in $\gamma$-alumina, in part (a) of the Figure.
The reconstructed time-domain signal in (d, blue curve) equates to the
Lorentzian line shape fitted in (c).

Figure 4a shows 1H relaxation rates at 30 ∘C for imbibed n-alkanes, measured
between 1 Hz and 5.5 kHz Larmor frequency using the sequence shown in Figure
4b. Due to excess noise in the magnetometer below 100 Hz (including mains
electricity noise and $1/f$ noise, see Figure 4c), fast field switching
between relaxation and detection events is the preferred measurement option to
probe the lowest fields, where the NMR signal is always detected at frequency
above 100 Hz. Above 100 Hz Larmor frequency, the noise floor is low enough to
detect NMR signals at the relaxation field, without switching. The measurable
NMR relaxation is limited in principle to rates
$T_{1,I}^{-1}<R_{\mathrm{op}}^{-1}$, where the latter is of order
$300\text{\,}{\mathrm{s}}^{-1}$. However, in practice, the limit is
$T_{1,I}^{-1}<T_{2,S}^{-1}$ or around $100\text{\,}{\mathrm{s}}^{-1}$ since
the atomic precession signal causes a $10\text{\,}\mathrm{ms}$ dead time
caused following the $\pi/2$ pulse (see Figure 4d).

The main feature of Figure 4a is the weak dispersion in $T_{1,I}^{-1}$ for
each alkane, and moreover between the two porous materials, across the
conventional FFC-NMR frequency range 10 kHz to 1 MHz[11]. The relaxation rate
for each alkane depends only slightly on the porous material, therefore bulk
effects dominate the relaxation process in this range. In contrast, relaxation
rates below 10 kHz depend strongly on the material and mechanisms related to
the surface are prominent, with the higher values being observed towards zero
field. The $T_{1,I}$ dispersion in titania is much weaker than in
$\gamma$-alumina; $T_{1,I}^{-1}$ reaches only around
$2\text{\,}{\mathrm{s}}^{-1}$ below 200 Hz, compared to
$30\text{\,}{\mathrm{s}}^{-1}$ for alumina. Although the two materials have
similar mean pore diameter and surface area/volume ratio, surface-induced
relaxation is not so active in the first material. It is known from electron
spin resonance spectroscopy[46] that the alumina contains a higher
concentration of paramagnetic impurity – [Fe3+] $\approx$
$2\text{\times}{10}^{16}\text{\,}{\mathrm{g}}^{-1}$ (i.e., ions per unit mass
of the dry porous material) in alumina vs.
$2\text{\times}{10}^{15}\text{\,}{\mathrm{g}}^{-1}$ in titania – suggesting
that the lower-frequency relaxation mechanism involves dipole-dipole coupling
between 1H and the surface spins, rather than surface-induced modulation of
intra-molecular 1H-1H spin couplings.

Between $\omega_{I}/(2\pi)=50$ Hz and 5000 Hz, the longitudinal relaxation in
$\gamma$-alumina obeys a power-law frequency dependence:
$T_{1,I}\propto\omega_{I}^{\xi}$. Fitted slopes
$-\mathrm{d}(\mathrm{log}\,T_{1,I}^{-1})/\mathrm{d}(\mathrm{log}\,\omega_{I})$
give exponents $\xi=0.50\pm 0.03$ for octane and $\xi=0.45\pm 0.03$ for
decane. Such values are consistent with simple numerical simulations in which
imbibed molecules randomly walk within a dilute matrix of non-mobile spins –
such as surface paramagnets – where the strength of dipole-dipole interactions
between the two spin species scales with the inverse cube of their
instantaneous separation[39]. This nonlinear dependence results in an example
of Lévy walk statistics. A detailed characterization of these effects in the
alumina system is ongoing work.

Although developing an analytical model for the surface dynamics is outside
the scope of this paper, for analysis of the correlation time it suffices to
fit the measured relaxation rates by a stretched Lorentzian function
$[T_{1,I,\mathrm{fit}}(\omega_{I})]^{-1}=[T_{1,I}(0)]^{-1}/(1+\tau_{\mathrm{c}}^{2}\omega_{I}^{2})^{\beta}+[T_{1,I}(\infty)]^{-1}$
with four independent fit parameters: $T_{1,I}(0)$, $T_{1,I}(\infty)$,
$\tau_{\mathrm{c}}$ and $\beta$. For $\omega_{I}\tau_{\mathrm{c}}\gg 1$ and
$T_{1,I,\mathrm{fit}}(\omega)\ll T_{1,I}(\infty)$, the function is
approximated by a power law with $\xi=2\beta$. The fitted curves are plotted
as solid lines in Figure 4a. The parameter $\tau_{c}$ for alkanes in alumina
is determined from the relaxation behavior below 50 Hz, where
$T_{1,I}(\omega_{I})$ changes from a power law frequency dependence to a
constant, i.e., towards a plateau at $T_{1,I}(0)$. Using the analysis
presented earlier, this indicates a maximum correlation time
($\tau_{c}=\tau_{\rm c,max}$) of around 20-30 ms, which is at least two orders
of magnitude longer than the maximum correlation time of more polar molecules
in porous confinement, such as water. Relative to octane, the plateau for
decane extends to a higher Larmor frequency, indicating a shorter $\tau_{\rm
c,max}$, despite octane having a higher self-diffusion coefficient as a bulk
liquid. However, at this point $\tau_{\rm c}$ is also of similar magnitude to
the longitudinal relaxation time. Under such conditions the assumptions of
standard NMR relaxation theories – such as the Wangsness-Bloch-Redfield theory
– are not strictly justified as valid, in particular the coarse graining of
time[35], where spin diffusion may be a part of the relaxation mechanism, or
set an upper limit for the relaxation rate in the plateau. Whether this is
true requires more information on the physical process responsible for spin
relaxation.

### High-resolution relaxometry

Figure 5:  NMR spectra of 13CH3OH versus field strength in the range
$25\text{\,}\mathrm{nT}$ to $100\text{\,}\mathrm{\SIUnitSymbolMicro T}$.
Horizontal lines show NMR spectra acquired as in Figure 2, offset vertically
by the field at which they were acquired (left scale), and horizontally so as
to place the uncoupled OH resonance at zero. Individual resonances for the
chemically distinct CH3 and OH spin groups are clearly visible and demonstrate
relative frequency shifts due to the transition between strong and weak
heteronuclear coupling regimes corresponding to low and high field,
respectively. The different peaks correspond to different spin combinations;
in high field the peaks correspond to single-spin transitions of 1H, while at
zero field peaks at ${}^{1}J_{\rm CH}\,=$ $140.1\text{\,}\mathrm{Hz}$ and
$2\times^{1}J_{\rm CH}\,=$ $280.2\text{\,}\mathrm{Hz}$ correspond respectively
to singlet-to-triplet and triplet-to-quintet transitions of the 13CH3.
Features marked with an asterisk ($\ast$), e.g. $50\text{\,}\mathrm{nT}$ and
$\approx$48\text{\,}\mathrm{Hz}$$ plus $\approx$98\text{\,}\mathrm{Hz}$$, are
artefacts due to $50\text{\,}\mathrm{Hz}$ line noise and harmonics. Dotted
curves show the predicted resonance frequencies as a result of the
heteronuclear coupling. Inset shows transverse decay rate $1/T_{2,I}^{\ast}$
for the CH3 signal peak (dotted orange curve) and OH signal peak (at
$0\text{\,}\mathrm{Hz}$) versus field strength. Error bars represent estimated
standard deviation errors obtained from least-squares regression analysis.

Field instability in traditional NMRD electromagnets is large compared to
spectroscopic dispersion from NMR chemical shifts or inter-spin couplings,
resulting in severe or even complete overlap of the signals from nuclei in
different chemical groups or different compounds in a mixture. Although
additional strategies may prove helpful to assign relaxation rates to distinct
chemical groups, (e.g. selective deuteration or other isotopic substitution,
inverse Laplace transforms), higher-resolution signal detection would be a
more general and direct solution.

Here we illustrate simultaneous measurement and independent fitting of
relaxation rates for two chemically distinct 1H environments in methanol
(CH3OH). A scalar coupling (${}^{1}J_{\rm CH}\,=$ $140.1\text{\,}\mathrm{Hz}$)
between 13C and 1H nuclei in the 13CH3 group shifts the corresponding NMR
signal by around $\pm 0.5J_{\rm CH}$ relative to that of the non-coupled OH,
when measured at fields $|B_{z,\rm I}|\gg|2\pi^{1}J_{\rm CH}/(\gamma_{\rm
H}-\gamma_{\rm C})|$. The latter criterion defines the well-known weak
heteronuclear coupling regime. Shifts by other multiples of ${}^{1}J_{\rm CH}$
between 1 and 2 occur at lower fields. Experimental spectra and simulated
positions of the NMR peaks are shown in Figure 5. Line widths are on the order
of 1 Hz, which should also allow spectral resolution of the CH3 groups in
methanol, acetone (CH3COCH3, ${}^{1}J_{\rm CH}\,=$ $127\text{\,}\mathrm{Hz}$),
acetic acid (CH3COOH, ${}^{1}J_{\rm CH}\,=$ $130\text{\,}\mathrm{Hz}$)
dimethylsulfoxide (CH3SOCH3, ${}^{1}J_{\rm CH}\,=$ $137\text{\,}\mathrm{Hz}$)
and other solvents. Isotopomers splittings that arise for couplings over more
than one chemical bond, e.g. –13C12CH3, for which ${}^{2}J_{\rm CH}\,=$ 5–30
$\mathrm{Hz}$, would also be resolvable.

By using the sequence shown in Figure 3a to provide a series of
$T_{1,I}$-weighted spectra, a fitted relaxation rate $1/T_{1,{1\rm H}}$ =
$0.44(5)\text{\,}{\mathrm{s}}^{-1}$ (not plotted) is obtained for the OH
subsystem. Within error, the value does not depend on field. Relaxation rates
for the CH3 subsystem are also field independent in the weak-coupling regime
above $B_{z,\rm I}=$10\text{\,}\mathrm{\SIUnitSymbolMicro T}$$, and are very
close to those of OH: $1/T_{1,{1\rm H}}$ =
$0.45(14)\text{\,}{\mathrm{s}}^{-1}$ (not plotted). Both sets of rates refer
to relaxation of the 1H spin species and are thus comparable with results
obtained via conventional field cycling NMRD.

At lower fields, however, L-S type effects of the scalar coupling between 13C
and 1H3 lead to a significant contrast in transverse relaxation rates. Of most
interest is the peak tending to frequency ${}^{1}J_{\rm CH}$ at zero field
(orange curve in Figure 5), which corresponds to singlet-to-triplet coherence
in the isolated manifold formed between 13C and the 1H3 state of total spin
quantum number 1/2. Here the transverse relaxation rate is around 2.5 to 3
times slower than for the non-coupled OH (Figure 5, inset). The 13CH3 system
therefore exhibits a type of long-lived spin order in the NMR ensemble[47].
Despite the presence of 1H–1H and 13C–1H dipole-dipole couplings, this
singlet-to-triplet coherence is long lived because it is less sensitive to
relaxation by fields that are correlated across the 13C and 1H spin groups.
This includes much of the intra-13CH3 dipole coupling as well as longer-range
couplings. The result has potential importance in applications to probe
dipole-dipole interactions at short distances away from the CH3, including
intermolecular interactions.

## Discussion and outlook

The study demonstrates that unique, important information about nuclear spin
relaxation in liquids can be obtained by fast field switching and tunable NMR
detection at ultralow magnetic fields.

A basic advantage of the low-field NMR detection is that it eliminates many
concerns about magnetic field homogeneity and stability, since magnetic fields
are accurately and precisely controlled. This is shown by the result
$T_{2}\sim T_{1}$ for the series of TEMPOL solutions. Additionally, as shown
for methanol, the Fourier-transform NMR spectrum line width is adequate to
resolve spin-spin couplings (even in the alumina system where line widths
exceed, $10\text{\,}\mathrm{Hz}$) and therefore components of liquid mixtures.
Ultralow-field NMRD may therefore be able to probe other interplay, such as
competitive adsorption between molecules and pore. Besides nonuniformity of
applied magnetic fields, conventional NMR is also confounded by sample
heterogeneity, especially in multi-phase samples with internal magnetic
susceptibility variation, including porous materials, or metal regions. The
$\tau_{\rm c}$ values obtained here for alkanes in porous alumina are
extremely long by FFC-NMR standards - comparable $\tau_{\rm c}$s are typically
probed in high field by pulsed-field gradient (PFG) diffusometry and rotating-
frame ($T_{1\rho,I}$) relaxometry techniques[48]. In most if not all
applications, both of the latter are highly susceptible to contamination by
poor field homogeneity and radiofrequency offset errors.

Compared to high-field inductive-detected NMRD, ultralow-field OPM-detected
NMRD currently has some limitations. A main limitation, resulting from the
Hartmann-Hahn matching condition, is that the OPM Faraday rotation signal
contains free-precession responses of the sensor atom and NMR sample spins at
the same frequency. The atomic response is at least two orders of magnitude
stronger than the NMR signal and can easily saturate the digitizer, which
leads to a “dead time” on the order of the optical pumping time (10 ms, see
Figure 4d). This currently hinders applications in chemical systems where
molecules interact more strongly, namely liquids in nanopores (e.g. zeolites,
shale) and interfaces with hydrogen bonding, where relaxation rates are
higher. In principle, Q-switching of the optical pumping beam[49] is a method
to accelerate magnetometer recovery after the magnetic field pulses and reduce
the dead time down to the field switching time, well below 1 ms, without
compromising sensitivity.

Ultralow-field FFC NMRD may also in the future offer new study paths when
enriched by nuclear spin hyperpolarization. As shown in Figure 2c, a few tens
of scans result in snr $>$20 dB, even though the spins are only prepolarized
to around 1 part in $10^{8}$ at the $20\text{\,}\mathrm{mT}$ starting field.
Nitroxide radical compounds such as TEMPOL are a source of higher electron
spin polarization, around 1 part in $10^{5}$ at $20\text{\,}\mathrm{mT}$, that
can be efficiently transferred to nuclei via the Overhauser effect at both
high[44] and ultralow[50] magnetic fields. Hyperpolarization via surface-
supported paramagnetic species and other spin transfer catalysts may also be
an option to study nuclear polarization buildup near pore surfaces, providing
information that may differ from relaxation decay. TEMPOL and other persistent
radicals are used to prepare hyperpolarized biochemical probes for clinically
relevant in-vivo observations of disease via MRI[51]. These systems could
profit from a knowledge of signal decay mechanisms at ultralow magnetic
fields, for new sources of image contrast or to minimize polarization losses
before imaging/detection.

## Methods

###  .1 Sample preparation

All samples studied in this work were contained in disposable glass vials (12
mm o.d., 20 mm length, 1.8 mL internal volume, 8–425 thread) sealed with a
silicone septum and finger-tight polypropylene screw cap.

Preparation of TEMPOL samples: a 10 mM stock solution of the radical
4-hydroxy-2,2,6,6-tetramethylpiperidin-1-oxyl (Sigma Aldrich, CAS: 2226-96-2)
was prepared in 5.0 mL deoxygenated milli-Q water and diluted to
concentrations of 0.5, 1, 2, 3, 4, 5 and 10 mM with deoxygenated milli-Q
water. The diluted solutions were not further de-gassed.

Preparation of porous materials samples: cylindrical extrudate pellets of
meso-porous $\gamma$-alumina (Alfa Aesar product 43855, lot Y04D039: 3 mm
diameter, 3 mm length, 9 nm BJH mean pore diameter, Langmuir surface area
$250\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{g}}^{-1}$) and anatase titania
(Alfa Aesar product 44429, lot Z05D026: 3 mm diameter, 4 mm length, 7–10 nm
mean pore size, Langmuir surface area
$150\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{g}}^{-1}$) were obtained
commercially. Pellets were oven dried at 120 ∘C for 12h to remove physisorbed
H2O and then imbibed in neat n-alkane for at least 12 h after recording the
dry mass. Excess liquid on the pellet outer surface was gently removed using
tissue paper. The pellets were then placed in a vial (see Figure 4a), sealed
with the cap and the combined mass of pellet and imbibed hydrocarbon was
recorded.

Preparation of methanol sample: 0.9 g of 13C-methanol (13CH3OH 99%, Sigma
Aldrich product 277177) was added to the sample vial without dilution, then
followed by N2 bubbling (2-3 min) to displace dissolved paramagnetic O2.

###  .2 Optical magnetometer

The magnetometer used to detect 1H precession signals in the NMR samples was
operated as follows. A cuboid borosilicate glass cell of inner dimensions
$5\times 5\times 8\,\mathrm{mm}^{3}$ contained a droplet of rubidium-87 metal
and $90\text{\,}\mathrm{kPa}$ N2 buffer gas (Twinleaf LLC). The cell was
electrically heated to 150 ∘C to vaporize the alkali metal. A circularly
polarized light beam along $z$ axis (3 mW, tuned to the center of the
collision-shifted D1 wavelength) optically pumped the atomic spin polarization
to $S_{z}\approx 0.5$. Faraday rotation in a second, linearly polarized light
beam (10 mW, $65\text{\,}\mathrm{GHz}$ red-shifted from the pump, along $x$
axis) was used to non-resonantly probe the $S_{x}$ component of atomic
polarization. The probe beam was linearly polarized and slightly detuned from
the 87Rb D1 transition, such that on passing through the cell along the $x$
axis its axis of polarization was optically rotated by an angle proportional
to $S_{x}$. The Faraday rotation was detected by polarimetry using a
differential photodetector (Thorlabs PDB210A), which produced an analog
voltage signal that was conditioned (amplified, filtered to eliminate high-
frequency noise and dc offset) and digitized (60 ksps, 16-bit $\pm
5\,\mathrm{V}$ ADC range) before storage and further processing on a computer.

###  .3 Magnetic coils and shielding

The vapor cell and heating assembly was placed as close as possible to the NMR
sample at a standoff distance $d_{1}=3.5\,\mathrm{mm}$ between outer walls of
the vial and atomic vapor cell (see Figure 2a). In order of increasing
distance away from the NMR sample, $d_{1}$ accounts for (i) S2 + S3 coil
windings (34 AWG enameled copper wire, solenoid length 13 cm, diameter 14 mm),
(ii) a carbon-fiber support structure, (iii) S1 coil windings (36 AWG enameled
copper wire, solenoid length 2.5 cm, single layer), (iv) a water-cooling
jacket (de-ionized water, flow rate
$1\text{\,}\mathrm{mL}\text{\,}{\mathrm{s}}^{-1}$) to remove heat deposited
when the polarizing coil is energized and to maintain a stable sample
temperature, (v) a PEEK support structure and (vi) an air gap for further
thermal insulation. The entire structure was operated within a cylindrical
magnetic shield (Twinleaf LLC, model MS-1F) of 20 cm outer diameter and length
30 cm. The main axis of the cylinder was co-axial with the pump beam axis.

Field-to-current ratios inside each coil were calibrated using the frequency
of 1H precession frequency in de-ionized water. These were S2:
$7.59(3)\text{\,}\mathrm{\SIUnitSymbolMicro T}\text{\,}{\mathrm{mA}}^{-1}$;
S3: $7.50(3)\text{\,}\mathrm{\SIUnitSymbolMicro
T}\text{\,}{\mathrm{mA}}^{-1}$; S4:
$150.1(5)\text{\,}\mathrm{nT}\text{\,}{\mathrm{mA}}^{-1}$. The atomic spin
precession frequency at the atomic vapor cell was used to calibrate the
external field of coils S2
$-11.1(3)\text{\,}\mathrm{nT}\text{\,}{\mathrm{mA}}^{-1}$ and S3:
$-4.4(2)\text{\,}\mathrm{nT}\text{\,}{\mathrm{mA}}^{-1}$, which equate to
$1460\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ and
$580\text{\,}\mathrm{p}\mathrm{p}\mathrm{m}$ of the field at the vial,
respectively.

###  .4 Magnetic noise spectra

An ac test signal of $\pm 8$ $\mathrm{pT}$ along the y axis was applied to
calibrate the Faraday response as a function of frequency and z bias field.
The calibration vs. frequency was used to scale the spectral response of the
balanced photodetector from units of $\mathrm{V}\mathrm{/}\sqrt{\mathrm{Hz}}$
into $\mathrm{T}\mathrm{/}\sqrt{\mathrm{Hz}}$. The maximum magnetic response
at a given bias field was confirmed to equal the tuning condition where the
atomic Larmor frequency matched the frequency of the ac signal, given the
prior calibration of the magnetic field at the sample vial and magnetometer
cell.

###  .5 Field switching

Timing of the NMR pulse sequences and data acquisition was controlled by a
microcontroller (Kinetis K20 series: time base
$2\text{\,}\mathrm{\SIUnitSymbolMicro s}$, precision $17\text{\,}\mathrm{ns}$,
CPU speed 120 MHz). Current to the polarizing coil was switched via a dual
H-bridge circuit with parallel flyback diodes and the switching time was
$<1.0$ $\mathrm{ms}$. Coils S2 and S4 were connected to a low-noise precision
current source (Twinleaf model CSB-10, 20-bit resolution over $\pm 10$
$\mathrm{mA}$) with a low-pass LC filter in series, resulting in a combined
switching and settling time of order 100 ms. The FFC solenoid coil S3 operated
at a current $<1\,\mathrm{mA}$ direct from the microcontroller digital-to-
analog converter (12-bit resolution, 0-1 $\mathrm{mA}$) for rapid and precise
field switching without feedback controls. Typical S3 switching times were
0.25 $\mathrm{ms}$ and the accuracy (determined from standard error in the
mean NMR center frequency over repeated scans at $\omega_{I}/(2\pi)=550$
$\mathrm{Hz}$, see Figure 4b inset) was around 1 $\mathrm{nT}$. The residual
interior field of the MuMetal shield along the $x$, $y$, and $z$ axes of
$\sim$ $10\text{\,}\mathrm{nT}$ was also compensated for.

Under steady-state conditions with the pre-polarizing coil turned off, the
cooling system maintained a temperature of 27–28 ∘C at a thermocouple attached
to the outside wall of the sample vial. The steady-state temperature rose to
30–31 ∘C when the polarizing coil was energized at 20 mT (2.2 A).

## Acknowledgments

The work described is funded by: EU H2020 Marie Skłodowska-Curie Actions
project ITN ZULF-NMR (Grant Agreement No. 766402); Spanish MINECO projects
OCARINA (Grant No. PGC2018-097056-B-I00), the Severo Ochoa program (Grant No.
SEV-2015-0522); Generalitat de Catalunya through the CERCA program; Agència de
Gestió d’Ajuts Universitaris i de Recerca Grant No. 2017-SGR-1354; Secretaria
d’Universitats i Recerca del Departament d’Empresa i Coneixement de la
Generalitat de Catalunya, co-funded by the European Union Regional Development
Fund within the ERDF Operational Program of Catalunya (project QuantumCat,
ref. 001-P-001644); Fundació Privada Cellex; Fundació Mir-Puig; MCD Tayler
acknowledges financial support through the Junior Leader Postdoctoral
Fellowship Programme from “La Caixa” Banking Foundation (project
LCF/BQ/PI19/11690021). The authors also thank Jordan Ward-Williams and Lynn
Gladden (University of Cambridge) for providing samples of porous alumina and
titania, and for discussions.

## Author contributions

MCD Tayler proposed the study. S Bodenstedt prepared the samples, measured and
analyzed the experimental data and together with MCD Tayler built the
experimental apparatus and made the theoretical interpretation. MCD Tayler
wrote the manuscript with input from all authors. All authors reviewed the
manuscript and suggested improvements. MCD Tayler and MW Mitchell supervised
the overall research effort.

## Data availability

The raw data generated in this study have been deposited in the OpenAIRE
database under accession code https://doi.org/10.5281/zenodo.4840653

## Competing interests

The authors declare no competing interests.

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# The cosmic coincidences of primordial-black-hole dark matter

Yi-Peng Wua Elena Pinettiabf Joseph Silkcde aLaboratoire de Physique Théorique
et Hautes Energies (LPTHE),
UMR 7589 CNRS and Sorbonne Université, 4 Place Jussieu, F-75252, Paris, France
bDipartimento di Fisica, Universitá di Torino and INFN, Sezione di Torino, via
P. Giuria 1, I-10125 Torino, Italy  cInstitut d’Astrophysique de Paris, UMR
7095 CNRS and Sorbonne Université, 98 bis boulevard Arago, F-75014 Paris,
France  dDepartment of Physics and Astronomy, The Johns Hopkins University,
3400 North Charles Street, Baltimore, Maryland 21218, USA  eBeecroft Institute
for Particle Astrophysics and Cosmology, University of Oxford, Keble Road,
Oxford OX1 3RH, United Kingdom  fTheoretical Astrophysics Department, Fermi
National Accelerator Laboratory, Batavia, Illinois, 60510, USA

###### Abstract

If primordial black holes (PBHs) contribute more than 10% of the dark matter
(DM) density, their energy density today is of the same order as that of the
baryons. Such a cosmic coincidence might hint at a mutual origin for the
formation scenario of PBHs and the baryon asymmetry of the Universe.
Baryogenesis can be triggered by a sharp transition of the rolling rate of
inflaton from slow-roll to (nearly) ultraslow-roll phases that produce large
curvature perturbations for PBH formation in single-field inflationary models.
We show that the baryogenesis requirement drives the PBH contribution to DM,
along with the inferred PBH mass range, the resulting stochastic gravitational
wave background frequency window, and the associated cosmic microwave
background tensor-to-scalar ratio amplitude, into potentially observable
regimes.

##  I Introduction

Primordial black holes (PBHs) are one of the most interesting dark matter (DM)
candidates that have been severely constrained by joint astrophysical and
cosmological observations [5, 1, 6, 2, 4, 7, 8, 11, 9, 3, 10, 12, 13, 14]. The
asteroid-mass window $M_{\rm PBH}/M_{\odot}\sim 10^{-16}-10^{-12}$ for PBHs to
be all DM [17, 18, 15, 16] will be tested in the near future by femtolensing
of gamma ray bursts [19, 21, 20], microlensing of x-ray pulsars [22], primary
photons measured by next-generation MeV detectors [7, 23], neutron star
disruption [24], white dwarf explosions [25], MeV photons from the Galactic
Center [26] and radio emission measurable with the next-generation radio
telescopes [27]. Even if PBHs only occupy a small fraction of the DM density
today, their existence could be (or could have been [28]) probed by
gravitational wave observations through binary mergers, either mutual or with
neutron stars (for recent reviews, see [29, 30, 29]).

If PBHs really constitute more than 10% of the DM density today, the PBH
density up to matter-radiation equality is of the same order as that of the
baryons, namely $\Omega_{\rm PBHeq}/\Omega_{\rm Beq}\sim\mathcal{O}(1)$. Such
a cosmic coincidence would hint at a mutual origin for PBHs and the baryon
asymmetry of the Universe. In this Letter, we argue that the PBH-baryon
density coincidence could be a natural consequence due to baryogenesis
triggered by inflation models for PBH formation. We show that the abundance of
PBHs and baryons are indirectly correlated to each other via the dynamics of
inflation, and, thus, the scenario is very different from previous suggestions
that existing PBHs create baryon asymmetry [31, 32, 33, 34, 35, 36, 37, 40,
39, 38] or account for cosmic coincidence [41, 43, 42]. (See, also, [44] for
cosmic coincidence from asymmetric DM [46, 47, 45] collapses into PBHs.)

##  II  PBHs from Inflation

Let us focus on single-field inflation for PBH formation [55, 48, 60, 49, 52,
50, 53, 75, 77, 57, 63, 74, 61, 59, 58, 51, 64, 56, 65, 49, 76, 54, 62]. The
generic assumption is that the inflaton $\phi$ receives a sudden deceleration
on comoving scales $k_{0}\sim 10^{12}-10^{15}$ Mpc-1 which leads to a sharp
decrease of the first slow-roll parameter $\epsilon_{H}\equiv-\dot{H}/H^{2}$
and, thus, largely enhances the power spectrum of the curvature perturbation
$P_{\zeta}$ [66]. Such an enhancement is due to the temporal dominance of the
entropy mode in the curvature perturbation [64, 67, 79, 68].

The key parameter for realizing the enhancement of $P_{\zeta}$ is the rate of
rolling:

$\delta=\frac{\ddot{\phi}}{H\dot{\phi}}<-3/2,\quad t_{0}<t<t_{\ast},$ (1)

where $\delta\rightarrow 0$ is the standard slow-roll inflation and
$\delta\rightarrow-3$ is the so-called ultraslow-roll (USR) limit [72, 71, 73,
69, 70]. With $\delta<-3$, $P_{\zeta}$ can exhibit a spiky peak at a desired
scale for producing a nearly monochromatic distribution of PBH mass.

We consider that $\delta$ takes constant values in different phases of
inflation. In terms of the $e$-folding numbers, $N\equiv\ln a$, where
$k_{0}/k_{\rm\ast}\approx a(t_{0})/a(t_{\ast})=e^{N_{\ast}-N_{0}}$ with
$N_{0}\equiv 0$, the duration $N_{\ast}$ (for $\delta$ having a negative
value) is the fundamental parameter that controls the spectral amplitude of
$P_{\zeta}$ (and, thus, the PBH abundance).

The analytic structure of the USR power spectrum at the end of inflation
($N=N_{\rm end}$) has been intensively investigated [74, 77, 75, 76, 79, 78].
Dilatation symmetry of the de Sitter background requires the momentum scaling
at each phase with different values of $\delta$ to satisfy [77, 79]

$\displaystyle P_{\zeta}=\left\\{\begin{array}[]{ll}A_{\rm CMB}&k<k_{\rm
min},\\\ A_{\rm PBH}(k/k_{0})^{4},&k_{\rm min}<k<k_{0},\\\ A_{\rm
PBH}(k/k_{0})^{6+2\delta},&k_{0}<k<k_{\rm end},\end{array}\right.$ (5)

where $A_{\rm CMB}\approx 2.2\times 10^{-9}$ measured on comis microwave
background (CMB) scales has negligible contribution to PBH formation. $k_{0}$
is the pivot scale for the enhancement and shall be fixed by the desired peak
scale $M_{\rm PBH}$ in the PBH mass function. $k_{\rm min}\approx k_{0}(A_{\rm
CMB}/A_{\rm PBH})^{1/4}$ is the beginning scale of the $k^{4}$ growth driven
by the Leach-Sasaki-Wands-Liddle mechanism [66] (sometimes also called the
steepest growth [74, 80]). The amplitude $A_{\rm PBH}$ is determined by USR
parameters as

$A_{\rm PBH}\approx A_{\rm
CMB}\left(\frac{k_{0}}{k_{\ast}}\right)^{6+4\delta}=A_{\rm
CMB}e^{-N_{\ast}(6+4\delta)}.$ (6)

Note that in the template (5) one should use the value of $\delta<-3$ found in
the deceleration phase $N_{0}<N<N_{\ast}$, since (1) must become positive for
$N>N_{\ast}$ to increase $\epsilon_{H}$ and terminate inflation. The positive
rolling rate in the final acceleration phase ($N>N_{\ast}$) is constrained by
$\delta$ in the deceleration phase (with respect to the conformal symmetry due
to nonviolation of the adiabatic condition [79]) so that the scaling of
$P_{\zeta}(k)$ for $k_{\ast}<k<k_{\rm end}$ is the same as $k_{0}<k<k_{\ast}$.

In the case of exact USR ($\delta\rightarrow-3$), the inflaton potential
$V(\phi)$ is completely flat so that $\phi$ is exactly massless, where quantum
diffusion led by short wavelength modes well inside the horizon may have an
important impact on the classical trajectory of $\phi$ [54, 57, 110]. A non-
Gaussian tail in the high-sigma limit of the probability distribution of
$\zeta$ can significantly raise the resulting PBH abundance from USR inflation
[57, 108, 110, 109, 111], indicating the real amplitude $A_{\rm PBH}$
estimated by the Gaussian spectrum (5) (based on the linear relation
$\zeta=-H/\dot{\phi}\delta\phi$) should be smaller than expected. To suppress
the effect of quantum diffusion, we adopt an upper bound $\delta<-3.1$, which
corresponds to an effective mass
$m_{\phi}\equiv(V_{\phi\phi})^{1/2}>H_{\ast}/2$ for the inflaton fluctuation
$\delta\phi$ [92].

##  III  Baryogenesis via inflation

Now, we show that baryogenesis can be triggered by USR inflation. Scalar
fields naturally develop large vacuum expectation values (VEVs) during
inflation due to the high energy background expansion at the scale of
$H_{\ast}$ (possibly as high as $10^{13-14}$ GeV [82, 81]). These large VEVs
provide suitable initial conditions for baryogenesis driven by the Affleck-
Dine (AD) mechanism [83, 84, 85, 87, 86]: (1) The stochastic nature of the
inflationary fluctuations always allows CP-violating VEVs arising from
theories with CP invariant Lagrangian [88, 86], and (2) the post-inflationary
relaxation of a $B$ or $B-L$ violating scalar condensate is an out-of-
equilibrium process.

A possible realization for the USR inflation to affect the dynamics of a
charged scalar $\sigma$ is given by

$\displaystyle\mathcal{L}=$
$\displaystyle\mathcal{L}_{\phi}+|\partial\sigma|^{2}+m_{\sigma}^{2}|\sigma|^{2}+\frac{c_{1}}{\Lambda}\left|\sigma^{2}\right|\square\phi$
(7)
$\displaystyle+\frac{c_{2}}{\Lambda}\partial_{\mu}\phi\left[\sigma\partial^{\mu}\sigma+\sigma^{\ast}\partial^{\mu}\sigma^{\ast}\right]+\mathcal{O}(\Lambda^{-2})\cdots,$

where $c_{1}$ and $c_{2}$ are real constants of $\mathcal{O}(1)$. CP
invariance is imposed on the Lagrangian for demonstrative purposes, yet it is
not a necessary condition for AD baryogenesis. Similar couplings for enhanced
charged scalar production from rolling inflaton as a chemical potential can be
found in [89, 90, 91]. We ask $H_{\ast}\ll\Lambda\leq M_{P}$ for the cutoff
$\Lambda$. Note that the $c_{2}$ term violates the conserved current
$j^{\mu}=i(\sigma^{\ast}\partial_{\mu}\sigma-\sigma\partial_{\mu}\sigma^{\ast})$,
which is identified as a baryon number for convenience.

In terms of the mass eigenstates $\sigma_{\pm}$, where
$\sigma\equiv(\sigma_{-}+i\sigma_{+})/\sqrt{2}$, the charged scalar is
decomposed into a pair of decoupled canonical real scalars,
$\mathcal{L}_{\sigma_{\pm}}=\frac{1}{2}(\partial\sigma_{\pm})^{2}+\frac{1}{2}m_{\pm}^{2}\sigma_{\pm}^{2}$,
with asymmetric (nondegenerate) masses as

$\displaystyle m_{\pm}^{2}=m_{\sigma}^{2}+\frac{c_{1}\pm
c_{2}}{\Lambda}\square\phi.$ (8)

One can see that the phase transition of $\delta$ for PBH formation also
changes the effective masses as
$\square\phi=-\ddot{\phi}-3H\dot{\phi}\approx-(\delta+3)\sqrt{2\epsilon_{H}}M_{P}H_{\ast}^{2}$.
For $m_{\sigma}\sim H_{\ast}$ and $\Lambda>0.1M_{P}$, the sharp decrease of
$\epsilon_{H}$ for the $P_{\zeta}$ enhancement usually leads to
$m_{\pm}\approx m_{\sigma}$ in the $\phi$-deceleration (USR) phase.

The sudden transition of $m_{\pm}$ from the primary slow-roll phase (with
$\delta\sim 0$) to the deceleration phase with $\delta<-3$ drives the original
VEVs of $\sigma_{\pm}$ out of equilibrium in their potential, triggering the
coherent motion of these scalar condensates. The analytic solutions for the
coherent motion of $\sigma_{\pm}$ are given in [92]. At the end of inflation,
the VEVs of the mass eigenstates are led by

$\displaystyle\sigma_{\pm}\sim e^{-\Delta_{\pm}^{-}N_{\rm
end}},\quad\dot{\sigma}_{\pm}\sim-\Delta_{\pm}^{-}e^{-\Delta_{\pm}^{-}N_{\rm
end}},$ (9)

where $\Delta_{\pm}^{-}\equiv 3/2-\sqrt{9/4-m_{\pm}^{2}/H_{\ast}^{2}}$ is
nothing but the negative branch of the conformal weight for a massive scalar
in de Sitter [93]. The late-time approximation used in (9) applies when
$m_{\pm}/H_{\ast}<3/2$.

Assuming the standard reheating process driven by the coherent oscillation of
$\phi$, one can numerically solve the relaxation of $\sigma_{\pm}$ from the
end of inflation to reheating completion (or radiation domination) [88, 92].
Here, we consider the decay of $\phi$ into radiation via a perturbative
channel with a decay width $\Gamma_{\phi}$. Thus, the approximated time scale
at the beginning of radiation domination is $t_{r}\sim 1/\Gamma_{\phi}$. The
final baryon asymmetry in radiation domination reads

$\displaystyle
Y_{B}=\frac{n_{B}(t_{r})}{s(t_{r})}=\frac{\sigma_{+}(t_{r})\dot{\sigma}_{-}(t_{r})-\sigma_{-}(t_{r})\dot{\sigma}_{+}(t_{r})}{s(t_{r})},$
(10)

where $s(t)\approx 2\pi^{2}g_{\ast}T^{3}(t)/45$ is the entropy production and
$T=(\frac{90}{\pi^{2}g_{\ast}}M_{p}^{2}H^{2})^{1/4}$ is the temperature. Note
that $Y_{B}=Y_{B}(\delta,N_{\ast},N_{\rm end})$ as those parameters of USR
inflation enter through the initial conditions (9).

Figure 1:  The final baryon asymmetry in radiation domination with
$\delta=-3.15$, $m_{\sigma}=H_{\ast}/2$, $\Lambda=0.3M_{P}$, where
$H_{\ast}=2.37\times 10^{13}$ GeV and $\Gamma_{\phi}=10^{13}$ GeV are used.

We highlight the generic property of the final baryon asymmetry with examples
given in Fig. 1. In general, $Y_{B}$ is sensitive to $N_{\rm end}$ since
initial conditions (9) are exponentially diluted by the $e$-fold numbers, but
it approaches a constant value when $N_{\ast}\sim\mathcal{O}(1)$ depending on
the value of $\delta$. For $\delta=-3.15$, examples in Fig. 1 indicate that
$Y_{B}\simeq$ const. when $N_{\ast}\gtrsim 2$, which corresponds to $A_{\rm
PBH}\gtrsim 10^{-3}$. This asymptotic constant behavior of $Y_{B}$ is the most
important property for resolving the coincidence problem for PBH DM.

##  IV PBH dark matter

As the inflaton decays during reheating, the enhanced curvature perturbation
on scales $k>k_{0}$ is inherited by the density perturbation of the radiation.
Soon after reentry into the horizon in radiation domination, PBHs are formed
at high-sigma peaks of the density contrast $\Delta$ smoothed over a given
comoving scale $R=1/(aH)=1/k$. The comoving scale $R$ can be expressed in
terms of the horizon mass parameter $M_{H}=\frac{4\pi}{3}H^{-3}\rho_{R}$, with
$\rho_{R}$ the energy density of the radiation dominated Universe, as

$R(M_{H})=\frac{1}{k_{\rm eq}}\left(\frac{M_{H}}{M_{\rm
eq}}\right)^{1/2}\left(\frac{g_{\ast}}{g_{\rm eq}}\right)^{1/6},$ (11)

where $M_{\rm eq}=2.9\times 10^{17}M_{\odot}$ and $g_{\rm eq}\approx 3$ are
the horizon mass and the number of relativistic degrees of freedom at matter-
radiation equality. We use $k_{\rm eq}=0.01\textrm{Mpc}^{-1}$ and
$g_{\ast}=106.75$ for $M_{H}<1.5\times 10^{-7}M_{\odot}$ where the temperature
of the Universe is higher than $300$ GeV.

The mass fraction $\beta(M_{\rm PBH},M_{H})$ of a flat Universe that collapses
into PBHs with mass $M_{\rm PBH}$ at a given horizon mass $M_{H}$ can be
obtained from the density parameter $\Omega_{\rm PBH}(R)$ of PBHs at the
corresponding scale $R(M_{H})$ as $\beta(M_{\rm PBH},M_{H})=d\Omega_{\rm
PBH}/d\ln M_{\rm PBH}$, where $\Omega_{\rm PBH}$ is usually estimated via
threshold statistics:

$\displaystyle\Omega_{\rm PBH}(R)=$
$\displaystyle\int\cdots\int_{\Delta_{c}}^{\infty}\frac{M_{\rm PBH}}{M_{H}}$
(12) $\displaystyle\times f_{c}(y_{i})P(\Delta,y_{i},\sigma_{i})d\Delta
dy_{1}\cdots dy_{i}.$

Here $P(\Delta,y_{i},\sigma_{i})$ is the joint probability distribution of
$\Delta$ and $y_{i}$ are components of its first and second order spatial
derivatives. $f_{c}(y_{i})$ describes spatial constraints to ensure the
selected peaks are local maxima in space [94, 96, 95, 97, 98]. All $y_{i}$ are
Gaussian random fields if $\Delta$ is Gaussian. $\sigma_{i}$ stands for the
$i$th spectral moment of the Gaussian field $\Delta$ smoothed by the window
function $W(kR)$. The form of $\sigma_{i}$ is defined as

$\sigma_{i}^{2}(R)=\int_{0}^{\infty}k^{2i}W^{2}(kR)P_{\Delta}(k)d\ln k,$ (13)

where $P_{\Delta}$ is the dimensionless power spectrum. $\Delta_{c}$ is the
threshold value above which the density contrast will collapse to form a PBH.
Therefore, the mass fraction $\beta=\beta(M_{H},\Delta_{c},\sigma_{i})$ is a
general function of the smoothed scale $R$ (or, namely, $M_{H}$), the
threshold $\Delta_{c}$, and the spectral moment $\sigma_{i}$.

Even for inflation close to the USR limit ($\delta\rightarrow-3$), we find
that $M_{\rm PBH}\approx M_{H}$ can be a good approximation for resolving the
PBH mass function $f(M_{\rm PBH})$ [92]. Such a monochromatic relation leads
to a simple expression of the PBH density at matter-radiation equality

$\Omega_{\rm PBHeq}=\int\beta(M_{H})\left(\frac{M_{\rm
eq}}{M_{H}}\right)^{1/2}d\ln M_{H},$ (14)

where $(M_{\rm eq}/M_{H})^{1/2}\sim a_{\rm eq}/a$ accounts for the relative
growth of PBH density during radiation domination. The PBH mass function
defined from the PBH-to-DM ratio,

$f_{\rm PBH}\equiv\frac{\Omega_{\rm PBHeq}}{\Omega_{\rm
DMeq}}=\frac{1}{\Omega_{\rm DMeq}}\int f(M_{H})d\ln M_{H},$ (15)

implies $f(M_{H})=\beta(M_{H})(M_{\rm eq}/M_{H})^{1/2}/\Omega_{\rm DMeq}$.

##  V The cosmic coincidence

In the standard $\Lambda$-cold dark matter (CDM) Universe [99], the cold dark
matter density today $\Omega_{\rm CDM0}=0.265$ and the redshift $z_{\rm
eq}=3402$ gives $\Omega_{\rm CDMeq}=0.42$ and $\Omega_{\rm Beq}=m_{B}n_{\rm
Beq}=0.08$. This shows that $\Omega_{\rm PBHeq}/\Omega_{\rm Beq}\simeq 0.5-5$
for $f_{\rm PBH}=0.1-1$. Here $m_{B}=0.938$ GeV is the averaged nucleon mass
and $n_{\rm Beq}=|Y_{B}|s(t_{\rm eq})$ is the baryon number density at matter-
radiation equality. Using $H_{\rm eq}=H_{0}\sqrt{\Omega_{\Lambda
0}+2\Omega_{m0}(a_{0}/a_{\rm eq})^{3}}$ with $\Omega_{\Lambda
0}=1-\Omega_{m0}=0.6847$ and $H_{0}=67.36$ km s-1 Mpc-1, we find an
expectation value $|Y_{B}|=6.25\times 10^{-11}$ at $t_{\rm eq}$.

An example of a parameter scan for the $|Y_{B}|$ given by (10) at the
beginning of radiation domination is given in Fig. 2 with $\delta=-3.15$,
$m_{\sigma}/H_{\ast}=0.5$ and $\Lambda/M_{P}=0.3$. $Y_{B}$ is assumed to be a
conserved quantity until matter-radiation equality. $Y_{B}\gtrsim 10^{-10}$
can be reached with $N_{\ast}>1.5$ (for $N_{\rm end}<18$), which translates to
$A_{\rm PBH}>4.4\times 10^{-5}$. Changing $A_{\rm PBH}$ by 1 order of
magnitude roughly corresponds to a $0.35$ variation in $N_{\ast}$.

Figure 2: (Upper panel-) Contours of the final baryon asymmetry $|Y_{B}|$ in
radiation domination with $\delta=-3.15$, $m_{\sigma}=H_{\ast}/2$,
$\Lambda=0.3M_{P}$, where $H_{\ast}=2.37\times 10^{13}$ GeV and
$\Gamma_{\phi}=10^{13}$ GeV are used. The region $2.506<N_{\ast}<2.515$
corresponds to the PBH-to-DM ratio $0.1<f_{\rm PBH}<1$ for $\delta=-3.15$ at
the pivot scale $k_{0}=9.46\times 10^{13}$ Mpc-1. (Lower panel-) $f_{\rm PBH}$
and $|Y_{B}|$ as functions of $N_{\ast}$ at $N_{\rm end}=20$ and
$\delta=-3.15$ from $f_{\rm PBH}=1$ at $N_{\ast}=2.515$ to $f_{\rm
PBH}=10^{-50}$ at $N_{\ast}=2.3$.

The PBH abundance is exponentially sensitive to the peak value
$\nu\equiv\Delta/\sigma_{0}$ in all statistical methods, which means that a
tiny change in $A_{\rm PBH}$ will result in a large difference to $\Omega_{\rm
PBHeq}$ or $f_{\rm PBH}$. As a result, the condition $f_{\rm PBH}>0.1$ for PBH
to be an important DM contributor, indeed, specifies a very precise parameter
space for the USR inflation.

To explore the fiducial parameter space for PBH DM, we adopt the standard
Press-Schechter (PS) method [100] based on the linear density relation
$P_{\Delta}=16/81(kR)^{4}P_{\zeta}$ to obtain the mass fraction $\beta_{\rm
PS}(M_{H})=\textrm{erfc}(\nu_{c}/\sqrt{2})$ in terms of inflation parameters
$\\{\delta,N_{\ast},N_{\rm end}\\}$ with a detailed analytic expression given
in [92]. The mass function $f(M_{H})$ based on $\beta_{\rm PS}(M_{H})$ with
various choices of $\delta$ is displayed in Fig. 3 (blue shadowed regions).

Figure 3:  The PBH mass function based on the fiducial Press-Schechter
statistics at $N_{\rm end}=20$ with various choices of $\delta$, where
$N_{\ast}$ is fixed by the condition $f_{\rm PBH}=1$. The pivot scale
$k_{0}=9.46\times 10^{13}$ Mpc-1 is used. Existing observational constraints
are also shown, using the publicly available code PLOTBOUNDS.

We compare the PBH abundance with the existing observational constraints in
the literature. The Galactic constraints are displayed in green. They include
the bounds from: the local flux of $e^{\pm}$ measured by Voyager1 [4], the MeV
diffuse flux observed by the INTEGRAL/SPI detector [10], the 511 keV line in
our Galaxy [9, 8], the primary photons detected by the Comptel experiment
[23]. The red curves refer to the extragalactic constraints, comprising the
diffuse neutrino background measured by Super-Kamiokande [11] and the
extragalactic background radiation [5]. The black lines denote the constraints
from the energy injection on the cosmic microwave background at recombination
[2, 1] as well as the evaporating constraints from the 21 cm signal observed
by EDGES [103, 104, 105]. The figure does not include the bounds from the
heating of the interstellar medium in dwarf galaxies [3], recently found in
the analysis of [102]. The dynamical constraints based on the destruction of
white dwarfs and neutrons stars by PBHs are displayed in purple. They are
shown with dash-dotted lines, since they are controversial [101, 106]. The
interested reader can find a comprehensive discussion on these constraints and
future prospects in [107, 12, 30].

The parameter space for $0.1<f_{\rm PBH}<1$ based on $\beta_{\rm PS}$ with
$\delta=-3.15$ is given in Fig.1. $f_{\rm PBH}$ is invariant with respect to
$N_{\rm end}$. The difference in $N_{\ast}$ for $f_{\rm PBH}=1$ and $f_{\rm
PBH}=0.1$ is $\mathcal{O}(10^{-2})$. The mass function computed by the peak
statistics [94] with additional spatial constraints [98, 96] shows a $10^{-2}$
difference in $N_{\ast}$. The uncertainty of $N_{\ast}$ due to nonlinear
effect between $\Delta$ and $\zeta$ could be as large as
$\mathcal{O}(10^{-1})$ [92]. $N_{\rm end}<16.5$ is excluded by the minimal
$e$-fold number for the pivot scale $k_{0}$ to pass the spatial curvature
constraint and the finite deviation of scale-invariant $P_{\zeta}$ in single-
field inflation [81].

Nonperturbative contributions to the curvature perturbation $\zeta$ due to
quantum diffusion of the inflaton dynamics can play an important role in the
resulting PBH abundance [57, 108, 110, 109, 111]. In general, the non-Gaussian
tail of $\zeta$ in the limit of USR inflation ($\delta\rightarrow-3$) can
raise the PBH abundance from the standard Gaussian prediction $\beta_{\rm PS}$
by some 10 orders of magnitude so that the real $N_{\ast}$ for $0.1<f_{\rm
PBH}<1$ might be shifted toward a smaller value. However, for the given
example in Fig. 2 with $\delta=-3.15$ and $k_{0}=9.46\times 10^{13}$ Mpc-1, we
find $\beta_{\rm PS}(N_{\ast}=2.5)/\beta_{\rm PS}(N_{\ast}=2.2)\gg 10^{100}$
for the viable range of $N_{\rm end}$. This implies that the effect of quantum
diffusion seems unlikely to shift the parameter space for $0.1<f_{\rm PBH}<1$
to $N_{\ast}<2$, leaving the $\mathcal{O}(1)$ ratio $\Omega_{\rm
PBHeq}/\Omega_{\rm Beq}$ nearly unchanged in the scenario.

##  VI Summary and discussion

PBHs fostered by the USR transition during inflation can contribute as a
significant DM component. We have shown that such an USR transition of the
inflationary background can trigger successful baryogenesis via the AD
mechanism. The resulting baryon asymmetry is asymptotically constant toward
the long USR duration limit ($N_{\ast}\gg 1$), allowing the cosmic coincidence
ratio $\Omega_{\rm CDM}/\Omega_{\rm B}\sim\mathcal{O}(1)$ to be realized over
large $\sim 10^{100}$ uncertainties in the PBH abundance.

The present scenario involves a plethora of observational tests, especially
via many of the future gravitational wave (GW) experiments. First, the
stochastic GW background sourced by enhanced curvature perturbations (at
second order) for PBH DM could be measured by space-based laser
interferometers [112, 113, 114, 116, 115, 117], where the maximal GW density
associated with the onset of the USR transition at the $k_{0}$ of interest
corresponds to the frequency band $f\sim 10^{-3}-1$ Hz [64]. Second, the high
scale inflation preferred by USR baryogenesis implies an observable tensor-to-
scalar ratio for the next generation CMB measurements close to the current
upper bound [82, 81]. Third, one of the consequences of AD baryogenesis is
that nontopological solitons ($Q$-balls) could have formed due to the
fragmentation of scalar condensates during relaxation [118], leading to an
enhanced stochastic GW background from second-order curvature perturbations
via temporal $Q$-ball domination [119]. Last but perhaps the most important of
all, PBH as a significant component of DM in the asteroid-mass window could be
verified or eliminated by any of the astrophysical projects mentioned at the
beginning of this Letter.

###### Acknowledgements.

We are grateful to Kalliopi Petraki for helpful discussions and the full
support on this project. E. P. is supported by: the Fermi Research Alliance,
LLC under Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy,
Office of High Energy Physics; Department of Excellent grant 2018-2022,
awarded by the Italian Ministry of Education, University and Research (MIUR);
Research grant of the Italo-French University, under Bando Vinci 2020. Y.-P.
W. was supported by the Agence Nationale de la Recherche (ANR) Accueil de
Chercheurs de Haut Niveau (ACHN) 2015 grant (“TheIntricateDark” project). The
project has received funding from the European Union’s Horizon 2020 research
and innovation programme under Grant Agreement No. 101002846 (ERC CoG
“CosmoChart”).

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