[a]Rebecca Diesing

# Galactic Cosmic Ray Acceleration with Steep Spectra

Damiano Caprioli

###### Abstract

Galactic cosmic rays (CRs) are accelerated by astrophysical shocks, primarily
supernova remnants (SNRs), via diffusive shock acceleration (DSA), an
efficient mechanism that predicts power-law energy distributions of CRs.
However, observations of both nonthermal SNR emission and Galactic CRs imply
CR spectra that are steeper than the standard DSA prediction, $\propto
E^{-2}$. Recent kinetic hybrid simulations suggest that such steep spectra may
be the result of a “postcursor”, or drift of CRs and magnetic structures with
respect to the thermal plasma behind the shock. Using a semi-analytic model of
non-linear DSA, we generalize this result to a wide range of astrophysical
shocks. By accounting for the presence of a postcursor, we produce CR energy
distributions that are substantially steeper than $E^{-2}$ and consistent with
observations. Our formalism reproduces both modestly steep spectra of Galactic
SNRs ($\propto E^{-2.2}$) and the very steep spectra of young radio supernovae
($\propto E^{-3}$).

##  1 Introduction

Understanding the origin of Galactic cosmic rays (CRs) with energies up to
$\sim 10^{8}$ GeV requires a complete paradigm for their acceleration and
propagation. The best source candidates for such acceleration are supernova
remnants (SNRs), which provide sufficient energy and an efficient acceleration
mechanism [49, 13, 64, 23]. In this mechanism, known as _diffusive shock
acceleration_ (DSA), particles are scattered by magnetic field perturbations,
resulting in diffusion across the SNR forward shock and an energy gain with
each crossing [44, 54, 6, 8, 15].

DSA predicts a power law momentum distribution of particles, $f_{\rm
sh}(p)\propto p^{-q_{\rm p}}$, where $f_{\rm sh}(p)$ is the instantaneous
momentum distribution of particles at the shock and $q_{\rm p}$ is set by the
balance between the energy gained with each crossing and the escape of
particles from the acceleration region [8]. Both of these quantities depend on
the shock hydrodynamics such that $q_{\rm p}$ can be written in terms of the
fluid compression ratio, $R=\rho_{2}/\rho_{0}$. Here, $\rho_{1}$ and
$\rho_{2}$ are the densities of the fluid in front of the shock (upstream) and
behind the shock (downstream) respectively. The relationship between $q_{\rm
p}$ and $R$ reads, $q_{\rm p}=(3R)/(R-1)$. For a strong shock with Mach number
$M\gg 1$, $R=4$ and we obtain $q_{\rm p}=4$. Equivalently, DSA predicts power-
law distributions in energy for relativistic particles, $\Phi_{\rm
sh}(E)\propto E^{-q}$, where $\Phi_{\rm sh}(E)$ is the instantaneous energy
distribution of particles at the shock. The relationship between $q$ and $R$
reads, $q=(R+2)/(R-1)$, with $q=2$ for a strong shock ($R=4$).

A modification to the standard DSA prediction arises when CRs carry a non-
negligible fraction of the shock’s energy. When this occurs, CRs can no longer
be treated as test-particles, resulting in modifications to the shock
hydrodynamics and thus particle spectra [e.g., 62, 61, 14, 50, 12, 58, 39, 40,
11]. In this non-linear DSA (NLDSA), the CR pressure produces a region in
front of the shock where the fluid is compressed, heated, and slowed. The
presence of this region, or _precursor_ , reduces the compression ratio near
the shock into a subshock with $R_{\rm sub}\equiv\rho_{2}/\rho_{1}<4$.
Meanwhile, the total compression ratio between the downstream and far upstream
becomes larger than the standard prediction: $R_{\rm
tot}\equiv\rho_{2}/\rho_{0}>4$. Note that, throughout this proceedings,
subscripts 0, 1, 2, and 3 are used to denote quantities at upstream infinity,
immediately upstream of the shock, immediately downstream of the shock, and
far downstream respectively.

As a result of these two compression ratios, NLDSA predicts concave CR
spectra. More specifically, particles with lower energies remain close to the
shock and probe $R_{\rm sub}<4$, while particles with higher energies diffuse
further upstream and probe $R_{\rm tot}>4$. Thus, low/high energy particles
are expected to exhibit spectra steeper/flatter than $E^{-2}$. The transition
between these regimes occurs at the lowest energy where CRs carry non-
negligible pressure, which is usually trans-relativistic. Since the nonthermal
emission in astrophysical environments is typically generated by relativistic
CRs, the classical NLDSA theory predicts that observations of non-thermal
emission from shock-powered sources should be explained by CR spectra flatter
than $E^{-2}$.

###  1.1 Theory vs. Observations

This prediction is readily testable via observations of the nonthermal
emission, e.g., from the relics of stellar explosions. However, the first GeV
observations of SNRs, combined with preexisting TeV data, did not confirm the
existence of concave spectra. On the contrary, they pointed toward CR
acceleration with spectra _steeper_ than $E^{-2}$ [21]. Notable examples
include historical remnants such as Tycho’s SNR [$q=2.3\pm 0.2$, 45, 5] and
Cassiopeia A [$q=2.36\pm 0.02$ above 17 GeV, 66].

Further evidence for steep spectra comes in the form of SNR radio emission,
particularly that of young, extragalactic supernovae (_radio SNe_). These
remnants exhibit synchrotron spectra that imply electron distributions as
steep as $E^{-3}$ [e.g., 33, 34, 71, 72, 51]. However, it is possible that
these synchrotron spectra probe the steep portion of a concave spectrum, since
the electrons responsible are likely sub-GeV [e.g., 41, 40, 73].

The CR spectrum measured at Earth also points toward CR acceleration with
spectra steeper than $E^{-2}$. In the standard picture of CR transport, this
measured CR spectrum goes as $E^{-(q+\delta)}$, where $\delta$ is the slope of
the CR residence time in the Galaxy: $\tau_{\rm res}\propto E^{-\delta}$.
Measurements of the CR anisotropy suggest $\delta\sim 0.3$ [16, 17].
Meanwhile, secondary to primary ratios suggest that $0.3\lesssim\delta\lesssim
0.4$, depending on the CR energy [e.g., 2]. Thus, fitting the observed
Galactic CR spectrum–which goes as $E^{-2.7}$–requires $2.3\lesssim q\lesssim
2.4$ [42, 43].

###  1.2 A Revised Theory of DSA

A number of explanations for steep CR spectra have been proposed in the
literature. For a detailed summary of these explanations and their
limitations, see [27].

One possible explanation considers the role of the magnetic fluctuations
responsible for CR scattering [e.g., 80, 21, 22, 53]. In the standard DSA
theory, particles are isotropized in both the upstream and downstream such
that they “feel" a head-on collision with each crossing of the shock. The
resulting energy gain per crossing thus depends on the difference in velocity
between the upstream and downstream plasma, $u_{1}-u_{2}$. In reality,
however, magnetic fluctuations–not thermal plasma–are responsible for particle
scattering, meaning that particles will be isotropized in the _fluctuation_
frame. The relative drift between the fluid and the fluctuations was already
present in the early DSA theory [8], but it has been usually neglected because
the fluctuation drift is roughly the Alfvén speed, much smaller than the fluid
speed in the shock frame. In the presence of CR-driven magnetic field
amplification, however, such a drift may be significantly enhanced; one can
argue that, in the upstream, these fluctuations move against the fluid with
the local Alfvén velocity in the amplified field, $v_{\rm A,1}$ [e.g., 22].
Thus, CRs experience a smaller energy gain per crossing $\propto u_{1}-v_{\rm
A,1}-u_{2}$ or, equivalently, they “feel" a compression ratio, $\tilde{R}$
that is smaller than that of the fluid,

$\tilde{R}=\frac{u_{1}-v_{\rm A,1}}{u_{2}}<R=\frac{u_{1}}{u_{2}}.$ (1)

This prescription may naturally lead to spectra that are steeper than $E^{-2}$
[21, 22], and it has been used, e.g., to model for the broadband emission of
Tycho’s SNR [60, 70] and of intracluster shocks [52].

The potential role of such drifts had not been validated by self-consistent
kinetic simulations until very recently, when [48] and [27] put forward
unprecedentedly-long hybrid simulations (i.e., particle-in-cell simulations
with kinetic ions and fluid electrons) that showed the onset of CR-modified
shocks. That being said, the presence of a precursor is insufficient to
explain the very steep spectra ($\propto E^{-3}$) of radio SNe, and its effect
may be limited if magnetic field amplification in the upstream is spatially-
dependent. In particular, if the local Alfvén speed decreases in the
precursor, particles with long diffusion lengths will probe a region with
reduced fluctuation drift, resulting in a flattening of the CR spectrum at the
highest energies.

However, [48] finds that not only does a precursor form in front of the shock,
in which self-generated fluctuations move at roughly $v_{\rm A,1}$ in the
amplified field, but also that the motion of magnetic structures behind the
shock leads to the formation of a _postcursor_. In this picture, CR-driven
magnetic fluctuations generated in the upstream retain their inertia over a
non-negligible distance (larger than the CR diffusion length) when advected
and compressed into the downstream. As a result, these fluctuations move away
from the shock faster than the background plasma, or more specifically, with
velocity $\tilde{u}_{2}=u_{2}+v_{\rm A,2}$ with respect to the shock.

Since CRs tend to isotropize with magnetic fluctuations, they too experience a
net drift equal to $v_{\rm A,2}$ relative to the background plasma [see Figure
6 in 48]. These drifts away from the shock lead to the removal of CR and
magnetic energy from the shock and thus an enhancement of the fluid
compression ratio and a steepening of the CR spectrum, as discussed in Section
5 of [48].

Equivalently, one can think of the postcursor as modifying the compression
ratio “felt" by CRs, just as the precursor modifies this ratio in [22]. In the
postcursor paradigm (ignoring, for now, the presence of a precursor), we have,

$\tilde{R}=\frac{u_{1}}{u_{2}+v_{\rm A,2}}=\frac{R}{1+\alpha},$ (2)

where $\alpha\equiv v_{\rm A,2}/u_{2}$. Thus, $q_{\rm p}$ depends only on $R$
and $\alpha$ or, equivalently, on $R$ and the magnetic pressure fraction
downstream, $\xi_{\rm B,2}\equiv B_{2}^{2}/(8\pi\rho_{0}v_{\rm sh}^{2})$:

$q_{\rm p}=\frac{3R}{R-1-\alpha}=\frac{3R}{R-1-\sqrt{2R\xi_{\rm B,2}}}.$ (3)

Note that the effect of the postcursor will dominate that of a precursor,
since compression of the magnetic field in the downstream leads to
$\alpha>v_{\rm A,1}/u_{1}$ [27]. In the case of efficient CR acceleration and
thus magnetic field amplification, [48] reports $\alpha\sim 0.6$, which is
sufficient to produce spectra steeper than $p^{-4}$, or $E^{-2}$ at
relativistic energies.

While these hybrid simulations provide a motivation and a physical explanation
for the modification of the standard DSA theory, quantifying the steepening of
the CR spectra in astrophysical systems requires additional calculations.
Namely, the postcursor paradigm implies that spectral steepening increases
with the downstream magnetic field strength, which, due to magnetic field
amplification via CR-driven instabilities, increases with the CR pressure
[e.g., 9, 35]. However, if spectra become too steep, the CR pressure will
drop, reducing magnetic field amplification and thus causing the steepening to
saturate.

In this proceedings, we use a semi-analytic model of NLDSA to generalize the
results of [27] and estimate $q$ for a wide range of SNR shocks. The results
presented in this proceedings can also be found in [38].

##  2 Method

To fully understand how a postcursor affects CR acceleration, we use a semi-
analytic formalism to model SNR shocks over a range of ambient number
densities, $n_{\rm ISM}$, ambient magnetic fields, $B_{0}$, and SN energies,
$E_{\rm SN}$. Herein we describe this formalism briefly, including our models
for SNR evolution, particle acceleration, and magnetic field amplification. A
more detailed description of our model, particularly our prescription for
particle acceleration, can be found in [22] and [37].

###  2.1 Shock Hydrodynamics

We model SNR shock hydrodynamics using the formalism described in [36], which
includes the effect of CR pressure on the evolution of the shock. More
specifically, SNR evolution is modeled through three stages spanning $\gtrsim
10^{5}$ yr: the _ejecta-dominated stage_ , in which the mass of the swept-up
ambient gas is less than that of the SN ejecta, the _Sedov stage_ , in which
the swept-up mass dominates the total mass and the SNR expands adiabatically,
and the _pressure-driven snowplow_ , in which the remnant cools due to
forbidden atomic transitions but continues to expand because its internal
pressure exceeds the ambient pressure. After this point, the remnant enters
the _momentum-driven snowplow_ , in which the internal pressure falls below
the ambient pressure and expansion continues due to momentum conservation.

All SNRs are assumed to eject $M_{\rm ej}=1M_{\odot}$ (1 solar mass) with
$E_{\rm SN}\in[10^{51},10^{52}]\ \rm erg$ into a uniform ambient medium of
density $n_{\rm ISM}\in[10^{-1},10^{5}]\ \rm cm^{-3}$ and magnetic field
$B_{0}\in[3,3000]\ \mu\rm G$.

###  2.2 Particle Acceleration

We model CR acceleration using a semi-analytic model of NLDSA described in
[26, 24, 22, 37] and references therein. This model self-consistently solves
the diffusion-advection equation for the transport of non-thermal particles in
a quasi-parallel, non-relativistic shock, including the dynamical backreaction
of accelerated particles and of CR-generated magnetic turbulence.

Particles above a threshold in momentum, $p_{\rm inj}$, are injected into the
acceleration process, with $p_{\rm inj}\equiv\xi_{\rm inj}m_{\rm p}v_{\rm
sh}/(1+R_{\rm tot}^{-1})$, consistent with the parameterization described in
[28], since $v_{\rm sh}/(1+R_{\rm tot}^{-1})$ is simply the velocity of the
upstream fluid in the downstream frame. In general, an increase in $\xi_{\rm
inj}$ corresponds to a decrease in the fraction of particles crossing the
shock that are injected into DSA. Here we neglect the dependence of injection
on the shock inclination and set an effective value of $\xi_{\rm inj}=3.8$,
which yields $\xi_{\rm CR}\equiv P_{\rm CR}/(\rho_{0}v_{\rm sh}^{2})\approx
0.1$ for a prototypical SNR ($n_{\rm ISM}=1$ cm-3, $B_{0}=3\mu$G, $E_{\rm
SN}=10^{51}$ erg, $M_{\rm ej}=1M_{\odot}$) after a few hundred years,
consistent with SNR observations. Note that $P_{\rm CR}$ refers to the CR
pressure.

To account for the effects of a precursor and postcursor, we introduce into
the diffusion-advection equation $\tilde{u}(x)\equiv u(x)\pm v_{\rm A}(x)$ ,
the effective fluid velocity as felt by the non-thermal particles which are
scattered by magnetic structures moving at $v_{\rm A}(x)$ relative to the
thermal plasma. Note that these structures move _against_ the fluid in the
upstream, but _with_ the fluid in the downstream [27]. Throughout this work,
we assume that the postcursor extends beyond the diffusion length of the
highest energy particles, i.e., behind the shock, $\tilde{u}(x)=u_{2}+v_{\rm
A,2}$.

The actual extent of the postcursor in astrophysical shocks is difficult to
quantify, even if high-resolution X-ray observations of individual SNRs with
Chandra suggest that the magnetic field remains amplified on a scale of
$1-5\%$ of the SNR radius [e.g., 74]. Physically speaking, since the maximum
CR energy $E_{\rm max}$ is controlled by the smallest between the upstream and
the downstream diffusion length [e.g., 61, 56, 18], the post-shock region with
high magnetic field must be at least as extended as the diffusion length of
particles with $E_{\rm max}$. It follows that the postcursor must be more
extended than the diffusion length of any particle, thereby leading to a
_global_ steepening of the CR spectrum. Note that, when only the Alfvénic
drift in the precursor is retained [á la 80, 22], a global steepening is only
possible if escaping CRs drive magnetic field amplification on all scales,
which is not guaranteed.

In practice, our formalism begins with an initial guess for the CR pressure,
which is used to solve the equations for conservation of mass, momentum, and
energy across a plane, nonrelativistic shock. The magnetic field pressure,
$P_{\rm B}$, is then calculated using the prescription described in 2.3, and
the resulting $u(x)$ and $P_{\rm B}$ are then used to solve the diffusion-
advection equation, which can be integrated to find a new guess for $P_{\rm
CR}$. In this manner, our formalism iteratively solves for the CR spectrum
while self-consistently accounting for the dynamical effect of accelerated
particles and the amplification of magnetic fields.

Once the proton spectrum has been calculated at each timestep of SNR
evolution, particle momenta are shifted and the instantaneous spectra are
weighted to account for adiabatic losses [see 23, 60, 37, for more details].
These weighted contributions are then added together to obtain a cumulative
spectrum.

###  2.3 Magnetic Field Amplification

The propagation of energetic particles ahead of the shock is expected to
excite streaming instabilities, [8, 9, 4, 20], which drive magnetic field
amplification and enhance CR diffusion [29, 30]. The result is magnetic field
perturbations with magnitudes that can exceed that of the ordered background
magnetic field. This magnetic field amplification has been observationally
inferred from the X-ray emission of many young SNRs, which exhibit narrow
X-ray rims due to synchrotron losses by relativistic electrons [e.g., 63, 7,
59, 65].

We model magnetic field amplification by assuming contributions from both the
resonant streaming instability [e.g., 55, 81, 67, 68, 69], and the non-
resonant hybrid instability [9]. A detailed discussion of these instabilities
and their saturation points can be found in [35].

In the resonant instability, CRs excite Alfvén waves with a wavelength
matching their gyroradius. The growth of this instability saturates when the
strength of magnetic perturbations reaches the level of the ordered background
field: $\delta B/B\sim 1$. More specifically, [3] derives this saturation
level to be $P_{\rm B1,res}=(P_{\rm CR,1})/(4M_{\rm A,0}),$ where $M_{\rm
A}\equiv v_{\rm sh}/v_{\rm A,0}$ is the Alfvénic Mach number.

For fast shocks typical of young SNRs, more significant is the non-resonant
hybrid instability. Driven by CR currents, $\bf{j}$, in the upstream, [9]
predicts that saturation occurs when tension in magnetic field lines becomes
sufficient to oppose the $\bf{j}\times\bf{B}$ force or, equivalently, when the
magnetic field pressure reaches approximate equipartition with the anisotropic
fraction of the CR pressure [also see 19],

$P_{\rm B1,Bell}=\frac{v_{\rm sh}}{2c}\frac{P_{\rm CR,1}}{\gamma_{\rm CR}-1}.$
(4)

Here, $c$ is the speed of light and $\gamma_{\rm CR}=4/3$ is the CR adiabatic
index. This saturation can lead to $\delta B/B_{0}\gg 1$ and has been
validated with hybrid simulations in [78].

To account for both the resonant and non-resonant instabilities, we pose here
that the upstream magnetic field pressure is given by $P_{\rm
B,1}=\sqrt{P_{\rm B1,res}^{2}+P_{\rm B1,Bell}^{2}}$. Assuming that all
components of the magnetic perturbations upstream are compressed, the
downstream magnetic field strength is $B_{2}\simeq R_{\rm sub}B_{1}$. For an
acceleration efficiency $\xi_{\rm CR}\approx 0.1$, our typical SNR parameters
give $B_{2}$ near a few hundred $\mu$G, in good agreement with X-ray
observations of young SNRs [77, 63, 25].

Figure 1: Downstream Alfvén speed, $v_{\rm A,2}$, as a function of shock
velocity, $v_{\rm sh}$ for a number of modeled SNR evolutions (dotted and
dashed lines). Each line corresponds to a single evolution with a fixed
ambient density (color scale) and ambient magnetic field (line style).
Overlaid are the SNR data aggregated in [79]. Our prescription for magnetic
field amplification produces modeled SNRs in good agreement with the measured
relationship between $v_{\rm A,2}$ and $v_{\rm sh}$.

For a more robust test of our prescription, we consider the relationship
between $v_{\rm sh}$ and $v_{\rm A,2}$. Specifically, our prescription
predicts a positive relationship between $v_{\rm sh}$ and $v_{\rm A,2}$ for
large $v_{\rm sh}$ (i.e., where the non-resonant instability dominates). At
lower $v_{\rm sh}$ (i.e., where the resonant instability dominates), we would
expect little to no correlation, since the resonant instability has a weaker
dependence on $v_{\rm sh}$ and depends on the ambient magnetic field, which
may vary. In Figure 1, we compare our predicted relationship between $v_{\rm
A,2}$ and $v_{\rm sh}$ to observational results compiled in [79]. As Figure 1
shows, our prescription yields a good agreement with observations. This
agreement also provides circumstantial evidence that the presence of a
postcursor is responsible for steep SNR spectra, particularly in light of the
fact that SNRs with large $v_{\rm sh}$ tend to have larger $q$ [e.g., 10].

##  3 Results

(a)  (b)

Figure 2: _Left:_ The modeled proton distribution, $\Phi(E)$, for a Tycho-like
SNR with precursor and postcursor drift included: $n_{\rm ISM}=1$ cm-3,
$B_{0}=3\mu$G, $E_{\rm SN}=10^{51}$ erg, and $M_{\rm ej}=1M_{\odot}$. The
black line shows the cumulative proton spectrum after $10^{5}$ yr, while the
colored lines show the contributions to this final spectrum from various
timesteps. Throughout the SNR’s evolution, protons are accelerated with
spectra steeper than $E^{-2}$. _Right:_ The modeled proton distribution of the
same Tycho-like SNR after 400 yr. Spectra are shown assuming traditional NLDSA
with no net drift of magnetic fluctuations (red dotted line), assuming net
drift in the precursor only (green dot-dashed line), assuming net drift in the
postcursor only (blue solid line), and assuming the net drift in both the
precursor and postcursor (purple dashed line). The inclusion of postcursor
drift produces a substantial spectral steepening relative to the traditional
NLDSA prediction. The addition of precursor drift further steepens the proton
spectrum, but its effect is subdominant.

Herein we present our modeled CR spectra and quantify the steepening resulting
from the modified shock dynamics–namely, the presence of a
postcursor–described in [48] and [27]. We also compare our results to
observations. Throughout this section, we estimate power-law slopes as
$q\equiv-\left<d\log{\Phi(E)}/d\log{E}\right>,$ where $\Phi(E)=dN(E)/dE$ is
the cumulative proton spectrum and $q$ is averaged between $10-10^{3}$ GeV.

###  3.1 Spectral Steepening

Our modeled spectrum of a “prototypical,” or Tycho-like SNR ($n_{\rm ISM}=1$
cm-3, $B_{0}=3\mu$G, $E_{\rm SN}=10^{51}$ erg, and $M_{\rm ej}=1M_{\odot}$) is
shown in Figure 2a, including the contributions of protons accelerated at
various stages of its evolution. These contributions are all steeper than
$E^{-2}$, resulting in a cumulative spectrum $\Phi(E)\propto E^{-2.23}$ by the
end of the SNR lifetime ($\sim 10^{5}$ yr).

(a)  (b)

Figure 3: Power-law slopes, $q$, of modeled proton spectra as a function of
shock velocity. The dotted vertical lines correspond to $v_{\rm sh}^{*}$, the
shock velocity where, for a given ambient density or magnetic field denoted by
the color scale, the dominant source of magnetic field amplification
transitions from the resonant to the non-resonant instability. _Left_ : The
ambient magnetic field is held fixed at $3\mu$G while density, denoted by the
color scale, is varied. _Right_ : The ambient density is held fixed at 1 cm-3
while ambient magnetic field, again denoted by the color scale, is varied. In
general, faster shocks give rise to larger magnetic field amplification and
thus steeper spectra. However, this dependence on shock velocity disappears at
low velocities where the resonant streaming instability is the primary source
of magnetic field amplification.

A more explicit quantification of the effect of the postcursor can be found in
Figure 2b. Here, we compare the cumulative spectrum of our Tycho-like SNR
after 400 yr to the traditional NLDSA result and to the case with a postcursor
but no net motion of magnetic structures in the precursor. As expected, the
NLDSA formalism produces a modestly concave spectrum that deviates slightly
from the standard $E^{-2}$ prediction. Meanwhile, the addition of a postcursor
softens this spectrum substantially to $E^{-2.30}$. The addition of a
precursor yields a slight increase in this steepening to $E^{-2.34}$, but its
effect is underdominant due to the fact that the upstream magnetic field is
decompressed such that $v_{\rm A,1}/u_{1}<v_{\rm A,2}/u_{2}$.

A summary of our results can be found in Figure 5, which shows the average
power law slope, $q$, as a function of shock velocity, $v_{\rm sh}$, for the
full range of modeled SNRs described in 2.1. To span a larger velocity range,
we include models with initial energy, $E_{\rm SN}$, between $10^{51}$ and
$10^{52}$ erg. Since increasing $E_{\rm SN}$ increases the shock velocity but
does not otherwise affect shock hydrodynamics, we do not visually distinguish
between different $E_{\rm SN}$ in Figure 5. A fast shock may therefore
correspond to a large $E_{\rm SN}$ or a young SNR; from the perspective of CR
acceleration and magnetic field amplification, the two scenarios are
equivalent. For this reason, our parameter range effectively spans different
ejecta masses as well. Namely, an increase in $M_{\rm ej}$ simply corresponds
to a decrease in $v_{\rm sh}$ for a given $E_{\rm SN}$. With the range of SNR
parameters described in 2.1, we obtain $2.1\lesssim q\lesssim 3$.

For large $v_{\rm sh}$ an increase in $v_{\rm sh}$ corresponds to an increase
in $q$, as one would expect when the Bell instability drives magnetic field
amplification. As suggested in Figure 2a, this dependence disappears when
$v_{\rm sh}$ becomes small enough that the resonant instability dominates. The
velocity where this transition occurs depends on the ambient density and
magnetic field, introducing a spread in the relationship between $q$ and
$v_{\rm sh}$ which, in the case of small ambient densities and large magnetic
fields, can extend up to high $v_{\rm sh}$ ($\gtrsim 10^{4}$ km s-1).

###  3.2 Comparison to Observations

By solving the equations for conservation of mass, momentum, and energy across
a postcursor-modified shock, one can predict the fluid compression ratio as a
function of the CR acceleration efficiency, $\xi_{\rm CR}$, and the magnetic
pressure fraction, $\xi_{\rm B}$. Thus, the postcursor paradigm predicts a
well-defined relationship between $q$, $\xi_{\rm CR}$, and $\xi_{\rm B}$.
Assuming magnetic field amplification is driven by the non-resonant
instability, $\xi_{\rm B}$ can be recast in terms of $\xi_{\rm CR}$ and
$v_{\rm sh}$, meaning that observational constraints on the shock velocity and
spectral slope correspond to constraints on the CR acceleration efficiency.
For reference, we summarize this relationship in Figure 4, assuming CRs probe
the full compression ratio from the far upstream to the downstream.

Figure 4: Predicted power law slope, $q$, denoted by color scale, as a
function of CR acceleration efficiency, $\xi_{\rm CR}$, and magnetic pressure
fraction downstream, $\xi_{\rm B,2}$. $q$ is calculated for a strong shock
assuming CRs probe the full compression ratio from the far upstream to the
downstream (regions 0 and 2 respectively. White lines denoting $\xi_{\rm B,2}$
as a function of $\xi_{\rm CR}$ for various shock velocities are overlaid,
assuming magnetic field amplification is dominated by the non-resonant
instability.

(a)  (b)

Figure 5: _Left:_ Power law slopes, $q$, of modeled proton spectra (dotted an
dashed lines) as a function of SNR age. The slopes of GeV spectra from the
Fermi LAT catalog [1] are overlaid. For simplicity, SNRs have been removed if
their GeV emission is likely leptonic in origin or exhibits a significant
spectral break (see text for details). The spectral information for Cassiopeia
A has been taken from [66]. The inclusion of a postcursor produces steep
proton spectra in good agreement with SNR observations. _Right_ : The modeled
proton distribution ($E^{3}\Phi(E)$) for a sample radio SN expanding into a
circumstellar wind ($n_{\rm ISM}\propto r^{-2}$; see text for details). The
black line shows the cumulative proton spectrum after 10 yr, while the colored
lines show the contributions to this spectrum from various timesteps. Our toy
model reproduces the very steep spectra characteristic of radio SNe; for this
setup, we obtain $q\simeq 2.99$.

Equivalently, we can test the validity of the postcursor paradigm by comparing
our predicted spectra to to observations, in particular the non-thermal
emission of Galactic remnants (including historical SNRs), and young
extragalactic supernovae (radio SNe). The SNRs in our Galaxy consist largely
of older, slower shocks [$v_{\rm sh}\ll 10^{4}$ km s-1, see, e.g., 46].
Assuming magnetic field amplification driven by the non-resonant instability,
we would therefore expect these SNRs–in the postcursor paradigm–to exhibit
only modestly steep spectra. To test this, we look to GeV observations
aggregated in [21] from the Fermi LAT source catalog [1]. To ensure accurate
estimates of $q$, we remove SNRs with GeV emission that is likely leptonic in
origin: RX J1713.7-3946 [e.g., 47] and Vela Jr. [e.g., 57]. For simplicity, we
also remove SNRs with breaks or cutoffs in the GeV range, which are typically
interpreted as due to reacceleration [e.g., W44, see 31]. When possible, we
use results from combined GeV-TeV analyses [e.g., for Cas A, 66], which
provide a more accurate representation of the full $\gamma$-ray slope.

Figure 5a compares the values of $q$ calculated for our modeled SNRs to those
in our sample. Our models are able to reproduce the full range of slopes
inferred from GeV and TeV observations of Galactic SNRs: $2.1\lesssim
q\lesssim 2.6$.

In addition to explaining the modestly steep spectra of Galactic SNRs, the
presence of a postcursor may also explain the very steep spectra of their
extragalactic counterparts: radio SNe. These young, fast remnants ($v_{\rm
sh}\gtrsim 10^{4}$ km s-1) typically expand into dense circumstellar winds
blown by the progenitor star [see, e.g., 34]. Their high $v_{\rm sh}$ and,
more explicitly, their large inferred postshock magnetic fields [$\sim$ 0.1-1
G, see, e.g., 32] imply strong magnetic field amplification, making them
excellent candidates for tests of postcursor physics.

Intriguingly, radio SNe exhibit synchrotron emission that suggest electron
distributions $\Phi(E)\propto E^{-3}$ or even steeper [see, e.g., 33, 71, 72,
51]. Assuming protons and electrons are accelerated with the same spectral
slope–a reasonable assumption given that DSA depends only on a particle’s
rigidity–and that synchrotron cooling is negligible at energies corresponding
to radio frequencies as discussed in [33], we can conclude that the proton
distribution must be similarly steep.

As we discussed above, postcursor physics can reproduce $q\simeq 3$ under the
right conditions: specifically, when $v_{\rm sh}$ is large and the magnetic
field is generated by the Bell instability. However, if our intent is to
describe a typical radio SN, the models presented above are rather rough
approximations, since they assume uniform ambient densities and include an
injection prescription tuned to observations of Galactic SNRs (i.e., $\xi_{\rm
inj}=3.8$ so that $\xi_{\rm CR}\approx 0.1$ for a prototypical Galactic
remnant). To more accurately approximate the proton distribution of a typical
radio SN, we produce a toy-model hydrodynamic evolution that follows an
ejecta-dominated radio SN expanding into a circumstellar wind for
approximately 10 years. We then use our semi-analytic formalism to self-
consistently calculate the corresponding proton spectrum.

More explicitly, we consider an energetic SN ($E_{\rm SN}=10^{52}$ erg) that
ejects $M_{\rm ej}=1M_{\odot}$ into the circumstellar medium. Since we only
model the first 10 years of evolution, the mass swept up by the shock is much
smaller than $M_{\rm ej}$ and we therefore use the approximation in Table 9 of
[75] for an ejecta-dominated SNR expanding into a wind: $v_{\rm sh}\propto
t^{-1/5}$. For our circumstellar density, we assume a wind profile given by
$\rho_{0}=\dot{M}/(4\pi v_{\rm w})$, where $\dot{M}$ is the mass-loss rate of
the progenitor and $v_{\rm w}$ is the wind velocity. As discussed in [33], we
assume typical paramenters for a Wolf-Rayet progenitor:
$\dot{M}=10^{-5}M_{\odot}$ yr-1 and $v_{\rm w}=1000$ km s-1. We choose an
ambient magnetic field that follows our density profile: $B_{0}/\text{G}\simeq
0.01\sqrt{n_{\rm ISM}/(5000\text{ cm}^{-3})}$ with normalization chosen such
that our magnetic field amplification prescription produces postshock fields
consistent with observations [$B_{2}\sim$ 0.1-1 G, e.g., 32]. Finally, since
$\xi_{\rm inj}=3.8$ gives extremely small acceleration efficiencies for our
toy model ($\xi_{\rm CR}<0.01$), we reduce $\xi_{\rm inj}$ slightly to 3.4.
With this adjustment, $\xi_{\rm CR}$ remains modest ($<0.05$). The decrease in
$\xi_{\rm inj}$ needed to produce acceleration efficiencies of 5-10% would
yield even steeper spectra.

Our model spectrum is shown in Figure 5b and has a slope of $q\simeq 2.99$;
note that to make this slope visually apparent, we plot $E^{3}\Phi(E)$. As
time passes, each new shell of protons contributes a slightly harder spectrum
due to the modest decrease in $v_{\rm sh}$, which leads to a reduction in
magnetic field amplification (for the parameters discussed here, the non-
resonant instability dominates). This behavior implies a simple physical
explanation for the discrepancy between the very steep spectral slopes of
radio SNe and the modestly steep slopes of Galactic SNRs. Namely, as young
remnants age and slow down, their postshock magnetic fields decrease, reducing
the strength of their postcursors and flattening their spectra.

Note that our model predicts CR spectra to be steep even at high energies,
while the classical concave-spectra explanation [e.g., 41, 40, 73] returns
rather flat spectra at TeV energies; therefore, X-ray and, possibly,
$\gamma$-ray observations may be able to distinguish between models.

##  4 Conclusion

In summary, we use a semi-analytic model of NLDSA to quantify the CR spectral
steepening in SNRs that arises from the presence of a _postcursor_ , i.e., a
region behind a shock in which magnetic fluctuations drift away from the shock
at the local Alfvén speed with respect to the background fluid. Since CRs
isotropize with these fluctuations, they too experience a net drift, leading
to a removal of CR energy from the system and thus a steepening of their
spectra relative to the standard DSA prediction ($\Phi(E)\propto E^{-2}$). The
formation of a postcursor has been validated with kinetic simulations [48, 27]
and provides a natural explanation for the steep CR spectra inferred from
observations of SNRs [e.g., 45, 5, 66] and Galactic CRs, once corrected for
propagation [e.g., 2, 42].

Because magnetic fluctuations drift with the local Alfvén speed, it is
important that we include a prescription for magnetic field amplification that
is not only theoretically motivated, but consistent with observations. In our
model, we implement a self-consistent prescription that incorporates the
saturation points of both the resonant [3] and non-resonant [9, 78] streaming
instabilities. This model yields magnetic fields that are consistent with
those inferred from X-ray observations of young SNRs [76, 77, 63, 25] and
reproduces the observed relationship between shock velocity and downstream
Alfvén speed reported in Zeng et al. [79].

With this prescription for magnetic field amplification [also see 35], our
model produces modestly steep spectra $\propto E^{-2.34}$ for a Tycho-like SNR
after 400 yr: $n_{\rm ISM}=1$ cm-3, $B_{0}=3\mu$G, $E_{\rm SN}=10^{51}$ erg,
and $M_{\rm ej}=1M_{\odot}$. We also confirm that the postcursor is the
dominant source of this steepening; neglecting the effect of the precursor
still yields spectra $\propto E^{-2.30}$.

As SNRs age and slow down, we find that this steepening diminishes, yielding a
power-law slope, $q\simeq 2.23$ for our prototypical SNR after $10^{5}$ yr.
Given observational constraints on the slope of the CR diffusion coefficient,
this slope is consistent with that needed to reproduce the spectrum of
Galactic CRs observed at Earth [e.g., 42, 43].

More generally, for large $v_{\rm sh}$, the nonresonant instability dominates
magnetic field amplification such that the magnetic pressure scales as $v_{\rm
sh}P_{\rm CR}$. As a result, the downstream Alfvén speed and thus the
steepening due to the postcursor diminish as the SNR slows. This dependence
largely disappears at lower $v_{\rm sh}$, when the resonant instability
dominates. The location of this transition depends on the ambient density and
magnetic field.

The relationship between $v_{\rm sh}$, magnetic field amplification, and $q$
that arises from postcursor physics provides a theoretically-motivated
explanation for the modestly steep spectra of Galactic SNRs ($\propto
E^{-2.2}$), the very steep spectra of radio SNe ($\propto E^{-3}$), and the
connection between them. More specifically, we use our formalism to model both
source classes and find that we are able to produce spectra in good agreement
with observations.

Our work represents the first generalization of postcursor physics to a wide
range of SNR shocks, as well as the first self-consistent quantification of
the spectral steepening that arises. The good agreement between our modeled
spectra and those inferred from the nonthermal emission of real SNRs implies
that the presence of a postcursor may resolve the tension between DSA
predictions and observations.

##  5 Acknowledgements

This research was partially supported by a Eugene and Niesje Parker Graduate
Student Fellowship, NASA (grants NNX17AG30G and 80NSSC18K1726) and the NSF
(grants AST-1909778, PHY-1748958, and PHY-2010240).

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# Extremal problems for GCDs

Ben Green  Mathematical Institute
Andrew Wiles Building
Radcliffe Observatory Quarter
Woodstock Rd
Oxford OX2 6QW<EMAIL_ADDRESS>and  Aled Walker  Trinity College
Trinity Street
Cambridge CB2 1TQ<EMAIL_ADDRESS>
###### Abstract.

We prove that if $A\subseteq[X,2X]$ and $B\subseteq[Y,2Y]$ are sets of
integers such that $\gcd(a,b)\geqslant D$ for at least $\delta|A||B|$ pairs
$(a,b)\in A\times B$ then
$|A||B|\ll_{\varepsilon}\delta^{-2-\varepsilon}XY/D^{2}$. This is a new result
even when $\delta=1$. The proof uses ideas of Koukoulopoulos and Maynard and
some additional combinatorial arguments.

###### 2000 Mathematics Subject Classification:

Primary

The first-named author is supported by a Simons Investigator Award and is
grateful to the Simons Foundation for their support. The second-named author
is supported by a Postdoctoral Fellowship with the Centre de Recherches
Mathématiques and by a Junior Research Fellowship from Trinity College
Cambridge. We would also like to thank Andrew Granville, Dimitris
Koukoulopoulos, and James Maynard for interesting conversations related to
this work.

##  1\. Introduction and proof strategy

Fix $\varepsilon\in(0,1)$ throughout the paper; implied constants and
thresholds may depend on $\varepsilon$ but are otherwise absolute. Let $p_{0}$
be a threshold which (at the start of Section 3) will be taken to be
sufficiently large. Given a finite set $S\subset\mathbf{N}$, write
$\mathscr{P}(S)$ for the set of primes dividing some element of $S$, and
$\mathscr{P}_{\operatorname{small}}(S)$ for the set of primes $p\leqslant
p_{0}$ dividing some element of $S$.

Our main result is the following.

######  Theorem 1.1.

Let $X,Y,D\in[1,\infty)$, and suppose that $D\leqslant\min(X,Y)$. Let
$\delta\in(0,1]$. Suppose that $A\subset[X,2X]$ and $B\subset[Y,2Y]$ have the
following property: for at least $\delta|A||B|$ pairs $(a,b)\in A\times B$,
$\gcd(a,b)\geqslant D$. Then we have the bound

$|A||B|\leqslant(1000)^{1+\\#\mathscr{P}_{\operatorname{small}}(A\cup
B)}\delta^{-2-\varepsilon}\frac{XY}{D^{2}}.$

Let us make some remarks on this theorem.

_1._ This obviously implies the cruder bound
$|A||B|\ll\delta^{-2-\varepsilon}XY/D^{2}$, mentioned in the abstract. The
more precise form we have stated seems of little additional interest in its
own right, but is critical for the proof. Perhaps the most natural case is
when $A=B$ and $X=Y$, when the result says the following: if, for a proportion
$\delta$ of all pairs $(a,a^{\prime})\in A\times A$ we have
$\gcd(a,a^{\prime})\geqslant D$, then $|A|\ll\delta^{-1-\varepsilon}X/D$.

_2._ We believe that the result is new even when $\delta=1$, that is to say
when $\gcd(a,b)\geqslant D$ for _all_ $a\in A$ and $b\in B$. In this case, the
result is clearly sharp up to a multiplicative constant. Indeed, assuming that
$D$ is an integer, we may take $A=\\{x\in[X,2X]:D|x\\}$ and
$B=\\{y\in[Y,2Y]:D|y\\}$. One might wonder whether all the tight examples have
approximately such a structure; however, Chow has constructed a different
family of tight examples for $\delta=1$ (see [1, Section 15]) in which there
is no single $d\gg D$ that divides a positive proportion of $A$ and $B$.

_3._ For $\delta\in(0,1)$ the result is also sharp for a wide range of
parameters, apart from the factor of $\delta^{-\varepsilon}$. To see this, let
$D\geqslant\delta^{-1}$ be given, set $D_{0}:=\lfloor\delta D\rfloor$, and
consider the sets $A=B=\\{x\in[X,2X]:D_{0}|x\\}$. Evidently
$|A||B|\sim\delta^{-2}X^{2}/D^{2}$. However, if
$x=D_{0}m,x^{\prime}=D_{0}m^{\prime}$ with $m,m^{\prime}\in[X/D_{0},2X/D_{0}]$
and $\gcd(m,m^{\prime})\geqslant D/D_{0}$ then $x,x^{\prime}\in A$ and
$\gcd(x,x^{\prime})\geqslant D$. The proportion of pairs of integers with gcd
$k$ is $1/k^{2}\zeta(2)$, and so the proportion of pairs of integers with gcd
$\geqslant k$ is $\gg 1/k$. It follows (at least if $X/D_{0}$ is big enough
compared to $D/D_{0}$) that the number of such pairs $(x,x^{\prime})$ is
$\gg\delta|A||B|$.

_4._ When $A=B$, $X=Y$ and $\delta=1$, the result says the following: if
$\gcd(a,a^{\prime})\geqslant D$ for all $a,a^{\prime}\in A$, then $|A|\ll
X/D$. However (we are rather embarrassed to admit) Zachary Chase pointed out
to the authors that this particular result is trivial, because the assumption
implies that $|a-a^{\prime}|\geqslant D$ whenever $a\neq a^{\prime}$. This
argument does not, however, appear to extend to the other cases.

_5._ A straightforward dyadic decomposition argument would allow one to
establish similar results under the assumption that $A\subset[X]$ and
$B\subset[Y]$. We leave the details to the reader.

_Notation._ Our notation is standard. If $p$ is a prime and $a\in\mathbf{Z}$,
we write $v_{p}(a)$ for the largest $k$ such that $p^{k}|a$. We extend this to
rationals by $v_{p}(a/b)=v_{p}(a)-v_{p}(b)$. Implied constants in the
$O(),\ll$ and $\gg$ notations are absolute (though they may depend on
$\varepsilon$, which is fixed throughout the paper).

_Strategy._ Our strategy for proving Theorem 1.1 is essentially to proceed by
induction on $\\#\mathscr{P}(A\cup B)$, but we will phrase the argument in
terms of a hypothetical counterexample with minimal $\\#\mathscr{P}(A\cup B)$.
The first main business is to show that such a minimal counterexample has a
very specific structure.

######  Proposition 1.2.

Suppose we have a counterexample to Theorem 1.1 with the set
$\mathscr{P}(A\cup B)$ minimal in size. Let $\Omega\subset A\times B$, with
$|\Omega|=\delta|A||B|$, be the set of pairs for which $\gcd(a,b)\geqslant D$.
Then there is $\Omega^{\prime}\subset\Omega$, with
$|\Omega^{\prime}|\geqslant\frac{1}{2}|\Omega|$, and an integer $N$ such that
the following is true. For all primes $p$ and for all
$(a,b)\in\Omega^{\prime}$ we have $|v_{p}(a/N)|+|v_{p}(b/N)|\leqslant 1$.

Though such a statement does not appear explicitly in their work, this
proposition should be considered essentially due to Koukoulopoulos and Maynard
[1]. We will give a fairly short, self-contained proof. On some level this is
equivalent to the argument of [1], but we phrase things rather differently.

To complete the proof of Theorem 1.1, we prove the following counterpart to
Proposition 1.2.

######  Proposition 1.3.

Suppose that $A\subset[X,2X]$, $B\subset[Y,2Y]$, $\Omega,D,\delta,N$ are as in
Proposition 1.2. Then $|A||B|\leqslant 1000\delta^{-2}XY/D^{2}$.

Evidently, this means that $A,B$ do not in fact give a counterexample to
Theorem 1.1. Combining Propositions 1.2 and 1.3 shows that no minimal
counterexample to Theorem 1.1 exists, so Theorem 1.1 is true.

The proof of Proposition 1.3 uses some combinatorial arguments and is not
found in [1].

##  2\. Concentrated measures on $\mathbf{Z}^{2}$

In this section we prove a result about concentration of probability measures
on $\mathbf{Z}^{2}$. It is the key technical ingredient in the proof of
Proposition 1.2, where it is used to concentrate the pair of valuation
functions $(v_{p}(a),v_{p}(b))$ around a diagonal pair $(k,k)$.

Here, as in the rest of the paper, we write $q=2+\varepsilon$ and write
$q^{\prime}$ for the conjugate index to $q$ (i.e.
$\frac{1}{q}+\frac{1}{q^{\prime}}=1$).

######  Lemma 2.1.

Let $c\leqslant 1$ and $\lambda\leqslant\frac{4}{5}$. Suppose that $\mu$ is a
finitely-supported probability measure on $\mathbf{Z}^{2}$. Suppose that there
are sequences $x=(x_{i})_{i\in\mathbf{Z}}$, $y=(y_{j})_{j\in\mathbf{Z}}$ of
non-negative reals such that
$\|x\|_{\ell^{q^{\prime}}(\mathbf{Z})}=\|y\|_{\ell^{q^{\prime}}(\mathbf{Z})}=1$,
and such that for all $(i,j)\in\mathbf{Z}^{2}$ we have

$\mu(i,j)\leqslant c\lambda^{|i-j|}x_{i}y_{j}.$ (2.1)

Then $c\geqslant\frac{1}{9}$, and $\mu$ is highly concentrated near some point
$(k,k)$:

$\sum_{|i-k|+|j-k|\geqslant 2}\mu(i,j)\ll\lambda^{q+\varepsilon}.$ (2.2)

###### Proof.

We first prove the lower bound on $c$. Using (2.1),
$\sum_{\mathbf{Z}^{2}}\mu(i,j)=1$,
$\sum_{m\in\mathbf{Z}}\lambda^{|m|}\leqslant 9$ and $q^{\prime}<2$, we have

$\frac{1}{c}\leqslant\sum_{i,j}\lambda^{|i-j|}x_{i}y_{j}\leqslant\sum_{i,j}\lambda^{|i-j|}(x_{i}y_{j})^{\frac{q^{\prime}}{2}}\leqslant
9\sup_{m}\sum_{i}(x_{i}y_{i+m})^{\frac{q^{\prime}}{2}}\leqslant 9,$

where the last step follows from the Cauchy-Schwarz inequality and the
assumption
$\|x\|_{\ell^{q^{\prime}}(\mathbf{Z})}=\|y\|_{\ell^{q^{\prime}}(\mathbf{Z})}=1$.
The lower bound on $c$ follows.

Turning to (2.2), write $\sup_{i}x_{i}y_{i}=1-\gamma$ for some
$\gamma\in[0,1]$, and suppose this supremum attained when $i=k$. Then
$x_{k},y_{k}\geqslant 1-\gamma$, so

$\sum_{i\neq k}x^{q^{\prime}}_{i},\sum_{j\neq
k}y^{q^{\prime}}_{j}\ll\gamma\;\mbox{and}\;x_{i},y_{j}\ll\gamma^{1/q^{\prime}}\;\mbox{when}\;i,j\neq
k.$ (2.3)

For $n=1,2,3,4,5,6$ write $\Sigma_{n}:=\sum_{(i,j)\in S_{n}}\mu(i,j)$, where
$S_{1},\dots,S_{6}$ are the following sets, which partition $\mathbf{Z}^{2}$:

$S_{1}:=\\{(i,j)\in\mathbf{Z}^{2}:i\neq j\neq k\neq i\\},\quad
S_{2}:=\\{(k,j):|j-k|\geqslant 2\\},$ $S_{3}:=\\{(i,k):|i-k|\geqslant
2\\},\quad S_{4}:=\\{(k,k\pm 1),(k\pm 1,k)\\},\quad S_{5}:=\\{(i,i):i\neq
k\\},$

and finally $S_{6}:=\\{(k,k)\\}$. We bound the $\Sigma_{n}$ in turn.

_Bound for $\Sigma_{1}$._ By (2.1) and (2.3) we have

$\Sigma_{1}\leqslant\sum_{\begin{subarray}{c}i,j\neq k\\\ i\neq
j\end{subarray}}\lambda^{|i-j|}x_{i}y_{j}\ll\gamma^{\frac{2}{q^{\prime}}(1-\frac{q^{\prime}}{2})}\sum_{\begin{subarray}{c}i,j\neq
k\\\ i\neq
j\end{subarray}}\lambda^{|i-j|}(x_{i}y_{j})^{\frac{q^{\prime}}{2}}=\gamma^{\frac{2}{q^{\prime}}-1}\sum_{m\neq
0}\lambda^{|m|}\\!\\!\\!\\!\sum_{i,i+m\neq
k}\\!\\!\\!(x_{i}y_{i+m})^{\frac{q^{\prime}}{2}}.$

By Cauchy-Schwarz and (2.3), for each fixed $m$ we have

$\sum_{i,i+m\neq k}(x_{i}y_{i+m})^{\frac{q^{\prime}}{2}}\leqslant(\sum_{i\neq
k}x_{i}^{q^{\prime}})^{1/2}(\sum_{j\neq k}y_{j}^{q^{\prime}})^{1/2}\ll\gamma.$

Since $\sum_{m\neq 0}\lambda^{|m|}\ll\lambda$, putting these together gives
$\Sigma_{1}\ll\lambda\gamma^{\frac{2}{q^{\prime}}}$.

_Bounds for $\Sigma_{2},\Sigma_{3}$._ For $\Sigma_{2}$, we use the trivial
bound $x_{k}\leqslant 1$ and (2.3) for $y_{j}$. This gives (using the
assumption that $\lambda\leqslant\frac{4}{5}$)

$\Sigma_{2}\leqslant\sum_{|j-k|\geqslant
2}(5\lambda/4)^{|j-k|}(4/5)^{|j-k|}y_{j}\ll\lambda^{2}\\!\\!\\!\sum_{|j-k|\geqslant
2}\\!\\!\\!(4/5)^{|j-k|}y_{j}\ll\gamma^{\frac{1}{q^{\prime}}(1-\frac{q^{\prime}}{2})}\lambda^{2}\\!\\!\\!\sum_{|j-k|\geqslant
2}\\!\\!\\!(4/5)^{|j-k|}y_{j}^{\frac{q^{\prime}}{2}}.$

By Cauchy-Schwarz and (2.3),

$\sum_{|j-k|\geqslant
2}(4/5)^{|j-k|}y_{j}^{\frac{q^{\prime}}{2}}\ll(\sum_{j\neq
k}(4/5)^{2|j-k|})^{1/2}(\sum_{j\neq
k}y_{j}^{q^{\prime}})^{1/2}\ll\gamma^{1/2}.$

Combining these bounds gives
$\Sigma_{2}\ll\gamma^{\frac{1}{q^{\prime}}}\lambda^{2}$, and an essentially
identical argument yields
$\Sigma_{3}\ll\gamma^{\frac{1}{q^{\prime}}}\lambda^{2}$.

_Bound for $\Sigma_{4}$._ From (2.1), (2.3) we immediately get
$\Sigma_{4}\ll\gamma^{\frac{1}{q^{\prime}}}\lambda$.

_Bound for $\Sigma_{5}$._ A trivial modification to the argument used for
$\Sigma_{1}$ (allowing $i=j$, which gives just a term with $m=0$) shows that
$\Sigma_{5}\ll\gamma^{\frac{2}{q^{\prime}}}$.

_Bound for $\Sigma_{5}+\Sigma_{6}$._ By (2.1) and the fact that
$\sup_{i}x_{i}y_{i}=1-\gamma$,

$\Sigma_{5}+\Sigma_{6}\leqslant\sum_{i}x_{i}y_{i}\leqslant(1-\gamma)^{1-\frac{q^{\prime}}{2}}\sum_{i}(x_{i}y_{i})^{\frac{q^{\prime}}{2}}\leqslant(1-\gamma)^{1-\frac{q^{\prime}}{2}}\leqslant
1-(1-\frac{q^{\prime}}{2})\gamma,$

where we used Cauchy-Schwarz yet again.

Putting all this together gives

$1=\sum_{n=1}^{6}\Sigma_{n}\leqslant
1-(1-\frac{q^{\prime}}{2})\gamma+O(\lambda\gamma^{\frac{1}{q^{\prime}}}).$

This implies that $\lambda\gg\gamma^{1-\frac{1}{q^{\prime}}}$, i.e.
$\gamma\ll\lambda^{q}$. Finally, we see that

$\sum_{|i-k|+|j-k|\geqslant
2}\mu(i,j)=\Sigma_{1}+\Sigma_{2}+\Sigma_{3}+\Sigma_{5}\ll\lambda^{\frac{2q}{q^{\prime}}}+\lambda^{2+\frac{q}{q^{\prime}}}\ll\lambda^{\frac{2q}{q^{\prime}}}$

(since $q\leqslant 3$). The result follows, noting that
$\frac{2q}{q^{\prime}}=q+\varepsilon$. ∎

##  3\. Properties of a minimal counterexample

We turn now to the proof of Proposition 1.2. We first reduce matters to the
following “local” statement at a single prime $p$.

######  Proposition 3.1.

Suppose we have a counterexample to Theorem 1.1 with the set
$\mathscr{P}(A,B)$ minimal in size. Let $\Omega\subset A\times B$, with
$|\Omega|=\delta|A||B|$, be the set of pairs for which $\gcd(a,b)\geqslant D$.
Let $p\in\mathscr{P}(A,B)$ be a prime. Then $p>p_{0}(\varepsilon)$, and there
is $k_{p}\in\mathbf{Z}_{\geqslant 0}$ and $\Omega_{p}\subset\Omega$ such that
for all $(a,b)\in\Omega_{p}$ we have
$|v_{p}(a)-k_{p}|+|v_{p}(b)-k_{p}|\leqslant 1$, and such that
$|\Omega\setminus\Omega_{p}|\ll p^{-1-\varepsilon/3}|\Omega|$ .

Proposition 1.2 follows quickly from this by taking $N=\prod_{p}p^{k_{p}}$ and
$\Omega^{\prime}:=\bigcap_{p}\Omega_{p}$. We have

$|\Omega^{\prime}|\geqslant|\Omega|\big{(}1-O\big{(}\sum_{p>p_{0}}p^{-1-\varepsilon/3}\big{)}\big{)}\geqslant\frac{1}{2}|\Omega|$

if $p_{0}$ is big enough (this is the point at which $p_{0}$ is constrained).

It remains, then, to establish Proposition 3.1. Fix, for the rest of this
section, the prime $p$. For $i,j\in\mathbf{Z}_{\geqslant 0}$, we define
$A_{i}:=\\{a\in A:v_{p}(a)=i\\}$, $B_{j}:=\\{b\in B:v_{p}(b)=j\\}$, and write
$\alpha_{i}:=\frac{|A_{i}|}{|A|}$ and $\beta_{j}:=\frac{|B_{j}|}{|B|}$ for the
relative densities of these sets. Write
$\mu(i,j):=\frac{|\Omega\cap(A_{i}\times B_{j})|}{|\Omega|}$, thus $\mu$ is a
finitely-supported probability measure on $\mathbf{Z}_{\geqslant 0}^{2}$.

For any $i,j$, consider the sets $\bar{A}_{i}:=p^{-i}\cdot A_{i}$ and
$\bar{B}_{j}:=p^{-j}\cdot B_{j}$. These are sets of integers, coprime to $p$,
with $\bar{A}_{i}\subset[\frac{X}{p^{i}},\frac{2X}{p^{i}}]$,
$\bar{B}_{j}\subset[\frac{Y}{p^{j}},\frac{2Y}{p^{j}}]$ and
$\gcd(x,y)\geqslant\frac{D}{p^{\min(i,j)}}$ whenever $x=p^{-i}a$, $y=p^{-j}b$
with $(a,b)\in\Omega$.

By the minimality assumption, these sets cannot be a counterexample to Theorem
1.1, and therefore we have the inequality

$|\bar{A}_{i}||\bar{B}_{j}|\leqslant(1000)^{1+\\#\mathscr{P}_{\operatorname{small}}(\bar{A}_{i},\bar{B}_{j})}(\frac{\delta\mu(i,j)}{\alpha_{i}\beta_{j}})^{-2-\varepsilon}\frac{\frac{X}{p^{i}}\frac{Y}{p^{j}}}{(\frac{D}{p^{\min(i,j)}})^{2}}.$
(3.1)

On the other hand,

$|\bar{A}_{i}||\bar{B}_{j}|=\alpha_{i}\beta_{j}|A||B|\geqslant(1000)^{1+\\#\mathscr{P}_{\operatorname{small}}(A,B)}\alpha_{i}\beta_{j}\delta^{-2-\varepsilon}\frac{XY}{D^{2}}.$
(3.2)

Note also that
$\mathscr{P}(\bar{A}_{i},\bar{B}_{j})\subset\mathscr{P}(A,B)\setminus\\{p\\}$,
and so

$\\#\mathscr{P}_{\operatorname{small}}(\bar{A}_{i},\bar{B}_{j})\leqslant\\#\mathscr{P}_{\operatorname{small}}(A,B)-1_{p\leqslant
p_{0}}.$ (3.3)

Comparing (3.1), (3.2) and (3.3) gives, for all $i$ and $j$,

$\mu(i,j)\leqslant 10^{-1_{p\leqslant
p_{0}}}(\alpha_{i}\beta_{j})^{\frac{1+\varepsilon}{2+\varepsilon}}p^{-\frac{|i-j|}{2+\varepsilon}},$
(3.4)

since $\varepsilon<1$. This puts us in the situation covered by Lemma 2.1,
with (in that lemma)

$q=2+\varepsilon,q^{\prime}=\frac{2+\varepsilon}{1+\varepsilon},\lambda=p^{-\frac{1}{q}},c=(\frac{1}{10})^{1_{p\leqslant
p_{0}}},x_{i}:=\alpha_{i}^{1/q^{\prime}},y_{j}:=\beta_{j}^{1/q^{\prime}}.$

The hypotheses of the lemma are satisfied, since $p^{-\frac{1}{q}}\leqslant
2^{-\frac{1}{3}}\leqslant 4/5$. The lemma implies, first of all, that
$c>\frac{1}{10}$; this immediately tells us that $p>p_{0}$. We conclude that
there is some $k$ such that

$\sum_{|i-k|+|j-k|\geqslant 2}\mu(i,j)\ll\lambda^{q+\varepsilon}\ll
p^{-1-\varepsilon/3}.$ (3.5)

This is precisely what is needed in Proposition 3.1, taking

$\Omega_{p}=\bigcup_{|i-k|+|j-k|\leqslant 1}(\Omega\cap(A_{i}\times B_{j})).$

##  4\. Finishing the argument

In this section we complete the proof of Theorem 1.1 by establishing
Proposition 1.3. That is, our task is as follows. Suppose that
$A\subset[X,2X]$, $B\subset[Y,2Y]$, that $\Omega\ \subset A\times B$ has size
$\frac{\delta}{2}|A||B|$, and that $\gcd(a,b)\geqslant D$ whenever
$(a,b)\in\Omega$. Suppose that there is some positive integer $N$ such that

$|v_{p}(a/N)|+|v_{p}(b/N)|\leqslant 1$ (4.1)

for all primes $p$ and for all $(a,b)\in\Omega$. We are to show that, under
these assumptions, we have the bound

$|A||B|\leqslant\frac{1000}{\delta^{2}}\frac{XY}{D^{2}}.$ (4.2)

Let us begin the proof. In the course of the argument it will be convenient to
use a little of the language of graph theory. Thus if $a\in A$ then we write
$\deg(a):=\\#\\{b\in B:(a,b)\in\Omega\\}$, and analogously for $b\in B$. Write
$A^{\prime}:=\\{a\in A:\deg(a)>0\\}$ and $B^{\prime}:=\\{b\in B:\deg(b)>0\\}$.

If $a\in A^{\prime}$ then, by (4.1), $v_{p}(a/N)\in\\{-1,0,1\\}$ for all
primes $p$. We define the _defect_ $a_{*}$ to be the product of all primes for
which $v_{p}(a/N)\neq 0$. Now we make the crucial observation that if
$(a,b)\in\Omega$ then

$a_{*}b_{*}=\frac{ab}{\gcd(a,b)^{2}}.$ (4.3)

To prove this, we take $p$-adic valuations. It is easily seen that

$v_{p}(ab/\gcd(a,b)^{2})=|v_{p}(a)-v_{p}(b)|=|v_{p}(a/N)-v_{p}(b/N)|,$

so we need only prove that

$v_{p}(a_{*})+v_{p}(b_{*})=|v_{p}(a/N)-v_{p}(b/N)|$ (4.4)

whenever $(a,b)\in\Omega$. This follows immediately from (4.1), noting that
$v_{p}(a_{*})=1$ if $v_{p}(a/N)=\pm 1$ and $v_{p}(a_{*})=0$ otherwise, and
similarly for $v_{p}(b_{*})$.

As a consequence of (4.3) and our assumptions, we see that

$a_{*}b_{*}\leqslant\frac{4XY}{D^{2}}$ (4.5)

whenever $(a,b)\in\Omega$. This would allow us to conclude very quickly, were
it not for the fact that the map $a\mapsto a_{*}$ need not be injective (see
Section 5 for some further remarks on this point). Fortunately, we have the
following substitute for injectivity.

######  Lemma 4.1.

Let $T\in\mathbb{R}_{>0}$. The number of $a\in A^{\prime}$ for which
$a_{*}\leqslant T$ is at most $2T$. Similarly, the number of $b\in B^{\prime}$
for which $b_{*}\leqslant T$ is at most $2T$.

###### Proof.

If $a\in A^{\prime}$, write $a_{+}$ for the product of all primes with
$v_{p}(a/N)=1$, and $a_{-}$ for the product of all primes with
$v_{p}(a/N)=-1$. Thus

$a_{*}=a_{+}a_{-}.$ (4.6)

Since $v_{p}(a/N)\in\\{-1,0,1\\}$, we have

$\frac{a_{+}}{a_{-}}=\frac{a}{N}.$ (4.7)

Since $A\subset[X,2X]$, it follows by multiplying (4.6) and (4.7) that if
$a_{*}\leqslant T$ then

$a_{+}\leqslant(\frac{aT}{N})^{1/2}\leqslant(\frac{2XT}{N})^{1/2}.$ (4.8)

Similarly, dividing (4.6) by (4.7), we see that if $a_{*}\leqslant T$ then

$a_{-}\leqslant(\frac{NT}{a})^{1/2}\leqslant(\frac{NT}{X})^{1/2}.$ (4.9)

It follows from (4.8), (4.9) that the number of choices for the pair
$(a_{+},a_{-})$ is at most $2T$. However, if we know $a_{+},a_{-}$ and $N$
then we can recover $a$ uniquely, so the map $a\mapsto(a_{+},a_{-})$ is
injective. The proof for $B^{\prime}$ is the same.∎

Now we finish the argument. By a standard averaging argument there is a set
$\tilde{A}\subset A$ with $|\tilde{A}|\geqslant\delta|A|/4$ such that
$\deg(a)\geqslant\delta|B|/4$ for all $a\in\tilde{A}$. Clearly
$\tilde{A}\subset A^{\prime}$, so by Lemma 4.1 there is some $a\in\tilde{A}$
such that

$a_{*}\geqslant\delta|A|/8.$ (4.10)

Set $\tilde{B}:=\\{b\in B:(a,b)\in\Omega\\}$. Thus
$|\tilde{B}|\geqslant\delta|B|/4$. Clearly $\tilde{B}\subset B^{\prime}$, so
by Lemma 4.1 there is some $b\in\tilde{B}$ such that

$b_{*}\geqslant\delta|B|/8.$ (4.11)

By construction we have $(a,b)\in\Omega$, so we have the upper bound (4.5).

Comparing (4.5), (4.10), (4.11) immediately yields (4.2).

##  5\. Further results and remarks

Suppose that $A$ and $B$ are finite sets of square-free positive integers. In
this instance, we may assume that the positive integer $N$ that satisfies
(4.1) is also square-free, and thus the map $a\mapsto a_{*}$ is injective,
since knowing $N$ and $a_{*}$ determines $a$. This enables us to circumvent
Lemma 4.1, and prove the following theorem.

######  Theorem 5.1.

Let $Q\in[1,\infty)$ and $\delta\in(0,1]$. Suppose that $A,B$ are finite sets
of square-free positive integers with the following property: for at least
$\delta|A||B|$ pairs $(a,b)\in A\times B$, $ab/\gcd(a,b)^{2}\leqslant Q$. Then
we have the bound

$|A||B|\leqslant(1000)^{1+\\#\mathscr{P}_{\operatorname{small}}(A\cup
B)}\delta^{-2-\varepsilon}\frac{Q}{4}.$

Of course this result implies the cruder bound
$|A||B|\ll\delta^{-2-\varepsilon}Q$. It also implies Theorem 1.1, upon taking
$Q=4XY/D^{2}$.

###### Proof.

The proof of Proposition 1.2 holds mutatis mutandis. Analysing the minimal
counterexample as before, we conclude that $a_{*}b_{*}\leqslant Q$ (by analogy
with (4.5)). Using graph theoretic language as before, there exists a set
$\tilde{A}\subset A$ with $|\tilde{A}|\geqslant\delta|A|/4$ such that
$\deg(a)\geqslant\delta|B|/4$ for all $a\in\tilde{A}$. Since $a\mapsto a_{*}$
is injective, there is some $a\in\tilde{A}$ for which
$a_{*}\geqslant\delta|A|/4$. Letting $\tilde{B}:=\\{b\in B:(a,b)\in\Omega\\}$,
we have $|\tilde{B}|\geqslant\delta|B|/4$ and for all $b\in\tilde{B}$ we have
$b_{*}\leqslant Q/a_{*}$. Therefore, since $b\mapsto b_{*}$ is injective, we
have

$\frac{\delta|B|}{4}\leqslant|\tilde{B}|\leqslant\max_{b_{*}:b\in\tilde{B}}b^{*}\leqslant\frac{Q}{a_{*}}\leqslant\frac{4Q}{\delta|A|}.$

This rearranges to $|A||B|\leqslant 16\delta^{-2}Q$, which shows that the
minimal counterexample is not in fact a counterexample, thus settling the
theorem. ∎

One might wonder whether the bound $|A||B|\ll\delta^{-2-\varepsilon}Q$ holds
for general finite sets of integers $A$ and $B$ (not just for square-frees).
However there is a counterexample to this assertion, even with $\delta=1$,
given by

$A=B=\Big{\\{}\Big{(}\prod\limits_{p\leqslant
X}p\Big{)}\frac{m}{n}:mn\leqslant
X,\,\mu^{2}(m)=\mu^{2}(n)=1,\,\gcd(m,n)=1\Big{\\}}.$

One may establish that for all $(a_{1},a_{2})\in A\times A$ one has the bound
$a_{1}a_{2}/\gcd(a_{1},a_{2})^{2}\leqslant X^{2}$. Yet $|A|\gg X\log X$.

By this, one notes that the use of dyadic ranges in the proof of Lemma 4.1 was
critical.

## References

  * [1] D. Koukoulopoulos and J. Maynard, _On the Duffin-Schaeffer Conjecture,_ Annals of Mathematics, 192(1), pp. 251-307.