# A new method to determine $H_{0}$ from cosmological energy-density
measurements

Alex Krolewski<EMAIL_ADDRESS>Will J. Percival  Waterloo Centre
for Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada
Department of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L
3G1, Canada  Perimeter Institute for Theoretical Physics, 31 Caroline St.
North, Waterloo, ON NL2 2Y5, Canada  Alex Woodfinden  Waterloo Centre for
Astrophysics, University of Waterloo, Waterloo, ON N2L 3G1, Canada  Department
of Physics and Astronomy, University of Waterloo, Waterloo, ON N2L 3G1, Canada

###### Abstract

We introduce a new method for measuring the Hubble parameter from low-redshift
large-scale observations that is independent of the comoving sound horizon.
The method uses the baryon-to-photon ratio determined by the primordial
deuterium abundance, together with Big Bang Nucleosynthesis (BBN) calculations
and the present-day CMB temperature to determine the physical baryon density
$\Omega_{b}h^{2}$. The baryon fraction $\Omega_{b}/\Omega_{m}$ is measured
using the relative amplitude of the baryonic signature in galaxy clustering
measured by the Baryon Oscillation Spectroscopic Survey, scaling the physical
baryon density to the physical matter density. The physical density
$\Omega_{m}h^{2}$ is then compared with the geometrical density $\Omega_{m}$
from Alcock-Paczynski measurements from Baryon Acoustic Oscillations (BAO) and
voids, to give $H_{0}$. Including type Ia supernovae and uncalibrated BAO, we
measure $H_{0}=67.1^{+6.3}_{-5.3}$ km s-1 Mpc-1. We find similar results when
varying analysis choices, such as measuring the baryon signature from the
reconstructed correlation function, or excluding supernovae or voids. This
measurement is currently consistent with both the distance-ladder and CMB
$H_{0}$ determinations, but near-future large-scale structure surveys will
obtain 3–4$\times$ tighter constraints.

††preprint: XXX

##  I Introduction

The standard model in cosmology, flat $\Lambda$CDM, has been the benchmark for
the last 25 years. Recently, however, tensions have started to appear between
the parameters of the model as measured from different data. The strongest of
these tensions is that the $H_{0}$ measurements from Cepheid variables [1]
disagree with those from the CMB [2] or Baryon Acoustic Oscillation (BAO) data
[3] analysed together with Big Bang nucleosynthesis (BBN) measurements. It is
imperative to understand tensions such as these in case they require
extensions to the standard flat $\Lambda$CDM model.

The Hubble constant is an absolute quantity with units of inverse time and
therefore needs to be constrained by an absolute measurement, which for
distance ladder measurements typically comes from geometrical measurements of
the distance to the LMC (e.g. [4]). For the CMB or BAO measurements, this
comes from the CMB temperature. Relative measurements of Cepheid brightness as
a function of redshift, or of angular anisotropies in the CMB, then allow us
to convert these absolute measurements to a measurement of $H_{0}$. The CMB
temperature is extremely well-measured, $T_{\gamma}=2.7255\pm 0.00057$ [5],
and the shift required to solve the Hubble tension is too large to be viable
(e.g. [6])

The most accurate way of measuring $H_{0}$ from CMB and large scale structure
is by using the sound horizon as a standard ruler. The comoving sound horizon
at the baryon drag epoch, $r_{d}$, is extremely well-determined in the
standard cosmological model, with the absolute baseline set by the CMB
temperature. But unlike the distance-ladder measurements, this requires an
assumption on the cosmological model. Therefore, a promising way to raise
$H_{0}$ from CMB and LSS is by modifying the cosmological model around
recombination, decreasing the sound horizon and increasing $H_{0}$ [7, 8, 9].
One model that accomplishes this is Early Dark Energy (EDE), which posits the
existence of an additional scalar field in the early Universe, leading to an
increase in the expansion rate around the baryon drag epoch [10, 11].

We present a new method to measure $H_{0}$ comparing low redshift physical and
geometrical matter densities without having to model the sound horizon. This
combines $\Omega_{b}h^{2}$ from the CMB temperature and light element
abundances together with the baryon fraction $\Omega_{b}/\Omega_{m}$ from the
amplitude of the baryon signal in the power spectrum. As discussed in [12],
this is a robust measurement that directly depends on the relative densities
of baryons and cold dark matter. Finally, adding a geometrical measurement of
the matter density $\Omega_{m}$ from the Alcock-Paczynski (AP) effect,
uncalibrated standard rulers, and uncalibrated standard candles is sufficient
to form a new measurement of $H_{0}$. To improve the information coming from
the AP effect, measurements from cosmological voids can be included [13, 14].
This signal has previously been included in other fits to galaxy clustering
data [15, 16], and there have been previous attempts to remove the dependence
on the sound horizon, marginalising over $r_{s}$ [17, 18, 19, 20, 21].
However, these $H_{0}$ measurements use the matter power spectrum turnover,
which occurs at the matter-radiation equality scale; since EDE alters the
shape of the power spectrum, these measurements are potentially unable to rule
out EDE despite the sound horizon freedom [22, 23]). We provide a new method
that extracts the amplitude of the baryon signal, sacrificing signal-to-noise
for simplicity of the underlying physics.

We apply this new method to data from the Sloan Digital Sky Survey [24] Baryon
Oscillation Spectroscopic Survey (BOSS) sample [25]. We include the
measurements of the baryon fraction made in our companion paper [12] and AP
measurements from voids from [14]. The method is discussed in more detail in
the next section, and then we consider our results, ending with a discussion
and a look forward to the improved large-scale structure measurements due from
the Dark Energy Spectroscopic Instrument (DESI [26]) and Euclid satellite
[27].

##  II The Hubble constant from galaxy surveys using energy densities

The relative density of baryons in the galaxy clustering signal is usually
marginalised in any BAO measurement, as the cosmological constraints it
provides are weak compared with those from the CMB. However, the amplitude can
be separated from other measurements and depends on the properties of the
Universe in an exceptionally clean way. To see this, consider the combined
baryon+CDM transfer function

$T(k)=\frac{\Omega_{b}}{\Omega_{bc}}T_{b}(k)+\frac{\Omega_{c}}{\Omega_{bc}}T_{c}(k)\,.$
(1)

where $\Omega_{bc}$ is the total matter density excluding massive neutrinos.
The physics of the relative weighting of the baryon ($T_{b}$) and CDM
($T_{c}$) transfer functions is simply that the growth of structure responds
to both components proportional to the energy-density in each. The baryon
transfer function can be approximated

$T_{b}(k)=\alpha_{b}\frac{\sin(kr_{d})}{kr_{d}}\cal{D}(k)\,,$ (2)

in the small-scale limit $ks>>1$ [28]. $\cal{D}(k)$ represents the effects of
Silk damping, and $\alpha_{b}$ is a suppression factor arising from the
adiabatic decrease in the sound speed, the growth suppression between matter-
radiation equality and the drag epoch before velocities decay due to
cosmological expansion (see equations 14 & 15 in [28], also [29]). For
cosmological models close to the Planck best-fit, $\alpha_{b}$ is very close
to constant, independent of $\Omega_{m}h^{2}$ (we investigate this further in
[12]). Thus, $T_{b}$ has a very different form than $T_{c}$, such that the
fraction of the clustering signal seen in galaxy surveys resulting from
$T_{b}$ can be measured, and provides a relatively clean measurement of
$\Omega_{b}/\Omega_{m}$, which crucially does not depend on $r_{d}$. We
present extensive tests on $N$-body mocks, and noiseless theory vectors in
both $\Lambda$CDM and EDE cosmologies in [12].

This leads to a new method to measure the Hubble parameter, which can be
understood as follows: we start with the CMB temperature, which provides a
strong (absolute) constraint on the present-day physical photon density
$\epsilon_{\gamma,0}\propto\Omega_{\gamma,0}h^{2}$. Observations of primordial
deuterium abundances together with BBN calculations tell us the photon-to-
baryon ratio, while the amplitude of the baryonic signal constrains the
baryon-to-matter density. The AP measurements from BAO and voids constrain the
matter density relative to the critical density
$\Omega_{m,0}\equiv\epsilon_{m,0}/\epsilon_{c,0}$ from geometrical
considerations at low redshift.

Together, the four observations constrain the Hubble parameter. The method
relies on having a Universe in which the CMB is a blackbody, BBN acts as
expected such that the relative abundances can be used to constrain the
baryon-to-radiation ratio, and the structure in the matter field depends on
the baryon and CDM components in proportion to their relative densities. We
also assume that the BAO and voids are isotropic, the BAO (SNe) are standard
rulers (standardizable candles), the Universe is flat, and the dark energy is
a cosmological constant, to convert the AP measurements to measurements of
$\Omega_{m}$. We consider the amount of curvature and dynamical dark energy
required to affect this measurement later. The BAO amplitude constrains the
ratio of the baryon and CDM densities (since galaxy clustering responds to the
power spectrum of matter perturbations, excluding neutrinos, rather than all
matter [30]) whereas the geometric measurements of $\Omega_{m}$ are also
sensitive to the small density of non-relativistic massive neutrinos
$\Omega_{\nu}$. However, the variation in $\sum m_{\nu}$ required to affect
this measurement is large compared to current uncertainties [2, 21, 3, 31, 32,
33].

The method is summarised in Eq. 3

$\frac{3c^{2}H_{0}^{2}}{8\pi
G}=\epsilon_{c}=\epsilon_{\gamma,0}\times\frac{\epsilon_{b,0}}{\epsilon_{\gamma,0}}\times\frac{\epsilon_{m,0}}{\epsilon_{b,0}}\times\frac{1}{\Omega_{m,0}}\,.$
(3)

The first term on the right hand side of this expression comes from the CMB
temperature, the second from BBN, the third from the amplitude of the baryon
signal and the fourth from the AP effect.

##  III Measurements

Between 2009 and 2014, the BOSS survey measured redshifts for 1.4 million
galaxies across $\sim$10,000 deg2 and publicly released large-scale structure
catalogues, $N$-body mocks, and clustering measurements at $0.2<z<0.75$ [34,
35]. As detailed in [12], we added a parameter quantifying the amplitude of
the baryon signature to two kinds of existing models: template-based models of
the power spectrum typically used to make BAO measurements [36], fitted to
data both before and after BAO reconstruction [37]; and the effective field
theory of large scale structure models of [38, 15]. These methods are quite
complementary: they use different galaxy statistics (correlation function vs.
power spectrum); make different assumptions about the broad-band (i.e. non-
BAO) power spectrum (fully modelling it within $\Lambda$CDM vs. using a fixed-
cosmology template with empirical polynomial terms to account for broadband
variations); and make different assumptions about the galaxy bias parameters.
Nevertheless, we find that they give very consistent constraints on the baryon
fraction, $f_{b}=0.173\pm 0.029$ ($0.153\pm 0.029$) from the post-
reconstruction (pre-reconstruction) correlation function fit with the template
model, and $f_{b}=0.154\pm 0.023$ from the EFT fits. Based on fits to $N$-body
simulations and noiseless theory vectors in EDE cosmologies, we place a
systematic error on $f_{b}$ of 0.013 [12].

We use the full-shape fits as our baseline results since the model is more
flexible and applicable in a wider range of situations. This choice does in
principle reduce the expected precision of the BAO amplitude from the post-
reconstruction correlation function, in which the BAO amplitude is sharpened.
However, in practice, the full-shape constraints on data from the
unreconstructed power spectrum are actually tighter (though they are similar
to the post-reconstruction constraints on mocks), and the post-reconstruction
template fits only provide $\sim 10\%$ tighter precision on the baryon
fraction than the pre-reconstruction template fits. When quoting the pre-
reconstruction results, we always combine with the best available standard
ruler measurements using reconstruction.

We use uncalibrated BAO measurements made by the BOSS team on the combination
of the post-reconstruction correlation function and power spectrum [39, 35,
40]. We use their best-fit combined $\alpha_{\parallel}$ and $\alpha_{\perp}$
measurements (see Eqs. 21–23 in [35]), and add the sound horizon $r_{d}$ as an
extra parameter which is marginalised over in our parameter fits, so we only
include relative measurements. A wide redshift range significantly increases
the constraining power of uncalibrated BAO on $\Omega_{m}$ [41, 42, 43, 44],
and hence we also use isotropic BAO measurements from the SDSS Main Galaxy
Survey (MGS) at $z=0.15$ [45]; $D_{M}$ and $D_{H}$ measurements from eBOSS
quasars at $z=1.48$ [3, 46, 47]; and $D_{M}$ and $D_{H}$ at $z=2.334$ from the
eBOSS Ly$\alpha$ forest autocorrelation and cross-correlation with quasars
[48].

We use the void isotropy measurements presented in [14] for MGS at
$0.07<z<0.2$ and BOSS at $0.2<z<0.6$. We do not use their highest-redshift
measurement, which combines BOSS and eBOSS at $0.6<z<1.0$, because we cannot
measure the covariance between this measurement at the galaxy power spectrum
at $0.5<z<0.75$, due to the lack of simulations with suitable geometry.

In our baseline analysis, we add supernova constraints using the Pantheon+
measurement of $\Omega_{m}=0.334\pm 0.018$ as an external prior (or
representing the supernova likelihood as a 2D or 3D gridded prior on
($\Omega_{m}$, $\Omega_{k}$) or ($\Omega_{m}$, $w_{0}$, $w_{a}$) in models
with curvature or varying dark energy equation of state) [49]. For the
extended models, we use the publicly available Pantheon+ chains. 111In detail,
we measure the $\Omega_{m}$-$\Omega_{k}$ or $\Omega_{m}$-$w_{0}$-$w_{a}$
priors from the “Pantheon+_SH0ES” chains, marginalizing over $H_{0}$. The
“Pantheon+_only” chains do not give consistent constraints on the dark energy
equation of state parameters. More recent measurements from Union3
($\Omega_{m}=0.36\pm 0.03$) [51] or DES ($\Omega_{m}=0.352\pm 0.017$) [52] are
modestly different from Pantheon+. However, replacing the Pantheon+ prior with
a UNION or DES prior does not appreciably change our $H_{0}$ constraints.

We use the recent updated BBN recommendations of [53],
$\Omega_{b}h^{2}=0.02218\pm 0.0055$ (assuming the standard model value of
$N_{\textrm{eff}}=3.046$), which rely on measurements as compiled by the
Particle Data Group [54] and nuclear rates from the PRyMordial code [55],
which adds marginalization over nuclear reaction rate uncertainties. The
observational measurements are challenging because the inferred abundances
must be primordial, i.e. not contaminated by stellar production or destruction
of helium and deuterium. The PDG uses the weighted mean of the 11 most precise
deuterium abundances from quasar absorption systems [56, 57, 58, 59, 60, 61,
62] and the 6 best helium abundance measurements from extragalactic HII
regions [63, 64, 65, 66, 67, 68]. There are modest differences when using
experimental or theoretical nuclear reaction rates, but the marginalization in
PRyMordial comfortably encompasses both experimental and ab-initio results.
Additional relativistic species $\Delta N_{\textrm{eff}}$ do not appreciably
change the central value or uncertainties. The systematic uncertainties on the
BBN measurement of $\Omega_{b}h^{2}$ are certainly quite formidable,
particularly on the astrophysical side (perhaps leading to outlying $Y_{p}$
measurements like [69]). However, they are completely subdominant to the
uncertainties in $\Omega_{m}$ and $\Omega_{b}/\Omega_{m}$ in our $H_{0}$
measurement.

The voids and galaxies overlap in redshift and thus have a nonzero covariance,
which must be taken into account. We measure the covariance using 1000 Multi-
Dark Patchy mocks [70]. For the template-based fits, we measure $\gamma_{B}$
using our code for all 1000 pre or post-reconstruction mock correlation
functions, and measure the covariance with the BOSS team’s 1000 fits combining
post-reconstruction correlation function and power spectrum. We then re-scale
the resulting covariance matrices to match the measured error on $\gamma_{B}$
on data (which is slightly larger than the spread of the mocks, as described
in [12]). We also measure the covariance with the void measurements using the
void-finding pipeline of [71] applied to the 1000 Patchy mocks. The template-
based fits thus consist of two stages, one in which $\gamma_{B}$,
$\alpha_{\parallel}$ and $\alpha_{\perp}$ are measured from the correlation
function on both data and the Patchy mocks, and another in which $H_{0}$ and
$\Omega_{m}$ are inferred from these measurements and the mock-based
covariance.

Data Combination | $f_{b}$ | $\Omega_{m}$ | $H_{0}$
---|---|---|---
Full Shape $P(k)$ | $0.153\pm 0.026$ | $0.313\pm 0.012$ | $67.1^{+6.3}_{-5.3}$
Post-Reconstruction Template $\xi(r)$ | $0.170\pm 0.030$ | $0.312\pm 0.012$ | $64.8^{+6.6}_{-5.7}$
Pre-Reconstruction Template $\xi(r)$ | $0.153\pm 0.032$ | $0.312\pm 0.012$ | $68.3^{+8.4}_{-6.7}$
Full Shape $P(k)$ \+ UNION SNe | $0.153\pm 0.026$ | $0.310\pm 0.014$ | $67.6^{+6.1}_{-5.6}$
Full Shape $P(k)$ \+ DES SNe | $0.153\pm 0.026$ | $0.323\pm 0.012$ | $66.1^{+6.1}_{-5.4}$
Full Shape $P(k)$ \+ no SNe | $0.153\pm 0.026$ | $0.296\pm 0.015$ | $69.1^{+6.1}_{-5.6}$
Full Shape $P(k)$ \+ no voids | $0.154\pm 0.026$ | $0.279\pm 0.015$ | $70.9^{+6.6}_{-5.8}$
Full Shape $P(k)$ \+ BOSS only | $0.150\pm 0.026$ | $0.385\pm 0.035$ | $61.0^{+6.1}_{-5.5}$
Post-Reconstruction Template $\xi(r)$ \+ $w_{0}w_{a}$ | $0.175\pm 0.030$ | $0.260^{+0.039}_{-0.054}$ | $71.0^{+11.0}_{-7.7}$
Post-Reconstruction Template $\xi(r)$ \+ $\Omega_{\mathrm{k}}$ | $0.175\pm 0.030$ | $0.273^{+0.021}_{-0.018}$ | $68.4^{+7.3}_{-5.8}$
Table 1: $H_{0}$ constraints in km s-1 Mpc-1 from the signature of baryons,
uncalibrated BAO positions and supernovae, and BBN. The fiducial $H_{0}$
constraint is the top (shaded) row. The errors on $f_{b}$ and $H_{0}$ include
the estimated systematic error of 0.013 in $f_{b}$ or 2.9 km s-1 Mpc-1in
$H_{0}$, added in quadrature. The next rows show various changes to the baryon
amplitude measurement, the geometrical datasets, or the late-time expansion
history, as explained in the text.

In the full-shape fits, we measure the covariance between the power spectrum
multipoles $P_{\ell}(k)$ and the AP parameters from both galaxies and voids.
An additional complication is that the power spectra are split between the
North Galactic Cap and the South Galactic Cap (owing to the slightly different
galaxy selection and thus bias parameters in the two regions), but the AP
parameters are measured on the combined sample. However, all of these
quantities can be measured on mocks and the covariance directly computed,
similar to [72].

We use the MCMC samplers cobaya [73, 74] for the template fits and MontePython
[75, 76] for the full-shape fits. Due to the dramatically faster likelihood
evaluation times, we run the template fits to Gelman-Rubin R-1$=0.01$ [77] and
the full-shape fits to Gelman-Rubin R-1$=0.05$, but we verify in both cases
that our results are consistent to stopping at slightly higher values of R-1 =
0.05 or 0.1. We analyze the resulting chains with GetDist [78].

##  IV Results

Figure 1: Constraints on $H_{0}$ (in km s-1 Mpc-1), $\Omega_{m}$ and
$\Omega_{b}h^{2}$ from the combination of Big Bang Nucleosynthesis;
uncalibrated standard rulers, voids, and Type Ia supernovae; and the amplitude
of the baryonic signature in BOSS galaxy clustering. Three variations are
shown corresponding to different methods for measuring the baryon fraction:
using the EFT of Large Scale Structure on the pre-reconstruction power
spectrum (blue); a damped exponential BAO model (with fixed power spectrum
template) applied to the pre-reconstruction correlation function (gray); and
the same model applied to the post-reconstruction correlation function (red).
These contours show statistical errors only, without contribution from
systematic errors.

In our baseline results, combining the amplitude of the baryonic signature,
BBN, uncalibrated standard rulers, and type Ia supernovae, we find
$H_{0}=67.1^{+6.3}_{-5.3}$, including the systematic error budget on $f_{b}$.
The statistical errors alone give $H_{0}=67.1^{+5.6}_{-4.4}$ km s-1 Mpc-1 with
an additional systematic error of $\pm 2.9$ km s-1 Mpc-1 (Fig. 1 and Table 1).
The systematic error is derived from the estimated systematic error on
$f_{b}$, scaling to $H_{0}$ since the $f_{b}$ errors dominate the errors on
$\Omega_{m}$ or $\Omega_{b}h^{2}$. Our result is robust to changing the method
used to measure the baryon fraction: we find a very similar result using the
template fit for the pre-reconstruction correlation function, though with a
slightly larger error ($68.3^{+8.4}_{-6.7}$ km s-1 Mpc-1); and a $0.5\sigma$
shift towards lower $H_{0}$ after reconstruction ($64.8^{+6.6}_{-5.7}$ km s-1
Mpc-1); in all cases systematic errors are added in quadrature. The difference
between the full-shape and post-reconstruction results is consistent with the
differences seen in $N$-body mocks [12].

Indeed, $H_{0}$ is more sensitive to the geometric dataset used to constrain
$\Omega_{m}$. Both the supernovae and voids prefer high $\Omega_{m}$, whereas
BAO prefers lower $\Omega_{m}$ (in this case, driven by the comparison of BOSS
galaxy and eBOSS Ly$\alpha$ distance measurements). While the supernovae
measurements are in $\sim$1$\sigma$ tension with each other, our result does
not change much when using UNION (DES), changing to $67.6^{+6.1}_{-5.6}$ km
s-1 Mpc-1 ($66.1^{+6.1}_{-5.4}$ km s-1 Mpc-1). Removing either supernovae or
voids lowers $\Omega_{m}$ and thus raises $H_{0}$ to $69.1^{+6.1}_{-5.6}$ km
s-1 Mpc-1 (no SNe) or $70.9^{+6.6}_{-5.8}$ km s-1 Mpc-1 (no voids). Finally,
when using BOSS data only (including voids), $\Omega_{m}$ increases by nearly
2$\sigma$, leading to a very low $H_{0}$ value of $61.0^{+6.1}_{-5.5}$ km s-1
Mpc-1. The errors are dominated by the errors on $f_{b}$ even when the
$\Omega_{m}$ error triples when using BOSS data alone.

Finally, this measurement is sensitive to different models for the late-time
expansion history. Adding curvature lowers $\Omega_{m}$ and weakens its
constraint, but doesn’t change $H_{0}$ significantly (to $68.4^{+7.3}_{-5.8}$
km s-1 Mpc-1), a 20% increase in the error. A varying dark energy equation of
state has a larger impact on $H_{0}$, significantly weakening the constraint
in a $w_{0}$ \+ $w_{a}$ cosmology to $71.0^{+11.0}_{-7.7}$ km s-1 Mpc-1. This
is largely due to the degeneracy between the poorly constrained $w_{a}$ and
$H_{0}$. While we do not explore the impact of varying the neutrino mass in
this work, a minimum $\Sigma m_{\nu}$=0.06 eV translates to
$\Omega_{\nu}=0.0013$. Using the 90% upper limit on $\Sigma m_{\nu}$ from
direct detection, 0.8 eV [33], $\Omega_{\nu}$ is still a considerably smaller
fraction of $\Omega_{m}$ (5%) than our $f_{b}$ fractional error (15%) and thus
will not substantially broaden the $H_{0}$ constraint.

##  V Discussion

Our $H_{0}$ measurement is consistent both with SH0ES ($73.04\pm 1.04$ km s-1
Mpc-1) [1] and Planck ($67.26\pm 0.49$ km s-1 Mpc-1) [31] (see also [32] and
[2]). In Fig. 2, we compare to other sound horizon-free $H_{0}$ measurements
from large-scale structure. [19], [22] and [79] infer cosmological parameters
from the shape of the matter power spectrum, with explicit marginalization
over the sound horizon. While they obtain tighter constraints on $H_{0}$,
these measurements are model-dependent; nonzero Early Dark Energy leads to a
different power spectrum shape than in $\Lambda$CDM, shifting the $H_{0}$
constraint [22, 23]. In contrast, our measurement has larger errors, but is
less model-dependent, as demonstrated in our companion paper [12]. For
comparison, we also show results from an alternative local calibration of the
distance ladder using the Tip of the Red Giant Branch rather than Cepheids.
Refs. [80, 81] achieve $H_{0}$ precision of $\sim$2 km s-1 Mpc-1, although
with a fairly large discrepancy between the two results perhaps suggesting
larger systematic errors.

Figure 2: $H_{0}$ measurements (in km s-1 Mpc-1) from Cepheids [1] and Planck
CMB measurements assuming a flat $\Lambda$CDM cosmology [2], shown in the gray
bands. The points show various measurements with minimal additional
assumptions: those presented in this paper (red, at top, with fiducial
measurement starred); other sound horizon free $H_{0}$ measurements from
large-scale structure; and local $H_{0}$ measurements from the Tip of the Red
Giant Branch. Our constraints on $H_{0}$ are currently weak and compatible
with both the local and CMB/$\Lambda$CDM measurements; but with DESI and
Euclid, our method will have sufficient constraining power to differentiate
Planck and SH0ES at $\sim$5$\sigma$.

While the current constraining power of this measurement is insufficient to
resolve the Hubble tension, ongoing large-scale structure surveys will allow
for significantly higher precision. DESI [26, 82] and Euclid [83] will both
survey roughly ten times the effective volume of BOSS. We show in Fig. 2 the
forecasted constraining power of these surveys, scaling our fiducial
constraint by the square root of the effective volume (as demonstrated in the
companion paper [12]) and centering at 70 km s-1 Mpc-1. The systematic errors
are dominated by prior effects from the large number of poorly constrained EFT
parameters, and hence are expected to improve in tandem with the statistical
errors.222In [12], we find $\sim 0.5\sigma$ biases on $H_{0}$ using an EDE
cosmology as truth and scaling the covariance by a factor of 10. This is
similar to the 0.5$\sigma$ biases that we find with an unscaled covariance,
but with a smaller $\sigma_{H_{0}}$ and thus smaller offset in $H_{0}$. If the
systematics remain under control, the combination of DESI and Euclid could
reach a precision of 1.2 km s-1 Mpc-1, sufficient to differentiate SH0ES and
Planck at $\sim$5$\sigma$. Moreover, sound horizon free measurements
constraining $H_{0}$ from the turnover of the matter power spectrum will
continue to improve [20]. The complementarity of these two large-scale
structure measurements could yield intriguing hints about potential solutions
to the Hubble tension, or put substantial pressure on currently-allowed models
that can resolve the tension by shifting the sound horizon.

###### Acknowledgements.

We thank Ashley Ross for useful conversations and for helping to dig out old
data. AK was supported as a CITA National Fellow by the Natural Sciences and
Engineering Research Council of Canada (NSERC), funding reference
#DIS-2022-568580. WP acknowledges support from the Natural Sciences and
Engineering Research Council of Canada (NSERC), [funding reference number
RGPIN-2019-03908] and from the Canadian Space Agency. Research at Perimeter
Institute is supported in part by the Government of Canada through the
Department of Innovation, Science and Economic Development Canada and by the
Province of Ontario through the Ministry of Colleges and Universities. This
research was enabled in part by support provided by Compute Ontario
(computeontario.ca) and the Digital Research Alliance of Canada
(alliancecan.ca).

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$H_{0}$ from cosmological energy-density measurements

H0 from cosmological energy-density measurements


Collective hydrodynamic transport of magnetic microrollers†

Gaspard Junot,a Andrejs Cebers,b and Pietro Tiernoacd

Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XX
First published on the web Xth XXXXXXXXXX 200X

DOI: xxxxx

We investigate the collective transport properties of microscopic magnetic
rollers that propel close to a surface due to a circularly polarized, rotating
magnetic field. The applied field exerts a torque to the particles, which
induces a net rolling motion close to a surface. The collective dynamics of
the particles result from the balance between magnetic dipolar interactions
and hydrodynamic ones. We show that, when hydrodynamics dominate, i.e. for
high particle spinning, the collective mean velocity linearly increases with
the particle density. In this regime we analyse the clustering kinetics, and
find that hydrodynamic interactions between the anisotropic, elongated
particles, induce preferential cluster growth along a direction perpendicular
to the driving one, leading to dynamic clusters that easily break and reform
during propulsion.

##  1 Introduction

††footnotetext: † Electronic Supplementary Information (ESI) available: Three
.WMV videos showing the dynamics of the hematite microroller at different
frequencies. See DOI: xxxx/b000000x/††footnotetext: a Departament de Física de
la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain E-mail:
<EMAIL_ADDRESS>b MMML Lab, Department of Physics, University of
Latvia, Jelgavas-3, Riga, LV-1004, Latvia.††footnotetext: c Universitat de
Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona,
Barcelona, Spain††footnotetext: d Institut de Nanociència i Nanotecnologia,
Universitat de Barcelona, 08028, Barcelona, Spain.

Microscopic surface rotors driven by a time-dependent magnetic field,
represent a versatile and simple, yet non trivial, model system to investigate
emergent collective phenomena, 1, 2, 3, 4 similar to those observed in
ensemble of active and self-propelling particles. 5, 6 An external, time-
dependent field is used to apply a magnetic torque to the particles, and to
induce a net spinning close to a substrate. 7, 8, 9, 10, 11, 12, 13, 14, 15
Since these particles have size within the micrometer range and speed of the
order of few $\rm{\mu ms^{-1}}$ (similar to that of bacteria or cells), their
dynamics occur at effectively low Reynolds number ($Re$). Under these
conditions, inertia is negligible and Navier-Stokes equations become time-
reversible. 16 In this situation propulsion is possible only when the motion
is non-reciprocal, namely based on a sequence of body/shape changes that are
not identical when reversed in time. 17 Rotating helical flagella in bacteria,
18, 19 flexibility in the tail 20, 21 or chemical reactions, 22, 23 represent
some ways to circumvent this constraint. In contrast, surface rollers present
a relative simple mechanism of motion, based on the rotational/translational
coupling with a close substrate. 24 Moreover, magnetic dipolar forces between
the microrollers can be tuned by an external field, a feature that allows to
investigate the role of such forces in the organization process and their
interplay with other interactions such as hydrodynamic ones.

In this context, some of us recently investigate the dynamics of pair of
propelling microrollers composed of anisotropic hematite particles that are
driven close to a plane by an external rotating magnetic field. 25, 26 In such
system, hydrodynamic interactions (HIs) between the particles induce the
formation of stable bound states, where the particles align tip to tip, namely
with their relative position parallel to the long axis, even if dipolar forces
are repulsive in this configuration. However, the collective effects generated
by several interacting microrollers and how the produced flow field alters the
global mean velocity remain still unclear.

In this article we investigate the collective dynamics of interacting
microrollers, and show that the average translational speed linearly raises
with density. A similar behaviour was recently reported with different
magnetic particles rolling close to a surface 47, a fact that enforces its
generic nature. For our system, this feature cannot be explained by
considering only dipolar forces. Furthermore, this speed up effect can be
controlled by tuning the driving frequency which set the strength of the
hydrodynamic interactions. We rationalize this observation by calculating the
flow field generated by a set of rotating point particles. Further, we analyse
the clustering process, and show that HIs lead to the formation of dynamic
clusters which preferentially grow perpendicular to the propulsion direction.

##  2 Experimental details

As shown in Fig.1, our magnetic microrollers are monodisperse hematite
particles with a peanut-like shape. This shape results from the particular
fabrication process, more details can be found in a previous work. 25 From the
analysis of images obtained with scanning electron microscopy, as shown in the
inset in Fig.1(b), we find that the hematite particles are characterized by
two connected lobes with a long (short) axis equal to $\alpha=2.6\,{\rm\mu m}$
($\beta=1.2\,{\rm\mu m}$). The hematite particles present a small permanent
dipole moment oriented perpendicular to their long axis. This feature can be
explained by considering the magnetic structure of hematite, that crystallizes
in the corundum form, where Fe cations are aligned antiferromagnetically. 27,
28 The magnetic moment of the hematite ellipsoids was previously determined by
measuring the particle relaxation under a static field, 29 and is of the order
$m\simeq 9\cdot 10^{-16}\,\rm{Am^{2}}$.

After synthesis, the particles are dispersed in pure water (Milli-Q system,
Millipore), with sodium dodecyl sulfate (SDS) at a concentration $1.4\cdot
10^{-3}\,\rm{gmL^{-1}}$, and let sediment above a plastic petridish. The SDS
gives a steric layer to the particles which avoids irreversible particle
sticking due to attractive Van der Waals interaction. Further, we raise the pH
of the solution between $9$ and $9.5$ by adding Tetramethylammonium Hydroxide.
The particles remain confined close to the plate due to gravity, where they
float at a distance $h$ there.

The system dynamics are visualized using a light microscope (Eclipse Ni,
Nikon) equipped with a CCD camera (Scout scA640-74f, Basler). Further, we
apply the external magnetic fields using a set of custom-made magnetic coils
mounted directly on the microscope stage. The rotating field in the
$(\hat{\bm{x}},\hat{\bm{z}})$ plane is obtained by connecting two set of coils
to a wave generator (TTi-TGA1244, TTi) feeding a power amplifier (IMG STA-800,
stage line).

Fig. 1: (a) Schematic showing a hematite particle rolling close to a plane due
to an external rotating magnetic field $\bm{H}$. (b) Scanning electron
microscope image of one hematite particle with the direction of the magnetic
moment superimposed in red. (c) Sequence of microscope images showing the
transport of several magnetic particles subjected to a rotating field with
amplitude $H=4400\,\rm{Am^{-1}}$, and driving frequency $f=100$Hz. The
particle trajectories over the last second are superimposed to the second
image. The colour bar shows the mean particle velocity along the trajectory.
(d) Single particle velocity along the propulsion direction $v_{x}$ (circles)
and perpendicular to it $v_{y}$ (squares) versus driving frequency $f$. The
continuous black and blue lines are fits to the synchronous and asynchronous
regimes, respectively.

##  3 Microroller propulsion

To propel the hematite particles we use a rotating magnetic field circularly
polarized in the $(\hat{\bm{x}},\hat{\bm{z}})$ plane, $\bm{H}=H(\cos{(2\pi
ft)}\hat{\bm{x}}-\sin{(2\pi ft)}\hat{\bm{z}})$, being $H$ the amplitude and
$f$ the driving frequency, see Fig.1(a). As an example, in Fig.1(c) we show
two microscope snapshots of a collection of hematite rollers that are driven
toward right by a rotating field with frequency $f=100$Hz, reaching an average
translational speed $\bar{v}_{x}=12.0\pm 4\,\rm{\mu ms^{-1}}$. We indicate
with $v_{i}$, the single particle velocities in the plane
$(\hat{\bm{x}},\hat{\bm{y}})$, and with $\bar{v}_{i}$ the collective speed,
where $i=(x,y)$. As the magnetic moments in each particle are directed along
their short axis, the particles roll around their long axis. Since we are
interested in the collective particle transport, in this work we keep fix the
field amplitude to $H=4400\,\rm{Am^{-1}}$, and vary mainly the driving
frequency $f$ and the particle density $\phi=N/A$, being $N$ the number of
particles in the observation region $A$. Further, we note that above
$\phi>0.04\rm{\,\mu m^{-2}}$ the magnetic rollers start to overlap and propel
also in the third dimension. This situation complicates the particle tracking,
thus we limit our measurements to lower particle densities.

Before exploring the collective transport, we start by showing in Fig.1(d) how
the single particle speed changes with $f$. For a rotating field polarized in
the $(\hat{\bm{x}},\hat{\bm{z}})$ plane, the hematite particles display a
negligible transverse velocity ($v_{y}\sim 0$) while the translational motion
along the $\hat{\bm{x}}$-axis can be characterized by two dynamic regimes.
Below a critical frequency $f_{c}=108.4$Hz, the magnetic moment of the
particle follows the applied field. In this synchronous regime the particle
rotational spinning coincides with the driving one, and the rolling velocity
is $v_{x}=2\pi a\gamma f$ where $a=\beta/2$, and $\gamma$ is a prefactor that
takes into account the close proximity of the wall. The continuous blue line
in Fig.1(d) shows the corresponding fit to the experimental data in this
linear regime, where we get $\gamma=(3.19\pm 0.09)\cdot 10^{-2}$. In contrast,
for $f>f_{c}$ the motion becomes asynchronous, and this produces a reduction
of the translational speed which can be described as $v_{x}=2\pi
a\gamma(f-\sqrt{f^{2}-f_{c}^{2}})$, 30, 31 as shown by the continuous blue
line in Fig.1(d). We note that, given the large density of the hematite
particles ($\rho=5.3\rm{g\,cm^{-3}}$ in the bulk), some rollers may become
close to the surface especially at low spinning frequencies ($f<<f_{c}$ and
$f>>f_{c}$), when the hydrodynamic flow generated by the spinning is low. In
such cases, there can be some effects due to surface friction. We discuss the
role of the different forces acting on the single propelling rotor in the
Appendix of the manuscript.

##  4 Collective transport.

At large density, the hematite rollers start to interact due to their magnetic
moments and the generated hydrodynamic flow. In agreement with previous
observations for pair of particles, 25 we find that close to $f_{c}$ the HIs
dominate over magnetic one, and the particle preferentially assemble tip to
tip. In this configuration magnetic dipolar interactions are repulsive, since
the hematite moments rotate along parallel planes. This effect can be
quantified if one consider the magnetic dipolar interaction $U_{m}$ between a
pair of particles $(i,j)$, at positions $\bm{r}_{i}$ and $\bm{r}_{j}$ and with
moments $\bm{m}_{i}$ and $\bm{m}_{j}$. Assuming point like particles, this
interaction reads

$U_{m}=\frac{\mu_{w}}{4\pi}\left(\frac{{\bm{m}}_{i}{\bm{m}}_{j}}{r_{ij}^{3}}-\frac{3({\bm{m}}_{i}\cdot{\bm{r}}_{ij})({\bm{m}}_{j}\cdot{\bm{r}}_{ij})}{r_{ij}^{5}}\right)$
(1)

were $\mu_{w}\sim\mu_{0}=4\pi\cdot 10^{-7}\,\rm{H\cdot m}$ is the magnetic
susceptibility of the medium (water), and $r_{ij}=|\bm{r}_{i}-\bm{r}_{j}|$ the
separation distance. Eq.1 shows that $U_{m}$ is maximally attractive
(repulsive) for particles with magnetic moments parallel (perpendicular) to
$\bm{r}_{ij}$. If one perform a time averaging, it is straightforward to show
that hematite particles rolling side by side are effectively attractive with
$\overline{U}_{m}=-\mu_{0}m^{2}/(8\pi(x+z)^{3})$. In contrast, when the
particles are aligned tip to tip on the same plane, the moments are always
perpendicular to their separation distance, and $U_{m}$ is repulsive,
$\overline{U}_{m}=\mu_{0}m^{2}/(4\pi y^{3})$.

Fig. 2:  Top: collective mean velocities $\bar{v}_{i}$ with $i=(x,y)$ along
($\bar{v}_{x}$, blue disks) and perpendicular to ($\bar{v}_{y}$, magenta
squares) the propulsion direction versus the roller’s particle density $\phi$
for $f=100$Hz. Scattered points are experimental data, the continuous red line
denotes the linear regression $\bar{v}_{x}=v_{0}+\alpha\phi$, with
$\alpha=1183\pm 12\,\rm{\mu m^{3}s^{-1}}$ and $v_{0}=7.08\pm 0.11\,\rm{\mu
ms^{-1}}$ Bottom: corresponding velocity fluctuations $\sigma_{vx}$ and
$\sigma_{vy}$.

From the experimental observations, we find that at low frequency $f<50$Hz,
the particles tend usually to form compact clusters that grow along the
propulsion direction, thus favouring chaining due to attractive dipolar force,
see VideoS1 in the Supporting Information. However, for higher frequency these
clusters form and break preferentially along the perpendicular direction, and
thus their formation is promoted by HIs, being $U_{m}$ repulsive, see VideoS2.

We now focus on the collective velocity as function of the particle density.
As shown in Fig. 2, $\bar{v}_{x}$ linearly raises with the particle density
$\phi$, while being negligible the transverse velocity. The fluctuations in
speed $\sigma_{vx,vy}$ also increase with $\phi$ along both directions. At
high frequencies, when HIs dominates, we find that such interactions induce
the formations of fragile clusters which grow perpendicular to the propulsion
directions, see supporting videos. These clusters continuously break and
reform, and this effect induces a raise of the speed fluctuations along the
propulsion directions due to their fragmentation process, and perpendicular to
it due to their anisotropic growth.

We explain this collective speed up effect by considering only HIs and taking
into account the proximity of the wall. We assume an homogeneous distribution
of point like particles with density $\phi$, each subjected to a magnetic
torque $\bm{T}_{m}=(0,T_{m},0)$. For a single particle at position
$(x^{\prime},y^{\prime},h)$ above a no-slip surface at $z=0$, the
translational velocity is given by

$v_{x}(x,y,z,x^{\prime},y^{\prime},h)=\frac{T_{m}}{8\pi\eta}\Bigl{(}\frac{z-h}{\Delta_{-}^{3/2}}-\frac{z-h}{\Delta_{+}^{3/2}}+\frac{6z(x-x^{\prime})^{2}}{\Delta_{+}^{5/2}}\Bigr{)}$
(2)

where $\Delta_{\pm}=(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+(z\pm h)^{2}$. The
velocity field created by the ensemble of rollers is obtained as:

$\bar{v}_{x}(x,y,z)=\int^{\infty}_{-\infty}dx^{\prime}\int^{\infty}_{-\infty}dy^{\prime}\phi
v_{x}(x,y,z,x^{\prime},y^{\prime},h)\,\,.$

Carrying out this integration gives rise to a discontinuous velocity profile
$\bar{v}_{x}=(T_{m}\phi)/(2\eta)$ for $z>h$ and $\bar{v}_{x}=0$ for $z<h$. The
obtained relationship shows that effectively the particle speed grows
proportionally to the density of rollers in a two dimensional plane. We note
that such density dependence was shown to give rise to the Burger equation for
the particle concentration. 32

To apply this result to our hematite rollers, we derive the translational
speed of the single particle subjected to the magnetic torque $\bm{T}$.
According to Ref.24, the linear $v_{x}$ and angular $\Omega$ velocities of a
force-free spherical particle near a solid wall under an external torque
$T_{m}$ must satisfy the conditions

$6\pi\eta aF^{*}_{t}v_{x}+6\pi\eta a^{2}F^{*}_{r}\Omega=0$ (3)

and

$T_{m}=8\pi\eta a^{2}T^{*}_{t}v_{x}+8\pi\eta a^{3}T^{*}_{r}\Omega$ (4)

where we consider the following friction coefficients approximating rotating
particles by spheres: 24

$\displaystyle F^{*}_{t}=-\frac{8}{15}\ln{(\delta)}+0.9588$ $\displaystyle
F^{*}_{r}=\frac{2}{15}\ln{(\delta)}+0.2526$ $\displaystyle
T^{*}_{r}=-\frac{2}{5}\ln{(\delta)}+0.3817$ $\displaystyle
T^{*}_{t}=\frac{1}{10}\ln{(\delta)}+0.1895\,.$

Here $\delta=h/a-1$ characterizes the thickness of the liquid layer between
the particle and the no-slip surface. Eqs. (3,4) give

$v_{x}\equiv v_{0}=\frac{T_{m}}{8\pi\eta
a^{2}}\frac{F^{*}_{r}}{F^{*}_{r}T^{*}_{t}-T^{*}_{r}F^{*}_{t}}$ (6)

and

$\bar{v}_{x}/v_{0}=4\phi\frac{F^{*}_{r}T^{*}_{t}-T^{*}_{r}F^{*}_{t}}{F^{*}_{r}}\,\,.$
(7)

We use Eq.7 to fit the experimental data in Fig.2, where we keep as sole
variable the particle elevation $\delta$ from the surface. From the linear
regression we determine the elevation $h=0.66\,\rm{\mu m}$ from the centre of
the particle to the plane. Since the hematite particles roll perpendicular to
their long axis, we obtain that the surface to surface elevation is given by
$h-a=200$nm, a reasonable value similar to that found in previous works with
other magnetic colloids floating close to a plane in water. 33, 34

Fig. 3: (a) Slope $\alpha$ of the collective velocity vs driving frequency
$f$. The dashed line is a guide to the eye. Inset shows the corresponding
collective mean velocities $\bar{v}_{x}$ vs $\phi$ for different frequencies.
Scattered points are experimental data, continuous lines are linear
regressions. (b) Three microscope images illustrating typical clusters
observed at $f=10$Hz (left), $f=100$Hz (middle) and $f=300$Hz (right). Scale
bar is $10\,\rm{\mu m}$ for all images. The corresponding videos can be found
in the Supporting Information.  Fig. 4: (a,b) Position $x$ versus time $t$ of
different hematite rollers during the aggregation (a) and fragmentation (b)
process of a cluster. The optical microscope images on the right with
superimposed the particle trajectories over the last $0.25$s, serve to
illustrate the corresponding dynamic processes. Scale bars are $10\,\rm{\mu
m}$ for all images. (c) Cluster size distribution $C_{N}$ versus number of
particles $N$ at a constant particle density $\phi\sim 10^{-2}\,\rm{\mu
m^{-2}}$. Empty symbols are different experiments while the continuous red
line is an exponential fit. The particles are driven by a rotating field with
amplitude $H=4400\,\rm{Am^{-1}}$ and frequency $f=100$Hz.

We note that even if our approach uses a mean-field like description,
neglecting any correlation effect between the particles, it captures very well
the experimental findings. Further, we note that in the case of stress-free
boundary at $z=0$ one can demonstrate that surface rollers under the same
torque $\bm{T}$ move along the opposite direction. Indeed using the method of
images 35 the velocity field is given by

$\bm{v}=\frac{\bm{T}\times\bm{r}}{8\pi\eta
r^{3}}-\frac{\bm{T}\times\bm{r}^{*}}{8\pi\eta r^{*3}}\,\,,$ (8)

where $\bm{r}^{*}=\bm{r}+2h\bm{z}$. In a similar way than Eq.(3), after
integration we obtain $\bar{v}_{x}=0$ for $z>h$ and
$\bar{v}_{x}=-(T\phi)/(2\eta)$ at $z<h$ satisfying the zero stress boundary
condition at $z=0$. Thus, rollers located on the free interface move along the
opposite direction with respect to the particles placed on a no-slip boundary.
This conclusion was experimentally confirmed with chain of paramagnetic
rotors.34

The data shown in Fig.2 were obtained at $f=100$Hz, thus very close to
$f_{c}$. However, we also find that the mean velocity increases with the
density for other frequencies but, as $f<<f_{c}$ or $f>>f_{c}$, this effect
becomes less pronounced. Fig.3(a) shows the slope
$\alpha=(\bar{v}_{x}-v_{0})/\phi$ obtained from the linear regression for $6$
different frequencies, inset Fig.3(a). For $f=100,150$ and $200$ (dashed line)
we find that $\alpha$ reaches a maximum constant value, where HIs completely
dominate over dipolar forces. In this situation, we observe the formation of
clusters that grow perpendicular to the $\hat{\bm{x}}$-axis, as shown in the
microscope image in the middle of Fig.3(b). In contrast, at low or high
frequencies, the slow particle spinning (Fig.1(d)) reduces the effect of HIs
favouring dipolar forces and thus the growth of clusters along the propulsion
direction ($\hat{\bm{x}}$), see left and right images in Fig.3(b). Now
attractive dipolar forces slow down the mean velocities of the clusters,
reducing the speed up effect. This is consistent with other cases of
clustering induced by dipolar interactions in one 34 and two dimensions 36
were the mean velocity of the rollers was observed to saturate with the number
of particles. Further, the competition between dipolar interactions and HIs do
not dissolve the clusters, but rather induce a precessional-like motion of the
particles and the formation of more irregular structures, as shown in VideoS3
of the supporting information.

##  5 Dynamic clustering

At large density the magnetic rollers tend to aggregate in form of small
dynamic clusters which continuously break and reform during propulsion. Two
typical situations are shown in Figs.4(a,b). The first one (Fig.4(a))
illustrates the formation of a three particle cluster that assembles along the
($\hat{\bm{y}}$)-direction. Once formed, the clusters usually propel at a
similar speed than the composing particles. The lifetime of the clusters is
relative short as interaction with other incoming particles can easily break
them. This is illustrated in Fig.4(b) where one particle is able of breaking a
four particle cluster. During this fragmentation process, the hematite
particles speed up, since they are literally pushed away by the flow field of
the incoming roller. The microrollers form rather ”fragile” hydrodynamic
states, in contrast to other clusters made of active or self-propelling
particle systems. 37, 38, 39, 40, 41 In Fig.4(c) we analyse the cluster size
distribution $C_{N}$ at a relative large density of particles, $\phi\sim
10^{-2}\,\rm{\mu m^{-2}}$. Even if the maximum cluster size observed is
limited to $N=7$, we find that the data follow a simple exponential law,
$C_{N}\sim\exp{(-N/N_{c})}$ which gives a characteristic value $N_{c}=0.78$.
This behaviour is different than the one observed in other active systems,
where the dynamic clustering process is usually describe by a mixed
exponential and power law function. 41 In our case, the absence of an initial
power law scaling in $C_{N}$ reflects the different cluster nature, where HIs
make these structures rather fragile. Indeed, clusters made of chemically
active particles 37, 42 or bacteria 43, 44, 45 can grow isotropically over a
much larger area and the composing units present, apart from the random
orientation during propulsion, other interactions competing with HIs. Instead
our driven microrollers are characterized by almost deterministic trajectories
with small thermal fluctuations, and the cluster aggregation/fragmentation
process is driven by the sole HIs.

##  6 Conclusions

In this work we investigate the collective transport properties of anisotropic
hematite rollers that are driven in a viscous fluid by an external rotating
magnetic field. We find that at high driving frequency, HIs dominate and
induce the formation of elongated clusters perpendicular to the direction of
motion, even if dipolar interactions are repulsive for such configuration.
Moreover the particle collective mean velocity linearly increases with the
density, a fact that we explain on the basis of Blake’s tensor formalism
applied to our colloidal rollers. 46

We note that Driscoll et al. 47 reported recently that a collection of
microscopic magnetic rollers made of spherical particles with protruding
hematite cubes, display a fingering instability produced by the hydrodynamic
flow generated by their spinning. In such work the authors also report that
the velocity of the rollers linearly grow with the particle density, and the
density-dependent mean velocity has been later used to predict the formation
of shock fronts in this system. 48 Our hematite colloidal rollers display
relative stronger magnetic forces, a feature that may be used to further
control dipolar interactions in contrast to hydrodynamic one. Indeed, it would
be interesting to investigate in the future weather HIs can play an important
role in the synchronization process of the magnetic rollers with the rotating
field.

##  7 Appendix

###  7.1 Single particle dynamics

Isolated hematite particles are subjected to different forces during motion.
Given the large density mismatch, $\Delta\rho=\rho_{p}-\rho_{w}$, between
hematite ($\rho_{p}=5.3\,\rm{gcm^{-3}}$ in bulk) and water
($\rho_{w}=1\,\rm{gcm^{-3}}$), the gravitational force acting on a spherical
particle of radius $a=\beta/2$ can be estimated as $F_{g}=\frac{4}{3}\pi
a^{3}\Delta\rho g\sim 0.06\rm{pN}$, thus much higher than lift force
$F_{l}\sim 60\rm{fN}$, calculated Rashidi2016, Helgessen2018 for a similar
spherical particle spinning at the maximum frequency of $f=100$Hz. Thus, in
absence of repulsive electrostatic interactions that float the particle at a
distance of $\sim 200$nm from the surface, the particles would touch the solid
surface and display a stick-slip dynamics.

Another effect which may affect the roller dynamics at very low and high
frequency, is the surface roughness. Its effect may be present in the
asynchronous regime (see VideoS3 of the Supporting Information) when the
particle perform a wobbling-like movement. We consider the particle roughness
Smart1989 in terms of the typical size of surface asperities or bumps,
$\varepsilon a$. The solid friction becomes important when the size of the
surface bumps $\varepsilon a$ is of the order of the thickness of the gap
between the particle and plane $(h-a)$ Smart1993. In this situation, the
hydrodynamic problem for a particle moving near the plane 24 is given by the
following force and torque balance equations

$F_{s}-6\pi\eta aF^{*}_{t}v_{x}-6\pi\eta a^{2}F^{*}_{r}\Omega=0$ (9)

and

$T_{m}+F_{s}a-8\pi\eta a^{3}T^{*}_{r}\Omega-8\pi\eta a^{2}T^{*}_{t}v_{x}=0$
(10)

where $F_{s}$ is the solid friction force and $T_{m}$ is the torque from the
rotating magnetic field. Two different situations may be considered - motion
without slip when $v_{x}=a\Omega$ and $F_{s}<F_{s}^{max}=kN$, where $k$ is a
solid friction coefficient and $N=F_{g}$ the normal force. A simple estimate
for the hematite particle under a rotating field with amplitude
$H=4000\rm{Am^{-1}}$ of the ratio $\tilde{T_{m}}=MHV/(\Delta\rho gVa)$ shows
that $\tilde{T_{m}}>>1$ and the particle is in slipping regime when
$F_{s}=F_{s}^{max}$. Scaling $v_{x}$ with the sedimentation velocity
$N/(6\pi\eta a)$ and the angular velocity by $N/(6\pi\eta a^{2})$ give

$F^{*}_{t}\tilde{v_{x}}+F^{*}_{r}\tilde{\Omega}=k$ (11)

and

$\tilde{T_{m}}=\frac{4}{3}T^{*}_{r}\tilde{\Omega}+\frac{4}{3}T^{*}_{t}\tilde{v_{x}}-F^{*}_{t}\tilde{v_{x}}-F^{*}_{r}\tilde{\Omega}$
(12)

From Eq.(11)

$\tilde{\Omega}=\frac{k}{F^{*}_{r}}-\frac{F^{*}_{t}\tilde{v_{x}}}{f^{*}_{r}}$
(13)

Inserting that in Eq.(12) we have

$\tilde{v_{x}}=\frac{3}{4}\tilde{T_{m}}\frac{F^{*}_{r}}{F^{*}_{r}T^{*}_{t}-T^{*}_{r}F^{*}_{t}+k(T^{*}_{r}-\frac{3}{4}F^{*}_{r})}$
(14)

In dimensional units this gives

$v_{x}=\frac{T_{m}}{8\pi\eta
a^{2}}\frac{F^{*}_{r}}{F^{*}_{r}T^{*}_{t}-T^{*}_{r}F^{*}_{t}+k(T_{r}^{*}-\frac{3}{4}F_{r})}$
(15)

As a result for the ratio of the collective velocity of ensemble
$v_{c}=T_{m}\phi/(2\eta)$ and velocity of the single particle $v_{x}$ we have

$\frac{v_{c}}{v_{x}}=4\varphi\frac{F^{*}_{r}}{F^{*}_{r}T^{*}_{t}-T^{*}_{r}F^{*}_{t}+k(T_{r}^{*}-\frac{3}{4}F_{r})}$
(16)

where $\varphi=\pi a^{2}\phi$ is the surface fraction covered by the magnetic
particles.

##  8 Acknowledgments

G. J. and P. T. acknowledge support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No.: 811234). A.C. acknowledge support by grant of Scientific
Council of Latvia lzp-2020/1-0149. P. T. acknowledges support from MICIU
(grant No. PID2019-108842GB-C21), AGAUR (grant No. 2017-SGR-1061) and from the
Generalitat de Catalunya via program ”Icrea Acadèmia”.

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