# Redshift-dependent RSD bias from Intrinsic Alignment with DESI Year 1
Spectra

Claire Lamman ,1 Daniel Eisenstein,1 Jessica Nicole Aguilar,2 Steven Ahlen ,3
David Brooks,4Todd Claybaugh,2 Axel de la Macorra ,5 Arjun Dey ,6 Biprateep
Dey ,7 Peter Doel,4 Simone Ferraro ,2,8Andreu Font-Ribera ,9 Jaime E. Forero-
Romero ,11,12 Satya Gontcho A Gontcho ,2 Julien Guy,2Robert Kehoe,13 Anthony
Kremin ,2 Laurent L. Le Guillou ,14 Michael Levi ,2 Marc Manera ,9Ramon
Miquel,9,15 Jeffrey A. Newman ,7 Jundan Nie ,16 Nathalie Palanque-Delabrouille
,17Francisco Prada ,18 Mehdi Rezaie ,19 Graziano Rossi,20 Eusebio Sanchez ,21
Michael Schubnell,22Seo Hee-Jong ,23 Gregory Tarlé ,22 Benjamin Alan Weaver,6
Zhimin Zhou 24

1Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
2Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA
3Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA
4Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
5Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México
6NSF’s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA
7Department of Physics & Astronomy and Pittsburgh Particle Physics,
Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941
O’Hara Street, Pittsburgh, PA 15260, USA
8University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA 94720,
USA
9Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain
10Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
11Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia
12Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia
13Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, TX 75275, USA
14Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France
15Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain
16National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China
17IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
18Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía,
s/n, E-18008 Granada, Spain
19Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA
20Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea
21CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain
22University of Michigan, Ann Arbor, MI 48109, USA
23Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA
24National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China E-mail:
<EMAIL_ADDRESS>0000-0002-6731-9329 0000-0001-6098-7247
0000-0002-1769-1640 0000-0002-4928-4003 0000-0002-5665-7912
0000-0003-4992-7854 0000-0002-3033-7312 0000-0002-2890-3725
0000-0003-3142-233X 0000-0001-6356-7424 0000-0001-7178-8868
0000-0003-1887-1018 0000-0003-4962-8934 0000-0001-8684-2222
0000-0001-6590-8122 0000-0003-3188-784X 0000-0001-7145-8674
0000-0001-5589-7116 0000-0002-9646-8198 0000-0002-6588-3508
0000-0003-1704-0781 0000-0002-4135-0977

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We estimate the redshift-dependent, anisotropic clustering signal in DESI’s
Year 1 Survey created by tidal alignments of Luminous Red Galaxies (LRGs) and
a selection-induced galaxy orientation bias. To this end, we measured the
correlation between LRG shapes and the tidal field with DESI’s Year 1
redshifts, as traced by LRGs and Emission-Line Galaxies (ELGs). We also
estimate the galaxy orientation bias of LRGs caused by DESI’s aperture-based
selection, and find it to increase by a factor of seven between redshifts
$0.4-1.1$ due to redder, fainter galaxies falling closer to DESI’s imaging
selection cuts. These effects combine to dampen measurements of the quadrupole
of the correlation function ($\xi_{2}$) caused by structure growth on scales
of 10-80 $h^{-1}$Mpc by about 0.15% for low redshifts (0.4$<$z$<$0.6) and 0.8%
for high (0.8$<$z$<$1.1). We provide estimates of the $\xi_{2}$ signal created
by intrinsic alignments that can be used to correct this effect, which is
necessary to meet DESI’s forecasted precision on measuring the growth rate of
structure. While imaging quality varies across DESI’s footprint, we find no
significant difference in this effect between imaging regions in the Legacy
Imaging Survey.

###### keywords:

methods: data analysis –cosmology: observations – large-scale structure of
Universe – – cosmology: dark energy

††pubyear: 2023††pagerange: Redshift-dependent RSD bias from Intrinsic
Alignment with DESI Year 1 Spectra–Redshift-dependent RSD bias from Intrinsic
Alignment with DESI Year 1 Spectra

##  1 Introduction

Measuring the growth of large-scale structure in the Universe informs us about
the components that drive it: gravity and dark energy. The main observable
used to measure this evolution is the Kaiser effect (Kaiser, 1987). As
structure grows, matter falls towards dense regions. This increases the
recessional velocity of matter between us and a dense region, while decreasing
it for matter falling in from the other side. The result is a “squashing"
effect in redshift-space. This is the dominant source of redshift-space
distortions (RSD) on scales larger than clusters (around $10h^{-1}$Mpc).
Clustering is quantified using the correlation function. This can be expressed
as a series of spherical harmonics, of which the quadrupole $\xi_{2}$
describes the anisotropic clustering that arises from RSD. On large scales,
the growth rate of structure is linearly related to $\xi_{2}$. This makes RSD
a powerful test of cosmological parameters and measuring it is one of the two
main science goals of the the Dark Energy Spectroscopic Instrument (DESI).

DESI is in the midst of a 5-year survey, measuring spectra of over 40 million
galaxies within 16,000 deg2 of the sky. The instrument is installed on the
4-meter Mayall telescope and can gather tens of thousands of extra-galactic
spectra in one night with its focal plane, which is comprised of 5000
individually controlled robots to position fibers onto galaxies (Levi et al.,
2013; DESI Collaboration et al., 2016a, b, 2022).

DESI is forecasting a 0.4-0.7% measurement of the growth rate of structure,
$f\sigma_{8}$. This is measured through the comparison of anisotropic and
isotropic clustering, $\xi_{2}$ and $\xi_{0}$. It is more difficult to obtain
high precision on $\xi_{2}$ than $\xi_{0}$, so errors on $f\sigma_{8}$ are
dominated by $\xi_{2}$. To meet DESI’s science goals, it’s imperative to
measure $\xi_{2}$ to a precision of at least 0.4-0.7%. A subtle effect that
could take up a significant fraction of this error budget is the bias in
$\xi_{2}$ due to a combination of two effects: Intrinsic Alignment (IA) and a
selection-induced orientation bias (Hirata, 2009).

IA refers to physical correlations between galaxy shapes and with galaxy
shapes to the underlying density. See Lamman et al. (2023a) for a pedagogical
guide to IA and Joachimi et al. (2015), Troxel & Ishak (2015) for detailed
reviews. It is historically measured as a contaminant of weak lensing, but IA
in upcoming surveys may provide novel constraints on galaxy formation and
cosmology (Chisari & Dvorkin, 2013; Okumura & Taruya, 2023; Kurita & Takada,
2023; Xu et al., 2023). For DESI, IA also needs to be understood as a bias in
measurements of anisotropic clustering.

This effect arises from the extent to which galaxy shapes are correlated with
the underlying tidal field. The primary axis of Luminous Red Galaxies (LRGs)
tend to be aligned along strands of density and point towards denser regions.
This creates a clustering bias when combined with DESI’s aperture-based target
selection. An elliptical galaxy with its primary axis pointed at the observer
will have a more concentrated light profile on the sky and a higher fraction
of its light will fall within the aperture. This makes DESI more likely to
observe galaxies which lie in density filaments that are parallel to the line
of sight (LOS). For a visualization of this effect, see Figure 1 in Lamman et
al. (2023b). Studies have explored the effects of orientation-dependent
selection in Sloan Digital Sky Survey (SDSS) catalogs with differing results
(Martens et al., 2018; Obuljen et al., 2020; Singh et al., 2021). We expect
this effect to be more pronounced with DESI, which has a smaller fiber
aperture of 1.5 arcsec in diameter, as opposed to SDSS’ 3 arcsec aperture.

A total-magnitude selection would remove this bias from DESI, but
spectroscopic success is highly dependent on the surface brightness of an
object. Especially for a survey which prioritizes speed, there will be a
surface-brightness dependence on the sample which is easier to impose
explicitly as a target cut (Zhou et al., 2022). This selection-induced bias in
galaxy orientation likely also affects DESI’s Emission-Line Galaxy (ELG)
sample. However, while predicted by simulations, there is currently no
observed shape alignment in ELG-like (spiral) galaxies and both effects need
to be present to create this RSD bias (Samuroff et al., 2019; Johnston et al.,
2019; Samuroff et al., 2023).

Measuring IA for the purpose of predicting an RSD bias has a few differences
from IA measured in the context of weak lensing, which requires very precise
shape measurements with well controlled systematic effects to measure
gravitational shear. We only require shape measurements which are more precise
than intrinsic shape variation. This is the case with LRGs in DESI’s Legacy
Imaging Survey, which are relatively large and bright. Therefore it is more
valuable for us to use the full redshift sample available than limit to a
region which overlaps with a deeper imaging survey, as will be done with other
DESI IA measurements (Lange et al. in prep). Also, since this study uses
spectroscopic redshifts, we can sufficiently isolate pairs with low separation
along the LOS to the degree that we are unconcerned about contamination from
weak lensing or across redshift bins.

Lamman et al. (2023b) used photometric redshifts from DESI’s imaging catalog
to estimate that this effect could lower DESI’s measurement of $\xi_{2}$ by
about 0.5% for LRGs. In this work, we use DESI’s Year-one spectra (DESI
Collaboration, in prep) to produce estimates which can be used to correct
DESI’s RSD measurements. We measure the tidal alignment of LRGs as traced by
LRGs and ELGs, assess the impacts of imaging on the IA measurement, and
estimate the redshift dependence of the selection-induced shape polarization.
We report the resulting redshift-dependent bias for DESI’s Year one RSD
results and discuss sources of systematic uncertainties.

##  2 DESI Catalogs

###  2.1 Imaging

DESI’s targets are chosen from DR9 of the Legacy Imaging Survey (Dey et al.,
2019; Myers et al., 2023). This contains imaging of sources in
$14,000\deg^{2}$ of the extra galactic sky from three different telescopes:
Mayall z-band Legacy Survey at the Mayall telescope at Kitt Peak (MzLS), the
DECam Legacy Survey from the Blanco telescope at Cerro Tololo (DECaLS), and
the Beijing-Arizona Sky Survey from the BOK telescope at Kitt Peak (BASS) (Zou
et al., 2017). A region in the South Galactic cap also contains imaging from
the DES survey.

Their shapes are fit using Tractor (Lang et al., 2016). After deconvolving
with a point-spread function (PSF), shapes are modeled at the pixel level with
several light profiles: exponential disk, de Vaucouleurs, Sersic, PSF, and
round-exponential. The default shape parameters are chosen based on a
marginalized $\chi^{2}$ criteria to avoid over-fitting bright targets as
round-exponentials. Measuring intrinsic alignment requires shape orientations,
so where galaxies are fit as circles (PSF and round-exponentials), we use
shape parameters from the best fit between exponential disk and de
Vaucouleurs. This will not affect our final results as the DESI target
selection does not depend on these derived shape parameters.

The parameters used to describe each projected galaxy image are its primary
axis, $a$, secondary axis, $b$, and orientation $\theta$ (Figure 1). This is
used to describe the shape of an ellipse relative to some direction using
$\epsilon_{+}$:

$\epsilon_{+}=\frac{a-b}{a+b}\cos(2\theta)$ (1)

Figure 1: The parameters used to describe the shape and orientation of the
ellipse created by projecting an elliptical galaxy. Here, the ellipticity is
measured relative to North. For our measurement, ellipticity is measured
relative to the positions of a tracer sample.

DESI’s LRG target selection from this catalog includes a cut based on the
expected flux which falls within a DESI fiber, which corresponds to a 1.5
arcsec diameter aperture. The $z$-band magnitude within the aperture is
limited to $z_{\text{fiber}}<21.61$ in the Northern Galactic Cap and
$z_{\text{fiber}}<21.60$ in the Southern Galactic Cap. For more information
DESI’s target selection, see Raichoor et al. (2020); Zhou et al. (2020, 2022).

This shape fitting and target selection is dependent upon imaging quality,
which varies across sky regions. To qualify the effect of imaging quality on
shape parameters, we separate the LRGs into three sky regions: The MzLS and
BASS region, the DECaLS region which does not contain DES imaging, and the DES
region. We compare the reported axis ratio, $b/a$, of the reported galaxy
shapes in each region in Figure 2. The MzLS and BASS region reports more
eccentric LRG shapes than the other regions, and the region with highest-
quality imaging, DES, reports the roundest shapes. While this may indicate an
over-correcting of the PSF in MzLS and BASS imaging, we measure the IA signal
independently in these regions and do not find any significant impact on final
results (Section 3.2).

Figure 2: The distribution of projected axis ratios for LRGs in three DESI
imaging regions: the North Galactic cap, which contains imaging from MzLS and
BASS, the portion of the South Galactic cap which contains DECaLS but no DES
imaging, and the DES region. Residuals from the mean are plotted below. These
reported shapes have been deconvolved with a point spread function to account
for imaging conditions. The region with the highest quality imaging, DES,
reports the least eccentric shape.

###  2.2 Spectroscopy

We used spectra from DESI’s internal data release, Iron, which is comprised of
data from commissioning through Year 1 of the survey (DESI Collaboration, in
prep). It contains spectra of 2.9 million LRGs and 4.0 million ELGs from
observations taken during December 14th, 2020 through June 13th 2022 (Zhou et
al., 2022; Raichoor et al., 2023; Guy et al., 2023). We used Iron’s large-
scale structure catalog which includes a veto on targets based on hardware and
imaging conditions. The catalog contains weights to account for completeness
based on the probability that each target was assigned. LRGs also have a
weight to account for redshift failures. From this catalog, we ran a basic
redshift-quality cut to obtain our final samples of 2.5 million LRGs and 3.1
million ELGs.

The projected correlation functions used to calibrate our measurements
(Section 3.1) were made with this Year 1 data. Due to internal blinding
policies, our estimation of DESI’s $\xi_{2}$ measurements are calibrated with
the smaller, publicly available spectroscopic catalog from DESI’s Survey
Validation (DESI Collaboration et al., 2023a, b; Lan et al., 2023). Our
determination of the $\xi_{2}$ signal which arises from IA is independent of
the RSD $\xi_{2}$ signal.

##  3 Intrinsic Alignment Measurement

###  3.1 IA Formalism

As in Lamman et al. (2023b), we measure the correlation between galaxy shapes
and density by averaging the ellipticity of each LRG relative to the
separation vector between it and nearby galaxies in the tracer sample111code
available here: github.com/cmlamman/ellipse_alignment.

$\mathcal{E}(r_{p})=\langle\epsilon_{+}(a,b,\theta)\rangle$ (2)

For a given galaxy-tracer pair, $a$ and $b$ are the axis lengths of the galaxy
shape and $\theta$ is the orientation of the galaxy relative to the separation
vector between it and the tracer. This is measured as a function of the
projected separation between them, $r_{p}$.

We limit the separation of pairs along the LOS, $r_{\Pi}$, to
$\pm\Pi_{\text{max}}=$ 30$h^{-1}$Mpc. This, along with clustering, is taken
into account in our model when estimating how far along the LOS the IA
measurement is averaged over.

For measuring the IA of our full LRG sample, we divide the tracer catalog into
100 sky regions based on right ascension and declination with an equal number
of galaxies in each. $\mathcal{E}(r_{p})$ is measured independently in each
region using its tracers and the full shape catalog, then averaged over every
pair. This average included the catalog weights described in Section 2.2 for
both the shape and tracer samples. The average of these 100 measurements and
standard error is our final measurement.

IA is often quantified using a form of correlation functions generalized to
include information about galaxy shapes (Mandelbaum et al., 2006). The IA
correlation function relating galaxy shapes and density is

$\xi_{g+}(r_{p},\Pi)=\frac{S_{+}D-S_{+}R_{D}}{R_{S}R_{D}}.$ (3)

$S_{+}D$ is the count of data-data pairs weighted by the orientation of
shapes, $S_{+}$, relative to a tracer sample $D$. Measuring this as function
of projected separation and averaging over each data pair,
$S_{+}D(r_{p})/DD(r_{p})$, is equivalent to $\mathcal{E}(r_{p})$. $S_{+}R$
represents the data shapes relative to a random sample, which has an
expectation value of 0. $R_{S}R_{D}$ is the random-random count. Integrating
$\xi_{g+}$ along the LOS direction, $\Pi$, produces the projected IA
correlation function:

$w_{g+}(r_{p})=\int_{-\Pi_{\text{max}}}^{\Pi_{\text{max}}}\mathrm{d}\Pi\,\xi_{g+}(r_{p},\Pi)$
(4)

For predicting the RSD bias that arises from IA, $\mathcal{E}(r_{p})$ is the
most direct relevant observable. Unlike $w_{g+}$, $\mathcal{E}(r_{p})$ is
normalized by data pairs, not randoms. $DD$ can be expressed as

$\begin{array}[]{l}DD(r_{p})=RR\int_{-\Pi_{\text{max}}}^{\Pi_{\text{max}}}d\Pi\frac{DD(r_{p},\Pi)}{RR}\\\
=RR\int_{-\Pi_{\text{max}}}^{\Pi_{\text{max}}}d\Pi(1+\xi(r_{p},\Pi))=RR(2\Pi_{\text{max}}+w_{p}(r_{p})).\end{array}$
(5)

Here $\xi$ and $w_{p}$ are the typical correlation function and projected
correlation function, as opposed to those weighted by shape alignments.
$w_{g+}$ can be expressed as

$w_{g+}(r_{p})=\frac{1}{RR}\int_{-\Pi_{\text{max}}}^{\Pi_{\text{max}}}d\Pi
S_{+}D(r_{p},\Pi)=\frac{S_{+}D(r_{p})}{RR}.$ (6)

Therefore, a given $w_{g+}$ and $\mathcal{E}$ made with the same $\Pi_{\rm
max}$ and same clustering $w_{p}$ are related as

$w_{g+}(r_{p})=(2\Pi+w_{p}(r_{p}))\mathcal{E}(r_{p})=L\mathcal{E}(r_{p})$ (7)

Here we have introduced $L$, which can be understood as the effective LOS
distance that $\mathcal{E}$ is measured over, adjusted to account for
clustering which decreases the average LOS-separation of pairs. $L$ is
included in our final model of the RSD bias (Section 5)222$L$ here is
equivalent to $L_{\text{eff}}$ in Lamman et al. (2023b). While $L\mathcal{E}$
is functionally equivalent to $w_{g+}$, we notate $L$ and $\mathcal{E}$
separately to be explicit about how the quantity was estimated.

Figure 3: The IA signal of LRGs over the entire redshift range, 0.4 $<$ z $<$
1.1, compared to the estimate made in (Lamman et al., 2023b) with photometric
distances. The photometric estimate was made with 17.5 million galaxies,
compared to 2.5 million LRGs for the spectroscopic sample, but necessarily
averaged over a larger radial distance. This is adjusted for here, which shows
the "relative" IA signal that has been calibrated by the effective radial
depth $L$. Figure 4: Measurement of the tidal alignment of LRG shapes made
independently in areas from three regions of DESI’s Legacy Imaging Survey
described in Section 2.2. DES has the highest quality imaging, but there is no
significant effect on our total averaged IA signal.

We can compare our spectroscopic IA measurement with a similar one made with
photometric data (Lamman et al., 2023b) by scaling by $L$, as shown in Figure
3. Although made with seven times fewer galaxies, the spectroscopic sample
from Iron can be measured in smaller LOS bins and provide us with similar
level of precision.

Spectroscopic data also allows us to better isolate the sample in radial bins
and explore redshift-dependence. To compare our IA signal between samples of
different target classes and redshift distributions, $\mathcal{E}(r_{p})$
needs to be calibrated by $L$ as well as the galaxy clustering bias, $b$. For
bias-independent comparisons, we scale by a relative bias. The bias of a
sample $2$ relative to sample $1$ is

$b_{\rm
rel}=\frac{b_{2}}{b_{1}}=\frac{D(z_{1})}{D(z_{2})}\bigg{(}\frac{w_{p2}}{w_{p1}}\bigg{)}^{1/2},$
(8)

where $D(z)$ is the linear growth function.

Therefore, when comparing IA measurements across samples we use the value
$(L/b_{\rm rel})\mathcal{E}(r_{p})$. While $L$ is taken into account when
estimating the final RSD bias, $b$ does not affect the final result. This is
because the amplitude of the power spectrum quadrupole effect arises from the
correlation of the galaxy density field and the selection-induced shape
polarization, the latter of which is independent of bias.

When calculating distances and the growth factor, we assume a flat
$\Lambda$CDM cosmology with $\Omega_{m}=0.286$, $\Omega_{\Lambda}=0.714$ and
$H_{0}=69.6\text{ km s}^{-1}\text{Mpc}^{-1}$.

###  3.2 Dependence on Imaging Region

The amplitude of IA can strongly depend upon imaging quality and the methods
used to estimate shapes. This is in part due to difficulties in accurately
modeling imaging processes, and in part due to isophotal twisting (Fasano &
Bonoli, 1989), which causes the outer regions of galaxies to have a stronger
alignment signal than the inner regions. This has been measured in BOSS LOWZ,
DES, and LSST (Singh et al., 2015; Zuntz et al., 2018; Leonard et al., 2018;
Georgiou et al., 2019; MacMahon & Leonard, 2023). This is a concern for our
imaging catalog, as imaging quality and shapes vary across region (Figure 2).
A lower signal due to poor imaging will result in an underestimate of our
final estimate of the RSD bias.

To qualify the impact of imaging quality, we compare our IA signal across the
three different imaging regions used in the Legacy Imaging Survey: DES,
DECaLS, and MzLS+BASS. Each region has varying survey completeness, so to
avoid edge effects we made these measurements in a limited area with the most
completeness in each region. The result and size of each sample is shown in
Figure 4. We do not find a significant difference in the signals, which is in
part due to measurement noise. Additionally, although the BASS + MzLS region
may be over-correcting for the PSF and producing more eccentric shapes,
systematic imaging effects are uncorrelated with the tidal field. A small
change in ellipticity doesn’t propagate as an order-unity error on this
signal, which is a very small response to the tidal shear. This may be still
be an issue for higher signal-to-noise detections beyond DESI Year 1.

As a null test, we reproduced this measurement using the cross-component of
shape, $\epsilon_{\times}=\frac{a-b}{a+b}\sin(2\theta)$ instead of
$\epsilon_{+}$ in Equation 2. This was consistent with 0 on all scales.

###  3.3 Dependence on Redshift and Tracer Sample

The redshift dependence of IA is unclear (Samuroff et al., 2021; Zhou et al.,
2023; Samuroff et al., 2023), and cannot be directly observed without
accounting for luminosity differences across redshift bins. DESI’s LRG sample
is designed to have a constant co-moving volume with redshift, which results
in more luminous, and therefore more aligned, galaxies in high redshift
samples. However, since we are only inferring a systematic bias and not any
physical trends, we only require the IA of each sample. The IA RSD bias is
proportional to the amplitude of this signal, so if not properly accounted
for, it could manifest in DESI’s results as a false evolution of the growth
rate as measured by the quadrupole of the correlation function. Therefore we
separate our LRG tracer sample into three sub-samples based on redshift and
measure the correlation of LRG shapes in each.

The samples are plotted in Figure 5 and displayed in Table 1. To compare the
strength of tidal alignment between redshifts, the signal is adjusted based on
the clustering in each sample, as described in Section 3.1. As expected, we
find the weakest signal for nearby galaxies (0.4 $<$ z $<$ 0.6).

Figure 5: Comparison of the intrinsic alignment of LRGs between spectroscopic
redshift bins. The y-axis is scaled by the effective depth of the measurement
$L$ and the galaxy bias $b_{\rm rel}$, which here is defined as $b_{\rm
rel}(z=0.7)=1$. These were calculated using the projected correlation function
from DESI’s Year one data. Errors here only include the statistical difference
of the signal between sky regions and not from $b$ or $L$. Nearby galaxies
broadly display a weaker alignment, though here we have not accounted for
luminosity differences across samples. Figure 6: Correlation between LRG
shapes and the underlying galaxy density, as traced by both LRGs and ELGs.
These samples are both in the redshift range $0.8<z<1.1$. For comparison, this
IA signal is scaled by the samples’ clustering, as described in 3.1. Figure 7:
The reduced covariance matrix of $\mathcal{E}$ between bins of transverse
separation for our IA measurement with LRG tracers across the full redshift
range. The identity matrix has been subtracted from this plot. This
demonstrates that there is no evidence for significant correlations between
the measurements of $\mathcal{E}$ in each bin of projected separation.

We also measured the alignment of LRGs to the tidal field as traced by ELGs,
as opposed to the same LRG sample (Figure 6). In the overlapping redshift
range of the LRG and ELG samples, $0.8<z<1.1$, we find a similar IA signal
once both samples are adjusted for clustering. Although some regions of DESI’s
Year one footprint are less complete for ELGs, this is accounted for in the
catalog completeness weights described in Section 2 and we find no impact of
this on our IA measurement.

A covariance matrix for the spectroscopic LRG measurement made over all
redshift ranges is shown in Figure 7 and demonstrates no correlation between
the bins of projected separation that the IA measurement $\mathcal{E}$ was
made in.

Figure 8: Estimates of shape polarization for each of the three redshift bins,
or the tendency for these samples to have shapes aligned with the LOS due to
an aperture magnitude cut. These are based on the model in (Lamman et al.,
2023b), shown in grey, which estimates the polarization in bins of color and
magnitude for a sample of LRGs without the aperture magnitude cut. The total
polarization estimate for each redshift bin is the average of each galaxy’s
polarization corresponding to its respective color bin. The highest redshift
sample contains redder, fainter galaxies which are closer to the survey cuts
and therefore more likely to have biased galaxy orientations.

##  4 Selection-Induced Shape Polarization

The final RSD bias is proportional to both the degree to which galaxies are
aligned along the tidal field (IA) and the degree to which galaxies are
aligned along the LOS due to target selection, or "shape polarization". For
DESI, the latter plays a strong role in the redshift dependence of the RSD
bias. Redder, fainter galaxies fall closer to the aperture-magnitude cut that
is used to select DESI targets. Therefore their orientation will have a
stronger impact on whether or not they are selected; an elongated galaxy
aligned with the LOS will have more light concentrated within a sky aperture.

Lamman et al. (2023b) estimated the shape polarization of DESI LRGs from a
parent sample without the aperture-magnitude cut. This was done by generating
many 3D light profiles for each galaxy based on the expected distribution of
triaxial shapes from Padilla & Strauss (2008). These light profiles were
assigned random orientations then put through an aperture-magnitude cut. The
average ellipticity of the selected shapes relative to the LOS,
$\epsilon_{\text{LOS}}$, is the selection-induced shape polarization.

As this selection is done on aperture magnitudes from an image deconvolved
with a point-spread function, the shape selection bias is relatively
independent of imaging quality. It matters more to model this effect with
imaging that most closely reflects the intrinsic galaxy shape. Therefore the
polarization estimate for the entire sample was made using the portion of
DESI’s footprint with the highest quality imaging, the DES region. Although
this results in a noisier measurement, only the average polarization of a
sample affects the final RSD bias.

To estimate the polarization of the LRG redshift samples, we averaged the
polarization estimates from the parent sample in bins of color and magnitude.
LRGs in the redshift sample were assigned a polarization based on the average
in their corresponding bin, and their total average is the polarization
estimate of that sample. This was not done for the ELG sample, which were only
used as tracers. A demonstration of this mapping can be see in Figure 8 and
the results are also displayed in Table 1. It is important to note that the
polarization varies more across redshift bins than the IA signal, meaning that
the redshift dependence of the final RSD bias is more dependent on survey
selection than physical alignments.

Tracer | z${}_{\text{min}}$ | z${}_{\text{max}}$ | N | $\sigma_{E1}^{2}$ | $\epsilon_{\text{LOS}}$ | $\bar{\mathcal{E}}(0<r_{p}<20)$ | $\bar{L}(0<r_{p}<20)$ | $\bar{\tau}(5<r_{p}<18)$ | $\bar{\xi}_{\text{2, gI}}(5<s<80)$
---|---|---|---|---|---|---|---|---|---
LRG | 0.4 | 0.6 | 529852 | 0.046 | 2.3 $\times 10^{-3}$ | 1.8 $\times 10^{-3}$ | 95.1 $h^{-1}{\rm Mpc}$ | 5.9 $\pm$ 0.5 $\times 10^{-3}$ | 0.044
LRG | 0.6 | 0.8 | 805181 | 0.033 | 6.7 $\times 10^{-3}$ | 2.1 $\times 10^{-3}$ | 94.9 $h^{-1}{\rm Mpc}$ | 7.0 $\pm$ 0.2 $\times 10^{-2}$ | 0.22
LRG | 0.8 | 1.1 | 896150 | 0.026 | 12.8 $\times 10^{-3}$ | 1.8 $\times 10^{-2}$ | 92.9 $h^{-1}{\rm Mpc}$ | 5.6 $\pm$ 0.2 $\times 10^{-2}$ | 0.41
ELG | 0.8 | 1.1 | 591687 | 0.026 | 12.8 $\times 10^{-3}$ | 1.9 $\times 10^{-3}$ | 73.2 $h^{-1}{\rm Mpc}$ | 4.3 $\pm$ 0.3 $\times 10^{-2}$ | 0.34
Table 1: Samples and values used to estimate the RSD bias for three LRG
redshift bins and the LRGxELG cross-correlation. $r_{p}$ and $s$ are given in
units of $h^{-1}{\rm Mpc}$. The tracer samples used in the top three rows were
also used as the shape sample. The last row uses ELG tracers with LRG shapes.
The table shows the redshift range and number $N$ of tracers used, and
properties of the shape sample: the variance of the real component of
ellipticities $\sigma_{E1}^{2}$ and the estimated selection-induced
polarization of shapes along the LOS, $\epsilon_{\rm LOS}$. We do not include
uncertainties for these columns as they have negligible statistical errors.
The IA signal $\mathcal{E}(r_{p})$ is measured as the ellipticity of shapes
relative to the tracer sample. $L(r_{p})$ is the effective LOS-distance that
$\mathcal{E}(r_{p})$ is averaged over. $\tau(r_{p})$ is defined in Equation 9
and is a combination of $\mathcal{E}(r_{p})$, $L(r_{p})$, and the power
spectrum. These are functions of transverse separation, $r_{p}$ and are shown
in this table as averages over the marked scales. The final column shows the
average amplitude of the anisotropic clustering created by IA; the quadrupole
of the correlation function without RSD effects. The full estimate of this
final result along with the statistical error is shown in Figure 9. Figure 9:
The anisotropic clustering signal arising from tidal alignment and a selection
bias, $\xi_{\text{2, gI}}$. Statistical errors are shown in the shaded bands,
although the total errors are dominated by systematic effects (Section 6). For
context, we have also plotted 1% of the expected $\xi_{2}$ signal from RSD.
This is well above DESI’s error budget for measuring the growth rate of
structure, which is 0.4-0.7% for LRGs and ELGs combined. Since the $\xi_{2}$
signal created by the growth of structure is opposite in sign to that created
by IA, we have multiplied the RSD $\xi_{2}$ by -1 for an easier comparison.
This plot demonstrates that IA will dampen DESI’s RSD measurements to a degree
that is a significant fraction of the error budget, particularly at higher
redshifts. Incorporating these estimates into the $\xi_{2}$ measurement will
mitigate this effect.

##  5 False RSD signature in DESI

To estimate the RSD bias created by the combination of IA and the selection-
induced polarization, we use a nonlinear tidal model adopted from Lamman et
al. (2023b). The full details can be found in this paper; we give only the
results here. We have made minor notation changes for clarity.

The IA signal $\mathcal{E}$ is combined with the effective LOS-distance $L$,
described in Section 3.1, and the nonlinear power spectrum $P$ as $\tau$:

$\tau=\frac{2L(r_{p})\mathcal{E}(r_{p})}{r_{p}\frac{d}{dr_{p}}\left[\frac{1}{r_{p}}\Psi\right]},$
(9)

$\Psi(R)=\int\frac{K\;dK}{2\pi}\frac{P(K)}{K}J_{1}(KR)$ (10)

Here, $K$ is 2D Fourier Space and $J_{1}$ is the first Bessel function. $\tau$
is measured independently in each bin of transverse separation, $r_{p}$. The
final variable used in our result, $\bar{\tau}$, is the average of these
determinations with standard error. The transverse bins we used for
determining $\tau$ were linear bins between $5-20h^{-1}$Mpc. Since these are
relatively large scales, the change from a linear to nonlinear power spectrum
had minimal effects on our final result, though it produced more consistent
values of $\tau$ across the transverse bins.

The "false" signature this produces in the quadrupole of the correlation
function, $\xi_{2}$, is

$\xi_{\text{2, gI}}(s)=\epsilon_{\rm
LOS}\frac{\bar{\tau}}{2\sigma_{E1}^{2}}\int\frac{q^{2}dq}{2\pi^{2}}P(q)j_{2}(qs).$
(11)

Here, $\epsilon_{\rm LOS}$ is the selection-induced shape polarization,
$\sigma_{E1}^{2}$ is the variance of the shape parameter $\epsilon_{+}$
described in Section 2, $j_{2}$ is the second spherical Bessel function, and
$s$ is 3D separation. The relations most relevant for this study can be
summarized as

$\xi_{\text{2, gI}}\propto\epsilon_{\rm
LRG}\frac{\bar{\tau}}{\sigma_{E1}^{2}}\propto\epsilon_{\rm
LRG}\frac{L\mathcal{E}}{\sigma_{E1}^{2}}.$ (12)

Note that this result is independent of the amplitude of the power spectrum
and galaxy bias, $b$. This is because $\xi_{\text{2, gI}}$ arises from the
correlation of the galaxy density field and the selection-induced shape
polarization, the latter of which is independent of bias. It does depend on
the projected correlation function $w_{p}$ through $L$. Also, since the IA
signal only affects $\xi_{\text{2, gI}}$ through $\bar{\tau}$, which can be
determined in transverse bins independently, we can forecast $\xi_{\text{2,
gI}}$ beyond the projected scales used to measure $\mathcal{E}$.

The variables measured for this estimate are listed in Table 1 and the final
quadrupole signature for all our samples is shown in Figure 9. To provide
context for this signal, we estimate the total quadrupole signatures $\xi_{2}$
expected for these galaxy samples. They are based on HOD catalogs made with
the AbacusSummit simulations (Hadzhiyska et al., 2021; Maksimova et al., 2021;
Yuan et al., prep), and scaled with measurements from DESI’s Survey Validation
(DESI Collaboration et al., 2023b). Figure 9 shows 1% of these estimates,
which is well above DESI’s total error budget for measuring $\xi_{2}$. Since
the $\xi_{2}$ signal created by the growth of structure is opposite in sign to
that created by IA, we have also multiplied this 1% line by -1 for a clear
comparison. On the scales used to measure $f\sigma_{8}$ (10 $<s<$ 80
$h^{-1}$Mpc), $\xi_{2}$ for LRGs will be dampened by around 0.15% between
redshifts 0.4-0.6, 0.53% between redshifts 0.6-0.8, and 0.80% between
redshifts 0.8-1.1. $\xi_{2}$ as measured by LRG x ELG cross-correlations will
be biased by around 0.83% between redshifts of 0.8-1.1.

We used a Nonlinear Alignment model, which has shown to be valid down to 6
$h^{-1}$Mpc for LRGs (Singh et al., 2015). In principle our estimate can be
extended down to the scales of the Fingers of God effect, where peculiar
velocities of galaxies create a "smearing" along the LOS, as opposed to the
"squashing" created by structure growth (Jackson, 1972). Here, the sign of
$\xi_{2}$ switches and this bias will result in an enhancement of the signal.
However, as nonlinear effects become more apparent here and and this effect is
less relevant for DESI’s main science goals, it is most valuable to interpret
the bias on large scales.

##  6 Conclusion

We measure the tidal alignment of LRGs with DESI Year 1 redshifts, using both
LRG and ELG tracers. We also estimate a redshift-dependent polarization in LRG
orientations relative to the LOS which arises from an aperture-based target
selection. Using a nonlinear tidal model, we calculate the signal this will
create in DESI’s measurements of the quadrupole of the correlation function.
It ranges from 0.2-1.1% of the quadrupole signal created by RSD, a significant
fraction of DESI’s full-survey error budget of around 0.4-0.7% for measuring
the growth rate with LRGs and ELGs combined.

The RSD bias is over five times larger in the highest redshift sample,
$0.8<z<1.1$, than the lowest, $0.4<z<0.6$. This is partially due to a stronger
alignment signal, but mostly due to the selection effect. Galaxies at higher
redshifts are redder and fainter, falling closer to the target selection cuts.
Therefore their orientation has a stronger influence on whether or not they
pass the aperture magnitude cut and the sample has a stronger orientation
polarization. If uncorrected for, this redshift-dependent bias will suppress
measurements of the growth rate at all scales, but especially higher
redshifts. Therefore it will also bias determinations of how the growth rate
evolves, a critical estimator for constraining cosmological models
(Kazantzidis & Perivolaropoulos, 2021). Our estimates of the quadrupole
signature created by IA presented here can be used to mitigate this bias.

These results agree with the previous work in Lamman et al. (2023a), which
measured the IA signal using photometric redshifts and estimated that it will
produce around a 0.5% decrease on measurements of $\xi_{2}$ with the full LRG
sample. While large, upcoming photometric surveys can provide constraints on
IA, for this effect it’s most important to understand the IA of our particular
sample. Additionally, redshifts are necessary to make clean distant cuts to
explore redshift dependence in both IA and the shape polarization.

The largest uncertainty in our final results comes from systematic effects in
the estimate of the selection-induced shape polarization. This is sensitive to
assumptions in the light profiles used for mock selection and the underlying
triaxial distribution of shapes, which is based on SDSS imaging (Padilla &
Strauss, 2008) and not in clear agreement with comparable galaxies in
hydrodynamic simulations (Bassett & Foster, 2019). This could be significantly
improved with a large imaging survey such as the Dark Energy Survey or the
upcoming Legacy Survey of Space and Time (Gatti et al., 2021; Ivezić et al.,
2019). The methods of imaging and shape fits have a known impact on the
inferred intrinsic shapes of galaxies (Georgiou et al., 2019; MacMahon &
Leonard, 2023).

While we do not expect a significant RSD bias from the shape alignment of
ELGs, their spins are known to correlate with the tidal field (Lee, 2011).
Their shapes could also be biased in DESI’s sample, as spectroscopic quality
depends upon disk orientation (Hirata, 2009). This bias could be explored in
DESI through correlations between the quadrupole of the correlation function
and fundamental plane residuals (Singh et al., 2021).

The remaining four years of DESI’s main survey will produce millions more
spectroscopic redshifts, allowing us to refine IA measurements and their
redshift dependence. These will also produce higher precision RSD
measurements, necessitating the need to incorporate the anisotropic clustering
effect caused by IA.

## Acknowledgements

CL thanks the DESI internal reviewers of this paper, Mustapha Ishak and
Benjamin Joachimi, for through feedback. CL also thanks Jonathan Blazek for
several helpful discussions.

This material is based upon work supported by the National Science Foundation
Graduate Research Fellowship under Grant No. DGE1745303, the U.S. Department
of Energy under grant DE-SC0013718, NASA under ROSES grant 12-EUCLID12-0004,
and the Simons Foundation.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of High-Energy Physics, under Contract No.
DE–AC02–05CH11231, and by the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility under the same contract.
Additional support for DESI was provided by the U.S. National Science
Foundation (NSF), Division of Astronomical Sciences under Contract No.
AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research
Laboratory; the Science and Technology Facilities Council of the United
Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation;
the French Alternative Energies and Atomic Energy Commission (CEA); the
National Council of Science and Technology of Mexico (CONACYT); the Ministry
of Science and Innovation of Spain (MICINN), and by the DESI Member
Institutions: https://www.desi.lbl.gov/collaborating-institutions.

The DESI Legacy Imaging Surveys consist of three individual and complementary
projects: the Dark Energy Camera Legacy Survey (DECaLS), the Beijing-Arizona
Sky Survey (BASS), and the Mayall z-band Legacy Survey (MzLS). DECaLS, BASS
and MzLS together include data obtained, respectively, at the Blanco
telescope, Cerro Tololo Inter-American Observatory, NSF’s NOIRLab; the Bok
telescope, Steward Observatory, University of Arizona; and the Mayall
telescope, Kitt Peak National Observatory, NOIRLab. NOIRLab is operated by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation. Pipeline
processing and analyses of the data were supported by NOIRLab and the Lawrence
Berkeley National Laboratory. Legacy Surveys also uses data products from the
Near-Earth Object Wide-field Infrared Survey Explorer (NEOWISE), a project of
the Jet Propulsion Laboratory/California Institute of Technology, funded by
the National Aeronautics and Space Administration. Legacy Surveys was
supported by: the Director, Office of Science, Office of High Energy Physics
of the U.S. Department of Energy; the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility; the U.S. National
Science Foundation, Division of Astronomical Sciences; the National
Astronomical Observatories of China, the Chinese Academy of Sciences and the
Chinese National Natural Science Foundation. LBNL is managed by the Regents of
the University of California under contract to the U.S. Department of Energy.
The complete acknowledgments can be found at https://www.legacysurvey.org/.

Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the U. S. National Science Foundation, the U. S. Department of Energy, or
any of the listed funding agencies.

The authors are honored to be permitted to conduct scientific research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation.

## Data Availability

The DESI Legacy Imaging Survey is publicly available at legacysurvey.org and
DESI’s Early Data Release is publicly available at
data.desi.lbl.gov/doc/releases/edr/. Iron covers the DESI Year 1 sample and
will be publicly released as part of DESI Data Release 1 (DR1). AbacusSummit
simulations are publicly available at abacusnbody.org. Code for projecting
ellipsoids and generating light profiles can be found at
github.com/cmlamman/ellipse_alignment.

Data plotted in this paper are available at zenodo.org/uploads/10162040.

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# Scaling of the entropy production rate in a $\varphi^{4}$ model of Active
Matter

Matteo Paoluzzi<EMAIL_ADDRESS>Departament de Física de la Matèria
Condensada, Universitat de Barcelona, C. Martí Franquès 1, 08028 Barcelona,
Spain.

###### Abstract

In active $\varphi^{4}$ field theories the nonequilibrium terms play an
important role in describing active phase separation; however, they are
irrelevant, in the renormalization group sense, at the critical point. Their
irrelevance makes the critical exponents the same as those of the Ising
universality class. Despite their irrelevance, they contribute to a nontrivial
scaling of the entropy production rate at criticality. We consider the
nonequilibrium dynamics of a nonconserved scalar field $\varphi$ (Model A)
driven out-of-equilibrium by a persistent noise that is correlated on a finite
timescale $\tau$, as in the case of active baths. We perform the computation
of the density of entropy production rate $\sigma$ and we study its scaling
near the critical point. We find that similar to the case of active Model A,
and although the nonlinearities responsible for nonvanishing entropy
production rates in the two models are quite different, the irrelevant
parameter $\tau$ makes the critical dynamics irreversible.

††preprint: APS/123-QED

## Introduction

Self-propelled particles can undergo a second-order phase transition whose
numerical estimate of the critical exponents are consistent with those of the
Ising universality class [1, 2]. This happens despite the microscopic
mechanism responsible for this non-equilibrium phase transition, i.e., the so-
called Motility-Induced Phase Separation, does not have any equilibrium
counterpart [3, 4]. Coarse-grained models of active particles suggest that
non-equilibrium gradient terms have to be taken into account for describing
the active system on a large scale, e.g., the so-called Active Model A, Active
Model B, etc… [5]. However, once we study the system around the critical
point, the non-equilibrium terms of active $\varphi^{4}$ field theories are
irrelevant in the Renormalization Group (RG) sense and thus they run to zero
under RG transformations [6]. It has been shown recently that, even though
irrelevant, the non-equilibrium gradient term $(\nabla\varphi)^{2}$ of Active
Model A/B causes an anomalous scaling of the density of entropy production
rate $\sigma$. This result suggests that non-equilibrium irrelevant terms have
to be carefully studied around the critical point. We can speculate that this
fact might open the way to the definition of new non-equilibrium universality
classes.

On the other hand, one can expect that the non-equilibrium terms considered in
Active Model A/B are not the only ingredients that one should take into
account for building an active $\varphi^{4}$ field theory. And thus, a natural
question is whether or not other non-equilibrium terms make the critical
dynamics effectively non-equilibrium. Usually, one assumes that the noise
acting on the coarse-grained fields has no memory, not in time, nor in space.
This assumption seems reasonable because, to describe the system on a large
scale, one has to maintain only the relevant degrees of freedom, e.g., the
order parameters, which are slow-varying variables. In this framework, the
noise represents the effect of the fast degrees of freedom on the slow ones
[7].

In the case of active systems, the noise is not necessarily delta-correlated,
as proved by experiments of passive beads in active baths [8, 9, 10].
Moreover, not only is the noise generated by the active bath correlated on a
finite time-scale $\tau$, but also it violates the Fluctuation-Dissipation
Theorem in a peculiar way [10], i.e., although the active bath exerts time-
correlated forces, linear response reveals an instantaneous friction kernel.

If we interpret the passive bead as a probe that captures the emerging
properties of the active bath, it is reasonable to assume that fluctuations
are not delta-correlated but they are characterized by a finite correlation
time. Moreover, recent numerical experiments suggest that even approaching the
MIPS critical point, i.e., in a situation where the critical slowing down
dominates over any other time scale, field theories driven by an
exponentially-correlated stochastic force well capture the non-thermal active
fluctuations [11]. It is known that, once we consider an exponentially
correlated noise in a system composed of discrete degrees of freedom, a non-
vanishing entropy production rate can be due only to non-linear interactions
[12, 13]. A natural question is what happens in the case of a field theory.

In this work, using the standard machinery of dynamical critical phenomena and
stochastic thermodynamics, we compute the entropy production rate in the case
of Model A driven by a persistent noise on a finite time scale $\tau$. We
focus our attention on the behavior of the system at the critical point and we
consider the situation where the transition is approached from the symmetric
phase, i.e., the system is not phase-separated, and thus there are no reasons
for including other non-equilibrium active terms. The computation does not
require any perturbation expansion neither in the persistence time $\tau$ nor
in the non-linear couplings of the field theory.

We obtain that all the non-linear terms (in principle infinite) appearing in
the effective Hamiltonian that describes the critical system contribute to the
entropy production. Moreover, focusing our attention on a $\varphi^{4}$
theory, we show that, in agreement with that observed in Active Model A [6],
the irrelevant parameter $\tau$ provides a source of entropy production at the
critical point.

## Model A driven by a persistent noise

As it has been recently shown in Ref. [2] (where one can find details about
the RG of Model B driven by a persistent noise), field theories with
exponentially correlated noise well reproduce the critical dynamics of MIPS
and are consistent with the Ising universality class. Here we perform the
computation of the entropy production rate to understand the impact of the
persistent noise on the critical dynamics. Following Ref. [7], we focus our
attention on the simplest case of Model A. We consider the non-equilibrium
dynamics of a non-conserved scalar field $\varphi=\varphi(x,t)$ in $d$ spatial
dimensions (with $x\equiv\mathbf{x}\in\mathbb{R}^{d}$) that is driven out-of-
equilibrium by a persistent stochastic force $\eta=\eta(x,t)$ that is
correlated on a finite time scale $\tau$. The stochastic dynamics of the two
fields are

$\displaystyle\dot{\varphi}$ $\displaystyle=-\Gamma_{0}\frac{\delta
H_{LG}}{\delta\varphi}+\eta$ (1) $\displaystyle\dot{\eta}$
$\displaystyle=-\frac{1}{\tau}\eta+\zeta\;,$ (2)

with the noise $\zeta=\zeta(x,t)$ satisfying

$\langle\zeta(x,t)\rangle=0\;,\;\;\langle\zeta(x,t)\zeta(y,s)\rangle=\frac{2D\Gamma_{0}}{\tau^{2}}\delta(t-s)\delta^{(d)}(x-y)\;.$
(3)

In the present work we set $\Gamma_{0}\\!=\\!1$. We now consider the most
general expression for the Landau-Ginzburg Hamiltonian $H_{LG}$ which is as
follows

$\displaystyle H_{LG}[\varphi]$ $\displaystyle=\int
dx\,\left(H_{0}+H_{I}\right)$ (4) $\displaystyle H_{0}$
$\displaystyle\equiv\frac{1}{2}\left[\mu(\nabla\varphi)^{2}+r\varphi^{2}\right]$
$\displaystyle H_{I}$
$\displaystyle\equiv\sum_{p=2}^{\infty}\frac{u_{p}}{(2p)!}\varphi^{2p}\;,$

where we adopt the notation $dx\equiv d^{d}x$. In Eq. (4) $H_{0}$ is the
Gaussian contribution and $H_{I}$ contains the non-linear interactions. The
coupling constant $r$ sets the distance from the mean-field critical point
that is located at $r\\!=\\!0$. We work under a condition such that the
critical point is always approached from from the disordered phase where
$r\geq 0$ and thus $\langle\varphi\rangle=0$. We stress that, since we are
interested in studying only the contribution of the persistent noise on the
entropy production rate around the critical point, we neglect other non-
equilibrium terms like those in Active Model A and B [5]. These terms surely
play an important role in active phase separation and thus have to be taken
into account for describing the spinodal decomposed active phases [14, 15]. To
establish the relevant parameters of the theory, we can perform the scaling
analysis of Eq. (1) that is obtained by considering Kadanoff transformations
with the scaling parameter $b$; i.e., we replace $x\to bx$, $t\to b^{z}t$, and
$\varphi\to b^{\chi}\varphi$ (more details about the scaling analysis of the
dynamical action can be found in Ref. [11]). We obtain the following scaling
transformations for the parameters of the theory

$\displaystyle r^{\prime}$ $\displaystyle=$ $\displaystyle rb^{z}$ (5)
$\displaystyle\mu^{\prime}$ $\displaystyle=$ $\displaystyle\mu b^{z-2}$
$\displaystyle u^{\prime}_{p}$ $\displaystyle=$ $\displaystyle
u_{p}b^{2(p-1)\chi+z}$ $\displaystyle D^{\prime}$ $\displaystyle=$
$\displaystyle Db^{z-2\chi-d}$ $\displaystyle\tau^{\prime}$ $\displaystyle=$
$\displaystyle\tau b^{-z}\;,$

where $z$ is the dynamic exponent and $\chi$ is the dimension of the scalar
field $\varphi$. Once we impose $\mu=\mu^{\prime}$, we get $z\\!=\\!2$, i.e.,
the usual $z$ exponent of Model A [7]. Because of that, $\tau$ is an
irrelevant parameter that runs to zero under RG transformations [16, 11] and
thus the critical exponents are the same as those of the Ising universality
class. However, non-universal quantities as the location of the critical point
might depend on $\tau$, as shown in Ref. [17]. From the scaling of the noise
strength $D$ we obtain that the natural dimension of the field is
$\chi=1-d/2$, i.e., which is obtained by setting $D\\!=\\!D^{\prime}$. The
scaling dimension of the couplings constants $u_{p}$ sets the (upper) critical
dimensions of each non-linear interaction, i.e., $d_{c}=4$ for $u_{2}$ (which
we name $u$), $d_{c}=3$ for $u_{3}$, etc… meaning that the $\varphi^{4}$
interaction can be neglected above $4$ and $\varphi^{6}$ interactions can be
neglected above $d=3$ [18]. As discussed in Ref. [6], non-equilibrium control
parameters might play a role at the critical point even if they are irrelevant
in the RG sense. To study the impact of $\tau$ on critical dynamics, we
compute the entropy production rate of the model.

Following stochastic thermodynamics [19, 20, 21], we compute the steady-state
entropy production rate $\mathcal{S}$

$\displaystyle\mathcal{S}$
$\displaystyle=\lim_{t_{F}\to\infty}\frac{1}{t_{F}}\left\langle\log\frac{P[\varphi]}{P_{R}[\varphi]}\right\rangle\;.$
(6)

with $P[\varphi]$ indicating the probability of a path
$\\{\varphi(x,t)\\}_{t_{0}\leq t\leq t_{F}}$, with $t_{0}$ and $t_{F}$ being
the initial time and final time, respectively. In Eq. (6), $P_{R}[\varphi]$ is
the probability for the time-reversed trajectory that can be computed
considering the transformations $t\to t_{F}-t$ and
$\varphi\to\varphi_{R}(x,t)=\varphi(x,t_{F}-t)$. The average
$\langle\cdot\rangle$ is performed with respect to the noise realizations.
Under suitable ergodicity assumptions that we make in the present work, when
it is necessary, we replace averages over independent noise realizations with
the average over a long trajectory.

As we shall see, the probability $P[\varphi]$ can be written in terms of a
dynamical action $A[\varphi]$, i.e., the Onsager–Machlup action, so that

$\displaystyle P[\varphi]$ $\displaystyle\propto$
$\displaystyle\exp{\left(-A[\varphi]\right)}$ (7) $\displaystyle A[\varphi]$
$\displaystyle=$ $\displaystyle\int dx\,dt\;\mathcal{L}[\varphi]\;.$ (8)

Here we have written the dynamical action $A[\varphi]$ in terms of the
”density” $\mathcal{L}[\varphi]$. Similarly for $P_{R}[\varphi]$ one has
$P_{R}[\varphi]\propto\exp{\left(-A_{R}[\varphi]\right)}$ so that

$\displaystyle\mathcal{S}$
$\displaystyle=\lim_{t_{F}\to\infty}\frac{1}{t_{F}}\int_{t_{0}}^{t_{0}+t_{F}}dt\,dx\,\left\\{\mathcal{L}_{R}[\varphi]-\mathcal{L}[\varphi]\right\\}\;.$
(9)

## Onsager-Machlup dynamical action

To proceed with the program illustrated above, it is convenient to write the
dynamics in the Fourier space. We consider the system closed in a
$d-$dimensional box of side $L$ and the volume is $V=L^{d}$ where periodic
boundary conditions are employed. We can thus write $\varphi(x,t)$ and
$\eta(x,t)$ in terms of their space-Fourier components as follows [22, 23, 18]

$\displaystyle\varphi(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{L^{d/2}}\sum_{|k|\in(0,\Lambda)}e^{ikx}\varphi_{k}(t)$
(10) $\displaystyle\eta(x,t)$ $\displaystyle=$
$\displaystyle\frac{1}{L^{d/2}}\sum_{|k|\in(0,\Lambda)}e^{ikx}\eta_{k}(t)$

where we have introduced the ultraviolet cutoff $\Lambda=\frac{2\pi}{a}$ and
$a$ is the lattice spacing (or the typical particle size) used as a regulator.
In the Fourier space, the Hamiltonian is

$\displaystyle H_{LG}$ $\displaystyle=H_{0}+H_{I}$ (11) $\displaystyle H_{0}$
$\displaystyle=\frac{1}{2}\sum_{k}\left(\mu k^{2}+r\right)|\varphi_{k}|^{2}$
$\displaystyle H_{I}$
$\displaystyle=\sum_{p=2}^{\infty}\frac{u_{p}}{(2p)!}\frac{1}{V^{-p+1}}\sum_{k_{1},...,k_{2p}}\varphi_{k_{1}}\varphi_{k_{2}}\dots\varphi_{k_{2p}}\delta_{k_{1}+\dots+k_{2p}}\;$

where $H_{I}$ contains all the non-linear interactions. In this way, the
dynamics is as follows:

$\displaystyle\dot{\varphi}_{k}$ $\displaystyle=$
$\displaystyle-\frac{\partial H_{LG}}{\partial\varphi_{k}}+\eta_{k}$ (12)
$\displaystyle\dot{\eta}_{k}$ $\displaystyle=$
$\displaystyle-\frac{1}{\tau}\eta_{k}+\zeta_{k}$ (13)

with

$\displaystyle\langle\zeta_{k}(t)\rangle=0,\;\langle\zeta_{k}(t)\zeta_{q}(s)\rangle=\frac{2D}{\tau^{2}}\delta_{k,-q}\delta(t-s)\;.$

For making progress, we perform the time derivative of Eq. (12) obtaining

$\displaystyle\ddot{\varphi}_{k}$
$\displaystyle+\mathbb{H}_{kq}\dot{\varphi}_{q}+\frac{1}{\tau}\left[\dot{\varphi}_{k}+F_{k}\right]=\zeta_{k}\;$
(14) $\displaystyle F_{k}$ $\displaystyle\equiv\frac{\partial
H_{LG}}{\partial\varphi_{k}}$ $\displaystyle\mathbb{H}_{kq}$
$\displaystyle\equiv\frac{\partial^{2}H_{LG}}{\partial\varphi_{k}\partial\varphi_{q}}\;,$

where we have adopted the Einstein summation convention. In order to obtain
the expression of the probability distribution $P[\varphi]$, we start with
introducing the expectation value $\langle\mathcal{O}\rangle$ of a generic
observable $\mathcal{O}[\varphi]$, which is

$\displaystyle\langle\mathcal{O}\rangle=\int\mathcal{D}[\zeta_{k}]\mathcal{D}[\varphi_{k}]\,\mathcal{P}[\zeta]\mathcal{O}[\varphi]\delta\left[\varphi-\varphi_{\zeta}\right]$
(15)

with $\varphi_{\zeta}$ indicating a solution of the Langevin equation (14) and
$\mathcal{P}[\zeta]$ the Gaussian distribution of the noise $\zeta_{k}$. We
now perform some standard manipulations [24, 25, 26, 27]. First, we perform
the change of variable

$\displaystyle\delta\left[\varphi-\varphi_{\xi}\right]$
$\displaystyle=\mathcal{J}[\varphi]\delta\left[\ddot{\varphi}_{k}+\mathbb{H}_{kq}\dot{\varphi}_{q}+\frac{1}{\tau}\left[\dot{\varphi}_{k}+F_{k}\right]-\zeta_{k}\right]$
$\displaystyle\mathcal{J}[\varphi]$
$\displaystyle\equiv|\det\frac{\delta\xi_{k}(t)}{\delta\varphi_{q}(t^{\prime})}|\;.$

Second, we represent the delta-functional through a set of response fields
$\hat{\varphi}_{k}$. Third, we perform the Gaussian integral over the noise
$\zeta_{k}$. In the following, we neglect the contribution of
$\mathcal{J}[\varphi]$ since it does not contribute to the entropy production
rate [28, 29]. These standard manipulations [25, 26, 27] lead to

$\displaystyle\langle\mathcal{O}\rangle$
$\displaystyle=\int\mathcal{D}[\varphi_{k}]\mathcal{D}[\hat{\varphi}_{k}]\,e^{-S[\hat{\varphi},\varphi]}\mathcal{O}[\varphi]\;,$
(16)

where we $S[\hat{\varphi},\varphi]$ is the so-called Janssen-De Dominicis
response functional [24, 30, 31]

$\displaystyle S[\hat{\varphi},\varphi]$ $\displaystyle\equiv$
$\displaystyle\frac{D}{2\tau^{2}}\sum_{k}\int_{t_{0}}^{t_{F}}dt\hat{\varphi}_{k}^{2}$
$\displaystyle-$
$\displaystyle\frac{1}{\tau}\sum_{k}\int_{t_{0}}^{t_{F}}dt\hat{\varphi}_{k}\left[\tau\ddot{\varphi}_{k}+M_{kq}\dot{\varphi}_{q}+F_{k}\right]$
$\displaystyle M_{kq}$ $\displaystyle\equiv$
$\displaystyle\delta_{kq}+\tau\frac{\partial^{2}H_{LG}}{\partial\varphi_{k}\partial\varphi_{q}}\;.$

After performing the Gaussian integration over $\hat{\varphi}_{k}$, we finally
obtain that the probability $P[\varphi]$ of a path
$\\{\varphi_{k}(t)\\}_{t_{0}\leq t\leq t_{F}}$ can be expressed in terms of
the following Onsager-Machlup action (the proportional symbol indicates that
we are neglecting factors that do not contribute to the computation of the
entropy production rate)

$\displaystyle P[\varphi]$ $\displaystyle\propto$
$\displaystyle\exp{\left(-A[\varphi]\right)}$ (18) $\displaystyle A[\varphi]$
$\displaystyle=$
$\displaystyle\frac{1}{2D}\sum_{k}\int_{t_{0}}^{t_{F}}dt\,\left[\tau\ddot{\varphi}_{k}+M_{kq}\dot{\varphi}_{q}+F_{k}\right]^{2}\;.$

Since we are interested in the critical behavior of the system by approaching
the critical point from the disordered phase where $\langle\varphi\rangle=0$,
the dynamical action $A[\varphi]$ is well-defined for every value of $\tau$.
In such a situation the Hessian matrix $\mathbb{H}_{kq}$ is positive-definite
and so $M_{kq}$ is positive-definite as well.

The probability of the time-reversed path can be obtained by employing the
time-reversal operator $t\to t_{F}-t$ that brings us to the following
expressions

$\displaystyle P_{R}[\varphi]$ $\displaystyle\propto$
$\displaystyle\exp{\left(-A_{R}[\varphi]\right)}$ (19) $\displaystyle
A_{R}[\varphi]$ $\displaystyle=$
$\displaystyle\frac{1}{2D}\sum_{k}\int_{t_{0}}^{t_{F}-t_{0}}dt\,\left[\tau\ddot{\varphi}_{k}-M_{kq}\dot{\varphi}_{q}+F_{k}\right]^{2}\,.$

## Entropy Production Rate

Now we can perform the computation of the entropy production rate using the
expression

$\displaystyle S$ $\displaystyle=$
$\displaystyle\lim_{t_{F}\to\infty}\frac{1}{2Dt_{F}}\sum_{k}\int_{t_{0}}^{t_{F}-t_{0}}dt\left\\{\left[\tau\ddot{\varphi}_{k}+M_{kq}\dot{\varphi}_{q}+F_{k}\right]^{2}\right.$
(20) $\displaystyle-$
$\displaystyle\left[\tau\ddot{\varphi}_{k}-M_{kq}\dot{\varphi}_{q}+F_{k}\right]^{2}\left.\right\\}\;,$

which brings us to

$\displaystyle\mathcal{S}$
$\displaystyle=\lim_{t_{F}\to\infty}\frac{1}{Dt_{F}}\sum_{q}\int_{t_{0}}^{t_{F}-t_{0}}dt\,\left[A_{q}+B_{q}\right]$
(21) $\displaystyle A_{q}$
$\displaystyle\equiv\dot{\varphi}_{q}\left[\tau\ddot{\varphi}_{q}+F_{q}+\tau\mathbb{H}_{kq}F_{k}\right]$
$\displaystyle B_{q}$
$\displaystyle\equiv\tau^{2}\ddot{\varphi}_{k}\mathbb{H}_{kq}\dot{\varphi}_{q}\;.$

The term $A_{q}$ can be written as

$\displaystyle A_{q}$ $\displaystyle=\frac{d}{dt}Q_{q}$ (22) $\displaystyle
Q_{q}$ $\displaystyle=\frac{\tau}{2}\dot{\varphi}_{q}^{2}+H_{LG}+\tau
F_{q}\;.$

The time integration of $A_{q}$ is a border term and thus vanishes in the
limit $t_{F}\to\infty$ because of the factor $t_{F}^{-1}$. After integrating
by parts $B_{q}$, we finally get

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle-\lim_{t_{F}\to\infty}\frac{\tau^{2}}{Dt_{F}}\int_{t_{0}}^{t_{F}-t_{0}}dt\,\mathbb{H}_{kq}\ddot{\varphi}_{k}\dot{\varphi}_{q}\;.$
(23)

Performing an integration by parts and neglecting vanishing boundary terms, we
obtain the following expression for the entropy production rate

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\frac{\tau^{2}}{2D}\left\langle\dot{\varphi}_{k}\dot{\varphi}_{q}\dot{\varphi}_{l}G_{kql}\right\rangle$
(24) $\displaystyle G_{kql}$ $\displaystyle\equiv$
$\displaystyle\frac{\partial^{3}H_{LG}}{\partial\varphi_{k}\partial\varphi_{q}\partial\varphi_{l}}=\frac{\partial^{3}H_{I}}{\partial\varphi_{k}\partial\varphi_{q}\partial\varphi_{l}}\;.$
(25)

This is a central result of this paper: From Eqs. (24) and (25) we immediately
realize that, in principle, all the non-linear terms in a Landau-Ginzburg
model contribute to the entropy production. Moreover, the expression obtained
here generalized to field theories the entropy production rate formula
obtained in the case of Active Ornstein-Uhlenbeck particles [13, 12]. Finally,
no matter how big $\tau$ is, we notice that for a Gaussian model one has
$\mathcal{S}=0$. Using Eqs. (24) and (25), $\mathcal{S}$ can be computed
numerically by solving the equations of motion. This can be done, for
instance, by discretizing the dynamics on a $d-$dimensional grid.

Once we perform the inverse Fourier transform, we get the following expression

$\displaystyle\mathcal{S}$ $\displaystyle=\int dx\,\sigma(x)$ (26)
$\displaystyle\sigma$
$\displaystyle\equiv\frac{\tau^{2}}{2D}\left\langle\dot{\varphi}(x)^{3}\frac{\delta^{3}H_{I}}{\delta\varphi(x)^{3}}\right\rangle\;,$

where $\sigma$ is the density of the entropy production rate.

### The critical scaling

We now study the scaling of $\sigma$ close to the critical point in the case
of the $\varphi^{4}$ theory. For doing that, since $\varphi$ satisfies the
equation of motion (1), we can write

$\displaystyle\sigma=\frac{\tau^{2}}{2D}\left\langle(-\frac{\delta
H_{LG}}{\delta\varphi}+\eta)^{3}\frac{\delta^{3}H_{LG}}{\delta\varphi^{3}}\right\rangle_{\eta}\;.$
(27)

Different from Active Model A, where averages involve stochastic trajectories
of $\varphi$ driven by a delta-correlated noise, here the stochastic force
$\eta$ is time-correlated. Because of that, we do not have, in general, a way
for evaluating analytically the scaling of $\sigma$. However, since we aim to
study the system in the vicinity of the critical point, we consider some
reasonable assumptions for making the problem tractable. For making progress,
we notice that, since $\tau$ is an irrelevant parameter, its renormalized
value goes to zero on the critical surface. Once we approach the critical
point we can thus replace the correlated noise over $\tau$ with a white noise
that is delta-correlated. It is worth noting that this approximation has to be
reconsidered in the case of large $\tau$ values. Moreover, as well as in the
case of Active Ornstein-Uhlenbeck particles [13], we replace the average over
trajectories with an average over the stationary distribution
$P_{s}[\varphi]$, i.e.,
$\langle\mathcal{O}[\varphi]\rangle_{\eta}=\int\mathcal{D}[\varphi]P_{s}[\varphi]\mathcal{O}[\varphi]$.
Again, because $\tau$ is an irrelevant parameter these averages can be done by
replacing the stationary distribution with the Boltzmann distribution, i.e.,
$P_{s}[\varphi]\simeq e^{-\beta H_{LG}}$, with the inverse temperature $\beta$
set to $1$ in the following. Once we perform these approximations, we obtain
that $\sigma$ results from four contributions (in the following, the
proportional symbol indicates that we are neglecting numerical factors that
are inessential for our purpose)

$\displaystyle\sigma$
$\displaystyle\propto\sigma_{1}+\sigma_{2}+\sigma_{3}+\sigma_{4}$ (28)
$\displaystyle\sigma_{1}$
$\displaystyle\equiv\frac{\tau^{2}ur^{3}}{D}\langle\varphi^{4}\rangle$
$\displaystyle\sigma_{2}$
$\displaystyle\equiv\frac{\tau^{2}u\mu^{3}}{D}\langle(\nabla^{2}\varphi)^{3}\varphi\rangle$
$\displaystyle\sigma_{3}$
$\displaystyle\equiv\frac{\tau^{2}r^{2}\mu}{D}\langle\varphi^{3}\nabla^{2}\varphi\rangle$
$\displaystyle\sigma_{4}$
$\displaystyle\equiv\frac{\tau^{2}ur\mu^{2}}{D}\langle(\nabla^{2}\varphi)^{2}\varphi^{2}\rangle\;.$

We first consider the four contributions above at the Gaussian level, i.e., by
performing averages using the Gaussian measure given by $P_{s}=e^{-H_{0}}$. In
the Gaussian model, we obtain that the only non-vanishing contribution in the
limit $k\to 0$ is provided by $\sigma_{1}$. In particular, we get

$\displaystyle\sigma_{1}\propto\frac{\tau^{2}u}{D}r^{1+d/2}\;$ (29)

indicating that, at the mean-field level, i.e., above the upper critical
dimension, the density of the entropy production rate tends to zero at the
critical point, i.e., $\sigma\to 0$ for $r\to 0$. More in general, by
performing the naive Gaussian scaling analysis of $\sigma$, we obtain that it
follows the natural scaling, i.e., $\sigma\to\sigma b^{-(d+z)}$, where the
arrow indicates how $\sigma$ scales under Kadanoff transformations. As
discussed in Ref. [6] in the case of Active Model A, the fact that $\sigma$
goes to zero at the critical point does not guarantee that the dynamics is
effectively reversible at criticality because what is really matter is the
scaling of the singular part of the observable we are interested in. The
suitable observable for doing that in the case of $\sigma$ is the density of
the entropy production rate per spacetime correlation volume $\psi$ defined as
$\psi\equiv\xi^{d+z}\sigma$ [6]. Since $\xi\sim r^{-\nu}$, and the mean-field
values are $\nu_{MF}=1/2$ and $z_{MF}=2$, we obtain that
$\psi=\xi^{d+2}\xi^{-\frac{1}{\nu_{MF}}\frac{d+2}{2}}=1$ meaning that, if we
look for a singular part of $\psi$ that behaves as $\psi\sim
r^{-\theta_{\sigma}}$, we get the mean-field value $\theta_{\sigma}=0$, which
is in agreement also with Active Models A/B [6]. As discussed in detail in
Ref. [6], a non-negative value of $\theta_{\sigma}$, i.e.,
$\theta_{\sigma}\geq 0$, indicates already that, although the universality
class of the model remains untouched, the dynamics remains irreversible even
at the critical point.

We shall now provide arguments that support an exponent $\theta_{\sigma}$ of
order $\epsilon$ around the upper critical dimension. First of all, we notice
that the only contribution to $\sigma$ that depends on a relevant operator is
$\sigma_{1}$. In particular, the other composite operators in $\sigma_{2,3,4}$
have naive dimension $m+n>4$, with $m$ being the number of fields and $n$
being the number of gradients. In order to understand the scaling of
$\sigma_{1}$, we consider the following perturbative RG transformation (at the
lowest order in $u$). Using perturbation theory, one has
$\langle\mathcal{O}[\varphi]\rangle=\int\mathcal{D}[\varphi]\left(1-H_{I}\right)e^{-H_{0}}\mathcal{O}[\varphi]$
[18]. Since the observable we are going to consider is
$\mathcal{O}=\varphi^{4}$, at the lowest order, one has
$\sigma_{1}\\!=\\!\frac{\tau^{2}ur^{3}}{D}\langle\varphi^{4}\rangle_{H_{0}}$.
Once we consider the scaling transformations (5), the standard one-loop RG
equations bring us to the following scaling [32, 24] (here we are using the
fact that, under RG transformations, $r=r^{\prime}b^{-(z+\epsilon/3)}$,
$r\tau\\!=\\!r^{\prime}\tau^{\prime}\\!=\\!1$ and
$D\\!=\\!D^{\prime}\\!=\\!1$)

$\displaystyle\sigma_{1}=b^{\epsilon/3}b^{-(z+d)}r^{\prime}u^{\prime}\langle(\varphi^{\prime})^{4}\rangle$
(30)

with $\epsilon=4-d$. As a result, in this framework, the scaling of $\psi$ is
$\psi\sim r^{-\theta_{\sigma}}$ with $\theta_{\sigma}=\frac{\nu\epsilon}{3}$
which is positive below the upper critical dimension.

## Discussion and Conclusions

In this work, motivated by the experimental and numerical evidence that active
baths always develop correlations on finite time scales [9, 10, 11], we have
computed the entropy production rate in the case of a scalar field theory
(with a non-conserved order parameter) driven out-of-equilibrium by a
persistent noise. The computation predicts a non-vanishing entropy production
rate only when non-linear interactions are taken into account (with a
vanishing entropy production rate at equilibrium, i.e., that is recovered for
$\tau\\!=\\!0$). We have performed a scaling analysis of the density of the
entropy production rate $\sigma$ near the critical point in a static picture
by replacing the averages over the persistent dynamics with averages over
equilibrium configurations. The computation provides evidence in favor of an
exponent $\theta_{\sigma}$ that is positive below the upper critical dimension
and negative above it. However, even in this simplified framework, to have a
better estimate of the critical exponent, the finest computations involving
the scaling of composite operators are required [22]. Moreover, it will be
crucial to test those predictions against numerical data in three dimensions.
Since the non-linear terms responsible for a non-vanishing $\sigma$ in the
model presented here are different from those in Active Model A, it is quite
natural to obtain different critical scaling of $\sigma$. In Active Model A/B
the leading non-equilibrium contribution is provided by a non-integrable
gradient term, while the stochastic force at the coarse-grained level is
considered equilibrium-like. Here we neglected non-equilibrium gradient terms
and we focused our attention on the role of a non-equilibrium noise (as a step
forward, it might be interesting to consider both contributions). It would be
also interesting to investigate possible connections between the scaling of
$\sigma$ and other thermodynamics anomalies observed at the microscale [33].

As a future direction, the computation of $\sigma$ presented here might be
extended to the case of Model B and in the presence of noise that is also
correlated in space [11]. It might be also interesting to compare the
phenomenological model presented here with coarse-graining descriptions of
active systems obtained from microscopic models [34].

## Acknowledgments

I am deeply grateful to Mattia Scandolo (as well as to all the CoBBS group in
Rome) for his comments and his critical reading of the manuscript. I also
thank Claudio Maggi and Andrea Puglisi for illuminating discussions and for
their critical reading of the early version of the manuscript. This work has
received funding from the European Union’s Horizon 2020 research and
innovation programme under the MSCA grant agreement No 801370 and by the
Secretary of Universities and Research of the Government of Catalonia through
Beatriu de Pinós program Grant No. BP 00088 (2018).

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