# Non-Gaussian assembly bias from a semi-analytic galaxy formation model

M. Marinucci,1 V. Desjacques,1,2 A. Benson,3
1Physics department, Technion, 3200003 Haifa, Israel
2Asher Space Research Institute, Technion, 3200003 Haifa, Israel
3Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use $z=1$ mock galaxy catalogues produced with the semi-analytic code
GALACTICUS to study the dependence of the non-Gaussian bias parameter
$b_{\phi}$ on the mass assembly history of the host halos. We generate large
sets of merger trees and measure the non-Gaussian assembly bias $\Delta
b_{\phi}$ for galaxies selected by color magnitude and emission line
luminosities. For galaxies selected by $g-r$ color, we find a large assembly
bias consistent with the analysis of Barreira et al. (2020) based on hydro-
dynamical simulations of galaxy formation. This effect arises from the fact
that a larger value of the normalization amplitude $\sigma_{8}$ implies a
faster mass assembly (at fixed halo mass) and, therefore, older and redder
galaxies. On the contrary, for galaxies selected by their H$\alpha$
luminosity, we do not detect a significant assembly bias, at least at $z=1$
and in the halo mass range $3\times 10^{10}<M<10^{12}\ M_{\odot}$ considered
here. This is presumably due to the fact that emission line strengths are
mainly sensitive to the instantaneous star formation rate, which appears to
depend weakly on $\sigma_{8}$ at $z=1$. This indicates that the non-Gaussian
assembly bias should be less of a concern for future emission line galaxy
surveys.

We also investigate, for the first time, the sensitivity of the non-Gaussian
assembly bias to a change in the parameters of the galaxy formation model that
control the AGN and stellar feedback as well as the star formation rate. When
these parameters change within a factor of two from their fiducial value, they
induce variations up to order unity in the measured $\Delta b_{\phi}$, but the
overall trends with color or luminosity remain the same. However, since these
results may be sensitive to the choice of galaxy formation model, it will be
prudent to extend this analysis to other semi-analytic models in addition to
halo mass and redshift.

††pubyear: 2023††pagerange: Non-Gaussian assembly bias from a semi-analytic
galaxy formation model–References

##  1 Introduction

Understanding the origin of the fluctuations in the initial conditions of the
density distribution remains one of the most prominent questions of modern
cosmology. A potentially detectable amount of primordial non-Gaussianity (PNG)
can be produced when (at least) one of the assumptions of single field, slow
roll inflation is violated (Salopek & Bond, 1990; Falk et al., 1992; Gangui et
al., 1994; Acquaviva et al., 2003; Maldacena, 2003; Creminelli & Zaldarriaga,
2004; Creminelli et al., 2011; Tanaka & Urakawa, 2011). In particular,
multifield models of inflation Linde & Mukhanov (1997); Lyth et al. (2003) can
generate a primordial bispectrum (3-point correlation function) which peaks in
the squeezed limit. A detection of this bispectrum shape parametrized by
$f_{\text{NL}}$ would thus rule out single field slow roll inflation. The
current best constraints on PNG of the local type is $f_{\text{NL}}=-0.9\pm
5.1$ at 68$\%$ C.L. and comes from the temperature anisotropies of the cosmic
microwave background (Akrami et al., 2020a). In the coming years galaxy
surveys (such as Euclid Laureijs et al. (2011); Amendola et al. (2018),
SPHEREx (Doré et al., 2014)) are expected to give competitive constraints
thanks to their large volumes, the redshifts covered and the higher number of
tracers observed.

The clustering of large scale structure tracers is sensitive to a local PNG
through a scale-dependent term proportional to the product
$f_{\text{NL}}b_{\phi}$, where $b_{\phi}$ is the non-Gaussian bias (Dalal et
al., 2008). Therefore, the tightest constraints on $f_{\text{NL}}$ will be
achieved if good priors on $b_{\phi}$ are available. The peak-background split
argument implies a relation between $b_{\phi}$, the galaxy number density
$\bar{n}_{g}$, and the normalisation amplitude $\sigma_{8}$ (Slosar et al.,
2008). However, while this prediction has been thoroughly tested at the level
of dark matter halos, it has been used only recently to measure $b_{\phi}$ for
mock galaxies extracted from detailed hydrodynamical simulations. Assuming
universality of the halo mass function, a "universal" relation
$b_{\phi}=2\delta_{c}(b_{1}-1)$ holds between $b_{\phi}$ and $b_{1}$, the
large scale galaxy bias, where $\delta_{c}$ is the critical overdensity. The
current analyses using galaxy clustering data are performed assuming this
$b_{\phi}(b_{1})$ relation which is expected to hold only for halos selected
by virial mass: ref. D’Amico et al. (2022) found $f_{\text{NL}}=-30\pm 29$ at
68$\%$ C.L while ref. Cabass et al. (2022) found $f_{\text{NL}}=-33\pm 28$ at
68$\%$ C.L assuming $b_{\phi}=2\delta_{c}(b_{1}-0.55)$. As emphasized by e.g.
Slosar et al. (2008); Reid et al. (2010); Barreira (2020, 2022d), constraints
on $f_{\text{NL}}$ are sensitive to the prior imposed on $b_{\phi}$ (wrong
priors can lead to large systematic biases in the inferred values of
$b_{\phi}$). The latter can significantly deviate from the $b_{\phi}(b_{1})$
expectation if there is significant assembly bias.

In the last decade, much attention has been devoted to the assembly bias,
which is the fact that the clustering of dark matter halos and galaxies is not
solely determined by the mass of the virialized halo at the formation epoch,
but also by additional factors such as the formation history and environment.
Assembly bias was first introduced by Sheth & Tormen (2004); Gao et al.
(2005), who noticed that older dark matter halos preferentially reside in
overdense regions and, therefore, tend to be more clustered than younger halos
with the same halo mass. This led to the conclusion that there exist
additional factors which influence the clustering of halos beyond their mass.
Since then, numerous studies have attempted to quantify and understand the
nature of assembly bias for dark matter halos and galaxies (e.g. Gao & White,
2007; Wechsler et al., 2006; Giocoli et al., 2007; Jing et al., 2007; Keselman
& Nusser, 2007; Croton et al., 2007; Desjacques, 2008; Codis et al., 2012;
Aung & Cohn, 2016; Chaves-Montero et al., 2016; Miyatake et al., 2016;
Paranjape & Padmanabhan, 2017; Paranjape et al., 2018; Zehavi et al., 2018;
Shi & Sheth, 2018; Ramakrishnan et al., 2019; Hellwing et al., 2021).

The first estimates of the assembly bias dependence of $b_{\phi}$ (Slosar et
al., 2008; Reid et al., 2010) relied on the excursion set approach to halo
formation (e.g. Bond et al., 1991; Lacey & Cole, 1993) and on N-body
simulations. In particular, Reid et al. (2010) found that, at fixed halo mass,
the non-Gaussian bias $b_{\phi}$ depends strongly on the halo formation time,
with older halos having a much larger $b_{\phi}$ than younger ones. Recently,
Lazeyras et al. (2023) showed that $b_{\phi}$ is also correlated with halo
properties such as the spin or concentration. Furthermore, Barreira et al.
(2020); Barreira (2022c) measured $b_{\phi}$ for mock galaxies extracted from
the Illustris TNG suite of hydrodynamical simulations (Barreira et al., 2020;
Barreira, 2022b, c) and found that their measurements can largely deviate from
the $b_{\phi}(b_{1})$ relation, not only due to the convolution with the halo
occupation distribution (HOD) but also because of assembly bias. While this
may be a concern for the accuracy of $f_{\text{NL}}$-constraints, one could
also take advantage of a large non-Gaussian assembly bias to construct galaxy
samples which enhance the information on $f_{\text{NL}}$ (see Barreira &
Krause, 2023, for a recent study).

In this paper, we use a semi-analytical approach to galaxy formation in order
to study the impact of mass assembly on the non-Gaussian bias of $z=1$
galaxies. We consider two different galaxy properties: colour magnitudes and
emission line strengths. These two observables depend on the formation history
of the galaxy in a different way. They are basic galaxy properties measured in
current and future galaxy surveys such as SDSS-IV/eBOSS, DESI, Euclid or
SPHEREx (see, e.g., Dawson et al., 2016; Laureijs et al., 2011; Aghamousa et
al., 2016a, b; Doré et al., 2014). Although semi-analytical models (or SAMs,
see Kauffmann et al., 1993; Cole et al., 2000; Croton et al., 2006; Somerville
et al., 2008; Benson, 2012, for instance) of galaxy formation are not as
realistic as full hydro-dynamical simulations, they are realistic and
versatile enough to allow for a thorough study of the assembly bias as will be
demonstrated below. Therefore, our study nicely complements the work of
Barreira et al. (2020); Barreira (2022d), who relied on detailed hydro-
dynamical simulations, and also extends it to include emission lines. Our
paper is organized as follows. We provide a quick theoretical overview and
describe our methodology in §2. We present our results in §3, and conclude in
§4.

##  2 Theoretical background and methodology

###  2.1 Primordial non-Gaussianity and galaxy bias

Local primordial non-Gaussianity can be conveniently parametrized through the
mapping (Salopek & Bond, 1990; Gangui et al., 1994; Verde et al., 2000;
Komatsu & Spergel, 2001)

$\phi(\textbf{\em x})=\phi_{G}(\textbf{\em
x})+f_{\text{NL}}\left[\phi_{G}(\textbf{\em
x})^{2}-\left\langle\phi_{G}(\textbf{\em x})^{2}\right\rangle\right],$ (1)

where the primordial gravitational potential $\phi(\textbf{\em
x})=(3/5)\mathcal{R}(\textbf{\em x})$ is defined as Bardeen’s curvature
perturbation immediately after equality, $\phi_{G}$ is a Gaussian distributed
random field, and $f_{\text{NL}}$ is a parameter that quantifies the level of
non-Gaussianity of the spatial distribution of the primordial potential
(Komatsu & Spergel, 2001). The power spectrum of $\phi_{G}$ is
$P_{\phi}(k)\propto k^{n_{s}-4}$, where $n_{s}<1$ is the scalar spectral
index.

Large scale structure (LSS) tracers are especially sensitive to local
primordial non-Gaussianity and can thus be used to constrain $f_{\text{NL}}$
(see Biagetti, 2019, for a recent review). In particular, local PNG introduces
a coupling between large and small scales in the primordial fluctuations which
leaves a broadband, scale-dependent imprint in the Fourier modes of LSS
tracers. This leads to a scale-dependent bias at linear level (Dalal et al.,
2008; Slosar et al., 2008; Matarrese & Verde, 2008; McDonald, 2008;
Giannantonio & Porciani, 2010; Schmidt & Kamionkowski, 2010; Desjacques et
al., 2011a), i.e.

$\delta_{g}(\textbf{\em k},z)=b_{1}(z)\delta_{m}(\textbf{\em
k},z)+b_{\phi}(z)f_{\text{NL}}\phi(\textbf{\em k})\;.$ (2)

Here, k is the wavenumber and $\delta_{m}$ is the evolved matter density
contrast. While the linear bias parameter $b_{1}$ arises also when the
primordial curvature perturbation is purely Gaussian (Kaiser, 1984; Bardeen et
al., 1986; Cole & Kaiser, 1989; Mo & White, 1996; Sheth & Tormen, 1999), the
non-Gaussian bias $b_{\phi}$ contributes to the observed galaxy clustering
only if $f_{\text{NL}}\neq 0$. The value of $b_{1}$ can be extracted from the
observed galaxy power spectrum solely by measuring $\delta_{g}(\textbf{\em
k},z)$ down to the mildly non-linear scales where the 1-loop terms
proportional to $b_{1}$ contribute (see Desjacques et al., 2018, for a review
about galaxy bias). Model-independent analyses based on the consistency
relations of large scale structure (Peloso & Pietroni, 2013; Kehagias &
Riotto, 2013; Creminelli et al., 2013; Peloso & Pietroni, 2014) offer a
complementary approach to robustly measure the large scale bias (Marinucci et
al., 2019, 2020).

The local mapping Eq. (1) translates into a modulation of the small-scale
primordial power spectrum $P_{\phi}(k_{s})$ by long-wavelength perturbations
$\phi_{l}(x)$ with $k_{l}\ll k_{s}$. A peak-background split argument shows
that $b_{\phi}$ is then given by the response of the galaxy number density
$n_{g}$ to a change in the primordial scalar amplitude $A_{s}$ or,
equivalently, to the normalisation amplitude $\sigma_{8}$ (Slosar et al.,
2008),

$b_{\phi}=2\frac{\partial{\rm ln}\bar{n}_{g}}{\partial{\rm ln}\sigma_{8}}\;.$
(3)

The primordial potential and the matter density contrast are related through
$\delta_{m}(\textbf{\em k},z)=\mathcal{M}(k,z)\phi(\textbf{\em k})$ with
$\mathcal{M}(k,z)=(2/3)k^{2}T_{m}(k)D_{\rm md}(z)/(\Omega_{m}H_{0}^{2})$.
Here, $D_{\rm md}(z)$ is the linear growth rate normalized to $a=(1+z)^{-1}$,
$T_{m}(k)$ is the matter transfer function, $\Omega_{m}$ the present-day mean
matter density and $H_{0}$ the Hubble constant. Eq. (2) makes clear that the
scale-dependent bias constrains the product $f_{\text{NL}}b_{\phi}$, and not
just $f_{\text{NL}}$.

Figure 1: A change in the normalization amplitude $\sigma_{8}$ affects
halo/subhalo masses, merging time etc. and produces smooth variations in the
structure of the trees generated by GALACTICUS . In particular, increasing
$\sigma_{8}$ (at fixed final halo mass, $M_{h}=10^{12}\mathrm{M}_{\odot}$)
leads to an earlier merging of the subhalos and, thereby, increases the mass
of the parent halo throughout its assembly history as illustrated in the
figure. We show two trees run with $\sigma_{8}=0.81$ (red) and
$\sigma_{8}=0.83$ (blue). Each circle represents a halo/subhalo and its size
is proportional to the mass; the length of the lines is proportional to the
merging epoch.

The clustering of dark matter halos is imprinted in the galaxy bias
parameters. In particular, the non-Gaussian bias of dark matter halos takes
the simple form $b_{\phi}^{h}(M_{h},z)=2\delta_{c}(b_{1}^{h}(M_{h},z)-1)$ for
a universal halo mass function, where $b_{1}^{h}$ is the linear halo bias and
$\delta_{c}\approx 1.683$ is the critical threshold for (spherical) collapse.
This relation is usually used within LSS analysis with the EFTofLSS to
constrain the amplitude of primordial non-gaussianities, which for local type
are parametrized by $f_{\rm NL}$. It was shown Reid et al. (2010); Lazeyras et
al. (2022) that the relation $b_{\phi}(b_{1})=2\delta_{c}(b_{1}-p)$, with
$p=1$ for halos, is not a good description of the PNG bias of halos and
galaxies selected by other properties beyond the halo mass, like the stellar
mass $M_{\star}$.

###  2.2 Modelling galaxy assembly bias

In the Halo Occupation Distribution (HOD) approach (see for instance Benson et
al., 2000; Scoccimarro et al., 2001; Berlind & Weinberg, 2002; Kravtsov et
al., 2004; Zheng et al., 2005), galaxy abundances are determined by the mass
function $\bar{n}_{h}(M,z)$ of host dark matter halos and by the Halo
Occupation Distribution (HOD). Therefore, as emphasized in Voivodic & Barreira
(2021), galaxy bias parameters can be computed from the change (or response)
of $\bar{n}_{h}(M,z)$ and the HOD to some long-wavelength perturbation.

In plain words, let

$n_{g}(X,z)=\int\\!dM_{h}\,\bar{n}_{h}(M_{h},z)\,\big{[}N_{c}(X|M_{h},z)+N_{s}(X|M_{h},z)\big{]}$
(4)

be the comoving number density of galaxies $n_{g}(X,z)$ at redshift $z$ with
observed property $X$. Here, $N_{c}(X|M_{h},z)$ (resp. $N_{s}(X|M_{h},z)$) is
the (average) number of central (satellite) galaxies with a given property $X$
residing in halos of mass $M_{h}$ at redshift $z$. These conditional means can
be further decomposed into

$N_{c,s}(X|M_{h},z)=\bar{N}_{c,s}(M_{h},z)\,P_{c,s}(X|M_{h},z)$ (5)

where $\bar{N}_{c}(M_{h},z)$ (resp. $\bar{N}_{c}(M_{h},z)$) is the average
number density of central (satellite) galaxies per halo, and
$P_{c}(X|M_{h},z)$ is the probability distribution function (PDF) of $X$
conditioned to a halo mass $M_{h}$ and redshift $z$ (and likewise for
satellites galaxies).

The non-Gaussian bias $b_{\phi}$ of this galaxy sample is given by

$\displaystyle b_{\phi}(X,z)$ $\displaystyle=2\frac{\partial{\rm
ln}n_{g}}{\partial{\rm ln}\sigma_{8}}(X,z)$ (6)
$\displaystyle=\bar{b}_{\phi}(X,z)+\Delta b_{\phi}(X,z)\;,$

where

$\displaystyle\bar{b}_{\phi}(X,z)=\frac{1}{\bar{n}_{g}}$
$\displaystyle\int\\!dM_{h}\,b_{\phi}^{h}(M_{h},z)\,\bar{n}_{h}(M_{h},z)$ (7)
$\displaystyle\times\big{[}N_{c}(X|M_{h},z)+N_{s}(X|M_{h},z)\big{]}$

is a weighted averaged of the non-Gaussian halo bias (the response of the halo
mass function) solely. The non-Gaussian halo bias is
$b_{\phi}^{h}(M_{h},z)=2\,\partial{\rm ln}{\bar{n}_{h}(M_{h},z)}/\partial{\rm
ln}{\sigma_{8}}$ by definition. Similarly,

$\displaystyle\Delta b_{\phi}(X,z)$
$\displaystyle=\frac{1}{\bar{n}_{g}}\int\\!dM_{h}\,\big{[}f_{c}\Delta
b_{\phi}^{c}(X|M_{h},z)+f_{s}\Delta b_{\phi}^{s}(X|M_{h},z)\big{]}$
$\displaystyle\qquad\times\bar{n}_{h}(M_{h},z)\,\big{[}N_{c}(X|M_{h},z)+N_{s}(X|M_{h},z)\big{]}$
(8)

is the contribution arising solely from the response of the HOD to a change in
$\sigma_{8}$. Here, $f_{c}=f_{c}(X|M_{h},z)$ and $f_{s}=f_{s}(X|M_{h},z)$ are
the fraction of central and satellite galaxies with properties $X$ and
residing in halos of mass $M_{h}$ at redshift $z$, whereas

$\displaystyle\Delta b_{\phi}^{c,s}(X|M_{h},z)$
$\displaystyle=2\frac{\partial{\rm ln}N_{c,s}}{\partial{\rm
ln}\sigma_{8}}(X|M_{h},z)$ (9) $\displaystyle=2\frac{\partial{\rm
ln}\bar{N}_{c,s}}{\partial{\rm ln}\sigma_{8}}(M_{h},z)+2\frac{\partial{\rm
ln}P_{c,s}}{\partial{\rm ln}\sigma_{8}}(X|M_{h},z)$

is the corresponding non-Gaussian assembly bias. $\Delta b_{\phi}^{c,s}$ is
the fractional change in the number of central or satellite galaxies as
$\sigma_{8}$ is varied. Our definitions are related to the response
$R_{\phi}^{g}(X|M_{h},z)$ introduced in Voivodic & Barreira (2021) through

$R_{\phi}^{g}(X|M_{h},z)=\Big{(}\bar{N}_{c}\,\Delta
b_{\phi}^{c}+\bar{N}_{s}\,\Delta b_{\phi}^{s}\Big{)}(X|M_{h},z)\;,$ (10)

which shows that $R_{\phi}^{g}(X|M_{h},z)$ is the weighted sum of the central
and satellite non-Gaussian assembly bias parameters.

We will focus on the non-Gaussian "assembly" bias $\Delta b_{\phi}^{c,s}(X,z)$
since the contribution $\bar{b}_{\phi}$ arising purely from the response of
the halo mass function $\bar{n}_{h}(M,z)$ can be extracted from pure N-body
simulations solely for generic primordial bispectrum shapes (e.g. Dalal et
al., 2008; Grossi et al., 2009; Pillepich et al., 2010; Tseliakhovich et al.,
2010; Desjacques & Seljak, 2010; Shandera et al., 2011; Smith & LoVerde, 2011;
Desjacques et al., 2011b; Wagner & Verde, 2012; Smith et al., 2012;
Scoccimarro et al., 2012; Biagetti et al., 2017; Chan et al., 2019).

###  2.3 A semi-analytical model of galaxy formation: GALACTICUS

Galaxy formation is a complex process involving gas inflow and cooling onto
dark matter halos, star formation and self-regulation via feedback from
supernovae, AGN etc. (see Mo et al., 2010; Somerville & Davé, 2015, reviews on
this topic). To model the process of galaxy formation we make use of the semi-
analytic galaxy formation model (SAM) GALACTICUS (Benson, 2012) rather than
more realistic but computationally expensive hydrodynamical simulations of the
large scale distribution of galaxies.

The GALACTICUS model is able to generate realistic merger histories for halos
of any given mass and redshfit (once given cosmological and power spectrum
parameters) following the algorithm of Parkinson et al. (2008), who calibrated
their method to match results from cosmological N-body simulations. It then
solves the physics of galaxy formation in the resulting merging hierarchy of
halos through a combination of differential evolution (to describe processes
such as gas cooling, star formation, and feedback), and impulsive events (such
as galaxy mergers). This results in realizations of galaxy populations for the
required halo masses and redshifts. These predictions include both physical
properties (e.g. stellar masses), and observable properties (e.g. broad-band
and emission line luminosities). GALACTICUS has previously been used to model
the population of emission line galaxies to be studied by the _Roman_
telescope (Merson et al., 2018; Merson et al., 2019; Zhai et al., 2019a,
2021a, 2021b).

The baryonic physics of GALACTICUS is described by 30 model parameters, which
have been constrained using a variety of observational datasets111See here for
the full list of datasets used.. The model for the H$\alpha$ luminosity
functions has been constrained from HiZELS (Sobral et al., 2013) and GAMA
(Gunawardhana et al., 2013) observations, while the g and r-band luminosity
functions of SDSS galaxies from Montero-Dorta & Prada (2009) provide
constraints on galaxy colours.

We run GALACTICUS using the Planck fiducial cosmology 222The fiducial
cosmological parameters are
$\\{H_{0},\Omega_{m},\Omega_{\Lambda},\Omega_{b},n_{s}\\}=\\{67.36,0.31530,0.04930,0.04930,0.9649\\}$.
Akrami et al. (2020b). Our fiducial normalization amplitude is
$\sigma_{8}^{\text{fid}}=0.81$.

###  2.4 Extracting the non-Gaussian galaxy bias

To investigate the impact of the galaxy assembly histories on the non-Gaussian
assembly bias $\Delta b_{\phi}(X,z)$ with GALACTICUS , we choose a final halo
mass $M_{h}$ and redshift $z$ and generate $N_{h}$ Monte-Carlo mass assembly
histories for a few different values of $\sigma_{8}$. This allows us to
(numerically) directly measure the responses $\frac{\partial{\rm
ln}N_{c}}{\partial{\rm ln}\sigma_{8}}$ and $\frac{\partial{\rm
ln}N_{s}}{\partial{\rm ln}\sigma_{8}}$ for a given $M_{h}$ and $z$. We will
consider a single redshift, $z=1$, but repeat this procedure for a few
different values of $M_{h}$ in the range $[3\times
10^{10},10^{12}]\,\mathrm{M}_{\odot}$, which brackets the characteristic mass
$M_{*}(z=1)\sim 10^{11}\ \mathrm{M}_{\odot}$ of halos virializing at redshift
$z=1$ in our fiducial cosmology.

We implement the peak-background split expectation eq. (9) separately for
central and satellite galaxies and compute $\Delta b_{\phi}^{c,s}(X|M_{h},z)$
from the random realizations of the merger trees as follows:

$\Delta
b_{\phi}^{c,s}(X|M_{h},z)=\frac{1}{|\delta_{\sigma_{8}}|}\left[\frac{N_{c,s}^{\text{high}}(X|M_{h},z)-N_{c,s}^{\text{low}}(X|M_{h},z)}{N_{c,s}^{\text{fid}}(X|M_{h},z)}\right]$
(11)

where $N_{c}^{\text{fid}}$ , $N_{c}^{\text{high}}$ and $N_{c}^{\text{low}}$
are the total number of central galaxies obtained from $N_{h}$ realizations of
halos with mass $M_{h}$ and redshift $z$ in the fiducial cosmology, and in two
"separate" universes with slightly different normalisation amplitudes:
$\sigma_{8}^{\text{high}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.83$ and
$\sigma_{8}^{\text{low}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.79$.
Hence, the fractional change $\delta_{\sigma_{8}}\leavevmode\nobreak\
=\leavevmode\nobreak\
(\sigma_{8}^{\text{high}}-\sigma_{8}^{\text{fid}})/\sigma_{8}^{\text{fid}}$ in
the normalization amplitude is $\simeq 2.5$%.

To reduce as much as possible the fluctuations due to shot noise, we run
$N_{h}=10^{5}$ trees for the halo masses $M_{h}=\\{3\times
10^{10},10^{11}\\}\mathrm{M}_{\odot}$ and $N_{h}=10^{4}$ for $M_{h}=\\{3\times
10^{11},10^{12}\\}\mathrm{M}_{\odot}$. The mass resolution for each run is
fixed to the minimum value $M_{h}=5\times 10^{9}\mathrm{M}_{\odot}$ since
galaxies living in halos with smaller masses are faint and not easily observed
due to their low stellar content. To study the convergence of the trees in our
case of study, we have run $N_{r}=10$ realizations with different random
seeds. The scatter among the different realizations was used to estimate the
errorbars for our measurements. They vary significantly among panels due to
different choices of binning and changes in the relative fraction of central
and satellite galaxies as a function of halo mass.

The primary galaxy observables $X$ we are interested in are the galaxy colours
defined by the $(g,r,i,\dots)$ filters, and the emission line strengths such
as the H$\alpha$ luminosity $L_{\alpha}$. They are among the basic quantities
directly measured in optical or in emission line galaxy surveys. For instance,
the BOSS 333https://www.sdss3.org/ galaxy samples (LOWZ and CMASS) are
constructed using colour-magnitude cuts (Padmanabhan et al., 2012), whereas
the Euclid 444https://sci.esa.int/web/euclid and WFIRST
555https://www.jpl.nasa.gov/missions/the-nancy-grace-roman-space-telescope
galaxy samples will be constructed from the measured H$\alpha$ line fluxes
(Bisigello et al., 2020; Wang et al., 2022). Furthermore, SPHEREx
666https://www.jpl.nasa.gov/missions/spherex will measure molecular and PAH
emission in the $\mu m$ range (Stickley et al., 2016).

We shall also explore the dependence of $\Delta b_{\phi}$ on secondary
properties such as the stellar mass $M_{\star}$ of the simulated galaxies, the
black hole mass/accretion rate $M_{\text{BH}}$, $\dot{M}_{\text{BH}}$ and
morphological measures such as the bulge-to-disk ratio
$s=R_{\text{sph}}/R_{\text{disk}}$ (where $R_{\text{sph}}$ and
$R_{\text{disk}}$ are the size of the bulge and the disk respectively).
Reliable morphology measurements cannot be obtained at high redshift $z\gtrsim
1$ from current state-of-the-art imaging data. Our results for some secondary
observables are summarized in Appendix §A.

##  3 Results

In this Section, we present results for the dependence of the non-Gaussian
assembly bias $\Delta b_{\phi}^{c,s}$ on different galaxy properties. We will
focus on the $g-r$ colour (used to distinguish between red and blue galaxies,
e.g. Bell et al., 2004) and the H$\alpha$ line luminosity (which is a prime
indicator of star formation, e.g. Kennicutt, 1983). Studying the dependence of
the non-Gaussian $b_{\phi}$ on these observables is of much interest for an
optimal, well motivated choice of priors.

###  3.1 Galaxies selected by colour magnitude

Figure 2: Probability density of $z=1$ galaxies as a function of the colour
$g-r$ for the various halo masses $M_{h}$ considered in our analysis. Galaxies
with $g-r\gtrsim 0.5$ are not produced in significant numbers for the range of
halo masses considered here (see text).

Fig. 2 shows the $g-r$ colour distribution at redshift $z=1$ for the range of
halo mass considered here. The color distribution of GALACTICUS was studied
within the CosmoDC2 project, see Korytov et al. (2019). The lack of relatively
"red" galaxies with $g-r\gtrsim 0.5$ originates from the fact that these are
either satellite galaxies in high mass halos $M_{h}\gtrsim
10^{13}\,\mathrm{M}_{\odot}$, the hot circum-galactic medium of which has cut-
off the supply of new gas so that satellites no longer form stars, or galaxies
residing in very low mass halos $M_{h}\lesssim 10^{9}\ \mathrm{M}_{\odot}$
that are unable to accrete gas (due to their shallow potential wells). While
the latter have very low stellar content and would likely fall below the
detection limit of realistic galaxy surveys, the former are detected by
current galaxy surveys, albeit in small numbers since their massive host halos
are rare. The halo mass range considered here and, thereby, the distributions
shown in Fig. 2 sample the bulk of the $g-r$ colour distribution at $z=1$. The
difference seen with Barreira et al. (2020), who report measurements for
galaxies with colour magnitude as large as $g-r\sim 1$, presumably arises
because their analysis includes halos with masses up to $M_{h}\sim 5\times
10^{14}\ \mathrm{M}_{\odot}$ which we do not simulate. Moreover, the lack of
galaxies, observable in fig. 2, with $g-r\gtrsim 0.2$ could be the result of
intrinsic differences between galaxy formation models used in SAMs codes and
hydro-dynamical simulations.

Figure 3: Non-Gaussian assembly bias $\Delta b_{\phi}^{c,s}$ for galaxies
selected by their $g-r$ colour magnitude. Results are shown for central (upper
panels) and satellite (lower panels) galaxies as a function of the host halo
mass $M_{h}$. The errorbars are computed from the scatter between random
realizations of $N_{h}=10^{5},10^{4}$ merger trees for $M_{h}=\\{3\times
10^{10},10^{11}\\}\mathrm{M}_{\odot}$ and $M_{h}=\\{3\times
10^{11},10^{12}\\}\mathrm{M}_{\odot}$ respectively.

Fig. 3 displays the non-Gaussian assembly bias $\Delta b_{\phi}^{c,s}(g-r)$ of
$z=1$ central and satellite galaxies for different $M_{h}$ as indicated in the
figure. We find that $\Delta b_{\phi}^{c,s}$ is positive (negative) when $g-r$
is larger (smaller) than the mean colour of the sample. This trend originates
from the fact that an increase in $\sigma_{8}$ results, on average, in a
faster mass assembly history. Consequently, the final dark matter halo hosts a
higher number of older, "red" galaxies and, correspondingly, a smaller number
of younger ones. This effect is visible both for central and satellite
galaxies. Furthermore, in light of the findings of Reid et al. (2010), we
expect it to be correlated with a change in the average formation redshift of
the host dark matter halos.

Overall, the non-Gaussian assembly bias can change by up to a factor of
$\mathcal{O}(10)$ depending on the colour, in agreement with the results of
Barreira et al. (2020) extracted from detailed hydro-dynamical simulations (of
a different galaxy formation model). Furthermore, in most panels there is a
prominent peak at $g-r\sim 0-0.1$ on top of the broad tilt produced by the
change in merging histories of dark matter halos and galaxies. The position of
the peak matches the position of the maximum of $P_{c,s}(g-r|M_{h},z)$, around
which $\Delta b_{\phi}^{c,s}$ changes rather abruptly. This effect is
partially erased when larger $g-r$ bins are adopted. There are other features
which appear to be robust to statistical uncertainties. Although they might be
caused by some feature in the stellar population spectra, their origin is
unclear.

It is instructive to contrast these measurements to the contribution
$\bar{b}_{\phi}$ given by Eq. (7), which arises from the halo clustering
solely. Assuming the halo mass function of ref. Tinker et al. (2010) and the
universality relation $\bar{b}_{\phi}=2\delta_{c}(b_{1}-1)$, we find
$\bar{b}_{\phi}(M_{h},z=1)=[-0.57,-0.24,0.20,0.95]$ for the halo masses
$M_{h}=[3\times 10^{10},10^{11},3\times 10^{11},10^{12}]\mathrm{M}_{\odot}$
considered here. In other words, the non-Gaussian assembly bias is
significantly larger than the pure "halo" contribution $\bar{b}_{\phi}$.
Although a realistic prediction would have to integrate $\Delta b_{\phi}$
across the range of redshift and halo masses probed by the galaxy survey under
consideration, these results confirm again that the assumption of a
universality relation $b_{\phi}=\bar{b}_{\phi}(b_{1})$ can be a poor fit for
galaxies selected by colour magnitude (see Barreira et al., 2020; Barreira,
2022b, c). Furthermore, suitable colour cuts could be applied in order to
maximize the information that can be extracted on $f_{\text{NL}}$ from the
non-Gaussian bias (Barreira & Krause, 2023). For a host halo mass
$M_{h}=3\times 10^{11}\mathrm{M}_{\odot}$ for instance, a higher signal-to-
noise for the non-Gaussian bias $f_{\text{NL}}b_{\phi}$ is obtained for
galaxies with a colour magnitude $g-r\simeq 0.1$ close to the peak of $\Delta
b_{\phi}^{c,s}$ although, in practice, the peak of the response is broadened
by the range of halo mass and redshift probed by the survey.

Finally, we have also checked that, at least for host halos of mass $M=3\times
10^{10}\ M_{\odot}$, the non-Gaussian assembly bias $\Delta
b_{\phi}^{c,s}(r-i)$ at fixed $r-i$ color magnitude is similar to $\Delta
b_{\phi}^{c,s}(g-r)$ shown here.

###  3.2 Galaxies selected by H$\alpha$ luminosity

Figure 4: Same as Fig. 3, for the H$\alpha$ luminosity $L_{\alpha}$. Note that
the range of H$\alpha$ luminosity of the mock galaxies increases significantly
with the mass of the host halo.

Fig. 4 summarizes our findings for the non-Gaussian assembly bias $\Delta
b_{\phi}(L_{\alpha})$ as a function of the H$\alpha$ luminosity. For this
purpose, GALACTICUS was calibrated against simulations and observations, see
Zhai et al. (2019b) for details. The H$\alpha$ luminosity is calculated at the
selected redshift and post-processed in order to include correction from dust
extinction in the target galaxy. For a survey such as Euclid, a H$\alpha$ line
flux limit of $2\times 10^{16}\ {\rm erg}\,{\rm s}^{-1}\ {\rm cm}^{-2}$
(Euclid Collaboration et al., 2022) corresponds to a minimum H$\alpha$
luminosity of $\sim 4\times 10^{41}\ {\rm erg}\ {\rm s}^{-1}$ at redshift
$z=1$, which is barely reached in our sample of satellite galaxies.

Figure 5: Effect of varying the galaxy formation model on the non-Gaussian
assembly bias $\Delta b_{\phi}$ (see text for details). Results are shown for
central galaxies populating $z=1$ halos of mass $M_{h}=3\times
10^{11}\mathrm{M}_{\odot}$ and selected either according to their $g-r$ colour
magnitude (top panel), or to their H$\alpha$ luminosity (bottom panels).

Overall, the assembly bias is much smaller for galaxies selected by H$\alpha$
luminosity than $g-r$ colour magnitude. The reason is that $L_{\alpha}$ mainly
depends on the instantaneous star formation rate $\dot{\rho}_{*}(z)$, which
turns out to be a weak function of the normalization amplitude $\sigma_{8}$ at
the redshift considered here. For central galaxies we have that the
$L_{\alpha}$ distribution of galaxies for the redshift and halo masses
considered here are narrowly peaked, as can be inferred by the luminosity
ranges shown in the upper plot of fig. 4. Unlike colour cuts, we do not detect
any clear assembly bias trend, even though the departure from $\Delta
b_{\phi}$ of order $\sim O(1)$ seen for, e.g., central galaxies in halos of
mass $M_{h}=3\times 10^{10}$ and $10^{11}\ \mathrm{M}_{\odot}$ (upper row)
appear robust to sampling variance.

Likewise, there is no indication of assembly bias for the satellite galaxies
except, possibly, at the lowest luminosities $L_{\alpha}\lesssim 10^{38-39}\
{\rm erg}\ {\rm s}^{-1}$ probed by the mock galaxy catalogs. We have checked
that the non-Gaussian assembly bias exhibits a similar behaviour when galaxies
are selected by their H$\beta$ luminosity (at least for halos of mass
$M=3\times 10^{11}\ M_{\odot}$). All this suggests that setting
$b_{\phi}=\bar{b}_{\phi}$ for emission line galaxies selected by H$\alpha$
luminosity should be a reasonable approximation for the non-Gaussian bias.
This, of course, does not preclude the existence of a Gaussian assembly bias
(i.e. at the level of $b_{1}$ etc.) as reported in Jiménez et al. (2021) for
instance. This being said, one should remain cautious and avoid taking these
results as definitive until a thorough study based on different galaxy
formation models (we vary some of the GALACTICUS parameters in section §3.3)
and merger trees probing a wider range of halo mass and redshift is carried
out.

###  3.3 Changing the galaxy formation model

The flexibility of GALACTICUS allows us to study the dependence of our
assembly bias measurements on the choice of galaxy formation model. For this
purpose, we consider three variations around the fiducial model adopted by
GALACTICUS . The first variation is a change the efficiency of the star
formation rate (SFR), which should have an impact on the observed galaxy
colour and luminosity. The resulting "low SFR" / "high SFR" models have star
formation efficiencies twice as low / large as the fiducial one. All the other
model parameters are held fixed to their fiducial value. For the second and
third set of models, we modify the heating efficiency of the black hole
component (i.e. the efficiency of the heating from the active galactic nuclei,
or AGN feedback), and the mean velocity of the stellar feedback outflows. Both
parameters control the amount of energy which is dumped into the surrounding
gaseous medium, thereby affecting the production of new stars. Here again, we
adopt parameter values twice as large / as small as the fiducial ones. For
convenience, the values of the parameters changed to produce our different
galaxy formation models are summarized in Appendix §B. These physical
parameters are expected to have the highest impact on the observed galaxy
colours and H$\alpha$ luminosities, as well as the other galaxy observables
presented in Appendix A.

Fig. 5 displays the results obtained from 6 samples of $10^{4}$ merger trees
of host halos of mass $M_{h}=3\times 10^{11}\mathrm{M}_{\odot}$ virializing at
redshift $z=1$. They are shown as a function of $g-r$ colour magnitude (top
panels) and H$\alpha$ luminosity (bottom panels) for central galaxies solely.
While the behaviour of $\Delta b_{\phi}^{c}(g-r)$ remains mostly unchanged,
the peak of the assembly bias can be slightly shifted. The effect is most
pronounced when a lower efficiency in the SFR is considered. This is due to
the fact that a lower star formation efficiency delays star formation and,
thereby, increases the relative fraction of younger stars. As a result, this
shifts $\Delta b_{\phi}^{c}$ toward bluer colours (i.e. large values of
$g-r>0$). Moreover, fig. 5 shows that, while variations in the AGN feedback
efficiency appear to have a negligible impact, variations in the stellar
feedback can affect the slope of $\Delta b_{\phi}^{c}(g-r)$. For all the
variations considered here however, the results are always compatible with the
trend seen for the fiducial model.

Likewise, we do not detect large variations in the predicted $\Delta
b_{\phi}^{c}(L_{\alpha})$ when the aforementioned model parameters are varied.
However, while most of the departures from the fiducial model prediction are
within the error bars, some differences can be observed when the SFR and the
stellar feedback are varied, mainly at high H$\alpha$ luminosities. Still,
investigating these effect further requires a thorough comparison of different
SAMs, which is beyond the scope of this work.

##  4 Conclusions

Detecting or constraining primordial non-gaussianity (PNG) is one of the key
goals of present and future galaxy surveys. The current constraints on the
local PNG parameter $f_{\text{NL}}$ obtained using galaxy clustering data come
mainly from the scale-dependent bias effect in the galaxy power spectrum. At
leading order, PNG of the local type give a broadband, scale-dependent
contribution $b_{\phi}f_{\text{NL}}/k^{2}$ to the observed galaxy overdensity,
where the non-Gaussian bias $b_{\phi}$ can be derived from a peak-background
split argument and associated to a change in the normalization amplitude
$\sigma_{8}$. Good priors on $b_{\phi}$ are necessary to minimize the
uncertainty on a measurement of $f_{\text{NL}}$ from galaxy clustering
statistics such as the power spectrum and bispectrum.

In this paper, we have studied the dependence of $b_{\phi}$ on survey
selection cuts and on the physics of galaxy formation using $z=1$ mock galaxy
samples produced with the code GALACTICUS , which implements a particular
semi-analytical model (SAM) for galaxy formation (Benson, 2012). The separate
universe approach allows us to directly measure $b_{\phi}$ as the response of
the number of galaxies to a variation in the amplitude of the primordial
gravitational potential. To measure the non-Gaussian assembly bias $\Delta
b_{\phi}$ separately, we have produced merger trees with varying $\sigma_{8}$
but fixed final halo mass, thereby removing the response of the halo mass
function (which can be easily calibrated with N-body simulations). We have
focused on measurements of $\Delta b_{\phi}$ for galaxies selected by the
$g-r$ colour magnitude and by the intensity of H$\alpha$ emission lines. For
$g-r$ cuts, our findings are consistent with the previous study of Barreira et
al. (2020) based on detailed hydro-dynamical simulations, that is, $\Delta
b_{\phi}(g-r)$ can reach values as large as $|\Delta b_{\phi}|\simeq 10$ – 20
for galaxies selected by $g-r$ colour magnitude (see fig. 3 for instance).
This is much larger than the non-Gaussian bias contribution arising from the
response of the halo mass function solely, which is $\bar{b}_{\phi}\simeq
O(1)$ for the halo masses and the redshift considered in our study. This
strong assembly bias reflects the large dependence of halo formation times on
$\sigma_{8}$ identified by Reid et al. (2010): at fixed final halo mass,
higher values of $\sigma_{8}$ cause an earlier collapse of halos and, thereby,
a earlier star formation so that the final galaxies are populated by older
(redder) stars.

We have also measured the non-Gaussian assembly bias for $z=1$ galaxies
selected by H$\alpha$ luminosity $L_{\alpha}$, which is relevant for
forthcoming emission line galaxy (ELG) surveys such as Euclid or SPHEREx.
Previous studies have focused on the linear bias $b_{1}$ of ELGs and found it
to be insensitive to the star formation rate (SFR) (Angulo et al., 2012;
Nusser et al., 2020). Unlike a $g-r$ colour selection, we have found a weaker
non-Gaussian assembly bias $\Delta b_{\phi}(L_{\alpha})$ for the halo masses
and the redshift considered here (see fig. 4 for instance). The reason
presumably is the fact that the strength of the H$\alpha$ line is mainly
sensitive to the instantaneous SFR, which is weakly affected by a change of
$\sigma_{8}$ for the redshift analyzed here. The validity of these results is,
however, restricted to the redshift and halo masses considered here. Extending
the analysis to higher redshift (where the impact of $\sigma_{8}$ on the SFR
might be larger) and a wider range of halo mass is left for future work. On
the one hand, a small value of $|\Delta b_{\phi}|\ll|\bar{b}_{\phi}|$ would be
a good news for future emission-line surveys such as Euclid or SPHEREx since
accurate priors on $\bar{b}_{\phi}$ would only require (sufficient) knowledge
of the halo occupation distributions $N_{c}(X|M_{h},z)$ and $N_{s}(X|M_{h},z)$
of central and satellite galaxies. On the other hand, a weaker assembly
dependence would prevent us from taking advantage of selection cuts to find
samples with large $b_{\phi}$ and, thereby, improve $f_{\text{NL}}$
constraints at the level achieved by the multi-tracer implementation of
Barreira & Krause (2023).

Furthermore, we have explored the effect of varying some of the galaxy
formation parameters on the non-Gaussian assembly bias (see fig. 5) to assess
the robustness of our measurements of $\Delta b_{\phi}$. We have focused on
the model parameters which likely have the largest impact on the galaxy
colours and emission line strengths, that is, the efficiency of the star
formation rate, and the AGN and stellar feedbacks. For the few variations
considered here, we have not detected major changes in $\Delta b_{\phi}$
although, because $|\bar{b}_{\phi}|$ is small across the halo mass range
probed by our merger trees, these variations could still matter for
constraints on $f_{\text{NL}}$. Therefore, these findings need to be confirmed
with a more exhaustive investigation. In addition to other colours (e.g.
$r-i$) and emission lines (e.g. OII), such a study should compare the outcome
of different SAMs and validate the results against hydro-dynamical
simulations, which are required to quantify the systematics arising from
excursion set or N-body merger trees.

## Acknowledgments

We thank Alex Barreira, Guido D’Amico, Elisabeth Krause, Adi Nusser, Massimo
Pietroni and Ravi Sheth for useful discussions and comments on this work. MM
and VD acknowledge support by the Israel Science Foundation (ISF) grant No.
2562/20.

##  Appendix A Secondary observables

Figure 6: Non-Gaussian assembly bias $\Delta b_{\phi}^{c,s}$ for $z=1$
galaxies selected by stellar mass.

In this Appendix, we present our measurements of the non-Gaussian assembly
bias $\Delta b_{\phi}$ for a few other galaxy properties already considered in
literature: the stellar mass $M_{\star}$, the central black hole mass
$M_{\text{BH}}$ and bulge-to-disk ratio $s$. The galaxy stellar mass is
usually inferred from the luminosity through a modelling of the spectral
energy distribution (SED). Black hole properties such as mass and accretion
rate are correlated with the activity of quasars, which are often used to set
observational limits on $f_{\text{NL}}$ as they are highly biased tracers of
the underlying matter distribution (see Slosar et al., 2008; Leistedt et al.,
2014; Castorina et al., 2019; Mueller et al., 2022). Furthermore, the current
quasar samples cover large comoving volumes at high redshift, thereby reducing
the statistical uncertainties. Finally, galaxy morphologies as measured by $s$
can be inferred from imaging surveys, at least at low redshift.

For galaxies selected by stellar mass, there is a strong assembly bias mainly
for central galaxies as is apparent from Fig. 6. The earlier collapse time
caused by a higher value of $\sigma_{8}$ produces galaxies with higher average
stellar mass, which explains the measured slope and sign of $\Delta
b_{\phi}(M_{\star})$.

For galaxies selected by the mass of the central black hole, there is a weak
transition from negative to positive value of $\Delta b_{\phi}$ for the
central galaxies, see 7. This trend, in qualitative agreement with the results
of Barreira (2022a), may be explained by a change in the formation time if
older galaxies form bigger bulges with larger velocity dispersion and,
thereby, larger central black holes (due to the black-hole $M-\sigma$
relation). Galaxies with a central black hole of higher mass $M_{\rm BH}$
present higher average values of $g-r$.

Figure 7: Same as Fig. 6 but for galaxies selected by the mass $M_{\text{BH}}$
of their central black hole.

The correlation between a change in $\sigma_{8}$ and the halo formation time
also appears to explain the dependence on the morphological parameter $s$
reported in ig. 8. Increasing $\sigma_{8}$ produces on average older galaxies,
which are thought to have smaller disks Jiang et al. (2019), whence the
observed positive values of $\Delta b_{\phi}$ on the lower side of the
$s$-range. Note that $s$ varies at most by 20% at fixed $M_{h}$ for the halo
masses considered here.

Figure 8: Same as Fig. 6 but for galaxies selected according to their bulge-
to-disk ratio $s$.

##  Appendix B Parameters for different galaxy formation model

GALACTICUS provides a specific model for the different components that enter
into galaxy formation, see Benson (2012). In this work we changed some of the
parameters to study their effect on assembly bias. In tab. 1 we show the
parameters for the different model. There, EH stands for efficiency heating,
ESF for efficiency of stellar formation and VSF indicates the characteristic
velocity for stellar feedback. The fiducial values for all the parameters used
in this analysis can be found on the GitHub repository of GALACTICUS 777For
the full list of fiducial parameters used see here..

Model | EH (BH component) | ESF (disk/spheroid) | VSF (disk/spheroid)
---|---|---|---
Fiducial | 0.0001576 | 0.2537 / 0.003064 | 49.96 / 41.53
Low AGN | 0.00007880 | 0.2537 / 0.003064 | 49.96 / 41.53
High AGN | 0.0003152 | 0.2537 / 0.003064 | 49.96 / 41.53
Low SF | 0.0001576 | 0.25367 / 0.003064 | 25.00 / 25.00
High SF | 0.0001576 | 0.25367 / 0.003064 | 99.91 / 83.06
Low SFR | 0.0001576 | 0.1268 / 0.001532 | 49.96 / 41.53
High SFR | 0.0001576 | 0.5073 / 0.006128 | 49.96 / 41.53
Table 1: Values for the parameters in the different galaxy formation models.
These parameters are used to produce the results presented in section 3.3.

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# Review cuprates – Anderson’s unhappy electrons and their fate

Navinder Singh  Theoretical Physics Division, Physical Research Laboratory,
Ahmedabad. India. PIN: 380009<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In cuprates, as doping $p$ is reduced from the overdoped side through the
quantum critical point $p^{*}$, a transition in Hall number density of
carriers is observed, in which this number of carriers reduces from $1+p$
holes per copper site to $p$ holes per copper site. The connection of this
$1+p$ to $p$ transition with pseudogap and with superconductivity is
discussed. A ”panoramic” view or a ”broad-brush” discussion is presented in
which Anderson’s ”unhappy” electrons take on a central stage.

  * 7/7/2023

Keywords: Cuprates; antiferromagnetic correlations; itinerant and localized
carriers; pseudogap etc

”He [Bardeen] felt that formalism could lead one astray, unless it was closely
tied to experimental and physical intuition” –Schrieffer on Bardeen.

P. W. Anderson (Figure 1) while discussing the crystal and electronic
structure of cuprates argued that both localized (magnetic) and mobile
(itinerant) degrees-of-freedom reside in the $CuO_{2}$ planes of lamellar
cuprate high-$T_{c}$ superconductors[1]. Spacer layers separating the
$CuO_{2}$ planes act like charge reservoirs. By changing the chemical
composition of these spacer layers hole doping or electron doping in $CuO_{2}$
planes can be achieved.

Figure 1: Author pays his tribute to P. W. Anderson (December 13, 1923 – March
29, 2020) for contributing some of the pioneering ideas, right way to approach
the problem, and for his very insightful articles on the subject,
especially[1].

Simple valence counting shows that in the undoped compounds (for example in
$La_{2}CuO_{4}$) copper atoms should have the nominal valence state of nine
electrons in the valence d-orbitals ($Cu^{++}:Ar[3d^{9}]$). He further argued
that due to crystal field effects (created by an octahedron geometry of the
cages of oxygen atoms around copper atoms) copper d orbitals of the planner
copper atoms in $CuO_{2}$ planes of cuprates split into two sub-groups $e_{g}$
and $t_{2g}$, whereas $t_{2g}$ forms a lower energy set that is filled first
and $e_{g}$ form a higher energy set that is filled last. Out of nine
electrons in the relevant copper $3d^{9}$ valence state, six electrons are
filled first in $t_{2g}$ orbitals i.e., in $d_{xy},~{}d_{yz},~{}d_{zx}$
orbitals. The remaining three electrons go into the $e_{g}$ set which contains
the $d_{x^{2}-y^{2}}$ and $d_{3z^{2}-r^{2}}$ orbitals. It turns out that the
orbitals $d_{3z^{2}-r^{2}}$ which have lobs perpendicular to the $CuO_{2}$
planes and pointing towards the apical oxygen atoms in the octahedron geometry
has lower energy as compared to that of $d_{x^{2}-y^{2}}$ and two electrons
are filled in it.111This splitting in energy between $d_{x^{2}-y^{2}}$ and
$d_{3z^{2}-r^{2}}$ orbitals is due to Jahn-Teller distortions of the
octahedron cage. Apical oxygen atoms are pushed out whereas planner ones are
pulled in making the octahedron pointy, especially in LSCO[1]. The remaining
electron goes into the highest energy $d_{x^{2}-y^{2}}$ orbital which has four
lobs pointing towards oxygen atoms in the $CuO_{2}$ plane. Anderson called the
lobs pointing towards planner oxygen atoms as unhappy lobs, and this lone (un-
paired) electron is an ”unhappy electron”. ”Unhappy”, because of its highest
energy state. Thus there is a hole and an electron in the highest energy
$d_{x^{2}-y^{2}}$ orbital. In these ”unhappy” electrons there lies the secret
of cuprates. It is the seat of magnetic degrees of freedom at zero doping and
at finite doping too (although long range magnetic order is ”melted” at a
finite doping of about 0.05). It is also the seat of mobile degrees of freedom
at finite doping as the system becomes conducting. It exhibits dual character
(completely localized at zero doping and quasi-localized at any finite doping
but below overdoping, more precisely below $p^{*}=0.19$ as discussed in detail
below). In a more accurate description d-p hybridization between copper d
orbitals and oxygen p orbitals leads to bonding and anti-bonding sets of
orbitals, and it turns out that the hybrid $Cu3d_{x^{2}-y^{2}}--O_{2p\sigma}$
orbitals constitute higher energy states and form a narrow-hybrid-d-p band in
which ”unhappy” electrons reside and where electronic correlations play a
dominant role.

In the following paragraphs, we sketch a physical picture in which the
behaviour of these ”unhappy” electrons as a function of doping and temperature
takes a central stage.

These ”unhappy” electrons are localized at zero doping (due to strong on-site
repulsion $U$) and exhibit AFM order (local moments form at $T\sim J/k_{B}\sim
1200~{}K$, but, full 3-D AFM order (via inter-layer interactions) sets at a
much lower temperature, at around $300~{}K$). When the system is doped with
holes (or when the electrons are removed from $CuO_{2}$ planes) some of these
”unhappy” electrons loose their prejudice of being localized on copper
sites222Anderson pointed out that the amount of distortion of the octahedron
for $Cu^{++}$ (one hole and one unpaired electron in $3d_{x^{2}-y^{2}}$
orbitals) and for $Cu^{+++}$ (two holes in $3d_{x^{2}-y^{2}}$ orbitals) states
is roughly the same and the doped hole can become mobile without taking the
”burden” of a lattice distortion along with it, i.e., when a doped hole
migrates from one site to another, it does not carry along with it a
distortion (lattice distortion) of the octahedron. Contrary to it $Cu^{+}$
(two electrons in $3d_{x^{2}-y^{2}}$ orbitals) state is of spherical symmetry
and involve a distortion of the octahedron and not very supportive for easy
conduction as the mobile hole has to carry a lattice distortion along with
it[1]. This might be the reason why, in the electron doped side, AFM order is
so much extended in the phase diagram and SC is so much weaker as compared to
that in the hole doped side.. How does this happen? It is not only ”unhappy”
electrons on copper sites which are affected by hole doping, populations on
oxygen sites do also change[2]. Conduction becomes possible (doped holes are
mobile objects). In this conducting ”metal” short range AFM correlations are
dominant in the underdoped regime and survive even beyond the optimal doping
regime so long as superconductivity is present (Figure 2).

Figure 2: AFM correlation length from neutron scattering in
$La_{2-x}Sr_{x}CuO_{4}$[4].

These short range AFM correlations (mediated through superexchange
interaction) are some sort of a relics of the full fledged AFM order that
existed at zero doping. Cuprates are ”naturally paired”, that is, there exists
short-ranged AFM correlations which evolve in time (a patch of local AFM order
extending over several lattice sites appears at a given instant of time, and
then it dies aways, and it appears again at a different location, and so on.
This process is incessant). This state is a fundamental state out of which
superconductivity emerges. Superconductivity in cuprates require two essential
ingredients: presence of mobile states and presence of short ranged AFM
correlations. Do the doped holes change the average local moments on copper
sites? The answer is yes. They diminish them and can provide answers to
neutron scattering experiments. Barzykin and Pines argued based on extensive
experimental analysis that both local degrees-of-freedom that is the cause of
magnetic moments (spin liquid component) and mobile degrees-of-freedom that is
the cause of conduction (Fermi liquid component) exist in underdopped and
optimally doped cuprates[3]. They argued that the strength of effect exchange
interaction in the spin liquid component weakens by increasing doping:

$J_{eff}(x)=Jf(x),~{}~{}~{}f(x)=1-x/0.2.$ (1)

That is, spin liquid component weakens with increasing doping whereas Fermi
liquid component becomes stronger with increasing doping. But beyond optimal
doping or more precisely beyond a critical doping of $p>p^{*}\simeq 0.19$
something very drastic happens.

In 2016, an important discovery[5] highlighted this drastic change of
behaviour across the pseudogap quantum critcal point at $p^{*}\simeq 0.19$.
Hall effect measurements were done on YBCO. These measurements showed that in
the underdoped regime below $p<0.1$ the number of carriers per Cu atom is
simply equal to $p$ and in the overdoped side ($p>0.19$) this number is $1+p$.
There is a ”sudden” transition somewhere between these two dopings. As the
doping is reduced from overdoping to underdoping, this change in the number of
carriers begins at $p=p^{*}$ and it gradually reduces towards $p$ holes per Cu
atom at and below $p=0.1$ as doping is reduced from overdoping to underdoping
(refer Figure 3).

Figure 3: $1+p$ to $p$ transition as doping is reduced through $p^{*}$.

Something drastic happens to Anderson’s unhappy electrons at around $p^{*}$
where these are at the verge of itineracy. Below $p^{*}~{}(p<p^{*})$ they are
localized and above $p^{*}$ they are mobile. At $p=p^{*}$ they are not able to
”decide” whether to localize or to be itinerant. We call this a ”split
personality state” of these ”unhappy” electrons. They become mobile for
$p>p^{*}$ giving total carrier number as $1+p$ per copper atom as has been
discussed by Louis Taillefer and collaborators[5] and below $p^{*}$ these
unhappy electrons gradually begin to localize on copper sites, and the carrier
number is equal to the number of doped holes. At this critical doping Fermi
surface volume changes from a bigger volume of $1+p$ carriers per copper atom
to a smaller volume of $p$ carriers per copper atom as the doping is reduced
through $p^{*}$. Experiments cannot be done at zero temperature and
extrapolation shows that this transformation occurs at $p^{*}$ (Figure 3). But
at finite temperature this transformation through $p^{*}$ is not sharp but is
a cross-over[5]. The question is: What is the role played by this transition
in the mechanism of superconductivity? And a connected question is: Is $1+p$
to $p$ transition around $p^{*}$ the root cause of the higher temperature
pseudogap (at which the static magnetic susceptibility shows a maximum)?
Answer to the latter question, with the available evidence, seems to be
affirmative. Static magnetic susceptibility shows a maximum when electrons are
at the verge of itineracy or when they are in a state not able to ”decide”
whether to localized or be itinerant. In fact, generalized susceptibilities
show divergences at critical points. Such a maximum has been observed in the
static magnetic susceptibility of underdoped cuprates. At higher temperatures,
thermal effects broaden the peak.

Let us ask the first question in a slightly different way: Are local moments
on copper sites must for superconductivity? How does the change of character
beyond the critical doping of $p^{*}$ affects superconductivity? Is it behind
the destruction of superconducting dome at overdopping? And why does $T_{c}$
maximizes around $p^{*}$ where the electrons are in ”split personality state”.
In fact $T_{c}$ is maximum around $p\simeq 0.16$ not around $p=0.19$. Is this
observation very crucial and fundamental? If this is a crucial observation
then SC is not a phenomena ONLY governed by what happens at $p^{*}$. It is
maximum when ”Goldilocks” situations arises. ”Goldilocks” situation occurs
when the system achieves the best compromise between the growing number of
carriers (good for superfluid density) as doping is increased and
simultaneously decreasing short ranged AFM correlations (bad for
superconductivity).333For a discussion and problems of theories based on these
lines, refer to[6]. At overdoping when AFM correlations become non-existent SC
disappears. But some of the heavy Fermion superconductors (HFSC) portray a
different picture. In these systems SC order emerges when long range magnetic
order is suppressed (by pressure or chemical doping). SC dome appears around
the magnetic quantum critical point. At a sufficiently high pressure (or when
system is tuned much away from QCP) SC disappears as magnetic correlations
weaken or disappear. How do we understand the difference between what happens
at $p\simeq 0.16$ and at $p\simeq 0.19$?

Let us ask a connected question: why does superconductivity occur at
overdopping (just beyond $p>0.19$ and below the end point of the dome) when
all the carriers become mobile? The very fact that superconductivity is
present at overdopping means that antiferromagnetic correlations still present
at over doping (but below the doping level at which SC dome ends. Refer to
Figure 4). They disappear only at overdopping when there is no SC and when
there is $T^{2}$ behaviour of DC resistivity. $T^{2}$ behaviour of DC
resistivity is a clear signature that short ranged AFM correlations become
sub-dominant in this overdoped regime, and the situation is well described by
Fermi liquid theory[6].

Let us change gears and discuss about the most vexed issue of the pseudogapped
(PG) state of cuprates. Two lines are generally drawn for the pseudogapped
state in underdopped Cuprates: one starting at high temperatures
(approximately around 1200 K at zero doping) and then approximately linearly
going down with increasing doping and meeting the doping axis at a critical
doing of $p=0.19$ in the phase diagram (refer to Figure 4), and the second
line is generally drawn starting from a lower temperature of 300 K or 400 K
(depending upon the cuprate family) at zero doping, just above the Neel
ordering temperature. This line also decreases linearly with increasing doping
and ends at the quantum critical point ($p=0.19$) or in some scenarios
becoming tangent to the dome at overdoping[7]. These lines are cross overs,
and we denote the upper line (high temperature PG) with $T^{*}$444The upper
crossover has been called an energy scale $E_{pg}$ (not a temperature scale)
and $T^{*}=\frac{E_{pg}}{k_{B}}$. This point is stressed by Tallon and
collaborators[8]. and the lower line (low temperature PG) with $T^{**}$.

The mechanism behind the high temperature PG is connected with Taillefer’s
transition of $p$ to $1+p$ holes per copper atom when doping is increased
through the quantum critical point. That is, it is connected with de-
localization to localization transition of Anderson’s ”unhappy” electrons.
When temperature $T$ is reduced through $T^{*}$ boundary, quasi-local moments
appear on copper sites. This high temperature PG ($T^{*}$ line) is a single
plane phenomena. It is a direct product of Mott type correlations which begins
when local moments appear on copper sites below $T^{*}$, and Mott correlations
remain active even at finite dopings (for $p<p^{*}$). In the heat capacity
experiments gap is observed to open at and below the critical doping level
$p^{*}$[8]. The low temperature part of it is connected with the Fermi surface
volume reduction when doping is reduced through $p^{*}$ (refer to Figure 3).
Also, it is the same temperature scale (generally marked as $T_{max}$) at
which static magnetic susceptibility shows a maximum.

Figure 4: The high temperature ($T^{*}$) and the low temperature ($T^{**}$)
pseudogaps in the the phase diagram.

The second line (low temperature PG) is very enigmatic. Experimental
signatures of it are: (1) Reduction in $\frac{1}{T_{1}^{63}T}$ relaxation rate
below $T^{**}$; (2) reduction (down-turn) in in-plane DC resistivity; (3)
upturn in c-axis resistivity etc. For a review refer to[9, 10].

Various mechanisms has been proposed to address this low temperature PG: (1)
pre-formed pairs and related ideas[7, 11, 12, 13, 14]; (2) competing
orders[15]; (3) It could be an out-come of inter-plane ”coherence” or 3D
effects beyond the single CuO2 planes. At zero doping in-plane local moments
on Cu sites appear around $\sim 1200~{}K$ and they exhibit AFM only around
$\sim 300~{}K$ when inter-plane correlations set-in (3D effects). The question
that we ask is this: Whether these zero doping inter-plane effects vanish when
some small doping is made in the CuO2 planes? 555True nature of this
interlayer coupling between fluctuating short range AFM order in one $CuO_{2}$
layer with another $CuO_{2}$ layer is elusive, but it could be that a newly
born patch of local short range AFM order in one $CuO_{2}$ layer aligns a
similar patch of local AFM order in adjacent $CuO_{2}$ layers just above or
below it. And this interlayer correlation is transient in time (coming and
going with time). What is the effect of pressure on $T^{**}$?; (4) $T^{**}$
and associated spin damping phenomena[16]; (5) Nematicity: $T^{**}$ it might
be that degeneracy between oxygen $p$ orbitals is lifted that leads to
nematicity. Recently, the connection between nematicity and lifting of
degeneracy between oxygen $p$ orbitals has been pointed out (energy difference
between $p_{x}$ and $p_{y}$ orbitals is found to be roughly $50~{}meV$)[17,
18].

Regarding nematicity, in 2019, an investigation[17] hinted towards a link
between vetigial density waves (DW) and vestigial nematicity associated with
the PG state. A conundrum was presented: Density wave state (DW with $Q\neq
0$) that could produce an energy gap in EDOS is only detectable at
temperatures much lower than that of the PG temperature. No new additional
energy gap is opened at the on-set of DWs. On the other hand, nematic state
($Q=0$) that appears at the PG temperature is not capable opening the gap! So
what is the mechanism behind the gap opening? It is argued that some sort of
the relics of the DWs (vestigial DWs) are responsible666Here vistigial
nematicity means both $C_{4}$ symmetry breaking (remnants of DW order) and
rotational symmetry breaking (remnants of AFM order).. However, the resolution
remains wide open. Trouble starts when we try to reconcile the above conundrum
with the wisdom gained (in early years of 2000) by David Pines and
collaborators regarding the lower PG where it is associated with some sort of
spin damping phenomena below $T^{**}$[10, 16]. Spins become localized below
$T^{*}$ and exhibit Mott type correlations. At a still lower temperatures,
below $T^{**}$, something again happens to spin system (sharp drop in NMR
relaxation rate). How do we achieve a comprehensive understanding? Clearly,
these are very hard questions and provoke compound by compound analysis.
Resolution among these scenarios is much needed. It puts a heavy demand of a
very careful analysis of literature which has grown tremendously large over
the years. A detailed and critical discussion will be presented in a
forthcoming article:777”The pseudogap debate: current status”, manuscript
under preparation. What is universal and what is compound specific will be
distinguished and analysed.

With this, we end our brief review/perspective. Coming back to the title of
this article, Anderson’s ”unhappy” electrons start their journey as being
localized at zero doping and becoming quasi-local as doping is increased.
Doped holes stay mobile but they diminish the magnitude of local moments.
Short ranged AFM correlations remain active. From the overdoped side as doping
is reduced, PG opens when some of them become quasi-localized at $Cu$ sites
below $p^{*}$. This quasi-localization triggers Mott type correlations and
that open up a gap (the high temperature pseudogap at $T^{*}$). Mechanism
behind the lower PG at $T^{**}$ remains elusive.

## Acknowledgments

Author would like to thank Yves Noat, Alain Mauger, and William Sacks for
illuminating discussions regarding the pseudogapped state.

## References

## References

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