# Diversity Multiplexing Trade-off and Selection Gain in Media-Based
Modulation

Ehsan Seifi† and Amir K. Khandani‡
‡ E&CE Dept. University of Waterloo, Waterloo, ON, Canada, † & Ciena Inc.
Ottawa, ON, Canada

###### Abstract

The idea of Media-based Modulation (MBM), introduced in [1][2], is to embed
information in the variations of the transmission media (channel states).
Using an RF closure with $w$ RF walls, MBM creates a set of $2^{w}$ select-
able states for the end-to-end channel. Each state represents an index of an
MBM constellation point. In each state, the wave (tone) emanating from the
transmit antenna experiences many pseudo-random back-and-forth reflections
within the RF closure. The RF signal, upon finally leaving the RF closure,
further propagates in the rich scattering environment to reach the receiver.
This results in an independent complex channel gain to each receive antenna.
As a result, coordinates of different MBM constellation points (vectors of
channel gains formed over received antennas) will be independent of each
other. This is unlike legacy transmission schemes where a fixed constellation
structure used at the transmitter will be multiplied by a fixed, but random,
channel gain. Due to this independence property, MBM offers several advantages
vs. legacy systems, including “additivity of information over multiple receive
antennas (regardless of the number of transmit antennas)”, and “inherent
diversity over a static fading channel”. This work studies the Diversity-
Multiplexing Trade-off (DMT) [3] of an MBM constellation. Analytical
expressions are provided that demonstrate the advantages of MBM vs. legacy
systems. In particular, it is shown that a $1\times N_{r}$ SIMO-MBM
constellation equipped with an MDS code (even with a relatively small code
length) significantly outperforms an $N_{r}\times N_{r}$ legacy MIMO.

##  I Introduction

Reference [1] shows that embedding part or all of the information in the
(intentional) variations of the transmission media (channel states) can offer
significant performance gains vs. traditional Single-Input Single-Output
(SISO), Single-Input Multiple-Output (SIMO) and Multiple-Input Multiple-Output
(MIMO) systems. This method, coined in [1] as Media-Based Modulation (MBM), is
in contrast with traditional wireless systems where data is embedded in the
variations of an RF source (prior to the transmit antenna) to propagate via
fixed propagation paths (media) to the destination. In particular, using
capacity arguments, reference [1] shows that by using a single transmit
antenna and a single or multiple receive antennas; MBM can significantly
outperform SBM. Following [1], reference [2] proves that, a $1\times K$ MBM
over a static multi-path channel asymptotically achieves the capacity of $K$
(complex) AWGN channels, where for each unit of energy over the single
transmit antenna, the effective energy for each of the $K$ AWGN channels is
the statistical average of channel fading. It is shown that significant gains
can be realized even in a SISO-MBM setup. An example for the practical
realization of the system using RF mirrors, accompanied with realistic RF and
ray tracing simulations, are presented.

MBM falls within the more general category of “index modulation” [4], which
includes a number of different approaches for modulating the RF carrier in the
spatial domain. In particular, spatial modulation and its generalizations rely
on multiple ON-OFF transmit antennas, wherein part of the data selects one of
the transmit antennas (to be ON) and the rest of the data modulates the
carrier to be transmitted through the selected antenna. As a result, the
number of data bits embedded in the channel state will be equal to $\log_{2}$
of the number of ON-OFF antennas. The primary difference between MBM and
spatial modulation is that MBM relies on the use of multiple ON-OFF RF mirrors
(parasitic RF elements with two states) placed around a transmitting antenna.
Due to the interaction among RF mirrors, each selection of ON-OFF pattern for
the mirrors results in a new pattern for the overall transmit antenna
structure. Consequently, the number of data bits embedded in the channel state
will be equal to the number of ON-OFF mirrors, resulting in a linear scaling
in rate vs. the logarithmic scaling in the case of spatial modulation. Due to
space limitations and sheer volume of publications on the topic of spatial
modulation, readers are referred to [4], and references therein, for more
details.

This article studies the advantages of MBM scheme with respect to the
achievable diversity and spatial multiplexing gains. While un-coded MBM offers
a good trade-off between diversity and multiplexing, using a maximum distance
separable (MDS) code with minimum distance $D$ increases diversity by a factor
of $D$ for a small reduction in multiplexing. Larger code lengths attain such
diversity gain with a smaller reduction in multiplexing. Next, we review the
problem formulation and the outline of the results given in the paper.

###  I-A Diversity and Multiplexing

A scheme is said to achieve spatial multiplexing gain $r$ and diversity gain
$d$ if it supports the data rate

$\displaystyle\lim_{\mathsf{snr}\to\infty}\frac{R(\mathsf{snr})}{\log\mathsf{snr}}=r\>\text{(bit/s/Hz)}$
(1)

with the average error probability

$\displaystyle\lim_{\mathsf{snr}\to\infty}\frac{\log
P_{e}(\mathsf{snr})}{\log\mathsf{snr}}=-d,$ (2)

where $\mathsf{snr}$ is the average signal-to-noise ratio (SNR) at each
receive antenna [3].

After providing the mathematical model for MBM system in section II, we obtain
the pair-wise error probability over the ensemble of MBM realizations in
section III. Subsequently, section IV shows that un-coded media-based
modulation achieves diversity gain $d=N_{r}-r$, where $N_{r}$ is the number of
receive antennas and $r,\ 0\leq r\leq N_{r}$ is the spatial multiplexing gain.
Furthermore, in section V we establish that coded media-based scheme using an
MDS code with maximum likelihood decoding, achieves diversity gain
$d=DN_{r}-{r}/{\tau}$, where $D$ is the code minimum distance and $\tau$ is
the MDS code dimensionless rate. More precisely, using an MDS code over
$GF(q)$ with parameters $(N,K,D=N-K+1)$ the average word error probability is
bounded by

$\displaystyle P_{w}\leq\frac{2^{N}}{\sqrt{2\pi
N_{r}}}\left(1+\frac{\mathsf{snr}}{2}\right)^{-(DN_{r}-{r}/{\tau})}.$ (3)

###  I-B Selection Gain

In MBM A coding gain may be achieved by means of a minimal feedback from the
receiver. If the transmitter is provided with the knowledge of the index of RF
mirror patterns that induce the lowest energy over the receive antennas, the
communication may be enhanced by removing such subset with low projected
energy. This will require using additional RF mirrors to maintain the same
rate of transmission. In section VI, we give analytical expressions for the
potential energy gain as a function of additional RF mirrors.

##  II System model

Figure 1: Media-based system.

Fig. 1 shows a $1\times N_{r}=n/2$ SIMO-MBM system model with $N_{r}$ receive
antennas and $n$ real dimensions at the receiver. There are $M$ messages and
$\log_{2}M$ RF mirrors. Each message index $m\in\\{0,...,M-1\\}$ selects a
unique ON-OFF pattern for the RF mirrors. Consequently, for each message a
different channel gain $\mathbf{h}_{m}$ will be realized at the receiver. The
projected components of channel gain along each receive dimension are
$h_{m,d},d=1,...,n$.

We denote the set of all channel gains (due to all possible ON-OFF
combinations of RF mirrors) by
${{\mathbf{h}}}^{M}:=\\{{\mathbf{h}}_{0},{\mathbf{h}}_{2},...,{\mathbf{h}}_{M-1}\\}$.
Throughout the paper we will refer to ${{\mathbf{h}}}^{M}$ as the set of MBM
constellation points. In a static Rayleigh fading channel, the elements of MBM
constellation set, $\mathbf{h}_{m}$, are independent and identically
distributed (i.i.d.) according to multivariate normal distribution
$\mathcal{N}(0,\mathsf{snr}I_{n})$. Here, $\mathsf{snr}$ is the statistical
average of signal energy at each receive dimension, i.e.
$\mathbb{E}h^{2}_{m,d}=\mathsf{snr}$. Even though the transmitter selects the
message index $m$ and consequently $\mathbf{h}_{m}$, it is oblivious to its
details. Receiver, however, knows $\mathbf{h}_{m}$. Receiver training is
achieved by sending a set of pilot symbols.

Assuming additive white Gaussian noise (AWGN), the noisy projected signals at
the receiver $\mathbf{y}$ is written as

$\displaystyle\mathbf{y}=\mathbf{h}_{m}+\mathbf{z}.$ (4)

AWGN noise vector $\mathbf{z}$ has i.i.d. components $z_{d},d=1,...,n$.
Without loss of generality, we assume noise variance per dimension is
normalized to one: $\mathbb{E}z^{2}_{d}=1$.

##  III Exact pairwise error probability

Using maximum likelihood decoding, given that message $m=0$ is transmitted,
(i.e. signal $\mathbf{h}_{0}$ is projected at the receiver), a decoding error
occurs if

$\displaystyle\left\\{\mathbf{h}_{0}\rightarrow\mathbf{h}_{i}\>|\>\mathbf{h}_{0}\right\\}=\big{\\{}\lVert\mathbf{y}-\mathbf{h}_{i}\rVert^{2}\leq\lVert\mathbf{y}-\mathbf{h}_{0}\rVert^{2}\big{\\}},$
(5)

for some $i\neq 0$. Let ${\phi}(\mathbf{h}_{0},\mathbf{h}_{i})$ denote the
pairwise error probability between $\mathbf{h}_{0}$ and $\mathbf{h}_{i}$ given
$\mathbf{h}_{0}$ is sent. Since $\mathbf{h}_{0}$ and $\mathbf{h}_{i}$ are
distributed normally, the squared Euclidean norm of their difference their
follows a chi-squared distribution with $n$ degrees of freedom. Therefore,
exact pairwise error probability averaged over ensemble of MBM realizations is
obtained via

$\displaystyle\overline{{\phi}(\mathbf{h}_{0},\mathbf{h}_{i})}=\int_{x=0}^{\infty}Q\left(\sqrt{\frac{\mathsf{snr}x}{2}}\right)f(x;n)\
\mathrm{d}x.$ (6)

Here, the bar indicates averaging over ensemble, $f(x;n)$ is the probability
density function for chi-squared distribution with $n$ degrees of freedom and
$Q$ is the tail distribution function of the standard normal distribution. In
[5] a closed form solution for this integral is provided according to

$\displaystyle\overline{{\phi}(\mathbf{h}_{0},\mathbf{h}_{i})}$
$\displaystyle=[P(c)]^{\frac{n}{2}}\sum_{k=0}^{\frac{n}{2}-1}\binom{\frac{n}{2}-1+k}{k}[1-P(c)]^{k}$
(7) $\displaystyle P(c)$
$\displaystyle:=\frac{1}{2}\left(1-\sqrt{\frac{c}{1+c}}\right),$ (8)

where $c$ is defined as $c:={\mathsf{snr}}/2$. Same definition for $c$ is used
throughout the article.

##  IV Un-coded Diversity and Multiplexing

We use the alternative form given in [5] to obtain a closed form bound on
pair-wise error probability which does not require summation. The alternative
form for $\overline{{\phi}(\mathbf{h}_{0},\mathbf{h}_{i})}$ is given by

$\displaystyle\frac{\sqrt{c/\pi}\Gamma(\frac{n+1}{2})}{2(1+c)^{\frac{n+1}{2}}\Gamma(\frac{n+2}{2})}\>_{2}F_{1}(1,\frac{n+1}{2};\frac{n+2}{2};\frac{1}{1+c}),$
(9)

where ${}_{2}F_{1}(.,.;.;)$ indicates Gauss hyper-geometric function. Applying
the bound given in [6] for Gauss hyper-geometric function, we get

$\displaystyle\overline{{\phi}(\mathbf{h}_{0},\mathbf{h}_{i})}\\!\leq\frac{\Gamma(\frac{n+1}{2})}{2\sqrt{\pi}\Gamma(\frac{n+2}{2})}(1+c)^{-\frac{n}{2}}\\!\left(\\!1+\\!\frac{n+1}{(n+2)c}\right).$
(10)

Using the union-bound and the fact that
$\overline{{\phi}(\mathbf{h}_{0},\mathbf{h}_{i})}$ is independent of
$\mathbf{h}_{0}$ and $\mathbf{h}_{i}$, the average probability of error over
the ensemble of media-based constellation is bounded by

$\displaystyle
P_{e}\leq\frac{(M-1)\Gamma(\frac{n+1}{2})}{2\sqrt{\pi}\Gamma(\frac{n+2}{2})}(1+c)^{-\frac{n}{2}}\left(1+\frac{n+1}{(n+2)c}\right).$
(11)

Fig. 2 provides a comparison of (11) vs. simulation. Plugging $\log
M=r\log\mathsf{snr}$, the average error probability at high SNR is closely
approximated by

$\displaystyle
P_{e}(\mathsf{snr})\approx\frac{2^{\frac{n}{2}}\Gamma(\frac{n+1}{2})}{\sqrt{\pi}\Gamma(\frac{n+2}{2})}(\mathsf{snr})^{-(\frac{n}{2}-r)}.$
(12)

Recall that $N_{r}=2n$ is the number of receive antennas. Therefore, un-coded
MBM achieves diversity gain $d=N_{r}-r$, while maintaining date rate $R=\log
M=r\log\mathsf{snr}$.

Figure 2: Monte-Carlo vs. bound (11) with $N_{r}=8$ receive antennas and
transmission rate $R=16$ bits/sec/Hz. ${E_{b}}/{N_{0}}:={\mathsf{snr}}/{R}$.

Figure 3: Error probability slope (i.e. diversity gain) for different rates
(i.e. spatial multiplexing gains) for $N_{r}=8$ receive antennas.

##  V Coded Diversity and Multiplexing

In this section, we follow the steps developed in [7] and specialize them to
media-based scheme to arrive at achievable diversity/multiplexing gains using
MDS codes. For completeness, we will provide all the details.

Let $\Delta$ denote and MDS code over $GF(q)$, where $q$ is a prime power. Let
$K$ and $N$, respectively, denote its dimension and block length. Then the
minimum distance of $\Delta$ is given by $D=N-K+1$ and the dimensionless rate
is $\tau={K}/{N}$.

###  V-A Encoder Mapping

The symbols of the code-words in $\Delta$ are mapped into sequences of MBM
constellation points. To make the analysis simpler, we shall assume that the
constellation set is partitioned into $N$ disjoint subsets. Each subset is
assigned to one coordinate of code-words
$\mathbf{u}=(u_{1},u_{2},...,u_{N})\in\Delta$. This is to make sure, even
identical symbols of $GF(q)$ in different coordinates of the code-word
sequence are mapped to different points of the constellation set. Hence, we
consider the constellation sets with cardinality greater than or equal to
$Nq$. More specifically, the Galois field elements in the $i$th coordinate of
a code-word are mapped to the $i$th subset of constellation points via
mappings $g_{i},i=1,...,N$. Then the projected code-words at the receiver has
the form

$\displaystyle\mathbf{h}=g(\mathbf{u}):=(g_{1}(u_{1}),...,g_{N}(u_{N})),$ (13)

for every $\mathbf{u}\in\Delta$.

In a nutshell, the encoder operations are: 1) Map a message from the set
${1,2,..,q^{K}}$ into an MDS code-word $\mathbf{u}\in\Delta$, 2) Map MDS code-
word $\mathbf{u}$ into a sequence of MBM constellation points,
$\mathbf{h}=g(\mathbf{u})$.

###  V-B Decoder Analysis

In these analysis we consider word error probability of MDS codes over random
ensemble of media-based constellation. Hence, for each fixed
$\mathbf{u}\in\Delta$, the code-word $g(\mathbf{u})$ and the error probability
${\phi}(\mathbf{u})$ given $\mathbf{u}$ is sent, become random variables
defined on ensemble, and we will use bar to indicate ensemble average.

The receiver is assumed to use maximum likelihood decoding. The noisy
projected coded vector at the receiver is of the form

$\displaystyle\mathbf{y}=\mathbf{h}+\mathbf{z},$ (14)

where

$\displaystyle\mathbf{h}=(\mathbf{h}_{1},...,\mathbf{h}_{N}),\
\mathbf{h}_{i}=(h_{i,1},...,h_{i,n})\>,\>1\leq i\leq N$ (15)
$\displaystyle\mathbf{y}=(\mathbf{y}_{1},...,\mathbf{y}_{N}),\
\mathbf{y}_{i}=(y_{i,1},...,y_{i,n})\>,\>1\leq i\leq N$ (16)

In i.i.d Rayleigh fading, elements $h_{i,j}$ of the constellation points are
sampled identically and independently from a normal distribution.

Let $S(n)=\\{1,2,...,N\\}$. For $I\subseteq S(N)$, let $\Delta(I)$ indicate
the subset of code-words that have non-zeros in the positions given by $I$ and
zeros outside these positions. For $I\geq D$, and upper bound on $|\Delta(I)|$
is the number of code-words in $\Delta$ that have zeros outside the positions
given by $I$. Since for every MDS code, any set of $K$ symbols form an
information set, we have,

$\displaystyle|\Delta(I)|\leq q^{|I|-D+1},\quad|I|\geq D.$ (17)

Using union bound the error probability given that $\mathbf{u}$ is sent is
bounded by

$\displaystyle\overline{\phi(\mathbf{u})}\leq\sum_{\mathbf{u}^{\prime}\in\Delta}\overline{\phi(\mathbf{u},\mathbf{u}^{\prime})}.$
(18)

Here, ${\phi}(\mathbf{u},\mathbf{u}^{\prime})$ is the pairwise error
probability between code-words $\mathbf{u}$ and $\mathbf{u}^{\prime}$.

To utilize the MDS code property that every set of $K$ symbols form an
information set, first partition the summation in (18) into summations over
$\mathbf{u}^{\prime}\in\Delta(I)+\mathbf{u}$.

$\displaystyle\overline{\phi(\mathbf{u})}\leq\sum_{\begin{subarray}{c}I\subseteq
S(N)\\\ D\leq|I|\leq
N\end{subarray}}\sum_{\mathbf{u}^{\prime}\in\mathbf{u}+\Delta(I)}\overline{\phi(\mathbf{u},\mathbf{u}^{\prime})}.$
(19)

Denoting by $d:=\lVert g(\mathbf{u})-g(\mathbf{u}^{\prime})\rVert$, the
Euclidean distance between the two code-words, we have

$\displaystyle\overline{\phi(\mathbf{u})}\leq\sum_{\begin{subarray}{c}I\subseteq
S(N)\\\ D\leq|I|\leq
N\end{subarray}}\sum_{\mathbf{u}^{\prime}\in\mathbf{u}+\Delta(I)}\mathbb{E}\left[Q\left(\frac{\sqrt{\mathsf{snr}}d(\mathbf{u},\mathbf{u}^{\prime})}{2}\right)\right].$
(20)

Because only the symbol positions with nonzero value in
$\mathbf{u}-\mathbf{u}^{\prime}$ contribute to
$d^{2}(\mathbf{u},\mathbf{u}^{\prime})$, it follows that for
$\mathbf{u}^{\prime}\in\Delta+\mathbf{u}$,

$\displaystyle d^{2}(\mathbf{u},\mathbf{u}^{\prime})=\lVert
g_{I}(\mathbf{u})-g_{I}(\mathbf{u}^{\prime})\rVert^{2}.$ (21)

Given that $\mathbf{u}^{\prime}\in\Delta+\mathbf{u}$, random variable
$d^{2}(\mathbf{u},\mathbf{u}^{\prime})/2$ is sampled from chi-squared
distribution with $n|I|$ degrees of freedom. This follows from the
independence of symbols in an information set. Consequently,
$\overline{\phi(\mathbf{u})}$ is bounded by

$\displaystyle\sum_{\begin{subarray}{c}I\subseteq S(N)\\\ D\leq|I|\leq
N\end{subarray}}\sum_{\mathbf{u}^{\prime}\in\mathbf{u}+\Delta(I)}\int
Q\left(\sqrt{\frac{\mathsf{snr}x}{2}}\right)f(x;n|I|)\ \mathrm{d}x$ (22)
$\displaystyle=\sum_{\begin{subarray}{c}I\subseteq S(N)\\\ D\leq|I|\leq
N\end{subarray}}|\Delta(I)|\int
Q\left(\sqrt{\frac{\mathsf{snr}x}{2}}\right)f(x;n|I|)\ \mathrm{d}x.$ (23)

Since the right hand-side of (23) is independent of $\mathbf{u}$, it is a
bound on the media-based ensemble probability of error and is independent of
probabilities with which the MDS code-words are used. Define
$P_{w}:=\overline{\phi(\mathbf{u})}$. Plugging (17) in (23) to upper bound
$|\Delta(I)|$ , it follows

$\displaystyle P_{w}$ $\displaystyle\leq\\!\sum_{\begin{subarray}{c}I\subseteq
S(N)\\\ D\leq|I|\leq
N\end{subarray}}\\!q^{|I|-D+1}\\!\int\\!Q\left(\sqrt{\frac{\mathsf{snr}x}{2}}\right)\\!f(x;n|I|)\mathrm{d}x$
(24) $\displaystyle=\sum_{i=D}^{N}\binom{N}{i}q^{i-D+1}\int
Q\left(\sqrt{\frac{\mathsf{snr}x}{2}}\right)f(x;ni)\mathrm{d}x.$ (25)

Let us use (10) to replace the integral. For large $c$, the term
$({n+1})/\big{(}{(n+2)c\big{)}}\leq 1$. Therefore,

$\displaystyle P_{w}$
$\displaystyle\leq\sum_{i=D}^{N}\binom{N}{i}q^{i-D+1}\frac{\Gamma(\frac{ni+1}{2})}{2\sqrt{\pi}\Gamma(\frac{ni+2}{2})}(1+c)^{-\frac{ni}{2}}$
(26) $\displaystyle\leq\frac{q^{-D+1}}{\sqrt{2\pi
n}}\sum_{i=D}^{N}\binom{N}{i}\left[q(1+c)^{-\frac{n}{2}}\right]^{i},$ (27)

where in (27) we have used the inequality
${\Gamma(i+0.5)}/{\Gamma(i+1)}\leq{1}/{\sqrt{i}}$ and inequality
$0<{1}/{\sqrt{i}}\leq 1$ for $i\geq 1$. If $q<(1+c)^{{n}/{2}}$, the largest
geometric term in above sum is obtained for $i=D$ . Hence,

$\displaystyle P_{w}$ $\displaystyle\leq\frac{q^{-D+1}}{\sqrt{2\pi
n}}\sum_{i=D}^{N}\binom{N}{i}\left[q(1+c)^{-\frac{n}{2}}\right]^{D}$ (28)
$\displaystyle\leq\frac{q^{-D+1}}{\sqrt{2\pi
n}}\left[q(1+c)^{-\frac{n}{2}}\right]^{D}\sum_{i=D}^{N}\binom{N}{i}$ (29)
$\displaystyle\leq\frac{2^{N}}{\sqrt{2\pi n}}\
q\left[1+c\right]^{-\frac{nD}{2}}.$ (30)

Step (30) uses the fact that $\sum_{i=D}^{N}\binom{N}{i}\leq 2^{N}$.

Let us choose the MDS code Galois field size according to $\log
q=({r}/{\tau})\log\left(1+{\mathsf{snr}}/{2}\right),0\leq{r}/{\tau}<N_{r}$.
Such scheme achieves spatial multiplexing gain $r$ while obtaining coded
diversity gain $DN_{r}-{r}/{\tau}$. This statement is verified noting,

$\displaystyle P_{w}\leq\frac{2^{N}}{\sqrt{2\pi
N_{r}}}\left(1+c\right)^{-(DN_{r}-{r}/{\tau})}.$ (31)

Fig. 4 compares the achievable diversity and multiplexing gains in MBM vs.
MIMO. A $1\times 8$ MBM equipped with an MDS code of minimum distance $D=8$
(even with a relatively small code length) outperforms an $8\times 8$ legacy
MIMO.

Figure 4: Diversity multiplexing trade-off, media-based vs. legacy MIMO.

##  VI Selection gain due to removing low energy points

Figure 5: Selection gain (based on expressions in (32)) for various number of
receive antennas.

Figure 6: Comparison of analytical expression (32) vs. simulated selection
gain.

In MBM removing constellation points with the smallest projected energy over
the receive dimensions provides coding gain. This is possible if the index of
the points with the lowest energy is provided to the transmitter. Removing a
subset of constellation points require using additional RF mirrors to maintain
the same transmission rate (Each additional RF mirror increase the number of
points by a factor of two). This section quantifies the potential energy gain
as a function of average additional RF mirrors required to obtain the same
rate.

For a given $E$, assume that the transmitter is provided with the index of the
points in the constellation set with overall energy less than $nE$.
Furthermore, the transmitter is instructed to remove those points from the
constellation set. Assuming a maximum likelihood decoder, two consequences
will follow:

  1. 1.

The required SNR to obtain the same average symbol error rate (averaged over
all MBM ensembles in static Rayleigh fading channel) will approximately reduce
by a factor of

$\displaystyle\gamma_{c}=\bigg{(}\frac{\Gamma(\frac{n}{2},\frac{n}{2}E)}{\Gamma(\frac{n}{2},nE)}\bigg{)}^{\frac{2}{n}},$
(32)

where $\Gamma(n,x):=\int_{x}^{\infty}t^{n-1}e^{-t}\mathrm{d}t$ indicates the
upper incomplete gamma function.

  2. 2.

An average of $M\times\big{(}1-F(nE;n)\big{)}$ points will be removed from the
constellation, where $F(x;n)$ is the cumulative distribution function for a
chi-squared random variable with $n$ degrees of freedom.

Appendix A provides proof of items 1 and 2.

SNR gain $\gamma_{c}$ can be thought of as a coding gain, and
$\delta:=\log_{2}\big{(}1-F(nE;n)\big{)}$ as the average number of additional
RF mirrors required to achieve an SNR gain specified in (32) and maintain same
rate of transmission. Fig. 5 illustrates savings in SNR (expressed in dB) as a
function of number of additional RF mirrors, for various number of receive
antennas. Fig. 6 compares the accuracy of the analytic expression in (32)
versus Monte-Carlo simulation for $N_{r}=2$ receive antennas.

##  Appendix A Selection Gain Analysis

Fix an MBM constellation point $\mathbf{h}$ with energy
$\lambda:=\sum_{i=1}^{n}x_{i}^{2}$. Given $\mathbf{h}$ is transmitted, the
error probability averaged over competing symbol $\mathbf{h}^{\prime}$ is
equal to

$\displaystyle
P_{e}(\mathbf{h})=\mathbb{E}_{\mathbf{h}^{\prime}}\left[Q\left(\sqrt{\frac{\mathsf{snr}}{4}}\lVert\mathbf{h}-\mathbf{h}^{\prime}\rVert\right)\right].$
(33)

Since $\mathbf{h}$ is given, random variable
$\lVert\mathbf{h}-\mathbf{h}^{\prime}\rVert^{2}$ follows a non-central chi-
squared distribution with non-centrality parameter $\lambda$. Because
$\mathbf{h}$ follows an isotropic Gaussian distribution, (33) only depends on
energy of the point, namely $\lambda$. In high SNR regime, we have

$\displaystyle P_{e}(\lambda)$ $\displaystyle:=P_{e}(\mathbf{h})$ (34)
$\displaystyle=\int_{z=0}^{\infty}Q\left(\sqrt{\frac{\mathsf{snr}z}{4}}\right)f(z;n,\lambda)\
\mathrm{d}z$ (35)
$\displaystyle=\sum_{i=0}^{\infty}\frac{({\lambda}/{2})^{i}}{i!}e^{-\frac{\lambda}{2}}\int_{z=0}^{\infty}Q\left(\sqrt{\frac{\mathsf{snr}z}{4}}\right)f(z;n+2i)\
\mathrm{d}z.$ (36)

Replacing the integral with the closed form solution, gives

$\displaystyle P_{e}(\lambda)$
$\displaystyle=e^{-\frac{\lambda}{2}}\sum_{i=0}^{\infty}\frac{({\lambda}/{2})^{i}}{i!}\frac{\sqrt{{\mathsf{snr}}/(4\pi)}}{2(1+{\mathsf{snr}}/{4})^{\frac{n+2i+1}{2}}}\frac{\Gamma(\frac{n+2i+1}{2})}{\Gamma(\frac{n+2i+2}{2})}\>_{2}F_{1}(1,\frac{n+2i+1}{2};\frac{n+2i+2}{2};\frac{1}{1+{\mathsf{snr}}/{4}})$
(37)
$\displaystyle\approx\frac{1}{2\sqrt{\pi}}e^{-\frac{\lambda}{2}}(1+\frac{\mathsf{snr}}{4})^{-\frac{n}{2}}\sum_{i=0}^{\infty}\frac{\left(\frac{\lambda}{2(1+{\mathsf{snr}}/{4})}\right)^{i}}{i!}\frac{\Gamma(\frac{n+2i+1}{2})}{\Gamma(\frac{n+2i+2}{2})}.$
(38)

The term in the summation is of the form of confluent hypergeometric functions
of the first kind, ${}_{1}F_{1}(.;.;.)$. Accordingly,

$\displaystyle P_{e}(\lambda)$ $\displaystyle=\frac{1}{\sqrt{2\pi
n}}e^{-\frac{\lambda}{2}}(1+\frac{\mathsf{snr}}{4})^{-\frac{n}{2}}\
_{1}F_{1}(\frac{n+1}{2};\frac{n+2}{2};\frac{\lambda}{2(1+{\mathsf{snr}}/{4})})$
(39) $\displaystyle\approx\frac{1}{\sqrt{2\pi
n}}e^{-\frac{\lambda}{2}}(1+\frac{\mathsf{snr}}{4})^{-\frac{n}{2}}(1+\frac{\lambda}{2(1+\mathsf{snr}/4)})\
$ (40) $\displaystyle\approx\frac{1}{\sqrt{2\pi
n}}e^{-\frac{\lambda}{2}}(1+\frac{\mathsf{snr}}{4})^{-\frac{n}{2}}.$ (41)

To achieve a selection gain the transmitter is instructed to remove any
constellation point with energy lower than $nE$. The average error probability
given that low energy points are removed is given by

$\displaystyle P_{e}(E)$ $\displaystyle=\mathbb{E}\big{[}P_{e}(\lambda)\
\big{|}\ \lambda\geq nE\big{]}$ (42)
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n}{2},\frac{nE}{2})}\int_{nE}^{\infty}P_{e}(\lambda)f(\lambda;n)\
\mathrm{d}\lambda.$ (43)

Substituting $P_{e}(\lambda)$ with (41), leads to

$\displaystyle P_{e}(E)$
$\displaystyle\approx\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n}{2},\frac{nE}{2})}\frac{1}{\sqrt{2\pi
n}}\left(1+\frac{\mathsf{snr}}{4}\right)^{-\frac{n}{2}}\int_{nE}^{\infty}e^{-\frac{\lambda}{2}}f(\lambda;n)\
\mathrm{d}\lambda$ (44)
$\displaystyle=\frac{\Gamma(\frac{n}{2})}{\Gamma(\frac{n}{2},\frac{nE}{2})}\frac{1}{\sqrt{2\pi
n}}\left(1+\frac{\mathsf{snr}}{4}\right)^{-\frac{n}{2}}\int_{nE}^{\infty}e^{-\lambda}{\frac{\lambda^{\frac{n}{2}-1}}{2^{\frac{n}{2}}\Gamma(\frac{n}{2})}}\
\mathrm{d}\lambda$ (45)
$\displaystyle={\frac{\Gamma(\frac{n}{2},nE)}{2^{\frac{n}{2}}\Gamma(\frac{n}{2},\frac{nE}{2})}}\frac{1}{\sqrt{2\pi
n}}\left(1+\frac{\mathsf{snr}}{4}\right)^{-\frac{n}{2}}$ (46)
$\displaystyle\approx{\frac{\Gamma(\frac{n}{2},nE)}{\Gamma(\frac{n}{2},\frac{nE}{2})}}\frac{1}{\sqrt{2\pi
n}}\left(\frac{\mathsf{snr}}{2}\right)^{-\frac{n}{2}}.$ (47)

Effectively, the required SNR to achieve the same symbol error probability is
reduced by a factor given by,

$\displaystyle\gamma_{c}\approx\left(\frac{\Gamma(\frac{n}{2},\frac{nE}{2})}{\Gamma(\frac{n}{2},nE)}\right)^{\frac{2}{n}}.$
(48)

The choice of $E$ determines the selection gain as well as the number of extra
RF mirrors required to maintain the same rate of transmission. Let $M$ be the
initial number of points, and $M^{\prime}$ the remaining number of points
after removing low energy ones. Since energy of an MBM constellation point
follows a chi-square distribution, the average number of points remaining in
the constellation set is given by the probability tail of chi-squared random
variable with $n$ degrees of freedom.

$\displaystyle M^{\prime}=MF(nE;n)$ (49)

Therefore, on average, $\delta=\log_{2}\left(1-F(nE;n)\right)$ additional RF
mirrors is required to support the same transmission rate.

## References

  * [1] A. K. Khandani, “Media-based modulation: A new approach to wireless transmission,” IEEE International Symposium on Information Theory (ISIT 2013), July 2013, pp.3050—3054
  * [2] A.K. Khandani, “Media-based Modulation: Converting Static Rayleigh Fading to AWGN,” IEEE International Symposium on Information Theory (ISIT 2014), June 2014, pp. 1549—1553
  * [3] Lizhong Zheng and D. N. C. Tse, ”Diversity and multiplexing: a fundamental tradeoff in multiple-antenna channels,” in IEEE Transactions on Information Theory, vol. 49, no. 5, pp. 1073-1096, May 2003, doi: 10.1109/TIT.2003.810646.
  * [4] N. Ishikawa, S. Sugiura and L. Hanzo, “50 Years of Permutation, Spatial and Index Modulation: From Classic RF to Visible Light Communication and Data Storage,” in IEEE Comm. Surveys & Tutorials, vol. 20, no. 3, pp. 1905-1938
  * [5] M.-S. Alouini and A. Goldsmith, “A unified approach for calculating error rates of linearly modulated signals over generalized fading channels,” IEEE Transactions on Communications, vol. 47, no. 9, pp. 1324–1334, 1999.
  * [6] R. Barnard, K. Richards, and H. Tiedeman, “A survey of some bounds for gauss’ hypergeometric function and related bi-variate means,” Journal of Mathematical Inequalities, vol. 4, pp. 45–52, 01 2010.
  * [7] C. Thommesen, “Error-correcting capabilities of concatenated codes with MDS outer codes on memory-less channels with maximum likelihood decoding,” IEEE Transactions on Information Theory, vol. 33, no. 5, pp. 632–640, 1987.


# Gas metallicity distributions in SDSS-IV MaNGA galaxies: what drives
gradients and local trends?

N. Boardman1, V. Wild1, T. Heckman2, S. F. Sanchez3, R. Riffel4,5, R. A.
Riffel5,6, G. Zasowski7
1School of Physics and Astronomy, University of St Andrews, North Haugh, St
Andrews KY16 9SS, UK
2The William H. Miller III Department of Physics & Astronomy, Johns Hopkins
University, Baltimore, MD 21218, United States
3Universidad Nacional Autónoma de México, Instituto de Astronomıa, A.P.
70-264, 04510, Mexico, D.F., Mexico
4jDepartamento de Astronomia, Instituto de Física, Universidade Federal do Rio
Grande do Sul, CP 15051, 91501-970, Porto Alegre, RS, Brazil
5Laboratório Interinstitucional de e-Astronomia - LIneA, Rua Gal. José
Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil
6Departamento de Física, CCNE, Universidade Federal de Santa Maria, 97105-900,
Santa Maria, RS, Brazil
7Department of Physics & Astronomy, University of Utah, Salt Lake City, UT,
84112, USA
E-mail<EMAIL_ADDRESS>
(Accepted X. Received X; in original form X)

###### Abstract

The gas metallicity distributions across individual galaxies and across galaxy
samples can teach us much about how galaxies evolve. Massive galaxies
typically possess negative metallicity gradients, and mass and metallicity are
tightly correlated on local scales over wide range of galaxy masses; however,
the precise origins of such trends remain elusive. Here, we employ data from
SDSS-IV MaNGA to explore how gas metallicity depends on local stellar mass
density and on galactocentric radius within individual galaxies. We also
consider how the strengths of these dependencies vary across the galaxy mass-
size plane. We find that radius is more predictive of local metallicity than
stellar mass density in extended lower mass galaxies, while we find density
and radius to be almost equally predictive in higher-mass and more compact
galaxies. Consistent with previous work, we find a mild connection between
metallicity gradients and large-scale environment; however, this is
insufficient to explain variations in gas metallicity behaviour across the
mass-size plane. We argue our results to be consistent with a scenario in
which extended galaxies have experienced smooth gas accretion histories,
producing negative metallicity gradients over time. We further argue that more
compact and more massive systems have experienced increased merging activity
that disrupts this process, leading to flatter metallicity gradients and more
dominant density-metallicity correlations within individual galaxies.

###### keywords:

galaxies: ISM – galaxies: structure – galaxies: general – ISM: general –
galaxies: statistics – ISM: abundances

††pubyear: 2022††pagerange: Gas metallicity distributions in SDSS-IV MaNGA
galaxies: what drives gradients and local trends?–B

##  1 Introduction

Gas-phase chemical abundances in galaxies are products of a range of physical
processes; thus, these abundances can teach us much about how galaxies form
and evolve. Modern integral-field unit (IFU) datasets – including CALIFA
(Sánchez et al., 2012a), SAMI (Croom et al., 2012), and MaNGA (Bundy et al.,
2015)– provide spatial distributions of abundances across large numbers of
galaxies and have led to a range of advances over the preceding decade.

Notably, the majority of massive galaxies possess negative gas metallicity
gradients. Sánchez et al. (2014), for instance, report a characteristic slope
of -0.1 dex per disc effective radius in their sample of CALIFA galaxies (see
also Sánchez et al., 2012b), and Sánchez-Menguiano et al. (2018) report a
similar slope in their MUSE galaxy sample. Gas metallicity gradients have been
reported from observations to correlate with a range of other galaxy
properties including stellar mass (Belfiore et al., 2017; Mingozzi et al.,
2020; Schaefer et al., 2020; Franchetto et al., 2021), size (Carton et al.,
2018), gas fraction (Franchetto et al., 2021), bulge-to-total (B/T) ratio
(Moran et al., 2012), and environment (Lian et al., 2019; Franchetto et al.,
2021). Boardman et al. (2021) report a tight trend in gradients across the
mass-size plane for MaNGA galaxies, wherein more extended galaxies display
more strongly negative gradients at a given stellar mass; this is similar to
what was reported for stellar metallicity gradients by Li et al. (2018), and
is found in Boardman et al. (2021) to be significantly tighter than gradient
trends with mass or size individually. Such a result is also analogous to the
apparent complex connection between metallicity gradient, mass and morphology
(Sánchez et al., 2021), due to the tight trend between galaxy size and
morphology at a given stellar mass (e.g. Fernández Lorenzo et al., 2013).

A number of local $\sim$kpc-scale relations involving gas metallicity have
also been reported in the literature. An example of this is the resolved mass-
metallicity relation (rMZR) between gas metallicity and stellar mass surface
density $\Sigma_{*}$ (e.g. Rosales-Ortega et al., 2012; Sánchez et al., 2013;
Barrera-Ballesteros et al., 2016), which can be understood as reflecting a
tight interplay between gas metallicities and local conditions (e.g. Barrera-
Ballesteros et al., 2018), though an additional metallicity dependence on
stellar mass is also apparent for lower-mass galaxies in particular (e.g.,
Barrera-Ballesteros et al., 2016; Gao et al., 2018; Hwang et al., 2019).
Barrera-Ballesteros et al. (2016) argue that the rMZR alone is sufficient to
reproduce metallicity gradients over stellar masses between $10^{9}$ and
$10^{11}M_{\odot}$, while Boardman et al. (2022, hereafter B22) find local
relations – involving metallicity, $\Sigma_{*}$, stellar mass $M_{*}$, stellar
age and the D4000 index (Bruzual, 1983) – to predict gradient trends across
the mass-size plane qualitatively similar to what is observed. Thus, gas
metallicity gradients can potentially be understood as arising from local
conditions across a galaxy, with inside-out formation naturally producing
negative metallicity gradients in star-forming galaxies (e.g. Franchetto et
al., 2021).

Chemical evolution models can provide us with additional insight into gas
metallicities. In the simple “closed-box" model (Schmidt, 1963), the
interstellar medium (ISM) is enriched by successive stellar generations and
the gas metallicity depends solely on the local gas fraction, with higher
metallicities associated with lower gas fractions. More complex models also
consider the effects of inflow and outflow, at which point the local escape
velocity also becomes relevant (e.g. Lilly et al., 2013; Zhu et al., 2017).
Recent observational studies have generally favoured both gas fraction and
escape velocity as being important in shaping the gas metallicity (Moran et
al., 2012; Carton et al., 2015; Barrera-Ballesteros et al., 2018). The rMZR
provides a tighter correlation overall than do gas fraction or escape
velocity, however, which could potentially be due to $\Sigma_{*}$ encoding
information on both of these parameters (Barrera-Ballesteros et al., 2018).

Typically, studies of the rMZR along with other local relations consider large
samples of star-forming regions observed over many separate galaxies. However,
further information can be gained by considering local trends within
individual galaxies. An example of this can be seen in Sánchez-Menguiano et
al. (2019), who investigate residual trends between gas metallicity and star-
formation rate within individual MaNGA galaxies. In particular, the
opportunity exists to determine whether the rMZR is the most fundamental local
relation within individual systems.

Here, we employ SDSS MaNGA data to assess the strength of the correlation
between metallicity and $\Sigma_{*}$ in individual galaxies, which we then
compare to the strength of correlation between metallicity and galactocentric
radius. Our direct use of radius is motivated by the steep gas metallicity
gradients measured in extended star-forming MaNGA galaxies, along with the
radial dependencies in both gas mass fractions (e.g. Carton et al., 2015) and
escape velocities within galaxies. We consider the gas metallicity as given by
the oxygen abundance, $\mathrm{12+log(O/H)}$. We then investigate whether our
results could be explained by variations in galaxies’ environments, motivated
primarily by the work of Franchetto et al. (2021).

This article is structured as follows. We present our sample and explain
relevant parameters in § 2. We present our results in § 3 and discuss our
findings in § 4, before summarising and concluding in § 5. Throughout this
work, we assume a Kroupa IMF (Kroupa, 2001; Kroupa & Weidner, 2003) and we
adopt the following standard $\Lambda$ Cold Dark Matter cosmology:
$\mathrm{H_{0}}=71$ km/s/Mpc, $\mathrm{\Omega_{M}}$ = 0.27,
$\mathrm{\Omega_{\Lambda}}$ = 0.73.

##  2 Sample and data

Figure 1: Effective radius, plotted against galaxy stellar mass, with data
points coloured by the radial gas metallicity gradients (left) and by the
Spearman correlation coefficients between $\Sigma_{*}$ and metallicity
(middle) and between radius and metallicity (right). We apply LOESS smoothing
in all panels. The middle panel uses a reversed colour scale; thus, blue
regions correspond to negative gradients, positive $\Sigma_{*}$-metallicity
correlations and negative radius-metallicity correlations. The contours
enclose $\sim$50% and $\sim$80% of galaxies.

We employ integral-field spectroscopic data from the SDSS-IV MaNGA survey. The
MaNGA survey observed galaxies with the BOSS spectrographs (Smee et al., 2013)
on the 2.5 meter telescope at Apache point observatory (Gunn et al., 2006).
The MaNGA galaxy sample covers redshifts over the approximate range of 0.01 to
0.15, and was selected to possess a roughly flat logarithmic mass distribution
(Yan et al., 2016b; Wake et al., 2017). The MaNGA IFUs consist of bundles of
19-127 optical fibres with 2′′ diameters (Law et al., 2016). The IFUs employ
hexagonal fibre configurations, and galaxies are observed with three-point
dither patterns to fully sample the field of view (Drory et al., 2015; Law et
al., 2015). The MaNGA data reduction pipeline (DRP; Law et al., 2016; Yan et
al., 2016a) reduces the data, producing $0.5^{\prime\prime}\times
0.5^{\prime\prime}$ spaxel datacubes with a median PSF full-width at half-
maximum (FWHM) of roughly 2.5′′ (Law et al., 2016). The MaNGA data analysis
pipeline (DAP; Belfiore et al., 2019a; Westfall et al., 2019) then computes
various mapped quantities relating to stellar and gaseous galaxy components.
MaNGA data and derived quantities are accessible through the SDSS Science
Archive Server111https://data.sdss.org/sas/, and can also be accessed through
the Marvin 222https://www.sdss.org/dr17/manga/marvin/ interface (Cherinka et
al., 2019). The reduced spectra have a resolution of $R\simeq 2000$ and cover
a wavelength range of of 3600–10000Å. The full MaNGA galaxy sample consists of
more than 10000 galaxies as of SDSS DR17 (Abdurro’uf et al., 2022); however,
only a subset of these will be suitable for studying gas metallicities and gas
metallicity gradients.

We selected a sample of MaNGA galaxies and spaxels in almost exactly the same
manner as in B22, employing values from the DAP and from Pipe3D (Sánchez et
al., 2016a, b, 2018). Our one difference from B22 is that we employ the DR17
versions of Pipe3D and the DAP. We therefore refer the reader to B22 for a
full description of the sample selection, though we also summarise it here. In
brief, we begin by selecting MaNGA galaxies from the MPL-10 data release
(which contains $\sim$9000 galaxies) with NASA-Sloan-Atlas (NSA; Blanton et
al., 2011) elliptical Petrosian axis ratios (b/a) of at least 0.6 to avoid
edge-on cases, and we cross-match with the 2X version of the GALEX-SDSS-WISE
Legacy catalog (GSWLC; Salim et al., 2016, 2018) to obtain stellar masses. We
identify star-forming spaxels with galactocentric radii between 0.5 $R_{e}$
and 2 $R_{e}$ using the BPT-NII diagnostic diagram (Baldwin et al., 1981) and
by requiring a minimum H$\alpha$ equivalent width of 10, and we require a
minimum S/N of 3 for the following emission features: H$\alpha$, H$\beta$,
$\mathrm{[OIII]}\lambda 5007$, $\mathrm{[NII]}\lambda 6583$ and
$\mathrm{[OII]}\lambda 3737,3729$. We further restrict to galaxies with at
least 20 star-forming spaxels. We employ Pipe3d $\Sigma_{*}$ values along with
the associated uncertainties. As in past works (e.g. Barrera-Ballesteros et
al., 2016; Boardman et al., 2022), we multiply observed $\Sigma_{*}$ values
(along with the errors) by b/a to correct for the effects of inclination.

We then calculate gas metallicity maps for each sample galaxy using DAP
emission line fluxes with the Marino et al. (2013, hereafter M13) calibrator.
The M13 calibrator estimates gas metallicity at
$\mathrm{12+log(O/H)}=8.533-0.214\ \mathrm{O3N2}$, where O3N2 is given as

$\mathrm{O3N2}=\log\left(\frac{\mathrm{[OIII]}\lambda
5007}{\mathrm{H\beta}}\times\frac{\mathrm{H\alpha}}{\mathrm{[NII]}\lambda
6583}\right)$ (1)

O3N2 calibrators have the advantage of being essentially unaffected by dust
extinction. However, these calibrators implicitely assume a fixed N/O–O/H
relation, which can bias metallicity gradient measurements in certain cases.
Thus, as in B22, we present in Appendix A results from the R2 calibrator
described in Pilyugin & Grebel (2016), though we will focus on the M13
calibrator in the main paper text. As explored further in Appendix A, we
obtain similar outcomes from both calibrators.

B22 generated ‘model’ metallicities from local relations, which they used to
predict gas metallicity gradients for comparisons with observed gradients. The
first of these model sets, termed ‘base models’ in B22, constructed
metallicity predictions using the three-way relation between $M_{*}$,
$\Sigma_{*}$ and $\mathrm{12+log(O/H)}$ (e.g. Gao et al., 2018). For the
second model set, termed ‘star formation history models’ (hereafter ‘SFH
models’, B22 corrected the base models for the two-dimensional residual
dependence on D4000 (Bruzual, 1983) and on light-weighted pipe3d stellar age.
We construct these two model sets in the exact same manner as described in
B22. We remove spaxels in sparsely-sampled regions of the parameter space at
this stage, and we remove galaxies (along with associated spaxels) that are
left with fewer than 20 star-forming spaxels afterwards. Following some
additional cuts as described in B22, we obtain a final sample consisting of
2127 galaxies containing 860214 individual star-forming spaxels.

The resulting sample is dominated by galaxies with significant ongoing star-
formation: 2027 (95% ) of the galaxies possess GSWLC-2X specific star
formation rates (sSFRs) above $10^{-11}yr^{-1}$, while only 15 (0.7% ) possess
sSFRs below $10^{-11.5}yr^{-1}$. We calculate gas metallicity gradients for
each galaxy in units of dex/$R_{e}$, as described in B22, where $R_{e}$ is the
elliptical Petrosian half-light radius from the NSA catalog.

We also obtain from Duckworth et al. (2019) a set of environmental parameters
calculated for MaNGA galaxies. The first two of these are
$\rho_{3\mathrm{Mpc}}$ and $\rho_{\mathrm{9Mpc}}$, which denote galaxy
environment densities smoothed with a Gaussian kernal over local scales (3
Mpc) and larger scales (9 Mpc). In addition to these, we consider galaxies’
$D_{skel}$ and $D_{node}$ parameters, which denote the distances to cosmic web
nodes and filaments respectively. As in Duckworth et al. (2019), we normalise
$D_{skel}$ and $D_{node}$ by the mean inter-galaxy separation at a given
redshift, $\langle D_{z}\rangle=n(z)^{-1/3}$ where $n(z)$ is the number
density; we calculate the number density as a function of redshift using the
Tempel et al. (2014) spectroscopic catalog of SDSS DR10 galaxies.

Figure 2: Histogram of errors in $\log(\Sigma_{*})$.

##  3 Results

###  3.1 Correlations of density and radius with gas metallicity

We begin by considering the strength of the correlations of $\Sigma_{*}$ and
normalised galactocentric radius $R/R_{e}$ with gas-phase metallicity, for
spaxels in each sample galaxy separately. We parametrize this strength using
the Spearman rank correlation coefficient $\rho$, computed from the IDL
R_CORRELATE procedure for each galaxy in turn; we denote the coefficient
between gas metallicity and $\Sigma_{*}$ as $\rho_{\Sigma}$, while for gas
metallicity and radius we use $\rho_{R}$.

In Figure 1, we plot these coefficients across the mass-size plane along with
the radial gas metallicity gradients ($\nabla[O/H]$) from B22. We employ
locally-weighted regression smoothing (LOESS; Cleveland & Devlin, 1988) as
implemented in IDL333available from http://www-
astro.physics.ox.ac.uk/~mxc/software/ when showing these figures, to more
cleanly show the overall mass-size trend. We compute the LOESS-smoothed value
for each data point using the closest 20% of data points with the rescale
keyword applied. We compute errors from the scatter in neighbouring points for
the purpose of the smoothing calculation.

It is apparent from Figure 1 – unsurprisingly so – that the coefficients trend
similarly to the gradients across the plane, with the strongest correlations
seen in more massive extended galaxies for both $\Sigma_{*}$ and $R$. We
further explore the connection between the gradients and correlation
coefficients in Appendix B.

From Figure 1, the absolute values of $\rho_{R}$ are typically as high or
higher than $\rho_{\Sigma}$ However, it is not truly fair to compare
$\rho_{\Sigma}$ and $\rho_{R}$ directly, since $\Sigma_{*}$ possesses non-
negligible measurement errors compared to $R$. Thus, before comparing these
two parameters further, we perturb the radii of spaxels to approximately match
the effect of the $\Sigma_{*}$ errors.

Figure 3: Difference in magnitudes of the Spearman correlation coefficient
between gas metallicity and stellar density, $|\rho_{\Sigma}|$, and the
coefficient between gas metallicity and galactocentric radius before
($|\rho_{R}|$; left panel) and after ($|\rho_{R,perturbed}|$; right panel)
perturbing spaxel radii. Each datapoint represents one galaxy, and LOESS
smoothing is applied; blue regions indicate where metallicity is more strongly
correlated with radius, while red regions indicate a stronger correlation with
density. The contours enclose $\sim$50% and $\sim$80% of galaxies. Figure 4:
Effective radius, plotted against galaxy stellar mass, with data points
coloured by the B22 ‘SFH model’ metallicity gradients (left) and by the
Spearman correlation coefficients between $\Sigma_{*}$ and model metallicity
($[\rho_{\Sigma}]_{SFH}$, middle) and between radius and model metallicity
($[\rho_{R}]_{SFH}$, right). We apply LOESS smoothing in all panels, and the
middle panel uses a reversed colour scale. The contours enclose $\sim$50% and
$\sim$80% of galaxies. We find that the SFH models do not reproduce the trends
shown in Figures 1 and 2 for observed gas metallicities: while individual
parameters behave similarly across the mass-size plane, we find local
metallicties to be correlated more strongly with stellar density than with
radius in most cases.

We perturb spaxels’ $R/R_{e}$ as follows. The error in $\Sigma_{*}$ is
typically well below 0.1 dex, as shown in Figure 2; thus, we adopt 0.1 dex as
a conservative estimate for the level of scatter in $\Sigma_{*}$ due to
measurement/fitting uncertainties. We then apply a Gaussian scatter to
spaxels’ $\log(R/R_{e})$ of
$\Delta_{R}=0.1\mathrm{dex}\times\sigma_{\log(r)}/\sigma_{\log(\Sigma)}$,
where $\sigma_{\log(r)}$ and $\sigma_{\log(\Sigma)}$ denote the
dispersions444These dispersions and all subsequent dispersions are calculated
using the ROBUST_SIGMA IDL procedure, available at
https://idlastro.gsfc.nasa.gov/ftp/pro/robust/robust_sigma.pro in
$\log(R/R_{e})$ and $\log(\Sigma_{*})$ over the full spaxel sample. We obtain
$\sigma_{\log(r)}=0.149\mathrm{dex}$ and
$\sigma_{\log(\Sigma)}=0.409\mathrm{dex}$, and we hence adopt
$\Delta_{R}=0.036\mathrm{dex}$ when perturbing spaxel radii. We calculate
Spearman correlation coefficients between gas metallicity and the perturbed
radii, and we denote these coefficients as $\rho_{R,perturbed}$.

Figure 5: Difference in magnitudes of the Spearman correlation coeffcient
between SFH model gas metallicity and stellar density and the coefficient
between model gas metallicity and galactocentric radius, before (left panel)
and after (right panel) perturbing spaxel radii. The contours enclose
$\sim$50% and $\sim$80% of galaxies. We further demonstrate here that the B22
SFH models do not predict the behaviour of observed metallicities shown in
Figure 3.

In Figure 3, we plot across the mass-size plane the difference between
$|\rho_{\Sigma}|$ and $|\rho_{R}|$ along with the difference between
$|\rho_{\Sigma}|$ and $|\rho_{R,perturbed}|$, with LOESS smoothing applied. We
see a clear trend in the two $\rho$ values: spaxels in extended galaxies with
low-to-intermediate masses have metallicities more closely correlated with R,
while $\Sigma_{*}$ and R are almost equally predictive of the metallicity in
higher-mass and more compact galaxies. We also find only mild differences in
our results from using $\rho_{R}$ or $\rho_{R,perturbed}$; thus, the choice of
$\rho_{R}$ or $\rho_{R,perturbed}$ does not significantly affect the final
outcome.

By construction, the B22 base models cannot predict the behaviour observed in
Figure 3, due to the predicted metallicity depending entirely on $\Sigma_{*}$
within any given galaxy. However, the B22 SFH models depend also on D4000 and
on stellar age, and so are worth considering further. We present across the
mass-size plane the resulting values of $\nabla[O/H]$, $\rho_{\Sigma}$ and
$\rho_{R}$ in Figure 4, and we present the corresponding values of
$|\rho_{\Sigma}|-|\rho_{R}|$ and $|\rho_{\Sigma}|-|\rho_{R,perturbed}|$ in
Figure 5. We find that the SFH models likewise do not predict the observed
metallicity behaviour in these regards, with $|\rho_{\Sigma}|$ greater than
$|\rho_{R,perturbed}|$ over almost the whole mass-size plane once smoothing is
applied. From this, we argue that radius-metallicity trends are not simply a
consequence of the local relations covered by the B22 models.

###  3.2 Gas metallicity as a 2D function of density and radius.

To further assess how spaxel gas metallicity varies with both $\Sigma_{*}$ and
$R$, we split our galaxy sample into subsamples based on their positions in
the mass-size plane, and we then consider all spaxels within a given subsample
together. Our procedure for selecting the subsamples is the same as in B22: we
split the sample in stellar mass so as to encompass 1/3 of the sample apiece,
and then further split the sample using the median mass-size relation
calculated over a series of mass bin. We demonstrate this process in Figure 6.
We deem galaxies “extended" or “compact" based on whether they lie above or
below the median mass-size relation, and we deem galaxies “low-mass", “mid-
mass" or “high-mass" depending on the mass bins they fall in.

Figure 6: Same as the left window of Figure 1, but with the positions of our
six mass-size bins overlaid. We split the galaxy sample into six subsamples
accordingly, and consider these subsamples over the remainder of Section 4.2
Figure 7: Mean observed spaxel gas metallicity in bins of stellar density and
radius, for the six mass-size subsamples. Each bin encompasses a minimum of 50
spaxels. We find gas metallicity to vary with both $\Sigma_{*}$ and $R/R_{e}$,
with the precise behaviour differing significantly across different mass-size
bins.

In Figure 7 we plot the spaxel gas metallicities as a combined function of
$\Sigma_{*}$ and $R/R_{e}$ for each of the six mass-size subsamples. Here, we
calculate the mean metallicity in bins of $\Sigma_{*}$ and $R/R_{e}$, with all
shown bins containing at least 50 spaxels. This is conceptually similar to
Figure 10 of Neumann et al. (2021), which presents Firefly stellar
metallicities in terms of $\Sigma_{*}$ and $R/R_{e}$ for galaxy subsamples
split simultaneously by mass and morphology. In general, simultaneous splits
by mass and morphology (and/or by mass and size) are a common way to explore
galaxy samples in the literature (e.g. García-Benito et al., 2017; Sánchez,
2020; Sánchez et al., 2021), and allow for additional insight compared to
splitting by any one property alone. We see in Figure 7 that spaxel gas
metallicities trend with both $\Sigma_{*}$ and radius, and that the shape of
these trends is not consistent between different mass-size bins. Metallicity
trends mainly with density in low-mass compact galaxies, for instance, while
at higher masses the metallicity both rises with density and falls with
radius.

Our findings here possess some differences to what Neumann et al. (2021)
report for stellar metallicities, In general, Neumann et al. (2021) find
stellar metallicity to trend chiefly with $\Sigma_{*}$ for most mass-
morphology bins. For low-to-intermediate mass galaxies and for massive
spirals, Neumann et al. (2021) then find stellar. metallicity to rise with
radius at fixed $\Sigma_{*}$. Neumann et al. (2021) argue this to require a
metallicity driver in addition to $\Sigma_{*}$, with the driver serving either
to raise metallicity in galaxies’ outer parts or else to dilute it in
galaxies’ inner regions. By contrast, we find that gas metallicity typically
falls with radius at fixed $\Sigma_{*}$, though we also find metallicity to
rise with $\Sigma_{*}$ at fixed radius; the only exception here is our low-
mass compact galaxy subsample, which behaves similarly to what Neumann et al.
(2021) report for stellar metallicities.

###  3.3 Connection with environment

Figure 8: Environment measures plotted across the mass-size plane with LEOSS
smoothing applied. Data points are coloured by local density calculated by a 3
Mpc (top left) or 9 Mpc (top right) Gaussian kernal, by the nearest node
distance (bottom left) and by the nearest fillament distance (bottom right).
We invert the color scale on the bottom two panels, such that darker colours
are associated with less dense environments.

From past work, the connection between gas metallicity gradients and galaxy
environment appears to be small but non-neglible. Few differences are found in
the gradients of satellite and central galaxies. Lian et al. (2019) only find
a significant effect for satellite galaxies with $M_{*}<10^{9.7}M_{\odot}$,
for instance, for which denser environments are associated with flatter
gradients. Schaefer et al. (2019) likewise only detect mild gradient
differences between galaxies with satellite/central classifications, with
those differences depending on the chosen gas metallicity calibrator. On the
other hand, cluster environments have been found to be associated with flatter
gradients than are field environment at a given mass, in observations
(Franchetto et al., 2021) as well as in simulations (Lara-López et al., 2022).
Denser environments have also been reported to be associated with smaller disc
galaxy sizes, particularly at lower stellar masses (e.g. Cebrián & Trujillo,
2014; Kuchner et al., 2017). Thus, it is worthwhile to consider whether our
gas metallicity results can be explained – at least in part – by variations in
galaxies’ environments.

We now cross-match our sample with the Duckworth et al. (2019) catalog, in
order to assess the importance of environment for understanding our results
reported up to now. We obtain environment measures for 1726 galaxies (81.1% of
our sample), and we analyse only these galaxies over the remainder of this
subsection. This reduction in sample size is due to Duckworth et al. (2019)
employing an earlier MaNGA data-release (MPL-6) that contains fewer galaxies
than MPL-10.

In Figure 8, we present the four environment quantities described in Section 2
across the mass-size plane, with LOESS smoothing applied. We see that all four
quantities primarily trend with mass, such that higher mass star-forming
galaxies are associated with lower environmental densities and greater
distances from cosmic web features; this is different from the trends
involving metallicity gradients, in which size is found to be a significant
factor. As such, we argue that environment variations are not sufficient to
explain the variation in metallicity gradients across the mass-size plane. The
finding of a mass-environment connection, we note, is likely a consequence of
our selected sample: the sample consists only of star-forming galaxies by
design, with the most massive star-forming galaxies then being more likely to
avoid quenching by residing in less dense environments.

We probe for a connection between mass and environment in the following
manner. We calculate the 10th and 90th percentiles of environment measures in
bins of stellar mass; for each bin, we compute the median and dispersion of
the gas metallicity gradient for all galaxies below the 10th percentile and
above the 90th percentile separately. This results in eight sets of median and
dispersion values for a given mass bin. We then plot the resulting values as a
function of mass in Figure 9, along with showing the full environment sample.
This is conceptually similar to Franchetto et al. (2021), who consider field
and cluster galaxies’ gas metallicity gradients as a function of stellar mass.

Figure 9: B22 gas metallicity gradient as a function of stellar mass. The blue
(red) squares show the median gradients for galaxies in the top (bottom) 10
percentiles in a given environment measure, as indicated by the panel titles:
local density with a 3 Mpc kernel (top left), local density with a 9 Mpc
kernel, nearest filament distance (bottom left) and nearest node distance
(bottom right).

It is evident from Figure 9 that galaxies in dense and sparse environments
have gradients that depend differently on mass, with sparser environments –
that is, larger values of $\rho_{3Mpc}$ and $\rho_{9Mpc}$ and/or smaller
values of $D_{skel}/\langle D_{z}\rangle$ and $D_{skel}/\langle D_{z}\rangle$
– associated with a slightly stronger mass-gradient dependencies; this
suggests that environment is a factor in gas metallicity gradients’ scatter at
a given $M_{*}$, regardless of how environment is quantified.

To summarise, we detect a mild connection between the metallicity properties
of star-forming galaxies and those galaxies’ large-scale environments. We find
dense galaxy environments to be associated with slightly higher overall
metallicities, consistent with the findings of Franchetto et al. (2021); thus,
we find that environment is indeed relevant to the evolution of gas
metallicity gradients. However, we find environment measures in our sample to
trend primarily with mass, as opposed to trending with both mass and size.
Thus, environment variations are not sufficient to explain the behaviour of
galaxies’ gas metallicity profiles across the mass-size plane.

##  4 Discussion

In the previous section, we employed MaNGA data to assess how metallicity
connects with $\Sigma_{*}$ and galactocentric radius on local ($\sim$1 kpc)
scales. Using the Spearman correlation coefficient, we found gas metallicity
to be correlated more strongly with radius than with $\Sigma_{*}$ in lower-
mass extended galaxies, while we found gas metallicity correlate roughly as
strongly with $\Sigma_{*}$ and with radius in more massive compact galaxies.
We also considered spaxel metallicities as a combined function of radius and
$\Sigma_{*}$, finding complex two-dimensional trends that vary with both
galaxy mass and galaxy size; these trends differ somewhat from what has
previously been reported for stellar metallicities (Neumann et al., 2021).
Finally, we found that large-scale environment variations are not sufficient
to explain our main findings, though we do find mild differences in gas
metallicity gradients for low-density and high-density galaxy environments.

Our results for lower-mass extended galaxies imply that position within the
galaxy, and not $\Sigma_{*}$, is the primary decider of gas metallicity within
these systems. This suggests that the rMZR computed across large spaxel
samples is not sufficient to fully capture the metallicity behaviour in these
particular galaxies. More complex local relations, such as those explored in
B22, are likewise unable to predict this particular finding. The apparent
importance of radius can potentially be explained as a result of the tight
radial dependencies both on gas fraction (which increases with radius; e.g.
Carton et al., 2015) and on escape velocity (which decreases with radius; e.g.
Barrera-Ballesteros et al., 2018), both of which influence gas metallicity
evolution in chemical evolution models (e.g. Lilly et al., 2013; Zhu et al.,
2017) while possessing far larger uncertainties than the radius itself. In
this picture, the outer parts of galaxies evolve slower (and hence take longer
to use their gas supply) and receive more inflowing metal-poor gas while also
being more susceptible to metal outflows, producing negative metallicity
gradient as a consequence of both points.

The above picture does not explicitly consider radial flows. Recent
simulations support a view in which gas primarily accretes onto the plane of a
galaxy’s disc and then gradually migrates towards a galaxies’ centre (e.g.
Trapp et al., 2022). Such a picture forms the basis of the ‘modified accretion
disc’ model presented in Wang & Lilly (2022a), which is argued in that paper
to be sufficient to reproduce the observed exponential structure of star-
forming discs. Wang & Lilly (2022b) subsequently found this model to be
capable of reproducing gas metallicity gradients: radially-inflowing gas
continuously enriches as it travels closer to a galaxies’ centre, naturally
giving rise to negative metallicity gradients. However, observational evidence
of significant inward gas flows remains scarce: though Schmidt et al. (2016)
reports significant inflow activity from direct HI observations, most such
studies report results consistent with little activity (Wong et al., 2004;
Trachternach et al., 2008; Di Teodoro & Peek, 2021).

Hwang et al. (2019) employed the $M_{*}$-$\Sigma_{*}$-O/H relation to identify
and study regions with anomalously low gas metallicities in MaNGA galaxies;
amongst other results, they found such regions to be preferentially located in
galaxies’ outer parts. Luo et al. (2021) subsequently studied the N/O
abundance ratios of star-forming gas and found anomalously-low metallicities
to be associated with elevated N/O ratios at a given metallicity, with the
greatest elevations seen at the largest radii; from simple models, they argue
this to be evidence for metal-poor gas inflows. Such results are consistent
with the Wang & Lilly (2022b) scenario, in which metal-poor gas migrates
inwards from the outer parts of a disc. More generally, these results
highlight the likely importance of metal-poor inflows in understanding gas
metallicity gradients, and they support a view in which inflows preferentially
occur at larger radii. Such a view is entirely consistent with our findings
for extended galaxies.

By contrast, more massive and more compact galaxies possess flatter gas
metallicity gradients and hence display weaker metallicity-radius
correlations, suggesting that other factors are relevant in shaping the
metallicity distributions of these systems. One possibility is that these
galaxies possess increased escape velocities due to their compactness,
resulting in reduced metal outflows and a higher relative dependence on
$\Sigma_{*}$ (which, in this picture, serves as a proxy for the gas content).
However, it can be seen from Figure 1 that $\rho_{\Sigma}$ actually weakens
somewhat within the most compact massive galaxies, making it worthwhile to
explore additional explanations.

At a given stellar mass, compact galaxies possess earlier-type morphology than
their more extended counterparts (e.g. Fernández Lorenzo et al., 2013;
Boardman et al., 2021). To explore this point further, we cross-match our
sample with the catalog of Simard et al. (2011) to obtain light-weighted
r-band B/T values for our galaxies, employing the fits in which the bulge
Sersic index was fixed to 4. We obtain values for 2013 galaxies (94.6 % of our
full sample), and we plot these with LOESS smoothing applied in Figure 10. As
expected, we find very low average B/T values amongst lower mass extended
galaxies, with B/T then rising for more massive and more compact galaxies.
Thus, low B/T ratios appear to be associated with strong metallicity-gradient
correlations and (particularly for higher-mass galaxies) with steeper gas
metallicity gradients in general. In turn, higher B/T ratios are then
associated with flatter gradients and a weaker metallicity-radius dependence.

Figure 10: Galaxy bulge-to-total ratios from Simard et al. (2011) plotted
across the mass-size plane with LOESS smoothing applied. The color scale is
such that blue regions indicate dominant disc components, with red regions
indicating increasingly prominent bulges.

Fu et al. (2013) have previously reported a significant connection between gas
metallicity gradients and B/T mass ratios: in both semi-analytic models and
data (Moran et al., 2012), smaller B/T values are associated with steeper gas
metallicity gradients. Fu et al. (2013) ascribe this to differences in
galaxies’ merger histories: model galaxies with steep metallicity gradients
are those which have continuously accreted gas in an undisturbed way, while
model galaxies with shallow/flat gradients have experienced greater merging
activity. In the Fu et al. (2013) models, mergers trigger central starbursts
and grow galaxies’ bulges while also flattening metallicity gradients, with
gradients then gradually steepening once smooth gas accretion resumes. Fu et
al. (2013) also consider the effects of inward radial flows, but find them to
have little impact on gradients. It should be noted that Yates et al. (2021)
find no such B/T-gradient dependence in their own semi-analytic models, which
employ a later version of the L-Galaxies code used by Fu et al. (2013);
however, Yates et al. (2021) select their sample to include only disk-
dominated (B/T < 0.3) galaxies, which is a possible factor in their different
findings.

Concerning the impact of mergers on gas metallicity profiles, various
hydrodynamical simulations have likewise found merging/interaction activity to
disrupt and flatten metallicity gradients (e.g. Rupke et al., 2010; Torrey et
al., 2012). Such a picture also has some observational support, with Kewley et
al. (2010) reporting flatter gradients in close galaxy pairs. A connection
between mergers and bulge-growth has also been reported in simulations (e.g.
Hopkins et al., 2010; Sauvaget et al., 2018). Mergers have been associated in
simulations with spin reorientation (Welker et al., 2014), with galaxies
transitioning from parallel to perpendicular with respect to their nearest
fillimant; a similar observational effect has been reported by Barsanti et al.
(2022) between spin alignment and bulge mass (see also Kraljic et al., 2021,
concerning the connection between alignment and morphology). Thus, we find it
plausible that different merging histories can explain differences in
metallicity gradient behaviour across the mass-size plane, particularly
amongst more massive galaxies.

As has been previously noted for MaNGA galaxies, we see a flattening in gas
metallicity gradients at low stellar masses. Such galaxies are metal-poorer
overall than their higher-mass counterparts (e.g., Lequeux et al., 1979;
Tremonti et al., 2004), and consequently can be expected to have evolved at a
slower rate (Asari et al., 2007; Camps-Fariña et al., 2021, 2022). Increased
metal outflows likely play a role in the flattening of low mass galaxies’
gradients (e.g. Belfiore et al., 2019b), along with wind recycling (e.g.
Belfiore et al., 2017, and references therein). Steller metallicity
distributions for low-mass spirals also appear consistent with little metal-
poor gas accretion, which is not the case for higher-mass spirals (Greener et
al., 2021). In addition, low-mass MaNGA galaxies have been found to possess
flatter gas density profiles than do higher-mass objects (Barrera-Ballesteros
et al., 2022), possibly due to shallower gravitational potentials in low-mass
objects. Thus, we argue that a reduced prominence of metal-poor inflows and an
increased prominence of metal outflows, along with shallower gravitational
potentials more generally, are what lead to flatter gas metallicity gradients
in low-mass galaxies.

A potential concern in this study is the effects of beam smearing from the
MaNGA PSF, particularly for galaxies observed with 19-bundle and 37-bundle
fibre-IFUs. Such galaxies have the smallest spatial extents on the sky, and
Belfiore et al. (2017) have shown this to be associated with flatter
calculated gas metallicity gradients in MaNGA galaxies. With this in mind,
Boardman et al. (2021) considered how MaNGA gas abundance varied with both
angular (in arcseconds) and physical (in kpc) galaxy size and found
significant trends between gas metallicity and physical size at most given
angular sizes; Boardman et al. (2021) thus argued that gradient trends across
the mass-size plane are not a result of MaNGA PSF effects, and we likewise
argue that PSF effects are not the cause of our results here. The importance
of PSF effects could be minimised by cutting explicitly on angular galaxy
size; however, this would bias our sample towards intrinsically larger
galaxies (Wake et al., 2017) and would reduce the coverage of our sample
across the mass-size plane, so we do not consider such an approach to be
optimal for our purposes.

The overall interpretation we advocate is as follows. At any given mass, the
most extended galaxies are those with the smoothest gas accretion histories.
These galaxies form in an inside-out manner, resulting in steep metallicity
gradients and large stellar discs. More compact galaxies have experienced more
significant merging activity, by contrast, and so possess more significant
bulges along with flatter metallicity gradients due to disruption of
previously-established metallicity profiles. Finally, the least massive
galaxies are more susceptable to significant metal outflows while also
experiencing less metal-poor inflow (as evidenced by their abundant low-
metallicity stars), resulting in flatter gradients overall.

##  5 Summary and conclusion

In this paper, we have employed MaNGA data to study local gaseous and stellar
metallicity as functions of stellar mass surface density and of galactocentric
radius. We considered star-forming spaxels across each individual sample
galaxy in turn, while also considering spaxels across galaxies grouped
according to their position within the mass–size plane.

Amongst lower mass extended galaxies, we found spaxels’ gas metallicity to
correlate more strongly with galactocentric radius than with density. The
local relations considered in B22 – which do not include radius – are not able
to replicate this finding. Thus, we argue that radius is an important
parameter in its own right for understanding gas metallicities on local
scales: metallicities are strongly related to the radius, or else are related
to one or more parameters strongly correlated with the radius. We argued our
findings to be consistent with a view in which extended galaxies experienced
comparatively smooth gas accretion histories, with metal-poor inflows and
metal outflows both preferentially affecting the galaxies’ outer parts; this,
along with the inside-out evolution of galaxies, naturally gives rise to
negative metallicity gradients. Our results here could also be explained by
the presence of significant inward radial flows, as formalised for instance by
the ‘modified accretion disc’ framework (Wang & Lilly, 2022a, b), but
observational support for this view remains limited at present.

More compact higher-mass galaxies, meanwhile, possess a reduced correlation
between gas metallicity and radius – something that is to be expected given
their relatively flattened gradients. In this case, stellar density provides a
stronger correlation than radius once the effect of uncertainties are
considered. This difference is associated with earlier-type morphologies –
that is, more prominent bulges – and we argue this to reflect the impact of
different merger histories for galaxies of different morphologies.

We also investigated the impact of environment, including position within the
Cosmic Web, finding only a limited connection between gas metallicity
gradients and local environment. We also detected no significant connections
between environment and galaxy size. We therefore argued that environment
variations are not sufficient to explain the findings described above.

Various potential extensions to this work remain. A machine learning approach
(e.g. Bluck et al., 2019, 2020) that includes spaxel radii along with other
local parameters would allow for a more thorough investigation of local
relations. Chemical evolution modelling for different classes of galaxy across
the mass-size plane could also prove fruitful.

## Acknowledgements

For the purpose of open access, the author has applied a Creative Commons
Attribution (CC BY) licence to any Author Accepted Manuscript version arising.
The support and resources from the Center for High Performance Computing at
the University of Utah are gratefully acknowledged. NFB and VW acknowledge
STFC grant ST/V000861/1.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions. SDSS-IV acknowledges support and resources from
the Center for High-Performance Computing at the University of Utah. The SDSS
web site is www.sdss.org. J.B-B thanks IA-100420 (DGAPA-PAPIIT, UNAM) and
CONACYT grant CF19-39578 support. RR thanks Conselho Nacional de
Desenvolvimento Científico e Tecnológico ( CNPq, Proj. 311223/2020-6,
304927/2017-1 and 400352/2016-8), Fundação de amparo ’a pesquisa do Rio Grande
do Sul (FAPERGS, Proj. 16/2551-0000251-7 and 19/1750-2), Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Proj. 0001). RAR
acknowledges financial support from Conselho Nacional de Desenvolvimento
Científico e Tecnológico (302280/2019-7).

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, the Chilean Participation Group, the French Participation Group,
Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de
Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley
National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-
Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für
Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik
(MPE), National Astronomical Observatories of China, New Mexico State
University, New York University, University of Notre Dame, Observatário
Nacional / MCTI, The Ohio State University, Pennsylvania State University,
Shanghai Astronomical Observatory, United Kingdom Participation Group,
Universidad Nacional Autónoma de México, University of Arizona, University of
Colorado Boulder, University of Oxford, University of Portsmouth, University
of Utah, University of Virginia, University of Washington, University of
Wisconsin, Vanderbilt University, and Yale University.

## Data Availability

All non-MaNGA data used here are publically available, as are all MaNGA data
as of SDSS DR17.

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##  Appendix A Results with R2 calibrator

Thus far, we have focussed on gas metallicity results from the O3N2 calibrator
of Marino et al. (2013). This calibrator has the advantage of simplicity, due
to it being essentially unaffected by dust attenuation. However, different
calibrators can lead to notably different outcomes in terms of gas
metallicities and gas metallicity gradients (e.g. Kewley & Ellison, 2008;
Schaefer et al., 2020), and the O3N2 calibrator implicitely assumes a fixed
N/O-O/H relation which, in practice, will not be true for all spaxels (e.g.
Luo et al., 2021). As such, we briefly present results from the R2 calibrator
of Pilyugin & Grebel (hereafter P16 2016), as was done in Appendix B of B22.

Figure 11: Effective radius plotted against galaxy stellar mass, with data
points coloured by the radial gas metallicity gradients (left) and by the
Spearman correlation coefficients between $\Sigma_{*}$ and metallicity
(middle) and between radius and metallicity (right); the gas metallicity here
is determined from the P16 calibrator. We apply LOESS smoothing in all panels.
The middle panel uses a reversed colour scale; thus, blue regions correspond
to negative gradients, positive $\Sigma_{*}$-metallicity correlations and
negative radius-metallicity correlations. The contours enclose $\sim$50% and
$\sim$80% of galaxies.

Similarly to the M13 calibrator, the P16 R2 calibrator was determined from
observational data; however, it employs the $\mathrm{[OII]}\lambda 3737,3729$
doublet in addition to the lines within the O3N2 indicator, making the R2
calibrator more resistant to biases from N/O variations. The R2 calibrator
employs the full [OIII] and [NII] doublets, so we assume a 1/3 ratio between
the dominant and subdominant lines of these doublets. We employ the same
spaxel/galaxy sample presented in the main paper text.

Figure 12: Difference in magnitudes of the Spearman correlation coefficient
between gas metallicity and stellar density, $|\rho_{\Sigma}|$, and the
coefficient between gas metallicity and galactocentric radius before
($|\rho_{R}|$; left panel) and after ($|\rho_{R,perturbed}|$; right panel)
perturbing spaxel radii. Each datapoint represents one galaxy, and LOESS
smoothing is applied; blue regions indicate where metallicity is more strongly
correlated with radius, while red regions indicate a stronger correlation with
density. The contours enclose $\sim$50% and $\sim$80% of galaxies.

We present in Figure 11 gas metallicity gradients, $\rho_{\Sigma}$ and
$\rho_{R}$ values across the mass-size plane. These were calculated in an
identical manner to what was described in the main paper text, but with P16
metallicities instead of M13 metallicities. We then present in Figure 12 a
comparison between $|\rho_{\Sigma}|$ and $|\rho_{R}|$ and between
$|\rho_{\Sigma}|$ and $|\rho_{R,perturbed}|$, with all values calculated in
the same way as before. Finally, we present in Figure 13 the behaviour of B22
‘SFH models’ as derived from the P16 calibrator.

Figure 13: Effective radius, plotted against galaxy stellar mass, with data
points coloured by the B22 ‘SFH model’ metallicity gradients (left) and by the
Spearman correlation coefficients between $\Sigma_{*}$ and model metallicity
($[\rho_{\Sigma}]_{SFH}$, middle) and between radius and model metallicity
(right). The model metallicities were build using the P16 R2 indicator in this
case. We apply LOESS smoothing in all panels, and the middle panel uses a
reversed colour scale. The contours enclose $\sim$50% and $\sim$80% of
galaxies. We find that the SFH models do not reproduce the trends shown in
Figures 1 and 2 for observed gas metallicities: while individual parameters
behave similarly across the mass-size plane, we find local metallicties to be
correlated more strongly with stellar density than with radius in most cases.

The correlations are found to be a little weaker here than was found from the
M13 calibrator; this is unsurprising, as B22 previously noted a larger scatter
in gas metallicity relations when the P16 R2 calibrator was applied.
Nonetheless, we obtain the same broad results that were presented and
discussed in the main paper text: $\rho_{\Sigma}$ and $\rho_{R}$ trend
similarly to the gradients across the mass-size plane, as is to be expected,
with $|\rho_{R}|$ greater than $|\rho_{\Sigma}|$ for less massive extended
galaxies. We further find that the SFH models remain unable to replicate this
behaviour, predicting stronger metallicity correlations with $\Sigma_{*}$ than
with radius. Thus, we find that the main results from this paper are not
simply due to the use of the M13 calibrator.

##  Appendix B Relation between gas metallicity gradients and correlation
coefficients

We now explore in more depth how the gas metallicity gradients relate to
$\rho_{R}$ and $\rho_{\Sigma}$, focussing specifically on the M13 calibrator.
In Figure 14, we plot both $\rho_{R}$ and $\rho_{\Sigma}$ against the radial
metallicity gradient; as was already apparent, steeper gradients are
associated more positive $\rho_{\Sigma}$ values and more negative $\rho_{R}$
values.

Figure 14: Radial gas metallicity gradients plotted against $\rho_{\Sigma}$
(left) and $\rho_{R}$(right), with gradients obtained using the M13
calibrator. A small number of anomalous gradient values are not shown for
presentational purposes.

Next, we consider how metallicity gradients vary with $\rho_{\Sigma}$ and
$\rho_{R}$ at different locations in the mass-size parameter space. For this,
we employ the six mass-size subsamples described in Section 3.2. We plot in
Figure 15 the gas metallicity gradients against $\rho_{\Sigma}$ for each of
the subsamples, while in Figure 16 we compare the metallicity gradients and
$\rho_{R}$ in the same manner.

Figure 15: Radial gas metallicity gradients plotted against $\rho_{\Sigma}$
for the six mass-size bins first described in Section 3.2. Figure 16: Radial
gas metallicity gradients plotted against $\rho_{R}$ for the six mass-size
bins first described in Section 3.2.

For both the sample as a whole and for the mass-size subsamples, it is
apparent that the gradient correlates tightly with $\rho_{\Sigma}$ and
especially with $\rho_{R}$. This is by construction: $\Sigma_{*}$ declines
with radius, meaning that steeper gradients can be immediately expected to
yield tighter metallicity correlations with both $\Sigma_{*}$ and radius.


$\displaystyle\max_{\mathbf{p}}\left(\min(m_{b},(\alpha_{*}-\gamma)\theta_{b})\right)$
$\displaystyle\geq\gamma,$
$\displaystyle\max_{\mathbf{p}}\left(\min(m_{t},3(\alpha_{*}-\gamma)\theta_{t})\right)$
$\displaystyle\geq\gamma.$

Numerical optimization shows that one can take $\gamma=0.142$.

For such $\gamma$ and for sufficiently small $\eta>0$, assuming further that
$\Delta\leq p^{-\gamma-o_{p}(1)}$, it follows from the LLL that there exists a
choice of projections $\pi_{v}$ for $v\in\mathcal{V}_{S}$ so that for all
$C\in\cal{C}$,

$b(C)\leq(600\Delta/\eta)^{-1},$

and

$t(C)\leq p^{3\gamma}.$ (6.1)

Furthermore, by Theorem 2.2, such projection maps can be constructed in time
$O(n\Delta k\log(1/\delta))$ with probability at least $1-\delta$.

For such a projection scheme, the properties (A1), (A3) and (A4) are easily
verified. We now show that (A2) is also satisfied. Let
$\kappa=12\log(3000(\Delta+100))$. Since $\zeta(C)\leq 2\Delta$, we have

$\displaystyle|\overline{\operatorname{vbl}}(C)|^{2}\kappa^{2}\cdot\zeta(C)\cdot\prod_{v\in\overline{\operatorname{vbl}}(C)}\left((1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]+e^{-\kappa/3}\right)$
$\displaystyle\leq(12\log(3000(\Delta+100)))^{2}\cdot
2\Delta\cdot|\overline{\operatorname{vbl}}(C)|^{2}\prod_{v\in\overline{\operatorname{vbl}}(C)}\left((1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]+e^{-\kappa/3}\right).$

By (A1) and our choice of $\kappa$, we have
$(1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]+e^{-\kappa/3}\leq\frac{3}{4}$
for all $C\in\mathcal{C}$ and $v\in\operatorname{\overline{vbl}}(C)$. We have
two cases.

(1) If there exists some $v\in\operatorname{\overline{vbl}}(C)$ for which
$e^{-\kappa/3}>\frac{1}{|\overline{\operatorname{vbl}}(C)|}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]$,
then

$\displaystyle(12\log(3000(\Delta+100)))^{2}\cdot
2\Delta\cdot|\overline{\operatorname{vbl}}(C)|^{2}\prod_{v\in\overline{\operatorname{vbl}}(C)}\left((1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]+e^{-\kappa/3}\right)$
$\displaystyle\leq(12\log(3000(\Delta+100)))^{2}\cdot 2\Delta\cdot
2|\overline{\operatorname{vbl}}(C)|^{2}\cdot(3/4)^{\overline{\operatorname{vbl}}(C)-1}\cdot
e^{-\kappa/3}$ $\displaystyle\leq(60000\Delta)^{-2},$

where the last inequality follows by our choice of $\kappa$.

(2) On the other hand, if for all $v\in\operatorname{\overline{vbl}}(C)$,
$e^{-\kappa/3}\leq\frac{1}{|\overline{\operatorname{vbl}}(C)|}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]$,
and further since

$\displaystyle 6b\Delta\leq\eta/100,$ (6.2)

we have

$\displaystyle(12\log(3000(\Delta+100)))^{2}\cdot
2\Delta\cdot|\overline{\operatorname{vbl}}(C)|^{2}\prod_{v\in\overline{\operatorname{vbl}}(C)}\left((1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]+e^{-\kappa/3}\right)$
$\displaystyle\leq(12\log(3000(\Delta+100)))^{2}\cdot
6\Delta\cdot|\overline{\operatorname{vbl}}(C)|^{2}\prod_{v\in\overline{\operatorname{vbl}}(C)}\left((1-3b)^{-\Delta}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]\right)$
$\displaystyle\leq(12\log(3000(\Delta+100)))^{2}\cdot\Delta\cdot
C_{\eta}\prod_{v\in\overline{\operatorname{vbl}}(C)}\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]^{1-\eta}.$
(6.3)

Here we have used the inequality $x^{2}(1-\delta)^{x}<C_{\delta}$, which holds
for all $\delta\in(0,1)$, $x\geq 1$, and for a sufficiently large $C_{\delta}$
depending only on $\delta$. Now, since
$\mathbb{P}_{\pi}[\operatorname{value}(v)=\boldsymbol{C}_{\pi}(v)]=\frac{|\pi_{v}^{-1}(\boldsymbol{C}_{\pi}(v))|}{|\Omega_{v}|}$,
it follows from 6.1 that Section 6 is at most $(60000\Delta)^{-2}$, which
verifies (A2). ∎

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