# Resource Adequacy and Capacity Procurement: Metrics and Decision Support
Analysis

Chris J. Dent1, Nestor Sanchez1, Aditi Shevni2,
Jim Q. Smith2, Amy L. Wilson1 and Xuewen Yu3

1School of Mathematics, University of Edinburgh, UK
2Department of Statistics, University of Warwick, UK
3MRC Biostatistics Unit, University of Cambridge, UK
Corresponding author<EMAIL_ADDRESS>
( )

###### Abstract

Resource adequacy studies typically use standard metrics such as Loss of Load
Expectation and Expected Energy Unserved to quantify the risk of supply
shortfalls. This paper critiques present approaches to adequacy assessment and
capacity procurement in terms of their relevance to decision maker interests,
before demonstrating alternatives including risk-averse metrics and
visualisations of wider risk profile. This is illustrated with results for a
Great Britain example, in which the risk profile varies substantially with the
installed capacity of wind generation. This paper goes beyond previous
literature through its critical discussion of how current practices reflect
decision maker interests; and how decision making can be improved using a
broader range of outputs available from standard models.

##  1 Introduction

Power system resource adequacy (RA) is the field of managing the risk that
there will be insufficient supply resource to meet electricity demand. Studies
vary as to the precise class of events in scope, but is generally taken to
encompass the balance of resources without considering fine detail of system
operation (see [1] for a survey of current issues). This remains a topic of
great interest, due to the need to maintain an appropriate level of system
reliability as the profile of resources evolves towards a lower carbon
portfolio – and hence it is a key topic in assessing the performance of energy
technologies within the system, and how technologies relevant to the net zero
transition complement each other (or not) at whole power system level.

The area of RA also provides an excellent example of wider issues in project
and policy appraisal, and in decision analysis, in that relevant decisions
typically involves balancing capital investment costs (which are relatively
concrete) against future reliability of the system (quantification of which is
both more uncertain, and is not a cashflow that can immediately be
incorporated in a monetised const-benefit analysis).

RA studies typically use a standard set of risk metrics to quantify system
reliability: Loss of Load Expectation (LOLE), the expected (in the statistical
sense) duration of shortfall in the future year or season under study;
Expected Energy Unserved (EEU), the expected volume of energy demand not
supplied; or if going beyond these, further summary statistics such as System
Average Interruption Duration/Frequency Index (SAIDI/SAIFI) [2]. Formal
decision analysis for capacity procurement tends involve either a risk level
target set with respect to one of these statistics, or a Cost-Benefit Analysis
(CBA) in terms of a per-MWh Value of Lost Load (VOLL) multiplied by EEU [3, 4]
– historically less formally justified standards in terms of a deterministic
measure of the margin of installed capacity over peak demand were common [1],
though these are now less widespread due to increased use of probabilistic
analysis and the difficulty in including new technologies such as renewables
without this.

There are issues in principle with the use of such summary statistics, which
can have substantial practical consequences.

  * •

Decision makers are likely to be risk averse, and interested in variability of
outcome in individual years, as well as an average over possible outcomes.
Expected value indices such as LOLE and EEU by definition do not reflect this,
and thus decisions based on current decision analysis approaches might not
properly reflect the concerns of decision makers.

  * •

One single-number index cannot capture the whole risk profile of a system, and
if the mix of supply and demand changes there may be changes in risk profile
that these indices do not capture. There is nothing fundamentally wrong with
the use of summary statistics, but they must be customised to and reflect the
needs of decision makers.

  * •

Standard approaches do not specify who the decision maker is. For instance,
stakeholders (including ultimate decision makers in governments) tend to be
very risk averse about electricity security of supply, due to the consequences
for society and the economy if confidence in the electricity system decreases.

  * •

The disruption costs of events are often assessed on a narrow basis of direct
costs to customers, and not considering points such as wider societal and
economic confidence in a robust electricity supply.

  * •

It may not be natural to trade off investment and disconnection costs on the
same (usually monetary) scale, i.e. they may not be _commensurate_ [5] (the
formal decision analysis term for whether quantities can naturally be compared
on the same numerical scale). For instance, many customers are unlikely to be
indifferent between having supply and being paid monetary compensation, though
this is implicitly assumed in many analyses.

Other works have looked for instance at use of multiple metrics [1], inclusion
of risk-averse metrics within optimization problems [6], and construction of
probability distributions of _ex post_ outcome metrics ([7] and references
therein, and the Belgian standard references within the international
comparison at [8]). This paper, however, goes beyond previous literature
through its critical discussion of how current practices reflect decision
maker interests; and how decision making can be improved using a wider range
of outputs available from standard risk model structures. For a general
reference on relevant principles of decision analysis on which this work is
based, including risk aversion, see [9].

This paper will first describe and critique present approaches to adequacy
assessment and capacity procurement in Section 2, before presenting in Section
3 alternatives such as risk-averse metrics and broader visualisations of risk
profile to support decision makers. This is illustrated with results for Great
Britain, considering a range of wind generation capacities. Section 4 provides
an extended discussion of issues of application and relevance to wider
decision appraisal, and finally Section 5 presents summary conclusions.

##  2 Standard picture of decision analysis for capacity procurement

###  2.1 Risk modelling for resource adequacy

This section provides a brief overview of the resource adequacy modelling on
which results presented, based on a GB system supplied by wind and
conventional generation. This is satisfactory for our purpose of demonstrating
use of model outputs in supporting decision making, where the key point is how
a changing penetration of wind energy changes the overall risk profile. Issues
of extension to other resources such as storage and interconnection to other
systems, and uncertainty management arising from limited data on extremes and
complexity of systems spanning multiple countries, will be discussed in
Section 4.

We denote random variables with uppercase and constants with lowercase.
$X_{t},Y_{t}$ and $D_{t}$ denote available conventional and renewable
generation, and demand, respectively at time $t$ in the future period under
study; then the surplus $Z_{t}=X_{t}+Y_{t}-D_{t}$. This section is consistent
with standard references such as [2, 10], but expressed in slightly different
notation.

####  2.1.1 Non-sequential model

Let the period for which an assessment is being made (e.g. a future peak
season) be divided into $n$ hours. LOLE, the expected number of hourly
shortfalls in the period is111For simplicity, an hourly time step is used, as
per available GB data – in N America this ‘hourly’ LOLE is usually called
LOLH. We will also use data from the GB peak winter (Nov-Mar) season – as
daily peak demands in winter are much higher than at other times of year, and
very low renewable output can occur at times of very high demand, the peak
demand season dominates annual shortfall risk in GB. This observation would
not apply if a large renewable capacity shifts the time of year at which the
highest values of (demand - renewables) occur – however, as this paper is
primarily about the choice of model _outputs_ , its conclusions are also
broadly relevant to such systems.:

$\mathrm{LOLE}=\mathbb{E}\left[\sum_{t=1}^{n}\mathbb{I}(Z_{t}<0)\right]=\sum_{t=1}^{n}\mathbb{P}(Z_{t}<0),$
(1)

and the EEU is defined as:

$\mathrm{EEU}=\mathbb{E}\left[\sum_{t=1}^{n}\max\\{0,-Z_{t}\\}\right].$ (2)

This is commonly referred to as a non-sequential model, as unless there are
technologies such as storage which explicitly link the system states at
different times, the terms in the sum may be evaluated separately, and the
times re-ordered without affecting the result.

For statistical modelling, it is often more convenient to work in a _time-
collapsed_ picture with the time-collapsed variable $Z$ representing surplus
at a randomly chosen point in time. For a system that does not have storage or
other technologies which link time periods, the LOLE is then specified as

$\mathrm{LOLE}=n\mathbb{P}(Z<0)=n\mathbb{P}(X<D^{\prime}),$ (3)

and an analogous formula applies for EEU.

The most common means of estimating the distribution of $D_{t}-Y_{t}$ is to
use the empirical historic data directly in predictive risk calculations,
sometimes referred to as hindcast [11, 2] The hindcast estimate of LOLE
conditional on a particular historic weather year $y$ is then

$\mathrm{LOLE}_{y}=\sum_{\tau\in T_{y}}\mathbb{P}(X_{\tau}<d_{\tau}-y_{\tau})$
(4)

where $T_{y}$ is the set of times in historic year $y$, $d_{\tau}-y_{\tau}$ is
written in lower case as it is historic data rather than a random variable,
and _historic_ times are indexed by $\tau$. LOLE conditional on no particular
weather year is then usually estimated as
$\mathrm{LOLE}=(1/n_{Y})\sum_{y}\mathrm{LOLE}_{y}$, where $n_{Y}$ is the
number of years of data (and similar for EEU). As discussed later, there can
be substantial uncertainty in any estimate of unconditional risk level, as the
estimae of the mean will often be dominated by a small proportion of the
historic years.

####  2.1.2 Time sequential model

For model outputs beyond the standard expected value indices, such as the
distribution of energy unserved, or the distribution of Loss of Load Duration
(LOLD, the random variable of which LOLE is the mean) a time sequential model
would be required. LOLD is specified as

$\mathrm{LOLD}=\sum_{t=1}^{n}\mathbb{I}(Z_{t}<0),$ (5)

and the energy unserved (EU) as

$\mathrm{EU}=\sum_{t=1}^{n}\max\\{0,-Z_{t}\\}.$ (6)

A wide range of other possible outputs may also be calculated, the usual
mechanism for doing so being Monte Carlo simulation.

Stochastic process models must then be estimated for $X_{t}$ and
$(D_{t},Y_{t})$. In practice, again a common way to proceed is to use a
hindcast estimate for the process of demand and wind, i.e.

$\mathrm{LOLD}_{y}=\sum_{\tau\in T_{y}}\mathbb{I}(X_{\tau}<d_{\tau}-y_{\tau})$
(7)

and similar expression for EU. The aggregate available conventional capacity
is usually specified as a sum of stochastic process models for each individual
unit, with the unit models commonly (as here) being two state Markov birth-
death processes.

###  2.2 Decision analysis for capacity procurement

It is common to look at capacity procurement for a single future year, which
for simplicity we will do here. This is the approach taken in the GB capacity
market, where a target risk level is set based on a cost-benefit analysis
(CBA) for the future year considered [12], or might represent a long run
equilibrium problem [13].

The standard CBA is then be expressed as an optimization problem [3]:

$\mathrm{min}~{}~{}~{}~{}~{}~{}c(R)+[\mathrm{VOLL}]\times[\mathrm{EEU}](R)$
(8)

over the possible sets $R$ of capacity-providing resources [3]. Procurement
cost $c$ and EEU are both functions of $R$. This is commonly simplified [4] to

$\mathrm{min}~{}~{}~{}~{}~{}~{}[\mathrm{CONE}]\times
r+[\mathrm{VOLL}]\times[\mathrm{EEU}]_{r},$ (9)

on the assumption that the additional procured capacity simply shifts the
probability distribution of surplus/deficit by the mean available capacity $r$
from the addition222This is justified either through convolution of an
independent addition with a distribution of deficit that is approximately
exponential in shape in the relevant region [14], or through the Central Limit
Theorem if a small independent addition is made to a large background of
independent units.. In practice, this assumption implies that the addition
should be small compared to the resource already present (generally the case
in capacity markets, where the volume of new capacity is usually limited); and
does not contain renewable generation or storage for which independence
between existing and additional units does not apply. Here Cost of New Entry
(CONE) and VOLL take fixed values, and $[\mathrm{EEU}]_{r}$ is the expected
energy unserved if the volume of capacity procured is $r$; it is
straightforward to generalise this to non-constant CONE and VOLL.

###  2.3 Consequences of choice of VOLL

At the value of $r$ which minimises (9),

$[\mathrm{LOLE}]_{r}=\frac{[\mathrm{CONE}]}{[\mathrm{VOLL}]}$ (10)

There are various challenges to taking this as an ‘optimal’ solution for the
real world, even if EEU is regarded as a sufficient summary statistic of risk
profile.

  * •

Studies usually implicitly assume that the control room can disconnect the
amount of load required and no more, with perfect foresight. This is not the
case in practice, which implies an increase in energy unserved.

  * •

An average VOLL across all customers is usually used, i.e. the interests of
customers who are relatively indifferent to disconnection are treated
interchangeably with interests of customers who are more averse to
disconnection, even if there is no discrimination as to who is disconnected
involuntarily. If one accepts the idea of monetising unserved energy using
VOLL, should customers’ interests be averaged in this way, or should more
weight placed on the interests of customers who are more inconvenienced by
being disconnected?

  * •

As in the introduction, one needs to consider whether reliability and
procurement costs are commensurate, i.e. whether they can be compared on the
same numerical scale.

This standard optimization picture in (9) typically recommends a level of
reliability similar to the GB standard of 3 hours/year LOLE, whereas a system
that unreliable would probably be deemed politically unacceptable. For
instance, in GB, a system margin warning can be a major news story [15],
whereas this lowest level of system warning actually means that some hours
ahead of real time the operator was not certain of having their usual real
time operating headroom, i.e. nowhere near a shortfall in real time – if
actual real time shortfalls happened in a substantial proportion of colder
winters, the reaction in public debate would be stronger still.

Using a higher (not average) VOLL, based on the second bullet above, would
push the reliability standard more towards a level that would be considered
acceptable in this wider sense but would not consider wider issues of societal
confidence in the electricity supply. Indeed if such factors beyond individual
customer damage are considered important, then for use in comparing different
capacity portfolios VOLL might be chosen such that (9) gives an acceptable
level of system-level reliability, rather than being based on customer
surveys. It is certainly the case that making different judgments associated
with the first two bullets can change the ‘optimal’ level of reliability from
(9) very substantially.

##  3 Beyond the conventional framework

###  3.1 Background and multicriteria formulation

Most formal decision analysis frameworks for capacity procurement assume the
approach described in the previous section, i.e. monetising future reliability
in terms of VOLL multiplied by expected energy unserved. Clearly capital costs
are naturally in terms of money, though there may be uncertainty in the
monetary sum, or one may wish to use a capacity price curve (i.e. as more is
procured, the unit cost of capacity increases) rather than a fixed CONE – we
have already discussed whether this can naturally be compared with reliability
on the same numerical scale. Moreover, expected monetary return is rarely a
utility function that reflects decision makers’ interests, particularly for
mitigation of rare high impact events, and thus introducing a degree of risk
aversion seems natural.

The simplest evolution of (9) would be to extend this framework to a multi-
criteria decision question, seeking a Pareto frontier on which EEU cannot
improved without disbenefit in terms of cost, and vice versa. However, for
this case mapping the Pareto frontier is in fact equivalent to a sensitivity
analysis on VOLL, so results for it are not presented here.

###  3.2 Data for examples

The following sections include results from an exemplar based on the Great
Britain (GB) system. The standard approaches described previously are used for
risk calculations. A comparison between scenarios of different installed wind
capacities is made by using the same portfolio of available conventional
capacity for each scenario; and for each scenario of installed wind capacity
shifting the supply-demand balance to give a common EEU of 3 GWh/year. For
instance, one might hypothesise that at higher penetrations of renewable
generation, for a given value of standard indices such as LOLE or EEU, greater
variability of supply leads to greater variability of outcome. This comparison
is controlled in the sense that it looks at overall risk profile for a range
of scenarios which have the same value of the headline EEU.

The scenario of installed conventional capacity is based on one originally
provided by National Grid ESO, with a small random element added to each
capacity due to the commercial sensitivity of the raw data. For sequential
models, availability probabilities from NGESO are supplemented with mean
repair time data from the IEEE Reliability Test System [16].

GB demand data for the 12 peak (winter) seasons 2005-17 are used, with an
estimate of historic available embedded renewable capacity added back on.
Demand data are rescaled to a common system scenario according to historic
values of the Average Cold Spell (ACS) Peak statistic, with the given scenario
defined by an ACS value.

The wind generation data used are from [17], and combine historic reanalysis
wind speed data with a future scenario of what is connected to the system.
Hourly capacity factors (CFs) for onshore and offshore wind in the ‘near term’
wind fleet from [17] are used, and for a given scenario of onshore and
offshore wind capacity connected to the system these hourly CFs are multiplied
by the respective GW installed capacities and added to give a total hourly
available capacity. The division between onshore and offshore for a given
level of total installed wind capacity is based on projections in [18].

Thus our exemplar system is generally representative of the GB system and will
suffice for our illustrative purpose, however for a fully applied GB study one
would need to use data that are specialised to the particular decision
question considered.

###  3.3 Risk averse metrics

####  3.3.1 Background

It is possible to define alternative single number metrics which give a
measure of risk aversion, for example the well known (Conditional) Value at
Risk (VaR and CVaR) [19], with respect to model outputs such as the
constructed distribution of energy unserved. This is defined as for a random
ariable $U$ as

$[\mathrm{CVaR}]_{\alpha}=\mathbb{E}[U|U>u^{\prime}]$ (11)

where $P(U\geq u^{\prime})=1-\alpha$ for a risk threshold parameter $alpha$.
Thus the mean of $U$ is the special case $[\mathrm{CVaR}]_{0}$, and if $U$ is
the energy unserved then CVaR is a generalisation of the standard EEU index –
while one can use other mappings of EU and LOLD to give risk averse utility
functions, this would not have the same attractive property of generalising
the expected value indices.

CVaR has the further beneficial property of convexity when embedded in a wide
range of optimization problems [19]. We do note however that where it is not
necessary to embed within an optimization model, there can be difficulties in
communicating CVaR results outside the specialist community, particularly as
for low degrees of risk aversion the risk threshold $u^{\prime}$ will sit in
the other tail of the distribution of $U$ from the one of interest. Another
diadvantage is that CVaR values with different $\alpha$ are not directly
comparable, despite having the same dimensions.

Calculating risk-averse indices will in general require time-sequential
modelling. The exception would be to work with VaR and CVaR with respect to
the snapshot LOLP or Expected Power Unserved (i.e. the probability of a
shortfall, or expected shortfall, at a randomly chosen point in time [14]).
This would, however, need to be interpreted carefully. For instance, VAR with
respect to snapshot LOLP may have a useful interpretation in terms of the
expected number of hours with surplus below a given level, but other
combinations of VAR/CVaR with LOLP or EPU might not be so interpretable.

####  3.3.2 Example

Figure 1: CVaR as a function of the risk aversion parameter $\alpha$ for a
range of installed wind capacities.

Fig. 1 shows CVaR with respect to EU as a function of $\alpha$, for a range of
installed wind capacities. As in all examples, for a controlled experiment the
scenarios of different installed wind capacity have the same EEU. As
anticipated in the section on data, as the wind capacity increases, the CVaR
for a given $\alpha$ also increases. This is consistent with a hypothesis that
greater variability of supply would lead to greater variability of outcome –
however the effect is not very large, and the next section will explore how
CVaR as a summary statistic does not reveal the most striking change in the
risk profile at higher wind capacities.

###  3.4 Visualisations for decision makers

Instead of attempting to define metrics in this way, one could instead provide
visualisations to decision makers of the consequences of particular decisions
in different scenarios of planning background, and let them decide on that
basis how much capacity to procure. Even if there is still a preference of
working with summary statistics for formal decision analysis, there is value
in supplementing this with a wider range of visualisations to understand more
broadly the system’s risk profile, or the consequences of results from formal
optimization models.

This is an attractive idea in principle, though to go with such visualisations
one needs the necessary skills in how to use them well – both on the part of
the analysts in terms of how to create visualisations, and also in terms of
how some technical statistical understanding on the part of decision makers
may be necessary. On the other hand, visualisations such as these may be more
tangible and easier to communicate than summary statistics such as CVaR.
Appropriate use of scenarios and supporting narratives can help here,
potentially using an interactive dashboard to provide model outputs and
narratives of consequences for decisions under consideration.

(a) Loss of load duration

(b) Energy unserved

(c) Number of days of shortfall

(d) EU within days of shortfall

Figure 2: Scenario with fixed EEU = 3 GWh/year

Fig. 2 provides an example of this for a scenario with fixed EEU of 3
GWh/year, and for a range of installed wind capacities. There is substantial
variation with installed wind capacity of the distribution of EU in our
experiment, but rather less variation in the distribution of LOLD. However
while the former explains the variation of CVaR with respect to EU, the CVaR
being a summary statistic of the distribution of EU, the difference between
scenarios of different wind capacities is less striking in the CVaR results
than in the underlying probability distribution.

One can look further at the overall risk profile through the distributions of
number of days with a shortfall, and of the probability distribution of EU
within a day conditional on that day having a shortfall. It is clear from the
lower two panels of Fig. 2 that at higher wind penetration the same EEU is
made up (in a probabilistic sense) of fewer days of greater shortfall. The
variation with wind capacity of these two distributions is rather greater than
that of the distributions of LOLD and EU – this difference may well be
material for decision making, demonstrating how one might go in to
considerable detail of risk model outputs to understand fully how the risk
profile is changing.

There are important caveats on these results as quantitative calculations for
the real GB system. We have carried out a calculation using a standard
approach to illustrate important issues for decision support analysis in the
model-world, but for a fully applied study one would need to specialise the
statistical estimation to the relevant scenario, and to consider whether the
modelling assumptions made give meaningful results on low probability events
with respect to the real world.

##  4 Discussion

###  4.1 Statistical issues

In addition to this paper’s technical work exploring the range of model
outputs available from standard calculations and how these might be used,
there are also important statistical and uncertainty management issues which
must be considered. These include estimation of model inputs, including having
a limited number of examples in the historic record of extreme conditions,
estimates of the availability properties of conventional generation at these
times, and modelling of very complex continental-scale systems.

It is common in reliability analysis to have limited direct historic data on
failure events, particularly in an area such as resource adequacy where there
is a specific external driver of stress events (i.e. the weather). The
consequent uncertainty in estimation of model outputs tends to increase in
systems with high renewables penetrations, where the highest values of (demand
minus renewables) are from times combining sufficiently high demand with low
renewable resource – this tends to concentrate risk in a smaller number of
events as compared to a system where risk is driven by the highest values of
demand.

Strikingly, these conditions that drive risk at hig renewable penetrations
would otherwise be regarded as extreme in any one location, where they would
appear to be a standard and benign winter day with fairly low temperature and
little wind. This dominance of risk profile by a limited number of years will
manifest itself in different ways depending on the way the demand and
renewable resource interact, and how much storage is connected to the system;
for an example with a high storage penetration see [20].

Figure 3: Proportion of estimated EEU arising from 2005 data.

This effect is illustrated for the example of this paper in Fig. 3. At low
penetrations of wind generation, 2005 does not make a substantial contribution
to the calculated EEU, as peak demand was quite modest that winter. However,
due to the very calm conditions on certain days of fairly high demand, at high
wind penetrations this becomes the most significant winter by far in driving
the outcome of the risk calculation.

Figure 4: Probability distribution of Energy Unserved (same experiment as Fig.
2(b) except that data from 2005-6 are omitted).

As a further illustration based on the experiment in this paper, Fig. 4
presents the same analysis as Fig. 2(b), except with the data from winter
2005-6 omitted333A comprehensive description of the experiment in this Figure
is as follows. For each wind capacity, the data from all years are normalised
to give an EEU of 3000 MWh/y in a calculation using data from 2006/7 onward,
and then those same years of data from 2006/7 onward are used in the
predictive risk calculation which outputs the distribution of EU.. At high
wind penetrations, the estimated probability of very severe outcomes is then
much lower than if data from the outlier winter 05-06 are included.

A particular manifestation of this will be in estimating tail event metrics
such as VaR or CVaR of energy unserved with a high threshold - for instance in
assessing what a 1 in $n$ year risk level might be. Even if one assumes that
past weather is statistically representative of the relevant future year, by
definition one will pick up a relevant weather year on average once every $n$
years. Historic data will thus be very sparse, and estimates of such metrics
(not conditional on particular weather) will be speculative.

Another key issue of uncertainty management is interconnection across very
wide (continental scale) areas. As GB is an island system, this manifests as
undersea connections to Ireland, Norway and the main continental European
system. The level of interconnection between GB and other systems is forecast
to rise rapidly – for instance one scenario study by the system operator has a
minimum of 20 GW of connections to other systems on GB peak demand which is
currently a little over 50 GW, making interconnector support a key
consideration in evaluating adequacy risk [21]. However there will be
considerable uncertainty in statistical characterisation of available support
from other systems, due to the volume of model assumptions and data involved
in continental scale assessments, and when carrying out a study knowledge of
other systems almost certainly being poorer than knowledge of one’s own. The
need to bring interconnection into assessments is recognised in other systems,
as discussed in [1].

###  4.2 Decision support for capacity procurement

There is broad recognition that planning for system adequacy needs to
recognise the changing needs of present and future systems with high renewable
penetrations [1]. In particular, it is necessary to recognise that if the
profile of supply and demand changes in a system, then it may be difficult to
track this using a small set of single-number metrics – this paper has
demonstrated by numerical experiment how this can be the case in a system with
increasing penetration of renewable generation. Decision support approaches
must both be transparent to the decision maker and not oversimplify the
situation.

Historically, resource adequacy standards have tended to be set in terms of a
single metric, usually a variant of LOLE – either the expected number of days
on which there is a shortfall (the classic LOLE standard in N America being
0.1 days/year LOLE [22]), or the expected number of hours of shortfall (e.g.
the GB standard of 3 hours/year LOLE; this metric is usually referred to in N
America as Loss of Load Hours, LOLH, though in Europe is more commonly
referred to as LOLE).

There is thus a tension between the need to reflect the increasing complexity
of patterns of supply and demand, and the natural desire to have simple,
transparent criteria for use in supporting procurement decisions. The
structure of working in this more complex environment would be simpler if
there is a single ‘controlling mind’ (i.e. a point of decision making in a
single organisation, maybe with the decision vested in a single individual)
which is able to take judgments as to how to balance different aspects of risk
profile in a decision on capacity procurement. In that case, standard
approaches to uncertainty management and multi-criteria decision analysis
could then be used to handle the issues described.

However when there is an industry and policy need for a clearly defined
standard, one must seek appropriate compromises which reflect the changing
nature of power systems and are sufficiently grounded in the relevant decision
science; there is unlikely to be one definitive best approach here applicable
in all situations. A further consideration is what actually _can_ be estimated
confidently – in particular, in many systems it may be possible to make
confident estimates of risk profile conditional on a particular weather year
and on a statistical characterisation of available support from
interconnectors, these being the principal sources of uncertainty in risk
model outputs.

One basis for a solution could therefore be to set a public facing risk
target, for use in capacity procurement, conditional on a single given weather
and interconnector scenario, or an assumption that the available time series
data characterise fully the range of future possibilities that might be faced.
This would allow performance of alternative resource portfolios to be
compared, but it might be necessary to check against other possible
interconnector or weather scenarios that the outcome is likely to be robust.
It may further be necessary to adjust either the risk target or scenario
choice from time to time, in order to maintain an appropriate level of risk as
the portfolio of technologies develops. A related challenge is that supply
capacity is not a simple additive commodity of the kind that is traded in
standard auctions; a fully discussion of the consequence of this may be found
in [3].

It may also be necessary to make some special consideration of very extreme
situations. If it is deemed infeasible to assign probabilities to events which
are too sparse in the historic record, then these might be treated through
scenario analysis – however if this means that there is a probability model
used for some classes of event, but the events which really matter are
excluded from this model, there might be logical gap in the framework.

Another situation might be where weather beyond a certain degree of severity
introduces distinct failure modes which are not seen at all in less extreme,
but still severe, conditions, for instance those of Texas in 2021 [23]. A
further example is the severe cold and snow in GB in 1962-62 [24], which would
bring very substantial changes in demand patterns over an extended period of
months if it occurred again. Apart from any difficulties with assigning
probabilities or mean return times, such conditions might require bespoke
studies of effect of temperature and snowfall on demand and the power system,
going well beyond the modelling that suffices for more normal circumstances.

Finally we note that the transparency of metrics and visualisations for
decision makers is a significant issue. Standard indices such as LOLE and EEU
are sometimes regarded outside the specialist modelling community as as being
hard to understand, and inside that specialist community there is recognition
of limits on the information that they contain. It is likely that non-
specialists will find some additional metrics such as CVaR more difficult
still to understand, so it is vital to take appropriate care in communicating
the information that they contain. It may be, however, that visualisations of
probability distributions are easier to communicate in that they contain all
the information on estimated statistics of a particular quantity, though
initially some might find the overt presentation of probability distributions
intimidating.

###  4.3 Relation to wider issues of project and policy appraisal

The questions discussed in this paper should also be seen in the wider context
of decision analysis and project appraisal. Indeed this area of resource
adequacy and capacity procurement often provides a very good exemplar of wider
issues, in that with a fairly simple system model (at least for a single power
system area), it presents a range of subtle statistical issues.

The master guide for such analysis in the public and regulated sector in GB is
the HM Treasury Green Book [25]; while this is explicitly about appraisal and
evaluation of policies and projects in central government, it is very
influential on decision analysis in wider contexts. One key principle is that
categories of assessment in an appraisal should be monetised if possible; if
monetisation is not possible they should be quantified on a different scale,
and if quantification is not possible they should be assessed qualitatively.

There is no dispute that RA risk is subject to quantification, albeit with
significant caveats on whether confident estimates can be made without
conditioning on assumptions about weather and interconnector support. However,
as mentioned earlier in this article, monetising RA risk is much more doubtful
– and the standard monetisation in terms of expected energy unserved
multiplied by a (fixed, survey based, averaged over customers) per MWh value
of lost load, tends to recommend unacceptably low level of reliability. In
addition to this top-down argument, there are bottom-up arguments against
averaging interests of all customers including that disconnections do not
discriminate between customers with different interests.

One could go beyond the standard monetisation by making the VOLL dependent on
the depth and length of shortfall, and more generally by eliciting decision
maker judgments as to how the economic value should be considered; or by
considering variability of economic damage about the mean. However, this still
runs into problems with scope: does one only include direct economic damage?
or does one wish to include in decision analysis wider issues such as the need
for wider economic and societal confidence in a reliable electricity supply?
The latter is likely to be of interest to high level decision makers.

This in turn is a wider example of the issue in appraisals of managing
imprecisely defined concepts such as social value, environmental capital and
cultural value; energy security may be regarded as an example of the former.
There will usually then be uncertainty arising from questions of scope, and of
the way that any monetisation is conceptualised. This means that uncertainty
in monetisations goes beyond considerations of input, parameter and modelling
uncertainty to uncertainty in the conceptualisation (essentially that equally
reasonable and expert people might come up with different broad approaches).
The problems with bringing such disparate criteria together into a single
number score has been noted previously in an energy context by Hammond and co-
authors [26, 27]. A very prominent example in the wider GB infrastructure
sector is the business case underpinning the present HS2 high speed rail
project, in which many factors are combined into a single monetary CBA [28].

Going back to the specific case of resource adequacy, consequence is clearly
quantifiable, and relative economic consequence between different scenarios is
likely also be quantifiable – but on the latter, questions of scope might mean
that this is naturally a multicriteria comparison. The problem comes with the
final step of bringing everything together into a single line item of money,
when procurement costs and reliability are not fully commensurate. In this and
a very wide range of case studies in other domains, we would recommend the
following:

  * •

If monetisation is used, it should come with a sufficiently broad
quantification of uncertainty, including consequences of social value etc.
being imprecisely defined quantities;

  * •

In parallel with monetisation, a multicriteria analysis should be performed,
recognising where quantities are not fully commensurate, i.e. they cannot
naturally be brought together in the same line item of money;

  * •

Visualisations such as a red-amber-green representation of the different
options against the range of criteria may be very helpful in presentation to
decision makers.

##  5 Conclusions

This paper has explored use of a range of risk model outputs as a basis for
resource adequacy assessment and capacity procurement, including extensions to
standard decision analysis pictures such as risk averse metrics and wider
visualisations of risk profile. For the GB example considered, as the capacity
of renewables connected to the system increases, not all changes to the risk
profile are captured by expected value metrics such as LOLE and EEU. If a
single summary statistic is required, then CVaR with respect to the
distribution of energy unserved or loss of load duration is attractive due to
the way CVaR generalises these standard expected value metrics – however this
may not reveal other aspects of risk profile such as how the same annual
aggregate model output can be made up of a larger/smaller number of less/more
severe days.

Overall, these results provide a strong argument for finding ways to balance
objectives in a transparent way that recognises the interests of decision
makers, and we provide examples of how this might be done, along with an
extended discussion of their use in practical application – a particular
challenge is the specification of transparent standards when it is not natural
to construct a monovariate utility function. For the necessary combination of
broad uncertainty management with decision analysis, a possible framework is a
Bayes network-based decision support system as described in [29]; any
alternative would require similar functionality for combining a wide range of
uncertainties.

## Acknowledgments

The authors acknowledge discussions with A. Dobbie, H. Wynn, S. Zachary, and
members of the IEEE RAWG and the RA activities of ESIG, EPRI and G-PST. They
also acknowledge meetings in which Colin Gibson and Andrew Wright provided
advice about decision maker interests based on their industry experience.

Author Chris Dent further expresses gratitude to the late Colin Gibson for
exchanges throughout CD’s career in energy systems, from which he learned much
about power system planning and operation.

The authors acknowledge the following funding for this work: CJD, AS, JQS, ALW
and XY from the Alan Turing Institute ‘Towards Turing 2.0’ programme under
EPSRC Grant EP/W037211/1; CJD, JQS and ALW from the Turing Fellow project
‘Managing Uncertainty in Government Modelling’; NS a PhD scholarship from the
Mexican Conacyt funding council. CJD, JQS and ALW also acknowledge Turing
Fellowships from the Turing Institute; and CJD and JQS would like to thank the
Isaac Newton Institute for Mathematical Sciences for support and hospitality
during the programme Mathematical and Statistical Foundations of Data Driven
Engineering when work on this paper was undertaken (supported by EPSRC Grant
Number EP/R014604/1), associated with which CJD was partially supported by a
grant from the Simons Foundation. CJD further acknowledges a grant from the
International Centre for Mathematical Sciences under its Knowledge Exchange
Catalyst scheme to work with the Global Power System Transformation
consortium.

## References

  * [1] Stenclik D, Bloom A, Cole W et al.  Quantifying Risk in an Uncertain Future: The Evolution of Resource Adequacy.  _IEEE Power and Energy Magazine_ 2021; 19(6): 29–36.  DOI:10.1109/MPE.2021.3104076.
  * [2] Keane A, Milligan M, Dent C et al.  Capacity value of wind power.  _IEEE Trans Power Syst_ 2011; 26: 564–572.  DOI:10.1109/TPWRS.2010.2062543.
  * [3] Zachary S, Wilson A and Dent CJ.  The Integration of Variable Generation and Storage into Electricity Capacity Markets.  _The Energy Journal_ 2022; 43(4).
  * [4] ACER.  Decision 23-2020: on the methodology for calculating VOLL, CONE, and the reliability standard.  [Online] Available: https://extranet.acer.europa.eu/Official_documents/
Acts_of_the_Agency/Pages/Individual-decision.aspx, 2020.  Accessed 4 Feb 2023.

  * [5] P Self.  Nonsense on Stilts: Cost-Benefit Analysis and the Roskill Commission.  _The Political Quarterly_ 1970; 41(3): 249–260.
  * [6] da Costa LC, Thomé FS, Garcia JD et al.  Reliability-Constrained Power System Expansion Planning: A Stochastic Risk-Averse Optimization Approach.  _IEEE Trans Power Syst_ 2021; 36(1): 97–106.  DOI:10.1109/TPWRS.2020.3007974.
  * [7] Sheehy S, Edwards G, Dent CJ et al.  Impact of High Wind Penetration on Variability of Unserved Energy in Power System Adequacy.  In _International Conference on Probabilistic Methods Applied to Power Systems (PMAPS) 2016_.  DOI:10.1109/PMAPS.2016.7764199.
  * [8] D Burke.  GB Security of Supply – International Study on Standards and Implementation.  IEEE Resource Adequacy Working Group, available at: https://cmte.ieee.org/pes-rawg/meetings/, 2016.  Accessed 17 Aug 2023.
  * [9] J Q Smith.  _Bayesian Decision Analysis_.  CUP, 2010.
  * [10] Singh C, Jirutitijaroen P and Mitra J.  _Electric Power Grid Reliability Evaluation Models and Methods_.  Wiley/IEEE, 2019.
  * [11] Wilson AL and Zachary S.  Using extreme value theory for the estimation of risk metrics for capacity adequacy assessment, 2019.  Pre-print, available at https://arxiv.org/abs/1907.13050.
  * [12] UK Government.  Capacity Market.  [Online] Available:
https://www.gov.uk/government/collections/electricity-market-reform-capacity-
market.  Accessed 4 Feb 2023.

  * [13] Bothwell C and Hobbs BF.  Crediting Wind and Solar Renewables in Electricity Capacity Markets: The Effects of Alternative Definitions upon Market Efficiency.  _The Energy Journal_ 2017; 38(SI1): 173–188.
  * [14] S Zachary and C J Dent.  Probability theory of capacity value of additional generation.  _Proc IMechE Pt O: J Risk and Reliability_ 2012; 226(1): 33–43.
  * [15] R Davies.  National Grid makes urgent call for companies to reduce electricity usage.  Guardian Newspaper, available at https://www.theguardian.com/business/2015/nov/04/national-grid-issues-urgent-call-for-extra-power, 2015.  Accessed 21 Aug 2023.
  * [16] Grigg C, Wong P, Albrecht P et al.  The IEEE Reliability Test System-1996.  _IEEE Trans Power Syst_ 1999; 14(3): 1010–1020.
  * [17] Pfenninger S and Staffell I.  Renewables Ninja.
https://www.renewables.ninja/, 2015.  Accessed 6 Nov 2022.

  * [18] N Sanchez.  _Power system adequacy: on two-area models and the capacity procurement decision process_.  PhD Thesis, University of Edinburgh, 2023.  Available at https://era.ed.ac.uk/handle/1842/39744.
  * [19] RT Rockafellar and SP Uryasev.  Optimization of conditional value-at-risk.  _Journal of Risk_ 2000; 2: 21–42.
  * [20] Zachary S.  Scheduling and dimensioning of heterogeneous energy stores, with applications to future GB storage needs.  Preprint available at: https://arxiv.org/abs/2112.00102, 2023.  Accessed 17 Aug 2023.
  * [21] National Grid ESO.  Network Options Assessment for Interconnectors.  Available at:
https://www.nationalgrideso.com/research-and-publications/network-options-
assessment-noa/noa-interconnectors, 2022.  Accessed 17 Aug 2023.

  * [22] Billinton R and Chu K.  Early History on the Application of Probability Methods in the Evaluation of Generating Capacity Requirements.  IEEE LOLEWG presentation, available at: https://cmte.ieee.org/pes-rawg/meetings/, 2014.  Accessed 18 Aug 2023.
  * [23] University of Texas at Austin.  The Timeline and Events of the February 2021 Texas Electric Grid Blackouts.  Available at: https://energy.utexas.edu/research/ercot-blackout-2021, 2021.  Accessed 18 Aug 2023.
  * [24] Royal Meteorological Society.  The Big Freeze of 1963: One of the coldest winters on record.  Available at: https://www.rmets.org/metmatters/big-freeze-1963-one-coldest-winters-record, 2013.  Accessed 18 Aug 2023.
  * [25] HM Treasury.  The Green Book: appraisal and evaluation in central government.  Available at: https://www.gov.uk/government/publications/the-green-book-appraisal-and-evaluation-in-central-governent/the-green-book-2020, 2022.  Accessed 17 Aug 2023.
  * [26] Allen SR, Hammond GP, Harajli HA et al.  Integrated appraisal of micro-generators: methods and applications.  _Proceedings of the Institution of Civil Engineers - Energy_ 2008; 161(2): 73–86.
  * [27] Hammond GP, Jones CI and O’Grady A.  Environmental Life Cycle Assessment (LCA) of Energy Systems.  In Chou SK, Wei YM and Yan J (eds.) _Handbook of Clean Energy Systems, vol. 6 (Sustainability of Energy Systems)_. Wiley, 2015.
  * [28] HS2 Ltd.  Economic Case for HS2: Updated appraisal of transport user benefits and wider economic benefits.  Available at: https://assets.publishing.service.gov.uk/government/uploads/
system/uploads/attachment_data/file/3650/hs2-economic-case-appraisal-
update.pdf, 2012.  Accessed 21 Aug 2023.

  * [29] Volodina V, Sonenberg N, Smith JQ et al.  Propagating uncertainty in a network of energy models.  In _International Conference on Probabilistic Methods Applied to Power Systems (PMAPS) 2022_.


# Constraints on Self-Interacting dark matter from relaxed galaxy groups

Gopika K<EMAIL_ADDRESS>Shantanu Desai  E-mail:
<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology,
Hyderabad, Telangana-502284, India

###### Abstract

Self-interacting dark matter (SIDM) has been proposed as an alternative to the
standard collisionless cold dark matter to explain the diversity of galactic
rotation curves and core-cusp problems seen at small scales. Here, we estimate
the constraints on SIDM for a sample of 11 relaxed galaxy groups with X-ray
observations from Chandra and XMM-Newton. We fit the dark matter density
distribution to the Einasto profile and use the estimated Einasto $\alpha$
parameter to constrain the SIDM cross-section, based on the empirical relation
between the two, which was obtained in Eckert et al. (2022). We obtain a non-
zero central estimate for the cross-section per unit mass ($\sigma/m$) for
seven groups, with the most precise estimate obtained for NGC 5044, given by
$\sigma/m=0.165\pm 0.025~{}\rm{cm^{2}/g}$, for dark matter velocity dispersion
of about 300 km/sec. For the remaining four groups, we obtain 95% c.l. upper
limits on $\sigma/m<0.16-6.61~{}\rm{cm^{2}/g}$ with dark matter velocity
dispersions between 200-500 km/sec, with the most stringent limit for our
sample obtained for the group MKW 4, given by $\sigma/m<0.16~{}\rm{cm^{2}/g}$
for dark matter velocity dispersion of about 350 km/sec.

##  I Introduction

The dark matter problem is one of the most vexing problems in modern physics
and astrophysics. Although dark matter constitutes about 25% of total energy
density of the universe and is a basic tenet of the $\Lambda$CDM model Aghanim
et al. (2020), its identity is still unknown, despite close to 100 years of
evidence Bertone and Hooper (2018). There is also no laboratory evidence for
some of the most well motivated dark matter candidates such as Weakly
Interacting Massive Particles (WIMPs) or axions or through indirect searches,
which have only reported null results Merritt (2017); Gaskins (2016); Desai et
al. (2004); Boveia and Doglioni (2018); Schumann (2019); Abdalla et al.
(2021); Acciari et al. (2022); Chan and Lee (2022). Furthermore, the currently
established $\Lambda$CDM model has also faced some tensions at scales smaller
than 1 Mpc, such as the core/cusp problem Flores and Primack (1994); Moore
(1994); Del Popolo and Le Delliou (2022), missing satellite problem Bullock
and Boylan-Kolchin (2017); Weinberg et al. (2015), too big to fail problem
Boylan-Kolchin et al. (2011), and satellites plane problem Banik and Zhao
(2022). Data on galactic scales for spiral galaxies have also revealed some
intriguing deterministic scaling relations or correlations such as the Radial
Acceleration Relation (RAR) McGaugh et al. (2016); Pradyumna et al. (2021),
mass acceleration discrepancy relation Famaey and McGaugh (2012), the
constancy of dark matter halo surface density Donato et al. (2009), linearity
of dark matter to baryonic matter ratio Lovas (2022), which remain an enigma,
although recent works have shown that some of these regularities could be
explained within $\Lambda$CDM Paranjape and Sheth (2021); Gopika et al.
(2023); Sales and Navarro (2023), while some other observations such as the
RAR Chan and Del Popolo (2020); Tian et al. (2020); Pradyumna et al. (2021);
Pradyumna and Desai (2021); Chan et al. (2022), constancy of dark matter
surface density Gopika and Desai (2020, 2021) and linearity of dark to
baryonic matter ratio Upadhyaya and Desai (2023) are not universal. Therefore
a large number of alternatives to the standard $\Lambda$CDM model have emerged
such as Self-Interacting Dark Matter (SIDM, hereafter) Adhikari et al. (2022),
Superfluid dark matter Berezhiani et al. (2018), Warm Dark Matter Colín et al.
(2000), Wave (or Fuzzy) Dark Matter Ferreira (2021), Flavor Mixed Dark Matter
Medvedev (2014), modified gravity (which obviates the need for dark matter)
Famaey and McGaugh (2012); Banik and Zhao (2022), etc.

The original motivation of SIDM about twenty years ago was to resolve the
core/cusp and missing satellite problems Spergel and Steinhardt (2000).
Although the missing satellite problem is no longer a major point of
contention given the discovery of many satellite galaxies of the Milky way
from wide-field sky surveys Nadler et al. (2021); Adhikari et al. (2022), the
core/cusp problem is still an issue Oh et al. (2011). Furthermore, another
acute problem is the diversity in the dark matter density profiles inferred
from rotation curves, which cannot be resolved using feedback models
Kaplinghat et al. (2020). SIDM can resolve these problems and also some of the
aforementioned anomalies at the galactic scale such as RAR and constant dark
matter surface density Tulin and Yu (2018); Kamada et al. (2017); Adhikari et
al. (2022); Ren et al. (2019). SIDM has been applied to a whole suite of
astrophysical observations from dwarf galaxies to galaxy clusters (See
Adhikari et al. (2022); Tulin and Yu (2018) for recent reviews). Current
observations are consistent with a velocity-dependent SIDM cross-sections with
values of $\sigma/m\sim 2~{}\rm{cm^{2}/g}$ on single galaxy scales to
$\sigma/m\sim 0.1~{}\rm{cm^{2}/g}$ on galaxy cluster scales Kaplinghat et al.
(2016).

A large number of works have obtained results on SIDM cross-sections using
galaxy clusters Meneghetti et al. (2001); Gnedin and Ostriker (2001); Mahdavi
et al. (2007); Randall et al. (2008); Bradač et al. (2008); Peter et al.
(2013); Kahlhoefer et al. (2015); Gastaldello et al. (2014); Massey et al.
(2015); Harvey et al. (2015); Robertson et al. (2017); Harvey et al. (2019);
Banerjee et al. (2020); Sagunski et al. (2021); Eckert et al. (2022) as well
as a combination of isolated galaxies and brightest cluster galaxies (BCGs)
within clusters Yang et al. (2023). These include looking at the shape
distribution of galaxy cluster arcs Meneghetti et al. (2001), dark matter-
galaxy offsets Randall et al. (2008); Robertson et al. (2017), spherical Jeans
modeling Sagunski et al. (2021), subhalo evaporation of elliptical galaxies in
clusters Gnedin and Ostriker (2001), fitting the dark profiles to cored
isothermal Andrade et al. (2019) or Einasto profiles Eckert et al. (2022),
offset between dark matter and stars Massey et al. (2015); Harvey et al.
(2015); Kahlhoefer et al. (2015), baryon-dark matter separation Gastaldello et
al. (2014), offset between X-ray gas and galaxies Bradač et al. (2008), offset
between X-ray center and optical center Cross et al. (2023), ellipticities of
clusters Peter et al. (2013), wobbling of BCG around the center of dark matter
Harvey et al. (2019), effect on the profiles at the outskirts near the
splashback radius Banerjee et al. (2020). The most stringent constraints come
from the core densities obtained using cluster strong lensing with limits
ranging from $\sigma/m<~{}\rm{0.13~{}cm^{2}/g}$ Andrade et al. (2022) to
$\sigma/m<~{}\rm{0.35~{}cm^{2}/g}$ Sagunski et al. (2021). We also note that
some of the earliest stringent limits obtained on the SIDM cross-section, for
example $\sigma/m<0.02~{}\rm{cm^{2}/g}$ from MS 2137-23 Miralda-Escudé (2002)
had to be revised with the advent of SIDM simulations, and subsequently became
much more relaxed with $\sigma/m<1~{}\rm{cm^{2}/g}$ Peter et al. (2013).
Evidence for non-zero cross-sections have been reported at cluster scales with
$\sigma/m\approx 0.19\pm 0.09~{}\rm{cm^{2}/g}$ Sagunski et al. (2021). Most
recently, an analysis of the offsets between the X-ray and central galaxy
position for 23 clusters from DES and SDSS is tentatively consistent with
$\sigma/m\sim 1~{}\rm{cm^{2}/g}$ Cross et al. (2023). Similarly, evidence for
a non-zero cross section from galaxy groups has also been found with
$\sigma/m=0.5\pm 0.2~{}\rm{cm^{2}/g}$ Sagunski et al. (2021).

In this work we obtain constraints on the SIDM cross-section using a sample of
relaxed galaxy groups and low mass galaxy clusters imaged with Chandra and
XMM-Newton Gastaldello et al. (2007). Galaxy groups are dark matter dominated
systems with masses in the range of $10^{13}-10^{14}M_{\odot}$ and containing
less than 50 galaxies Sun (2012); Lovisari et al. (2021); Oppenheimer et al.
(2021). The exact demarcation between a galaxy cluster and galaxy group is not
well defined Lovisari et al. (2021); Oppenheimer et al. (2021). On group
scales the expected values of $\sigma/m$ are expected to be
$0.1-1~{}\rm{cm^{2}/g}$ Kaplinghat et al. (2016); Sagunski et al. (2021). We
have previously used these same systems to test the invariance of dark matter
halo surface density and to test the RAR Gopika and Desai (2021). To obtain
constraints on SIDM cross-sections, we follow the same methodology as Eckert
et al. (2022) (E22 hereafter).

The outline of this manuscript is as follows. We recap the E22 analysis in
Sect. II. The data sample used for this work along with analysis procedure and
results are outlined in Sect. III. We conclude in Sect. IV.

##  II E22 analysis

Here, we provide an abridged summary of the method used in E22 to obtain
limits on SIDM cross-section using galaxy clusters. For their analysis, they
considered mock clusters with $M_{200}>3\times 10^{14}M_{\odot}$ from the
Bahamas-SIDM hydrodynamical cosmological simulations describing cluster
formation McCarthy et al. (2017); Robertson et al. (2019). These simulations
have been carried out for four distinct values of $\sigma/m$ ranging from 0
(corresponding to CDM) to $1.0~{}\rm{cm^{2}/g}$ with a total of 230 clusters.
These simulations also include baryonic feedback effects such as cooling, star
formation, stellar as well as AGN feedback McCarthy et al. (2017). However,
since there is no ab-initio understanding of how the first black holes form,
for these simulations black hole seeds are injected by hand into the dark
matter halos, and the growth of black holes is governed by Bondi accretion
following the prescription in Booth and Schaye (2009). These simulations do
not model the cold interstellar medium, which could underestimate the
accretion rate onto black holes. These simulations have been shown to
reproduce the galactic stellar mass fraction including dependence on galaxy
type, while also matching the hot gas fraction in groups and clusters McCarthy
et al. (2017). However, in these simulations, the stellar density profiles are
sensitive to the resolution and details of the AGN feedback processes
Robertson et al. (2019). A full listing of all the successes and shortcomings
of these simulations have been summarized in McCarthy et al. (2017); Robertson
et al. (2019). The synthetic clusters from these simulations were then fitted
to an Einasto profile Einasto (1965):

$\rho_{Ein}(r)=\rho_{s}\exp\bigg{[}-\frac{2}{\alpha}\bigg{(}\bigg{(}\frac{r}{r_{s}}\bigg{)}^{\alpha}-1\bigg{)}\bigg{]}$
(1)

where $r$ is the radial distance; $r_{s}$ is the scale scale radius Ettori et
al. (2019); $\rho_{s}$ is the scale density corresponding to the mass density
at $r=r_{s}$; and $\alpha$ is the Einasto index. The Einasto profile provided
a good fit to the synthetic clusters. Based on these fits E22 constructed an
empirical relation between $\alpha$ and $\sigma/m$ which can be written as
follows:

$\alpha=\alpha_{0}+\alpha_{1}\left(\frac{\sigma/m}{1cm^{2}/g}\right)^{\gamma}$
(2)

where $\alpha_{0}$ is the mean value obtained for CDM, while $\alpha_{1}$ and
$\gamma$ denote the scaling parameters that encode the dependence of $\alpha$
on the SIDM cross section/unit mass. For relaxed clusters in hydrostatic
equilibrium, E22 obtained $\alpha_{0}=0.178$, $\alpha_{1}=0.20$, and
$\gamma=0.63$. This relation was found to be robust with respect to the choice
of subgrid physics, which incorporates different models of cooling, star
formation, AGN and supernova feedback McCarthy et al. (2017). This empirical
relation was then applied to the clusters in the XMM-Newton Cluster Outskirts
Project (X-COP) sample Eckert et al. (2017), and a combined best fit value of
$\alpha$ was used to constrain $\sigma/m$. The estimated value of $\sigma/m$
from this stacking analysis was found by E22 to be
$\sigma/M<0.19~{}\rm{cm^{2}/g}$ at 95% c.l Eckert et al. (2022) at an assumed
dark matter collision velocity of 1000 km/sec. We should also point out that
the isothermal Jeans profile Kaplinghat et al. (2016) could also adequately
model the simulated SIDM halos Robertson et al. (2021). However, in order to
obtain constraints on SIDM cross-sections from isothermal profiles, one needs
to make additional assumptions such as the age of the halo, etc Robertson et
al. (2021).

##  III Analysis and Results

We use X-ray observations of 17 relaxed galaxy groups imaged with Chandra
and/or XMM-Newton with redshifts up to z = 0.08. The observational details for
A2589 can be found in Zappacosta et al. (2006), while details for all the
remaining groups can be found in Gastaldello et al. (2007). These groups have
masses between ($10^{13}-10^{14}M_{\odot}$) and span the range between
galaxies and clusters, with temperatures in the range 1-3 keV. These suites of
galaxy groups were used to test the constancy of dark matter density and
radial acceleration relation for groups in Gopika and Desai (2021) as well as
the MOND paradigm Angus et al. (2008).

###  III.1 Einasto fits to DM halos

The reconstruction of the dark matter density profiles for the group sample
considered in this work can be found in Gopika and Desai (2021). We only
provide a brief summary here. We first estimate the total group mass after
assuming that the groups are in hydrostatic equilibrium, since we are dealing
with relaxed systems. The gas mass was estimated from the X-ray temperature
and surface-brightness data. We then obtain the dark matter mass by
subtracting the gas mass (using the fitted gas density profiles) and the
stellar mass obtained from the $K$-band luminosity for the brightest group
galaxy. The dark matter density profile was then estimated from the dark
matter mass assuming spherical symmetry. The errors in the density profile
have been obtained using error propagation based on the errors in the
temperature. This error budget does not include systematic errors due to
hydrostatic equilibrium and hydrostatic bias, spherical symmetry as well as
due to uncertainties in the stellar mass. These hydrostatic bias values range
from no bias Liu et al. (2023) to about 40% Salvati et al. (2018) (see Ref.
Wicker et al. (2023) for a recent compilation of hydrostatic bias measurements
in literature.) The latest state of the art cosmological simulations predict a
bias of about 20% Gianfagna et al. (2021). In order to estimate the
hydrostatic bias on a group by group basis, we would need robust lensing
masses for all our objects, which are not available at the moment. E22 has
shown that that the Einsato $\alpha$ parameter gets overestimated by 2-8%
based on the hydrostatic equilibrium assumption, depending on the model for
non-thermal pressure support. The dynamical modeling in Gastaldello et al.
(2007) as well as in Angus et al. (2008) have been done assuming spherical
symmetry. A detailed discussion of the validity of the spherical symmetrical
assumption is discussed in one of our previous works and could cause
systematic errors of about 5% in the total mass determination Gopika and Desai
(2020). It is also not straightforward to estimate this error for every group,
since the X-ray images are intrinsically two dimensional in nature. Finally,
since the stellar mass is sub-dominant compared to the gas and dark matter
contributions, we neglect the uncertainties in the estimates of stellar mass.
Other causes of systematic errors could be due to inadequate background
modelling and subtraction and whose magnitude could be about 5% Gastaldello et
al. (2007).

Similar to E22, we fit these density profile data for the groups to the
Einasto profile (Eq. 1) using three free parameters: $\rho_{s}$, $r_{s}$, and
$\alpha$. We maximize the log-likelihood function using the emcee MCMC sampler
Foreman-Mackey et al. (2013). The likelihood used for getting the best-fit
parameters can be found in Eq. 5 in the Appendix. The priors for $\rho_{s}$
are in the range $10^{11}<\rho_{s}<10^{16}M_{\odot}/\rm{Mpc^{-3}}$, $r_{s}$
between 0 kpc and 300 kpc, and $\alpha$ spans from 0 to 0.5. For the MCMC
runs, the number of walkers was set to 200 and the total number of iterations
to 5000, which attained a mean acceptance fraction of approximately 0.52 for
these fits, where the acceptance fraction is defined as the fraction of the
total parameter space, which gets accepted for sampling the posterior pdf. The
corner plots showing the 68%, 90%, and 99% credible intervals for the best-fit
parameters associated with the 11 galaxy groups used in this analysis can be
found in the Appendix. We have excluded the groups ESO 3060170, MS
0116.3-0115, NGC 1550, NGC 4325, NGC 533, and RX J1159.8+5531 in this analysis
as their density profiles could not be fitted by marginalized closed contours
for all the three Einasto parameters. For these groups, at least one of the
parameters showed only one-sided marginalized contours at 68% credible
intervals. The dark matter density profiles for the remaining galaxy groups
along with the best-fit Einasto parameters is shown in Figure 1. For every
group, we also show the normalized residuals in the bottom panel, where the
residuals are normalized by the error in each data point. We find that two
groups have a reduced $\chi^{2}>2$. Table 1 summarizes the parameters obtained
from fitting the dark matter density profiles of the galaxy groups with an
Einasto model along with their reduced $\chi^{2}$, which gives a measure of
the efficacy of the fit. A graphical summary of the values of $\alpha$ for
every group can be found in Fig. 3. The estimated values of $\alpha$ for all
the systems are in the range of approximately 0.12-0.49. This agrees with the
values obtained in E22 for the X-COP sample, which had found $\alpha\sim
0.19$.

Figure 1: Dark matter density profile for the galaxy groups fitted with an
Einasto model. The top panel shows the dark matter density profile $\rho$ as a
function of radius ($r$) from the group center along with the best-fit Einasto
model parameters given in Table 1. The bottom panel in each plot shows the
normalized residual given by the difference in the data and model divided by
the error in the data.

Figure 2: Dark matter density profile for the galaxy groups fitted with an Einasto model (contd). The plot description is the same as for the other groups in previous page. Figure 3: The Einasto shape parameter ($\alpha$) calculated for all the galaxy groups analyzed in this work. Group |  | $\mathbf{\rho_{s}(10^{14}M_{\odot}Mpc^{-3})}$ |  | $\mathbf{r_{s}(kpc)}$ |  | $\mathbf{\alpha}$ |  | DOF |  | $\mathbf{\chi^{2}_{red}}$
---|---|---|---|---|---|---|---|---|---|---
A2589 |  | $6.66\pm 0.35$ |  | $130.74\pm 3.94$ |  | $0.29\pm 0.02$ |  | 6 |  | 1.34
A262 |  | $4.39\pm 0.75$ |  | $161.16\pm 17.16$ |  | $0.25\pm 0.02$ |  | 16 |  | 0.002
A2717 |  | $2.22\pm 0.25$ |  | $253.23\pm 13.38$ |  | $0.29\pm 0.03$ |  | 4 |  | 0.90
AWM 4 |  | $3.88\pm 0.98$ |  | $204.42\pm 29.99$ |  | $0.31\pm 0.05$ |  | 5 |  | 0.05
ESO 5520200 |  | $2.88\pm 0.31$ |  | $208.57\pm 12.32$ |  | $0.21\pm 0.04$ |  | 4 |  | 0.27
MKW 4 |  | $17.7\pm 2.02$ |  | $68.24\pm 3.78$ |  | $0.18\pm 0.02$ |  | 13 |  | 1.74
NGC 2563 |  | $11.1\pm 3.05$ |  | $61.47\pm 7.71$ |  | $0.19\pm 0.06$ |  | 4 |  | 9.02
NGC 5129 |  | $7.75\pm 1.15$ |  | $71.86\pm 5.38$ |  | $0.49\pm 0.12$ |  | 2 |  | 2.21
RGH 80 |  | $8.76\pm 0.30$ |  | $70.24\pm 1.22$ |  | $0.23\pm 0.01$ |  | 9 |  | 1.01
IC 1860 |  | $5.60\pm 0.75$ |  | $117.53\pm 9.07$ |  | $0.40\pm 0.04$ |  | 4 |  | 1.16
NGC 5044 |  | $8.62\pm 0.1$ |  | $84.42\pm 0.18$ |  | $0.24\pm 0.003$ |  | 11 |  | 1.19
Table 1:  This table shows the best-fit parameter values for the Einasto
profile (cf. Eq. 1) to the group halos along with the degrees of freedom
(DOF). The efficacy of the fit can be quantified by the reduced $\chi^{2}$
(shown in the last column). The data for A2589 was obtained from Zappacosta et
al. (2006) and for all other groups from Gastaldello et al. (2007).

###  III.2 Results for $\sigma/m$

Once we have determined the best-fit $\alpha$ for each group, we obtain an
estimate for $\sigma/m$ by comparing with Eq. 2. We assume that this equation
is also valid for low mass clusters and groups with masses in the range
$10^{13}-10^{14}M_{\odot}$, as no new physics kicks in below
$10^{14}M_{\odot}$. Furthermore, our current sample also contains five objects
with $M_{vir}>10^{14}M_{\odot}$. If the observed $\alpha$ for a given group is
consistent (within errors) we set an upper limit on $\sigma/m$. Else one could
get a central bounded estimate for $\sigma/m$. For this purpose, we calculate
$\chi^{2}$ as a function of $\sigma/m$ as follows:

$\chi^{2}=\left[\frac{\alpha_{obs}-\alpha_{th}(\alpha_{0},\alpha_{1},\gamma,\sigma)}{\sigma_{\alpha}}\right]^{2},$
(3)

where $\alpha_{th}(\alpha_{0},\alpha_{1},\gamma,\sigma)$ is given by Eq. 2,
$\alpha_{obs}$ is the observed value for each group and $\sigma_{\alpha}$ its
associated error. If the $\chi^{2}$ functional is shaped like a parabola with
a well-defined minimum, one could get bounded c.l. intervals for $\sigma/m$
based on fixed $\Delta\chi^{2}$ values compared to the minimum Press et al.
(1992), as long as the lower point of intersection is greater than 0.

For the cases when the reduced $\chi^{2}>1$ in the estimation of the SIDM
cross-section, we assume that there are unaccounted for systematic effects,
and rescale $\sigma_{\alpha}$ (in Eq. 3) by $\chi^{2}_{red}$ following Ref.
Workman and Others (2022); Desai et al. (2012); Tarafdar et al. (2022); Erler
and Ferro-Hernández (2020).111This procedure has also been criticized in
literature Mohayaee et al. (2021). In the case of linear regression, one can
show that the errors in the estimated parameters need to be rescaled by
$\sqrt{\chi^{2}_{red}}$ Press et al. (1992). Although such a rescaling may not
be exact for non-linear regression or if the errors are not Gaussian, this
rescaling procedure has been applied in a number of studies from Particle
Physics (where it is routinely used by Particle Data Group to rescale the
errors in the masses or lifetimes of elementary particles Workman and Others
(2022); Erler and Ferro-Hernández (2020)) to pulsar timing, (where the errors
in the pulsar times of arrival have been rescaled based on reduced $\chi^{2}$
Tarafdar et al. (2022).) Therefore, for our analysis, in case the reduced
$\chi^{2}$ for any group is greater than 1, for estimating the SIDM cross-
section, we rescaled the $\sigma_{\alpha}$ (in Eq. 3) by
$\sqrt{\chi^{2}_{red}}$. The $\chi^{2}$ curves as a function of $\sigma/m$ are
shown in Fig. 4. We have a total of two groups with reduced $\chi^{2}>2$. So
this rescaling would at most drastically affect the results of only two
groups.

Since all other works in literature have obtained limits on SIDM at 95% c.l,
we also obtain limits (or central estimates) at 95% c.l. These are obtained by
finding the maximum value of $\sigma/m$ for which $\Delta\chi^{2}<4$ Press et
al. (1992). In case the $\chi^{2}$ curve shows a minimum for $\sigma/m>0$, we
report 95% central estimate if the lower value of $\sigma/m$ for which
$\Delta\chi^{2}=4$ is greater than 0. A comprehensive summary of our results
for the 11 groups can be found in Table 2. We find non-zero central estimates
for $\sigma$/m for seven groups, whereas for the remaining four groups we
report upper limits at 95% c.l. However among these seven groups, only one
group, viz. NGC 5044 has fractional error $<$ 20%, given by $\sigma/m=0.165\pm
0.025~{}\rm{cm^{2}/g}$. The upper limits on $\sigma/m$ which we obtain for the
remaining groups are between $0.16-6.61~{}\rm{cm^{2}/g}$. The most stringent
limit is obtained for the group MKW 4, which has
$\sigma/m<0.16~{}\rm{cm^{2}/g}$.

In addition to estimating the limits (or central intervals) on the SIDM cross-
section, we have also estimated the dark matter collision velocity for each of
the groups, so that a comparison can be made with the currently viable models
of SIDM cross-section decreasing with velocity Adhikari et al. (2022). For
this purpose, we used the following scaling relation between line of sight
velocity dispersion of galaxies ($\sigma_{LOS}$) in the cluster and $M_{200}$
from Munari et al. (2013), (which was also used in Sagunski et al. (2021)):

$\log(h(z)M_{200})=13.98+2.75\log\left[\frac{\sigma_{LOS}}{500~{}\rm{(km/sec)}}\right]$
(4)

To estimate $\sigma_{LOS}$ and its uncertainty, we used the values of
$M_{200}$ and its error from Gastaldello et al. (2007); Zappacosta et al.
(2006). We also note that there are also variations of order 10% reported in
Munari et al. (2013) in the exponent and slope in Eq. 4, depending on the
simulation set used. We do not take this into account while estimating the
uncertainty. However, this uncertainty would uniformly affect all the groups.
These values of $\sigma_{LOS}$ can be found in Table 2. Therefore, the groups
with 95% upper limits between $0.16-6.61~{}\rm{cm^{2}/g}$ have dark matter
collision velocity between approximately 200-500 km/sec with the most
stringent limit for MKW 4 ($\sigma/m<0.16~{}\rm{cm^{2}/g}$) at a velocity
dispersion of $\approx$ 350 km/sec. Conversely, the group with the most
precise non-zero estimate for $\sigma$/m, viz. NGC 5044 has dark matter
velocity dispersion of about 300 km/sec.

Therefore, our results are consistent with the constraints on SIDM using
galaxy groups obtained in Sagunski et al. (2021) and agree with predictions of
velocity-dependent SIDM cross-sections on group/cluster scales Kaplinghat et
al. (2016). The constraints on $\sigma/m$, which we obtain are within the
ballpark of $0.1-1~{}\rm{cm^{2}/g}$ needed to solve the core-cusp and too big
to fail problems Tulin and Yu (2018).

Figure 4: Distribution of $\chi^{2}$ as a function of the dark matter self-
interaction cross-section $\sigma/m$ for all the 11 groups in our sample.

Figure 5: Distribution of $\chi^{2}$ as a function of the dark matter self-interaction cross-section $\sigma/m$ for groups (contd.)  Group | Limit (Estimate) on $\sigma/m$ | $\sigma_{LOS}$
---|---|---
| $\rm{(cm^{2}/g})$ | (km/sec)
A262 | $0.25\pm 0.21$ | 436 $\pm$ 32
A2717 | $0.47\pm 0.35$ | 523 $\pm$ 26
AWM 4 | $0.67\pm 0.63$ | 504 $\pm$ 45
ESO 5520200 | $<0.39$ | 463 $\pm$ 54
RGH 80 | $0.11\pm 0.038$ | 267 $\pm$ 11
IC 1860 | $1.63\pm 0.87$ | 329 $\pm$ 23
NGC 5044 | $0.165\pm 0.025$ | 295 $\pm$ 8
A2589 | $0.46\pm 0.31$ | 694 $\pm$ 53
MKW 4 | $<0.16$ | 358 $\pm$ 15
NGC 2563 | $<2.61$ | 259 $\pm$ 42
NGC 5129 | $<6.61$ | 216 $\pm$ 24
Table 2:  95% c.l. upper limits (or central intervals) on SIDM cross-section
for the galaxy groups considered in this work along with the line of sight
velocity dispersion of galaxies obtained using the scaling relation in Eq. 4
Munari et al. (2013) and based on the $M_{200}$ values in Gastaldello et al.
(2007); Zappacosta et al. (2006). For all galaxy groups with reduced
$\chi^{2}>1$ the upper limits were determined by re-scaling the errors in
$\alpha$ by $\sqrt{\chi^{2}_{red}}$.

##  IV Conclusions

In this work, we obtain constraints on the SIDM cross-sections using a sample
of 11 relaxed galaxy groups with X-ray observations, which we have previously
used to test the invariance of the dark matter halo surface density and RAR
Gopika and Desai (2021). For this purpose, we follow the same prescription as
E22, that derived an empirical relation between the SIDM cross-selection
($\sigma/m$) and the $\alpha$ parameter of the Einasto profile using simulated
SIDM cluster halos from the Bahamas-SIDM set of simulations Eckert et al.
(2022). This empirical relation between the Einasto $\alpha$ parameter and the
SIDM $\sigma/m$ can be found in Eq. 2.

We tried to fit the density profiles of 17 galaxy groups to the Einasto
parameterization. We were able to obtain closed contours for the Einasto
parameters for 11 of these groups. The best-fit Einasto parameters for all
these groups along with the reduced $\chi^{2}$ can be found in Table 1. The
values of $\alpha$ which we get are between 0.12-0.49. To obtain constraints
on the SIDM cross section, we used the aforementioned 11 groups and rescaled
the errors in $\alpha$ by $\sqrt{\chi^{2}_{red}}$ for the groups with
$\chi^{2}_{red}>1$. Possible reasons for the large reduced $\chi^{2}$ for some
of these groups could be due to AGN feedback in the core of the group or
incomplete relaxation Gastaldello et al. (2007). We then obtain the 95% c.l.
upper limits (or central estimates) on the SIDM cross-section, by finding the
maximum value of $\sigma/m$ for which $\Delta\chi^{2}=4$, where $\chi^{2}$ has
been obtained using Eq. 3.

Our results for SIDM cross-sections for the 11 relaxed groups can be found in
Table 2. We obtain a non-zero value for seven groups, with the most precise
estimate (fractional error $<$ 20%) for $\sigma/m$ for one group (NGC 5044),
given by $\sigma/m=0.165\pm 0.025~{}\rm{cm^{2}/g}$ at 95% c.l. at a velocity
dispersion of about 300 km/sec. For the remaining four other groups, we
estimate 95% c.l. upper limits on $\sigma/m$ in the range of
$0.16-6.61~{}\rm{cm^{2}/g}$ with dark matter velocity dispersions in the range
of 200-500 km/sec. The most stringent limit which we obtain is for MKW 4 for
which $\sigma/m<0.16~{}\rm{cm^{2}/g}$ at velocity dispersion of $\approx$ 350
km/sec. Our results for $\sigma/m$ for all the groups in our sample are in the
same ballpark as those found in Sagunski et al. (2021), (which were consistent
with $\sigma/m=0.5\pm 0.2~{}\rm{cm^{2}/g}$ or $\sigma/m<0.9~{}\rm{cm^{2}/g}$,
if interpreted as an upper limit). Our results are also consistent with the
predictions of velocity-dependent SIDM cross-sections at group/cluster scales
Kaplinghat et al. (2016).

## Acknowledgements

We are grateful to Fabio Gastaldello and Dominique Eckert for useful
correspondence related to the galaxy group data used for this work and E22,
respectively. We also thank the anonymous referee for several useful and
constructive comments on the manuscript. GK also acknowledges the Ministry of
Education (MoE), Government of India for the Senior Research Fellowship.

## References

  * Eckert et al. (2022) D. Eckert, S. Ettori, A. Robertson, R. Massey, E. Pointecouteau, D. Harvey, and I. G. McCarthy, Astron. & Astrophys. 666, A41 (2022), eprint 2205.01123.
  * Aghanim et al. (2020) N. Aghanim et al. (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], eprint 1807.06209.
  * Bertone and Hooper (2018) G. Bertone and D. Hooper, Reviews of Modern Physics 90, 045002 (2018), eprint 1605.04909.
  * Merritt (2017) D. Merritt, Studies in the History and Philosophy of Modern Physics 57, 41 (2017), eprint 1703.02389.
  * Gaskins (2016) J. M. Gaskins, Contemporary Physics 57, 496 (2016), eprint 1604.00014.
  * Desai et al. (2004) S. Desai et al. (Super-Kamiokande), Phys. Rev. D 70, 083523 (2004), eprint hep-ex/0404025.
  * Boveia and Doglioni (2018) A. Boveia and C. Doglioni, Annual Review of Nuclear and Particle Science 68, 429 (2018), eprint 1810.12238.
  * Schumann (2019) M. Schumann, Journal of Physics G Nuclear Physics 46, 103003 (2019), eprint 1903.03026.
  * Abdalla et al. (2021) H. Abdalla, F. Aharonian, F. Ait Benkhali, E. O. Angüner, C. Arcaro, C. Armand, T. Armstrong, H. Ashkar, M. Backes, V. Baghmanyan, et al., Astrophys. J. 918, 17 (2021), eprint 2106.00551.
  * Acciari et al. (2022) V. A. Acciari, S. Ansoldi, L. A. Antonelli, A. Arbet Engels, M. Artero, K. Asano, D. Baack, A. Babić, A. Baquero, U. Barres de Almeida, et al., Physics of the Dark Universe 35, 100912 (2022), eprint 2111.15009.
  * Chan and Lee (2022) M. H. Chan and C. M. Lee, Phys. Rev. D 105, 123006 (2022), eprint 2205.08692.
  * Flores and Primack (1994) R. A. Flores and J. R. Primack, Astrophys. J. Lett. 427, L1 (1994), eprint astro-ph/9402004.
  * Moore (1994) B. Moore, Nature (London) 370, 629 (1994).
  * Del Popolo and Le Delliou (2022) A. Del Popolo and M. Le Delliou, arXiv e-prints arXiv:2209.14151 (2022), eprint 2209.14151.
  * Bullock and Boylan-Kolchin (2017) J. S. Bullock and M. Boylan-Kolchin, Ann. Rev. Astron. Astrophys. 55, 343 (2017), eprint 1707.04256.
  * Weinberg et al. (2015) D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray, and A. H. G. Peter, Proceedings of the National Academy of Science 112, 12249 (2015), eprint 1306.0913.
  * Boylan-Kolchin et al. (2011) M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat, Mon. Not. R. Astron. Soc. 415, L40 (2011), eprint 1103.0007.
  * Banik and Zhao (2022) I. Banik and H. Zhao, Symmetry 14, 1331 (2022), eprint 2110.06936.
  * McGaugh et al. (2016) S. S. McGaugh, F. Lelli, and J. M. Schombert, Phys. Rev. Lett. 117, 201101 (2016), eprint 1609.05917.
  * Pradyumna et al. (2021) S. Pradyumna, S. Gupta, S. Seeram, and S. Desai, Physics of the Dark Universe 31, 100765 (2021), eprint 2011.06421.
  * Famaey and McGaugh (2012) B. Famaey and S. S. McGaugh, Living Reviews in Relativity 15, 10 (2012), eprint 1112.3960.
  * Donato et al. (2009) F. Donato, G. Gentile, P. Salucci, C. Frigerio Martins, M. I. Wilkinson, G. Gilmore, E. K. Grebel, A. Koch, and R. Wyse, Mon. Not. R. Astron. Soc. 397, 1169 (2009), eprint 0904.4054.
  * Lovas (2022) S. Lovas, Mon. Not. R. Astron. Soc. 514, L56 (2022), eprint 2206.11431.
  * Paranjape and Sheth (2021) A. Paranjape and R. K. Sheth, Mon. Not. R. Astron. Soc. 507, 632 (2021), eprint 2102.13116.
  * Gopika et al. (2023) K. Gopika, S. Desai, and A. Paranjape, Mon. Not. R. Astron. Soc. 523, 1718 (2023), eprint 2303.12859.
  * Sales and Navarro (2023) L. V. Sales and J. F. Navarro, Nature Astronomy (2023), eprint 2303.10190.
  * Chan and Del Popolo (2020) M. H. Chan and A. Del Popolo, Mon. Not. R. Astron. Soc. 492, 5865 (2020), eprint 2001.06141.
  * Tian et al. (2020) Y. Tian, K. Umetsu, C.-M. Ko, M. Donahue, and I. N. Chiu, Astrophys. J. 896, 70 (2020), eprint 2001.08340.
  * Pradyumna and Desai (2021) S. Pradyumna and S. Desai, Physics of the Dark Universe 33, 100854 (2021), eprint 2107.05845.
  * Chan et al. (2022) M. H. Chan, S. Desai, and A. Del Popolo, PASJ 74, 1441 (2022), eprint 2205.07515.
  * Gopika and Desai (2020) K. Gopika and S. Desai, Physics of the Dark Universe 30, 100707 (2020), eprint 2006.12320.
  * Gopika and Desai (2021) K. Gopika and S. Desai, Physics of the Dark Universe 33, 100874 (2021), eprint 2106.07294.
  * Upadhyaya and Desai (2023) V. Upadhyaya and S. Desai, Physics of the Dark Universe 40, 101182 (2023), eprint 2212.02326.
  * Adhikari et al. (2022) S. Adhikari, A. Banerjee, K. K. Boddy, F.-Y. Cyr-Racine, H. Desmond, C. Dvorkin, B. Jain, F. Kahlhoefer, M. Kaplinghat, A. Nierenberg, et al., arXiv e-prints arXiv:2207.10638 (2022), eprint 2207.10638.
  * Berezhiani et al. (2018) L. Berezhiani, B. Famaey, and J. Khoury, JCAP 2018, 021 (2018), eprint 1711.05748.
  * Colín et al. (2000) P. Colín, V. Avila-Reese, and O. Valenzuela, Astrophys. J. 542, 622 (2000), eprint astro-ph/0004115.
  * Ferreira (2021) E. G. M. Ferreira, Astronomy and Astrophysics Review 29, 7 (2021), eprint 2005.03254.
  * Medvedev (2014) M. V. Medvedev, Phys. Rev. Lett. 113, 071303 (2014), eprint 1305.1307.
  * Spergel and Steinhardt (2000) D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84, 3760 (2000), eprint astro-ph/9909386.
  * Nadler et al. (2021) E. O. Nadler, A. Drlica-Wagner, K. Bechtol, S. Mau, R. H. Wechsler, V. Gluscevic, K. Boddy, A. B. Pace, T. S. Li, M. McNanna, et al., Phys. Rev. Lett. 126, 091101 (2021), eprint 2008.00022.
  * Oh et al. (2011) S.-H. Oh, C. Brook, F. Governato, E. Brinks, L. Mayer, W. J. G. de Blok, A. Brooks, and F. Walter, Astron. J. 142, 24 (2011), eprint 1011.2777.
  * Kaplinghat et al. (2020) M. Kaplinghat, T. Ren, and H.-B. Yu, JCAP 2020, 027 (2020), eprint 1911.00544.
  * Tulin and Yu (2018) S. Tulin and H.-B. Yu, Physics Reports 730, 1 (2018), eprint 1705.02358.
  * Kamada et al. (2017) A. Kamada, M. Kaplinghat, A. B. Pace, and H.-B. Yu, Phys. Rev. Lett. 119, 111102 (2017), eprint 1611.02716.
  * Ren et al. (2019) T. Ren, A. Kwa, M. Kaplinghat, and H.-B. Yu, Physical Review X 9, 031020 (2019), eprint 1808.05695.
  * Kaplinghat et al. (2016) M. Kaplinghat, S. Tulin, and H.-B. Yu, Phys. Rev. Lett. 116, 041302 (2016), eprint 1508.03339.
  * Meneghetti et al. (2001) M. Meneghetti, N. Yoshida, M. Bartelmann, L. Moscardini, V. Springel, G. Tormen, and S. D. M. White, Mon. Not. R. Astron. Soc. 325, 435 (2001), eprint astro-ph/0011405.
  * Gnedin and Ostriker (2001) O. Y. Gnedin and J. P. Ostriker, Astrophys. J. 561, 61 (2001), eprint astro-ph/0010436.
  * Mahdavi et al. (2007) A. Mahdavi, H. Hoekstra, A. Babul, J. Sievers, S. T. Myers, and J. P. Henry, Astrophys. J. 664, 162 (2007), eprint astro-ph/0703372.
  * Randall et al. (2008) S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradač, Astrophys. J. 679, 1173 (2008), eprint 0704.0261.
  * Bradač et al. (2008) M. Bradač, S. W. Allen, T. Treu, H. Ebeling, R. Massey, R. G. Morris, A. von der Linden, and D. Applegate, Astrophys. J. 687, 959 (2008), eprint 0806.2320.
  * Peter et al. (2013) A. H. G. Peter, M. Rocha, J. S. Bullock, and M. Kaplinghat, Mon. Not. R. Astron. Soc. 430, 105 (2013), eprint 1208.3026.
  * Kahlhoefer et al. (2015) F. Kahlhoefer, K. Schmidt-Hoberg, J. Kummer, and S. Sarkar, Mon. Not. R. Astron. Soc. 452, L54 (2015), eprint 1504.06576.
  * Gastaldello et al. (2014) F. Gastaldello, M. Limousin, G. Foex, R. P. Munoz, T. Verdugo, V. Motta, A. More, R. Cabanac, D. A. Buote, D. Eckert, et al., Mon. Not. R. Astron. Soc. 442, L76 (2014), eprint 1404.5633.
  * Massey et al. (2015) R. Massey, L. Williams, R. Smit, M. Swinbank, T. D. Kitching, D. Harvey, M. Jauzac, H. Israel, D. Clowe, A. Edge, et al., Mon. Not. R. Astron. Soc. 449, 3393 (2015), eprint 1504.03388.
  * Harvey et al. (2015) D. Harvey, R. Massey, T. Kitching, A. Taylor, and E. Tittley, Science 347, 1462 (2015), eprint 1503.07675.
  * Robertson et al. (2017) A. Robertson, R. Massey, and V. Eke, Mon. Not. R. Astron. Soc. 465, 569 (2017), eprint 1605.04307.
  * Harvey et al. (2019) D. Harvey, A. Robertson, R. Massey, and I. G. McCarthy, Mon. Not. R. Astron. Soc. 488, 1572 (2019), eprint 1812.06981.
  * Banerjee et al. (2020) A. Banerjee, S. Adhikari, N. Dalal, S. More, and A. Kravtsov, JCAP 2020, 024 (2020), eprint 1906.12026.
  * Sagunski et al. (2021) L. Sagunski, S. Gad-Nasr, B. Colquhoun, A. Robertson, and S. Tulin, JCAP 2021, 024 (2021), eprint 2006.12515.
  * Yang et al. (2023) S. Yang, F. Jiang, A. Benson, Y.-M. Zhong, C. Mace, X. Du, Z. Carton Zeng, and A. H. G. Peter, arXiv e-prints arXiv:2305.05067 (2023), eprint 2305.05067.
  * Andrade et al. (2019) K. E. Andrade, Q. Minor, A. Nierenberg, and M. Kaplinghat , Mon. Not. R. Astron. Soc. 487, 1905 (2019), eprint 1901.00507.
  * Cross et al. (2023) D. Cross, G. Thoron, T. Jeltema, A. Swart, D. Hollowood, S. Adhikari, S. Bocquet, O. Eiger, S. Everett, and J. Jobel, arXiv e-prints arXiv:2304.10128 (2023), eprint 2304.10128.
  * Andrade et al. (2022) K. E. Andrade, J. Fuson, S. Gad-Nasr, D. Kong, Q. Minor, M. G. Roberts, and M. Kaplinghat, Mon. Not. R. Astron. Soc. 510, 54 (2022), eprint 2012.06611.
  * Miralda-Escudé (2002) J. Miralda-Escudé, Astrophys. J. 564, 60 (2002), eprint astro-ph/0002050.
  * Gastaldello et al. (2007) F. Gastaldello, D. A. Buote, P. J. Humphrey, L. Zappacosta, J. S. Bullock, F. Brighenti, and W. G. Mathews, Astrophys. J. 669, 158 (2007), eprint astro-ph/0610134.
  * Sun (2012) M. Sun, New Journal of Physics 14, 045004 (2012), eprint 1203.4228.
  * Lovisari et al. (2021) L. Lovisari, S. Ettori, M. Gaspari, and P. A. Giles, Universe 7, 139 (2021), eprint 2106.13256.
  * Oppenheimer et al. (2021) B. D. Oppenheimer, A. Babul, Y. Bahé, I. S. Butsky, and I. G. McCarthy, Universe 7, 209 (2021), eprint 2106.13257.
  * McCarthy et al. (2017) I. G. McCarthy, J. Schaye, S. Bird, and A. M. C. Le Brun, Mon. Not. R. Astron. Soc. 465, 2936 (2017), eprint 1603.02702.
  * Robertson et al. (2019) A. Robertson, D. Harvey, R. Massey, V. Eke, I. G. McCarthy, M. Jauzac, B. Li, and J. Schaye, Mon. Not. R. Astron. Soc. 488, 3646 (2019), eprint 1810.05649.
  * Booth and Schaye (2009) C. M. Booth and J. Schaye, Mon. Not. R. Astron. Soc. 398, 53 (2009), eprint 0904.2572.
  * Einasto (1965) J. Einasto, Trudy Astrofizicheskogo Instituta Alma-Ata 5, 87 (1965).
  * Ettori et al. (2019) S. Ettori, V. Ghirardini, D. Eckert, E. Pointecouteau, F. Gastaldello, M. Sereno, M. Gaspari, S. Ghizzardi, M. Roncarelli, and M. Rossetti, Astron. & Astrophys. 621, A39 (2019), eprint 1805.00035.
  * Eckert et al. (2017) D. Eckert, S. Ettori, E. Pointecouteau, S. Molendi, S. Paltani, and C. Tchernin, Astronomische Nachrichten 338, 293 (2017), eprint 1611.05051.
  * Robertson et al. (2021) A. Robertson, R. Massey, V. Eke, J. Schaye, and T. Theuns, Mon. Not. R. Astron. Soc. 501, 4610 (2021), eprint 2009.07844.
  * Zappacosta et al. (2006) L. Zappacosta, D. A. Buote, F. Gastaldello, P. J. Humphrey, J. Bullock, F. Brighenti, and W. Mathews, Astrophys. J. 650, 777 (2006), eprint astro-ph/0602613.
  * Angus et al. (2008) G. W. Angus, B. Famaey, and D. A. Buote, Mon. Not. R. Astron. Soc. 387, 1470 (2008), eprint 0709.0108.
  * Liu et al. (2023) A. Liu, E. Bulbul, M. E. Ramos-Ceja, J. S. Sanders, V. Ghirardini, Y. E. Bahar, M. Yeung, E. Gatuzz, M. Freyberg, C. Garrel, et al., Astron. & Astrophys. 670, A96 (2023), eprint 2210.00633.
  * Salvati et al. (2018) L. Salvati, M. Douspis, and N. Aghanim, Astron. & Astrophys. 614, A13 (2018), eprint 1708.00697.
  * Wicker et al. (2023) R. Wicker, M. Douspis, L. Salvati, and N. Aghanim, Astron. & Astrophys. 674, A48 (2023), eprint 2204.12823.
  * Gianfagna et al. (2021) G. Gianfagna, M. De Petris, G. Yepes, F. De Luca, F. Sembolini, W. Cui, V. Biffi, F. Kéruzoré, J. Macías-Pérez, F. Mayet, et al., Mon. Not. R. Astron. Soc. 502, 5115 (2021), eprint 2010.03634.
  * Foreman-Mackey et al. (2013) D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Publ. Astron. Soc. Pac. 125, 306 (2013), eprint 1202.3665.
  * Press et al. (1992) W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, _Numerical recipes in FORTRAN. The art of scientific computing_ (1992).
  * Workman and Others (2022) R. L. Workman and Others (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Desai et al. (2012) S. Desai, R. Armstrong, J. J. Mohr, D. R. Semler, J. Liu, E. Bertin, S. S. Allam, W. A. Barkhouse, G. Bazin, E. J. Buckley-Geer, et al., Astrophys. J. 757, 83 (2012), eprint 1204.1210.
  * Tarafdar et al. (2022) P. Tarafdar, K. Nobleson, P. Rana, J. Singha, M. A. Krishnakumar, B. C. Joshi, A. K. Paladi, N. Kolhe, N. D. Batra, N. Agarwal, et al., Pub. Astro. Soc. Aust. 39, e053 (2022), eprint 2206.09289.
  * Erler and Ferro-Hernández (2020) J. Erler and R. Ferro-Hernández, European Physical Journal C 80, 541 (2020), eprint 2004.01219.
  * Mohayaee et al. (2021) R. Mohayaee, M. Rameez, and S. Sarkar, European Physical Journal Special Topics 230, 2067 (2021), eprint 2106.03119.
  * Munari et al. (2013) E. Munari, A. Biviano, S. Borgani, G. Murante, and D. Fabjan, Mon. Not. R. Astron. Soc. 430, 2638 (2013), eprint 1301.1682.

##  Appendix A Einasto fits to Galaxy Groups

In this section, we shall discuss the fitting of the Einasto profiles to the
dark matter profiles derived from the X-ray brightness and temperature data.
The Einasto model was fitted to the group density profiles by maximizing the
log-likelihood function given below:

$\displaystyle\ln\mathcal{L}=-0.5\sum\bigg{[}\bigg{(}\frac{\rho_{Einasto}-\rho_{data}}{\sigma}\bigg{)}^{2}+\ln(\sigma^{2})\bigg{]},$
(5)

where $\rho_{data}$ and $\sigma$ denote the group density profiles and their
errors, respectively.

The MCMC corner plots displaying the posterior distributions with 68%, 90% and
99% credible intervals of the free parameters for the Einasto model, viz
$r_{s}$, $\rho_{s}$, and $\alpha$ are shown in Figure 6. The posterior
distribution medians and their 1-$\sigma$ uncertainties are represented by
vertical dashed lines in the histograms shown diagonally in the corner plots.

(a) A2589

(b) A262

(c) A2717

(d) AWM 4

Figure 6: Corner plots for the Einasto fit to the galaxy group data. The plot
shows the 68%, 90%, and 99% credible intervals. Here, $\rho_{s}$ has units of
$M_{\odot}/Mpc^{3}$ and $r_{s}$ is in $kpc$.

(a) ESO 5520200

(b) MKW 4

(c) NGC 2563

(d) NGC 5129

Figure 7: Corner plots for the Einasto fit to the galaxy group data. The plot
shows the 68%, 90%, and 99% credible intervals. (cont.)

(a) RGH 80

(b) IC 1860

(c) NGC 5044. The error in $\alpha$ is 0.003.

Figure 8: Corner plots for the Einasto fit to the galaxy group data. The plot
shows the 68%, 90%, and 99% credible intervals. (cont.)