# Designing a Socio-Technical Business Process for Analyzing Information
Quality Requirements: Experience Report

Mohamad Gharib<EMAIL_ADDRESS>DiMaI, University of Florence Viale
Morgagni 65FlorenceItaly50134

###### Abstract.

Although many BPs involve critical activities that demand high-quality
information for their successful enactment, most available BP approaches focus
mainly on control-flow, and either ignore the Information Quality (IQ)
perspective or consider it as a mere technical issue, instead of a social and
organizational one. This leaves a BP subject to different types of social and
organizational IQ vulnerabilities that may negatively impact, or even abort
the BP enactment. To tackle this problem, a Socio-Technical BP (STBP), namely
a Workflow-net with Actors (WFA-net) has been developed. WFA-net allows for
capturing IQ requirements in their social and organizational context. This
paper reports on the experience gained, findings and lessons learned while
developing the WFA-net.

Information Quality, IQ, Socio-Technical Business Process, WF-nets, Goal Model

††copyright: none††ccs: Software and its engineering Requirements
analysis††ccs: Software and its engineering Model-driven software engineering

##  1\. Introduction

Although many BPs involve critical activities that demand high-quality
information for their successful enactment, most available BP approaches focus
mainly on control-flow, and either ignore the Information Quality (IQ)
perspective, or consider it as a mere technical issue, instead of a social and
organizational one (Gharib et al., 2018). Accordingly, a BP is subject to
different types of social and organizational IQ vulnerabilities that may
negatively impact, or even abort the BP enactment. Therefore, capturing IQ
requirements of BPs in their social and organizational context is essential to
detect and address such vulnerabilities (Gharib et al., 2018).

There are two main threads of research that might tackle this problem: 1-
Social Business Process Management (SBPM), in which essential elements of
social software are applied to the different stages of BPM (Schmidt and
Nurcan, 2008). However, the few existing approaches do not focus on
information and none of them consider IQ to the best of my knowledge. 2- Goal-
based approaches, in which some techniques for capturing an execution order
between goals/tasks have been proposed.

Although some goal-based approaches are an appropriate solution for designing
socio-technical systems, where the stakeholders along with their social
dependencies are considered as a fundamental component of the system. They
lack the formality to appropriately capture a control-flow among their
goals/tasks. On the other hand, several BP approaches offer concrete semantics
for capturing control-flow among activities of BPs, but offer no semantics for
capturing the social and organizational context where BPs are enacted.

To this end, the idea for proposing a Socio-Technical BP (STBP), namely a
Workflow-net with Actors (WFA-net), for modeling and analyzing IQ requirements
in their social and organizational context was depending on a goal-based
modeling approach for capturing the social and organizational context where a
STBP is enacted, and on a Workflow modeling approach for capturing the
control-flow among the activities of a STBP.

The rest of this report is organized as follows; Sect. 2 presents the case
study, and Sect. 3 discusses the requirements for designing a STBP approach.
In Sect. 4, the approach is presented, and in Sect. 5 its implementation and
evaluation are briefly discussed. Sect. 6 lists and discusses the findings and
lessons learned, and Sect. 7 concludes the report.

##  2\. Case study: The Flash Crash

The Flash Crash was a trillion-dollar stock market crash, where a stock market
is a physical or virtual place where stock traders trade securities (Gharib
and Giorgini, 2014). Traders can be classified into Market Makers (MMs) that
facilitate trading on specific securities, and they can trade a large number
of such securities; High-Frequency Traders (HFTs) can trade a huge number of
securities with a very high-frequency; and traders who trade a few securities
are called Small traders. Usually, stock markets analyze the trading
environment and apply their Circuit Breakers (CBs) to slow or even pause
trading to prevent a market crash.

The chronology of the Flash Crash events. The following chronology of events
is based on the joint report of the Commodity Futures Trading Commission
(CFTC) and the Securities and Exchange Commission (SEC) regarding the Flash
Crash (U.S. Commodity Futures Trading Commission and Securities and Exchange
Commission and others, 2010):

  * •

On the 6 of May, after the U.S. stock market opened, it started trending down
influenced by concerns about the debt crisis in Europe.

  * •

By 2:30 PM, the value of the Dow Jones Industrial Average (DJIA) dropped down
by about 2.5%.

  * •

At 2:32 PM, a big sell order of 75,000 E-mini S&P 500 contracts (with an
approximate value of $4.1 Billion) was placed, which was mainly bought by
HFTs.

  * •

Between 2:41 and 2:44 PM, the price of the E-Mini contracts drop down by
around 5%.

  * •

At 2:45:28 PM, the Chicago Mercantile Exchange (CME) applied its CB to pause
trading on the E-Mini contracts for 5 seconds to stop a further price
declines.

  * •

At 2:45:33 PM, prices stabilized again when trading resumed.

Main reasons for the Flash Crash. Several theories have been proposed to
explain this crash, including:

  * •

A human error when placing the order. However, it was confirmed that the trade
under suspicion of triggering the crash was not a result of a human error.

  * •

The loosely-coupled nature of the market and the inefficient coordination
mechanisms among the trading venues concerning the work of their CBs. More
specifically, trading venues need to coordinate their CBs, otherwise, HFTs can
continue trading in any venue that did not apply its CB. This is exactly what
happened during the Flash Crash as CME employs its CB while New York Stock
Exchange (NYSE) did not.

  * •

Intentionally falsified orders that have been provided by some traders. For
example, some HFTs have issued flickering quotes that last for a very short
time, which makes them unavailable for most traders. Flickering quotes can be
used by HFTs to drive the market prices up or down before HFTs start their
actual trades. On the other hand, some MMs have issued stub quotes to fulfill
their obligations concerning in facilitate trading on certain securities. Such
orders are issued with prices far away from the actual market prices, i.e.,
they are not intended to be traded. Throughout the Flash Crash, the great
majority of the trades (more than 98%) were executed with prices about 10% of
their market values because of stub orders, i.e., stub quotes contributed
significantly to the severity of the crash.

Several key factors that contributed to this crash were due to IQ related
vulnerabilities that manifested themselves in the social and organizational
context of the system (Gharib and Giorgini, 2015c). For instance, the
incompetent coordination among several markets concerning their CBs was
because of relying on incomplete and inconsistent information. Moreover, both
flickering and stub quotes can be considered untrustworthy, unbelievable
and/or inaccurate information; since such orders were not intended to be
executed. If markets analyze these quotes; they might be able to detect
falsified ones and mitigate their harmful effect. Similarly, if markets relied
on complete and consistent information for their CBs, such a crash might be
avoided.

Being a real case study of a complex system that fails mainly due to IQ
issues, makes the Flash Crash a good example for illustrating the importance
of capturing and addressing IQ requirements in their social and organizational
context where a STBP is enacted.

##  3\. Identifying requirements for designing a STBP for IQ

Based on a thorough analysis of the Flash Crash, the STBP should be able to
analyze IQ requirements based on several IQ dimensions (e.g., accuracy,
believability, trustworthiness, consistency), and some of these dimensions
should be analyzed in their social and organizational context. Moreover, the
STBP should be able to capture control-flow as well as information-flow among
its activities. Therefore, the main existing BP approaches have been surveyed
to identify the best BP language that can be adopted to develop a STBP that
satisfies the previously mentioned requirements. Existing BP approaches can be
broadly classified into four main classes:

Activity-driven approaches are the most existing and adopted approaches (e.g.,
Petri-nets, workflow-nets, BPMN 2.0), and they mainly focus on capturing the
control flow. Therefore, the emphasis of such approaches is on how a process
should be enacted.

Data-driven approaches (Müller et al., 2008) focus mainly on identifying data
entities that are used by activities of the process, how data flows among such
activities, and also specifying pre- and post-constraints/conditions on the
use of such data.

Event-driven approaches (Van Der Aalst, 1999) are mainly used for modeling
processes that their activities might be initiated due to some events (e.g.,
information modifications, time trigger, etc.). In such approaches, an event
captures when an activity should be initiated.

Role-driven modeling approaches (Saidani and Nurcan, 2006) focus on the
identification of roles involved in the process along with their interactions.
That is why they are used for modeling communication-based processes, i.e.,
the emphasis is laid on who is responsible for executing an activity.

These approaches can model how, what, and when activities of a BP are
required, yet none of them captures IQ requirements in their social and
organizational context, which can be solved by adopting a goal-based approach.
To this end, the idea was adopting one of the existing Activity-driven
approaches that propose concepts for capturing control and information flow.
Then, extending such BP approach with concepts for analyzing IQ in the social
and organizational context of the system. However, this solution was not
practical due to the complexity of the resulting language. Therefore, a
decision was made to model the social and organizational context where the
STBP is enacted and the control flow among the activities of the STBP at two
different abstraction levels.

##  4\. Approach for designing STBP for analyzing IQ requirements

The process underlining the approach for designing STBP that is specialized
for modeling and analyzing IQ requirements is shown in Figure 1. The process
consists of four main phases: 1- analyzing IQ requirements, 2- modeling IQ
requirements in their social and organizational context relying on an extended
goal-based modeling language; 3- mapping, in which leaf goals of the goal
model are mapped into activities of WFA-net (STBP); and 4- analysis, in which
the correctness of the STBP is verified. Each of these phases is described in
the rest of this section:

Figure 1.  The process for creating the STBP for modeling and analyzing IQ
requirements

###  4.1. Analyzing IQ requirements

Numerous models for analyzing IQ based on various dimensions have been
proposed, yet most of them ignore the social and organizational aspects that
may underlie some of these dimensions leaving the analysis incomplete for
important vulnerabilities. Therefore, we proposed a multi-dimensional model
for analyzing IQ requirements in their social and organizational context based
on seven IQ dimensions (Gharib and Giorgini, 2015b). These dimensions have
been chosen based on the requirements of the stock market system, and they
might need to be extended or reduced for other systems. In particular,
accuracy, completeness, timeliness, and consistency are considered since they
are the key IQ dimensions to tackle the vulnerabilities that led to the crash.
Additionally, believability and trustworthiness have been considered as they
can be used for analyzing accuracy. Finally, accessibility has been considered
to analyze information availability to perform a task. In what follows, each
of the chosen dimensions is defined:

1\. Accessibility:

measures the extent to which information is available for use as well as the
permissions over it that might be required for performing a specific task.

2\. Accuracy:

measures the extent to which information is true concerning some known value.
Accuracy is analyzed based on trustworthiness and believability since they
allow for analyzing the intentional aspects underlying accuracy, i.e., dealing
with situations when actors provide falsified information (e.g., flickering
and stub quotes).

3\. Believability:

measures the extent to which information is perceived as true.

4\. Trustworthiness:

measures the extent to which information is credible, and it is analyzed
relying on the trustworthiness of the source and trustworthiness of the
provision.

5\. Completeness:

measures the extent to which information is complete, and it can be analyzed
based on i- value completeness measures the extent to which the integrity of
information is preserved, and ii- purpose of use completeness measures the
extent to which information is complete for performing a specific task, which
cannot be analyzed properly unless the information has been considered in its
organizational context. For instance, a main reason for the incompetent
coordination among the CBs was because markets depend only on their local
trading information to manage their CB, which can be considered incomplete for
coordinating the application of their CBs with other markets. In other words,
the overall system design did not consider the need of markets to coordinate
their CBs, thus, markets depend on incomplete information.

6\. Timeliness:

measures the extent to which information is valid for performing a task.

7\. Consistency:

measures the extent to which all multiple records of the same information are
the same across time. Consistency has been considered to guarantee that actors
will be able to coordinate their interdependent activities, and it cannot be
analyzed properly unless the information has been considered in its
organizational context. For example, information that is used for coordinating
the CBs of markets also needs to be consistent among markets, since it is used
for interdependent activities.

###  4.2. Modeling Phase

To capture the stakeholders’ IQ requirements, the goal-based modeling
framework we proposed in (Gharib and Giorgini, 2015c, b) has been adopted
since it is the only one that proposes concepts for capturing IQ requirements.
Figure 2 shows a partial diagram of the stock market system represented with
our modeling language. The language introduces primitives for modeling actors
in terms of agents and roles they are playing. Goals that represent actors’
strategic interests, and they can be refined through and/or-decomposition into
finer sub-goals. Refining a root-goal into sub-goals through and-decomposition
implies that all sub-goals need to be achieved to achieve the parent goal.
While refining them through or-decomposition implies achieving any of them
achieve the root-goal. Information has a volatility attribute that represents
the change rate of its value. Information can be composed of more than one
part, and the part of is used to represent such relationship. Moreover, an
actor can be an information owner, which enables it to control the permissions
over its usage.

Goals can produce, read, modify and send information. The Produce relationship
has an attribute that denotes whether such relation applies a believability
check or not (shown as B/NB respectively) when producing such information. The
Read relationship has three attributes, the first one is read type that can be
either Optional or Required for goal achievement. The second captures whether
the read relationship applies believability check, and the third captures the
purpose-of-use. The Send relationship has an attribute that specifies the
destination and another one for specifying the acceptable transmission time.

Delegation captures the transfer of goals and permissions between actors.
While provision models information transmission between actors, and it has two
attributes, the first describes the transmission time, and the second
specifies the provision type, which can be normal Provision (P) or Integrity-
Preserving (IP). Trust/distrust (T/DT) captures the actors’ expectations in
one another regarding their delegated entitlements and authorities, and also
for produced information (TP/DP).

In what follows, the concepts for analyzing IQ requirements are discussed:

Figure 2. A partial goal model concerning the stock market structure

1\. Accessibility:

is analyzed relying on information availability, and [P]roduce, [R]ead,
[M]odify and [S]end permissions over such information.

2\. Accuracy:

is analyzed based on believability and trustworthiness of the
production/trustworthiness of the provenance for produced/read information.

3\. Believability:

arises when information is being produced or read. Therefore, believability is
analyzed in produce and read relationships by examining whether such
relationships apply a believability check, if yes information is considered
believable.

4\. Trustworthiness:

is analyzed based on i- trustworthiness of the source that can be analyzed
based on the trust/distrust produce relationship; and ii- trustworthiness of
the provision that is analyzed considering the operations, which have been
applied to information starting from its source until reaching its final
destination, and whether such operations were authorized.

5\. Completeness:

is analyzed based on i- value completeness, whether the integrity of
information is preserved, i.e., its value completeness is preserved if it has
been provided only through IP provision, otherwise, its value completeness is
not guaranteed; and ii- purpose of use completeness, whether information has
all its sub-parts for performing a specific task, which can be analyzed with
the help of the PartOf relationship.

6\. Timeliness,:

only the read and send relationships are influenced by time aspects. Read
timeliness is analyzed by comparing the currency of information with its
volatility, and if its currency is smaller, information is valid. Send
timeliness is analyzed by comparing the send time of information and its read
time at its destination. Then, information is valid only if the read time is
smaller than the send timeliness.

7\. Consistency:

is analyzed relying on the interdependent readers concept, which identifies
actors that read the same information for the exact same purpose-of-use. If
all interdependent readers have the same read time for such information, it is
considered consistent among them.

###  4.3. Mapping phase

This section starts by describing how the semantics of WFA-net have been
defined, and then discusses the developed mechanisms for mapping the goal-
based model into WFA-net.

Workflow net with Actors (WFA-net). To develop the WFA-net111For more
information about the semantics of WFA-nets please refer to (Gharib et al.,
2018), social actors and their IQ requirements need to be integrated into the
BP design. Thus, the WF-net has been adopted as a baseline and its semantics
has been extended with the notion of social actor and IQ requirements. In
particular, a WFA-net (shown in Figure 3) is a WF-net, in which each activity
$T$ is described by an actor that is responsible (res) for its achievement,
sets of information that the activity produce (pd), read (rd), modify (md)
and/or send (sd).

Figure 3. A partial WFA-net concerning a stock investor process for trading
securities

The WFA-net configuration has been defined to capture the workflow, which
consists of a marking and an activity state that can be evaluated either to
true ($\top$) or false ($\bot$). An activity of a WFA-net might be enabled at
a configuration iff the activity is enabled at a marking (activity flow), and
the activity state is evaluated to true ($\top$), which means that the
information-flow and IQ requirements are achieved. An activity may fire when
it is enabled, and its firing may enable a set of successor configurations. In
addition, the reachability property for WFA-net has been defined to check
whether some configuration (e.g., $c^{{}^{\prime}}$) can be reached from
another one (e.g., $c$). Moreover, the firing of a single transition for WFA-
net has been extended to the firing of a sequence of transitions. Finally, the
soundness property for WFA-net has been defined to verify whether the final
configuration of a WFA-net is reachable from any configuration.

Mapping goal-based model into WFA-net. The rules and constraints that are used
to guarantee a correct mapping of goal-based models into WFA-net are
illustrated in Figure 4, and they are described as follows:

Figure 4. Mapping rules and constraints

Building blocks are used to identify constructs of the goal-based model that
can be mapped into activities of WFA-net. The main idea of identifying
building blocks is guaranteeing that no key organizational nor social aspects
will be lost during the mapping process. Three rules for identifying building
blocks have been defined: (1) A goal that is not and/or-decomposed of or into
any other goal is considered as a building block that can be mapped into an
activity of WFA-net. (2) A goal that is and-decomposed is considered as a
building block in terms of all its sub-goals, which are mapped into sequencing
activities. Mapping these sub-goals into sequencing activities is derived from
the semantics of the and-decomposition relationship, and mapping them into
sequencing activities of WFA-net implies that all of them should be achieved.
(3) A goal that is or-decomposed is considered as a building block in terms of
its all sub-goals, which are mapped into parallel activities of the WFA-net.
Similarly, the mapping of the or-decomposed sub-goals into parallel activities
is derived from the semantics of or-decomposition relationship, which implies
that it is enough to achieve any one of these parallel activities in the WFA-
net.

Consistency constraints are used to assure a correct mapping between building
blocks and activities of WFA-net. Three consistency constraints have been
defined: (1) No goal is allowed to be mapped unless it can be considered as a
complete building block. (2) Mapping is allowed for leaf goals only, i.e.,
root-goals are only mapped in terms of their sub-goals. (3) No information is
allowed in the WFA-net unless the goal that produces it, is already
represented in the WFA-net. This enables for analyzing accessibility and other
IQ dimensions.

Sequencing constraints are used to assure an appropriate ordering of the
activities of WFA-net. Two sequencing constraints have been defined to
guarantee that: (1) Activities of WFA-net should be consistent with their
sequencing order in their building blocks, and (2) If an activity depends on
the outcome of another one, it should appear after the later.

Refinement constraints are used to guarantee a correct sequencing of the
activities and places of a WFA-net. Two refinement constraints have been
defined: (1) No two consequent places can exist in WFA-net without an activity
separating them, and (2) No two consequent activities can exist in a WFA-net
without a place separating them.

###  4.4. Analysis Phase

After modeling (mapping) the desired WFA-net, its correctness should be
verified. Therefore, a formal framework to underlies the modeling language has
been developed based on Disjunctive Datalog formal language. This allows for
converting constructs of the graphical model into formal predicates, and helps
in verifying the correctness of the WFA-net222The formalization of the
concepts/axioms can be found at https://bit.ly/3fldgKc. Additionally, a set of
21 properties of the design has been defined333For more information about the
properties of the design please refer to (Gharib et al., 2018), which specify
constraints for verifying the correctness of the mapping, control-flow,
information-flow and IQ requirements of the WFA-net model.

##  5\. Implementation and evaluation

The proposed approach was evaluated using a simulation method, i.e.,
developing a prototype implementation tool444The prototype tool is available
at http://goo.gl/Iy1BjR and test its applicability with realistic data. More
specifically, the approach has been evaluated by showing its utility and
efficacy in modeling and analyzing the IQ requirements in STBPs by applying it
to two realistic scenarios abstracted from the Flash Crash case study. The
approach was able to models and effectively analyze (e.g., detect violations
to the properties of the design) the IQ requirements of the two scenarios.

##  6\. Findings and lessons learned

While developing this approach, several challenges have been faced. The main
challenges are summarized along with findings and lessons learned as follows:

A model for analyzing IQ. There is no general agreement on the number or which
dimensions should be considered for analyzing IQ. Therefore, choosing the
appropriate set of IQ dimensions is not an easy task. Like many researchers, I
believe that IQ dimensions should be chosen with respect to the application
domain (i.e., domain-specific). A more complex problem is how the selected IQ
dimensions should be analyzed as researchers have proposed various
methods/metrics for their analysis. I also believe that the analysis of IQ
dimensions should be domain-specific with the help of domain experts if
possible since the same dimension might be interpreted differently in
different domains. For example, most of the existing models do not propose
analysis for accuracy that can tackle the problem of intentionally falsified
information. Therefore, accuracy has been analyzed in this approach in a way
that allows for capturing the intentional aspects underlying it.

Modeling IQ requirements in a BP. 1- Adopting an i* based language (Gharib and
Giorgini, 2015c, b, 2019) for modeling IQ requirements as such languages allow
for dealing with requirements considering their social and organizational
aspects. Moreover, some of these modeling languages are supported by formal
frameworks that allow for performing different kinds of analysis to verify the
requirements model. 2- Adopting a WF-net as a base BP modeling language since
it proposes a simple but powerful semantics, which can be extended to
integrate almost any property of interest.

Adopting two layers of abstraction. Capturing the social and organizational
context where the STBP is enacted, and the control flow among its activities
at two different levels of abstraction offers several advantages: 1- modeling
and verifying the goal-based model allows for detecting and resolving any
vulnerability at the social and organizational level concerning IQ
requirements before modeling the STBP; 2- relying on a verified goal model,
you can model and analyze almost any STBP without influencing the base goal
model; 3- most of the properties of the design can be derived either from the
semantics of the goal-based model or the WFA-net model. In WFA-net, most of
the properties for the satisfaction of IQ requirements were derived from the
goal-based model semantics, while most of the properties for the control and
information flow were derived from the WFA-net semantics; and 4- allows for
extending the goal-based modeling language to consider new IQ dimensions.
Then, easily reflecting such extensions into the WFA-net.

##  7\. Conclusions and Future work

This paper reported on experience gained while developing an STBP, namely WFA-
net (Gharib and Giorgini, 2015c; Gharib et al., 2018), which integrates goal-
based and WF-net approaches. The focus was laid on making the approach easy to
be understood and followed; each of its steps was accompanied by a detailed
description of how it was performed. This may help other scholars while
dealing with IQ for BP. The proposed approach has several limitations and
threats to its validity, which opens opportunities for future work. For
example, the approach considers binary IQ requirement satisfaction (e.g., a
requirement is satisfied or denied), it cannot deal with more than one STBP at
the same time, and it considers that all IQ dimensions have the same
priority/importance to the system. Finally, the approach has been applied only
to the stock market domain, which threatens the generalization of the
findings.

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mydate []

# Breaking the Single Clock Symmetry: measuring single-field inflation non-
Gaussian features.

Daniele Bertacca  Dipartimento di Fisica e Astronomia Galileo Galilei,
Università di Padova, 35131 Padova, Italy  INFN, Sezione di Padova, via F
Marzolo 8, I-35131 Padova, Italy.  INAF - Osservatorio Astronomico di Padova,
vicolo dell Osservatorio 5, I-35122 Padova, Italy<EMAIL_ADDRESS>Raul Jimenez  ICC, University of Barcelona, Marti i Franques 1, 08028
Barcelona, Spain.  ICREA, Pg. Lluis Companys 23, Barcelona, E-08010, Spain.
Sabino Matarrese  Dipartimento di Fisica e Astronomia Galileo Galilei,
Università di Padova, 35131 Padova, Italy  INFN, Sezione di Padova, via F
Marzolo 8, I-35131 Padova, Italy.  INAF - Osservatorio Astronomico di Padova,
vicolo dell Osservatorio 5, I-35122 Padova, Italy.  Gran Sasso Science
Institute, viale F. Crispi 7, I-67100 L’Aquila, Italy.  Licia Verde  ICC,
University of Barcelona, Marti i Franques 1, 08028 Barcelona, Spain.  ICREA,
Pg. Lluis Companys 23, Barcelona, E-08010, Spain.

###### Abstract

The Universe is not just cold dark matter and dark energy, it also contains
baryons, radiation and neutrinos. The presence of these components, beyond the
pressure-less cold dark matter and the quasi-uniform dark energy ones, imply
that the single clock assumption from inflation is no longer preserved. Here
we quantify this effect and show that the single-clock symmetry is ensured
only on scales where baryonic effects, neutrinos effects, or sound speed are
zero. These scales depend on the cosmic epoch and the Universe composition.
Hence for all use and purposes of interpreting state-of-the-art and possibly
forthcoming surveys, in the accessible scales, single clock symmetry cannot be
said to be satisfied. Breaking the single-clock symmetry has key consequences
for the study of non-Gaussian features generated by pure single-field
inflation which arise from non-linearities in the metric yielding non-
Gaussianities of the local type: the $n_{s}-1$ and the relativistic $-5/3$
term.

##  I introduction

The most economical model to describe the early Universe is a quasi-de Sitter
state, i.e., inflation driven by a single field that is undergoing slow-
rolling. Strong empirical support for inflation comes from two observational
facts: the discovery of super-horizon fluctuations WMAP03  and the near-scale-
invariant spectrum of scalar fluctuations with a red tilt WMAP03+ ;
Planck:2018vyg . In the simplest inflationary scenario, one single field
dominates the quasi-exponential expansion of the space-time. This can happen
because, at the energy scale of inflation, no other fields are light enough as
to be able to provide enough e-folds on the expansion and are effectively
frozen. This lack of other fields that are relevant during the inflation
period indicates that one could describe inflation as a single clock with
translational invariance of the field111For a description of the role of time
during inflation see Ref. Gomez:2021yhd . Under this symmetry, it is natural
that one expects no coupling between the different fields and thus no
deviation from Gaussian fluctuations as produced by the modes leaving the
horizon without interaction. Self-coupling of the inflaton will produce a
negligible non-Gaussian signal, of order ${\cal O}(\epsilon^{2})$ where
$\epsilon$ denotes the slow roll parameter. However, non-linearities of the
single field paint a different story.

Non-linearities in the metric induced by the non-linearities of the single-
field while slow rolling, give rise to two very well known effects
Gangui:1993tt ; Gangui:1999vg ; Wang:1999vf ; Acquaviva:2002ud ;
Maldacena:2002vr . The first is a non-Gaussianity contribution with a local
shape component with amplitude parameter $f_{\rm NL}^{\rm Local}\propto
n_{s}-1$ and an equilateral shape component of amplitude parameter $f^{\rm
equil.}_{\rm NL}\propto\epsilon$. The second effect is due to General
Relativity and translates effectively into a non-Gaussian contribution of the
local type with $f_{\rm NL,eff.}^{\rm local}=-5/3$.

The inclusion of gravity in the analysis is crucial as this non-Gaussian
effect is generated by the non-linearities in the metric. In Ref.
Bellomo:2018lew  a method was recently proposed to look at the effect of
graviton interactions with scalar fluctuations of the inflaton field. This
opens the possibility to study observational signatures of single field
inflation via the generation of gravitons, which is again a pure gravity
effect that can only happen during a phase of accelerated expansion.

Each of the above effects, primordial non-Gaussianity and relativistic
effects, if measured, would open a new window into the physics of the early
Universe. Despite some early claims that GR-effects driven non-Gaussian
signatures were simply gauge artifacts Tanaka_2011 ; dePutter:2015vga ;
Pajer:2013ana ; Cabass:2016cgp , extensive work e.g., Bertacca:2015mca ;
VerdeSasaki ; Sasaki ; Rollo , has shown that indeed these non-Gaussianities
are measurable and physical222One interesting and intuitive argument that
shows that these higher order correlations cannot be gauged away is that if
this was the case, one could also gauge away the 2-point correlation function
(the power spectrum). This is very simply to see by using the Mukhanov-Sasaki
variable in the in-in formalism with the proposed gauge transformations in
Ref. Tanaka_2011 ; dePutter:2015vga ; Pajer:2013ana ; Cabass:2016cgp .. In
addition, it has been shown, from a quantum information theory point of view
QFNG  that single field non-Gaussianities cannot be gauge artifacts. Only in
the unphysical case of exact $k=0$ can these non-Gaussianities be gauged away.

In this work we take the calculation a step further and show that, in addition
to the above arguments, the presence in the Universe of pressure(full) terms
(e.g. baryons, radiation, neutrinos etc…) do break the single clock symmetry
and automatically imply that the above non-Gaussianities cannot be gauged away
during the entire history of the evolution of the patch i.e., on scales that
are not significantly larger than the sound horizon at radiation drag. This
impossibility to synchronize clocks across cosmic epochs and separate patches,
which size approaches the size of the current horizon, has important
consequences for the detectability of primordial non-Gaussianity signatures.

##  II Methodology

Let us start by considering a Universe with a Friedmann-Lema$\rm\hat{i}$tre-
Robertson-Walker (FLRW) space-time at the background level, setting to zero
the constant curvature $K$ (for simplicity), and assume General Relativity
(GR) as the theory of gravity. The unperturbed energy-momentum tensor
$T_{\mu\nu}$ of matter contains ordinary (baryonic) matter, radiation,
neutrinos, Cold Dark Matter (CDM) and Dark Energy (DE) or, the cosmological
constant $\Lambda$. At the background level, $T_{\mu\nu}$ describes a perfect
fluid where the total energy density $\rho^{(0)}$, and the total pressure
$p^{(0)}$ depend only on time 333The superscript $(0)$ denotes quantities at
background.. The Einstein and continuity background equations are

$\mathcal{H}^{2}={8\pi G\over
3}a^{2}\rho^{(0)}\;,\quad\quad\mathcal{H}^{2}-\mathcal{H}^{\prime}=4\pi
Ga^{2}(\rho^{(0)}+p^{(0)})\quad{\rm
and}\quad{\rho^{(0)}}^{\prime}=-3\mathcal{H}(p^{(0)}+\rho^{(0)})\;,$ (1)

where $\mathcal{H}=a^{\prime}/a=aH$ is the comoving Hubble scale and prime
denotes the derivative with respect to $\eta$. Here $\eta$ is the conformal
time, i.e., $\eta=\int{{\rm d}t/a(t)}$, where $a$ is the scale factor. It is
useful to define (see Ref. Mukhanov:1990me ; Mukhanov:2005sc ) a quantity
obtained by the ratio between $0-0$ and $i-j$ components of Einstein’s
equations

$\theta_{\rm M}={1\over
a}{1\over\left(1+p^{(0)}/\rho^{(0)}\right)^{1/2}}={{\rho^{(0)}}^{1/2}\over
a}\exp{\left[-{3\over 2}\int^{\eta}(1+\tilde{c}_{s}^{2})\tilde{\cal H}\;{\rm
d}\tilde{\eta}\right]}\;,$ (2)

that depends (at the functional level, i.e. $\theta_{\rm M}=\theta_{\rm
M}[c_{s}]$) on the speed of sound
$c_{s}^{2}={p^{(0)}}^{\prime}/{\rho^{(0)}}^{\prime}$. Here we define two
models: the spatially flat $\Lambda$CDM model, which components are, as it is
customary, ordinary (baryonic) matter, radiation, neutrinos, Cold Dark Matter
(CDM) and the cosmological constant; we also define a pressureless counterpart
of this model, the 0p$\Lambda$CDM model, which does not contain (or contains
only a fully negligible contribution of) baryons, radiation and neutrinos.
Hence the speed of sound just introduced is related to any possible effect
that describes the physics beyond the pressure-less 0p$\Lambda$CDM
model444Here we have defined $\tilde{c}_{s}^{2}=c_{s}^{2}(\tilde{\eta})$ and
$\tilde{\cal H}={\cal H}(\tilde{\eta})$. Following Ref. Mukhanov:2005sc , the
constant of integration in Eq. (2) corresponds to an unphysical solution and,
from now, it will be discarded.. Using Eq. (2) we note, immediately, that
$\mathcal{H}^{2}-\mathcal{H}^{\prime}=(3/2){\cal H}^{2}/a^{2}\theta_{\rm
M}^{2}$.

We expand the background around the pressureless case; using the bar to imply
quantities in the the corresponding 0p$\Lambda$CDM model, we have
$\bar{\mathcal{H}}^{2}=8\pi G\bar{a}^{2}\bar{\rho}^{(0)}/3$,
$\bar{\mathcal{H}}^{2}-\bar{\mathcal{H}}^{\prime}=4\pi
G\bar{a}^{2}(\bar{\rho}^{(0)}+\bar{p}^{(0)})=4\pi
G\bar{a}^{2}\bar{\rho}^{(0)}_{\rm DM}$ and
${{\bar{\rho}}^{(0)^{\prime}}}=\bar{\rho}^{(0)^{\prime}}_{\rm
DM}=-3\bar{\mathcal{H}}(\bar{p}^{(0)}+\bar{\rho}^{(0)})=3\bar{\mathcal{H}}\bar{\rho}^{(0)}_{\rm
DM}$, where $\bar{\rho}^{(0)}=\bar{\rho}^{(0)}_{\rm DM}+\bar{\rho}_{\Lambda}$,
$\bar{\rho}^{(0)}_{\rm DM}\propto\bar{a}^{-3}$ and
$\bar{p}^{(0)}=\bar{p}_{\Lambda}=-\bar{\rho}_{\Lambda}$. (Note that in general
if today $a(\eta=\eta_{0})=a_{0}=1$, then $\bar{a}_{0}\neq 1$). Of course, in
a $\Lambda$CDM model with massless neutrinos, one could adjust the values of
the model’s parameters to ensure that $\Omega_{\rm CDM}$ in the 0p$\Lambda$CDM
coincides with $\Omega_{\rm m}$ in the $\Lambda$CDM one, where baryons and
cold dark matter are all included in the matter density parameter. In this
case the scale factor evolution after recombination and for negligible amount
of radiation, would effectively be the same and $\bar{a}\simeq a$. Even in
this hyper-simplified $\Lambda$CDM, $c_{s}$ is still non zero, but the effects
can only be seen in the perturbations (see below) and not in the background.
However for non-massless neutrinos, different neutrino masses transition from
relativistic to non-relativistic at different times so if at a starting point,
even if well after matter radiation equality, we set $a=\bar{a}$, in general
at later times this will not hold in details. Moreover, if the dark energy
were not to be a cosmological constant, the dark energy component could also
contribute to $c_{s}$ in principle. For our analytical calculation, we have to
consider the initial conditions, which we denote with subscript (in). We set
initial conditions in the matter dominated era, and we consider two cases, one
well before recombination and one well after recombination.

In addition, at the background level (in FLRW space-time), we assume that the
parameter expansion $a$ at $\eta_{\rm in}$ coincides in the 0p$\Lambda$CDM and
$\Lambda$CDM model555In principle we could use an alternative condition, i.e.,
the parameter expansion could coincide both for 0p$\Lambda$CDM and
$\Lambda$CDM model today at $z=0$. In the main text we have used the condition
during matter epoch in order to match our considerations with approach used in
separate universe technique. See also the discussion at the end of the next
section..

For a generic quantity, let us call it $C^{(0)}$ which in this section we take
to be a background quantity, we define, at the linear level, the deviations
from 0p$\Lambda$CDM due to the presence of a small pressure component as

$\textswab{d}C^{(0)}=C^{(0)}-\bar{C}^{(0)}\;,\quad{\rm
where}\quad\left|\textswab{d}C^{(0)}/\bar{C}^{(0)}\right|\ll 1\,.$ (3)

Note that for the pressure component to be small, the size of the patch of the
Universe under consideration needs to be much larger than the largest scale
reached by the sound horizon (the Jean’s length) from the initial conditions
on. Hence, for initial conditions set before recombination, that corresponds
to scales much larger than the sound horizon at radiation drag. If we instead
set initial conditions after recombination, this scales needs to be larger
than massive neutrinos free streaming length, or for massless neutrinos,
larger than the scale where baryonic effects become important. Recall that the
sound horizon at radiation drag is $\sim 150$ Mpc, the neutrinos free
streaming length corresponds to $k\sim 0.01h$ Mpc-1, and baryonic effects are
important on scales of few Mpc.

Immediately we note that $\textswab{d}C^{(0)}$ depends on $c_{s}^{2}$ and it
is possible to identify explicitly $\textswab{d}p^{(0)}$ from the definition
of the speed of sound. Indeed we obtain

$\textswab{d}p^{(0)}[c_{s}^{2}]=\textswab{d}p^{(0)}_{\rm(in)}+\int_{\eta_{\rm(in)}}^{\eta}\tilde{c}_{s}^{2}{\tilde{\rho}}^{(0)^{\prime}}~{}{\rm
d}\tilde{\eta}.$ (4)

From $a=\bar{a}+\textswab{d}a$

$\mathcal{H}^{2}=\bar{\mathcal{H}}^{2}\left(1+2{\textswab{d}\mathcal{H}\over\bar{\mathcal{H}}}\right)={8\pi
G\over
3}\bar{a}^{2}\bar{\rho}^{(0)}\left(1+{\textswab{d}a\over\bar{a}}\right)\left(1+{\textswab{d}\rho^{(0)}\over\bar{\rho}^{(0)}}\right)$
(5)

and

$\mathcal{H}=(\bar{\mathcal{H}}+\textswab{d}\mathcal{H})=\left(1-{\textswab{d}a\over\bar{a}}\right)\left(\bar{\mathcal{H}}+{\textswab{d}a^{\prime}\over\bar{a}}\right)\;,$
(6)

we have the following relations for $\textswab{d}\mathcal{H}$

$\displaystyle\textswab{d}\mathcal{H}={\bar{\mathcal{H}}\over
2}\left({\textswab{d}\rho^{(0)}\over\bar{\rho}^{(0)}}+2{\textswab{d}a\over\bar{a}}\right)\;,\quad\quad\textswab{d}\mathcal{H}={\textswab{d}a^{\prime}\over\bar{a}}-\bar{\mathcal{H}}{\textswab{d}a\over\bar{a}}$
(7)

or, equivalently, combining the above relations

$\displaystyle
2\textswab{d}\mathcal{H}={\textswab{d}a^{\prime}\over\bar{a}}+{\bar{\mathcal{H}}\over
2}{\textswab{d}\rho^{(0)}\over\bar{\rho}^{(0)}}\;,\quad\quad{{\rm d}\over{\rm
d}\bar{a}}\left(\textswab{d}a\over\bar{a}^{2}\right)={1\over
2\bar{a}^{2}}{\textswab{d}\rho^{(0)}\over\bar{\rho}^{(0)}}\;.$ (8)

From the continuity equation we find

${\textswab{d}\rho^{(0)}}^{\prime}+3\bar{\mathcal{H}}\left(\textswab{d}p^{(0)}+\textswab{d}\rho^{(0)}\right)=-3(\bar{p}^{(0)}+\bar{\rho}^{(0)})\textswab{d}\mathcal{H}\,.$
(9)

Here we note that as long as we are only concerned with times well after
neutrino decoupling, it is possible to go from the 0p$\Lambda$CDM model to the
$\Lambda$CDM model in a continuous and differentiable way, i.e. the background
quantities $C$ are continuous and differentiable.

This equation, using Eq. (8), can be rewritten as

$\frac{{\rm d}^{2}\textswab{d}\rho^{(0)}}{{\rm d}\bar{a}^{2}}+{3\over
2\bar{a}}(\bar{w}+5)\frac{{\rm d}\textswab{d}\rho^{(0)}}{{\rm
d}\bar{a}}+{3\over
2\bar{a}^{2}}[4+3(\bar{w}+1)^{2}]\textswab{d}\rho^{(0)}+{6\over\bar{a}^{2}}\textswab{d}p^{(0)}+{3\over\bar{a}}\frac{{\rm
d}\textswab{d}p^{(0)}}{{\rm d}\bar{a}}=0\;,$ (10)

where we defined $\bar{w}=\bar{p}^{(0)}/\bar{\rho}^{(0)}$. This is the
complete and correct equation to solve in the generic case. However it has no
analytic solution and is therefore not too transparent. We proceed by
considering a symplified case, which, however, still captures all the
interesting aspects. For example in the $\Lambda$CDM case we can directly get
$\textswab{d}\rho^{(0)}=(\rho^{(0)}-\bar{\rho}^{(0)})$ from the Friedmann
equation

$\displaystyle\rho^{(0)}(z)$ $\displaystyle=$ $\displaystyle{3H_{0}^{2}\over
8\pi G}\Bigg{\\{}\left(\Omega_{{\rm CDM}0}+\Omega_{{\rm
b}0}\right)(1+z)^{3}+\Omega_{\gamma 0}(1+z)^{4}\left[1+0.2271N_{\rm
eff}f\left({m_{\nu}\over[(1+z)T_{\nu 0}]}\right)\right]$ (11)
$\displaystyle+\Omega_{\Lambda 0}\Bigg{\\}},\quad{\rm
and}\quad\bar{\rho}^{(0)}(z)={3\bar{H}_{0}^{2}\over 8\pi G}\Big{[}\Omega_{{\rm
CDM}0}(1+z)^{3}+\Omega_{\Lambda 0}\Big{]}\;,$

where $f$ is a suitable function in which $(\Omega_{\nu}/a^{4})0.2271N_{\rm
eff}f(m_{\nu}a/T_{\nu 0})\to\Omega_{\nu}/a^{3}$ for $a\to\infty$ (e.g. see
WMAP:2010qai ). In the massless neutrinos case $f=1$. Here the values of
different cosmological parameters for the $\Lambda$CDM model are constrained
in Ref. Planck:2018vyg  and for the 0p$\Lambda$CDM the reader can think of
absorbing the baryonic component into the cold component to keep $\Omega_{m}$
the same.

In general, if we choose $\textswab{d}a_{\rm(in)}=0$, using Eq. (7), we obtain

$\textswab{d}\mathcal{H}_{\rm(in)}={\bar{\mathcal{H}}_{\rm(in)}\over
2}{\textswab{d}\rho^{(0)}_{\rm(in)}\over\bar{\rho}^{(0)}_{\rm(in)}}\;.$ (12)

Using Eq. (8), we find

$\left({\textswab{d}a\over\bar{a}^{2}}\right)={1\over
2}\int^{a}{1\over\tilde{a}^{2}}{\textswab{d}\tilde{\rho}^{(0)}\over\tilde{\rho}^{(0)}}{\rm
d}\tilde{a}\;,$ (13)

and, taking into account Eq. (9), we have

$\textswab{d}p^{(0)}_{\rm(in)}=-{(3+\bar{w}_{\rm(in)})\over
2}\textswab{d}\rho^{(0)}_{\rm(in)}-{\textswab{d}\rho^{(0)^{\prime}}_{\rm(in)}\over
3\bar{\mathcal{H}}_{\rm(in)}}\;.$ (14)

Finally, from Eq. (2), the correction for $\theta_{\rm M}$ becomes

$\textswab{d}\theta_{\rm M}=\bar{\theta}_{\rm M}\left[{1\over
2}\left({\textswab{d}\rho^{(0)}\over\bar{\rho}^{(0)}}-{\textswab{d}p^{(0)}+\textswab{d}\rho^{(0)}\over\bar{p}^{(0)}+\bar{\rho}^{(0)}}\right)-{\textswab{d}a\over\bar{a}}\right]$
(15)

where

$\bar{\theta}_{\rm
M}={1\over\bar{a}}{1\over\left(1+\bar{p}^{(0)}/\bar{\rho}^{(0)}\right)^{1/2}}={1\over\bar{a}}\left({\bar{\rho}^{(0)}\over{\bar{\rho}}_{\rm
CDM}^{(0)}}\right)^{1/2}\;.$ (16)

Here ${\bar{\rho}}^{(0)}_{\rm CDM}$ denotes the density of cold dark matter
and we have assumed that for dark energy $p=-\rho$.

While at the background level baryons, radiation and neutrinos (massless,
fully relativistic or massive and non-relativistic) have a negligible effect,
it is important to bear in mind that at the perturbation level massive
neutrinos and baryons have important effects even at linear scales. Baryons
are known to have effects on the mildly non-linear regime (the highly feared
baryonic effects) which, however at linear scales are at the few percent
level. Massive neutrinos on the other hand, have non zero velocity dispersion
which acts as an effective sound speed; because of this, they have effects on
the power spectrum at scales $k>0.01h$ Mpc-1 comparable to the free streaming
length; the effect is scale- and redshift-dependent. Even in the linear
regime, at $k\sim 0.1h$ Mpc-1 neutrinos produce a suppression on the power
spectrum of $\sim 6f_{\nu}$ where $f_{\nu}$ denotes the neutrino mass
fraction666For the minimum neutrino mass allowed by oscillation experiments,
and for the concordance cosmology, $f_{\nu}$ is $\sim 0.01$.. At the minimum
neutrino mass imposed by oscillations experiments, the linear power spectrum
suppression is $\sim 6\%$ at $z=0$. It is well known that massive neutrinos
also introduce a scale-dependent bias Shiveshwarkar et al. (2021) and affect
peculiar velocities. Hence their effects likely cannot be neglected in the
perturbations. In particular while in the absence of massive neutrinos the
matter density perturbations in the linear regime evolve $\propto a$, in the
presence of massive neutrinos evolve as $a^{1-3/5f_{\nu}}$.

###  II.1 Corrections on cosmological perturbations

Consider the correction in the scalar metric perturbations in the longitudinal
gauge Bardeen:1980kt ; Kodama:1985bj ; Bertschinger:1993xt ; Malik:2008im  (or
also called Conformal Newtonian Gauge Mukhanov:1990me ; Ma:1995ey ;
Mukhanov:2005sc ). In this gauge777Here $\Phi=\Phi_{A}Q^{(0)}$ and
$\Psi=-\Phi_{H}Q^{(0)}$, where $\Phi_{A}$ and $\Phi_{H}$ are the well known
Bardeen potentials Bardeen:1980kt .

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$ $\displaystyle
a(\eta)^{2}\left[-\left(1+2\Phi\right){\rm
d}\eta^{2}+\delta_{ij}\left(1-2\Psi\right){\rm d}x^{i}{\rm d}x^{j}\right].$
(17)

Let us introduce the following variables (already used in Mukhanov:1990me ;
Mukhanov:2005sc ):888The constants of integration arising in these formulae
correspond to unphysical solution that can be removed.

$\displaystyle u_{\rm M}$ $\displaystyle=$ $\displaystyle\exp\left[{3\over
2}\int^{\eta}{\rm
d}\tilde{\eta}\;\left(1+\tilde{c}_{s}^{2}\right)\tilde{\mathcal{H}}\right]\Psi={a\theta_{\rm
M}\over\left(\rho^{(0)}\right)^{1/2}}\Psi\;,$ (18) $\displaystyle\zeta_{\rm
M}$ $\displaystyle=$
$\displaystyle\Psi-{\mathcal{H}^{2}\over\mathcal{H}^{\prime}-\mathcal{H}^{2}}\left(\Psi+{\Psi^{\prime}\over\mathcal{H}}\right)={2\over
3}\left({3\over 8\pi G}\right)^{1/2}\theta_{\rm M}^{2}\left({u_{\rm
M}\over\theta_{\rm M}}\right)^{\prime}\;,$ (19)

and the comoving curvature perturbation that can be written in terms of
$\zeta_{\rm M}$ and $D_{\rm M}$

$\mathcal{R}=\Psi-{\mathcal{H}^{2}\over\mathcal{H}^{\prime}-\mathcal{H}^{2}}\left(\Phi+{\Psi^{\prime}\over\mathcal{H}}\right)=\zeta_{\rm
M}+{2\over 3}a^{2}\theta_{\rm M}^{2}D_{\rm M}\;,$ (20)

where $D_{\rm M}=\Phi-\Psi$. (Of course, in 0p$\Lambda$CDM model, we have
$D_{\rm M}=0$ and $\mathcal{R}$ coincides with $\zeta_{\rm M}$). Using the
linear perturbation of the spatial part of the stress energy tensor

$T^{i(1)}_{j}=p^{(1)}\delta^{i}_{j}+\mathcal{D}^{i}_{j}\Pi\;,$

where $\mathcal{D}_{ij}=\partial_{i}\partial_{j}-\delta_{ij}\nabla^{2}/3$,
$p^{(1)}$ is the perturbation of isotropic pressure999Here $p^{(1)}$ is not
the same $\delta p$ defined in Mukhanov:1990me . and $\Pi$ is the trace-free
scalar part of total anisotropic stress tensor Malik:2008im , thorough the
Einstein equation we can write

$D_{\rm M}=-8\pi Ga^{2}\Pi\;.$ (21)

Finally, using (19), Eq. (20) becomes

$\mathcal{R}=\zeta_{\rm M}-{16\pi G\over
3}a^{2}{\rho^{(0)}\over\left(p^{(0)}+\rho^{(0)}\right)}\Pi\;.$ (22)

Deviations from the isotropic pressure can be split in the adiabatic
perturbations (which is proportional to $c_{\rm s}^{2}$) and the intrinsic
non-adiabatic pressure perturbation101010Note that $p^{(1)}$ is the pressure
perturbation in the longitudinal gauge. Bardeen:1980kt ; Kodama:1985bj  where
$\Gamma$ is the non-adiabatic component of the equation of state and the
$\delta$ is the adiabatic component.

$p^{(1)}=p^{(0)}\left(\Gamma+{c_{\rm s}^{2}\over w}\delta\right)\;,$ (23)

and we can easily generalise the prescription made in Ref. Mukhanov:1990me ;
Mukhanov:2005sc . Indeed, with some algebra, we can obtain

$\left[\theta_{\rm M}^{2}\left({u_{\rm M}\over\theta_{\rm
M}}\right)^{\prime}+\left(8\pi G\over 3\right)^{1/2}a^{2}\theta^{2}D_{\rm
M}\right]^{\prime}=c_{\rm s}^{2}\theta_{\rm M}^{2}\nabla^{2}\left({u_{\rm
M}\over\theta_{\rm M}}\right)-{a^{2}\theta_{\rm M}^{2}\over
3{\rho^{(0)}}^{1/2}}\nabla^{2}D_{\rm M}+4\pi G{a^{3}\theta_{\rm
M}^{2}p^{(0)}\over{\rho^{(0)}}^{1/2}}\Gamma\,.$ (24)

Without $D_{M}$ and $\Gamma$ this is identical to the results of
Mukhanov:2005sc , hence this is the generalization of that calculation in
presence of pressure terms.

Using Eqs. (18), (19), (22), (21) and the gauge-invariant definition of the
Newtonian Poisson equation

$\nabla^{2}\Psi=4\pi Ga^{2}\rho^{(0)}\Delta_{\rm com}=4\pi
Ga^{2}\rho^{(1)}_{\rm com}\;,$ (25)

where $\Delta_{\rm com}$ is the density contrast and $\rho^{(1)}_{\rm com}$ is
density perturbation in the comoving orthogonal gauge, we obtain the usual
relation

${\left(p^{(0)}+\rho^{(0)}\right)\over\mathcal{H}}\mathcal{R}^{\prime}=c_{\rm
s}^{2}\,\rho^{(1)}_{\rm com}+{2\over 3}\nabla^{2}\Pi+p^{(0)}\Gamma\;.$ (26)

The time evolution of $\mathcal{R}$ in a spatially flat background is dictated
by three components: the sound speed modulated by the comoving density
perturbation, the spatial variation (second derivative) of the shear and non-
adiabatic pressure perturbations: $\mathcal{R}^{\prime}=0$ on scales and
epochs where $c_{\rm s}$, $\Gamma$ and $\nabla^{2}\Pi$ are negligible. Before
recombination this happens on scales $\gg$ 150 Mpc, after recombination
depending on the neutrino mass the scales are well above tens or few
megaparsecs (respectively).

Some other important comments are in order here:

  * •

Due to the breaking of the Single Clock Symmetry, for each point in space we
have a non zero acceleration due to $c_{s}$, $\Pi$ and/or $\Gamma$ and we
cannot build an exact observers proper reference frame described by the local
(normal) coordinate within the patch along all the observer world line.

  * •

If we wanted to build a local volume expansion it is necessary to describe the
physics inside this patch with the coordinates of an observer that are
comoving with matter. For this purpose, the comoving gauge is the most
appropriate.

In this gauge the curvature potential is $\mathcal{R}$ and the lapse
perturbation are described by the $00$ metric perturbation
$\xi=g_{00}^{(1)}|_{\rm com}$ which can be written in terms of pressure
perturbation and anisotropic stress

$\left(p^{(0)}+\rho^{(0)}\right)\xi=c_{\rm s}^{2}\,\rho^{(1)}_{\rm
com}+{2\over 3}\nabla^{2}\Pi+p^{(0)}\Gamma\;.$ (27)

Then, using Eq. (26), we have $\xi=\mathcal{R}^{\prime}/\mathcal{H}$. We note
immediately that we can follow the separate Universe prescription (e.g., see
also Hu:2016wfa ) and using the above equations we are able to compute the
curvature evolution equation within the patch. We find Hu:2016wfa

${K_{\rm patch}^{\prime}\over\mathcal{H}}=-{2\over 3}\nabla^{2}\xi=-{2\over
3\left(p^{(0)}+\rho^{(0)}\right)}\nabla^{2}\left[c_{\rm
s}^{2}\,\rho^{(1)}_{\rm com}+{2\over 3}\nabla^{2}\Pi+p^{(0)}\Gamma\right]\;.$
(28)

Note that $K_{\rm patch}$ is not conserved if there is flow of material in or
out of the patch. For example, for a patch of $5$ Mpc corresponding roughly to
the Lagrangian radius of a massive elliptical, non-linear evolution would
yield a lot of material to enter this radius through the surrounding
filaments, even for a 0p$\Lambda$CDM Universe. Therefore the curvature $K_{\rm
patch}$ is conserved if the size of the patch is large enough and/or for
$p^{(1)}_{\rm com}$ and $\Pi$ equal to zero, i.e. in the 0p$\Lambda$CDM model.
Consequently, if $K_{\rm patch}$ is not conserved the lapse perturbation is
not zero. This is another way to see that the single clock symmetry is broken.

  * •

As an example, it is useful quantify the contribution of $\Pi$. For example
the well-know effect due to neutrinos is

$D_{\rm M}=-{2\over 5}R_{\nu}\Phi=-{2R_{\nu}/5\over 1+2R_{\nu}/5}\Psi\;,$ (29)

where WMAP:2010qai

$R_{\nu}={0.2271N_{\rm eff}f\left({m_{\nu}a/T_{\nu 0}}\right)\over
1+0.2271N_{\rm eff}f\left({m_{\nu}a/T_{\nu 0}}\right)}\;.$ (30)

Then using the Newtonian Poisson equation Eq. (25) we find

${2\over 3}\nabla^{2}\Pi={1\over 3}\left({2R_{\nu}/5\over
1+2R_{\nu}/5}\right)\rho_{\rm com}^{(1)}$ (31)

and we immediately note that the the single clock is explicitly broken due to
the neutrinos

$\left(p^{(0)}+\rho^{(0)}\right)\xi=\left[c_{\rm s}^{2}+{1\over
3}\left({2R_{\nu}/5\over 1+2R_{\nu}/5}\right)\right]\rho_{\rm
com}^{(1)}+p^{(0)}\Gamma\;.$ (32)

These points have a fundamental consequence. In order to quantify the size of
the patch which preserves the curvature $K_{\rm patch}$ at a given point of
the space we have to consider the entire evolution of the separate Universe
from $a_{\rm in}$, until today. (Note that our initial condition can be chosen
so that $a_{\rm patch}|_{\rm in}=a_{\rm in}$, where $a_{\rm patch}$ is the
local scale factor of the patch local Universe, see also Hu:2016wfa ). The
patch radius $R_{\rm patch}$ enclosing constant mass has to be above the Jeans
scale along all the observer’s world line. For example, assuming massive
neutrino, we should take scales with $k<k_{\rm nr}$. Here, for individual
neutrino masses $m_{\nu}$ from $0.046$ to $0.46$ eV, the scale $k_{\rm nr}$
ranges from $2.1\times 10^{-3}h$Mpc-1 to $6.7\times 10^{-3}h$ Mpc-1
Lesgourgues:2012uu  (see also Garay et al. (2017);
TopicalConvenersKNAbazajianJECarlstromATLee:2013bxd ). Of course at
recombination the Jeans length for baryons is the sound horizon which is
comparable to the horizon. This point confirms what was previously discussed
earlier, i.e. these patches are the only places where clocks can be
synchronized, but they are the ones with the most cosmic variance.

In the next subsection we analytically calculate the corrections that break
the validity of the separate Universe prescription (see also Ref. Hu:2016wfa
). As we will see, this implies that the single clock from inflation is no
longer preserved.

####  II.1.1 Correction to linear order perturbations

Denoting again the concordance model quantities with an over-bar, we split the
linear order perturbations, e.g. let us call $C^{(1)}$, in the following way

$C^{(1)}=\bar{C}^{(1)}+\textswab{d}C^{(1)}\;,$ (33)

where

$\displaystyle\textswab{d}C^{(1)}=\int{\textswab{d}\rho^{(0)}}\left({\delta
C^{(1)}\over\delta{\textswab{d}\rho^{(0)}}}\right){\rm d}\tilde{\eta}$
$\displaystyle+$ $\displaystyle\int c_{\rm s}^{2}\left({\delta
C^{(1)}\over\delta c_{\rm s}^{2}}\right){\rm d}\tilde{\eta}$ $\displaystyle+$
$\displaystyle\int{\Gamma}\left({\delta C^{(1)}\over\delta\Gamma}\right){\rm
d}\tilde{\eta}\,{\rm d}^{3}\tilde{\bf x}+\int{\Pi}\left({\delta
C^{(1)}\over\delta{\Pi}}\right){\rm d}\tilde{\eta}\,{\rm d}^{3}\tilde{\bf
x}+{\rm...}\,.$

Here, for simplicity, we are assuming that the correction, i.e., the
contribution due to
$\\{\textswab{d}\rho^{(0)}=\rho^{(0)}-\bar{\rho}^{(0)},c_{\rm
s}^{2},\Gamma,\Pi\\}$, is only linear. In principle, it is easy generalise
this point. Immediately we note that from Eq. (26) we have
$\bar{\mathcal{R}}^{\prime}=0$,

$\displaystyle\textswab{d}\mathcal{R}=\textswab{d}\mathcal{R}^{\rm(in)}+{3\over
2}$ $\displaystyle\left({3\over 8\pi G}\right)^{1/2}\int_{\rm
in}\tilde{c}_{\rm s}^{2}\bar{\theta}_{\rm
M}(\tilde{\eta})\nabla^{2}\bar{u}_{\rm M}(\tilde{\eta},{{\bf x}})\,{\rm
d}\tilde{\eta}$ (35) $\displaystyle+\int_{\rm
in}{\mathcal{H}(\tilde{\eta})\over\left(\bar{p}^{(0)}(\tilde{\eta})+\bar{\rho}^{(0)}(\tilde{\eta})\right)}\Bigg{[}{2\over
3}\nabla^{2}\Pi(\tilde{\eta},{{\bf x}})+\bar{p}^{(0)}\Gamma(\tilde{\eta},{{\bf
x}})\Bigg{]}{\rm d}\tilde{\eta}$

and, using Eqs. (20) and (22), we obtain

$\bar{\zeta}_{\rm M}=\bar{\mathcal{R}}\quad\quad{\rm
and}\quad\quad\textswab{d}\zeta_{\rm M}=\textswab{d}\mathcal{R}-{2\over
3}{\bar{\rho}^{(0)}\over\left(\bar{p}^{(0)}+\bar{\rho}^{(0)}\right)}D_{\rm
M}=\textswab{d}\mathcal{R}+{16\pi G\over
3}\bar{a}^{2}{\bar{\rho}^{(0)}\over\left(\bar{p}^{(0)}+\bar{\rho}^{(0)}\right)}\Pi\;.$
(36)

The leading contribution to $\Pi$ depends again on the scale, the cosmic epoch
and the Universe composition. Before recombination is dominant inside the
horizon. After recombination, for massive neutrinos, free streaming is not
negligible even on linear scales; for baryons, it is important on cluster
scales and below.

We note that Eq. (36) is our main result, and at the end of this section we
compute the initial conditions. It is transparent that it is not possible to
completely eliminate the $c_{s}$ dependence from the perturbed variable,
except in the (academic) case of infinite wavelength $k=0$.

Now, splitting each term in Eq. (19), we find immediately the solution for
$\textswab{d}u_{\rm M}$:

$\displaystyle\bar{u}_{\rm M}$ $\displaystyle=$ $\displaystyle{3\over
2}\left({3\over 8\pi G}\right)^{1/2}\bar{\theta}_{\rm M}\int_{\rm
in}{\bar{\zeta}_{\rm M}(\tilde{\eta},{{\bf x}})\over\bar{\theta}^{2}_{\rm
M}(\tilde{\eta})}{\rm d}\tilde{\eta}+{\bar{\theta}\over\bar{\theta}_{\rm
M}^{\rm(in)}}{\bar{u}_{\rm M}}^{\rm(in)}\;,\quad{\rm for}~{}\Lambda{\rm
CDM~{}model,~{}~{}and}$ $\displaystyle\textswab{d}u_{\rm M}$ $\displaystyle=$
$\displaystyle{\textswab{d}\theta_{\rm M}\over\bar{\theta}_{\rm
M}}\bar{u}_{\rm M}+{3\over 2}\left({3\over 8\pi
G}\right)^{1/2}\bar{\theta}_{\rm M}\int_{\rm
in}\left(-2{{\textswab{d}\theta_{\rm M}}(\tilde{\eta})\over\bar{\theta}_{\rm
M}^{3}(\tilde{\eta})}\bar{\zeta}_{\rm M}(\tilde{\eta},{\bf
x})+{\textswab{d}\zeta_{\rm M}(\tilde{\eta},{\bf x})\over\bar{\theta}_{\rm
M}^{2}(\tilde{\eta})}\right){\rm d}\tilde{\eta}+\bar{\theta}_{\rm
M}\Bigg{(}{{\textswab{d}u_{\rm M}}^{\rm(in)}\over{\bar{\theta}_{\rm
M}}^{\rm(in)}}-{{\textswab{d}\theta_{\rm
M}}^{\rm(in)}\over\left({\bar{\theta}_{\rm
M}}^{\rm(in)}\right)^{2}}{\bar{u}_{\rm M}}^{\rm(in)}\Bigg{)}\;.$

From Eq. (18), we get

$\bar{\Psi}={\left(\bar{\rho}^{(0)}\right)^{1/2}\over\bar{a}\bar{\theta}_{\rm
M}}\bar{u}_{\rm
M}=\left(\bar{p}^{(0)}+\bar{\rho}^{(0)}\right)^{1/2}\bar{u}_{\rm
M}\quad\quad{\rm
and}\quad\quad\textswab{d}\Psi=\left(\bar{p}^{(0)}+\bar{\rho}^{(0)}\right)^{1/2}\textswab{d}u_{\rm
M}+{1\over
2}{\textswab{d}p^{(0)}+\textswab{d}\rho^{(0)}\over\left(\bar{p}^{(0)}+\bar{\rho}^{(0)}\right)^{1/2}}\bar{u}_{\rm
M}\;,$ (38)

and from Eq. (21), we obtain

$\textswab{d}\Phi=\textswab{d}\Psi+D_{\rm M}=\textswab{d}\Psi-8\pi
G\bar{a}^{2}\Pi\;.$ (39)

Finally from the Newtonian Poisson Eq. (25) we deduce

$\bar{\Delta}_{\rm com}={1\over 4\pi
G\bar{a}^{2}\bar{\rho}^{(0)}}\nabla^{2}\bar{\Psi}\quad\quad{\rm
and}\quad\quad\textswab{d}\Delta_{\rm com}={1\over 4\pi
G\bar{a}^{2}\bar{\rho}^{(0)}}\nabla^{2}\textswab{d}\Psi-\left(2{\textswab{d}a\over\bar{a}}+{\textswab{d}\rho\over\bar{\rho}^{(0)}}\right)\bar{\Delta}_{\rm
com}\;.$ (40)

We suggest to set the initial conditions by the following approach. From
$\textswab{d}a_{\rm in}=0$, we can choose
$\textswab{d}\Psi^{\rm(in)}\equiv\Psi^{\rm(in)}-\bar{\Psi}^{\rm(in)}$ where
$\Psi^{\rm(in)}=4\pi
G\bar{a}_{\rm(in)}^{2}\rho^{(0)}_{\rm(in)}\nabla^{-2}\Delta_{\rm
com}^{\rm(in)}$ and $\bar{\Psi}^{\rm(in)}=4\pi
G\bar{a}_{\rm(in)}^{2}\bar{\rho}^{(0)}_{\rm(in)}\nabla^{-2}\bar{\Delta}_{\rm
com}^{\rm(in)}$. Using these relations and Eqs. (36)-(39), we get immediately
$\textswab{d}u_{\rm M}^{\rm(in)}$, $\textswab{d}\zeta_{\rm M}^{\rm(in)}$ and
$\textswab{d}\mathcal{R}^{\rm(in)}$.

##  III Conclusions

We have explicitly computed, for the first time, at the perturbation level,
the effect of pressure from baryons, radiation, neutrinos and any general
equation of state with sound speed $c_{s}$ different from zero on the
perturbations that arise from single-field inflation which include (small)
non-linearities in the metric. We have shown that $c_{s},\Pi,\Gamma$ appear in
the resulting perturbations thus, on scales/epochs when they are non zero
these terms induce breaking the single clock symmetry from single field
inflation.

The claims of Ref. Tanaka_2011 ; dePutter:2015vga ; Pajer:2013ana ;
Cabass:2016cgp  translated in asserting that one could synchronize (observers)
clocks to the inflaton one in patches of arbitrary size in the Universe. This
in itself is very counter intuitive as General Relativity forbids this in the
presence of curvature (gravity) as explicitely shown in Ref. Rollo . What we
have shown here is that, in addition, the presence of pressure also breaks the
synchronization of clocks in patches smaller than the Jeans length. These
patches can be very large depending on the time chosen to set the initial
conditions. As a general rule of thumb, patches must be larger than those
where baryonic acoustic oscillations are measurable, which translates on about
$150$ comoving Mpc for initial conditions set before recombination. For
initial conditions set after recombination, the patches must be much larger
than tens of Mpcs and larger than neutrinos free streaming length in the
presence of massive neutrinos. This impossibility to synchronize clocks across
separate patches opens up the possibility to measure the primordial non-
Gaussianity from single-field inflation of the local shape which include a
term $\sim$ $n_{s}-1$ and the General Relativity $\sim-5/3$ term arising from
modifications of the Poisson equation at the horizon scale. Current and
ongoing surveys like DESI, Spherex and LSST are forecasted to have enough
statistical power to detect a non-gaussian contribution of the local type of
an amplitude $f_{\rm NL}^{\rm loc}\sim 1$ Carbone ; Karagiannis  making them
well suited to access the inflationary signature.

###### Acknowledgements.

DB and SM acknowledge partial financial support by ASI Grant No. 2016-24-H.0.
Funding for this work was partially provided by project PGC2018-098866- B-I00
MCIN/AEI/10.13039/501100011033 y FEDER “Una manera de hacer Europa”, and the
“Center of Excellence Maria de Maeztu 2020-2023” award to the ICCUB (CEX2019-
000918-M funded by MCIN/AEI/10.13039/501100011033). LV acknowledges support
from the European Union Horizon 2020 research and innovation program ERC
(BePreSySe, grant agreement 725327).

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