2022

[1]Celia Escamilla-Rivera

[1]Aldo Gamboa

1]Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Circuito Exterior C.U., 04510, México D.F., México

# A New Parameterized Interacting Holographic Dark Energy

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###### Abstract

We study a cosmological model based on the holographic principle that allows
an interaction between dark energy and dark matter with a Hubble infrared
cutoff. We adopt an agnostic point of view with respect to the form of the
interaction between these dark components by using redshift parameterizations
of the dark energy equation of state (DE EoS). Following this approach, we
obtain an analytical expression of the Hubble parameter and we test it with
four different parameterizations of the DE EoS using currently Pantheon
supernovae, Cosmic Chronometers and Galaxy Clustering data sets. Our analyses
show that the considered parameterizations on this agnostic proposal allow a
late accelerated cosmic expansion in comparison to the results from previous
works where specific forms of the interaction are employed but fail to explain
this behaviour. Furthermore, we find that our proposal solves the so-called
coincidence problem in Cosmology and in this direction shed some light on the
problem of the Hubble tension.

###### keywords:

Dark Energy, Holographic Principle, Dark Matter, Cosmology

##  1 Introduction

The discovery of the accelerated expansion of the Universe
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns  gave rise to
one of the most fundamental problems in physics: the nature of Dark Energy
(DE). The standard pivot model, the concordance model $\Lambda$CDM, introduces
a cosmological constant as the source of this dark component, whose equation
of state (EoS) is $w_{\text{de}}=p_{\text{de}}/\rho_{\text{de}}=-1$, where
$p_{\text{de}}$ and $\rho_{\text{de}}$ are the pressure and energy density of
DE, respectively, reproducing the observed repulsive force needed to explain
the accelerated cosmic expansion. So far, this vanilla model has the best
performance fitting presently observational data Perivolaropoulos:2021jda .
However, the nature of one of the main characters of this concordance model,
the cosmological constant $\Lambda$, has not been clarified yet. In
particular, if we attempt to identify the DE density with the predicted
quantum vacuum energy, then a difference in several orders of magnitude arises
between the inferred and predicted values weinberg ; carroll ; Rugh:2000ji ;
Stenflo:2020zsc . Additionally, within the $\Lambda$CDM model, the DE density
$\rho_{\text{de},0}$ and Dark Matter (DM) density $\rho_{\text{dm},0}$ at the
present epoch $z=0$, are of the same order of magnitude,
$\rho_{\text{de},0}/\rho_{\text{dm},0}\sim\mathcal{O}(1)$, which seems to
indicate that we are living in a special period of the Universe since these
densities evolve differently over time. Nevertheless, this fact poses a
problem because it requires very specific initial conditions in the early
Universe. This is the so-called coincidence problem Velten:2014nra ;
Cunillera:2021izz . Our uncertainty about the nature of the cosmological
constant and the associated theoretical problems have motivated an extensive
study of alternative DE models Yang:2021hxg ; Schoneberg:2021qvd ;
DiValentino:2019ffd .

A particularly interesting model which couples the features of DE with quantum
gravity is the holographic dark energy (HDE). As the name suggests, this model
is based on the holographic principle (HP), which is an expected property of
any quantum gravity theory hp_intro . This HDE model has been successfully
applied to several fields of physics, e.g in nuclear physics Liu_2007 ,
condensed matter Hartnoll:2009sz  and cosmology Strominger:2001pn . According
to the HP, the number of degrees of freedom in a physical system scales with
the area of its boundary bekenstein ; tHooft:1993dmi ; Susskind:1994vu ; hp .
In this context, in Cohen_hde  was suggested that, for an effective quantum
field theory in a box of size $L$, its ultra-violet cutoff is related to the
infrared (IR) cutoff due to a limit set by the formation of a black hole,
which results in an upper bound to the zero-point energy density of the box.
Motivated by the cosmological constant problem, in original_hde  these results
were applied to the Universe to obtain an expression for the DE density which
saturates the mentioned upper limit of avoiding the collapse into a black
hole. The resulting expression is

$\rho_{\text{de}}=3C^{2}M_{\text{Pl}}^{2}L^{-2},$ (1)

where $C$ is a constant, $M_{\text{Pl}}=\sqrt{1/8\pi}$ is the reduced Planck
mass111In this paper we will use geometrical units so that $c=1$ and $G=1$,
unless otherwise specified. and $L$ is the IR cutoff which represents a
characteristic length scale of the Universe. The problem of DE now reduces to
find $C$ and to choose an appropriate IR cutoff. The first natural choice for
$L$ is the inverse of the Hubble parameter, $L=H^{-1}$ (the so-called Hubble
IR cutoff). Nevertheless, this choice implies that $w_{\text{de}}=0$, which is
an EoS that does not allow an accelerated expansion of the Universe within the
HDE model HSU200413 ; original_hde . Another choice for the length scale is
the future event horizon original_hde ,
$L=a\int_{t^{\prime}}^{\infty}dt^{\prime}/a(t^{\prime})$, where $a$ is the
scale factor and $t^{\prime}$ is the cosmic time. With this choice, the value
of the constant $C$ in (1), must be estimated with observations. Overall,
depending on the choice of the IR cutoff, the HDE proposal gives a very
competitive cosmological model. For example, within this framework, the
presence of curvature in the Universe has been analysed Huang_2004 , different
bounds for the neutrinos masses have been estimated zhang_neutrino ;
wang_neutrino , the inflationary period has been studied CHEN2007256  and also
several modified theories of gravity have been explored review_hde .

Another issue analysed using the HDE model is the Hubble constant tension, an
important problem in cosmology that arises due to the discrepancy between the
estimated values of the Hubble constant, $H_{0}$, at the early and late
Universe epochs DiValentino:2020zio ; Riess:2021jrx . The measurements of the
cosmic microwave background (CMB) radiation from Planck satellite
Planck:2018vyg  give the value $H_{0}=67.4\pm 0.5{\rm\,km\,s^{-1}\,Mpc^{-1}}$,
whereas a combination of different local measurements Verde:2019ivm  yields a
representative value of $H_{0}=73.1\pm 0.9{\rm\,km\,s^{-1}\,Mpc^{-1}}$, which
is $5.7\sigma$ discrepant from the Planck result. Recently, the Pantheon
collaboration has reanalyzed the calibration used for the photometry of the
supernovae type Ia sample, including the calibration used to obtain the SH0ES
results Brout:2021mpj ; Scolnic:2021amr , adding a new value from
Pantheon$+$Cepheid-Supernovae Ia baseline $H_{0}=73.04\pm
1.04{\rm\,km\,s^{-1}\,Mpc^{-1}}$ Riess:2021jrx . The contribution to the
Hubble constant from the systematic supernovae calibration is smaller than
$\sim 0.2{\rm\,km\,s^{-1}\,Mpc^{-1}}$, so that the tension cannot be
attributed to this type of systematics.

Several mechanisms and explanations have been proposed in order to alleviate
the Hubble tension DiValentino:2021izs ; DiValentino:2020zio . In particular,
in h0_HDE  it was shown that, within the HDE model with the future event
horizon as the IR cutoff, and depending on the value of $C$, the EoS of HDE
satisfies $w_{\text{de}}>-1$ at an early epoch ($z\gtrsim 1$), and
$w_{\text{de}}<-1$ at a later epoch ($z\lesssim 1$). Therefore, the Universe
has a smaller acceleration earlier on and faster acceleration afterwards. This
delayed acceleration successfully alleviates the Hubble tension because the
value of $H_{0}$ estimated from local measurements is higher than the value
calculated with the observations of the CMB.

Another fundamental problem in cosmology is the nature of DM dm_review ;
Salucci:2018hqu . There are several cosmological and astrophysical scenarios
whose theoretical explanations require the presence of this exotic entity in
order to fit the observational data. For example, the measurements of galaxy
rotation curves Brownstein_2006 , the dispersion velocities of galaxies in
clusters Faltenbacher:2006rb , the presence of gravitational lensing
Massey:2010hh , large scale structure formation Primack:1997av  and the
distribution of anisotropies in the CMB Planck:2018vyg . However, despite the
big theoretical and experimental efforts, we still do not arrive to a unique
agreement on the physics of DM, and its unknown nature has led us to explore
several possible candidates and exotic models Bergstrom:2009ib .

Among these models, exists the possibility that DM is coupled to DE. These
proposals are generically known as interacting DE (IDE) models Wang_2016 ;
DiValentino:2019ffd . Usually, this coupling is realised through an
interaction term $Q$ included in the continuity equations of DE and DM [see
Eqs. (3) and (4)]. Several specific forms of $Q$ have been considered in the
literature, but in general it is assumed to be a function of
$\rho_{\text{de}}$ and $\rho_{\text{dm}}$. The three most widely used options
are $Q=\Gamma_{1}H\rho_{\text{de}}$, $Q=\Gamma_{2}H\rho_{\text{dm}}$ and
$Q=\Gamma_{1}H\rho_{\text{de}}+\Gamma_{2}H\rho_{\text{dm}}$, where $H$ is the
Hubble parameter and $\Gamma_{1}$, $\Gamma_{2}$ are constants, although there
have been considered more intricate forms of the interaction delCampoo ;
Wang_2016 . However, an important theoretical caveat to these models is that
the interaction term is introduced in a phenomenological way, i.e. it is not
directly derived from an action, making the choice of the form of $Q$ somewhat
arbitrary. The coupling of DE and DM has already been studied in the context
of the HDE Wang:2005jx ; PAVON2005206 ; Ma:2007pd ; Zadeh:2016vgc ; delcampo ,
however, the specific couplings in these interacting HDE (IHDE) models do not
come from an action principle either. In particular, depending on the choice
of the interaction term and the IR cutoff, we could have a universe with
Nayak:2019njd  or without Zadeh:2016vgc  accelerated expansion. Therefore, a
more general framework is needed in order to address the viability of the IHDE
model without depending on the explicit form of the interaction.

In this paper, we present a generic treatment of the coupling between DE and
DM in the IHDE model with a Hubble IR cutoff. As we mentioned earlier, if
there is no coupling with DM, then this cutoff choice is discarded because it
yields a wrong DE EoS. As we will show, the interaction allows us to
reconsider this IR cutoff because it predicts an EoS which indeed allows an
accelerated expansion. This treatment is achieved by quantifying the
interaction $Q$ through a DE EoS redshift parameterization, instead of
choosing a specific coupling of the dark sector. We demonstrate the viability
of this parameterized IHDE model by doing statistical analyses with
observational data for various DE EoS parameterizations.

This paper is organised as follows: in Sec. 2, we present the cosmological
equations used for the parameterized IHDE scenario and we use them with the
Hubble IR cutoff to obtain an analytical expression for the Hubble parameter.
In Sec. 3, we use the data from certain cosmological observations and various
DE EoS parameterizations to study the consequences of the IHDE model in a
Universe dominated by matter and DE. Finally, in Sec. 4 we present our
conclusions and suggestions for future work.

##  2 A New Interacting Holographic Dark Energy

According to the IHDE model, the cosmological evolution is determined by:

  1. 1.

the DE density obtained from the HP with a certain choice of the IR cutoff,

  2. 2.

the standard Friedmann equation,

  3. 3.

and the coupling term for DM and DE densities.

Combining these three ingredients, we can obtain an analytical or numerical
solution of the Hubble parameter, which will allow us to calculate $H_{0}$
with standard parameter statistics estimation techniques.

We start with the Friedmann equation,

$3M_{\text{Pl}}^{2}H^{2}=\sum_{j}\rho_{j},$ (2)

written in terms of the Hubble parameter $H\coloneqq\dot{a}/a$, where $a$ is
the scale factor and the dot denote derivatives with respect to cosmic time
$t$, and the energy densities of radiation, baryonic matter, cold DM, DE and
curvature $k$, denoted collectively as $\rho_{j}$, where
$j=\\{\text{r},\text{b},\text{dm},\text{de},k\\}$, respectively, with
$\rho_{k}=-3M_{\text{Pl}}^{2}k/a^{2}$.

The radiation and baryon densities have the standard continuity equations. In
contrast, the modified version of the continuity equations for DE and DM are

$\displaystyle\dot{\rho}_{\text{de}}+3H\rho_{\text{de}}(1+w_{\text{de}})=Q,$
(3) $\displaystyle\dot{\rho}_{\text{dm}}+3H\rho_{\text{dm}}=-Q,$ (4)

where $Q$ is the interaction term which allows an energy transfer between the
two dark components. The sign of $Q$ is chosen so that if $Q>0$ then DM is
transferring energy to DE, and otherwise for $Q<0$.

Differentiating Eq. (2) with respect to the redshift $z=a^{-1}-1$ (so that
$\frac{d}{dt}=-(1+z)H\frac{d}{dz}$) we get an equation relating the
derivatives of the energy densities. Therefore, substituting Eqs. (3) and (4),
and the standard continuity equations for all the other components, we obtain
the evolution of the Hubble parameter as

$H^{\prime}(z)=\frac{3H}{2(1+z)}+\frac{\rho_{\text{r}}-\rho_{k}+3\rho_{\text{de}}w_{\text{de}}}{6M_{\text{Pl}}^{2}(1+z)H},$
(5)

where the prime denotes derivative with respect to $z$. Notice that the
interaction between DE and DM is implicitly contained in the EoS
$w_{\text{de}}$ [see Eq. (9)].

We will choose the Hubble horizon as the IR cutoff, $L=H^{-1}$. We start
evaluating Eq. (1) at present time $t_{0}$. Using the definition of the
fractional densities $\Omega_{j}\coloneqq\rho_{j}/3H^{2}M_{\text{Pl}}^{2}$, we
find

$C^{2}=\Omega_{\text{de},0}=\Omega_{\text{de}},$ (6)

where the subscript $0$ denotes evaluation at time $t_{0}$. Therefore, with
the Hubble IR cutoff, the value of $C$ comes directly from the theory (unlike
the future event horizon IR cutoff which requires that $C$ must be inferred
from cosmological observations). Remarkably, the fractional DE density will
remain the same during the evolution of the Universe. Next, from Eqs. (1) and
(2), we have

$\sum_{j}\rho_{j}=\frac{\rho_{\text{de}}}{\Omega_{\text{de},0}},$ (7)

so that

$\frac{\Omega_{\text{r}}+\Omega_{\text{b}}+\Omega_{\text{dm}}+\Omega_{k}}{\Omega_{\text{de}}}=\frac{1-\Omega_{\text{de},0}}{\Omega_{\text{de},0}}=\text{constant}.$
(8)

Notice that the coincidence problem is solved because the fractional densities
in this model must satisfy a constant constriction. In particular, for a DE
and DM dominated Universe, we have
$\Omega_{\text{dm}}/\Omega_{\text{de}}\approx\text{constant}$. The fact that
the ratio of DM and DE densities is a constant in an IHDE model with
$L=H^{-1}$ was previously identified in PAVON2005206 ; delcampo , nevertheless
in our work we unambiguously identify this constant and generalise it to
include other energy components.

To obtain an expression that relates the interaction term $Q$ with the DE EoS
$w_{\text{de}}$, we differentiate Eq. (7) and substitute the continuity
equations for all the components. After a straightforward calculation we find

$\frac{Q}{H}=3\rho_{\text{de}}w_{\text{de}}(1-\Omega_{\text{de},0})+(\rho_{k}-\rho_{\text{r}})\Omega_{\text{de},0}.$
(9)

Notice that this expression can be rewritten as
$Q\propto\Gamma\rho_{\text{de}}$, (with $\Gamma$ an interaction rate with
units of the Hubble constant; in our case $\Gamma=H$), which has been the
specific form of the interaction in some previous works delcampo ; Ma:2007pd ;
review_hde ; PAVON2005206 ; Nayak:2019njd ; Wang_2016 . Furthermore, if $Q=0$,
then from Eq. (9) we obtain

$w_{\text{de}}=\frac{\rho_{\text{r}}-\rho_{k}}{3\rho_{\text{de}}}\frac{\Omega_{\text{de},0}}{1-\Omega_{\text{de},0}}.$
(10)

In this latter expression we recognise the known fact that a non-interacting
dark sector ($Q=0$) in a flat late Universe yields an incorrect DE EoS
($w_{\text{de}}=0$) when the HP is applied with $L=H^{-1}$ original_hde .
Therefore, the presence of $Q$ allow us to reconsider this case.

Using Eq. (1), the evolution equation for the Hubble parameter (5) changes as

$H^{\prime}(z)=\frac{3H}{2}\frac{1+\Omega_{\text{de},0}\,w_{\text{de}}}{1+z}+\frac{\rho_{\text{r}}-\rho_{k}}{6M_{\text{Pl}}^{2}(1+z)H}.$
(11)

To overcome the lack of enlightenment about the exact form of the interaction
term (or, equivalently, the form of $w_{\text{de}}$), we parameterize the DE
EoS in terms of the redshift $z$ to account for the effects of $Q$. Then,
multiplying Eq. (11) by $H$, we obtain a first order linear ordinary
differential equation for $H^{2}$. The corresponding analytical solution is

$\frac{H(z)}{H_{0}}=\alpha(z)^{-\frac{1}{2}}\sqrt{1+\int_{0}^{z}\alpha(z^{\prime})\left[\Omega_{\text{r},0}(1+z^{\prime})^{3}+\frac{k(1+z^{\prime})}{H_{0}^{2}}\right]dz^{\prime}},$
(12)

where we have used that $\Omega_{\text{r}}=\Omega_{\text{r},0}(1+z)^{4}$,
$\rho_{k}=-3M_{\text{Pl}}^{2}k(1+z)^{2}$ and we have defined

$\alpha(z)\coloneqq\exp\left(-3\int_{0}^{z}\frac{1+\Omega_{\text{de},0}\,w_{\text{de}}(z^{\prime})}{1+z^{\prime}}\,dz^{\prime}\right).$
(13)

Therefore, with Eq. (12) and a given parameterization of the DE EoS, we can
calculate the Hubble parameter for the IHDE model with a Hubble IR cutoff in
the presence of radiation, baryonic matter, CDM, DE and curvature.

On the other hand, if we choose to parameterize directly the interaction term
$Q=Q(z)$, then one must solve Eq. (9) for $w_{\text{de}}$ in terms of $Q$, and
then substitute back into Eq. (11). Doing this, we obtain the differential
equation

$\frac{1}{3}\left(H^{3}\right)^{\prime}=\frac{3H^{3}}{2(1+z)}+\frac{Q(z)+H(\rho_{\text{r}}-\rho_{k})}{6M_{\text{Pl}}^{2}(1+z)(1-\Omega_{\text{de},0})},$
(14)

which does not have an analytical solution unless we consider
$\rho_{\text{r}}=\rho_{k}=0$. However, since it is common to parameterize the
DE EoS, we will use Eq. (12) throughout this work.

Finally, as a special case, the $\Lambda$CDM limit can be recovered by
equating the expression in (12) with the Hubble parameter $H_{\Lambda}$ of the
$\Lambda$CDM model, which is given by

$\frac{H_{\Lambda}(z)}{H_{0}}\coloneqq
E_{\Lambda}(z)=\sqrt{\Omega_{\text{r},0}(1+z)^{4}+\Omega_{\text{m},0}(1+z)^{3}+\Omega_{k,0}(1+z)^{2}+\Omega_{\text{de},0}}\,,$
(15)

where $\Omega_{\text{m},0}\coloneqq\Omega_{\text{b},0}+\Omega_{\text{dm},0}$.
Performing this, we obtain an expression for $w_{\text{de}}$, which we denote
as $w_{\text{de},\Lambda}$, and is given by

$w_{\text{de},\Lambda}(z)\coloneqq-\frac{1}{E_{\Lambda}(z)^{2}}.$ (16)

Thus, by construction, this EoS give us the Hubble parameter of $\Lambda$CDM
model within the framework of IHDE with $L=H^{-1}$.

##  3 Observational test on the New Interacting Holographic Dark Energy

In order to test the cosmological viability of this parameterized model of DE,
we will use current late time observations of the Universe to see if this
model can reproduce an accelerated expansion. As we mentioned, the Hubble IR
cutoff was discarded in the non-interacting case since it fails to reproduce
an accelerated expansion. Nevertheless, a parameterized interacting dark
sector allows us to reconsider the simple Hubble IR cutoff since it yields an
optimal data fitting behaviour.

For the following analysis, we will assume a flat Universe dominated by matter
(baryonic + CDM) and DE. Thus, $k=0$ and $\Omega_{\text{r},0}=0$, and Eq. (12)
takes the form

$\frac{H(z)}{H_{0}}=\alpha(z)^{-\frac{1}{2}}=\exp\left(\frac{3}{2}\int_{0}^{z}\frac{1+\Omega_{\text{de},0}\,w_{\text{de}}(z^{\prime})}{1+z^{\prime}}dz^{\prime}\right).$
(17)

Moreover, from Eq. (9), we obtain an expression for the interaction term

$Q=9M_{\text{Pl}}^{2}H^{3}w_{\text{de}}(z)\,\Omega_{\text{m},0}\Omega_{\text{de},0},$
(18)

where
$\Omega_{\text{m},0}=\Omega_{\text{b},0}+\Omega_{\text{dm},0}=1-\Omega_{\text{de},0}$.

###  3.1 Data analysis

For the statistical analysis, we perform a Markov chain Monte Carlo (MCMC)
Bayesian inference procedure with a modification of the Python implementation
emcee emcee . The likelihood function of the parameter set $\theta$ and their
best-fit values, will be determined with the $\chi^{2}$ statistics.

Observations indicate that the DE domination of the Universe and its
accelerated expansion is a recent phenomenon in a cosmological time scale
Sami:2013ssa . Therefore, if we want to test the capabilities of the
parameterized IHDE model to reproduce this behaviour, then we must use low
redshift data. In this way, we will use supernovae Type Ia (SNe Ia) data as
standarisable candles (objects whose intrinsic luminosity is well determined),
and observational Hubble data (OHD) composed by cosmic chronometers (CC) and
baryon acoustic oscillations (BAO) measurements as standard rulers (objects
whose comoving size is well determined). In what follows, we describe our
database employed.

####  3.1.1 Pantheon SNe Ia data set

We use the Pantheon SNe Ia compilation Pan-STARRS1:2017jku  which contains
1048 data points distributed over the redshift interval $0.01<z<2.26$.

To perform the statistical analysis of the Pantheon data set, we employ the
theoretical expression of the distance modulus

$\mu_{\text{th}}(z)=5\log_{10}\left(\frac{(1+z)c}{h}\int_{0}^{z}\frac{1}{E(z^{\prime})}\,dz^{\prime}\right)+\mu_{0},$
(19)

where $c$ is the speed of light, $E(z)\equiv H(z)/H_{0}$ is the normalized
Hubble parameter and $h\equiv H_{0}/(100\text{ km s}^{-1}\,\text{Mpc}^{-1})$
is the reduced Hubble constant. For our analysis, we allow the nuisance
parameter of the distance estimator $\mu_{0}$ be fixed according to the prior
$H_{0}=73.24\pm 1.74\text{ km s}^{-1}\,\text{Mpc}^{-1}$ Riess:2016jrr  and the
cosmological density matter $\Omega_{\text{m}}$ to vary.

The $\chi^{2}$ function for the best-fit parameter estimation is

$\chi_{\mathrm{Pantheon}}^{2}(\mathbf{\theta})=\sum_{i=1}^{1048}\frac{(\mu_{\text{obs}}(z_{i})-\mu_{\text{th}}(z_{i}))^{2}}{\sigma_{\mu,i}^{2}},$
(20)

where $\sigma_{\mu,i}$ are the associated measurement errors,
$\mu_{\text{obs}}$ is the observed distance modulus and $\mu_{\text{th}}$ is
given by Eq. (19).

####  3.1.2 Observational Hubble data set

We employ a compilation of 31 CC $H(z)$ data points Stern_2010 ; Moresco_2012
; Moresco_2016 ; 10.1093/mnrasl/slv037 ; Zhang_2014 ; 10.1093/mnras/stx301
measured with the differential age method proposed in Jimenez_2002 , and 20
galaxy clustering (GC) $H(z)$ data points obtained from baryon acoustic
oscillations (BAO) measurements under a $\Lambda$CDM prior bao1 ; bao2 ; bao3
; bao4 ; bao5 ; bao6 ; bao7 ; bao8 ; bao9 ; bao10 . The entire OHD sample
(CC+GC) consists of 51 data points and it is distributed over the redshift
interval $0.07<z<2.36$, and it is shown explicitly in CC_GC . We also follow
the assumption of CC_GC  that the BAO data is constructed with independent
measurements (not correlated).

The $\chi^{2}$ for the OHD is given by

$\chi_{\mathrm{OHD}}^{2}(\mathbf{\theta})=\sum_{i=1}^{51}\frac{(H_{\text{obs}}(z_{i})-H_{\text{th}}(z_{i}))^{2}}{\sigma_{H,i}^{2}},$
(21)

where $\sigma_{H,i}$ are the associated measurement errors, $H_{\text{obs}}$
is the measured Hubble parameter and $H_{\text{th}}$ is the theoretical Hubble
parameter of each considered model.

###  3.2 Statistical Analysis

In the analysis of the parameterized IHDE model, we employ the sets “Pantheon”
and “Pantheon+OHD”, whose $\chi^{2}$ functions are given by

$\displaystyle\chi^{2}=\left\\{\begin{array}[]{ll}\chi_{\mathrm{Pantheon}}^{2},&\\\
\chi_{\mathrm{Pantheon}}^{2}+\chi_{\mathrm{OHD}}^{2}.&\end{array}\right.$ (24)

Furthermore, we use the gaussian log-likelihood function for the data set
$\mathbf{d}$ with measurement errors $\sigma_{\mathbf{d}}$ given the parameter
set $\theta$,

$\log{\mathcal{L}}(\mathbf{d\mid\theta})=-\frac{1}{2}\chi^{2}+\log(\sigma_{\mathbf{d}}),$
(25)

whose maximization gives the values of the best-fit parameters.

The different models for the statistical analysis are constructed with the
following DE EoS (see Table 1):

  * •

$w_{0}$: a constant EoS.

  * •

$w_{\text{CPL}}$: Chevallier-Polarski-Linder Chevallier:2000qy ; linder .

  * •

$w_{\text{BA}}$: Barboza-Alcaniz Barboza:2008rh .

  * •

$w_{\Lambda}$: an EoS to measure deviations from $\Lambda$CDM within the IHDE
model, inspired by Eq. (16).

The constant parameterization is used as a first ansatz due to its simplicity.
The CPL and BA parameterizations are used because their two parameters contain
information about the present value of the DE EoS and its overall time
evolution Escamilla-Rivera:2016qwv . The associated Hubble parameters of each
DE EoS are calculated with Eq. (17) and are also shown in Table 1.

Equation of state $w_{\text{de}}$ | Dimensionless Hubble parameter $H(z)/H_{0}$
---|---
$w_{0}$ | $(1+z)^{\frac{3}{2}(1+\Omega_{\text{de},0}w_{0})}$
$w_{\text{CPL}}=w_{0}+\frac{z}{1+z}w_{1}$ | $(1+z)^{\frac{3}{2}[1+\Omega_{\text{de},0}(w_{0}+w_{1})]}\exp\left[-\frac{3}{2}\frac{\Omega_{\text{de},0}\,w_{1}\,z}{1+z}\right]$
$w_{\text{BA}}=w_{0}+\frac{z(1+z)}{1+z^{2}}w_{1}$ | $(1+z)^{\frac{3}{2}(1+\Omega_{\text{de},0}w_{0})}(1+z^{2})^{\frac{3}{4}\Omega_{\text{de},0}w_{1}}$
$w_{\Lambda}=\frac{w_{0}-w_{1}}{\Omega_{\text{m},0}(1+z)^{3}+\Omega_{\text{de},0}}+w_{1}$ | $(1+z)^{\frac{3}{2}(1+w_{0}-\Omega_{\text{m},0}w_{1})}[\Omega_{\text{m},0}(1+z)^{3}+\Omega_{\text{de},0}]^{\frac{w_{1}-w_{0}}{2}}$

Table 1: DE EoS used in our analysis and their corresponding dimensionless
Hubble parameters calculated with Eq. (17).

Our best fit values of the parameters are shown in Table 2 of Appendix A. Note
that in Eq. (17) the parameter $\Omega_{\text{de},0}$ and $w_{\text{de}}$
appear just one time in the form of a product. Therefore, the parameters of
conventional EoS parameterization will have a degeneracy with
$\Omega_{\text{de},0}$ that cannot be disentangled with the parameter
inference procedure (this degeneracy can be seen explicitly in the Hubble
parameters of $w_{0}$, $w_{\text{CPL}}$ and $w_{\text{BA}}$ in Table 1). For
this reason, we gave the best fit values of the products
$\Omega_{\text{de},0}w_{0}$ and $\Omega_{\text{de},0}w_{1}$ in Table 2. This
is not the case for the proposed EoS $w_{\Lambda}$, since it has a dependency
on $\Omega_{\text{de},0}$ which breaks the degeneracy, as can be seen in its
Hubble parameter and in our results given in Table 2.

In Figures 1 and 2 we show the modulus distance and the Hubble parameter for
the four selected parameterizations, respectively. As we can notice, the
evolution of the modulus distance of the four parameterizations follow the
same trend of the supernovae data. Nevertheless, in Figure 2 we can see that
the constant parameterization yields a fitting curve that deviates
significantly from the trend of the OHD, therefore we will exclude this
parameterization from next analyses. On the other hand, the complexity of the
other EoS produce qualitatively better fitting curves. Therefore, our analysis
will only use the CPL, BA and $w_{\Lambda}$ parameterizations.

Figure 1: Supernovae data with Pantheon R19 calibration (gray points) and the
best-fit curves of the modulus distance $\mu$ as a function of the redshift
$z$ [Eq. (19)] for the four parameterizations analyzed (see Table 1), each
color representing a different parameterization. The continuous (dashed)
curves are calculated with a Bayesian analysis using the Pantheon
(Pantheon+OHD) data. Figure 2: Observational Hubble data (gray points) and the
best-fit curves of the Hubble parameter $H$ as a function of the redshift $z$
for the four parameterizations analyzed (see Table 1), each color representing
a different parameterization. The continuous (dashed) curves are calculated
with a Bayesian analysis using the Pantheon (Pantheon+OHD) data.

In Figure 3 we present the behaviour of $w_{\text{CPL}}$, $w_{\text{BA}}$ and
$w_{\Lambda}$ multiplied by $\Omega_{\text{de},0}$ (to deal with the
degeneracy in the $w_{\text{CPL}}$ and $w_{\text{BA}}$ parameterizations). We
notice that all the analyzed parameterizations allow an accelerated expansion
of the Universe. However, the behaviour of $w_{\text{de}}$ changes allowing a
positive value of $w_{\text{de}}$ for certain parameterizations at higher
redshifts. Note that for the $w_{\Lambda}$ parameterization (where there is no
degeneracy in its parameters) with the Pantheon$+$OHD data, we have
$w_{\text{de}}(0)=-1.70\pm 0.12$ (see Table 2), which is relatively higher in
comparison to the $\Lambda$CDM EoS $w_{\text{de}}=-1$. We expect a similar
result in the CPL and BA parameterization (where there is a degeneracy): we
have $\Omega_{\text{de},0}w_{\text{de}}(0)\sim-0.8$, but if
$\Omega_{\text{de},0}\sim 0.7$, then $w_{\text{de}}(0)\lesssim-1$. Therefore,
the considered DE EoS induce a delayed acceleration, causing a smaller (or
null) acceleration earlier on, and a faster acceleration at late periods,
compared with the $\Lambda$CDM model.

Figure 4 describes the rate behaviour $Q/H$ (multiplied by a constant). This
term is a measure of the interaction compared with the rate of the expansion
of the Universe. Since $Q/H<0$ today and $H>0$, then $Q<0$. Therefore, from
Eq. (3), we can conclude that DM is receiving energy from DE at these late
stages of the Universe. Nevertheless, depending on the EoS, there may have
been a time where the opposite transformation of energy happened ($Q>0$).
Overall, the sign of $w_{\text{de}}$ determines the sign of $Q$, as can be
seen from the plots and from Eq. (18).

Finally, in Figure 5 we show the marginalized distributions of the reduced
Hubble constant $h$. For comparison, we have included the benchmark
$\Lambda$CDM model (15) with $\Omega_{\text{r},0}=0$ and $k=0$. First, we
notice that the inclusion of the OHD moves the posteriors towards smaller
values of $h$ as a consequence of the functional form of the Hubble parameters
obtained from the three parameterizations employed (see Table 1): in order to
fit the data in Figure 2, the right end of the continuous curves must be moved
to produce the dashed lines, moreover, with the cosmic evolution, the left end
(which determines $H_{0}$) is also affected. Other EoS parameterizations may
introduce a different behaviour. Despite this shift towards smaller values of
$h$, the CPL and $w_{\Lambda}$ parameterizations give a slightly higher value
of $h$ with respect to the matter-DE dominated $\Lambda$CDM universe.
Nevertheless, this change alone is not enough to give a value of $h$
consistent with late Universe observations, which generally estimate a value
of $h\gtrsim 0.73$ Verde:2019ivm . We expect that information about the DE EoS
at higher redshifts (or, equivalently through Eq. (14), information about the
interaction term at higher $z$) would be needed to increase the value of $h$
obtained from the statistical analysis of the parameterized IHDE model.
Indeed, early Universe observations (high and median redshifts of the CMB and
galaxy BAO data) were needed in the HDE model of h0_HDE  (with the future
event horizon as the IR cutoff) to alleviate the Hubble tension with a delayed
acceleration, which we expect is the same mechanism operating in this IHDE
model (see Figure 3). The aforementioned information about the DE EoS could be
obtained, for example, with the cosmography approach which allows to
reconstruct the kinematic evolution of the Universe acceleration at higher
redshifts (up to $z\sim 10$ Escamilla-Rivera:2021vyw ) without assuming a
cosmological model Escamilla-Rivera:2019aol ; Capozziello:2013wha ;
Bolotin:2018xtq .

Figure 3: The product $\Omega_{\text{de},0}w_{\text{de}}$ as a function of the
redshift $z$ for the four parameterizations analysed (see Table 1), each color
representing a different DE parameterization. The continuous (dashed) curves
are calculated with the Pantheon (Pantheon+OHD) data. The vertical dashed line
corresponds to $z=2.36$ which is the highest redshift in the data; the
information beyond this line is an extrapolation of the observational data to
show the expected behaviour of the DE EoS at higher redshifts in each model.
Figure 4: Rate of the interaction term $Q$ to the Hubble parameter $H$ as a
function of the redshift $z$ [Eq. (18)] for the four parameterizations
analysed (see Table 1), each color representing a different parameterization:
CPL (red), BA (yellow) and $w_{\Lambda}$ (blue). The continuous (dashed)
curves are calculated with the Pantheon (Pantheon+OHD) data. Figure 5:
Posterior distributions of the reduced Hubble constant $h$ for the four
parameterizations analysed (see Table 1) and for the standard $\Lambda$CDM
model. The continuous (dashed) curves are calculated with the Pantheon
(Pantheon+OHD) data.

##  4 Conclusions

In this paper we studied a new parameterized version of the IHDE model: a non-
standard cosmological model which is based on the holographic principle and
that allows an interaction between DE and DM. Our proposal uses a Hubble IR
cutoff and a DE EoS redshift parameterization to quantify the interaction,
instead of assuming it with a specific form. Our simple model solves the so-
called coincidence problem of Cosmology.

Furthermore, using standard Bayesian inference procedures applied to
observations coming from Pantheon, Cosmic Chronometers and Galaxy Clustering
data sets, we showed that this simple model allows an accelerated expansion of
the Universe in comparison to the results found in previous work where certain
forms of the interaction were considered. Therefore, our work suggests that we
can explain the late Universe observations with this simple parameterized
version of the IHDE model.

Many specific features discussed in the analysis of our results depend on the
chosen DE EoS parameterization. For this reason, we used the well-known CPL
and Barboza-Alcaniz parameterizations. Nevertheless, with these choices we
found a degeneracy in the Hubble parameter of this model that restricted our
knowledge about some of the parameters involved. Therefore, we proposed a new
DE EoS parameterization (motivated by the $\Lambda$CDM limit in this model)
that allowed us to break the degeneracy in the parameters. More DE EoS of this
kind could be used in future analyses.

We showed through plots the behaviour of the DE EoS, the interaction between
DE and DM, and the posterior distribution functions of the reduced Hubble
constant for the parameterizations considered. Depending on the redshift,
these parameterizations allow an energy transfer from DE to DM, or vice versa.
Furthermore, some of these parameterizations predict a slightly higher value
of the Hubble constant, compared with the $\Lambda$CDM model. In fact, this
model allows a faster accelerated expansion at later stages of the Universe,
and a possibly smaller (or null) accelerated expansion at earlier times. This
delayed acceleration (first discussed in h0_HDE ) may alleviate the Hubble
tension. Nevertheless, to test that hypothesis in this model, we would need a
specific form of the interaction term (or a DE EoS) valid at higher redshifts
to perform the statistical analysis. The cosmography paradigm may give us the
required information at higher redshifts, up to $z\sim 10$.

Natural suggestions for future work are the inclusion of more
parameterizations of the DE EoS to keep studying this model in the low-
redshift regime. Another option would be the use of a direct parameterization
of the interaction term; the corresponding Hubble parameter could be
calculated through Eq. (14). This redshift parameterization of the interaction
may offer a more direct insight of the coupling between DE and DM. The next
big step would be an analysis with higher redshifts (possibly through the
cosmography approach) to study the Hubble tension in the IHDE model with a
Hubble IR cutoff. This will require an interaction term that behaves
qualitatively similar to those presented here, in order to fit the late
Universe observations. Finally, the idea of parameterizing the DE EoS or the
interaction term could be used with other IR cutoffs to maintain the agnostic
position with respect to the interaction.

*Acknowledgments CE-R is supported by PAPIIT UNAM Projects IA100220 and TA100122 and acknowledges the Royal Astronomical Society as FRAS 10147. This work is part of the Cosmostatistics National Group (CosmoNag) project. AG acknowledges financial support from “Consejo Nacional de Ciencia y Tecnología” (CONACyT) graduate grants program.

##  Appendix A Best-fit parameters

| Constant $w_{0}$ | $w_{\text{CPL}}$ | $w_{\text{BA}}$ | $w_{\Lambda}$
---|---|---|---|---
| Pantheon | Pantheon+OHD | Pantheon | Pantheon+OHD | Pantheon | Pantheon+OHD | Pantheon | Pantheon+OHD
$h$ | $0.691\pm 0.003$ | $0.672\pm 0.003$ | $0.704\pm 0.005$ | $0.701\pm 0.004$ | $0.703\pm 0.004$ | $0.699\pm 0.004$ | $0.703\pm 0.004$ | $0.702\pm 0.004$
$\Omega_{\text{de},0}w_{0}$ | $-0.569\pm 0.021$ | $-0.433\pm 0.011$ | $-0.825\pm 0.066$ | $-0.763\pm 0.028$ | $-0.778\pm 0.055$ | $-0.719\pm 0.024$ | — | —
$\Omega_{\text{de},0}w_{1}$ | — | — | $1.257\pm 0.308$ | $0.976\pm 0.073$ | $0.624\pm 0.155$ | $0.472\pm 0.035$ | — | —
$\Omega_{\text{m},0}$ | — | — | — | — | — | — | $0.203^{+0.028}_{-0.034}$ | $0.531^{+0.036}_{-0.030}$
$w_{0}$ | — | — | — | — | — | — | $-0.988^{+0.084}_{-0.034}$ | $-1.70\pm 0.12$
$w_{1}$ | — | — | — | — | — | — | $0.470\pm 0.058$ | $-0.237^{+0.072}_{-0.063}$

Table 2: Best-fit values for the four parameterizations considered (see Table
1), for both Pantheon and Pantheon+OHD data.

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# Impact of AI-tooling on the Engineering workspace

Lena Chretien
Jellyfish.co
Boston, MA
<EMAIL_ADDRESS>
Nikolas Albarran
Jellyfish.co
Boston, MA
<EMAIL_ADDRESS>

###### Abstract

To understand the impacts of AI-driven coding tools on engineers’ workflow and
work environment, we utilize the Jellyfish platform to analyze indicators of
change. Key indicators are derived from Allocations, Coding Fraction vs. PR
Fraction, Lifecycle Phases, Cycle Time, Jira ticket size, PR pickup time, PR
comments, PR comment count, interactions, and coding languages. Significant
changes were observed in coding time fractions among Copilot users, with an
average decrease of 3% with individual decreases as large as 15%. Ticket sizes
decreased by an average of 16% across four companies, accompanied by an 8%
decrease in cycle times, whereas the control group showed no change.
Additionally, the PR process evolved with Copilot usage, featuring longer and
more comprehensive comments, despite the weekly number of PRs reviewed
remaining constant. Not all hypothesized changes were observed across all
participating companies. However, some companies experienced a decrease in PR
pickup times by up to 33%, indicating reduced workflow bottlenecks, and one
company experienced a shift of up to 17% of effort from maintenance and
support work towards product growth initiatives.
This study is the first to utilize data from more than one company and goes
beyond simple productivity and satisfaction measures, considering real-world
engineering settings instead. By doing so, we highlight that some companies
seem to benefit more than others from the use of Copilot and that changes can
be subtle when investigating aggregates rather than specific aspects of
engineering work and workflows - something that will be further investigated
in the future.

## Introduction

The emergence of artificial intelligence (AI)-enabled coding tools represents
a paradigm shift in contemporary software development and engineering
workflows. Prominent among these innovations is GitHub Copilot, an AI-driven
code completion tool collaboratively developed by OpenAI and GitHub.
Leveraging the capabilities of the Codex model, which is trained on extensive
corpora of publicly accessible code, GitHub Copilot aids developers by
recommending code snippets and even complete functions derived from natural
language descriptions.

A substantial body of research has addressed the robustness and security
implications of this tool [1, 2, 3, 4, 5]. Concurrently, several empirical
investigations have evaluated the professional utility and productivity
impacts of GitHub Copilot [6, 7, 8]. Collectively, these studies underscore
the potential benefits that such tools offer to software developers and
engineers, particularly when employed within appropriate contexts.
Nonetheless, they also caution against potential pitfalls— specifically, the
risk that developers lacking sufficient contextual understanding or coding
expertise may inadvertently accept flawed code suggestions.

Most studies to understand the impact of Copilot on engineering satisfaction
were conducted through surveys which highlighted the overall satisfaction of
engineers when able to use Copilot [9, 10], with some increases up to 50%.

One recent study was conducted to determine the enhancement in coding velocity
and the potential significant productivity advancements ([8]). This
investigation, conducted within a controlled setting, involved 95 freelance
engineers tasked with a specific coding challenge. Half of the engineers
received assistance from GitHub Copilot while the other half did not. The
results revealed a 55% reduction in task completion time when the AI tool was
utilized.

Despite these findings, a recurring theme across these studies, as well as
additional small pilot studies within smaller companies, is the prevalent
focus on productivity metrics, while neglecting a holistic examination of
daily engineering workflows, behavioral shifts, and broader engineering
dynamics. This together with the lack of data driven insights from a large
number of Copilot users are the two main factors that are currently limiting
our understanding of impacts of AI-driven coding tools on the engineering
workspace.

This paper seeks to bridge this gap by examining the impacts of the AI tool
GitHub Copilot on engineers’ workflows, productivity, and daily interactions.
Rather than relying on surveys, we leverage empirical data sourced from the
Jellyfish platform, encompassing GitHub, Jira, allocations, metrics, and more.
This dataset comprises real-world activities of 133 engineers from four
different companies who have integrated GitHub Copilot into their operations
over periods ranging from one to eight months. We conduct a comprehensive
before-and-after analysis of these engineers’ work and behavior, supplemented
by comparative evaluations with a control group of engineers who did not use
GitHub Copilot. Hence, we provide insights into more than productivity
metrics, considering the real-world applications of the tool in a day-to-day
setting.

## Hypothesis

This study proposes several transformative changes in the engineering
workflow, engineering operations, and strategic alignment within software
organizations due to the integration of the AI-based coding assistant,
Copilot. Specifically, we formulate eight hypotheses delineated in Figure 1.

Figure 1: Hypotheses of changes observed due to usage of Copilot and
indicators that can be used to investigate these changes.

Each hypothesis will be evaluated using a unique set of indicators derived
from the extensive Jellyfish dataset (Figure 1), which includes GitHub
metadata on commits, pull requests, reviews, Jira Metadata on tickets,
statuses, assignee, comments, as well as allocations, time, and effort spent
on tasks.

For instance, Hypothesis 1 suggests that a decrease in time spent on
repetitive tasks will manifest as an increase in allocations towards high-
value ’Growth’ work. Conversely, this should logically entail a decrease in
’Other’ categories such as KTLO, Support, etc.
Repetitive tasks and coding that still takes place will be done with the
assistance of an AI coding tool and so (Hypothesis 2) coding will become
quicker, while the amount of review for each piece of work will increase to
ensure correctness of AI-implemented code (or partial code).
As Copilot is built to enhance coding and deployment efficiencies (Hypothesis
3) it is anticipated that more resources will be directed towards the creative
tasks of refinement. This shift will be evident in the lifecycle phases of
tickets, with the noted accompanying increase in review activities.
Furthermore, as coding becomes a less taxing component of a developer’s
workload, the nature of task definition is expected to evolve (Hypothesis 4).
This evolution will manifest in smaller work units (effort per ticket) and
reduced cycle times per ticket.
With the acceleration of coding processes and a heightened focus on review
activities, new procedural bottlenecks are anticipated (Hypothesis 5) as would
be observable in extended PR Pickup times.
The emphasis on more rigorous review processes is also expected to result in
more thorough and detailed PRs and PR comments (Hypothesis 6) as evident in an
increased number and length of comments.
Regarding the day-to-day interactions of engineers, the integration of an
automated coding assistant is predicted to result in reduced collaboration
(Hypothesis 7), as reflected in the decreased number of interactions such as
comments, edits on peers’ work, etc. Concurrently, the removal of barriers to
acquiring new skills and utilizing new coding languages (Hypothesis 8), will
be evidenced by an increase in the diversity of coding languages employed.

Some of these hypotheses can be thought of as "leading impacts" - operational
metrics like coding time, cycle time, collaboration, review times, and others
are "lagging impacts" such as bottlenecks, effort allocations, and new skills.
In other words, for some areas, impacts might be stronger and observed sooner
than in others.

## Study setup

To better understand the evolution of the engineering workspace, our
investigation explores the impact of Copilot usage in everyday environments,
rather than a controlled or semi-controlled experiment setup as referenced in
prior studies [8, 10]. Our study was conducted in collaboration with four
distinct companies from our customer base, involving 133 engineers who
utilized Copilot in conjunction with the Jellyfish platform. The study was
conducted using a retrospective design, ensuring that the engineers’ behavior
and workflows remained unaffected by the awareness of the study taking place.
For each engineer, we obtained the initial date of Copilot usage, along with
comprehensive data from the Jellyfish platform, including Jira tickets, GitHub
activity, and Allocations. With this, we investigate changes in work behavior
and workflows for the period before Copilot adoption and the period of Copilot
usage.

Figure 2: Copilot seats over time, by company

The companies in this study integrated Copilot into their operations as early
as March 2023. Initially, there was a limited number of active seats, which
subsequently expanded to between 17 to 45 seats per company by May 2024
(Figure 2).

Our methodology for this study employed a comparative analysis for each
Copilot user’s performance, evaluating the three months prior to and
subsequent to the adoption of the AI tool. Furthermore, in order to isolate
the effects of the Copilot adoption, we conducted a comparative assessment
between the Copilot-user group and a control group. This control group
comprised 750 engineers from the same companies who, while active on the
Jellyfish platform, did not use Copilot.

We investigated the hypothesis mentioned in Figure 1 by examining indicators
for each, with data from the Jellyfish platform. The data and indicators are
detailed in the subsequent section.

### Data

We investigate indicators derived from Jellyfish data, including: Allocations,
Coding fraction vs PR fraction, Lifecycle Phases, Cycle time, Size of Jira
tickets, PR Pickup time, PR comments, PR comment count, Interactions, and
coding languages.

Allocations:
This metric quantifies how a software engineer’s efforts are distributed
across different areas of work: KTLO (Keeping the lights on), Growth, Support,
and Other [11]. The specific nature of these categories is customized to the
needs of each customer and captured and measured via Jellyfish. Allocations
are quantified in full-time engineers (FTEs), a standard unit that mitigates
inconsistencies found in other measures such as story points or issue counts
[12].

Coding fraction vs. PR fraction, and Cycle time:
By integrating insights from Jira issues with GitHub activities, we avoid
merely calculating cycle time in isolation. Instead, we can get a more nuanced
view and consider work only if it involved coding (at least one commit was
associated with a Jira ticket), as well as split cycle time into a) time spent
coding, vs b) the time spent under Review. To do this we capture the first
time instance of when a ticket was moved into the Indeterminate phase (this is
the phase that signals active work on a ticket). The time spent coding is the
time until the first PR was opened for this work. The time in review, on the
other hand, is the time from the first PR to the merging of this PR. Each
phase’s duration (coding and review) is expressed as a fraction of total work
length, standardizing variations of ticket size and work segmentation.
The total length of the ticket - that is the coding time plus the review time
- is referred to here as cycle time.

PR Pickup time, number of PRs, and PR comments:
The duration a ticket remains in the PR review phase depends not only on the
assigned engineer but also on the timeliness of PR reviews. PR Pickup time is
defined as the interval from PR opening to the first review. This metric is
captured by keeping track of the times of PRs and any associated activity such
as comments, reviews, etc. Furthermore, by analyzing data on the number of
pull requests (PRs) created and reviewed by each engineer, as well as the
comments made on these PRs, we are able to quantitatively assess individual PR
activity. Additionally, the aggregate length of comments on each PR provides a
metric for evaluating the thoroughness and comprehensiveness of the reviews
conducted.

Size of Jira tickets:
In addition to cycle time per ticket - a good indicator of how large a piece
of work might have been - we also explore the effort that was associated with
each ticket. This is measured by accumulating actual work signals (e.g.
comments, edits, commits, PRs, etc.). We do this to eliminate any outside
factors that might contribute to extended cycle times, such as bottlenecks, or
blocks because of dependencies in work. In particular, cycle time could be
large while effort was not if an engineer moved a ticket into the
Indeterminate status in Jira and started work but was consequently blocked or
pulled into different work which interrupted their focus. In this case, the
clock for cycle time would keep counting up, while effort - defined as actual
signals from Jira or GitHub activity - would not be detected and remain low.

Lifecycle Phases:
The lifecycle explorer is a feature within the Jellyfish platform designed to
provide insights into the software development process by breaking work down
into distinct phases. The phases track the progress of the work from
definition to completion, helping engineering leaders visualize and understand
how time and resources are allocated throughout the development cycle. There
are four phases in the lifecycle of work: Refinement, Work, Review, and
Deployment.

Interactions:
Identifying connections between engineers is accomplished through counting
upstream and downstream interactions through the count of comments and edits
made or received between team members on Confluence pages, PRs, and Jira
issues. This allows insights into the collaboration or isolation of engineers,
and provides a better understanding of team dynamics and knowledge sharing.

Coding Languages:
Although Jellyfish does not directly access the code committed to GitHub, the
extension on files that were worked on serves as proxies for understanding the
variety of coding languages employed over time.

This data-centric approach allows us to rigorously analyse the real-world
impacts of AI tools on software engineering workflows and productivity.

## Results

In our comparative analysis of Copilot users and non-users, significant
differences were observed particularly in Hypothesis 2, 4, and 6 (Figure
3(a)). Conversely, the indicators related to the other hypotheses did not
exhibit significant changes when compared to those observed in the control
group (Figure 3(b)).

(a) Hypothesis with changes

(b) Hypothesis without changes

Figure 3: Relative changes [%] in the before and after of Copilot users (blue)
vs. non-Copilot users (purple) for Hypothesis 2, 4, and 6 (a) and Hypothesis
1, 5, 7, and 8 (b) (Figure 1)

Significant changes were observed in coding time fractions. Specifically, the
fractions that Copilot users spent coding per task ranged from a decrease of
15% to an increase of 1%, with an average of 3% decrease. In contrast, the
control group showed changes ranging from a decrease of 6% to an increase of
3%, with no average change. This difference is statistically significant with
a p-value of 0.01, indicating that the two groups originate from distinct
populations. The average coding fraction per ticket remained at 34.25% ($\pm$
8% and 9% for the before and after timeframe, respectively) for the control
group but decreased from 31.5 $\pm$ 6% to 30 $\pm$ 2%.

Work definition (Hypothesis 4) showed notable changes as well, particularly in
how work was scoped. Knowing that a Jira ticket - a piece of work - can vary
in size, the FTE count from the Jellyfish platform provides a reliable measure
of the actual work associated with each ticket. Using this data, we observed
that post-Copilot adoption, ticket size decreased across all four companies
with an average reduction of 16%. This was accompanied by a nearly 8% decrease
in cycle times, while the control group saw no change for the former, and a
slight increase for the latter. In other words, work was not just defined in
smaller chunks - i.e. less work associated with each Jira ticket - but this
work was also completed more quickly, as evidence by the decreasing cycle
time. The decrease in ticket size and the subsequent scoping of work into
smaller chunks align with current CI/CD best practices. For some companies, we
observe coding ticket sizes decrease by as much as 33%.

Additionally, the PR process seemed to evolve with the usage of Copilot,
characterized by longer and more comprehensive comments. This was consistent
across all four companies, although the weekly number of PRs reviewed remained
unchanged. Conversely, the control group experienced a decrease in number and
length of comments.

As mentioned above, while significant changes aligned with Hypotheses 2,4, and
6 for Copilot users, the indicators for other hypotheses did not show any
conclusive differences in the change from the before and after periods when
compared to the control group. However, while not consistently observed across
all companies, in some cases PR pickup time decreased up to 33%, suggesting a
reduction in potential bottlenecks in developers’ workflows. In addition, in
some cases a shift of upwards of up to 17% of effort from KTLO (Keeping the
lights on) and support work toward product roadmap and growth work was
observed.

## Conclusion

In our study, three out of the eight hypothesized changes were substantiated
by the empirical data, while we saw no impact of Copilot on the other areas of
engineering work. This suggests that while Copilot influences certain specific
aspects of engineering workflows, its impact on the other dimensions remains
comparable to the standard practices of non-users

A nuanced reflection of the day-to-day operations of individual engineers,
teams, and entire organizations, sheds light on why the data did not reveal
more changes. AI-driven coding tools are built to accelerate coding tasks;
however, coding constitutes merely a fraction of an engineer’s daily
activities. Consequently, the pronounced transformations observed in the
controlled experiment conducted by [8] are mitigated in a typical work
environment, where non-coding tasks and workflows, such as reviews,
bottlenecks, dependencies, unplanned work, planning, meetings are predominant.
This is especially pronounced when aggregating data at the team level - coding
tasks become even less prominent, thereby diminishing the observable impacts.
Therefore, the influence of AI tooling is more discernible at an individual
level than at a team or organizational level, where the multiplicity of non-
coding activities dilutes the enhancements provided by such tools.

Figure 4: Hypotheses introduced in Figure 1 but sorted by Individual Developer
Impacts, and Team/Group/Organizational Impacts

The presented hypotheses can be categorized based on their impact on
individual versus aggregated levels, such as multiple individuals and teams
(Figure 4). Processes such as coding versus pull request fractions on each
task, task definition, and the review process are primarily individual-centric
activities that are likely to be directly impacted by an AI coding tool.

While one might consider "New Skills" in this individual-centric category, we
argue that this is not appropriate. An engineer’s role within a team or
organization typically does not change overnight. Consequently, the type of
work (e.g. front-end, back-end, infrastructure, research, etc.) remains
relatively constant. As a result, the adoption of new coding languages is
expected to occur at similar rates among both Copilot users and non-users.
Instead, it might be more pertinent to investigate the onboarding rate of new
employees with and without the use of Copilot. It is reasonable to assume that
new employees and junior engineers are able to take on more challenging tasks
quickly with the help of AI-coding tools.

Conversely, bottlenecks, Allocations, and interaction are dependent on the
team as a whole, the team’s processes, and the team’s role within the
organization. Changes in these areas might be observed if an entire team
adopted Copilot; however, influence from other parts of the organization are
likely to attenuate these impacts even then. Therefore, it is unsurprising
that changes for these hypotheses were not evident, given that only a small
fraction of engineers in each company are Copilot users.

Changes are subtle and even non-existent in many places, which we believe is
due to the noise and low resolution of the data. Early findings from a follow-
up study using the Copilot API for high-resolution usage data show that
Copilot is more heavily used in some areas versus others. We expect to see
different levels of impact when aggregating data by e.g. type of work, coding
language, seniority of engineer, area of work, etc.

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