Also at<EMAIL_ADDRESS>
Department of Informatics,
University of Sussex, Falmer, UK.

# Accelerated Particle Detectors with Modified Dispersion Relations.

Paul C.W. Davies<EMAIL_ADDRESS>Philip Tee  [<EMAIL_ADDRESS>The Beyond
Center for Fundamental Science, Arizona State University, Tempe AZ 85287, USA

###### Abstract

There is increasing interest in discrete or ‘pixelated’ spacetime models as a
foundation for a satisfactory theory of quantum gravity. If spacetime
possesses a cellular structure, there should be observable consequences: for
example, the vacuum becomes a dispersive medium. Of obvious interest are the
implications for the thermodynamic properties of quantum black holes. As a
first step to investigating that topic, we present here a calculation of the
response of a uniformly accelerating particle detector in the (modified)
quantum vacuum of a background pixelated spacetime, which is well known to
mimic some features of the Hawking effect. To investigate the detector
response we use the standard DeWitt’s treatment, with a two-point function
modified to incorporate the dispersion. We use dispersion relations taken from
the so-called doubly special relativity (DSR) and Hor̆ava-Lifshitz gravity. We
find that the correction terms retain the Planckian nature of particle
detection, but only for propagation faster than the speed of light, a
possibility that arises in this treatment because the dispersion relations
violate Lorentz invariance. A fully Lorentz-invariant theory requires
additional features; however, we believe the thermal response will be
preserved in the more elaborate treatment.

††preprint: APS/123-QED

##  I Introduction and Background

Much of theoretical physics is formulated on the assumption that spacetime is
continuous and maps to the real numbers. While this is an obvious
idealization, convenient for calculation, it rarely presents difficulties.
However, in quantum field theory, spacetime continuity leads to divergences
that must be evaded by renormalization – an ad hoc procedure. Worse still,
many formulations of quantum gravity are non-renormalizable.

What happens if spacetime continuity is discarded? There is a rich history of
models in which spacetime emerges in the macroscopic limit from some sort of
discrete or pixelated substructure [1, 2, 3, 4, 5, 6, 7]. Quantum gravity
defines a fundamental length scale, the Planck length
$l_{p}\equiv\sqrt{\frac{\hbar G}{c^{3}}}=1.616\times 10^{-35}~{}m$, which
provides a natural measure of the ‘pixel’ size, and a natural cut-off energy
for any incipient ultra-violet divergence. Although ‘pixelating’ spacetime,
i.e. replacing real numbers by a countable infinity, is quite probably a
placeholder for some more nuanced micro-structure, or pre-geometry [8], it is
interesting to explore the impact of this simple modification to determine
whether there are any observable consequences. One of the most basic spacetime
structures, and the most intensively studied, is the black hole. How might
spacetime discreteness affect its properties, such as Hawking radiation? As a
first step to addressing this question, we here investigate the response of an
accelerating particle detector, the so-called Davies-Fulling-Unruh effect,
known to mimic Hawking radiation, in a pixelated spacetime background.

Although the existence of a fixed fundamental length scale violates Lorentz
invariance, Minkowski space may be retained if momentum space acquires non-
zero curvature, a construction known as Doubly Special Relativity (DSR) [9,
10, 1]. DSR may be generalized to curved spacetime by replacing locally
Minkowski space with locally de Sitter space [11].

The effect of introducing curvature in momentum space is to render the vacuum
a dispersive medium in which different frequency light waves propagate at
different speeds. This can be modelled by adding successively higher powers of
the three-momentum $p$ to the standard energy-momentum dispersion relations,

$E^{2}=p^{2}+m^{2}+\frac{\kappa_{1}}{M_{p}}p^{3}+\frac{\kappa_{2}}{M^{2}_{p}}p^{4}+\dots\text{,}$
(1)

where the dimensionless kappa coefficients are arbitrary at this stage. The
dispersion relation Eq. 1 on its own puts the theory in violation of Lorentz
invariance, a defect that may, however, be remedied in a more elaborate
treatment that involves changing the momentum measure [12, 13]. We defer the
complete DSR treatment in this paper and restrict our analysis to the effect
of the dispersion relation alone on the response of a particle detector.

If the vacuum really is dispersive, then there should be observable effects.
Indeed, it is known that the $p^{3}$ term produces a correction to the black
hole entropy which is usually discarded by setting $\kappa_{1}$ to zero [14],
a practice to which we will adhere in this paper, focusing instead on the
consequences of non-zero $p^{4}$ and $p^{6}$ terms (the latter arising in
Hor̆ava-Lifshitz theories of gravity (HLG) [15]). Modifications to the
dispersion relations in turn alter the Green’s functions of any quantized
field theory, which we have previously explored in the context of the effect
of these changes on the bending of light by massive bodies [16]. The same
Green’s functions can be adapted for use in calculating the response of a
DeWitt-Unruh particle detector. The principle results of this calculation are
presented in Section IV, with the details covered in the appendices.

A similar calculation was performed by Rinaldi [17], but crucially only the
case of positive $\kappa_{2}$ was considered, and higher order terms in $p$
were not considered. The approach taken there was to treat the higher order
terms in the dispersion relationship perturbatively, whereas our calculation
is more direct. For positive $\kappa$ we obtain a similar result to Rinaldi,
but our method is more general and able to accommodate negative as well as
positive correction terms.

The layout of the paper starts with an overview of the effect of extra terms
on the propagation of waves through the vacuum in Section II, and establish
that $\kappa$ controls whether propagation is sub or superluminal. In Section
III we present the modified Hadamard functions that we use in Section IV to
compute the corrections to the detector response functions. We then briefly
consider the effect of higher order terms in the dispersion relations in
Section V, before concluding with a discussion of the impact of these results
on other semi-classical phenomena in Section VI.

##  II Modified dispersion relation and refractive index

We restrict the discussion here to the modified dispersion relation,

$E^{2}=p^{2}+\kappa\eta^{2}p^{4}+m^{2}\text{,}$ (2)

where $\eta$ is a characteristic length scale related to the pixelation scale,
and $\kappa$ controls the sign of the correction. In the massless limit with
units $\hbar=1,E=\omega,$ and $p=k$ we note that a scalar field $\phi(t,x)$
obeys the modified free space wave equation,

$\partialderivative[2]{\phi}{t}-\partialderivative[2]{\phi}{x}+\kappa\eta^{2}\partialderivative[4]{\phi}{x}=0\text{,}$
(3)

which has a set of solutions $\phi(t,x)=e^{\pm(\omega t-k\dotproduct
x)},e^{\pm i(\omega t-k\dotproduct x)}$, shown to be complete by computation
of the Wronskian. The key result is that, although Eq. 3 has wavelike
solutions, the vacuum itself is dispersive and possesses a refractive index
$n(k)$ given by,

$n(k)=\frac{\sqrt{1+\kappa\eta^{2}k^{2}}}{1+2\kappa\eta^{2}k^{2}}\text{,}$ (4)

a result that extends to spin 1 (photons) and (linearized) spin 2 (gravitons)
too, and broadly follows the analysis by Myers et al [18]. The fact that the
speed of wave propagation of the waves depends on the frequency introduces
some peculiar features. Note that for $\kappa>0$ high frequency radiation
propagates faster than low frequency radiation, and all waves propagate at a
speed $>1$, i.e. faster than the speed of light in the unmodified case. For
the case $\kappa\leq 0$ the propagation speed is $\leq 1$ and slows as $k$
increases, albeit slowly. There is a singularity at
$k=\frac{1}{\eta\sqrt{2\kappa}}$, which is determined by the pixalation
length, which might naturally be associated with the Planck length. However,
for the purposes of this paper, we shall sidestep the choice of length scale,
as the key issue from the above discussion is the sign rather than the value
of $\kappa$.

There is the obvious problem of reconciling superluminal propagation with
relativistic causality. This is of course expected since the dispersion
relations from DSR and HLG explicitly violate Lorentz invariance. Locally
Lorentz invariant discrete spacetimes theories can be constructed by
introducing curvature in momentum space [19]. We think our principal result
will remain true in the more general case.

##  III Green’s functions in position space

In the massive case, Eq. 2 leads to a modified propagator in n-dimensional
momentum space, and associated Feynman diagram [16],

prules (55,40) o1 o2 dashes_arrow, label=$p$o1,o2 o1
$=\frac{1}{p_{0}^{2}-p^{2}(1+\kappa\eta^{2}p^{2})-m^{2}+i\epsilon}$,o2

and the position space propagator,

$G(t,x;t^{\prime},x^{\prime})=\int\limits_{-\infty}^{\infty}\frac{\differential^{n}p}{(2\pi)^{n}}\frac{e^{-i[p_{0}(t-t^{\prime})-\overrightarrow{\mathbf{p}}\dotproduct(\overrightarrow{\mathbf{x}}-\overrightarrow{\mathbf{x^{\prime}}})]}}{p_{0}^{2}-p^{2}(1+\kappa\eta^{2}p^{2})-m^{2}}\text{.}$
(5)

The $p_{0}$ integral is performed using a contour integral noting that the
poles occur at $p_{0}^{2}=p^{2}(1+\kappa\eta^{2}p^{2})+m^{2}$, with the choice
of contour related to the relevant two-point functions in the standard manner
[20]. For our treatment we require the positive and negative frequency
Hadamard functions,

$G(t,x;t^{\prime},x^{\prime})=\frac{1}{(2\pi)^{n-1}}\int\limits_{-\infty}^{\infty}\frac{\differential^{n-1}p}{2E_{p}}e^{-i[E_{p}(t-t^{\prime})-\overrightarrow{\mathbf{p}}\dotproduct(\overrightarrow{\mathbf{x}}-\overrightarrow{\mathbf{x^{\prime}}})]}\text{,}$
(6)

where $E_{p}=\pm\sqrt{p^{2}(1+\kappa\eta^{2}p^{2})+m^{2}}$.

Note that we use the standard Lorentz invariant measure from conventional
quantum field theory in this integral, and not one modified by attributing a
curvature to momentum space, in accordance with the simpler Lorentz violating
version of the theory we are using. The integration in Eq. 6 is complicated,
but we require only the small corrections $\mathcal{O}(\kappa\eta^{2})$ here.
The details of these calculations are contained in the Appendices. In the
massless limit, $3+1$ case, we find for time-like separation,

$D^{+}(t,r)=\frac{-\theta(\sigma^{2})}{4\pi^{2}(\sigma^{2}-i\epsilon)}\left\\{1-\frac{\kappa\eta^{2}}{(\sigma^{2}-i\epsilon)}\right\\}\text{,}$
(7)

and for the (retarded) space-like case,

$D^{-}(t,r)=\frac{\theta(-\sigma^{2})}{4\pi^{2}(-\sigma^{2}+i\epsilon)}\left\\{1+\frac{\kappa\eta^{2}}{(-\sigma^{2}+i\epsilon)}\right\\}\text{.}$
(8)

Here we denote by $\sigma^{2}=t^{2}-r^{2}$ the invariant interval, with
$\theta(x)$ being the normal sign function. In the case of $\eta=0$ we recover
the standard result [21].

##  IV Accelerated Detectors

Following §3.3 of [20], we can use the correction to the propagator to
investigate the response of an accelerated particle in pixelated spacetime by
leveraging the modifications to dispersion relations  from DSR. We consider a
particle detector under uniform inverse acceleration $\alpha$, executing a
hyperbolic path in Minkowski space parameterized by proper time $\tau$
according to,

$\displaystyle x$
$\displaystyle=y=0,~{}z=(t^{2}+\alpha^{2})^{\frac{1}{2}},~{}\alpha=\text{const,
such that,}$ $\displaystyle z$
$\displaystyle=\alpha\cosh{\frac{\tau}{\alpha}},~{}t=\alpha\sinh{\frac{\tau}{\alpha}}\text{,}$

The standard DeWitt-Unruh detector [22, 23, 24] leads to the transition
probability per unit proper time,

$c^{2}\sum\limits_{E}\absolutevalue{\bra{E}m(0)\ket{E_{0}}}^{2}\int\limits^{\infty}_{-\infty}\differential(\Delta\tau)~{}e^{i(E-E_{0})\Delta\tau}D^{+}(\Delta\tau)\text{,}$
(9)

and the detector response function,

$\mathcal{F}(E)=\int\limits^{\infty}_{-\infty}\differential(\Delta\tau)~{}e^{i(E-E_{0})\Delta\tau}D^{+}(\Delta\tau)\text{.}$
(10)

The response function is modified by the correction term in the propagator
(see Appendix B) to yield,

$\mathcal{F}(E)=\left\\{\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}\right)\frac{E-E_{0}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\kappa\eta^{2}}{6}\right)\frac{(E-E_{0})^{3}}{e^{2\pi(E-E_{0})\alpha}-1}\right\\}\text{}$
(11)

which retains a thermal character. We also show (Appendix B) that the modified
Green’s function remains periodic in imaginary time with period $2\pi\alpha$,
as in the standard (unmodified) treatment.

Although the modification to the detector response looks innocuous at first
sight, it contains some troublesome features. If $\kappa<0$ (subluminal
propagation) the transition probability becomes negative for
$\absolutevalue{\kappa}>\frac{6\alpha^{2}}{\eta^{2}}$. This may point to a
breakdown of unitarity or some other pathology with the theory, but only in
cases of extreme acceleration (very small $\alpha$). Our results would still
be valid over a wide range of accelerations. Alternatively, one might simply
rule out values of $\kappa<0$, corresponding to subluminal propagation, as
unphysical.

In the case of $\kappa>0$, (superluminal propagation),the detector response
function has no worrying negative terms that would produce a pathological
transition probability. In some sense this is consistent with the fact that
the calculation is not Lorentz invariant, and so by definition places no
restriction of propagation velocities. This was essentially the case
considered by Rinaldi [17], and we need to perform the full Lorentz invariant
calculation to investigate whether this behavior survives.

##  V Effect of higher terms on the result

We now briefly consider higher order terms in the dispersion relations. It was
remarked in Section I that (HLG) can introduce a $p^{6}$ term into the
dispersion relations. The central idea of HLG gravity is an anisotropic
scaling of space and time, achieved by introducing a scaling parameter $b$ and
critical exponent $z$, such that $\overrightarrow{\mathbf{x}}\rightarrow
b\overrightarrow{\mathbf{x}}$, and $t\rightarrow b^{z}t$. As a quantized
theory this becomes power counting renormalizable at a value of $z=3$ [15],
and we focus on the modifications this makes to the dispersion relations. The
anisotropic scaling introduces new diffeomorphisms, that in turn introduce
additional terms to the gravitational action, in particular a potential term
that alters the dispersion relations. At the high energy UV limit these
modifications suggest the following form of the dispersion relations,

$E^{2}=p^{2}+m^{2}+\kappa\eta^{2}p^{4}-\kappa\eta^{4}p^{6}\text{.}$ (12)

The additional $p^{6}$ term is explicitly set to have the opposite sign to the
$p^{4}$ term, as required to ensure that the sign of $\kappa$ dictates whether
propagation is super or subluminal [19, 25].

We can formulate an associated equation of motion to the dispersion
relationship Eq. 12 to determine the refractive index of the vacuum. This
modifies the equation of motion Eq. 3 to

$\partialderivative[2]{\phi}{t}-\partialderivative[2]{\phi}{x}+\kappa\eta^{2}\partialderivative[4]{\phi}{x}-\kappa\eta^{4}\partialderivative[6]{\phi}{x}=0\text{,}$
(13)

with the corresponding momentum-dependent refractive index,

$n(k)=\frac{\sqrt{1+\kappa\eta^{2}k^{2}+\kappa\eta^{4}k^{4}}}{1+2\kappa\eta^{2}k^{2}+3\kappa\eta^{4}k^{4}}\text{.}$
(14)

As before, for $\kappa>0$ propagation is superluminal, and $\kappa<0$ restores
subluminal propagation.

The impact of this new dispersion relation on the detector response function
is straightforward to calculate as detailed in Appendix C and we obtain the
following modification to the detector response function,

$\mathcal{F}(E)=\left\\{\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}-\frac{3\kappa^{\prime}\eta^{4}}{5\alpha^{4}}\right)\frac{E-E_{0}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\eta^{2}[\kappa+12\kappa^{\prime}\eta^{2}]}{6}\right)\frac{(E-E_{0})^{3}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\kappa^{\prime}\eta^{4}}{10}\right)\frac{(E-E_{0})^{5}}{e^{2\pi(E-E_{0})\alpha}-1}\right\\}\text{.}$
(15)

The calculation introduces a new sign term
$\kappa^{\prime}=\kappa\left(\frac{3}{4}\kappa+1\right)$, but for $\kappa=\pm
1$ it has the same polarity as the original term (i.e. for $\kappa=\pm
1,\kappa^{\prime}=\pm 1$).

The first thing to note is that the additional contributions arising from the
$p^{6}$ term retain the Planck factor, and so preserve the thermal nature of
the response. However, as in Eq. 11, we have problematic features that arise
in the case of subluminal propagation where $\kappa,\kappa^{\prime}<0$. The
new terms in the detector function pick up a coefficient of $\eta^{4}$ from
the dispersion relation, and are therefore much smaller in magnitude than the
$p^{4}$ terms. The only hope for restoring positive definiteness would arise
from the $\frac{3\kappa^{\prime}\eta^{4}}{5\alpha^{4}}$ coefficient of the
first term overwhelming the $\frac{\kappa\eta^{2}}{6\alpha^{2}}$ from the
original calculation. If we assume $\eta\sim\frac{1}{M_{p}}$, this would only
be the case for very small accelerations in the order of a few $ms^{-2}$ or
$\alpha\sim c$. At this point the absence of a factor of $\alpha$ in the
second Planck term would cause that to dominate the sign of $\mathcal{F}(E)$
and give it an overall negative value. The converse argument for superluminal
propagation ($\kappa,\kappa^{\prime}>1$) confirms that the detector response
function remains positive across the full range of $\alpha$ in this case.

We can therefore conclude that the detector response function remains
potentially problematic for subluminal propagation, and the possibility exists
that the transition probability will then also be negative. On dimensional
grounds, for higher powers of $p$ in the dispersion relation, the corrections
will acquire increasing powers of $\eta$ and follow the same pattern for their
contribution to the response function. One would expect that similar arguments
would apply to those corrections, and the detector response function will
cease to be positive definite in the case of subluminal propagation. We
conclude that any route to cancellation of the $p^{4}$ term from higher order
terms in dispersion relations appears shut off.

##  VI Conclusion and discussion

The principal result of this paper is the discovery of severe pathologies in
the calculation of the Davies-Fulling-Unruh effect in the case of discrete
spacetime theories, unless we are willing to accept superluminal propagation.
Although we use the modified dispersion relations derived from DSR, a fully
Lorentz-invariant treatment that accommodates a fundamental length, requires
additional features in the theory that we shall address in a future paper. The
close relationship between the Davies-Fulling-Unruh effect, systems of
accelerating mirrors and Hawking radiation makes this result even more
interesting. Adding higher order terms in the dispersion relations does not
provide a remedy to the issue. In fact due to the relative size of
contribution at higher powers of $p$ and the appearance of powers of inverse
acceleration, the likelihood of cancellation from higher order momentum terms
in the dispersion relationship appears low. One is led to the conclusion that
the coarse-grained structure of spacetime may significantly modify, or even
suppress, thermal vacuum effects for certain parameter ranges. It is well
recognized that introducing a simple cut-off in trans-Planckian modes
occasioned by the existence of a fundamental length is problematic for some
derivations of the Hawking effect [26, 27]. Further investigation of how
vacuum dispersion affects the calculation of the Bogoliubov transformation in
the black hole radiance calculation may clarify these issues.

In particular our principal conclusion concerning pathologies in the detector
response function arise when considering subluminal propagation may not
survive the re-imposition of Lorentz invariance. The question of whether this
is indeed the case is the subject of ongoing investigation.

##  Appendix A Computation of position space Green’s function in 3+1 spacetime

###  A.1 First method of computation

We start with the massless propagator as defined in Eq. 6, which we convert to
$n$-dimensional spherical polar momentum space to obtain,

$D(t,r)=\frac{2(\pi^{\frac{n-2}{2}})}{(2\pi)^{n-1}\Gamma(\frac{n-2}{2})}\int\limits_{0}^{\pi}\sin^{n-3}\theta~{}\differential\theta\int\limits_{0}^{\infty}\frac{\differential^{n}p}{2E_{p}}e^{-i(E_{p}t-pr)}\text{.}$

The angular integral can be performed by analytic continuation of the standard
result from [28] (pg. 346. 3.387 (1) ),

$\int\limits_{0}^{\pi}\sin^{k}\theta
e^{ipr\cos\theta}~{}\differential\theta=\sqrt{\pi}\left(\frac{2}{pr}\right)^{\frac{k}{2}}\Gamma\left(\frac{k+1}{2}\right)J_{\frac{k}{2}}(pr)\text{,}$

where $J_{n}(x)$ are the Bessel functions of order $n$. To remove clutter in
what follows, we define,

$\zeta=\frac{1}{2(2\pi)^{(n-1)/2}r^{(n-3)/2}}\text{.}$ (16)

Using this result and inserting our modified form for $E_{p}$, we have the
following integral to perform,

$D(t,r)=\zeta\int\limits_{0}^{\infty}\differential
p~{}\frac{p^{\frac{n-3}{2}}}{(1+\kappa\eta^{2}p^{2})^{1/2}}J_{\frac{n-3}{2}}(pr)e^{-ip(1+\kappa\eta^{2}p^{2})^{1/2}t}\text{.}$
(17)

Exploiting the fact that $\kappa\eta^{2}$ is very small we can expand the
numerator $(1+\kappa\eta^{2}p^{2})^{-\frac{1}{2}}$ to first order in
$\kappa\eta^{2}$, to obtain,

$\begin{split}D(t,r)=&\zeta\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(pr)e^{-ip(1+\eta^{2}p^{2})^{1/2}t}\\\
-&\zeta\frac{\kappa\eta^{2}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n-3}{2}}(pr)e^{-ip(1+\eta^{2}p^{2})^{1/2}t}\end{split}$

One can now expand the $(1+\kappa\eta^{2}p^{2})^{\frac{1}{2}}$ in the
exponential,

$e^{-ip(1+\kappa\eta^{2}p^{2})^{1/2}t}=e^{-ipt}e^{-i\frac{\kappa\eta^{2}}{2}p^{3}t}\text{.}$

For the second factor for small $\kappa\eta^{2}$ we can further expand the
exponential to first order, obtaining
$e^{-i\frac{\kappa\eta^{2}}{2}p^{3}t}=1-\frac{\kappa\eta^{2}}{2}it$. This
would give a term proportional to $t$, which we are free to disregard as it is
not an invariant quantity and we are free to rotate to a frame where
$r=\theta(-\sigma^{2})\sqrt{r^{2}-t^{2}+i\epsilon},t=0$ as in the $1+1$
propagator case. This will compute the retarded or time-like massless
propagator $D^{-}(t,r)$.

We are left with the following integrals to evaluate,

$\displaystyle D^{-}(t,r)=$
$\displaystyle\zeta\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}$
$\displaystyle-\zeta\frac{\kappa\eta^{2}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}$

To solve these we analytically continue the standard integral from [28] (pg.
694. 6.623 (1)),

$\int\limits_{0}^{\infty}e^{-\gamma x}J_{\nu}(\beta x)x^{\nu}~{}\differential
x=\frac{(2\beta)^{\nu}\Gamma(\nu+\frac{1}{2})}{\sqrt{\pi}(\gamma^{2}+\beta^{2})^{\nu+\frac{1}{2}}}\text{.}$
(18)

It will be convenient to subtract a small imaginary component $i\epsilon$ to
$t$ to allow convergence of the integral later, replacing $t^{2}$ with
$t^{2}+i\epsilon$, after discarding the term in $\mathcal{O}(\epsilon^{2})$.

For the first integral we have,

$\zeta\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}=\frac{\Gamma(\frac{n}{2}-1)}{4(\pi)^{\frac{n}{2}}(r^{2}-t^{2}+i\epsilon)^{\frac{n}{2}-1}}\text{.}$
(19)

The second integral is more problematic, but can be massaged into standard
forms by taking advantage of the differential recurrence relationship of the
Bessel functions and integrating by parts. We note,

$\derivative{x}\left\\{x^{\eta}J_{\eta}(\beta x)\right\\}=\beta
x^{\eta}J_{\eta-1}(\beta x)\text{.}$ (20)

One can now write the integrand as,

$\begin{split}&\zeta\frac{\kappa\eta^{2}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}\\\
&=\zeta\frac{\kappa\eta^{2}}{2r}\int\limits_{0}^{\infty}\differential
p~{}pe^{-ipt}\derivative{p}\left\\{p^{\frac{n-1}{2}}J_{\frac{n-1}{2}}(pr)\right\\}\text{.}\end{split}$

This is in the form of $\int u\differential v=uv-\int v\differential u$, with
$u=pe^{-ipt}$ and $v=p^{\frac{n-1}{2}}J_{\frac{n-1}{2}}(pr)$. Rotation of the
time axis by an arbitrary small angle $\delta\leq\pi/2$, such that $t->\infty
e^{-i\delta}$ in the upper limit of the integral, ensures the convergence of
$[uv]^{\infty}_{0}$, and we are left with,

$\displaystyle\zeta\frac{\kappa\eta^{2}}{2r}\int\limits_{0}^{\infty}\differential
p~{}pe^{-ipt}\derivative{p}\left\\{p^{\frac{n-1}{2}}J_{\frac{n-1}{2}}(pr)\right\\}=$
$\displaystyle-\zeta\frac{\kappa\eta^{2}}{2r}\int\limits_{0}^{\infty}\differential
p~{}e^{-ipt}p^{\frac{n-1}{2}}J_{\frac{n-1}{2}}(pr)$
$\displaystyle+\zeta\frac{\kappa\eta^{2}}{2r}it\int\limits_{0}^{\infty}\differential
p~{}e^{-ipt}p^{\frac{n+1}{2}}J_{\frac{n-1}{2}}(pr)\text{.}$

The second of these integrals can be discarded due to the leading factor of
$it$ as above, and the first is in the standard form used earlier. Using the
formula Eq. 18 we have the result for this correction to order $\eta^{2}$,

$\zeta\frac{\kappa\eta^{2}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}=\frac{\kappa\eta^{2}\Gamma(\frac{n}{2})}{4(\pi)^{\frac{n}{2}}(r^{2}-t^{2}+i\epsilon)^{\frac{n}{2}}}\text{.}$
(21)

Substituting in $n=4$ into Eqs. 19 and 21, with $\sigma^{2}=t^{2}-r^{2}$ being
the invariant interval, we obtain our final result for the propagator to first
order in $\kappa$,

$D^{-}(t,r)=\frac{\theta(-\sigma^{2})}{4\pi^{2}(-\sigma^{2}+i\epsilon)}\left\\{1+\frac{\kappa\eta^{2}}{(-\sigma^{2}+i\epsilon)}\right\\}\text{.}$
(22)

This has the corresponding advanced or time-like counterpart,

$D^{+}(t,r)=\frac{-\theta(\sigma^{2})}{4\pi^{2}(\sigma^{2}-i\epsilon)}\left\\{1-\frac{\kappa\eta^{2}}{(\sigma^{2}-i\epsilon)}\right\\}\text{.}$
(23)

###  A.2 Second method of computation

We start with Eq. 17, however we proceed differently. Instead we make the
simplification $p(1+\kappa\eta^{2}p^{2})^{1/2}\sim\sqrt{\kappa}\eta p^{2}$ in
the exponential, we can also expand the $(1+\kappa\eta^{2}p^{2})$ for small
$\kappa\eta^{2}$ in the denominator to
$(1-\frac{\kappa\eta^{2}}{2}p^{2}+\dots+\mathcal{O}(\kappa^{2}\eta^{4}))$, and
following a change of variable $x=p^{2}$, we have,

$D(t,r)=\frac{\zeta}{2}\int\limits_{0}^{\infty}\differential
x~{}\left(x^{\frac{n-5}{4}}-\frac{\kappa\eta^{2}}{2}x^{\frac{n-1}{4}}\right)J_{\frac{n-3}{2}}(r\sqrt{x})e^{-i\sqrt{\kappa}\eta
xt}\text{.}$

These gives two standard integrals [28] (pg. 701. 6.643(i)) of the form,

$\begin{split}\int\limits_{0}^{\infty}\differential
x~{}x^{\mu-1/2}e^{-ax}J_{2\nu}(2b\sqrt{x})&=\\\
\frac{\Gamma(\mu+\nu+1/2)}{b\Gamma(2\nu+1)}\exp\frac{-b^{2}}{2a}a^{-\mu}M_{\mu,v}\left(\frac{b^{2}}{a}\right)\text{,}\end{split}$
(24)

where $M_{\mu,v}(x)$ is the Whittaker special function, and we analytically
continue to imaginary exponents. For our integrals,

$\displaystyle a$ $\displaystyle=i\sqrt{\kappa}\eta t\text{,}$ $\displaystyle
b$ $\displaystyle=\frac{r}{2}\text{,}$

and we have and two pairs of parameters $\mu,\nu$. These are

$\displaystyle\mu$
$\displaystyle=\frac{n-3}{4}\text{,}\nu=\frac{n-3}{4}\text{,
for}~{}x^{\frac{n-5}{4}}\text{, and,}$ $\displaystyle\mu$
$\displaystyle=\frac{n+1}{4}\text{,}\nu=\frac{n-3}{4}\text{,
for}~{}x^{\frac{n-1}{4}}\text{.}$

Completing the substitutions we have for our Green’s function,

$\begin{split}D(t,r)&=\frac{\zeta}{r}\exp{\frac{-r^{2}}{8i\sqrt{\kappa}\eta
t}}\times\\\
&\left[\frac{\Gamma(\frac{n-2}{2})}{\Gamma(\frac{n-1}{2})}(i\sqrt{\kappa}\eta
t)^{-(\frac{n-3}{4})}M_{\frac{n-3}{4},\frac{n-3}{4}}\left(\frac{r^{2}}{4i\sqrt{\kappa}\eta
t}\right)-\frac{\kappa\eta^{2}\Gamma(\frac{n}{2})}{2\Gamma(\frac{n-1}{2})}(i\sqrt{\kappa}\eta
t)^{-(\frac{n+1}{4})}M_{\frac{n+1}{4},\frac{n-3}{4}}\left(\frac{r^{2}}{4i\sqrt{\kappa}\eta
t}\right)\right]\end{split}$ (25)

This unpleasant looking result collapses rather nicely in the limit of
$t\rightarrow 0$, when the argument of the Whittaker function becomes
infinite. The Whittaker function $M_{\mu,\nu}(z)$ has the following asymptotic
behavior in the case of $z\rightarrow\infty$,

$\lim_{z->\infty}M_{\mu,\nu}(z)\sim\frac{\Gamma(1+2\nu)}{\Gamma(\frac{1}{2}+\nu-\mu)}e^{\frac{1}{2}z}z^{-\mu}\text{.}$
(26)

The exponential term is the reciprocal of the $\exp{\frac{-r^{2}}{8i\eta t}}$
term before the brackets, and the $z^{-\mu}$ cancels the $(i\sqrt{\kappa}\eta
t)^{\mu}$ terms preceding the Whittaker functions, leaving the
$\frac{r^{2}}{4}$ factors. In this limit therefore, the Green’s function
becomes,

$\begin{split}&D(t,r)=\\\
&\frac{\zeta}{r}\left[\frac{\Gamma(\frac{n-2}{2})}{\Gamma(\frac{1}{2})}\left(\frac{r^{2}}{4}\right)^{-(\frac{n-3}{4})}-\frac{\kappa\eta^{2}}{2}\frac{\Gamma(\frac{n}{2})}{\Gamma(-\frac{1}{2})}\left(\frac{r^{2}}{4}\right)^{-(\frac{n+1}{4})}\right]\text{.}\end{split}$
(27)

When taking the limit $t\rightarrow 0$, we should interpret $r^{2}$ as being
the spacelike interval, $r^{2}=\theta(-\sigma^{2})(-\sigma^{2}+i\epsilon)$,
and the propagator as the retarded or space-like propagator $D^{-}(t,r)$.
Setting $n=4$ and making this substitution for $r$ we have,

$D^{-}(t,r)=\frac{\theta(-\sigma^{2})}{4\pi^{2}(-\sigma^{2}+i\epsilon)}\left[1+\frac{\kappa\eta^{2}}{(-\sigma^{2}+i\epsilon)}\right]\text{.}$
(28)

We can further make the substitution
$(-\sigma^{2}+i\epsilon)\rightarrow-(\sigma^{2}-i\epsilon)$ to recover the
advanced or time-like propagator,

$D^{+}(t,r)=\frac{-\theta(\sigma^{2})}{4\pi^{2}(\sigma^{2}-i\epsilon)}\left[1-\frac{\kappa\eta^{2}}{(\sigma^{2}-i\epsilon)}\right]\text{.}$
(29)

It will be noted that this result is identical to that obtained using the
first method in Section A.1.

##  Appendix B Detector function calculation

To compute the transition probability we need to perform the integral in Eq.
10, and follow a calculation similar to that in [20]. To make progress we use
the modified $3+1$ propagator Eq. 8, choosing the advanced propagator (space-
like) form,

$\begin{split}D^{+}(\Delta t,\Delta z)=&-\frac{1}{4\pi^{2}(\Delta t^{2}-\Delta
z^{2}-i\epsilon)}\times\\\ &\left\\{1-\frac{\kappa\eta^{2}}{(\Delta
t^{2}-\Delta z^{2}-i\epsilon)}\right\\}\text{.}\end{split}$

Using the elementary properties of the hyperbolic functions, we have for
$\Delta t^{2}$ and $\Delta z^{2}$,

$\displaystyle\Delta t$
$\displaystyle=2\alpha\cosh{\left(\frac{\tau+\tau^{\prime}}{2\alpha}\right)}\sinh{\left(\frac{\tau-\tau^{\prime}}{2\alpha}\right)}$
$\displaystyle\Delta z$
$\displaystyle=2\alpha\sinh{\left(\frac{\tau+\tau^{\prime}}{2\alpha}\right)}\sinh{\left(\frac{\tau-\tau^{\prime}}{2\alpha}\right)}$

Inserting these back into the propagator we have,

$\begin{split}D^{+}(\Delta\tau)=&-\frac{1}{16\pi^{2}\alpha^{2}}\sinh^{-2}\left(\frac{\Delta\tau}{2\alpha}-\frac{i\epsilon}{\alpha}\right)\times\\\
&\left[1-\frac{\kappa\eta^{2}}{4\alpha^{2}}\sinh^{-2}\left(\frac{\Delta\tau}{2\alpha}-\frac{i\epsilon}{\alpha}\right)\right]\text{.}\end{split}$
(30)

One can make use of the identities,

$\displaystyle\sum\limits_{k=-\infty}^{\infty}(x-\pi ik)^{-2}$
$\displaystyle=\sinh^{-2}x\text{,}$
$\displaystyle\sum\limits_{k=-\infty}^{\infty}(x-\pi ik)^{-4}$
$\displaystyle=\frac{1}{3}(\tanh^{-2}x\sinh^{-2}x+\sinh^{-4}x)\text{,}$

to allow us to restate the $\sinh^{-4}$ term as,

$\sinh^{-4}x=\sum\limits_{k=-\infty}^{\infty}(x-\pi
ik)^{-4}-\frac{2}{3}\sum\limits_{k=-\infty}^{\infty}(x-\pi ik)^{-2}\text{,}$
(31)

and then write the propagator as,

$D^{+}(\Delta\tau)=-\frac{1}{4\pi^{2}}\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}\right)\sum\limits_{k=-\infty}^{\infty}(\Delta\tau-2i\epsilon-2\pi
i\alpha
k)^{-2}+\frac{\kappa\eta^{2}}{4\pi^{2}}\sum\limits_{k=-\infty}^{\infty}(\Delta\tau-2i\epsilon-2\pi
i\alpha k)^{-4}$ (32)

This expression for the propagator can be substituted back into Eq. 9, and
evaluated by reversing the order of the sum and the integral. To evaluate the
integral we choose a semi-circular contour around the multiple pole at
$\Delta\tau=2\pi i\alpha k$, closed in the upper half plane. The integral then
evaluates to $2\pi i$ times the residue at $2\pi i\alpha k$. These residues
are found by elementary means to be $-i(E-E_{0})e^{2\pi\alpha(E-E_{0})k}$, and
$\frac{i}{6}(E-E_{0})e^{2\pi\alpha(E-E_{0})k}$ respectively. We have for our
two integrals the values,

$\displaystyle\int\limits_{-\infty}^{\infty}\differential(\Delta\tau)\frac{e^{-i(E-E_{0})\Delta\tau}}{(\Delta\tau-2i\epsilon-2\pi
i\alpha k)^{2}}=2\pi(E-E_{0})e^{2\pi(E-E_{0})\alpha k}$
$\displaystyle\int\limits_{-\infty}^{\infty}\differential(\Delta\tau)\frac{e^{-i(E-E_{0})\Delta\tau}}{(\Delta\tau-2i\epsilon-2\pi
i\alpha k)^{4}}=-\frac{\pi}{3}(E-E_{0})^{3}e^{2\pi(E-E_{0})\alpha k}$

The sum over $k$ is from $k=-\infty$ to $k=\infty$, but our choice of contour
for the integral subtracts the contribution to the sum from $k\in[-\infty,0)$
leaving the result as the sum of a geometric series. One obtains for the
transition probability,

$\frac{c^{2}}{2\pi}\sum\limits_{E}\absolutevalue{\bra{E}m(0)\ket{E_{0}}}^{2}\left\\{\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}\right)\frac{E-E_{0}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\kappa\eta^{2}}{6}\right)\frac{(E-E_{0})^{3}}{e^{2\pi(E-E_{0})\alpha}-1}\right\\}\text{.}$
(33)

And correspondingly for the detector response function,

$\mathcal{F}(E)=\left\\{\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}\right)\frac{E-E_{0}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\kappa\eta^{2}}{6}\right)\frac{(E-E_{0})^{3}}{e^{2\pi(E-E_{0})\alpha}-1}\right\\}\text{.}$
(34)

The Planck factor is unaffected by the modification to the propagator, and the
presence of this factor suggests that the particles detected by the Unruh
detector are essentially thermal. This can be further reinforced by examining
the periodic behavior of the Green’s function. It is well known [20] that a
thermal Green’s function $G_{\beta}$ is periodic in imaginary time. In
particular it satisfies
$G_{\beta}^{\pm}(t,\mathbf{x};t^{\prime},\mathbf{x}^{\prime})=G_{\beta}^{\mp}(t+i\beta,\mathbf{x};t^{\prime},\mathbf{x}^{\prime})$,
where $\beta=(kT)^{-1}$. It will be noted that for our smoothly accelerating
detector the propagator Eq. 30 is composed of terms in $\sinh^{-2}(\Delta t)$.
It is well known that the hyperbolic functions are periodic with an imaginary
period of $2\pi i$, however, both of the $\sinh$ terms are squared. This has
the effect of changing the period to $\pi i$. Inspection of Eq. 30, reveals
that the period of the propagator is $2\pi\alpha$, indicating that
$\beta=2\pi\alpha$ as suggested by the Planck factor in Eq. 11.

##  Appendix C Computation of Hor̆ava-Lifshitz Corrections

As mentioned in Section I, and discussed in Section V, the Hor̆ava-Lifshitz
theory of gravity can introduce higher order terms in $p$ into the dispersion
relationship. In general the $p^{6}$ term has opposite sign to the $p^{4}$
term, and so to asses the effect of light speed propagation on the result
obtained in Section IV, we propose for a massless particle the following
dispersion relation,

$E^{2}=p^{2}+\kappa\eta^{2}p^{4}-\kappa\eta^{4}p^{6}\text{,}$ (35)

where $\eta^{4}$ is introduced on dimensional grounds.

We repeat our calculation in $3+1$ dimensions to obtain the $D^{+}(t,r)$
propagator as outlined in Section A.1. The crux of the calculation involves
expanding the denominator of the integral measure,
$E_{p}=\sqrt{p^{2}+\kappa\eta^{2}p^{4}-\kappa\eta^{4}p^{6}}$, for small
$\kappa\eta^{2}$. We proceed as before, but collect terms up to $p^{4}$, and
as such we now have,

$(E_{p})^{-1}\simeq
p\left[1-\frac{1}{2}\kappa\eta^{2}p^{2}+\frac{1}{2}\kappa\left(\frac{3}{4}\kappa+1\right)\eta^{4}p^{4}+\dots\right]\text{.}$

For brevity we will refer to
$\kappa^{\prime}=\kappa\left(\frac{3}{4}\kappa+1\right)$, but note for
$\kappa=1,\kappa^{\prime}=\frac{7}{4}$, and
$\kappa=-1,\kappa^{\prime}=-\frac{1}{4}$, preserving the opposite polarity of
the effect to the $p^{4}$ term.

Following the analysis in Section A.1, the expression for the space-like
propagator acquires an additional integral contribution,

$\displaystyle D^{-}(t,r)=~{}$
$\displaystyle\zeta\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n-3}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}$
$\displaystyle-\zeta\frac{\kappa\eta^{2}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}$
$\displaystyle+\zeta\frac{\kappa^{\prime}\eta^{4}}{2}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+5}{2}}J_{\frac{n-3}{2}}(pr)e^{-ipt}\text{,}$

after discarding the irrelevant contribution from expanding $E_{p}$ in the
exponential.

We can proceed identically for the third integral as undertaken for the
second, making use of the Bessel function recursion relationship Eq. 20 twice.
After manipulation the third integral is reduced to,

$\frac{3\zeta\kappa^{\prime}\eta^{4}}{2r^{2}}\int\limits_{0}^{\infty}\differential
p~{}p^{\frac{n+1}{2}}J_{\frac{n+1}{2}}(pr)e^{-ipt}\text{,}$

whose value can be read off from the standard integral Eq. 18, as,

$\frac{3\Gamma(\frac{n+2}{2})\kappa^{\prime}\eta^{4}}{2(\pi)^{\frac{n}{2}}(r^{2}-t^{2}+i\epsilon)^{\frac{n+2}{2}}}\text{.}$

Setting $n=4$, this can now be inserted into the expression for the
propagator, and for the time-like advanced propagator we have,

$D^{+}(t,r)=\frac{-\theta(\sigma^{2})}{4\pi^{2}(\sigma^{2}-i\epsilon)}\left\\{1-\frac{\kappa\eta^{2}}{(\sigma^{2}-i\epsilon)}+\frac{12\kappa^{\prime}\eta^{4}}{(\sigma^{2}-i\epsilon)^{2}}\right\\}\text{.}$
(36)

We note the positive sign for the $(\sigma^{2}-i\epsilon)^{-2}$ term, which
arises from the conversion of $(-\sigma^{2}+i\epsilon)$ to
$(\sigma^{2}-i\epsilon)$ inside an overall cubic denominator.

Turning to the computation of the detector response function, we follow the
calculation in Appendix B, noting that upon inserting the smoothly
accelerating trajectory we had an additional term in the propagator,

$\begin{split}&D^{+}(\Delta\tau)=-\frac{1}{16\pi^{2}\alpha^{2}}\sinh^{-2}\left(\frac{\Delta\tau}{2\alpha}-\frac{i\epsilon}{\alpha}\right)\times\\\
&\left[1-\frac{\kappa\eta^{2}}{4\alpha^{2}}\sinh^{-2}\left(\frac{\Delta\tau}{2\alpha}-\frac{i\epsilon}{\alpha}\right)+\frac{3\kappa^{\prime}\eta^{4}}{4\alpha^{4}}\sinh^{-4}\left(\frac{\Delta\tau}{2\alpha}-\frac{i\epsilon}{\alpha}\right)\right]\text{.}\end{split}$
(37)

Similar considerations to Eq. 31 allow us to substitute for the $\sinh^{-6}$
term the following infinite sum,

$\sinh^{-6}x=\sum\limits_{k=-\infty}^{\infty}(x-\pi
ik)^{-6}-\sum\limits_{k=-\infty}^{\infty}(x-\pi
ik)^{-4}-\frac{4}{5}\sum\limits_{k=-\infty}^{\infty}(x-\pi ik)^{-2}\text{.}$
(38)

The calculation proceeds identically and yields a similar result with
additional correction terms at all orders of $(E-E_{0})$ in the detector
response function,

$\mathcal{F}(E)=\left\\{\left(1+\frac{\kappa\eta^{2}}{6\alpha^{2}}-\frac{3\kappa^{\prime}\eta^{4}}{5\alpha^{4}}\right)\frac{E-E_{0}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\eta^{2}[\kappa+12\kappa^{\prime}\eta^{2}]}{6}\right)\frac{(E-E_{0})^{3}}{e^{2\pi(E-E_{0})\alpha}-1}+\left(\frac{\kappa^{\prime}\eta^{4}}{10}\right)\frac{(E-E_{0})^{5}}{e^{2\pi(E-E_{0})\alpha}-1}\right\\}\text{.}$
(39)

It was remarked in Appendix B that the propagator, being stated in terms of
$\sinh^{-2}$, is periodic with a period of $\pi i$, which strengthens the
interpretation of $2\pi\alpha$ as the temperature of the radiation. Although
we now have a term in $\sinh^{-4}$, the period is the same and we can conclude
that the propagator Eq. 37 satisfies
$G_{\beta}^{\pm}(t,\mathbf{x};t^{\prime},\mathbf{x}^{\prime})=G_{\beta}^{\mp}(t+i\beta,\mathbf{x};t^{\prime},\mathbf{x}^{\prime})$,
where $\beta=(kT)^{-1}=2\pi\alpha$.

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# Practice meets Principle: Tracking Software and Data Citations to Zenodo
DOIs

Stephanie van de Sandt
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland
<EMAIL_ADDRESS>Lars Holm Nielsen
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland
<EMAIL_ADDRESS>Alexandros Ioannidis
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland Dr. August Muench
American Astronomical Society
1667 K Street NW
Suite 800 Washington, DC 20006, USA Edwin Henneken
Center for Astrophysics
Harvard & Smithsonian
60 Garden Street
Cambridge, MA 02138, USA Dr. Alberto Accomazzi
Center for Astrophysics
Harvard & Smithsonian
60 Garden Street
Cambridge, MA 02138, USA Chiara Bigarella
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland Jose Benito Gonzalez Lopez
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland Sünje Dallmeier-Tiessen
CERN
Espl. des Particules 1
1211 Meyrin, Switzerland

###### Abstract

Data and software citations are crucial for the transparency of research
results and for the transmission of credit. But they are hard to track,
because of the absence of a common citation standard. As a consequence, the
FORCE11 recently proposed data and software citation principles as guidance
for authors.
Zenodo is recognized for the implementation of DOIs for software on a large
scale. The minting of complementary DOIs for the version and concept allows
measuring the impact of dynamic software.
This article investigates characteristics of 5,456 citations to Zenodo data
and software that were captured by the Asclepias Broker in January 2019. We
analyzed the current state of data and software citation practices and the
quality of software citation recommendations with regard to the impact of
recent standardization efforts. Our findings prove that current citation
practices and recommendations do not match proposed citation standards. We
consequently suggest practical first steps towards the implementation of the
software citation principles.

##  1 Introduction

Citations are recognized as one of the key incentives to make data and
software available, as researchers receive credit for their efforts (Fecher et
al. (2015), Huang et al. (2012), Wallis et al. (2013), Tenopir et al. (2011)).
However, the implementation of data and software citation capture as means of
accreditation is challenging, as data and software citations tend to be
idiosyncratic. Only little is known about how authors cite data and software
in publications and how the tracking of data and software usage can be
improved.
Empirical studies of data and software citation practices show a low coverage
of data citation databases such as the Data Citation Index (Peters et al.
(2016), Peters et al. (2017), Park & Wolfram (2019)) and a coexistence of
direct software citation practices and citations to software related research
publications (Li et al. (2019)). Thus, tracking citations is challenging
because of the great diversity of citable objects related to or representing
software. Additionally, citable software papers gain attention (Hong et al.
(2013)) and thus increase the diversity of targeted objects. This is an issue
for tracking use impact, as citation discovery systems track citation counts
to individual targeted objects, which often reflects only a fraction of real
software usage. Because of this “dilution of citations” over many individual
targets, citation counts in citation discovery services cannot reflect the
full impact of software. Besides this citation dilution, recent text analysis
studies have also demonstrated an inconsistency of data and software citation
formats (Yoon et al. (2019), Li et al. (2016), Li et al. (2017), Hwang et al.
(2017)) and a practice of informal data and software mentions in the main text
(Park et al. (2018), Li & Yan (2018), Howison & Bullard (2016), Pan et al.
(2015), Pia et al. (2009)). This increases the challenge of tracking data and
software usage, as informal mentions remain invisible to citation discovery
services. The absence of a widely adopted citation standard is stated as one
of the major blockers of direct data and software citations (Mooney & Newton
(2012)). To solve this, efforts towards the endorsement of citation principles
are being made. The joint declaration of data citation principles (Data
Citation Synthesis Group (2014)) covers purpose, function, and attribution of
data citations, but does not provide specific implementation mechanisms. For
software citations, the FORCE11 Software Citation Working Group published six
software citation principles (Smith et al. (2016)) as a base for future
concrete implementation plans.
However, despite all efforts, the tracking of the impact of data and software
based on individual citation counts remains problematic.

In January 2019, the CERN-hosted general-purpose repository Zenodo announced
the implementation of the Asclepias Broker (Nielsen (2019)) together with the
American Astronomical Society (AAS) and the NASA/SAO Astrophysics Data System
(ADS). The Asclepias Broker is at its core a citation links store that is
capable of harvesting discovery services such as the Astrophysics Data System,
Crossref Event Data and Europe PMC for citation links via their open public
APIs. The citation links found in the discovery services are either deposited
(Crossref Event Data) or text-mined from the reference section (ADS and Europe
PMC).
The tracking of citations to Zenodo objects (software, data, literature) is of
special interest, as Zenodo is widely recognized for hosting software (Brown
et al. (2018)) and because of Zenodo’s unique approach to versioning that
enables aggregating citations to multiple individual objects. According to
DataCite, Zenodo hosted more than 70% of all persistently citable software in
2018 and thus became a pioneer and key player for the DOI registration for
software (Fenner, Katz, Nielsen & Smith (2018)). Since 2014, software on
GitHub can be integrated in Zenodo in order to retrieve a DOI (GitHub (2016)).
Zenodo’s unique approach of minting a DOI for each version of an object and an
additional “concept DOI” representing all versions is especially suited for
dynamic software that often have complex versioning schemes. This feature
further facilitates tracking the impact of software, as citation counts to
multiple individual objects can be aggregated using the concept DOI and its
metadata links to the individual versions.
Thus, data on Zenodo’s citation counts is especially qualified for the
exploration of the impact of software citation recommendations (i.e. an
author-provided preferred citation) on actual citations.
This article explores the state of data and software citation practices by
analyzing the 5,456 citation links to Zenodo resulting from the implementation
of the Asclepias Broker. For the evaluation of the impact of recent software
citation recommendation standardization efforts, we compare our findings to
the quality of software citation recommendations.

##  2 Results

###  2.1 The state of data and software citation practices

Trackable data and software citations are not in common practice yet and we
can only observe a fraction of real usage. We observe multiple issues that
indicate how bad the state of data and software citations really is.

(a) dataset related DOIs (b) software related DOIs

Figure 1: Citation coverage of dataset and software related DOIs Figure 2:
Type of cited object Figure 3: Citation intensity Figure 4: Concept DOI
citation rate

We observe a very low citation coverage of Zenodo. Only 0.33% of all DOIs,
1.16% of dataset DOIs and 3.24% of all software DOIs registered by Zenodo are
traceably cited at least once (see Figure 1). 66.7% of all citations target
software, while only 16.8% target datasets and 16.6% target literature (see
Figure 2). Besides a low citation coverage, we also observe a low citation
intensity, as 87.2% of all cited DOIs were only cited once (see Figure 3).
Only a few DOIs of broadly reusable software libraries such as LMfit (Newville
et al. (2019)) received high citation counts up to 97. The average citation
count per cited DOI is 1.33. The majority of citations targeted version-
specific DOIs, which is in compliance with the software citation principles
(Smith et al. (2016)). The concept DOI was targeted by 8.31% of all citations
to objects with a concept DOI (see Figure 4), despite the fact that the
concept DOI is not very visible on Zenodo landing pages.

###  2.2 Most software and data citations in Zenodo are self-citations

In order to understand the context and circumstances of data and software
citations, we analyzed the citation speed and self-citation rate. The citation
speed measures the time between the publication dates of the cited objects and
the citing objects (i.e. how long does it take for an object to be cited).
We found varying citation speeds for different types of objects. For datasets
the average citation speed is 0.45 years while for software it is 0.86 years.
This means that datasets and software uploaded to Zenodo are likely to be
cited within the first year after publication.
We estimate with 95% confidence that 82.1% of all citations with a citation
speed between 0-1 years are self-citations (3,333 of 4,060 citations). The
estimated self-citation rate for datasets is much higher compared to other
types, as 98.5% of all analyzed citations to datasets proved to be self-
citations.

###  2.3 Uniqueness of citation data sources

All three above observations in our data indicate that we can only track
fractions of real data and software (re)usage. Issues with tracking citations
significantly decrease the coverage of citation discovery providers. It was
certainly expected that each of the discovery services (ADS, Crossref Event
Data, and Europe PMC) index citation links in content that is related or
characteristic for the community they serve. But in fact, the overlap of
citation links between services is so poor that we consider it almost non-
existent, as demonstrated in Figure 5. Unsurprisingly, only multi-disciplinary
open access journals with open reference lists qualify to be captured by all
discovery services. Thus, 18 of the 22 overlapping Zenodo data and software
citing articles are published with Springer Nature’s “Scientific Reports”.
However, the real overlap may be estimated to be much bigger.

Figure 5: Intersection of citing literature discovered by different data
providers

###  2.4 The potential impact of software citation recommendations

We analyzed the status and potential impact of citation recommendations (i.e.
author-provided “preferred citations”) for all 25 software projects archived
on Zenodo that received 10 or more citations. Citation recommendations are
meant to provide guidance and encourage citing software directly using
persistent identifiers (PIDs). Therefore, it is interesting to see if the
recommended identifier(s) were cited more frequently.

Recommended citations were often hard to find or when found outdated,
confusing, or nonexistent. We frequently had difficulties finding what seemed
to be “hidden” citation recommendations. Besides the GitHub ReadMe, software
may be documented in various places: compiled documentation (e.g. “Read the
Docs”), PDF manuals, etc. Given that we had to invest some time to find
information on how to give credit, these hidden citation recommendations are
vulnerable to be missed by authors. It becomes especially complicated when
developers change the name of the GitHub project or provide BibTeX snippets
with contradictory DOI and URL metadata fields, as it happened in the case of
“triangle / corner.py” (Foreman-Mackey et al. (2016)).

No trend towards changing recommendations in favor of citing the software
directly via Zenodo DOIs was found. Once a research article is published,
developers often recommend citing the article DOI instead of a Zenodo software
DOI. In the case of “SHTools” (Wieczorek et al. (2019)), the developers listed
both article and software DOIs as preferred citations for version 4.2, but
subsequent releases dropped the direct software DOI from the citation block.
On the other hand, some projects avoid dynamism as they recommend citation of
an old version, e.g. “LMfit” (Newville et al. (2019)). Six out of twenty-five
analyzed projects did not provide a recommendation at all. The Zenodo DOI
badge, which is a link the developer can add to the GitHub ReadMe, was not
counted as a recommended citation because it is not expressly declared as such
and may easily be overseen or misinterpreted.

Mixed forms of recommendations propose more than one identifier or include
contradictory statements. For example the software project “Flavio” (Straub et
al. (2019)) recommends to cite an article DOI, but also highlights a Zenodo
version DOI as a citable object.
Table 1 is a summary of our categorization of software citation
recommendations. The total number of citations is the sum over citations to
all versions and the concept DOI. The maximum age of the cited software
measures the currentness of the cited object. A low number indicates that a
recently published software version was cited. A high number indicates either
that a potentially outdated version has been cited or that the software
project was not updated for years. The diversity of a cited object expresses
how many identifiers out of the total number of identifiers related to a
software project were actually cited. The higher the count, the higher the
dilution of citations across many individual objects.
Overall, there is little evidence that citation recommendations have an impact
on the identifier choice. A more common phenomenon is that early version
identifiers hardly ever become extinct, independent from any recommendation.
Citation metadata in reference managers, once collected does not seem to be
updated regularly and early adopters keep citing the first version identifier.

Recommendation category (August 2019) | Name |  Citation count |  Max. age of cited object (in years) |  | Diversity of
---
cited object
| cited
---
DOIs
| existing
---
DOIs
Citation recommendation in place | Publication | GoaTools | 11 | 3 | 1 | 7
SHTools | 10 | 4 | 4 | 12
Software paper | Triangle/ Corner.py | 86 | 4 | 4 | 4
Concept DOI | sncosmo | 10 | 4 | 1 | 4
Most recent version DOI | Lasagne | 76 | 3 | 1 | 1
Hyperspy | 25 | 2 | 9 | 25
Python-Fsps | 17 | 4 | 1 | 1
Pv | 15 | 4 | 2 | 8
py-sphviewer | 10 | 3 | 1 | 1
Firehose | 10 | 3 | 1 | 1
Loprop for Dalton | 10 | 4 | 1 | 1
Old version DOI | LMfit | 99 | 4 | 3 | 22
| The version
---
that was used
Trackpy | 37 | 4 | 7 | 11
Mixed forms | Matplot | 20 | 2 | 5 | 40
Gatspy | 15 | 3 | 2 | 5
MixSIAR | 14 | 2 | 2 | 7
Pybinding | 13 | 2 | 2 | 10
Flavio | 12 | 2 | 8 | 50
NEST | 25 | 3 | 4 | 9
No recommendation |  DOI badge visible on GitHub ReadMe | Seaborn | 88 | 4 | 6 | 9
Plp | 20 | 4 | 3 | 10
Cantera | 18 | 1 | 1 | 3
Nothing | Hellwig | 18 | 3 | 1 | 1
Uves_Popler | 15 | 2 | 3 | 4
| Eddy-Covariance
---
Software TK3
11 | 3 | 1 | 1

Table 1: Software citation recommendation categorization

There are cases such as “LMfit”, for which 97 of 99 citations were to the
recommended but outdated version 0.8.0 DOI even after 4 years and the release
of 22 additional versioned software DOIs. This impression is supported by
Figure 6 and Figure 7, which demonstrates how an identifier choice evolves
over time for other packages. Even after many years and more recent versions
to cite, the first version identifier keeps getting cited. We speculate that
this latency is due more to static reference management systems (“BibTeX
latency”) than it is to a continued (re)use of old software. This speculation
is further justified when we find authors citing old software, but listing
more recent (and citable) versions in the full text.

Figure 6: Diversity of cited DOIs for “Trackpy” (Allan et al. (2019)) Figure
7: Diversity of cited DOIs for “Hyperspy” (Peña et al. (2019))

##  3 Discussion

Citation recommendations are meant to solve the problem of varying and thus
hard to catch citation formats. But our results indicate that citation
recommendations are in bad shape. They may even decrease the quality of
software citations.
Software developers and researchers often do not receive guidance to provide
high qualitative citation recommendations or how to use them. Despite the
efforts of the FORCE11 to establish standardized software citation principles,
software citation recommendations do not seem to change for the better. The
software citation principles clearly recommend citing a PID that indicates
which version of the software was actually used. In case there is an
accompanying software paper, it may be cited alongside the version PID. But so
far, the reality of software recommendations found in the wild greatly fails
this ideal. Recommendations proved to be outdated, hidden, rather recommend to
cite publications or do not even showcase a Zenodo DOI badge. Even if
developers are enthusiastic enough to provide recommendations, it is
tremendously hard to maintain them manually over the years.
In the end, this means that the right people will not receive credit for their
work, because citations are either not trackable or citations are attributed
to wrong versions with different author lists. Developer teams are just as
dynamic as the software they are working on, thus citing the correct version
ensures that the right team members receive credit instead of only e.g. the
initial authors.

The low citation coverage of Zenodo and the detected high self-citation rate
point out that the majority of data and software usage indications are still
hidden in informal mentions and indirect citations to related publications.
This also leads to developers not receiving credit for their work, because the
lack of ironclad citations means that informal mentions are not trackable.

The obstacles of citing the specific software version in publications may
still be too high to overcome, as authors may be overwhelmed with the variety
of potential identifiers or lack experience in citing software directly.
Reliable citation recommendations as guidance are crucial to help authors cite
software directly and thus increase the number of trackable citations, which
eventually leads to the right developers receiving credit.
Therefore, we suggest the following as a minimal first step to achieve
reliable and trackable citations and ultimately give credit to the right
people:

  * •

As a practical implementation, developers should recommend to cite one
persistent identifier that refers to the software. This identifier ideally
represents the whole software project, which is less prone to impermanence.
This recommendation does not require maintenance effort on the developer site,
but ensures that the software project is cited directly in a trackable way.

This first step is not compliant with the software citation principles, and is
prone to that the right people do not receive credit, but it achieves the
purpose of making the citation easily trackable. As the next level, developers
should recommend to cite the persistent identifier of the version that was
used. Additionally, if the software is cited in a general context only,
developers should recommend to cite the persistent identifier that represents
the whole software project.

However, enhanced standards alone will not lead to better citation practices.
Many players are involved in the dilution of citations. A citation evolves
through a complex chain link system, which is extremely vulnerable to errors.
Especially incorrect or ambiguous data and software citations are likely to be
missed out, as the chain link system is not prepared to deal with non-
traditional citations.

From the perspective of a discovery system, we observe that the technical
presentation of data and software citations often lacks quality and precision,
which complicates the parsing. During our analysis we have uncovered a number
of issues associated with the formatting of citations in the source papers.
Some of them indicate careless citation practices by the authors of the
papers, while others suggest problems in the editorial supply chain system.
Ambiguous citations that refer to more than one identifier cannot be assigned
to more than one cited object. Incorrectly structured data delivered by
editorial systems prevents the DOI in reference lists from being discovered.
Also, we observed that the publication dates of citing and cited objects are
not always trustworthy, as versioned objects were often associated with
incorrect metadata. All these technical issues decrease the capture rate of
each citation data provider, as many data and software citations are missed
out. The discovered technical issues complicate further the associated social
issues with software citation practices, which is beyond the limitations of
our paper. Overall, this again leads to the right people not receiving credit.

Incorrect and ambiguous citations have the potential of being trackable and
thus to highlight that credit has been given. However, this can only be
achieved by rigorous editorial and typesetting efforts. But, publishing
workflows are complicated and it is not trivial to reach out to every
individual involved in the process. Dedicated training must be provided not
only to editorial teams, but also to the publishing platform’s production
teams and conversion teams, which are almost beyond reach. After the
publication process, discovery systems are just as vulnerable to miss
citations and as a final step, repositories may fail to provide correct
citation metrics to the end users.
Thus, the overall picture is too complex to be changed by recommendations of
any kind only. However, we must understand the full extent of the problem
before we can apply changes to the whole chain link system, and before credit
is ultimately given where long due.

##  4 Methods

###  4.1 Citation links harvesting

We harvested citation links from three discovery systems: the NASA/SAO
Astrophysics Datasystem (ADS), Crossref Event Data and Europe PMC. Only
citation links with a target DOI in the DOI prefix 10.5281 (Zenodo’s DOI
prefix) were kept. Citation links were expressed in the Scholix Exchange
Format as a citation relationship between two persistent identifiers (e.g.
“arXiv ID cites DOI”).
Both ADS and Europe PMC do parse the reference sections of publications to
extract citation links. Europe PMC only extracts persistent identifiers from
the full-text, but does not validate the persistent identifier leading to
their data to contain invalid persistent identifiers (e.g “10.5281/zenodo.”).
ADS validates the persistent identifiers, which also allows them to detect
e.g. URLs in the reference strings and match them to a DOI. The collection we
used from Crossref Event Data depends on the reference lists that publishers
decide to make available and self-deposit for journal articles.

###  4.2 Deduplication of citation links

We collected metadata about title, authors, publisher, and publication date
from ADS, Crossref, DataCite, and Europe PMC for each persistent identifier in
a citation link relationship. Most importantly we also collected alternate
persistent identifiers. This allowed us to deduplicate citation links from
different providers. For instance, one provider might express a citation link
between a bibcode (an ADS identifier) and a DOI, while another provider might
express the exact same relationship between a DOI and DOI.

###  4.3 Cumulative citation counts

We further annotated each deduplicated citation link with its target “concept
DOI” if it existed in Zenodo. The “concept DOI” was obtained by querying
Zenodo’s REST API (the concept DOI is also expressed in the DataCite metadata
registered for Zenodo DOIs). This allowed us e.g. to investigate cumulative
citation counts for multiple persistent identifiers for the same software
project.

###  4.4 Software

The harvesting and deduplication of citation links explained above, was done
using an installation at CERN of the Asclepias Broker (Ioannidis et al.
(2019)), which was developed as part of the Asclepias Project. The project is
a Sloan-funded collaboration between American Astronomical Society, NASA/SAO
Astrophysics Data System and CERN / Zenodo (further information can be found
in (Henneken et al. (2017))). The core of the Asclepias Broker is a store for
citation links, the harvesting of citation links and metadata, as well as the
deduplication and cumulative citation counts as explained in the above
sections. The Asclepias Broker is open-source licensed under the MIT license.

###  4.5 Dataset

The dataset was retrieved from the Asclepias Broker early January 2019 after
having performed a full harvesting and deduplication cycle from a clean
database with zero citation links. The resulting 5,477 citation links were
cleansed for our analysis. Misspelled DOI citations were corrected and the
resulting duplicates deleted. 5,456 unique publication-to-software,
publication-to-data, and publication-to-literature links remained. Each row of
our dataset represents one unique link between a citing object and a Zenodo
DOI.

The dataset contains information on the provenance of the citation links
(field “providers”). A link can either be captured by Europe PMC, ADS
(NASA/SAO Astrophysics Data System), Crossref Event Data (crossref) or by a
combination of these providers.
Generally, the dataset contains metadata about the cited object (the target
object) and the citing object (the source object).

The citing object is identified by a “source identifier”, which is either a
DOI or an arXiv ID. All of our citing objects are of the “source type”
literature. Because every journal article or preprint can cite multiple Zenodo
objects, an identifier can appear multiple times as more than one citation
link was created. The “source publisher” contains information on the publisher
of the citing article or preprint and the “source publication year” indicates
the year the publication was published.
We made use of the re3data subject classification in order to classify the
disciplinary content of the citing object (“source discipline”). The subject
classification used at re3data offers four hierarchy levels of
(sub)disciplines of Engineering Sciences, Humanities and Social Sciences, Life
Sciences, and Natural Sciences. We used the second hierarchy level (14
categories) for our subject classification matching. The DOI prefix of the
citing object indicates the journal an article was published in (e.g.
“10.1103/PhysRevD” is exclusive for the journal Physical Review D). The
description of the journal’s aim and scope was then used as based for our
classification. arXiv preprints were classified using the arXiv subjects.

Equal information can be found on the cited object. The “target doi”’ contains
the identifier of the targeted object of the citation, which is always a
Zenodo DOI. The DOI may either be a version-specific DOI or a “concept DOI”,
which represents all versions of a project. The “target type” contains the
type of the cited object. This value can either be a “dataset”, “literature”,
or “software”. The “target title” includes the name string of the cited object
and the “target publication year” describes the publication year of the Zenodo
object. The publication year is defined by the user and does not necessarily
reflect the upload date on Zenodo. The “target version id” is a hash value
that refers to the concept of the project. Each version DOI and the concept
DOI of a project all refer to the same version identifier value. Citation
counts that are spread across versions can be summed up using the version id
value. The “citation speed” value was generated by subtracting the publication
date of the citing object from the publication date of the cited object. This
value indicates how many years it took for an object on Zenodo to get cited.

Additionally, we generated a second dataset with context information. We used
a Python script to harvest metadata via the Zenodo REST API and via the GitHub
REST API. This data was needed for the evaluation of potential correlations
between citations and usage metrics. For every cited unique Zenodo DOI, we
fetched basic metadata from Zenodo.
The “cited doi” value contains the DOI of the cited object. Every row of the
dataset represents information on a cited object DOI (in contrast to the
original dataset that is based on citation events instead of being object-
centered). The “concept doi” is either empty or contains the concept DOI.
Empty values mean that there is no concept DOI to be fetched due to the fact
that the concept DOI was introduced later. The column “citations” contains the
cumulated number of citations to a unique DOI. The number was generated by
summing up the number of citation links that contained identical target DOIs.
The “resource type” reflects the type of the cited object, which can be a
“dataset”, “publication”, or “literature”.

Usage metrics reflect how often an object may have been used. Zenodo delivers
information on downloads and views: “unique downloads”, “unique views”,
“version downloads”, “version views”, “version unique downloads”, “version
unique views”. Unique downloads and views count usage within one hour-sessions
of a unique visitor ID only once. Views and download statistics can be
aggregated to version usage using the concept ID. The version usage statistics
can be narrowed down to unique views and downloads per visitor ID per one-hour
session. Additionally, the “volume” and the “version volume” indicate the size
of an object. Zenodo counts usage actions by humans or machines, but excludes
double-clicks and robots.

For every cited DOI on Zenodo that is linked to a GitHub repository, we
fetched metadata from GitHub. The “github url” contains the link to the GitHub
project page. This value is empty for those Zenodo objects that do not link to
GitHub.
The dataset contains the following GitHub related usage metrics: “github
downloads”, “github forks”, “github stars”, “github subscribers”, and “github
watchers”. The variable GitHub downloads tells how many times a repository was
downloaded. GitHub forks count the number of a repository’s copies made by
users. GitHub stars describe how many users starred the repository.
Subscribers count the number of subscriptions to conversations in issues, pull
requests and team discussions of a GitHub repository. GitHub watchers are
users who gets notified of activity in a repository, but are not
collaborators.
Additionally, “github languages”, “github licenses”, and “github topics”
contain information about the software. GitHub languages contain the
(programming or markup) languages of a repository. A repository may contain
one or several languages. Each value is separated by a comma. In case a
license was assigned to the project, the value according to the GitHub license
classification is given in “github license”. A repository has only one or none
licenses. The content of a GitHub project may be classified by the developer
using topics. The content of the field is written in square brackets, which
can be empty or contain one or multiple comma-separated values.

###  4.6 Data analysis

We calculated basic statistics for our dataset such as the citation coverage,
the citation frequency, and the average citation count and the distribution of
citations across types.
The citation coverage of Zenodo was calculated using the number of unique DOIs
(4,108) that were cited in comparison to the total number of Zenodo DOIs
registered at DataCite in June 2019 (1,236,264). We used the number of
citations to each type (dataset, literature, software) for the distribution of
citations across types. The citation intensity describes how often a Zenodo
DOI was cited. The citation counts for each cited Zenodo DOI were calculated
by summing up the number of rows that contain the same Zenodo DOI (the number
of duplicates). Out of 4,108 Zenodo DOIs that were cited at least once, 3,582
DOIs were cited only once. The average citation count was calculated using the
arithmetic mean of the accumulated citation counts. We used a Python script to
harvest metadata from the Zenodo REST API and retrieved information on how
many cited Zenodo DOIs are linked to a concept DOI. Out of 4,108 Zenodo DOIs,
2,395 had a value for the concept DOI. In 233 cases, the string of the cited
Zenodo DOI was identical to the concept DOI, which means that the concept DOI
was cited.

The self-citation rate was estimated on the base of the “citation speed”. The
citation speed indicates how many years passed between the publication date of
the cited object on Zenodo and the publication date of the citing object. This
measure was calculated by subtracting the year of the publication date of the
cited object from the publication date of the citing object (because citations
generally look backward). Bigger numbers mean that more years have passed
after the cited object was published on Zenodo. We assume that it takes time
for third-parties to reuse datasets of software for different projects, as
they have to fully understand what was done before and then need some time for
a new analysis and for writing an article. Therefore, citations with a low
citation speed qualify to be potential self-citation candidates. We selected
all 4,060 rows with a citation speed with 0 or 1 (which means that the Zenodo
object was cited in the same year or within a year after the publication on
Zenodo) and analyzed a representative sample on their self-citation rate. The
minimum size of a representative sample was determined on a 95% confidence
interval. In our case, the minimum sample size is 352. The authors of the
citing articles of these 352 citations were manually compared to the authors
of the cited object on Zenodo. In case of a self-citation, both author sets
share at least one value. 289 out of 352 proved to be self-citations according
to our definition. This value allows us to estimate with 95% confidence on the
whole population.

The correlation between citation rate and usage metrics were calculated using
metadata retrieved from Zenodo and GitHub via their REST APIs.
We calculated the Spearman’s rank correlation coefficient for each usage
metric and the number of citations to a unique Zenodo DOI. We chose Spearman’s
rank correlation because it compares the rank values between two variables
instead of assessing linear relationships. This fitted our sample well,
because it contains many extreme values and outliers as few general purpose
software libraries were highly cited.
As usage metrics, we chose the unique view and unique download counts provided
by Zenodo, which are computed based on the COUNTER Project Code of Practice
for Research Data (Fenner, Lowenberg, Jones, Needham, Vieglais, Abrams, Cruse
& Chodacki (2018)). This value is deduplicated by accumulating actions of a
unique visitor within one-hour sessions. We calculated the coefficient using
all cited objects and then run our calculations on every type (dataset,
literature, software) separately.
We calculated the Spearman’s rank correlation coefficient for the accumulated
citation count of every cited Zenodo DOI that is linked to Github and usage
statistics retrieved from GitHub: forks, stars, subscribers, and watchers.

The overlap of the indexes of every citation discovery service (ADS, Europe
PMC), was visualized using a venn diagram. The set of each provider consists
of the citing object’s identifier. Thus, the overlap of sets indicate how many
citing articles are found by more than one citation discovery service.

For our analysis of the state of software citation recommendations, we
accumulated the citation counts for every concept using the “target version
id” hash value of our dataset. We considered all software concepts that had a
citation count of 10 or higher. For these 25 software concepts, we categorized
the state of their recommendations as it was presented to us in August 2019.
We always considered the text on the ReadMe page of the project’s master
branch and read every documentation we found. Precisely, we followed links
given on GitHub and googled the software. For every citation recommendation we
found, we additionally tracked the change history across every version release
in order to make statements on their dynamism.
Because of the concept DOI and its metadata links to the individual versions
on Zenodo, we were able to analyze the diversity of cited DOIs and observe the
time course of every version DOI’s usage. The existing DOIs were counted on
Zenodo. The count includes all version DOIs and the concept DOI. Manual effort
was needed for the accumulation, as some projects have more than one concept
DOI. Contrasting, we counted the number of actually cited DOIs.
The maximum age of the cited object indicates how old the cited software
version was.

## Data Availability

The unprocessed dataset that was retrieved from the Asclepias Broker in early
January 2019 is publicly available on Zenodo (Ioannidis (2019)).
The processed dataset we used for our analysis and the dataset containing
usage metrics are published on Zenodo as well (van de Sandt, Ioannidis &
Nielsen (2019)).

## Code Availability

The open source code of the Asclepias Broker is publicly available on Zenodo
(Ioannidis et al. (2019)).
The Python script used for the venn diagram and for fetching metadata from
Zenodo and Github can be found on Zenodo (van de Sandt, Nielsen & Ioannidis
(2019)).

## Acknowledgement

The Asclepias project is funded by the Alfred P. Sloan Foundation.

This work has been sponsored by the Wolfgang Gentner Programme of the German
Federal Ministry of Education and Research (grant no. 05E15CHA).

Zenodo has received funding from the European Union’s Horizon 2020 research
and innovation programme under grant agreements (OpenAIRE-Advance) no. 777541
and (OpenAIRE-Connect) no. 731011

## Competing interests statement

The authors have no competing interests.

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