hep-ph/***

FTUAM-23-xx IFT-UAM/CSIC-23-133

Where the cosmological constant comes from in unimodular gravity.

Enrique Álvarez, Jesús Anero and Irene Sánchez-Ruiz.

Departamento de Física Teórica and Instituto de Física Teórica, IFT-UAM/CSIC,

Universidad Autónoma de Madrid, Cantoblanco, 28049, Madrid, Spain

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>and
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###### Abstract

It is well-known that in unimodular gravity the cosmological constant is not
sourced by a constant energy density, but rather appears as some sort of
integration constant. In this work we try to flesh this out by studying in
some detail a couple of examples, one from cosmology and the other from
gravitational collapse.

###### Contents

  1. 1 Introduction and conclusions.
  2. 2 Cosmological solutions
    1. 2.1 Vacuum solutions.
    2. 2.2 The line $p+\rho=C$.
    3. 2.3 Unimodular gravity versus General Relativity.
  3. 3 Spherical collapse
    1. 3.1 A short review of the GR spherical collapse.
    2. 3.2 Unimodular presureless collapse.
      1. 3.2.1 Expansion in ${\Lambda\over\kappa^{2}\rho_{0}}$
      2. 3.2.2 Matching with the exterior solution.
    3. 3.3 UG with incompressible fluid (constant density)
      1. 3.3.1 Matching with the exterior solution.
  4. 4 Acknowledgements

##  1 Introduction and conclusions.

Unimodular Gravity (UG) is a modification of General Relativity (GR) where
only unimodular metrics (determinant, $g\equiv\left|det\,g_{\mu\nu}\right|=1$)
are considered. Even in the path integral we are instructed to integrate over
unimodular metrics only. The equations of motion (EM) of UG are not identical
to GR Einstein’s ones with the same source. In fact the second Noether theorem
(Bianchi identities) allows to recover the lost trace as an integration
constant. What happens then is that given a fixed value for the cosmological
constant (CC) (for example $\Lambda=0$) any solution of GR is also a solution
of UG, but the converse is untrue. If we represent by $E$ the space of
classical solutions of the EM, then

$E_{GR(CC=\Lambda)}\subset E_{UG}$ (1)

and, somewhat symbolically,

$\sum_{\Lambda\in\mathbb{R}}E_{GR(CC=\Lambda)}\equiv E_{UG}$ (2)

From the physical point of view, UG is interesting because the value of the CC
is not determined by the constant vacuum energy, which does not weigh at all
in UG. The natural question is then, what determines the value of the CC in
UG?.

In this paper we want to study this question under two different aspects, the
first one dealing with cosmological solutions and the second one dealing with
spherical collapse.

To begin with, in [1] we asked the same question in the framework of standard
cosmology. We found there (and we review here) that in vacuum there are two
solutions: one in which the scale factor is constant (which corresponds to
flat space)

$b(t)=b_{0}$ (3)

and another where

$b(t)=(3t-t_{0})^{4/3}$ (4)

In this case at least, the divide between these two solutions lies in whether
the initial condition on the derivative of the scale factor $\dot{b}(t_{0})$
is different from zero or not. This initial condition on a vacuum solution is
admissible only in UG.

Our second subject is the detailed study of gravitational collapse. This is of
course a complicated aspect, but it is believed that all $j\geq 2$ multipoles
of the matter are eventually radiated away, and that the final state is
stationary, with axial symmetry [5]. In this paper we shall consider the
simplest models of spherical collapse of a compact matter: the one originally
proposed by Schwarzschild [9], where the source is assumed to be an
incompressible perfect fluid (that is, with constant density) and another
where the source is assumed to be a presureless perfect fluid (a.k.a.
dust)[11].

What we find is that in GR the energy density of the collapsing cloud $\rho$
is bound to be related to the curvature by

$\rho=-{R\over\kappa^{2}}$ (5)

owing to Einstein’s equations. In UG this is not the case, and in fact

$\rho=-{R\over\kappa^{2}}+\text{constant}$ (6)

(note, however that the derivative $\dot{\rho}$ obeys the same restriction as
in GR). In fact this constant is intimately related to the value of the CC.

##  2 Cosmological solutions

The Friedmann-Lemaître metric in the unimodular gauge of GR (which is the only
admissible metric in UG cf. for example [2]) reads

$ds^{2}=b(t)^{-3/2}\,dt^{2}-b(t)^{1/2}\,\delta_{ij}dx^{i}dx^{j}$ (7)

where the function $b$ depends on time only, $b=b(t)$. The cosmic normalized
four velocity vector field, $u^{\mu}u_{\mu}=1$, is given explicitly by

$u^{\mu}=\left(b^{3/4},0,0,0\right)$ (8)

It is easy to check that this congruence is geodesic

$\dot{u}^{\mu}\equiv u^{\nu}\nabla_{\nu}u^{\mu}=0$ (9)

and the expansion reads

$\theta\equiv\nabla_{\mu}u^{\mu}=\frac{3}{4}b^{-1/4}\frac{db}{dt}$ (10)

The equation of motion in UG (the traceless piece of Einstein’s) reads

$R_{\mu\nu}-\dfrac{1}{4}R\,g_{\mu\nu}=\kappa^{2}\left(T_{\mu\nu}-\dfrac{1}{4}\,T\,g_{\mu\nu}\right)$
(11)

where the scalar curvature

$R=-\frac{3}{8\sqrt{b}}\left[\left(\frac{db}{dt}\right)^{2}+4b\frac{d^{2}b}{dt^{2}}\right]$
(12)

and we assume matter as a perfect fluid, that is

$T_{\mu\nu}\equiv\left(\rho+p\right)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (13)

the energy-momentum conservation, $\nabla_{\nu}T^{\mu\nu}=0$ is then
equivalent to

$\dot{\rho}+\left(\rho+p\right)\theta=0$ (14)

Using the equation of motion (11) and the property (9), Raychaudhuri’s
equation [8] reduces to

$\dot{\theta}+\frac{1}{n-1}\theta^{2}+\sigma_{\alpha\beta}\sigma^{\alpha\beta}-\omega_{\alpha\beta}\omega^{\alpha\beta}+\frac{1}{n}R+\frac{2(n-1)}{n}\kappa^{2}(\rho+p)=0$
(15)

The scalar curvature reads

$R=-2u^{\mu}\nabla_{\mu}{\theta}-\frac{4}{3}\theta^{2}$ (16)

Assuming as usual for simplicity vanishing shear and rotation,
$\sigma_{\alpha\beta}=\omega_{\alpha\beta}=0$, and in the physical dimension
$n=4$

$\dot{\theta}+3\kappa^{2}(\rho+p)=0$ (17)

It is worth remarking that it is not possible to express $R$ in terms of $T$.

We can use Ellis’ clever trick [6] to define a length scale through

$\theta=3{\dot{l}\over l}$ (18)

Then

$b\sim l^{4}.$ (19)

Finally we can write Raychaudhuri’s equation (17) as

$u^{\mu}u^{\nu}\Big{[}l\nabla_{\nu}\nabla_{\mu}{l}-\nabla_{\mu}l\nabla_{\nu}l\Big{]}+\kappa^{2}(\rho+p)l^{2}=0$
(20)

###  2.1 Vacuum solutions.

Vacuum implies $p=\rho=0$ and Raychaudhuri’s equation reduces to

$\dot{\theta}=0$ (21)

In our case

$\dot{\theta}=u^{\nu}\nabla_{\nu}\nabla_{\mu}u^{\mu}=-\frac{3}{16\sqrt{b}}\left[\left(\frac{db}{dt}\right)^{2}-4b\frac{d^{2}b}{dt^{2}}\right]$
(22)

It follows that

$\left(\frac{db}{dt}\right)^{2}-4b\frac{d^{2}b}{dt^{2}}=0$ (23)

Its general solution is given by either a constant

$b=H_{0}^{4\over 3}$ (24)

which corresponds to flat space; or else

$b(t)=H_{0}^{4\over 3}\left(3t-t_{0}\right)^{4\over 3}$ (25)

which corresponds to de Sitter 111 In unimodular coordinates, the maximally
symmetric, constant curvature de Sitter spacetime reads
$ds^{2}=\left({dt\over 3Ht}\right)^{2}-(3Ht)^{2/3}\delta_{ij}\,dx^{i}dx^{j}$
(26) that is, precisely  $b(t)\sim\,t^{4\over 3}$ (27) space; $H_{0}\equiv
3\theta$ being the constant expansion. In this solution it is arbitrary,
because there is no physical scale in the problem that determines it. It is to
be emphasized that this solution depends on two parameters, whereas the
constant solution depends only on one, being thus less generic.

This could be anticipated, because the vacuum EM in unimodular gravity are
just Einstein spaces

$R_{\mu\nu}={1\over 4}Rg_{\mu\nu}$ (28)

Flat space is just a quite particular solution; constant curvature space-times
[12] are a more generic one.

###  2.2 The line $p+\rho=C$.

Let us examine the inhomogeneous equation of state $\rho+p=C$. This is one of
the most interesting results of UG, where physics depends on the value of the
constant $C$ only.

Indeed, depending on the value of the constant $C$, it is possible that both
$p$ and $\rho$ are positive. Only in the case $C=0$ is this situation strictly
equivalent to a vacuum energy density.The differential equation (20) then
reads

$\frac{1}{l}{d^{2}l\over dt^{2}}-\frac{1}{l^{2}}\left({dl\over
dt}\right)^{2}+C\kappa^{2}l^{-4}=0$ (29)

whose general solution reads

$l_{C}(t)=e^{-\sqrt{C_{2}}(t+C_{3})}\Big{\\{}\frac{1}{6C_{2}}\Big{[}e^{6\sqrt{C_{2}}(t+C_{3})}-3C\kappa^{2}C_{2}\Big{]}\Big{\\}}^{1/3}$
(30)

Obviously when $C=0$ this solution reduces to the vacuum solution

$l_{\Lambda}(t)=l_{0}e^{H_{0}t}$ (31)

with $C_{3}=0$ and $C_{2}^{2}=H_{0}$.

More interestingly, this solution is an atractor asymptotically when
$t\rightarrow\infty$. Any solution tends asymptotically to de Sitter.

For $CC_{2}>0$ there is an origin of time $t_{0}$. For earlier times $t<t_{0}$
the solution becames unphysical. To be specific,

$t_{0}={1\over 6\sqrt{C_{2}}}\log\,\left(3C\kappa^{2}C_{2}\right)-C_{3}$ (32)

###  2.3 Unimodular gravity versus General Relativity.

The unimodular gauge of General Relativity (GR) is of course fully equivalent
to the usual formulation of GR in comoving coordinates [4, 10] where the
metric reads

$ds^{2}=d\tau^{2}-a(\tau)^{2}\sum\delta_{ij}dx^{i}dx^{j}$ (33)

with a four velocity

$u^{\mu}=(1,0,0,0)$ (34)

and

$\dot{u}^{\mu}=0$ (35)

so that in this case

$\theta=3\frac{1}{a}\frac{da}{dt}$ (36)

We insist that he only difference between GR and UG stems from the EM. Let us
spell this out in some detail.

Now the equation of motion is the usual Einstein one

$R_{\mu\nu}-\dfrac{1}{2}R\,g_{\mu\nu}=\kappa^{2}T_{\mu\nu}$ (37)

in this case the scalar of curvature reads

$R=-\frac{6}{a^{2}}\left[\left(\frac{da}{dt}\right)^{2}+a\frac{d^{2}a}{dt^{2}}\right]$
(38)

Then Raychaudhuri’s equation in comoving coordinates yields

$-3\left(\frac{da}{dt}\right)^{2}+2a^{2}\kappa^{2}\rho=0$ (39)

In this case, the vaccum solution reduces to

$\frac{da}{dt}=0$ (40)

i.e $\theta=0$ which is just flat spacetime, it is a subset of the UG result,
$\dot{\theta}=0$

##  3 Spherical collapse

We shall present two very simplified analysis. The first one corresponds to
what astrophysicists call dust (that is pressureless matter) cf. for example
[11]. The second analysis follows closely the classic approach pioneered by
Schwarzschild [9] and later by Oppenheimer and Volkoff [7] for an
incompressible fluid (which means constant density).

The main idea is the following. It is assumed the existence of a radial
coordinate, $r$, such that at a given value $r=a$ a matching can be performed
between the interior solution sourced by the energy momentum tensor of the
perfect fluid, and an exterior solution, which according to Birkoff’s theorem
[3] must be Schwarzschild’s.222 In fact a careful statement of Birkhoff’s
theorem [3] allows for Schwarzschild-de Sitter (SdS) or Schwarzschild-anti de
Sitter (SadS) as exterior solutions. The first thing to note is that the
interior solution is not Ricci flat (in fact the trace of the GR EM implies
that the curvature scalar is given by $R=-\kappa^{2}T$, where $T$ is the trace
of the energy-momentum tensor. For the equation of state this is just

$R=-\kappa^{2}\left(\rho-3p\right).$ (41)

This is at variance with the exterior solution, which is indeed Ricci flat
according to Birkhoff’s theorem. Then in the matching there is necessarily a
discontinuity in the second derivatives of the metric.

As we shall see in the next paragraph, this is not so in UG, where the
interior metric is allowed to be Schwarzschild-(anti)-de Sitter, so that

$R=-\kappa^{2}\rho(p)-4\Lambda$ (42)

where $\Lambda$ is the cosmological constant. What happens is that the UG EM
are traceless, which loosens somewhat the constraint on the scalar curvature,
allowing it to be modified by a constant term, precisely related to the CC.
The main purpose of our work is to understand better the physical meaning of
the CC from the collapsing matter viewpoint.

Let us first analyze some general properties of the energy-momentum tensor.
For matter which can be modelled as a perfect fluid

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (43)

Covariant conservation of the energy-momentum tensor
$\nabla_{\nu}T^{\mu\nu}=0$ is equivalent to

$\nabla^{\lambda}p=\left(\dot{\rho}+\dot{p}\right)u^{\lambda}+\left(\rho+p\right)\left(\theta
u^{\lambda}+\dot{u}^{\lambda}\right)$ (44)

In the synchronous gauge this implies

$\dot{\rho}+(\rho+p)\theta=0$ (45)

###  3.1 A short review of the GR spherical collapse.

  * •

When dealing with an incompressible fluid, where $\rho=\rho_{0}$, the metric
reads [9]

$ds^{2}=f_{4}dx_{4}^{2}-f_{1}dx_{1}^{2}-f_{2}\Big{[}\frac{dx_{2}^{2}}{1-x_{2}^{2}}+(1-x_{2}^{2})dx_{3}^{2}\Big{]}$
(46)

where the functions $f_{i}$ depend on $x_{1}$. If we choose to work in the
unimodular gauge of GR (as Schwarzschild did) we have to impose

$f_{4}f_{1}f_{2}^{2}=1$ (47)

The conservation of energy-momentum tensor reads

$\left(\rho_{0}+p\right)\sqrt{f_{4}}=\gamma\,\text{(constant)}$ (48)

The interior solution turns out to be

$\displaystyle
ds^{2}=\left(\frac{3\cos\chi_{a}-\cos\chi}{2}\right)^{2}dt^{2}-\frac{3}{\kappa^{2}\rho_{0}+\Lambda}\left[d\chi^{2}+\sin^{2}\chi\Big{[}\frac{dx_{2}^{2}}{1-x_{2}^{2}}+(1-x_{2}^{2})dx_{3}^{2}\Big{]}\right]$
(49)

whereas the exterior solution reads

$ds^{2}=\Big{[}1-\frac{r_{s}}{\bar{r}}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{\Big{[}1-\frac{r_{s}}{\bar{r}}\Big{]}}-\bar{r}^{2}\Big{[}\frac{dx_{2}^{2}}{1-x_{2}^{2}}+(1-x_{2}^{2})dx_{3}^{2}\Big{]}$
(50)

Matching both solutions, at the surface of the sphere $\bar{r}_{a}$, leads to

$\cos^{2}\chi_{a}=1-\frac{r_{s}}{\bar{r}_{a}}$ (51)

where

$\bar{r}_{a}=\sqrt{\frac{3}{\kappa^{2}\rho_{0}}}\sin\chi_{a}$ (52)

The total mass of our sphere will be

$M=\frac{3}{4\kappa^{2}}\sqrt{\frac{3}{\kappa^{2}\rho_{0}}}\left(\chi_{a}-\frac{1}{2}\sin
2\chi_{a}\right)$ (53)

  * •

Assume now that the collapsing cloud is made of dust ($p=0$), with synchronous
metric [11]

$ds^{2}=dt^{2}-f(r)S^{2}(t)dr^{2}-S^{2}(t)r^{2}d\Omega_{2}^{2}$ (54)

We have

$\displaystyle\theta=3{\dot{S}\over S}$ $\displaystyle\dot{u}^{\lambda}=0$
(55)

Conservation of the energy-momentum tensor implies that

$\rho(t)=\rho(0)S^{-3}(t)$ (56)

The interior solution, valid for $r\leq a$, reads

$ds^{2}=dt^{2}-S^{2}(t)\Big{[}\frac{dr^{2}}{1-\lambda
r^{2}}+r^{2}d\Omega^{2}_{2}\Big{]}$ (57)

where $S(t)$ stands for

$\displaystyle S[\psi[t]]=\frac{1}{2}\left(1+\cos[\psi[t]]\right)$
$\displaystyle t[\psi]=\left(\frac{\psi+\sin\psi}{2\sqrt{C}}\right)$ (58)

with $C=\frac{\kappa^{2}\rho_{0}}{3}$

The metric outside the sphere $r\geq a$ to be matched with this interior one
must be the usual Schwarzschild metric owing to Birkhoff’s theorem

$ds^{2}=\Big{[}1-\frac{r_{s}}{\bar{r}}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{\Big{[}1-\frac{r_{s}}{\bar{r}}\Big{]}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(59)

The actual matching of both solutions, in the surface of the sphere
$\bar{r}=S(t)a$, leads to

$M=\rho_{0}\frac{4\pi}{3}a^{3}$ (60)

This metric can be easily written in the unimodular gauge by using new
coordinates $x$ and $\tau$ such that the dependence between new and old
coordinates reads $r(x)$ and $t(\tau)$. Then $dr=r^{\prime}(x)dx$ and
$dt=t^{\prime}(\tau)d\tau$ the metric becomes

$ds^{2}=\left[t^{\prime}(\tau)\right]^{2}d\tau^{2}-S^{2}[t(\tau)]\Big{[}\frac{\left[r^{\prime}(x)\right]^{2}}{1-\lambda
r^{2}(x)}dx^{2}+r^{2}(x)d\Omega^{2}\Big{]}$ (61)

with the unimodular condition

$1=\left[t^{\prime}(\tau)\right]^{2}S^{6}[t(\tau)]\frac{\left[r^{\prime}(x)\right]^{2}}{1-\lambda
r^{2}(x)}r^{4}(x)$ (62)

we split in two equations

$\displaystyle\left[t^{\prime}(\tau)\right]^{2}S^{6}[t(\tau)]=C$
$\displaystyle\frac{\left[r^{\prime}(x)\right]^{2}}{1-\lambda
r^{2}(x)}r^{4}(x)=\frac{1}{C}$ (63)

Therefore the unimodular metric, with $C=1$, is

$ds^{2}=\frac{d\tau^{2}}{S^{6}[t(\tau)]}-S^{2}[t(\tau)]\Big{[}\frac{dx^{2}}{r^{4}(x)}+r^{2}(x)d\Omega^{2}\Big{]}$
(64)

It should be noted that the trace of Einstein’s equations

$R+\kappa^{2}\rho-3p=0$ (65)

is automatically enforced at all points, so the density at the surface (the
point where the interior solution should be matched with the exterior
solution) is not arbitrary, but determined by the geometry through the trace
equation, as we have pointed out already.

###  3.2 Unimodular presureless collapse.

After this short review, let us repeat the analysis in UG using the same
simplifying hypothesis on the matter source as before.

First, we assume $\rho(r,t)$ and $p(r,t)=0$ and the unimodular metric to be

$ds^{2}=A^{2}(r,t)dt^{2}-B^{2}(r,t)dr^{2}-\frac{1}{A(r,t)B(r,t)}\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(66)

In this case, the conservation of energy-momentum tensor reduces to

$\displaystyle\nabla_{\mu}T^{\mu\nu}=u^{\nu}u^{\mu}\nabla_{\mu}\rho+\rho\left[u^{\nu}\nabla_{\mu}u^{\mu}+u^{\mu}\nabla_{\mu}u^{\nu}\right]$

The timelike component yields

$\displaystyle\dot{\rho}-\frac{\dot{A}}{A}\rho=0$ (68)

and the radial one

$A^{\prime}=0$ (69)

implies $A(r,t)=A(t)$ so that the general solution reads

$\rho(r,t)=\rho_{0}(r)A(t)$ (70)

The UG EM read

$R_{\mu\nu}-\frac{1}{4}g_{\mu\nu}R=\kappa^{2}\left(T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\right)$
(71)

therefore the Bianchi identity yields

$R+\kappa^{2}\rho=-4\Lambda$ (72)

Assume again a separable solution

$\displaystyle A(t)=S^{\alpha}(t)\hskip 56.9055ptB(r,t)=b(r)S^{\beta}(t)$ (73)

the diagonal part of the EM has two properties, spherical symmetry and
tracelessness, to wit

$\displaystyle(\alpha(4+5\alpha)+6\alpha\beta+9\beta^{2})\left[S^{\prime}(t)\right]^{2}-4\alpha
S(t)S^{\prime\prime}(t)=0$
$\displaystyle(\alpha(4+7\alpha)+2(4+9\alpha)\beta+3\beta^{2})\left[S^{\prime}(t)\right]^{2}-4(\alpha+2\beta)S(t)S^{\prime\prime}(t)=0$
$\displaystyle-(\alpha^{2}+6\alpha\beta+(4-3\beta)\beta)\left[S^{\prime}(t)\right]^{2}+4\beta
S(t)S^{\prime\prime}(t)=0$ (74)

We have to remember that only two of these three equations are independent. On
the other hand the $rt$ component

$\left(3\beta+\alpha\right)\frac{b^{\prime}}{b}=0$ (75)

Let us now postulate, as usual, $\alpha=-3$ and $\beta=1$ [10].

$\displaystyle A(r,t)=\frac{1}{S^{3}[t(\tau)]}$ $\displaystyle
B(r,t)=\frac{S[t(\tau)]}{b^{2}(r)}$ (76)

i.e the metric is

$ds^{2}=\frac{d\tau^{2}}{S^{6}[t(\tau)]}-\frac{S^{2}[t(\tau)]}{b^{4}(r)}dr^{2}-b^{2}(r)S^{2}[t(\tau)]\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(77)

If we combine the equation of motion and the Bianchi identity, we obtain

$S^{2}[t(\tau)]\Big{\\{}\Lambda+\frac{\kappa^{2}\rho_{0}}{S^{3}[t(\tau)]}-3S^{4}[t(\tau)][S^{\prime}[t(\tau)]]^{2}\left[t^{\prime}(\tau)\right]^{2}\Big{\\}}=\frac{3}{b^{2}(r)}\left[1-b^{4}(r)[b^{\prime}(r)]^{2}\right]$
(78)

where we denote $S^{\prime}[t(\tau)]=\frac{S[t(\tau)]}{dt}$ and the relation

$1+b^{4}(r)[b^{\prime}(r)]^{2}+b^{5}(r)b^{\prime\prime}(r)=0$ (79)

The unimodular condition (• ‣ 3.1)

$\displaystyle\left[t^{\prime}(\tau)\right]^{2}S^{6}[t(\tau)]=1$ (80)

Then

$\Lambda
S^{2}[t(\tau)]+\frac{\kappa^{2}\rho_{0}}{S[t(\tau)]}-3[S^{\prime}[t(\tau)]]^{2}=\frac{3}{b^{2}(r)}\left[1-b^{4}(r)[b^{\prime}(r)]^{2}\right]$
(81)

with

$\frac{3}{b^{2}(r)}\left[1-b^{4}(r)[b^{\prime}(r)]^{2}\right]=\kappa^{2}\rho_{0}$
(82)

Eventually the complete equation reads

$\Lambda
S^{2}[t]+\frac{\kappa^{2}\rho_{0}}{S[t]}-3[S^{\prime}[t]]^{2}=\kappa^{2}\rho_{0}$
(83)

It is plain that in the case $\Lambda=0$, we recover the GR solution.

It is easy to find a formal solution of

$\Lambda
S^{2}[\psi[t]]+\frac{\kappa^{2}\rho_{0}}{S[\psi[t]]}-3[S^{\prime}[\psi[t]]]^{2}=\kappa^{2}\rho_{0}$
(84)

namely

$\displaystyle S[\psi[t]]=\frac{1}{2}\left(1+\cos[\psi[t]]\right)$ (85)

and

$[\psi^{\prime}[t]]^{2}=\frac{\kappa^{2}\rho_{0}}{3}\sec^{4}\left[\frac{\psi[t]}{2}\right]+\frac{\Lambda}{3}\cot^{2}\left[\frac{\psi[t]}{2}\right]$
(86)

whose implicit solution reads

$\displaystyle
t=\pm\sqrt{3}\int_{1}^{\psi}dx\frac{\sqrt{2+\cos[x]-2\cos[2x]-\cos[3x]}}{\sqrt{16\kappa^{2}\rho_{0}\left(1-\cos[x]\right)+\Lambda\left(10+15\cos[x]+6\cos[2x]+\cos[3x]\right)}}$
(87)

On the other hand, the proper energy density

$\rho=\frac{\rho_{0}}{S^{3}[\psi[t]]}$ (88)

Therefore when $\psi=\pi$ the density $\rho$ diverges.

####  3.2.1 Expansion in ${\Lambda\over\kappa^{2}\rho_{0}}$

Let us expand in ${\Lambda\over\kappa^{2}\rho_{0}}$ in order to get a grasp of
the physical properties of our UG solution

$\Lambda
S^{2}[t]+\frac{\kappa^{2}\rho_{0}}{S[t]}-3[S^{\prime}[t]]^{2}=\kappa^{2}\rho_{0}$
(89)

with

$S[t]=S_{0}[t]+\Lambda f[t]$ (90)

Introducing in (89) the GR solution which is valid at order zero in the
expansion,

$\displaystyle[S^{\prime}_{0}[t]]^{2}=\frac{\kappa^{2}\rho_{0}}{3}\Big{[}\frac{1}{S_{0}[t]}-1\Big{]}$
(91)

At first order,

$S_{0}^{2}[t]-\frac{\kappa^{2}\rho_{0}f[t]}{S_{0}^{2}[t]}-6f^{\prime}[t]S_{0}^{\prime}[t]=0$
(92)

Let us now perform the change of variables

$S_{0}^{2}[\psi[t]]-\frac{\kappa^{2}\rho_{0}f[\psi[t]]}{S_{0}^{2}[\psi[t]]}-6f^{\prime}[\psi[t]]S_{0}^{\prime}[\psi[t]][\psi^{\prime}[t]]^{2}=0$
(93)

in terms of the the GR solution

$\displaystyle S_{0}[\psi[t]]=\frac{1}{2}\left(1+\cos[\psi[t]]\right)$
$\displaystyle
t[\psi]=\left(\frac{\psi+\sin\psi}{2\sqrt{\frac{\kappa^{2}\rho_{0}}{3}}}\right)$
(94)

Substituting (3.2.1) into (93) we get:

$\displaystyle\cos^{8}\left[\frac{\psi[t]}{2}\right]-\kappa^{2}\rho_{0}\Big{(}f[\psi[t]]-\sin[\psi[t]]f^{\prime}[\psi[t]]\Big{)}=0$
(95)

where $\psi[t]$ is given by (3.2.1). Then

$f[\psi[t]]=C_{1}\tan\left(\frac{\psi[t]}{2}\right)+\frac{560t+512\cot(\tfrac{\psi[t]}{2})+376\sin(\psi[t])+40\sin(2\psi[t])+\tfrac{8}{3}\sin(3\psi[t])}{512\kappa^{2}\rho_{0}\cot(\tfrac{\psi[t]}{2})}$
(96)

The proper energy density is given by

$\rho=\frac{\rho_{0}}{S^{3}(\psi[t])}=\frac{\rho_{0}}{\left(S_{0}(\psi[t])+\Lambda
f(\psi[t])\right)^{3}}=\frac{\rho_{0}}{S_{0}^{3}(\psi[t])}-\frac{3\rho_{0}f(\psi[t])}{S_{0}^{4}(\psi[t])}\Lambda+\mathcal{O}(\Lambda^{2})$
(97)

in such a way that when $\psi=\pi$, using (3.2.1)

$T=\frac{\pi}{2}\sqrt{\frac{3}{\kappa^{2}\rho_{0}}}$ (98)

the density $\rho$ diverges.

####  3.2.2 Matching with the exterior solution.

The generalized Birkhoff’s theorem ensures that the metric outside the sphere
$r=a$ must be the usual Schwarzschild-(anti)de Sitter

$ds^{2}=\Big{[}1-\frac{r_{s}}{\bar{r}}-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{\Big{[}1-\frac{r_{s}}{\bar{r}}-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(99)

Inside the sphere $r\leq a$ we have

$ds^{2}=\frac{d\tau^{2}}{S^{6}[t(\tau)]}-\frac{S^{2}[t(\tau)]}{b^{4}(r)}dr^{2}-b^{2}(r)S^{2}[t(\tau)]\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(100)

In order to match both solutions at the surface, is necessary to redefine
$\bar{r}=S[t(\tau)]b(r)$ and the interior metric reads

$ds^{2}=\Bigg{[}\frac{b^{2}(r)(b^{\prime}(r))^{2}-[S^{\prime}[t(\tau)]]^{2}}{b^{2}(r)(b^{\prime}(r))^{2}S^{6}[t(\tau)]}\Bigg{]}d\tau^{2}-\frac{d\bar{r}^{2}}{b^{4}(r)(b^{\prime}(r))^{2}}+\frac{2S^{\prime}[t(\tau)]t^{\prime}(\tau)d\bar{r}dt}{b^{3}(r)(b^{\prime}(r))^{2}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(101)

To remove the cross term of the metric, we write

$ds^{2}=C(r,\tau)d\tau^{2}-D(r,\tau)d\bar{r}^{2}-2E(r,\tau)d\bar{r}d\tau-\bar{r}^{2}d\bar{\Omega}^{2}$
(102)

by defining a new time

$d\bar{t}=f(r,\tau)\left[C(r,\tau)d\tau-E(r,\tau)d\bar{r}\right]$ (103)

where $f$ is an integrating factor defined to make the right-hand side a
perfect differential. The metric reads

$\displaystyle
ds^{2}=\frac{1}{b^{2}(r)(b^{\prime}(r))^{2}-[S^{\prime}[t(\tau)]^{2}}\Bigg{\\{}\frac{b^{2}(r)(b^{\prime}(r))^{2}S^{6}[t(\tau)]d\bar{t}^{2}}{f^{2}(r,\tau)}-\frac{d\bar{r}^{2}}{b^{2}(r)}\Bigg{\\}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(104)

At the surface of the sphere $r=a$, i.e. $\bar{r}=b(a)S(t)$, implies

$f^{2}(a,t)=\frac{(b^{\prime}(a))^{2}S^{6}[t(\tau)]}{\Big{[}b^{2}(a)(b^{\prime}(a))^{2}-[S^{\prime}[t(\tau)]^{2}\Big{]}^{2}}$
(105)

therefore the metric reads

$\displaystyle
ds^{2}=b^{2}(a)\Big{[}b^{2}(a)(b^{\prime}(a))^{2}-[S^{\prime}[t(\tau)]^{2}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{b^{2}(a)\Big{[}b^{2}(a)(b^{\prime}(a))^{2}-[S^{\prime}[t(\tau)]^{2}\Big{]}}-\bar{r}^{2}d\bar{\Omega}^{2}$

with the equation of motion

$\Lambda
S^{2}[t(\tau)]+\frac{\kappa^{2}\rho_{0}}{S[t(\tau)]}-3[S^{\prime}[t(\tau)]]^{2}=\kappa^{2}\rho_{0}$
(107)

and

$\frac{3}{b^{2}(r)}\left[1-b^{4}(r)[b^{\prime}(r)]^{2}\right]=\kappa^{2}\rho_{0}$
(108)

This is not a surprise; here the function $b(r)$ acts as a radial coordinate
of sorts. Eventually

$ds^{2}=\Big{[}1-\frac{\kappa^{2}\rho_{0}}{3\bar{r}}b^{3}(a)-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{\Big{[}1-\frac{\kappa^{2}\rho_{0}}{3\bar{r}}b^{3}(a)-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(109)

where $r_{s}=2MG$ and the mass is given by

$M=\rho_{0}\frac{4\pi}{3}b^{3}(a)$ (110)

###  3.3 UG with incompressible fluid (constant density)

In this final section, we assume that the density is constant,
$\rho(r,t)=\rho_{0}$ whereas the pressure is to be calculated $p(r,t)$. We
further assume that the unimodular metric only depends on the radial
coordinate

$ds^{2}=A^{2}(r)dt^{2}-B^{2}(r)dr^{2}-r^{2}-{1\over
A(r)B(r)}\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(111)

In this case, the conservation of energy-momentum tensor reduces to

$\displaystyle\nabla_{\mu}T^{\mu\nu}=u^{\nu}u^{\mu}\nabla_{\mu}p+\left(\rho_{0}+p\right)u^{\mu}\nabla_{\mu}u^{\nu}-\nabla^{\nu}p=0$
(112)

because $\theta=0$. The timelike components tells us that

$\displaystyle\nabla_{\mu}T^{\mu
0}=\frac{1}{A^{2}}\dot{p}-\frac{1}{A^{2}}\dot{p}=0$ (113)

which is identically satisfied. The radial component

$\displaystyle\nabla_{\mu}T^{\mu
1}=\left(\rho_{0}+p\right)\frac{1}{A^{2}}\Gamma^{1}_{00}+\frac{1}{B^{2}}p^{\prime}=\left(\rho_{0}+p\right)\frac{A^{\prime}}{A}+p^{\prime}=0$
(114)

implies

$\rho_{0}+p(r,t)=\frac{\gamma}{A(r)}$ (115)

where $\gamma$ is a constant.

The EM read

$\hat{R}_{\mu\nu}-\frac{1}{4}g_{\mu\nu}\hat{R}=\kappa^{2}\left(T_{\mu\nu}-\frac{1}{4}g_{\mu\nu}T\right)$
(116)

To be specific, this is equivalent to the set

$\displaystyle\frac{4A^{3}B^{5}-11B^{2}A^{\prime
2}+2AB\left(-9A^{\prime}B^{\prime}+4BA^{\prime\prime}\right)+A^{2}\left(-11B^{\prime
2}+4BB^{\prime\prime}\right)}{8A^{2}B^{4}}=\frac{3\kappa^{2}(\rho_{0}+p(r))}{4}$
$\displaystyle-\frac{4A^{3}B^{5}+9B^{2}A^{\prime
2}+6ABA^{\prime}B^{\prime}+A^{2}\left(9B^{\prime
2}-4BB^{\prime\prime}\right)}{8A^{2}B^{4}}=\frac{\kappa^{2}(\rho_{0}+p(r))}{4}$
$\displaystyle-\frac{-4A^{3}B^{5}+B^{2}A^{\prime 2}+A^{2}B^{\prime
2}+2AB\left(3A^{\prime}B^{\prime}-2BA^{\prime\prime}\right)}{8A^{2}B^{4}}=\frac{\kappa^{2}(\rho_{0}+p(r))}{4}$
(117)

Bianchi’s identity implies

$\hat{R}+\kappa^{2}\left(\rho_{0}-3p(r)\right)=-4\Lambda$ (118)

Assume now (cf. [9])

$\displaystyle A^{2}[r]\equiv\zeta[r]\eta^{-1/3}[r]$ $\displaystyle
B^{2}[r]\equiv\frac{1}{\zeta[r]\eta[r]}$ (119)

Then the EM reduces to

$\displaystyle
2\zeta\eta^{\prime\prime}=-3\kappa^{2}\gamma\zeta^{-1/2}\eta^{1/6}$
$\displaystyle\zeta^{\prime\prime}=-2\eta^{-\frac{5}{3}}+3\kappa^{2}\gamma\eta^{-\frac{5}{6}}\eta^{-\frac{1}{2}}+\frac{5}{3}\zeta\eta^{-1}\eta^{\prime\prime}$
(120)

and the Bianchi identity, with the expression of $\zeta^{\prime\prime}$ (3.3)

$\displaystyle
2\zeta\eta^{\prime\prime}+\zeta^{\prime}\eta^{\prime}=3\eta^{-2/3}-3\kappa^{2}\rho_{0}-3\Lambda$
(121)

In fact, there is an integrating factor $\eta^{\prime}$ in such a way that

$\displaystyle\zeta(\eta^{\prime})^{2}=9\eta^{1/3}-3\kappa^{2}\rho_{0}\eta-3\Lambda\eta+D$
(122)

where $D$ is an integration constant (which will eventually vanish). It easily
follows that

$\displaystyle\frac{2\eta^{\prime\prime}}{(\eta^{\prime})^{3}}=-\frac{3\kappa^{2}\gamma\eta^{1/6}}{\left(9\eta^{1/3}-3\kappa^{2}\rho_{0}\eta-3\Lambda\eta+D\right)^{3/2}}$
(123)

This is a differential equation for $\eta$, and with $\eta[r]$ and (122) we
also get $\zeta[r]$.

To summarize, we can express the interior solution like

$\displaystyle
ds^{2}=\left(\frac{3\cos\chi_{a}-\cos\chi}{2}\right)^{2}dt^{2}-\frac{3}{\kappa^{2}\rho_{0}+\Lambda}\left[d\chi^{2}+\sin^{2}\chi\left(\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\right)\right]$
(124)

where

$\displaystyle\eta\equiv\left(\frac{3}{\kappa^{2}\rho_{0}+\Lambda}\right)^{3/2}\sin^{3}\chi$
$\displaystyle\zeta=\sqrt{\frac{3}{\kappa^{2}\rho_{0}+\Lambda}}\sin\chi\left(\frac{3\cos\chi_{a}-\cos\chi}{2}\right)^{2}$
(125)

It is plain that when $\Lambda=0$ we go back to Schwarzschild’s spacetime.

####  3.3.1 Matching with the exterior solution.

The metric outside the sphere must still be the usual Schwarzschild-(anti)de
Sitter

$ds^{2}=\Big{[}1-\frac{r_{s}}{\bar{r}}-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}d\bar{t}^{2}-\frac{d\bar{r}^{2}}{\Big{[}1-\frac{r_{s}}{\bar{r}}-\frac{\bar{r}^{2}\Lambda}{3}\Big{]}}-\bar{r}^{2}d\bar{\Omega}^{2}$
(126)

Inside the sphere we have the solution (124), In order to match solutions at
the surface, we need to redefine
$\bar{r}^{2}=\frac{3}{\kappa^{2}\rho_{0}+\Lambda}\sin^{2}\chi$, so that

$ds^{2}=\left(\frac{3\cos\chi_{a}-\cos\chi}{2}\right)^{2}dt^{2}-\frac{1}{\cos^{2}\chi}d\bar{r}^{2}-\bar{r}^{2}\Big{[}\frac{d\psi^{2}}{1-\psi^{2}}+(1-\psi^{2})d\phi^{2}\Big{]}$
(127)

At the surface of the sphere $\bar{r}_{a}$

$\cos^{2}\chi_{a}=1-\frac{r_{s}}{\bar{r}_{a}}-\frac{\bar{r}_{a}^{2}\Lambda}{3}$
(128)

where

$\bar{r}_{a}=\sqrt{\frac{3}{\kappa^{2}\rho_{0}+\Lambda}}\sin\chi_{a}$ (129)

##  4 Acknowledgements

We have enjoyed discussions with Eduardo Velasco-Aja. We acknowledge partial
financial support by the Spanish MINECO through the Centro de excelencia
Severo Ochoa Program under Grant CEX2020-001007-S funded by
MCIN/AEI/10.13039/501100011033.

We also acknowledge partial financial support by the Spanish Research Agency
(Agencia Estatal de Investigación) through the grant PID2022-137127NB-I00
funded by MCIN/AEI/10.13039/501100011033/ FEDER, UE

All authors acknowledge the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No
860881-HIDDeN and also byGrant PID2019-108892RB-I00 funded by MCIN/AEI/
10.13039/501100011033 and by “ERDF A way of making Europe”.

## References

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Petrov, A. Z. (1969). “Einstein spaces ”. Oxford: Pergamon Press.
Besse, A. L. (1987). “Einstein manifolds”. Berlin: Springer-Verlag.


§ INTRODUCTION

§ TECHNICAL BACKGROUND AND METHODS

§ RESULTS

§ DISCUSSION

§ ACKNOWLEDGEMENTS

This work has been funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant No 810367), project EMC2 (J.-P. P. and Y.M.)

§ APPENDIX


* Papamakarios et al., (2017) Papamakarios, G., Pavlakou, T., and Murray, I. (2017).  Masked autoregressive flow for density estimation.  In Advances in Neural Information Processing Systems, pages 2338–2347.
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  * Tran et al., (2019) Tran, D., Vafa, K., Agrawal, K., Dinh, L., and Poole, B. (2019).  Discrete flows: Invertible generative models of discrete data.  In Advances in Neural Information Processing Systems, pages 14719–14728.
  * Uehara et al., (2016) Uehara, M., Sato, I., Suzuki, M., Nakayama, K., and Matsuo, Y. (2016).  Generative adversarial nets from a density ratio estimation perspective.  arXiv preprint arXiv:1610.02920.
  * Wang et al., (2020) Wang, L., Jones, D. E., and Meng, X.-L. (2020).  Warp bridge sampling: The next generation.  Journal of the American Statistical Association, (just-accepted):1–31.
  * Wong et al., (2020) Wong, J. S., Forster, J. J., and Smith, P. W. (2020).  Properties of the bridge sampler with a focus on splitting the mcmc sample.  Statistics and Computing, pages 1–18.


# GenPlot: Increasing the Scale and Diversity of Chart Derendering Data

Brendan Artley
Independent Researcher
Vancouver, BC, Canada
<EMAIL_ADDRESS>

###### Abstract

Vertical bars, horizontal bars, dot, scatter, and line plots provide a diverse
set of visualizations to represent data. To understand these plots, one must
be able to recognize textual components, locate data points in a plot, and
process diverse visual contexts to extract information. In recent works such
as Pix2Struct, Matcha, and Deplot, OCR-free chart-to-text translation has
achieved state-of-the-art results on visual language tasks. These results
outline the importance of chart-derendering as a pre-training objective, yet
existing datasets provide a fixed set of training examples. In this paper, we
propose GenPlot; a plot generator that can generate billions of additional
plots for chart-derendering using synthetic data.

##  1 Introduction

Traditionally, OCR-aware methods such as LayoutLM (Xu et al., 2020), PresSTU
(Kil et al., 2022), PaLI (Chen et al., 2023), and ChartBERT (Akhtar et al.,
2023) have been used to extract information from plots. While these models can
accurately extract text, they require a dataset of labeled components which
can be expensive to obtain. Additionally, plots do not always represent
numerical components exactly (ie. Scientific Notation, Percent, etc.), and
therefore post-processing is required to extract numerical information.

In recent works like Donut (Kim et al., 2022), Pix2Struct (Lee et al., 2022),
Matcha (Liu et al., 2023b), and Deplot (Liu et al., 2023a), OCR-free chart-to-
text translation methods are used. Donut focuses on document understanding,
whereas pix2struct aims to provide a generic pre-trained checkpoint for many
downstream visual language tasks. Matcha and Deplot are concurrent to
Pix2Struct as the models use the same underlying architecture. The models all
require comprehensive datasets for pre-training tasks to achieve state-of-the-
art results on the PlotQA (Methani et al., 2020) and ChartQA (Masry et al.,
2022) benchmarks. It is computationally expensive to obtain these large
datasets for pre-training, and this is where GenPlot can help.

GenPlot provides a framework to generate billions of possible plot
combinations for chart derendering tasks. GenPlot is a standalone Python
script built using Matplotlib (Hunter, 2007) that can generate bar, scatter,
line, and dot plots. The configuration of the hyperparameters in the script
can be modified as each user sees fit. By default, we use similar categorical
labels determined using GloVe (Pennington et al., 2014) embeddings and
randomly generated numerical data. We also provide a pre-generated set of
500,000 plots that can be used in place of the generator. GenPlot provides a
way for researchers to quickly increase the scale of data for chart
derendering tasks.

##  2 Related Work

Recent works such as PlotQA (Methani et al., 2020), ChartQA (Masry et al.,
2022), Matcha (Liu et al., 2023b), and DVQA (Kafle et al., 2018) generate
plots with a focus on Visual Question Answering (VQA) (Agrawal et al., 2016).
PlotQA provides bar, line, and scatter plots. CharQA provides bar, line, and
pie plots, and DVQA provides bar plots. It is unclear which plot types are
generated for Matcha pretraining, as the data is not available in the public
domain. The result of these works is high-quality question-answer pairs,
rather than a means to generate large-scale datasets.

An existing work for chart generation is FigureQA (Kahou et al., 2018). This
source provides synthetically generated data, bounding boxes, and question-
answer pairs. FigureQA also provides 4 different plot types, and the authors
have released the code for data generation. The generation process is limited
to 100 colors, and fixed chart components such as gridlines, labels, and
legends.

GenPlot extends the work of FigureQA by providing a means to generate a larger
and more diverse set of plots. It does this through random color sampling, and
variability in margins, grids, ticks, labels, plot sizes, and more. To our
knowledge, there is no existing framework to generate plots with such a high
degree of variability, which motivated us to build this framework. In table 1,
we compare our dataset with existing works. Generate+ indicates whether the
source provides the ability to generate new plots, "#" and Tables is the
number of plots in the dataset. For "#" Plot Types we count the number of
unique plot types (ie. Bar, Line, Plot, Pie, etc.). Orientation of the chart
type is not considered. For example, horizontal bar plots and vertical bar
plots are counted as one.

Dataset | Generate+ | # Tables | Bar | Line | Pie | Scatter | Dot | Unique Plot Types
---|---|---|---|---|---|---|---|---
ChartQA | No | 22k | 1 | 1 | 1 | 0 | 0 | 3
PlotQA | No | 224k | 1 | 1 | 0 | 1 | 0 | 3
MATCHA | No | 270k | 1 | 1 | 1 | 0 | 0 | 3
DVQA | No | 300k | 1 | 0 | 0 | 0 | 0 | 1
FigureQA | No | 100k | 1 | 1 | 1 | 1 | 0 | 5
GenPlot (ours) | Yes | 500k | 1 | 1 | 0 | 1 | 1 | 5
Table 1: Dataset Statistics

##  3 Metadata and Generation

Metadata for each plot is stored in a similar format to pix2struct chart-to-table models (Lee et al., 2022). For example, the string may look like “0 | 1 <0x0A> 1 | 2 <0x0A> 2 | 7”. In this string, there are 3 data points, (0,1), (1,2), and (2,7). The first value in each pair is the x value, and the second value is the y value. The x and y values are separated by the “|” character in the string. Each pair is then separated by the “<0x0A>” character sequence. All data points are stored in order from left to right, with the expectation of horizontal bar plots which are stored in a top-down fashion. Regardless of the data type on each axis, the format of the metadata stays the same.

###  3.1 Text Generation

Plots can contain text in the title, subtitles, and as categorical labels. To
obtain groups of related words we utilized GloVe (Pennington et al., 2014)
embeddings. First, we set a predefined list of common objects like
“toothbrush”, “coffee”, “notebook”, etc. Then, we sampled the 25 most similar
words for each of the objects and added these to a vocabulary list. Any words
that contained non-ASCII characters were not included. We performed this step
twice which resulted in 42744 groups of similar words. We sampled in this way
to ensure that label groups were related rather than randomly selected words.

For each plot, the main title and x-axis title contains a sample of 3 to 7
words from a random group. The y-axis title contains a sample of 1-4 words. We
set this range to reduce the chance of title and tick label overlap during the
plot generation process.

For categorical labels, we use a list of place names, months, days, and part-
numerical strings. The place names include a list of 2123 countries, regions,
counties, and states. Months and days can be generated in numerous ways and
occasionally contain numerical values. For example, the months and days could
appear in the following formats: “Jan”, “December”, “Jan-Feb”, “Apr-10”,
“Tues”, “Friday”, etc. The part-numerical strings appear as follows: “10-20”,
“30-40”, etc. We included part-numerical strings as categorical labels to
create "difficult" examples.

###  3.2 Numerical Generation

The numerical data generation is done using a random polynomial sampler and a
random linear sampler implemented in Numpy (Harris et al., 2020). Each sampler
generates sequences of integers or floating point numbers and adds a small
degree of Gaussian noise to the set of values. The values are then scaled by a
factor between 0.01 and 1,000,000. Occasionally, random points are scaled
again to simulate outliers in the data. Plot-specific modifications are also
made to ensure the data was suitable for each plot type.

##  4 Plot Types and Styles

In this section, we outline the default settings and parameter combinations in
the plot generation process. Moreover, each plot is generated using a
programming tool called Matplotlib (Hunter, 2007). We select 8 unique styles,
and 9 font families as default parameters. At the time of generation, the
style and font family are randomly selected. We also randomly remove ticks,
grids, and spines from each plot.

To generate the data point colors, we randomly sample RGB values between 40
and 200. This ensures that the data points can be seen on light and dark
backgrounds, and gives more color variability to the plots. Finally, we
enforce graph conventions from the Benetech - Making Graphs Accessible
Competition (Andrews et al., 2023). These conventions were in place so that
generated plots are representative of non-generated plots.

###  4.1 Bar plot

There are 200,000 bar plots in the pre-generated data. Half of these are
vertical bar plots, and the other half are horizontal bars. These plot types
are very similar, with only the axes being flipped. We discuss the bar plots
in the context of vertical bars for the remainder of this section. The number
of generated bars is between 2 and 20, with oversampling to sizes of around 6
bars. Variation is added to the spacing between bars and the plot margins. The
X-axis labels are integers or strings, and the Y-axis values are numerical. No
value on the Y-axis can be 200 times greater than the minimum value in each
set of values. Each X-axis label corresponds to a single bar in the plot.

Figure 1: Vertical Bar Plot Figure 2: Horizontal Bar Plot

###  4.2 Scatter plot

There are 100,000 scatter plots in the pre-generated data. The number of
generated points in each plot is between 3 and 86. The two main variations of
the scatter plot are randomly sampled points, and points that follow a path-
like pattern. For a high number of randomly sampled points, values likely
overlap. Therefore we implement a custom sampler to ensure that any two points
do not completely overlap. The line-like scatter plot is generated using the
polynomial sampler and provides a version of the plot type similar to the line
plot. The X-axis labels and the Y-axis labels are numerical.

Figure 3: Scatter Plot

###  4.3 Line plot

There are 100,000 line plots in the pre-generated data. The number of
generated points is between 2 and 20, with oversampling to sizes of around 7
values. Values found on the X-axis are either dates or integer values in
ascending order. The Y-axis is strictly numerical. One unique characteristic
of the line plot is that X-axis labels do not always correspond to data
points. This is done to ensure that OCR-free systems do not just read the data
labels, and instead learn from the entire context. Additionally, dots are
randomly added to the line, and random smoothing is occasionally applied. This
is done to increase variability.

Figure 4: Line Plot

###  4.4 Dot Plot

There are 100,000 dot plots in the pre-generated data. The number of generated
points is between 2 and 21 values. Values on the Y-axis are always integers
between 1 and 10, and the Y-axis values can be strings or numerical. A unique
characteristic of the dot plots is that numerical X-axis labels are not always
present, as they can be inferred from surrounding labels. Additionally, Y-axis
ticks are sometimes completely removed, as counts can be inferred from the
plot. See table 5 for an example of this. Also, as of this writing, there is
no standardized Matplotlib function for dot plots, so we implement a custom
dot plot function using the scatter function.

Figure 5: Dot Plot

##  5 Conclusion

We have proposed a framework for large-scale plot generation for chart-
derendering. We use a variety of coloring, spacing, and sampling options to
yield a diverse generation for 4 plot types. We provide a pre-generated set of
500,000 plots which exceeds the size of existing datasets. The code for
GenPlot can be found on GitHub (link), along with instructions on how to get
started. The dataset containing the pre-generated plots can be found on
Kaggle.

Links: Source Code, Pre-Generated Data

###  5.1 Limitations

Though we built a diverse plot generator, there are still many plot types that
we did not implement. For example, we did not include pie plots, histograms,
or treemaps. It remains up for debate whether adding more plot types would be
beneficial for chart-derendering as a pre-training task. Additionally, we
added safeguards to reduce the likelihood of label overlap, but there is still
a possibility of this happening in newly generated plots. We were unable to
find a way to validate this without a human in the loop.

###  5.2 Ethics Statement

During the data generation process, we considered several ethical issues.
Firstly, we acknowledge Gender and Race bias found in GloVe embeddings.
Occupations and degrees of similarity between words are stereotyped. We do not
condone the stereotypes found in this model, and recognize how this is a
result of its training data.

## References

  * Agrawal et al. (2016) Aishwarya Agrawal, Jiasen Lu, Stanislaw Antol, Margaret Mitchell, C. Lawrence Zitnick, Dhruv Batra, and Devi Parikh. 2016.  Vqa: Visual question answering.
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