Dark Energy from the Fifth Dimension

A<EMAIL_ADDRESS>and Chuang<EMAIL_ADDRESS>

Department of Physics, University of Alabama,

Tuscaloosa, Alabama 35487, USA

###### Abstract

After extending the Regge-Teitelboim formulation of gravity to include the
case where the background embedding space is not flat, we examine the dynamics
of the four-dimensional $k=0$ Robertson-Walker (RW) manifold embedded in
various five-dimensional backgrounds. We find that when the background is
five-dimensional de Sitter space, the RW manifold undergoes a transition from
a de-accelerating phase to an accelerating phase. This occurs before the
inclusion of matter, radiation or cosmological constant sources, and thus does
not require a balance of different components. We obtain a reasonable two-
parameter fit of this model to the Hubble parameter data.

Regge-Teitelboim (RT) gravity is an alternative to standard general relativity
where the dynamical degrees of freedom are associated with the embeddings of
the space-time manifold in a fixed higher dimensional background.[1, 2, 3, 4,
5, 6, 7] Solutions to Einstein equations satisfy the RT field equations. More
generally, the RT formulation of gravity can effectively produce source terms
in the standard Einstein equations that are not attributable to the energy-
momentum tensor, but rather are a result of the embedding. These RT source
terms have the potential of providing an explanation for certain cosmological
phenomena, such as for cosmic acceleration. This idea was entertained in [8],
where a toy model was presented.333See also [9].  There it was shown that RT
gravity can generate cosmic acceleration for a simple class of embeddings of
the $k=-1$ Robertson-Walker (RW) metric in a flat five-dimensional background.
Moreover, a transition from a de-accelerating to accelerating phase could be
observed in a specific, but not physical, example. No cosmological
acceleration was found for the case of the $k=1$ RW metric, and the currently
favored $k=0$ case was not considered.

This article attempts to apply RT gravity in the direction of a more realistic
model of the observed cosmological acceleration. The approach taken here
relies on embedding the RW manifold in a curved background. While previous
discussions of RT gravity have been restricted to flat backgrounds, the
formalism can easily be extended to curved background spaces, as is
demonstrated here. After developing the formalism, we then apply it to
cosmology by embedding the four-dimensional RW manifold in three different
five-dimensional background spaces. We specialize to $k=0$, although the other
cases can also be considered as well. The backgrounds we consider are: $i)$
$R^{4,1}$, $ii)$ $AdS_{5}$ and $iii)$ $dS_{5}$. As a first approximation, we
obtain the evolution of the scale factor on the RW manifold in the absence of
matter, radiation or cosmological constant sources. We get that the
acceleration of the scale factor is negative for all time for cases $i)$ and
$ii)$. On the other hand, for case $iii)$ we find that a transition from the
de-accelerating phase to an accelerating phase occurs at a finite time. The
evolution in this case is determined by two free parameters, the curvature of
the background de Sitter space and the strength of the RT source term. The two
parameters allow for a fit to the Hubble parameter data. Unlike in the
$\Lambda$CDM model, neither the matter density nor the cosmological constant
play a role in the fit, meaning that their contributions should be
significantly weaker than the RT source term, and furthermore, that they can
have arbitrary strength relative to each other. So here we are able to avoid
the coincidence puzzle of the $\Lambda$CDM model, where the matter
contribution to Einstein equations, coincidentally, is of the same order of
magnitude as the cosmological constant contribution at the current time.

We begin with a very brief discussion of RT gravity, or more precisely, its
generalization to the case where the $d-$dimensional background space ${\bf
M}_{d}$, $d>4$, is not necessarily flat. We denote a local set of coordinates
on ${\bf M}_{d}$ by $Y^{a}$, $a,b,\cdots=0,...,d-1$, and its associated metric
tensor metric by ${\tt\bf g}_{ab}(Y)$. Next embed a four-dimensional space-
time manifold ${\cal M}_{4}$ in ${\bf M}_{d}$. This can be done by introducing
the set of functions $Y^{a}=Y^{a}(x)$, where $x^{\mu}$,
$\mu,\nu,\cdots=0,...,3$, span ${\cal M}_{4}$. The metric tensor
$g_{\mu\nu}(x)$ on ${\cal M}_{4}$ is defined to be induced from ${\tt\bf
g}_{ab}(Y)$. So

$g_{\mu\nu}(x)={\tt\bf g}_{ab}(Y)\partial_{\mu}Y^{a}\partial_{\nu}Y^{b}\;,$
(1)

$\partial_{\mu}$ denoting differentiation with respect to $x^{\mu}$. As is
usual $g_{\nu\lambda}$ is required to be invertible, and metric compatible on
${\cal M}_{4}$, $\nabla_{\mu}g_{\nu\lambda}=0$, $\nabla_{\mu}$ being the
covariant derivative on ${\cal M}_{4}$. The latter leads to the identity:

${\tt\bf
g}_{ab}\nabla_{\lambda}\partial_{\mu}Y^{a}\,\partial_{\nu}Y^{b}+\frac{1}{2}\frac{\partial{\tt\bf
g}_{ab}}{\partial
Y^{c}}\left(\partial_{\mu}Y^{a}\partial_{\nu}Y^{b}\partial_{\lambda}Y^{c}+\partial_{\nu}Y^{a}\partial_{\lambda}Y^{b}\partial_{\mu}Y^{c}-\partial_{\lambda}Y^{a}\partial_{\mu}Y^{b}\partial_{\nu}Y^{c}\right)=0$
(2)

To derive this compute
$\nabla_{\lambda}g_{\mu\nu}+\nabla_{\mu}g_{\nu\lambda}-\nabla_{\nu}g_{\lambda\mu}$
using (1), and apply metric compatibility and the Leibniz rule.

RT gravity assumes the usual Einstein-Hilbert action $S_{EH}$ for the
gravitational field, however the dynamical degrees of freedom are the
embedding functions, not $g_{\mu\nu}$. So upon including source terms
$S_{source}$, one has

$S=S_{EH}+S_{source},\qquad S_{EH}=\frac{1}{16\pi
G}\int_{M}d^{4}x\sqrt{|g|}R\;\;,$ (3)

with the scalar curvature constructed from (1). Field dynamics is obtained
from variations of $Y^{a}$. This gives

$\partial_{\mu}\left(\sqrt{|g|}E^{\mu\nu}{\tt\bf
g}_{ab}\partial_{\nu}Y^{b}\right)-\frac{1}{2}\sqrt{|g|}E^{\mu\nu}\frac{\partial{\tt\bf
g}_{bc}}{\partial Y^{a}}\partial_{\mu}Y^{b}\partial_{\nu}Y^{c}=0\;,$
$\qquad\quad E^{\mu\nu}=G^{\mu\nu}-8\pi G\,T^{\mu\nu}\;,$ (4)

$G^{\mu\nu}$ and $T^{\mu\nu}$ being the Einstein tensor and stress-energy
tensor, respectively. As in Einstein gravity, $T^{\mu\nu}$ must be covariantly
conserved. To see this one can first re-write the field equations as

$\nabla_{\mu}(E^{\mu\nu}{\tt\bf
g}_{ab}\partial_{\nu}Y^{b})-\frac{1}{2}E^{\mu\nu}\frac{\partial{\tt\bf
g}_{bc}}{\partial Y^{a}}\partial_{\mu}Y^{b}\partial_{\nu}Y^{c}=0\;,$ (5)

and then expand the first term using the Bianchi identity to obtain

$-8\pi G\,\nabla_{\mu}T^{\mu\nu}\,{\tt\bf
g}_{ab}\partial_{\nu}Y^{b}+E^{\mu\nu}\left(\nabla_{\mu}({\tt\bf
g}_{ab}\partial_{\nu}Y^{b})-\frac{1}{2}\frac{\partial{\tt\bf g}_{bc}}{\partial
Y^{a}}\partial_{\mu}Y^{b}\partial_{\nu}Y^{c}\right)=0$ (6)

Finally contract with $\partial_{\lambda}Y^{a}$ and apply (2) to get
$\nabla_{\mu}T^{\mu}_{\;\;\,\lambda}=0$. The field equations (4) are obviously
satisfied for solutions to Einstein equations, $E^{\mu\nu}=0$. More generally,
$E^{\mu\nu}$ need not vanish. Alternatively, we can argue that the Einstein
equations effectively pick up additional source terms, which we denote by
$T_{{}_{\rm RT}}^{\mu\nu}$, which are not associated with the standard stress-
energy tensor but rather are due to the embedding in the background space,

$G^{\mu\nu}=8\pi G\,(T^{\mu\nu}+T_{{}_{\rm RT}}^{\mu\nu})\;,$ (7)

Obviously, $T_{{}_{\rm RT}}^{\mu\nu}$ is covariantly conserved since
$T^{\mu\nu}$ is.

Next we want to apply this dynamical system to the case where the embedded
manifold ${\cal M}_{4}$ is that of standard cosmology, i.e., it is given by
the RW metric tensor. Here we will specialize to the currently favored case of
$k=0$

$ds^{2}=-dt^{2}+a(t)^{2}dx^{i}dx^{i}\;,$ (8)

where $t=x^{0}$ and $a(t)$ is the scale factor. As a first approximation let
us consider source free RT gravity, i.e., $T^{\mu\nu}=0$. From (7) we know
that the Einstein tensor need not vanish. $T_{{}_{\rm RT}}^{\mu\nu}$ in (7)
needs to be computed from the particular choice of embedding, however from
consistency with homogeneity and isotropy, we anticipate that its form should
be analogous to that of a perfect fluid in the co-moving frame

$T^{00}_{{}_{\rm RT}}=\rho_{{}_{\rm RT}}\qquad T^{11}_{{}_{\rm
RT}}=T^{22}_{{}_{\rm RT}}=T^{33}_{{}_{\rm RT}}=a(t)^{2}p_{{}_{\rm RT}}\;,$ (9)

with $\rho_{{}_{\rm RT}}$ and $p_{{}_{\rm RT}}$ being functions of $t$. Since
it is covariantly conserved we have

$\displaystyle\dot{\rho}_{{}_{\rm RT}}+3\frac{\dot{a}}{a}(\rho_{{}_{\rm
RT}}+p_{{}_{\rm RT}})$ $\displaystyle=$ $\displaystyle 0\;,$ (10)

the dot denoting a $t-$derivative.

Substituting(8) into (4) gives

$\partial_{t}\left(F_{1}(t){\tt\bf
g}_{ab}\partial_{t}Y^{b}\right)-\frac{1}{2}F_{1}(t)\partial_{t}Y^{b}\partial_{t}Y^{c}\frac{\partial{\tt\bf
g}_{bc}}{\partial Y^{a}}=F_{2}(t)\left(\partial_{i}\left({\tt\bf
g}_{ab}\partial_{i}Y^{b}\right)-\frac{1}{2}\partial_{i}Y^{b}\partial_{i}Y^{c}\frac{\partial{\tt\bf
g}_{bc}}{\partial Y^{a}}\right)\;,\quad$ (11)

where

$F_{1}(t)=3a\dot{a}^{2}\qquad\quad F_{2}(t)=2\ddot{a}+\frac{\dot{a}^{2}}{a}$
(12)

(11) can produce equations for $\dot{a}$ and $\ddot{a}$ which can be written
in the form of $k=0$ Friedmann equations

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}\rho_{\tt RT}$ (13)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$ $\displaystyle-\frac{4\pi
G}{3}(\rho_{{}_{\rm RT}}+3p_{{}_{\rm RT}})\;,$ (15)

allowing us to identify $\rho_{\tt RT}$ and $p_{{}_{\rm RT}}$ in (9). The
resulting expressions for $\rho_{\tt RT}$ and $p_{{}_{\rm RT}}$ will in
general depend on the background space and the choice of embedding, as we
illustrate in the examples that follow. As stated previously, the background
spaces we consider are $R^{4,1}$, $AdS_{5}$ and $dS_{5}$. We use the same
expression for the embedding in all three cases:

$\begin{pmatrix}Y^{0}\\\ Y^{1}\\\ Y^{2}\\\ Y^{3}\\\
Y^{4}\end{pmatrix}=\begin{pmatrix}b(t)\\\ x^{1}\\\ x^{2}\\\ x^{3}\\\
h(t)\end{pmatrix}\;,$ (16)

where the functions $b(t)$ and $h(t)$ need to satisfy certain constraints in
order to recover the $k=0$ Robertson-Walker metric on the embedded four-
dimensional manifold.

We next deduce $\rho_{\tt RT}$ and $p_{{}_{\rm RT}}$ for the three different
cases.

  1. 1.

Flat $5-$dimensional background $R^{4,1}$

A trivial system results if one chooses Cartesian coordinates for $R^{4,1}$
and maps to ${\cal M}_{4}$ using (16), as this restricts the scale factor in
(8) to be one. Alternatively, a nontrivial function $a(t)$ can result from a
different coordinatization on $R^{4,1}$ , such as is in [10], [11] where

$(ds^{2})_{R^{4,1}}=-(dY^{0})^{2}+(Y^{0}+Y^{4})^{2}\left((dY^{1})^{2}+(dY^{2})^{2}+(dY^{3})^{2}\right)+(dY^{4})^{2}$
(17)

It can be checked that the five-dimensional curvature resulting from this
metric is zero. Now using (16) to map to (8) one gets that $b(t)$ and $h(t)$
should satisfy

$b(t)+h(t)=a(t)\qquad\quad\dot{b}^{2}-\dot{h}^{2}=1\;$ (18)

Now substituting (16) in (11) gives

$\displaystyle\partial_{t}(\dot{b}F_{1})$ $\displaystyle=$
$\displaystyle\;\;\,3F_{2}a$ (19) $\displaystyle\partial_{t}(\dot{h}F_{1})$
$\displaystyle=$ $\displaystyle-3F_{2}a$ (20)

The sum of these two equations leads to a constant of motion
$\partial_{t}(\dot{a}F_{1})=0$, from which we get the following expression for
$\rho_{\tt RT}$

$\rho_{\tt RT}=\frac{c_{0}}{a^{3}\dot{a}}\;,$ (21)

$c_{0}$ being a constant. The Friedmann equation (13) then gives
$\dot{a}^{3}\propto\frac{1}{a}$, and so there is no acceleration as $a$
increases. One gets a simple solution for the scale factor in this case:
$a(t)\propto t^{3/4}$ for $a(0)=0$. This coincides with the time evolution of
the scale factor in the presence of a perfect fluid with equation of state
$p=-\frac{1}{9}\rho$. The same result was observed in [9] for a different
choice of embedding.

  2. 2.

$AdS_{5}$ background

Here we cover a patch of $AdS_{5}$ using Poincaré coordinates. The background
metric is

$(ds^{2})_{AdS_{5}}=-\frac{(Y^{4})^{2}}{L^{2}}(dY^{0})^{2}+\frac{(Y^{4})^{2}}{L^{2}}\Bigl{(}(dY^{1})^{2}+(dY^{2})^{2}+(dY^{3})^{2}\Bigr{)}+\frac{L^{2}(dY^{4})^{2}}{(Y^{4})^{2}}\;,$
(22)

the constant $L$ denoting the $AdS_{5}$ radius of curvature. Utilizing the
embedding (16), the $k=0$ RW metric (8) is recovered provided that

$h=L\,a\qquad\quad a^{2}\dot{b}^{2}-L^{2}\frac{\dot{a}^{2}}{a^{2}}=1$ (23)

Substituting (16) in (11) gives

$\displaystyle\partial_{t}(a^{2}\dot{b}F_{1})$ $\displaystyle=$ $\displaystyle
0$ (24) $\displaystyle
L^{2}\partial_{t}\left(\frac{\dot{a}}{a^{2}}F_{1}\right)+\left(a\dot{b}^{2}+L^{2}\frac{\dot{a}^{2}}{a^{3}}\right)F_{1}$
$\displaystyle=$ $\displaystyle-3aF_{2}$ (26)

From (13) and (24) we then get

$\rho_{\tt RT}=\frac{c_{0}}{a^{3}\sqrt{L^{2}\dot{a}^{2}+a^{2}}}\;,$ (27)

Note that the form (32) resulting from the flat background is recovered in the
limit $L\rightarrow\infty$, or more precisely when
$|\frac{\dot{a}}{a}|>>\frac{1}{L}$.

  3. 3.

$dS_{5}$ background

Using the so-called flat slicing the metric for $dS_{5}$ is

$(ds^{2})_{dS_{5}}=-(dY^{0})^{2}+e^{2Y^{0}/L}\Bigl{(}(dY^{1})^{2}+(dY^{2})^{2}+(dY^{3})^{2}\Bigr{)}+e^{2Y^{0}/L}(dY^{4})^{2}\;,$
(28)

$L$ again being the radius of curvature. Now (8) is recovered from the
embedding (16) for

$e^{b/L}=a\qquad\quad L^{2}\frac{\dot{a}^{2}}{a^{2}}-a^{2}\dot{h}^{2}=1$ (29)

After substituting (16) in (11)

$\displaystyle
L^{2}\partial_{t}\left(\frac{\dot{a}}{a}F_{1}\right)+a^{2}\dot{h}^{2}F_{1}-3a^{2}F_{2}$
$\displaystyle=$ $\displaystyle 0$ (30)
$\displaystyle\partial_{t}(a^{2}\dot{h}F_{1})$ $\displaystyle=$ $\displaystyle
0$ (31)

From (13) and (31) we then get

$\rho_{\tt RT}=\frac{c_{0}}{a^{3}\sqrt{L^{2}{\dot{a}^{2}}-{a^{2}}}}\;$ (32)

$c_{0}$ is real which means we need that $|\frac{\dot{a}}{a}|>\frac{1}{L}$.
The expression (32) is once again recovered for
$|\frac{\dot{a}}{a}|>>\frac{1}{L}$.

To summarize, the source term $\rho_{\tt RT}$ for the three different
backgrounds has the form444 Here we have done a rescaling of the constant
$c_{0}$ for the case $k_{5}=0$.

$\rho_{\tt RT}=\frac{c_{0}}{a^{3}\sqrt{L^{2}{\dot{a}^{2}}-k_{5}{a^{2}}}}\;,$
(33)

where $k_{5}$ defines the curvature of the five-dimensional background space:
$k_{5}=0,-1,1$ for $R^{4,1}$, $AdS_{5}$ and $dS_{5}$, respectively. Moreover,
from (13) one has that

${\dot{a}^{2}}a\sqrt{L^{2}{\dot{a}^{2}}-k_{5}{a^{2}}}={\rm constant}$ (34)

$p_{\tt RT}$ can be determined from the conservation law (10) leading to a
time-dependent555 The case $k_{5}=0$ is an exception. After using (34) one
gets the simple relation $p_{{}_{\rm RT}}=-\frac{1}{9}\rho_{\tt RT}$. equation
of state

$p_{{}_{\rm RT}}=-\frac{a}{3\dot{a}}\dot{\rho}_{{}_{\rm RT}}-\rho_{{}_{\rm
RT}}=\frac{a\left(L^{2}\ddot{a}-k_{5}a\right)}{3\left(L^{2}\dot{a}^{2}-k_{5}a^{2}\right)}\rho_{\tt
RT}$ (35)

The evolution of the scale factor for the three cases $k_{5}=1,0,-1$ is
obtained from (34). As stated previously, for $k_{5}=0$ one gets $a(t)\propto
t^{3/4}$. We resort to numerical integration to obtain solutions for the other
two cases, $k_{5}=\pm 1$. The results for all three cases are plotted in Fig.
1, using the initial condition $a(0)=0$. All three cases agree for small $t$,
i.e. $a(t)\propto t^{3/4}$ as $L\frac{\dot{a}}{a}\rightarrow\infty$, and so
$\ddot{a}<0$. For cases $k_{5}=0$ and $-1$, we find that $\ddot{a}<0$, for all
$t$. The situation is more interesting for $k_{5}=1$, corresponding to the de
Sitter background. In this case, $\ddot{a}$ vanishes at finite $t$, when
$L\frac{\dot{a}}{a}=\sqrt{2}$, thus signaling a transition from the de-
accelerating phase to an accelerating phase. We get that $L\frac{\dot{a}}{a}$
goes asymptotically to one in the $t\rightarrow\infty$ limit, where the scale
factor undergoes an exponential expansion at leading order,

$a(t)\rightarrow a_{1}e^{t/L}\left(1-a_{2}e^{-8t/L}+\cdots\right)\;,\qquad{\rm
as}\;t\rightarrow\infty\;,$ (36)

$a_{1}$ and $a_{2}$ being positive constants. From (35) we can obtain the
equation of state for the RT source as a function of time. The ratio
$p_{{}_{\rm RT}}/\rho_{{}_{\rm RT}}$, standardly denoted by $w$, goes from
$-\frac{1}{9}$, near $t=0$, to $-\frac{1}{3}$, at the transition, to $-1$, in
the limit $t\rightarrow\infty$. Note that unlike in the $\Lambda$CDM model,
here we get a transition from the de-accelerating phase to an accelerating
phase even without the inclusion of a matter component or cosmological
constant component to the Friedmann equations.

Figure 1: Plot of $t$ vs $a$ for three different five-dimensional background
spaces: $R^{4,1}$, $AdS_{5}$ and $dS_{5}$. (Here we set $L=1$.)

(a) $H$ versus $z$

(b) $H/(1+z)$ versus $z$

Figure 2:  The solid purple curve in figure (a) represents a fit of eq. (37)
with the Hubble parameter data, while the dashed red curve is $\Lambda$CDM.
$H$ is given in units of ${\rm km\,s}^{-1}{\rm Mpc}^{-1}$. The best fit occurs
for $\tilde{c}_{0}\approx.26$ and $1/L\approx 72\,{\rm km\,s}^{-1}{\rm
Mpc}^{-1}$. From figure (b) the minimum of $H/(1+z)$ for the best fit occurs
at $z\approx.675$, corresponding to the transition from a de-accelerating
phase to an acceleration phase.

Finally, we proceed with a fit of the $k_{5}=1$ case to observational data.
(34) gives an algebraic relation between the Hubble parameter $H=\dot{a}/a$
and the redshift parameter $z=a_{0}/a-1$, where $a_{0}$ is the scale parameter
at the current time. It is

$L^{2}H^{2}{\sqrt{L^{2}H^{2}-1}}={\tilde{c}_{0}\,(1+z)^{4}}\;,$ (37)

where $\tilde{c}_{0}=\frac{8\pi G}{3}L^{2}a_{0}^{-4}c_{0}$. In Fig. 2(a) we
fit the real solution to eq. (37) to observed results for $H$ versus $z$ using
the data in Table 1. The best fit occurs for $\tilde{c}_{0}\approx.26$ and
$1/L\approx 72\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$. For $H$ evaluated at $z=0$
one gets $H(0)\approx 74\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$. Our fit in Fig.
2(a) is compared to that of $\Lambda$CDM, where the expression for the Hubble
parameter is given by $H=H_{0}\sqrt{(1+z)^{3}\Omega_{m}+\Omega_{\Lambda}}$,
with $\Omega_{m}=.3$, $\Omega_{\Lambda}=.7$ and $H_{0}\approx 68.92\,{\rm
km\,s}^{-1}{\rm Mpc}^{-1}$. $H/(1+z)$ (which is proportional to $\dot{a}$)
versus $z$ is plotted in Fig. 2(b), using our fit for $H$ in Fig 2(a). It
shows that the transition from a de-acceleration phase to acceleration phase
occurs at $z\approx.675$, which is similar to the value predicted by
$\Lambda$CDM.

We now summarize some of the features of this model. After generalizing RT
gravity to curved backgrounds, we found universal formulas for the effective
density and pressure, (33) and (35), respectively, resulting from embedding
the $k=0$ RW manifold in three different five-dimensional background spaces.
We suspect that the results found here are dependent on the choice of
embedding (in addition to the choice of background space), although we have
not found specific examples of this.

A reasonable fit to the Hubble parameter data was obtained in the case where
the background was de Sitter space. This is true even without considering the
usual stress-energy contributions to the Einstein equations, which on the
other hand, play an essential role for $\Lambda$CDM. Such components can
easily be included in our model by adding appropriate terms to (12) and
consequent equations. For the case of nonrelativistic matter, one ends up with
the following modification to (37):

$\frac{L^{2}H^{2}}{(1+z)^{3}}-\frac{\tilde{c}_{0}(1+z)}{\sqrt{L^{2}H^{2}-1}}={\tilde{c}_{1}}\;,$
(38)

where $\tilde{c}_{1}$ is an additional constant which quantifies the
nonrelativistic matter component. The inclusion of the additional parameter
$\tilde{c}_{1}$ does not appear to improve the previous fit in any significant
manner.

The presence of the square root in (37) [and also in (38)] gives a lower bound
on the Hubble parameter, $H(z)>1/L$, which is in agreement with observation.

The fit we obtained to the Hubble parameter data holds for values of $z$ up to
approximately $2.36$. Concerning $z>2.36$, the deviation of our fit in Fig. 2
with that of $\Lambda$CDM grows when extrapolating to higher $z$. However, our
fit did not include contributions from the stress-energy tensor, which can
play a more significant role at large $z$. For example, if one considers the
matter density $\rho_{m}$ which is proportional to $a^{-3}$, then its relative
contribution is $\rho_{m}/\rho_{\tt RT}\propto\sqrt{L^{2}H^{2}-1}/(z+1)$,
which grows like $LH/z$ for large $z$. Also, there is no reason to assume that
the $5$d de Sitter background is valid for all $z$. This among other issues is
open for further investigation/speculation.

$z$ | $H$ | $\sigma_{H}$ |
---|---|---|---
0 | $74.03$ | $1.42$ | [12]
.17 | $83$ | $8$ | [13]
.1791 | $75$ | $4$ | [14]
.1993 | $75$ | $5$ | [14]
.38 | $81.5$ | $1.9$ | [15]
.4783 | $80.9$ | $9$ | [16]
.51 | $90.4$ | $1.9$ | [15]
.5929 | $104$ | $13$ | [14]
.61 | $97.3$ | $2.1$ | [15]
.6797 | $92$ | $8$ | [14]
.7812 | $105$ | $12$ | [14]
.8754 | $125$ | $17$ | [14]
1.037 | $154$ | $20$ | [14]
1.3 | $168$ | $17$ | [13]
1.43 | $177$ | $18$ | [13]
1.53 | $140$ | $14$ | [13]
2.34 | $222$ | $7$ | [17]
2.36 | $226$ | $8$ | [18]
Table 1: Data used for fit in Fig. 2. Columns 1-4 are $z$, $H$ , error in $H$
and citation respectively. Columns 2&3 are in units of ${\rm km\,s}^{-1}{\rm
Mpc}^{-1}$. Data was selected with $\sigma_{H}<.15H$.

## References

  * [1] T. Regge and C. Teitelboim, “General Relativity \‘a la string: a progress report,” in Proceedings of the First Marcel Grossmann Meeting (Trieste, Italy,1975), 77-87, ed. by R. Ruffini, North-Holland, Amsterdam, 1977; [arXiv:1612.05256 [hep-th]].
  * [2] S. Deser, F. A. E. Pirani and D. C. Robinson, “Imbedding the G-String,” Phys. Rev. D 14, 3301-3303 (1976).
  * [3] M. Pavsic, “On the Quantization of Gravity by Embedding Space-time in a Higher Dimensional Space,” Class. Quant. Grav. 2, 869 (1985).
  * [4] V. Tapia, “GRAVITATION A LA STRING,” Class. Quant. Grav. 6, L49 (1989).
  * [5] M. D. Maia, “On the Integrability Conditions for Extended Objects,” Class. Quant. Grav. 6, 173-183 (1989).
  * [6] I. A. Bandos, “String - like description of gravity and possible applications for F theory,” Mod. Phys. Lett. A 12, 799-810 (1997).
  * [7] R. Capovilla, A. Escalante, J. Guven and E. Rojas, “Hamiltonian dynamics of extended objects: Regge-Teitelboim model,” [arXiv:gr-qc/0603126 [gr-qc]].
  * [8] S. Fabi, A. Stern and C. Xu, “Cosmic acceleration in Regge-Teitelboim gravity,” [arXiv:2202.09453 [gr-qc]].
  * [9] A. Davidson, “Lambda = 0 cosmology of a brane - like universe,” Class. Quant. Grav. 16, 653-659 (1999).
  * [10] I. Robinson, “On plane waves and nullicles,” in From SU(2) to Gravity: Festschrift in Honor of Yuval Ne’eman , Cambridge Press, (1985).
  * [11] M. M. Akbar, “Embedding FLRW geometries in pseudo-Euclidean and anti–de Sitter spaces,” Phys. Rev. D 95, no.6, 064058 (2017).
  * [12] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, “Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond $\Lambda$CDM,” Astrophys. J. 876, no.1, 85 (2019).
  * [13] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S.A. Stanford, “Cosmic chronometers: constraining the equation of state of dark energy. I: $H(z)$ measurements” JCAP 1002, 008 (2010).
  * [14] M. Moresco, A. Cimatti, R. Jimenez, L. Pozzetti, G. Zamorani, M. Bolzonella, J. Dunlop, F. Lamareille, M. Mignoli and H. Pearce, et al. “Improved constraints on the expansion rate of the Universe up to z~1.1 from the spectroscopic evolution of cosmic chronometers,” JCAP 08, 006 (2012).
  * [15] S. Alam et al. [BOSS], “The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample,” Mon. Not. Roy. Astron. Soc. 470, no.3, 2617-2652 (2017).
  * [16] M. Moresco, L. Pozzetti, A. Cimatti, R. Jimenez, C. Maraston, L. Verde, D. Thomas, A. Citro, R. Tojeiro and D. Wilkinson, “A 6% measurement of the Hubble parameter at $z\sim 0.45$: direct evidence of the epoch of cosmic re-acceleration,” JCAP 05, 014 (2016).
  * [17] T. Delubac et al. [BOSS], “Baryon acoustic oscillations in the Ly$\alpha$ forest of BOSS DR11 quasars,” Astron. Astrophys. 574, A59 (2015).
  * [18] A. Font-Ribera et al. [BOSS], “Quasar-Lyman $\alpha$ Forest Cross-Correlation from BOSS DR11 : Baryon Acoustic Oscillations,” JCAP 05, 027 (2014).


&\,-\frac{1}{2}\log^{2}\left(1-\frac{1}{z}\right)+\frac{7\pi^{2}}{6}\Bigg{]}+\cal
O(\epsilon^{3})\,.\end{split}$ (157)

##  Appendix C Results for the sunrise graph

We write here the analytical expressions for the continued sunrise integrals
in the 8 regions defined in eq. (129). For each region
$r=\\{0a,0b,1a,1b,2a,2b,3a,3b\\}$, we define $\tau=\gamma_{r}\cdot\tau_{r}$
and we find:

$\begin{split}\mathcal{T}^{(0a)}_{1}(t(\tau_{0a}))&\,=\frac{i(\tau_{0a}-1)}{\pi}\Big{[}5I(\small{\texteisfont{a}}_{3,6,1,1};\tau_{0a})-I(\small{\texteisfont{a}}_{3,6,1,2};\tau_{0a})-5I(\small{\texteisfont{a}}_{3,6,2,1};\tau_{0a})\\\
&+5iI(\small{\texteisfont{b}}_{3,6,1,1};\tau_{0a})+iI(\small{\texteisfont{b}}_{3,6,1,2};\tau_{0a})+5iI(\small{\texteisfont{b}}_{3,6,2,1};\tau_{0a})\Big{]}\\\
&+10I(1,\small{\texteisfont{a}}_{3,6,1,1};\tau_{0a})-2I(1,\small{\texteisfont{a}}_{3,6,1,2};\tau_{0a})-10I(1,\small{\texteisfont{a}}_{3,6,2,1};\tau_{0a})\\\
&+2i\Big{[}5I(1,\small{\texteisfont{b}}_{3,6,1,1};\tau_{0a})+I(1,\small{\texteisfont{b}}_{3,6,1,2};\tau_{0a})+5I(1,\small{\texteisfont{b}}_{3,6,2,1};\tau_{0a})\Big{]}+\frac{i\pi}{8}\,,\\\
\mathcal{T}^{(0a)}_{2}(t(\tau_{0a}))&\,=\frac{(2\tau_{0a}-1)}{\pi}\Big{[}-5iI(\small{\texteisfont{a}}_{3,6,1,1};\tau_{0a})+iI(\small{\texteisfont{a}}_{3,6,1,2};\tau_{0a})+5iI(\small{\texteisfont{a}}_{3,6,2,1};\tau_{0a})\\\
&+5I(\small{\texteisfont{b}}_{3,6,1,1};\tau_{0a})+I(\small{\texteisfont{b}}_{3,6,1,2};\tau_{0a})+5I(\small{\texteisfont{b}}_{3,6,2,1};\tau_{0a})\Big{]}\\\
&-20I(1,\small{\texteisfont{a}}_{3,6,1,1};\tau_{0a})+4I(1,\small{\texteisfont{a}}_{3,6,1,2};\tau_{0a})+20I(1,\small{\texteisfont{a}}_{3,6,2,1};\tau_{0a})\\\
&-4i\Big{[}5I(1,\small{\texteisfont{b}}_{3,6,1,1};\tau_{0a})+I(1,\small{\texteisfont{b}}_{3,6,1,2};\tau_{0a})+5I(1,\small{\texteisfont{b}}_{3,6,2,1};\tau_{0a})\Big{]}-\frac{i\pi}{12}\,,\end{split}$
(158)

$\begin{split}\mathcal{T}^{(0b)}_{1}(t(\tau_{0b}))&\,=\frac{1}{\pi}\Big{[}iI(\small{\texteisfont{a}}_{3,6,1,1};\tau_{0b})-5iI(\small{\texteisfont{a}}_{3,6,1,2};\tau_{0b})+5iI(\small{\texteisfont{a}}_{3,6,2,1};\tau_{0b})\\\
&+I(\small{\texteisfont{b}}_{3,6,1,1};\tau_{0b})+5I(\small{\texteisfont{b}}_{3,6,1,2};\tau_{0b})-5I(\small{\texteisfont{b}}_{3,6,2,1};\tau_{0b})\big{]}\,,\\\
\mathcal{T}^{(0b)}_{2}(t(\tau_{0b}))&\,=\frac{(\tau_{0b}-1)}{\pi}\Big{[}iI(\small{\texteisfont{a}}_{3,6,1,1};\tau_{0b})-5iI(\small{\texteisfont{a}}_{3,6,1,2};\tau_{0b})+5iI(\small{\texteisfont{a}}_{3,6,2,1};\tau_{0b})\\\
&+I(\small{\texteisfont{b}}_{3,6,1,1};\tau_{0b})+5I(\small{\texteisfont{b}}_{3,6,1,2};\tau_{0b})-5I(\small{\texteisfont{b}}_{3,6,2,1};\tau_{0b})\Big{]}\\\
&+2I(1,\small{\texteisfont{a}}_{3,6,1,1};\tau_{0b})-10I(1,\small{\texteisfont{a}}_{3,6,1,2};\tau_{0b})+10I(1,\small{\texteisfont{a}}_{3,6,2,1};\tau_{0b})\\\
&-2i\Big{[}I(1,\small{\texteisfont{b}}_{3,6,1,1};\tau_{0b})+5I(1,\small{\texteisfont{b}}_{3,6,1,2};\tau_{0b})-5I(1,\small{\texteisfont{b}}_{3,6,2,1};\tau_{0b})\Big{]}-\frac{i\pi}{8}\,,\end{split}$
(159)

$\begin{split}\mathcal{T}^{(2a)}_{1}(t(\tau_{2a}))&\,=\frac{1}{\pi}\Big{[}-5I(\small{\texteisfont{b}}_{3,6,0,1};\tau_{2a})+I(\small{\texteisfont{b}}_{3,6,3,1};\tau_{2a})+5I(\small{\texteisfont{b}}_{3,6,3,2};\tau_{2a})\Big{]}\,,\\\
\mathcal{T}^{(2a)}_{2}(t(\tau_{2a}))&\,=\frac{(\tau_{2a}-3)}{\pi}\Big{[}-5I(\small{\texteisfont{b}}_{3,6,0,1};\tau_{2a})+I(\small{\texteisfont{b}}_{3,6,3,1};\tau_{2a})+5I(\small{\texteisfont{b}}_{3,6,3,2};\tau_{2a})\Big{]}\\\
&+2i(5I(1,\small{\texteisfont{b}}_{3,6,0,1};\tau_{2a})-I(1,\small{\texteisfont{b}}_{3,6,3,1};\tau_{2a})-5I(1,\small{\texteisfont{b}}_{3,6,3,2};\tau_{2a}))-\frac{i\pi}{8}\,,\end{split}$
(160)

$\begin{split}\mathcal{T}^{(2b)}_{1}(t(\tau_{2b}))&\,=10I(\small{\texteisfont{a}}_{3,6,1,0},1;\tau_{2b})-2I(\small{\texteisfont{a}}_{3,6,1,3},1;\tau_{2b})-10I(\small{\texteisfont{a}}_{3,6,2,3},1;\tau_{2b})\\\
&-\frac{5{\rm Cl}_{2}(\pi/3)}{2\pi}\,,\\\
\mathcal{T}^{(2b)}_{2}(t(\tau_{2b}))&\,=\frac{i}{\pi}\Big{[}5I(\small{\texteisfont{a}}_{3,6,1,0};\tau_{2b})-I(\small{\texteisfont{a}}_{3,6,1,3};\tau_{2b})-5I(\small{\texteisfont{a}}_{3,6,2,3};\tau_{2b})\Big{]}\\\
&-30I(\small{\texteisfont{a}}_{3,6,1,0},1;\tau_{2b})+6I(\small{\texteisfont{a}}_{3,6,1,3},1;\tau_{2b})+30I(\small{\texteisfont{a}}_{3,6,2,3},1;\tau_{2b})\\\
&+\frac{15{\rm Cl}_{2}(\pi/3)-i\pi^{2}}{2\pi}\,,\end{split}$ (161)

$\begin{split}\mathcal{T}^{(3a)}_{1}(t(\tau_{3a}))&\,=-\frac{i(\tau_{3a}-1)}{\pi}\Big{[}5I(\small{\texteisfont{a}}_{3,6,1,0};\tau_{3a})+5I(\small{\texteisfont{a}}_{3,6,1,3};\tau_{3a})+I(\small{\texteisfont{a}}_{3,6,2,3};\tau_{3a})\Big{]}\\\
&-10\Big{[}I(1,\small{\texteisfont{a}}_{3,6,1,0};\tau_{3a})+I(1,\small{\texteisfont{a}}_{3,6,1,3};\tau_{3a})\Big{]}-2I(1,\small{\texteisfont{a}}_{3,6,2,3};\tau_{3a})\\\
&-\frac{5{\rm Cl}_{2}(\pi/3)}{2\pi}\,,\\\
\mathcal{T}^{(3a)}_{2}(t(\tau_{3a}))&\,=\frac{i(3\tau_{3a}-2)}{\pi}\Big{[}5I(\small{\texteisfont{a}}_{3,6,1,0};\tau_{3a})+5I(\small{\texteisfont{a}}_{3,6,1,3};\tau_{3a})+I(\small{\texteisfont{a}}_{3,6,2,3};\tau_{3a})\Big{]}\\\
&+30\Big{[}I(1,\small{\texteisfont{a}}_{3,6,1,0};\tau_{3a})+I(1,\small{\texteisfont{a}}_{3,6,1,3};\tau_{3a})\Big{]}+6I(1,\small{\texteisfont{a}}_{3,6,2,3};\tau_{3a})\\\
&+\frac{15{\rm Cl}_{2}(\pi/3)-i\pi^{2}}{2\pi}\,,\end{split}$ (162)

$\begin{split}\mathcal{T}^{(3b)}_{1}(t(\tau_{3b}))&\,=\frac{(\tau_{3b}+1)}{\pi}\Big{[}-5I(\small{\texteisfont{b}}_{3,6,0,1};\tau_{3b})-5I(\small{\texteisfont{b}}_{3,6,3,1};\tau_{3b})-I(\small{\texteisfont{b}}_{3,6,3,2};\tau_{3b})\Big{]}\\\
&+2i\Big{[}5I(1,\small{\texteisfont{b}}_{3,6,0,1};\tau_{3b})+5I(1,\small{\texteisfont{b}}_{3,6,3,1};\tau_{3b})+I(1,\small{\texteisfont{b}}_{3,6,3,2};\tau_{3b})\Big{]}+\frac{i\pi}{8}\,,\\\
\mathcal{T}^{(3b)}_{2}(t(\tau_{3b}))&\,=\frac{(2\tau_{3b}+3)}{\pi}\Big{[}5I(\small{\texteisfont{b}}_{3,6,0,1};\tau_{3b})+5I(\small{\texteisfont{b}}_{3,6,3,1};\tau_{3b})+I(\small{\texteisfont{b}}_{3,6,3,2};\tau_{3b})\Big{]}\\\
&-4i\Big{[}5I(1,\small{\texteisfont{b}}_{3,6,0,1};\tau_{3b})+5I(1,\small{\texteisfont{b}}_{3,6,3,1};\tau_{3b})+I(1,\small{\texteisfont{b}}_{3,6,3,2};\tau_{3b})\Big{]}-\frac{3i\pi}{4}\,.\end{split}$
(163)

## References

  * (1) N. Nielsen, _Der Eulersche Dilogarithmus und seine Verallgemeinerungen_ , _Nova Acta Leopoldina (Halle)_ 90 (1909) .
  * (2) E. E. Kummer, _Über die Transcendenten, welche aus wiederholten Integrationen rationaler Formeln entstehen_ , _J. reine ang. Mathematik_ 21 (1840) 74.
  * (3) E. Remiddi and J. A. M. Vermaseren, _Harmonic polylogarithms_ , _Int. J. Mod. Phys._ A15 (2000) 725 [hep-ph/9905237].
  * (4) T. Gehrmann and E. Remiddi, _Two loop master integrals for $\gamma^{*}\to 3$ jets: The Planar topologies_, _Nucl.Phys._ B601 (2001) 248 [hep-ph/0008287].
  * (5) A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, _Classical Polylogarithms for Amplitudes and Wilson Loops_ , _Phys. Rev. Lett._ 105 (2010) 151605 [1006.5703].
  * (6) A. B. Goncharov, _Multiple polylogarithms and mixed Tate motives_ , math/0103059.
  * (7) C. Duhr, H. Gangl and J. R. Rhodes, _From polygons and symbols to polylogarithmic functions_ , _JHEP_ 1210 (2012) 075 [1110.0458].
  * (8) C. Duhr, _Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes_ , _JHEP_ 08 (2012) 043 [1203.0454].
  * (9) T. Gehrmann and E. Remiddi, _Numerical evaluation of harmonic polylogarithms_ , _Comput. Phys. Commun._ 141 (2001) 296 [hep-ph/0107173].
  * (10) T. Gehrmann and E. Remiddi, _Numerical evaluation of two-dimensional harmonic polylogarithms_ , _Comput. Phys. Commun._ 144 (2002) 200 [hep-ph/0111255].
  * (11) S. Buehler and C. Duhr, _CHAPLIN - Complex Harmonic Polylogarithms in Fortran_ , _Comput. Phys. Commun._ 185 (2014) 2703 [1106.5739].
  * (12) J. Vollinga and S. Weinzierl, _Numerical evaluation of multiple polylogarithms_ , _Comput. Phys. Commun._ 167 (2005) 177 [hep-ph/0410259].
  * (13) H. Frellesvig, D. Tommasini and C. Wever, _On the reduction of generalized polylogarithms to $\text{Li}_{n}$ and $\text{Li}_{2,2}$ and on the evaluation thereof_, _JHEP_ 03 (2016) 189 [1601.02649].
  * (14) J. Ablinger, J. Blümlein, M. Round and C. Schneider, _Numerical Implementation of Harmonic Polylogarithms to Weight w = 8_ , _Comput. Phys. Commun._ 240 (2019) 189 [1809.07084].
  * (15) T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, _The two-loop master integrals for $q\overline{q}\to VV$_, _JHEP_ 1406 (2014) 032 [1404.4853].
  * (16) T. Gehrmann, A. von Manteuffel and L. Tancredi, _The two-loop helicity amplitudes for $q\overline{q}^{\prime}\to{V}_{1}{V}_{2}\to 4$ leptons_, _JHEP_ 09 (2015) 128 [1503.04812].
  * (17) A. von Manteuffel and L. Tancredi, _The two-loop helicity amplitudes for $gg\to V_{1}V_{2}\to 4~{}\mathrm{leptons}$_, _JHEP_ 06 (2015) 197 [1503.08835].
  * (18) A. Sabry, _Fourth order spectral functions for the electron propagator_ , _Nucl. Phys._ 33 (1962) 401.
  * (19) M. Czakon, P. Fiedler and A. Mitov, _Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through $O(?\frac{4}{S})$_, _Phys. Rev. Lett._ 110 (2013) 252004 [1303.6254].
  * (20) D. J. Broadhurst, _The Master Two Loop Diagram With Masses_ , _Z. Phys._ C47 (1990) 115.
  * (21) S. Bauberger, F. A. Berends, M. Bohm and M. Buza, _Analytical and numerical methods for massive two loop selfenergy diagrams_ , _Nucl. Phys._ B434 (1995) 383 [hep-ph/9409388].
  * (22) S. Bauberger and M. Bohm, _Simple one-dimensional integral representations for two loop selfenergies: The Master diagram_ , _Nucl. Phys._ B445 (1995) 25 [hep-ph/9501201].
  * (23) S. Müller-Stach, S. Weinzierl and R. Zayadeh, _A Second-Order Differential Equation for the Two-Loop Sunrise Graph with Arbitrary Masses_ , _Commun.Num.Theor.Phys._ 6 (2012) 203 [1112.4360].
  * (24) S. Laporta and E. Remiddi, _Analytic treatment of the two loop equal mass sunrise graph_ , _Nucl. Phys._ B704 (2005) 349 [hep-ph/0406160].
  * (25) M. Caffo, H. Czyz, M. Gunia and E. Remiddi, _BOKASUN: A Fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams_ , _Comput. Phys. Commun._ 180 (2009) 427 [0807.1959].
  * (26) E. Remiddi and L. Tancredi, _Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral_ , _Nucl. Phys._ B907 (2016) 400 [1602.01481].
  * (27) J. Broedel, C. Duhr, F. Dulat and L. Tancredi, _Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral_ , _Phys. Rev._ D97 (2018) 116009 [1712.07095].
  * (28) E. Remiddi and L. Tancredi, _Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph_ , _Nucl. Phys._ B880 (2014) 343 [1311.3342].
  * (29) L. Adams, C. Bogner and S. Weinzierl, _The iterated structure of the all-order result for the two-loop sunrise integral_ , _J. Math. Phys._ 57 (2016) 032304 [1512.05630].
  * (30) M. Caffo, H. Czyz, S. Laporta and E. Remiddi, _The Master differential equations for the two loop sunrise selfmass amplitudes_ , _Nuovo Cim._ A111 (1998) 365 [hep-th/9805118].
  * (31) L. Adams, C. Bogner and S. Weinzierl, _The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms_ , _J. Math. Phys._ 55 (2014) 102301 [1405.5640].
  * (32) L. Adams, C. Bogner and S. Weinzierl, _The two-loop sunrise graph with arbitrary masses_ , _J. Math. Phys._ 54 (2013) 052303 [1302.7004].
  * (33) L. Adams, C. Bogner and S. Weinzierl, _The two-loop sunrise graph with arbitrary masses_ , _J.Math.Phys._ 54 (2013) 052303 [1302.7004].
  * (34) L. Adams, C. Bogner and S. Weinzierl, _The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case_ , _J. Math. Phys._ 56 (2015) 072303 [1504.03255].
  * (35) S. Bloch, M. Kerr and P. Vanhove, _Local mirror symmetry and the sunset Feynman integral_ , _Adv. Theor. Math. Phys._ 21 (2017) 1373 [1601.08181].
  * (36) S. Bloch and P. Vanhove, _The elliptic dilogarithm for the sunset graph_ , _J. Number Theor._ 148 (2015) 328 [1309.5865].
  * (37) B. A. Kniehl, A. V. Kotikov, A. Onishchenko and O. Veretin, _Two-loop sunset diagrams with three massive lines_ , _Nucl. Phys._ B738 (2006) 306 [hep-ph/0510235].
  * (38) C. Bogner, S. Müller-Stach and S. Weinzierl, _The unequal mass sunrise integral expressed through iterated integrals on $\overline{\mathcal{M}}_{1,3}$_, 1907.01251.
  * (39) A. Levin and G. Racinet, _Towards multiple elliptic polylogarithms_ , math/0703237.
  * (40) F. Brown and A. Levin, _Multiple Elliptic Polylogarithms_ , 1110.6917.
  * (41) J. Broedel, C. R. Mafra, N. Matthes and O. Schlotterer, _Elliptic multiple zeta values and one-loop superstring amplitudes_ , _JHEP_ 07 (2015) 112 [1412.5535].
  * (42) J. Broedel, O. Schlotterer and F. Zerbini, _From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop_ , 1803.00527.
  * (43) J. Broedel, N. Matthes and O. Schlotterer, _Relations between elliptic multiple zeta values and a special derivation algebra_ , _J. Phys._ A49 (2016) 155203 [1507.02254].
  * (44) J. Broedel, N. Matthes, G. Richter and O. Schlotterer, _Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes_ , _J. Phys._ A51 (2018) 285401 [1704.03449].
  * (45) J. Broedel, C. Duhr, F. Dulat and L. Tancredi, _Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism_ , _JHEP_ 05 (2018) 093 [1712.07089].
  * (46) J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, _Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series_ , _JHEP_ 08 (2018) 014 [1803.10256].
  * (47) J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, _Elliptic Feynman integrals and pure functions_ , _JHEP_ 01 (2019) 023 [1809.10698].
  * (48) J. Blümlein, _Iterative Non-iterative Integrals in Quantum Field Theory_ , in _KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Zeuthen, Germany, October 23-26, 2017_ , 2018, 1808.08128.
  * (49) J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, _Elliptic polylogarithms and Feynman parameter integrals_ , _JHEP_ 05 (2019) 120 [1902.09971].
  * (50) L. Adams and S. Weinzierl, _On a class of Feynman integrals evaluating to iterated integrals of modular forms_ , in _KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Zeuthen, Germany, October 23-26, 2017_ , 2018, 1807.01007.
  * (51) J. Broedel, C. Duhr, F. Dulat, B. Penante and L. Tancredi, _From modular forms to differential equations for Feynman integrals_ , in _KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory Zeuthen, Germany, October 23-26, 2017_ , 2018, 1807.00842.
  * (52) D. Zagier, _Elliptic Modular Forms and Their Applications_ , in _The 1-2-3 of Modular Forms_ , Springer, 2008.
  * (53) L. Adams and S. Weinzierl, _Feynman integrals and iterated integrals of modular forms_ , _Commun. Num. Theor. Phys._ 12 (2018) 193 [1704.08895].
  * (54) Y. I. Manin, _Iterated integrals of modular forms and noncommutative modular symbols_ , in _Algebraic geometry and number theory_ , vol. 253 of _Progr. Math._ , (Boston), pp. 565–597, Birkhäuser Boston, 2006, math/0502576.
  * (55) A. Primo and L. Tancredi, _Maximal cuts and differential equations for Feynman integrals. An application to the three-loop massive banana graph_ , _Nucl. Phys._ B921 (2017) 316 [1704.05465].
  * (56) A. von Manteuffel and L. Tancredi, _A non-planar two-loop three-point function beyond multiple polylogarithms_ , _JHEP_ 06 (2017) 127 [1701.05905].
  * (57) C. Bogner, A. Schweitzer and S. Weinzierl, _Analytic continuation and numerical evaluation of the kite integral and the equal mass sunrise integral_ , _Nucl. Phys._ B922 (2017) 528 [1705.08952].
  * (58) H. Frellesvig, M. Hidding, L. Maestri, F. Moriello and G. Salvatori, _The complete set of two-loop master integrals for Higgs + jet production in QCD_ , 1911.06308.
  * (59) J. Broedel, C. Duhr, F. Dulat, R. Marzucca, B. Penante and L. Tancredi, _An analytic solution for the equal-mass banana graph_ , _JHEP_ 09 (2019) 112 [1907.03787].
  * (60) F. Diamond and J. Shurman, _A First Course in Modular Forms_ , vol. 228 of _Graduate Texts in Mathematics_. Springer.
  * (61) F. Brown, _Multiple modular values and the relative completion of the fundamental group of $\mathcal{M}_{1,1}$_, 1407.5167v4.
  * (62) K. T. Chen, _Iterated path integrals_ , _Bull. Amer. Math. Soc._ 83 (1977) 831.
  * (63) N. Arkani-Hamed, J. L. Bourjaily, F. Cachazo and J. Trnka, _Local Integrals for Planar Scattering Amplitudes_ , _JHEP_ 1206 (2012) 125 [1012.6032].
  * (64) A. Goncharov, _Galois symmetries of fundamental groupoids and noncommutative geometry_ , _Duke Math.J._ 128 (2005) 209 [math/0208144].
  * (65) R. S. Maier, _On Rationally Parametrized Modular Equations_ , 2006, math/0611041.
  * (66) Y. Yang, _Transformation formulas for generlized dedekind eta functions_ , _Bull. London Math. Soc._ 36 (2004) 671.
  * (67) F. Brown, _From the Deligne-Ihara conjecture to Multiple Modular Values_ , 1904.00178.
  * (68) O. V. Tarasov, _Connection between Feynman integrals having different values of the space-time dimension_ , _Phys. Rev._ D54 (1996) 6479 [hep-th/9606018].
  * (69) R. N. Lee, _Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D_ , _Nucl. Phys._ B830 (2010) 474 [0911.0252].
  * (70) K. Chetyrkin and F. Tkachov, _Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops_ , _Nucl.Phys._ B192 (1981) 159.
  * (71) F. V. Tkachov, _A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions_ , _Phys. Lett._ B100 (1981) 65.
  * (72) A. Primo and L. Tancredi, _On the maximal cut of Feynman integrals and the solution of their differential equations_ , _Nucl. Phys._ B916 (2017) 94 [1610.08397].
  * (73) S. Borowka, G. Heinrich, S. P. Jones, M. Kerner, J. Schlenk and T. Zirke, _SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop_ , _Comput. Phys. Commun._ 196 (2015) 470 [1502.06595].
  * (74) S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, J. Schlenk et al., _pySecDec: a toolbox for the numerical evaluation of multi-scale integrals_ , _Comput. Phys. Commun._ 222 (2018) 313 [1703.09692].
  * (75) S. Bloch, M. Kerr and P. Vanhove, _A Feynman integral via higher normal functions_ , _Compos. Math._ 151 (2015) 2329 [1406.2664].