Also at ]Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional
de La Plata, Paseo del Bosque s/n, 1900 La Plata, Buenos Aires, Argentina

# Dynamical black hole in a bouncing universe

Daniela Pérez<EMAIL_ADDRESS>Instituto Argentino de
Radioastronomía (IAR, CONICET/CIC/UNLP), C.C.5, (1894) Villa Elisa, Buenos
Aires, Argentina  Santiago E. Perez Bergliaffa  Departamento de Física
Teórica, Instituto de Física, Universidade do Estado de Rio de Janeiro, CEP
20550-013, Rio de Janeiro, Brazil  Gustavo E. Romero  [  Instituto Argentino
de Radioastronomía (IAR, CONICET/CIC/UNLP), C.C.5, (1894) Villa Elisa, Buenos
Aires, Argentina

###### Abstract

We analyze the causal structure of McVittie spacetime for a classical bouncing
cosmological model. In particular, we compute the trapping horizons of the
metric and integrate the trajectories of radial null geodesics before, during,
and after the bounce takes place. In the contracting phase up to the
occurrence of the bounce, a dynamical black hole is present. When the universe
reaches a certain minimum scale, the trapping horizons disappear and the black
hole ceases to exist. After the bounce, the central weak singularity becomes
naked. In the expanding phase, for large positive values of the cosmic time,
the behaviour of null geodesics indicates that the solution contains a black
hole. These results suggest that neither a contracting nor an expanding
universe can accommodate a black hole at all times.

††preprint: APS/123-QED

##  I Introduction

Our current understanding of the evolution of the Universe is expressed in the
so-called $\Lambda$CDM model that includes gravity as described by the
Einstein’s field equations with a cosmological term, the standard model of
particle physics, a component of cold dark matter, and a hot initial phase.
This model is frequently complemented with an inflationary stage that would
have occurred immediately before the grand unification epoch. Inflation
requires additional physics beyond the standard model. With or without
inflation the standard cosmological model is singular [1]. An initial
cosmological singularity is a very undesirable feature that points out an
irreparable deficiency in the representation of the underlying physical
processes [2]. Bouncing cosmologies try to offer an alternative to overcome
such problems.

There is a wide variety of proposals, either classical or quantum, for a
cosmological bounce [3]. In all these models the universe starts from a very
diluted phase and proceeds to contract. The contraction then smoothly evolves
into a bounce that leads to the current phase of expansion as described by the
$\Lambda$CDM model. These models solve the problem of the initial cosmological
singularity. As the cosmic fluid contracts most structure is erased and the
universe becomes smooth [4]. Black holes, however, might survive the bounce
and play some role in the subsequent expanding universe [5, 6, 7]. Since a
black hole is essentially a region of spacetime with particular curvature, the
overall contraction and expansion of space during the bounce should have a
global-to-local effect upon its horizons. The whole process is dynamical, and
hence cannot be investigated using the standard static solutions.

An exact solution for a central inhomogeneity in a cosmological setting was
first found by McVittie long ago. It is now clear that such a solution
describes a black hole. McVittie solution has been investigated, so far, for
standard prescriptions of the scale factor of the universe. In this work we
extend the research to models that allow for a bounce. We discuss whether the
solution includes a black hole before the bounce and what happens with the
horizons along the cosmological history of a bouncing universe. The results we
have found, we hope, will help to obtain a better understanding of both the
McVittie solution and the fate of a dynamical black hole through a cosmic
bounce.

##  II Dynamical spacetimes

###  II.1 McVittie spacetime

In 1933, McVittie [8] discovered an exact solution of Einstein’s field
equations that describes an inhomogeneity embedded in a Friedmann-Lemaître-
Robertson–Walker (FLRW) cosmological background. In isotropic coordinates
$(t,r,\theta,\phi)$ its line element takes the form:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\frac{\left(1-\frac{m_{0}}{2ra(t)}\right)^{2}}{\left(1+\frac{m_{0}}{2ra(t)}\right)^{2}}dt^{2}+a^{2}(t)\left(1+\frac{m_{0}}{2ra(t)}\right)^{4}$
(1) $\displaystyle\times$
$\displaystyle\left[dr^{2}+r^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right)\right].$

Here, $a(t)$ is the scale factor of the background cosmological model and
$m_{0}$ is a non-negative constant. Setting $a(t)\equiv 1$, Eq. (1) reduces to
the Schwarzschild line element in isotropic coordinates, and in the limit
$m_{0}\rightarrow 0$ the FLRW metric is recovered. For the upcoming
discussion, it is convenient to express Eq. (1) in terms of $R$, the areal
radius coordinate 111The areal radius coordinate $R$ is defined by
$R:=\sqrt{\mathcal{A}/4\pi}$, where $\mathcal{A}$ is the area of the 2-sphere
of symmetry, and where
$d^{2}\Omega_{(2)}:={d\theta}^{2}+{\sin{\theta}}^{2}{d\phi}^{2},$ (2) is the
line element on the unit 2-sphere [40].:

$R\equiv a(t)r\left(1+\frac{m_{0}}{2ra(t)}\right)^{2}.$ (3)

Using this expression, the McVittie line element can be written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-f(t,R)dt^{2}-\frac{2H(t)R}{\sqrt{1-2m_{0}/R}}dtdR+\frac{dR^{2}}{1-2m_{0}/R}$
(4) $\displaystyle+$ $\displaystyle
R^{2}\left(d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right),$

where

$f(t,R)\equiv 1-2m_{0}/R-H(t)^{2}R^{2}.$ (5)

Here, $H(t)\equiv\dot{a}(t)/a(t)$ is the Hubble factor corresponding to the
background cosmological model. There are two key assumptions on which
McVittie’s solution is based. The first one is that the matter represented in
the field equations is described by a perfect fluid with density $\rho$ and
isotropic pressure $p$. The second one is that the fluid is at rest with
respect to the chosen reference frame. There are no additional hypothesis
regarding the properties of the matter that sources the geometry. In
particular, an equation of state, for instance of the form $p=p(\rho)$, is not
assumed. The relation between $\rho$ and $p$ is obtained a posteriori by
solving Einstein field equations when the scale function $a(t)$ is specified.
For further details on the derivation of the McVittie solution see Refs. [10,
11, 12].

There has been a long debate in the literature about the physical
interpretation of the McVittie spacetime, focused mainly on deciding whether
the solution characterizes a black hole in an expanding universe or not. A
recent series of works [11, 12, 13, 14, 15] has been crucial to establish that
the McVittie metric represents a dynamical black hole embedded in a
cosmological background. The details of the solution and its possible
analytical extension depend of the behaviour of $H(t)$ for
$t\rightarrow\infty$ [15]. The solution displays a curvature singularity at
$R=2m_{0}$ for finite values of $t$, as evidenced by the Ricci scalar, given
by:

${\cal R}=12H^{2}+\frac{6\dot{H}}{\sqrt{1-\frac{2m_{0}}{R}}}.$ (6)

The singularity is spacelike and, as shown by Nolan [13], gravitationally weak
[16].

The key feature associated with black holes is the presence of an event
horizon, i.e. a boundary between two regions of spacetime that are causally
disconnected. Events inside the black hole are separated from events in the
global external future of spacetime. In dynamical spacetimes, in order to
identify the event horizon, we would need to know the entire spacetime
manifold to future infinity, which is impossible if the metric is not
completely known.

In order to determine whether a black hole is embedded in a dynamical
background, a full analysis of the causal structure of the spacetime is
necessary. This includes studying the existence of trapping horizons, the
determination of regular trapped and anti-trapped regions, and the computation
of the trajectories of ingoing and outgoing radial null geodesics. In what
follows, we briefly review the main results of Refs. [11, 12, 13, 14, 15] that
prove the assertion that the McVittie spacetime describes a black hole in a
cosmological environment.

The trapping horizons of a spacetime are defined as the surfaces where null
geodesics change their focusing properties [17]. Mathematically, this kind of
horizon is determined by the condition

$\theta_{\mathrm{in}}\theta_{\mathrm{out}}=0,$ (7)

where $\theta_{\mathrm{in}}$ stands for the expansion of ingoing radial null
geodesics while $\theta_{\mathrm{out}}$ denotes the expansion of outgoing
radial null geodesics, respectively. Regions where
$\theta_{\mathrm{in}}\theta_{\mathrm{out}}<0$ are called regular. In the
opposite case, $\theta_{\mathrm{in}}\theta_{\mathrm{out}}>0$, the region is
called anti-trapped if $\theta_{\mathrm{in}}>0$ and $\theta_{\mathrm{out}}>0$,
and trapped if $\theta_{\mathrm{in}}<0$ and $\theta_{\mathrm{out}}<0$. In this
section, we are only considering expanding spacetimes, i.e. $H(t)>0$. The
trapping horizons for the McVittie metric occur for $\theta_{\mathrm{in}}=0$,
and null outgoing geodesics are always expanding ($\theta_{\mathrm{out}}>0$)
[14, 15].

We show in Figures 1 and 2 the location of the trapping horizons in the
McVittie spacetime for the $\Lambda$CDM model 222We are considering the
minimal 6-parameter $\Lambda$CDM model; we assume that the curvature density
parameter $\Omega_{k}=0$ and the equation of state parameter $w=-1$. The
radiation density is neglected, $\Omega_{\rm rad}=0$. Thus
$\Omega_{m}+\Omega_{\Lambda}=1$, being $\Lambda>0$. and for a dust background.
In the first case, the corresponding Hubble factor is
$H(t)=H_{0}\coth{(3/2H_{0}t)}$, where $H_{0}$ is the Hubble constant, while in
the second case H(t) = 2/3t [15]. The essential difference between these two
cosmological models is the asymptotic behaviour of the Hubble factor in the
future: $H(t)\rightarrow H_{0}$ for $t\rightarrow\infty$ in the $\Lambda$CDM
model whereas $H(t)\rightarrow 0$ for $t\rightarrow\infty$ in a dust-dominated
background.

There is a moment in time, denoted $t_{\star}$, when just one trapping horizon
exists. It can be computed by solving the following equation:

$m_{0}H(t_{\star})=\frac{1}{3\sqrt{3}}.$ (8)

If the cosmological background corresponds to the $\Lambda$CDM model,

${t_{\star}}_{\mathrm{\Lambda
CDM}}=\frac{2}{3H_{0}}\mathrm{arcoth}\left(\frac{1}{3\sqrt{3}m_{0}H_{0}}\right),$
(9)

and when the cosmological background is dust-dominated,

${t_{\star}}_{\mathrm{dust}}=2\sqrt{3}m_{0}.$ (10)

Inspection of Figures 1 and 2 reveals that when $t>t_{\star}$ there are two
trapping horizons: an inner (denoted $R_{-}$) and an outer one (denoted
$R_{+}$), such that $R_{+}>R_{-}$. No trapping horizons are present for
$t<t_{\star}$. In both figures, the regular region of the spacetime is
indicated in white while the anti-trapped region is painted in light pink. The
dot-dashed curve indicates the location of the singularity.

Due to spherical symmetry, the equation for the ingoing and outgoing radial
geodesics can be derived by setting $d\theta=d\phi=0$ in $ds^{2}=0$, thus
obtaining

$\frac{dR}{dt}=\sqrt{1-2m_{0}/R}\left(HR\pm\sqrt{1-2m_{0}/R}\right),$ (11)

where the “$-$” (“$+$”) corresponds to the ingoing (outgoing) case. We see
that $dR/dt>0$ for the outgoing branch, i.e, such geodesics are always
diverging. This result is consistent with the fact that
$\theta_{\mathrm{out}}>0$, as mentioned above. We plot these trajectories in
Figures 3 and 4.

Figure 1:  The black line indicates the location of the trapping horizons in
McVittie spacetime for the $\Lambda$CDM model. The white zone corresponds to
the regular region while the light pink corresponds to the anti-trapped
region, respectively. The dot dashed line denotes the location of the
singularity. Here, we fixed $m_{0}=0.479$ and $H_{0}=1/3$ as an example.
Figure 2:  The black line indicates the location of the trapping horizons in
McVittie spacetime for a dust background model. The white zone corresponds to
the regular region while the light pink corresponds to the anti-trapped
region, respectively. The dot dashed line denotes the location of the
singularity. Here, we fixed $m_{0}=0.479$ as an example. Figure 3:  Radial
outgoing null geodesics in McVittie spacetime for the $\Lambda$CDM model. The
black line indicates the location of the trapping horizons and the dot-dashed
line denotes the singular surface $R=2m_{0}$. Here, we fixed $m_{0}=0.479$ and
$H_{0}=1/3$ as an example. Figure 4:  Radial outgoing null geodesics in
McVittie spacetime for a dust cosmological background. The black line
indicates the location of the trapping horizons and the dot-dashed line
denotes the singular surface $R=2m_{0}$. Here, we fixed $m_{0}=0.479$ as an
example.

As seen from Eq. (11), only radial ingoing null geodesics have a turning
point, defined by $dR/dt=0$, and specified by those values of the coordinates
$R$ and $t$ that obey the following equation:

$\tilde{f}(t,R)=H(t)^{2}R^{3}-R+2m_{0}=0.$ (12)

Clearly, Eq. (12) is equivalent to $\theta_{\mathrm{in}}=0$. From Figures 5
and 6 we see that the radial ingoing geodesics are expanding in the anti-
trapped region ($\theta_{\mathrm{in}}>0$) and they converge in the regular
zone ($\theta_{\mathrm{in}}<0$). When they cross the trapping horizon
($\theta_{\mathrm{in}}=0$ or equivalently $\tilde{f}(t,R)=0$), their
convergence changes sign.

Figure 5:  Radial ingoing null geodesics in McVittie spacetime for the
$\Lambda$CDM model. The black line indicates the location of the trapping
horizons and the dot-dashed curve denotes the singular surface $R=2m_{0}$.
Here, we fixed $m_{0}=0.479$ and $H_{0}=1/3$ as an example. Figure 6:  Radial
ingoing null geodesics in McVittie spacetime for a dust cosmological
background. The black line indicates the location of the trapping horizons and
the dot-dashed curve denotes the singular surface $R=2m_{0}$. Here, we fixed
$m_{0}=0.479$ as an example.

We emphasise that trapping horizons are not equivalent to event horizons in
the context of dynamical spacetimes. As shown in Figures 3 and 4, some
outgoing geodesics cross both $R_{-}$ and $R_{+}$. In Figure 5, the green,
yellow, brown, and blue geodesics cross $R_{+}$ and enter the regular region
of the spacetime; as time goes by, they get closer and closer to the surface
$R_{-}$. We stress that for finite values of the time coordinate, $R_{-}$ is
just a trapping horizon. Only in the limit $t\rightarrow\infty$, $R_{-}$
becomes an event horizon and outgoing geodesics cannot travel out of the black
hole. In fact, Kaloper and collaborators [14] proved that, under certain
assumptions, the analysis of the behavior of ingoing null geodesics in the
limit $t\rightarrow\infty$ reveals the presence of an event horizon.

Assuming that $H(t\rightarrow\infty)\rightarrow H_{0}$=constant, two surfaces
are particularly relevant in this spacetime [14]:

  * •

A null surface at $R=R_{-}$, $t\rightarrow\infty$, where $R_{-}$ is the
smaller positive root of
$\tilde{f}(t\rightarrow\infty,R)=H_{0}^{2}R^{3}-R+2m_{0}=0.$

  * •

A null surface at $R=R_{+}$, $t\rightarrow{\infty}$, where $R_{+}$ is the
larger positive root of
$\tilde{f}(t\rightarrow\infty,R)=H_{0}^{2}R^{3}-R+2m_{0}=0.$

If the null energy condition is satisfied, it was demonstrated in Ref. [14]
that:

  1. 1.

Null ingoing geodesics in the regular region of the spacetime cross the
surface $R=R_{-}$, $t\rightarrow{\infty}$ at a finite value of the affine
parameter.

  2. 2.

Such surface is regular (_i.e._ all the squared curvature invariants
constructed with the Riemann tensor and its contractions are finite on it)333
Lake and Abdelqader [15] proved that, in the case $H_{0}=0$, $R=R_{-}=2m_{0}$
for $t\rightarrow\infty$ is also a regular surface in the sense defined above.
.

  3. 3.

Once the geodesics transverse the surface $R=R_{-}$, $t=t_{\infty}$, they are
in a trapped region since $\theta_{\mathrm{in}}\theta_{\mathrm{out}}>0$ there.

Consequently, those geodesics that go through the surface $R=R_{-}$,
$t\rightarrow{\infty}$, will not cross it again in the opposite direction.
This is precisely the situation in the presence of an event horizon. The
conclusion is that the solution represents a black hole with an horizon at
$R=R_{-}$ at large times.

Using a similar procedure as the one describe above, Lake and Abdelqader (see
Appendix D in [15]) also demonstrated that in the case
$H(t\rightarrow\infty)\rightarrow H_{0}$=0 (for instance, the dust-dominated
background previously discussed), the McVittie metric represents a black hole
in the future.

Having presented the main features of the McVittie solution, we introduce in
the next section the cosmological background model that we adopt in this work.

##  III Scale factor for a bouncing cosmological model

Cosmological models that display a bounce solve by construction the initial
singularity problem, as well as the horizon and flatness problems of the
standard cosmological model 444See Ref. [3] for a review. Such models can also
produce primordial cosmological perturbations from vacuum fluctuations, with
an almost scale-invariant spectrum [21], and can be viewed either as an
alternative or a complement to inflation (see for instance Ref. [22]).
Typically, models with a bounce join a contracting phase, in which the
Universe was very large and almost flat initially, to a subsequent expanding
phase. The bounce can be either generated classically (see e.g. Refs. [23, 24,
25]), or by quantum effects (see e.g. Refs. [21, 26, 27, 28]). Since our aim
is to investigate the effects of the bounce on the McVittie solution, with no
intention at this stage to build a complete cosmological model, our choice of
the regular model will be guided by simplicity. The expression for the scale
factor we adopt as background in our work, given by

$a(t)=a_{0}\left[1+\left(\frac{t}{t_{0}}\right)^{2}\right]^{1/3},$ (13)

was found in [29] by considering quantum corrections to the classical
evolution of the scale factor. The corrections were obtained by solving the
Wheeler-deWitt equation in the presence of a single perfect fluid, in the
framework of the de Broglie-Bohm quantum theory [30]. Other quantization
methods yield the same evolution for the scale factor, see Refs. [31, 32, 33].
Notice that the scale factor reduces to that of dust for $t>>t_{0}$, and leads
to an evolution that is dominated near the bounce by an effective fluid with
negative energy density that scales as $a^{-6}$, as can be seen from
Friedman’s equation555 Limits on the parameters $a_{0}$ and $t_{0}$ can be
found for instance in [38] .

##  IV McVittie spacetime in a bouncing cosmological model

Our goal is to compute and analyze the causal structure of the McVittie
spacetime in a classical cosmological bouncing model. As we discussed above,
under certain assumptions, the McVittie metric represents a black hole. In
what follows, we will examine how this solution behaves before, during, and
after the bounce, and whether a black hole is present in any of these stages
of cosmological evolution.

###  IV.1 Trapping horizons and null geodesics

We begin by computing the trapping horizons using Eq. (12) and the Hubble
factor

$H(\tilde{t})=\frac{2}{3t_{0}}\frac{\tilde{t}}{1+\tilde{t}^{2}},$ (14)

in the time interval $-\infty<\tilde{t}<+\infty$ 666Notice that, contrary to
the cases studied in [15], $H$ goes to zero at $t=0$.. In Eq. (14), the new
variable $\tilde{t}$ is defined as $\tilde{t}\equiv t/t_{0}$. In what follows,
to simplify the notation, we replace $\tilde{t}$ by $t$. We rewrite Eq. (12)
recovering the corresponding units:

$\frac{H(t)^{2}}{c^{2}}R^{3}-R+2\frac{Gm_{0}}{c^{2}}=0.$ (15)

Defining a new dimensionless variable as $x\equiv R/(Gm_{0}/c^{2})$, the
latter equation takes the form

$\alpha^{2}H(t)^{2}x^{3}-x+2=0,$ (16)

where $\alpha=Gm_{0}/c^{3}$. In order to solve this cubic equation, we express
(16) as [36]

$\alpha^{2}H(t)^{2}x^{3}-3\alpha^{2}H(t)^{2}\delta^{2}x+2=0.$ (17)

Here, $\delta^{2}\equiv 1/(3\alpha^{2}H(t)^{2})$. By performing the change of
variables $x=2\delta\sin{\phi}$, and after some simple algebraic
manipulations, Eq. (17) becomes

$\displaystyle
2\alpha^{2}H(t)^{2}\delta^{3}\left(4\sin^{3}{\phi}-3\sin{\phi}\right)+2$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle-\alpha^{2}H(t)^{2}\delta^{3}\sin{3\phi}+1$ $\displaystyle=$
$\displaystyle 0.$ (18)

Trapping horizons exist only if $0<\sin{3\phi}<1$. In particular, let us focus
on the case where the radial coordinate of the trapping horizons coincides;
this occurs for $\sin{3\phi}=1$, which in terms of Eq. (18) takes the form

$3\sqrt{3}\frac{Gm_{0}}{c^{3}}H(t)=1,\Rightarrow
t^{2}-2\sqrt{3}\frac{Gm_{0}}{c^{3}t_{0}}t+1=0.$ (19)

The solutions for (19) are

$t=\sqrt{3}\frac{Gm_{0}}{c^{3}t_{0}}\pm\sqrt{\left(\sqrt{3}\frac{Gm_{0}}{c^{3}t_{0}}\right)^{2}-1}.$
(20)

Thus, depending on the value of the mass of the central source $m_{0}$ and
$t_{0}$, which fixes the time scale of the bounce, we have the following three
cases777Since we are now working with the radial dimensionless coordinate $x$,
we denote the inner and outer radial coordinate of the trapping horizons as
$X_{-}\equiv R_{-}/(Gm_{0}/c^{2})$ and $X_{+}\equiv R_{+}/(Gm_{0}/c^{2})$.:

  1. 1.

If $\frac{\sqrt{3}Gm_{0}}{c^{3}}>t_{0}$, then there are two values of $t$
where $X_{-}=X_{+}$.

  2. 2.

If $\frac{\sqrt{3}Gm_{0}}{c^{3}}=t_{0}$, then there is only one value of $t$
where $X_{-}=X_{+}$.

  3. 3.

If $\frac{\sqrt{3}Gm_{0}}{c^{3}}<t_{0}$, then there is no value of $t$ where
$X_{-}=X_{+}$.

According to Frion and collaborators [38],
$10^{3}\;t_{\mathrm{Planck}}<t_{0}<10^{40}\;t_{\mathrm{Planck}}$, that is
$t^{\mathrm{min}}_{0}=10^{-41}\;\mathrm{s}<t_{0}<t^{\mathrm{max}}_{0}=10^{-4}\;\mathrm{s}$.
We show in Figure 7 the range of allowed values for $m_{0}$ (in units of solar
masses $M_{\odot}$) according to the three conditions described above: the
black line indicates the condition $\sqrt{3}Gm_{0}/c^{3}=t_{0}$. The light
yellow zone corresponds to values of $m_{0}$ and $t_{0}$ such that
$\sqrt{3}Gm_{0}/c^{3}>t_{0}$, while the light red region satisfies the
condition $\sqrt{3}Gm_{0}/c^{3}<t_{0}$.

From a physical point of view, it seems more attractive to consider a bounce
time scale close to the allowed upper limit
($t^{\mathrm{max}}_{0}=10^{-4}\;\mathrm{s}$), that is, in the limit between a
classical and quantum bounce. In such scenario, matter inhomogeneities that
existed in the contracting phase might be able to go through the bounce and
also be present in the expanding epoch. On the other hand, Carr and Kühnel
[39] have recently showed that primordial black holes in the mass range
$10\;M_{\odot}<M<10^{2}\;M_{\odot}$ could be relevant to provide a fraction of
the dark matter in the universe as well to explain the observed LIGO/Virgo
coalescence events in the mass range $\mathcal{O}(10)\;M_{\odot}$. Therefore,
in what follows we choose for $m_{0}$ and $t_{0}$, $m_{0}=50M_{\odot}$ and
$t_{0}=5\times 10^{-5}$ s. The structure of the trapping horizons that will be
analysed in this work corresponds to case 1.

Figure 7:  The black line indicates the condition
$\sqrt{3}Gm_{0}/c^{3}=t_{0}$. The light yellow zone corresponds to values of
$m_{0}$ and $t_{0}$ such that $\sqrt{3}Gm_{0}/c^{3}>t_{0}$ while the light red
region satisfy the condition $\sqrt{3}Gm_{0}/c^{3}<t_{0}$. Here, $m_{0}$ is in
units of solar masses $M_{\odot}$.

The trapping horizons are plotted in Figure 8: the lower plot zooms into the
region near the bounce. Very close to $t=0$, just before and after the bounce,
there is a trapping horizon. This surface is absent, for instance, in McVittie
spacetime for the dust-dominated background (see Figure 2). In Section VII, we
argue that this horizon is a particular feature related to the presence of the
bounce.

There are two additional trapping horizons for negative and positive values of
the cosmic time, respectively. The shape of these surfaces is qualitatively
similar compared to the trapping horizons in the McVittie solution for the
dust background (see Figure 2): for $t>0$ there is a moment in time when a
single trapping horizon begins to exist and immediately after an inner $X_{-}$
and outer $X_{+}$ trapping horizons emerge. In the limit $t\rightarrow\infty$,
$X_{-}\rightarrow 2$.

The second root of Eq. (12), namely $X_{+}$, is given by [40]

$X_{+}=\frac{c^{3}}{Gm_{0}}\left[\frac{1}{H(t)}\cos{\psi(t)}-\frac{1}{\sqrt{3}H(t)}\sin{\psi(t)}\right]$
(21)

where $\sin{3\psi(t)}=3\sqrt{3}Gm_{0}H(t)/c^{3}$. For the Hubble factor given
by Eq. (14), $X_{+}\rightarrow\infty$ when $t\rightarrow\pm\infty$, and thus
$X_{+}$ becomes a FLRW null infinity [14].

The shape of the curve of the horizons is symmetric with respect to the axis
$t=0$. In the limit $t\rightarrow-\infty$, $X_{-}\rightarrow 2$. As we
approach the bounce, $X_{-}$ increases while $X_{+}$ decreases up to they
merge. The symmetry for the trapping horizons with respect to $t=0$ is rooted
in the equation that defines these surfaces (see Eq. (12) or (16)), which is
quadratic in the Hubble factor.

The white zones in Figure 8 indicate the regular regions of the spacetime
($\theta_{\mathrm{in}}\theta_{\mathrm{out}}<0$), the light pink zone
corresponds to the anti-trapped region
($\theta_{\mathrm{in}}\theta_{\mathrm{out}}>0$, being $\theta_{\mathrm{in}}>0$
and $\theta_{\mathrm{out}}>0$), and the light blue zone marks the trapped
region ($\theta_{\mathrm{in}}\theta_{\mathrm{out}}>0$, being
$\theta_{\mathrm{in}}<0$ and $\theta_{\mathrm{out}}<0$). The dot-dashed curve
marks the location of the singularity $x=2$ for $t$ finite.

Figure 8:  The black lines indicate the location of the trapping horizons. The
white zones correspond the regular regions, the light pink zone indicates the
anti-trapped region and the light blue zone denotes the trapped region. The
dot dashed curve denotes the location of the singularity. The lower plot zooms
into the region near the bounce. Here, $m_{0}=50\;M_{\odot}$, and
$t_{0}=5\times 10^{-5}$ s

.

We also compute the trajectories of ingoing and outgoing radial null geodesics
by integrating Eq. (11). The behaviour of the geodesics changes before and
after the bounce:

  * •

Outgoing null geodesics are always expanding ($(dx/dt)_{\mathrm{out}}>0$) for
$t>0$, as can be seen in Figure 9. Radial ingoing geodesics expand in the
anti-trapped region until they cross the trapping horizon; once in the regular
region of the spacetime ${(dx/dt)}_{\mathrm{in}}<0$, they all seem to tend
asymptotically to the surface $X_{-}=2,\;t=\infty$, as shown in Figure 10.
Below, we will provide a more detail analysis of the properties of the ingoing
null geodesics in the limit $t\rightarrow\infty$, which is essential to
establish whether a black hole is present.

  * •

Ingoing and outgoing radial null geodesics reverse their character for $t<0$.
Now, all ingoing trajectories are expanding to the past of the bounce (for
increasing negative values of the $t$ coordinate). This is in Figure 10.
Outgoing null geodesics converge in the trapped region. To the past of these
geodesics, they seem to come from the surface $X_{-}=2$, $t=-\infty$ (see Fig.
9). In Section VI we offer a possible interpretation of the McVittie solution
before the bounce takes place.

In Figures 11 and 12 we offer a close up of the region near the bounce showing
the behaviour of the outgoing and ingoing null geodesics. Some null geodesics
(painted in blue and red in Figure 11) start at the remote past near $x=2$, go
through the bounce and expand getting away from the central inhomogeneity. The
same occurs for some ingoing geodesics. This does not happen in any McVittie
model without a bounce analyzed so far in the literature; in those spacetimes
the singularity at $x=2m_{0}$, $t$ finite, lies in the causal past of all
events, and thus it is regarded as a cosmological big bang singularity [14].
In the present model, the existence of a cosmological bounce renders only some
spacetime trajectories geodesically incomplete.

The equation for the ingoing radial geodesics in terms of the Hubble factor
(14) takes the form

$\left.\frac{dx}{dt}\right|_{\mathrm{in}}=\frac{1}{\alpha}\sqrt{1-2/x}\left(\frac{2\;\alpha}{3}\frac{t}{1+t^{2}}x-\sqrt{1-2/x}\right),$
(22)

where $\alpha=Gm_{0}/(c^{3}t_{0})$. Before the bounce, the Hubble factor can
be rewritten as $H(t)=-2t/(3t_{0}(1+t^{2}))$ where $t>0$. Replacing into (22):

$\displaystyle\left.\frac{dx}{dt}\right|_{\mathrm{in}}$ $\displaystyle=$
$\displaystyle-\left[\frac{1}{\alpha}\sqrt{1-2/x}\left(\frac{2\;\alpha}{3}\frac{t}{1+t^{2}}x+\sqrt{1-2/x}\right)\right]$
(23) $\displaystyle=$
$\displaystyle-\left.\frac{dx}{dt}\right|_{\mathrm{out}},$

for $t>0$. Thus, we can see that the trajectories of ingoing null geodesics,
before the bounce, are the reflection of the trajectories of outgoing null
geodesics after the bounce.

In the same way, we express the equation for the outgoing null geodesics
before the bounce as:

$\displaystyle\left.\frac{dx}{dt}\right|_{\mathrm{out}}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\sqrt{1-2/x}\left(-\frac{2\;\alpha}{3}\frac{t}{1+t^{2}}x+\sqrt{1-2/x}\right)$
(24) $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\left[\sqrt{1-2/x}\left(\frac{2\;\alpha}{3}\frac{t}{1+t^{2}}x-\sqrt{1-2/x}\right)\right]$
$\displaystyle=$ $\displaystyle-\left.\frac{dx}{dt}\right|_{\mathrm{in}},$

for $t>0$. We conclude that the congruence properties of outgoing null
geodesics before the bounce are the same that for ingoing null geodesics after
the bounce.

Figure 9:  Radial outgoing geodesics in McVittie spacetime for a bouncing
cosmological background. The black line indicates the location of the trapping
horizons while the dot dashed curve denotes the location of the singularity.
The lower plot zooms into the region near the bounce. Here,
$m_{0}=50\;M_{\odot}$, and $t_{0}=5\times 10^{-5}$ s. Figure 10:  Radial
ingoing geodesics in McVittie spacetime for a bouncing cosmological
background. The black line indicates the location of the trapping horizons
while the dot dashed curve denotes the location of the singularity. The lower
plot zooms into the region near the bounce. Here, $m_{0}=50\;M_{\odot}$, and
$t_{0}=5\times 10^{-5}$ s. Figure 11: Radial outgoing geodesics in McVittie
spacetime for a bouncing cosmological background in the region close to the
bounce. The black line indicates the location of the trapping horizons while
the dot dashed curve denotes the location of the singularity. Here,
$m_{0}=50\;M_{\odot}$, and $t_{0}=5\times 10^{-5}$ s. Figure 12: Radial
ingoing geodesics in McVittie spacetime for a bouncing cosmological background
in the region close to the bounce. The black line indicates the location of
the trapping horizons while the dot dashed curve denotes the location of the
singularity. Here, $m_{0}=50\;M_{\odot}$, and $t_{0}=5\times 10^{-5}$ s.

##  V Spacetime structure after the bounce

In order to prove that the surface $X_{-}=2$, $t=\infty$ is an event horizon
the following conditions must be fulfilled: a) null ingoing radial geodesics
reach $X_{-}=2$, $t=\infty$ in a finite interval of an affine parameter; b)
the surface $X_{-}=2$, $t=\infty$ is regular.

To check if condition (a) is met, we make the following change of variables
[15]:

$\displaystyle z$ $\displaystyle=$ $\displaystyle\sqrt{1-\frac{2}{x}},$ (25)
$\displaystyle l$ $\displaystyle=$ $\displaystyle\frac{1}{1+\alpha H(t)},$
(26)

where $0\leq z\leq 1$ ($2\leq x<\infty$) and $0<l<1$.

Since we are interested in the behavior of the geodesics for large values of
$t$, we restrict the integration in terms of $l$ in the interval $0.75\leq
l<1$.

Given the expressions (25) and (26), the equations for the ingoing geodesics
(11) take the form

$\displaystyle\frac{dz}{dl}$ $\displaystyle=$
$\displaystyle\frac{\left(1-z^{2}\right)^{2}}{4}\left[\frac{2(1-l)}{l\left(1-z^{2}\right)}-z\right]\frac{dt}{dl},$
(27) $\displaystyle\frac{dl}{dt}$ $\displaystyle=$
$\displaystyle\frac{9+l\left[-\beta+\alpha\sqrt{-9+\beta
l}\right]}{3\;l^{4}},$ (28) $\displaystyle\beta$ $\displaystyle=$
$\displaystyle 18+\left(\alpha^{2}-9\right)l.$ (29)

Figure 13:  Trajectories of radial ingoing null geodesics in the $z-l$ plane
for McVittie spacetime in a bouncing cosmological model. Here,
$m_{0}=50\;M_{\odot}$, and $t_{0}=5\times 10^{-5}$ s.

In Figure 13, we plot the solution of the numerical integration of Eq. (27)
for five different initial conditions. Clearly, radial ingoing null geodesics
reach the surface $x=2$ ($z=0$) for $l=1$.

Figure 14: Closeup in the $z-l$ plane showing the trajectories of radial
ingoing null geodesics approaching the surface $x=2$. Expression (32) is
represented by the dotted black curve. Here,$m_{0}=50\;M_{\odot}$, and
$t_{0}=5\times 10^{-5}$ s.

One way to decide whether condition (b), i.e. that $X_{-}$ is regular at large
values of $t$, is valid or not is to calculate the components of the Riemann
tensor using an appropriate _Vierbein_ , and evaluate them on null geodesics
approaching the surface. Since the model given by Eq.(14) behaves as a model
dominated by dust for large $t$, namely

$H(t)\approx\frac{2}{3\;t},$ (30)

we can use the result (valid for large values of $t$) obtained in [14]:

$t=\frac{2x}{3\sqrt{1-\frac{2}{x}}},$ (31)

or, in terms of the variables $z$ and $l$ :

$lz^{3}-lz-2l+2=0.$ (32)

Figure 14 shows that the asymptotic form of the null ingoing geodesics (given
by Eq.(32)) describes extremely well the result of the exact numerical
integration. Using GRTENSOR II 888http://grtensor.phy.queensu.ca/ , we have
calculated the components of the Riemann tensor in the following _Vierbein_ :

$[e^{j}_{\>\mu}]=\begin{pmatrix}\sqrt{H(t)^{2}r^{2}+f}&0&0&0\\\
H(t)r&\frac{1}{\sqrt{1-\frac{2m_{0}}{r}}}&0&0\\\ 0&0&r&0\\\
0&0&0&r\sin\theta\end{pmatrix}$ (33)

The evaluation of the components on the null ingoing geodesics using (31)
leads to finite values. Hence, the surface is regular. We have also evaluated,
in the same limit, invariants built with first or second derivatives of
tensors associated with curvature, such as $(\nabla_{\mu}R)(\nabla^{\mu}R)$
and $(\nabla_{\mu}\nabla_{\nu}R)(\nabla^{\mu}\nabla^{\nu}R)$, using RGTC
999http://www.inp.demokritos.gr/∼sbonano/RGTC/. While invariants of the first
kind yield a finite result, those of the second kind are divergent, in
agreement with the result found in [14]. Such a divergence does not influence
the finiteness of the tidal forces, and is probably a consequence of the
finite differentiability of the quantities describing the fluid [43].

We conclude that the solution contains a black hole after the bounce.

Figure 15:  Trajectory of a future-directed outgoing null geodesic that
emerges from the singular point $(l,z)=(0,0)$ and reaches $z=1$
($x\rightarrow\infty$) in a finite interval of time.

Using the change of variables given by Eqs. (25) and (26), we also integrate
the trajectories of outgoing null geodesics. We show in Figure 15 the
trajectory of a future-directed outgoing null geodesic that emerges from the
singular point $(l,z)=(0,0)$, that is $(t,x)=(0,2)$ and reaches $z=1$
($x\rightarrow\infty$) in a finite interval of time. Thus, we see that a naked
singularity is present in the expanding phase up to the formation of the black
hole.

##  VI Spacetime structure before the bounce

We showed in Figure 8 the existence of a trapped region, that is
$\theta_{\mathrm{in}}\theta_{\mathrm{out}}>0$ where $\theta_{\mathrm{in}}<0$
and $\theta_{\mathrm{out}}<0$, for $t<0$. This region is bounded from below by
inner $(X_{-})$ and outer $(X_{+})$ trapping horizons for which
$\theta_{\mathrm{out}}=0$. As we approach the bounce, both horizons get closer
and at

$t=t_{\ast}=-\left(\sqrt{3}\frac{Gm_{0}}{c^{3}t_{0}}+\sqrt{\left(\sqrt{3}\frac{Gm_{0}}{c^{3}t_{0}}\right)^{2}-1}\right),$
(34)

only a trapping horizon exist for $X_{-}=X_{+}=3$. If we choose
$m_{0}=50\;M_{\odot}$, and $t_{0}=5\times 10^{-5}$ s, then $t_{\ast}=-16.9$.

In Section IV.1, we proved that $X_{+}$ is a null infinity
($X_{+}\rightarrow\infty$ for $t\rightarrow-\infty$). The inner horizon
$X_{-}$, defined for $2<X_{-}\leq 3$, covers the singularity and encloses a
trapped region. We illustrate this situation in Figure 16 where the light cone
structure for small and negative values of $t$ is plotted. The light cones
have the trapping horizon in their local future . Both ingoing and outgoing
null rays that cross $X_{-}$ enter the trapped region and are unable to turn
around and escape. This horizon, thus, acts as a one way membrane, hiding the
singularity at $x=2$. This analysis leads us to conclude that in the time
interval $-\infty<t<t_{\ast}$ the solution contains a black hole.

Figure 16: Light cone structure in McVittie spacetime for a bouncing
cosmological model in the region $t<0$. The dotted curves represent the null
ingoing geodesics while the dot dashed curves the null outgoing geodesics. The
regions shaded in grey show some light cones and the black arrow indicates the
future direction.

##  VII Discussion

Given the analysis of the causal structure of the spacetime offered in the
previous sections, we now provide a description of the evolution of the
solution through cosmic time.

A black hole is present since the beginning of the contracting phase. The
inner trapping horizon $X_{-}$ increases its radius as the contraction gathers
pace. The range of values of the radial coordinate for $X_{-}$ is $2<X_{-}\leq
3$. Ingoing and outgoing null geodesics that cross the surface $X_{-}$, enter
the trapped zone of the spacetime interior to the black hole. Close to the
bounce, at $t_{\ast}$ (see Eq. (34)), the inner $X_{-}$ and outer $X_{+}$
trapping horizons merge and the black hole ceases to exist. This situation
lasts for a short time. Trapping horizons appear again right before the
bounce, and vanish right after it. These horizons are absent in other McVittie
models (for instance, compare Figures 1 and 2 with 8). We associate these
surfaces with the peculiarities of the cosmological background model, and more
specifically to the presence of the bounce. There is no salient feature
associated with these horizons: there, null geodesics just change their
convergence properties.

Afterwards, the universe begins to expand and an inner $X_{-}$ and outer
$X_{+}$ trapping horizons appear. For $t\rightarrow\infty$, the outer horizon
becomes a FLRW null infinity, while $X_{-}$ becomes an event horizon. Hence,
as the universe expands a black hole starts to form. We show in Figure 17 the
light cone structure for the spacetime after the bounce. We see that the
surface $x=2$ is not in the local future of those light cones: in the process
of black hole formation (the event horizon is not settled down) some geodesics
are able to escape from the central source. As the universe expands, however,
those outgoing geodesics that start at $x<X_{-}$ have a smaller and smaller
slope. In the limit $t\rightarrow\infty$, ${dx/dt}\rightarrow 0$ for outgoing
geodesics, as computed from Eq. (11). This implies that outgoing null
geodesics cannot leave the surface $x=2$ and the region contained by such
boundary becomes trapped. In the distant future, the McVittie solution for a
bouncing cosmological model harbors a black hole.

Figure 17:  Light cone structure in McVittie spacetime for a bouncing
cosmological model in the region $t>0$. The dotted curves represent the null
ingoing geodesics while the dot dashed curves the null outgoing geodesics. The
grey shadow regions show some light cones and the black arrow indicates the
future direction.

We show in Figures 18 and 19 qualitative Penrose diagrams of the McVittie
spacetime for a bouncing cosmological model in the expanding and contracting
phase, respectively. The dotted lines represent the trajectories of ingoing
null geodesics and the dashed lines the trajectories of outgoing null
geodesics. The black thick curve displays the trapping inner ($X_{-}$) and
outer ($X_{+}$) horizons. The straight red line named $\mathcal{H^{+}}$
($\mathcal{H^{-}}$) represents $X_{-}\rightarrow 2,\;\;t\rightarrow\infty$
($X_{-}\rightarrow 2,\;\;t\rightarrow-\infty$). There is a horizontal straight
black line that characterises the region where the bounce occurs, that is
$t=0$ and $x>2$. To the left, a dashed black and yellow line represents the
singular surface $x=2$, $t$ finite. As usual, $\mathcal{J^{+}}$
($\mathcal{J^{-}}$) denotes the future (past) null infinity.

In the expanding phase, all ingoing geodesics, regardless of the initial
conditions, reach the surface $\mathcal{H^{+}}$. Those outgoing null geodesics
that start at the bounce extend to $\mathcal{J^{+}}$. There are also future
directed outgoing radial null geodesics that emerge from the singular surface
$x=2$, $t$ finite (this includes $(t,x)=(0,2)$). The latter evidences the
existence of a spacelike naked singularity that lasts up to the formation of
the black hole which corresponds to $\mathcal{H^{+}}$ in Figure 18.

As shown in Section IV A, in the contracting phase, ingoing and outgoing null
geodesics reverse their character. Some ingoing geodesics end up in the
singular surface while some others make it to the bounce. The same fate share
outgoing null geodesics; all of them begin in $\mathcal{H^{-}}$. A black hole
is present since the beginning of the contracting phase until
$X_{-}=X_{+}.$The zone shaded in grey indicates the black hole region.

Figure 18: Qualitative Penrose diagram of the McVittie spacetime for a
bouncing cosmological model in the expanding region ($t>0$). The dotted lines
represent null ingoing geodesics while the dashed lines null outgoing
geodesics. Here, $\mathcal{H^{+}}=X_{-}\rightarrow 2,\;\;t\rightarrow\infty$,
and $\mathcal{J}^{+}$ is the future null infinity. Figure 19: Qualitative
Penrose diagram of the McVittie spacetime for a bouncing cosmological model in
the contracting region ($t<0$). The dotted lines represent null ingoing
geodesics while the dashed lines null outgoing geodesics. Here,
$\mathcal{H^{-}}=X_{-}\rightarrow 2,\;\;t\rightarrow-\infty$, and
$\mathcal{J}^{-}$ is the past null infinity.

##  VIII Conclusions

In this work the causal structure of McVittie spacetime for a bouncing
cosmological background is analyzed. The location of the trapping horizons is
computed, and the trajectories of null radial ingoing and outgoing null
geodesics through cosmic time are obtained by numerical integration. A
detailed study of the asymptotic behaviour of the metric is provided. Our main
result is that the solution represents a dynamical black hole since the
beginning of the contracting phase up to shortly before the bounce, and also
in the distant future. Just before the bounce, the inner and outer trapping
horizons merge and the black hole character of the solution is lost. After the
bounce, the central inhomogeneity starts to act again, and for large and
positive values of the cosmic time, a black hole is formed. Thus, we see that
the global dynamical state of the universe directly affects the conditions for
the existence of black holes. In particular, for a contracting universe, black
hole solutions are possible up to certain minimum scales.

Unlike all other McVittie models analyzed in the literature, there is no
cosmological big singularity in the present metric. In fact, the solution
admits trajectories that never encounter a singularity, that is, they are
geodesically complete. This peculiar feature of the model is related to the
occurrence of the bounce.

This work is a first step towards a better understanding of black holes
embedded in a bouncing cosmological background; the current solution does not
take into account the accretion of cosmological fluid by the central source.
The Generalized McVittie metric naturally incorporates this effect by assuming
an energy-momentum tensor of an imperfect fluid [40]. It remains an open issue
whether the Generalized McVittie spacetime might contain a black hole for a
bouncing cosmological background. We shall explore this issue in a future
work.

###### Acknowledgements.

This work was supported by the Argentine agency CONICET (PIP 2014-00338) and
the Spanish Ministerio de Ciencia e Innovación (MICINN) under grant
PID2019-105510GBC31 and through the ”Center of Excellence María de Maeztu
2020-2023” award to the ICCUB (CEX2019-000918-M). G.E.R. acknowledges support
from the _Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil
(CAPES)-Codigo de Financiamento 001_. Both D.P. and G.E.R. are very grateful
to the Department of Physics of UERJ for kind hospitality. We are very
grateful to two anonymous referees for their insightful comments.

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$\operatorname{gen}_{R^{\prime}_{\mathbb{C}}}(B^{\prime}_{\mathbb{C}})\geqslant
r+1$. Recall that by our choice of $r$, we have $i\leqslant d$. After
replacing $R^{\prime}$ by the polynomial ring
$R=R^{\prime}[x_{1},\ldots,x_{d-i}]$ and $B^{\prime}$ by
$B=B^{\prime}\otimes_{R^{\prime}}R$, we arrive an an Azumaya algebra $B$ over
$R$ such that $\operatorname{Kdim}R=d$ and
$\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})=\operatorname{gen}_{R^{\prime}_{\mathbb{C}}}(B^{\prime}_{\mathbb{C}})\geqslant
r+1$. In particular, $\operatorname{gen}_{R}(B)\geqslant r+1$; see Lemma 10.1.
This completes the proof of Theorem 1.5(b) in the case, where $k=\mathbb{Q}$.

In the general case, where $k$ is an arbitrary field of characteristic $0$ and
not necessarily embedded in $\mathbb{C}$, we invoke the Lefschetz principle of
Lemma 10.1. Consider the ring $R$ and Azumaya algebra $B$ produced in the case
of $\mathbb{Q}$. Set $R_{k}=R\otimes_{\mathbb{Q}}k$ and
$B_{k}=R\otimes_{\mathbb{Q}}k$. Clearly
$\operatorname{Kdim}R_{k}=\operatorname{Kdim}R=d$. Moreover,

$\operatorname{gen}_{R_{k}}(B_{k})\geqslant\operatorname{gen}_{R_{\mathbb{C}}}(B_{\mathbb{C}})\geqslant
r+1=\left\lfloor{\frac{d}{2(s-1)}}\right\rfloor+2$

by Lemma 10.1(c). ∎

##  Appendix A The equivariant Thom–Gysin sequence for varieties

This appendix proves a number of results about the Borel-equivariant singular
cohomology of a $k$-variety, $\mathrm{H}^{*}_{G}(X,\mathbb{Z})$, when
$k\subset\mathbb{C}$. These are used in Sections 11, 12 and 14. Many of them
are well known, and we observe in passing some of the places in the literature
where they are used, but clear statements and proofs are not easy to find.

We have made use of the following technical lemma in a few places in the text,
and will use it again in this appendix.

######  Lemma A.1.

Let $X$ be a smooth variety. If $Z\to X$ is a possibly-nonsmooth closed
subvariety of codimension $d$—that is, if each irreducible component of $Z$ is
of codimension at least $d$ in $X$—and if $U=X-Z$, then the pullback map
$\mathrm{H}^{i}(X)\to\mathrm{H}^{i}(U)$ is an isomorphism when $i<2d-1$ and is
injective when $i=2d-1$.

This is standard using Borel–Moore homology. Specifically, the exact sequence
[Ive86, IX.2 (2.1)] relates $\mathrm{H}^{\rm BM}_{\ast}(X)$, $\mathrm{H}^{\rm
BM}_{\ast}(U)$ and $\mathrm{H}^{\rm BM}_{\ast}(Z)$; the groups
$\mathrm{H}^{\rm BM}_{i}(Z)$ vanish unless $0\leqslant i\leqslant\dim Z$; and
there are isomorphisms $\mathrm{H}_{i}^{BM}(X)\cong\mathrm{H}^{\dim X-i}(X)$
and similarly for $U$.

### The Thom–Gysin sequence

If $i\colon Z\to X$ is the inclusion of a closed smooth subvariety of constant
codimension $c$ and if $j\colon X-Z\to X$ is the inclusion of the complement,
then there is a long exact Thom–Gysin sequence

(A.1)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}^{*-2c}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\textstyle{\mathrm{H}^{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}\;\;}$$\textstyle{\mathrm{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\;\;\partial}$$\textstyle{\cdots.}$

This sequence is widely known, appearing in [Gor75], [MFK94, p. 167] for
instance. The special case where $X$ is the total space of a vector bundle and
$Z$ the zero section is the Gysin sequence of [MS74, Theorem 12.2]. In the
general case, one can find tubular neighbourhood $\nu$ of $Z$ in $X$, and
deduce the Thom–Gysin sequence for $Z$ in $X$ from the case of $Z$ in $\nu$
using the excision theorem. Alternatively, one may derive the sequence (A.1)
from the long exact sequence of local cohomology [Ive86, Prop. II.9.2] along
with the isomorphism
$\mathrm{H}^{\ast}_{Z}(X;\mathbb{Z})\to\mathrm{H}^{\ast-2c}(Z;\mathbb{Z})$
implied by [Ive86, Theorem IX.4.7].

In order to articulate the naturality of the Thom–Gysin sequence, we recall
the concept of a transversal square. Let

(A.2)
$\textstyle{Z^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\scriptstyle{f}$$\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{X}$

be a cartesian square of smooth varieties, where $i$ is a closed embedding.
Let $N^{\prime}$ denote the normal bundle of $Z^{\prime}$ in $X^{\prime}$ and
$N$ the normal bundle of $Z$ in $X$. Recall from [Gro67, §17.13] that the
square is transversal if the induced map $N^{\prime}\to f^{*}N$ is an
isomorphism.

The most important case is when $X^{\prime}\to X$ and $Z\to X$ are both closed
embeddings. In this case, $X^{\prime}$ and $Z$ are said to intersect
transversely, and $Z^{\prime}$ is their transverse intersection.

Intersecting transversely is a condition on normal bundles. Using the normal
exact sequence [Har77, p. 182], we can say the intersection is transverse if

$\dim T_{p}X=\dim T_{p}X^{\prime}+\dim T_{p}Z-\dim T_{p}Z^{\prime}$

for all closed points $p$ of the smooth variety $Z^{\prime}$.

######  Remark A.2.

A square is transversal if a certain map of locally free sheaves is an
isomorphism. This holds for the square defined by $Z\subseteq X$ and
$X^{\prime}\to X$ if and only if it holds for the square obtained by pulling
back along a faithfully flat map $U\to X$.

If we are given a transversal square, then there is an induced natural
transformation of Thom–Gysin sequences.

(A.3)
$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}^{*-2c}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{*}}$$\scriptstyle{f^{*}}$$\textstyle{\mathrm{H}^{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\scriptstyle{g^{*}}$$\textstyle{\mathrm{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}(X^{\prime}-Z^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}^{*-2c}(Z^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}_{*}}$$\textstyle{\mathrm{H}^{*}(X^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}^{*}(X^{\prime}-Z^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$

This and other useful properties of Thom–Gysin sequence are listed in [Pan09,
Section 2.2].

######  Remark A.3.

If $s:X\to V$ is the zero-section of a rank-$m$ vector bundle on a smooth
variety, then the composite of the Gysin map $s_{*}$ with $s^{*}$

$\mathrm{H}^{0}(X)\overset{s_{*}}{\to}\mathrm{H}^{2m}(V)\overset{s^{*}}{\to}\mathrm{H}^{2m}(X)$

sends $1_{X}$ to the Euler class $e(V)$. We sketch the argument: In the case
of a zero-section of a bundle, the definition in [Pan09, Section 2.2] of the
Gysin map coincides with a composite of taking a cup-product with the Thom
class of the bundle, followed by extension of support:

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
21.36539pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{*-2m}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
22.68599pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\smile\tau}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 45.36539pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
45.36539pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{*}_{X}(V)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
101.49875pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
101.49875pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{H}^{*}(V)}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Composing further with $s^{*}$ gives the definition of Euler class in [Bre93,
Definition 12.1].

It is also the case that
$\mathrm{H}^{*}(\mathbb{P}^{n})\cong\mathbb{Z}[\vartheta]/(\vartheta^{n+1})$,
where $\vartheta$ is the first Chern class of $\mathcal{O}(1)$, so that
$|\vartheta|=2$.

### Equivariant Thom–Gysin sequence

There exists an equivariant Thom–Gysin sequence. Versions of this appear in
[Kir84, p. 170] and [CH16], for instance. In its Borel–Moore homology
incarnation, it is implied by the remarks on [EG98, Sec. 2.8]. In fact, this
appendix consists of an explication of some of what is implicit in [EG98, Sec.
2.8].

Let $G$ be an algebraic group. If $Z\to X$ is a closed $G$-invariant embedding
of constant codimension $c$ where $Z$ and $X$ are both smooth varieties, then
in any range of dimensions we may find some representation $V$ of $G$ and open
subspace $U\subset V$ in order to calculate both
$\mathrm{H}^{i}_{G}(Z)=\mathrm{H}^{i}(Z\times^{G}U)$ and
$\mathrm{H}^{*}_{G}(X)=\mathrm{H}^{i}(X\times^{G}U)$. The inclusion $(Z\times
U)/G\to(X\times U)/G$ is a closed embedding of smooth varieties, again of
codimension $c$. This means there is a Gysin map

$\mathrm{H}^{*-2c}_{G}(Z)\to\mathrm{H}^{*}_{G}(X)$

exactly as in the non-equivariant case.

######  Lemma A.4.

Let $G$ be an algebraic group acting freely on a smooth variety $X$ in such a
way that $X\to X/G$ is a $G$-torsor in the category of smooth varieties.
Suppose $Y$ and $Z$ are $G$-invariant closed smooth subvarieties of $X$ that
intersect transversely. Then $Y/G$ and $Z/G$ intersect transversely in $X/G$.

###### Proof.

By using the observations of Remark A.2, we can check for the transversality
of $Y/G$ and $Z/G$ after faithfully flat pullback to $X$, but these pullbacks
are $Y$ and $Z$, which intersect transversely. ∎

######  Lemma A.5.

Let $G$ be an algebraic group acting on a smooth variety $X$. Suppose $Y$ and
$Z$ are smooth closed $G$-subvarieties intersecting transversely, and $Y$ is
of constant codimension $c$. Then there is a natural transformation of
equivariant Thom–Gysin sequences

$\textstyle{\mathrm{H}_{G}^{*}(X-Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}_{G}^{*-2c}(Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{G}^{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\mathrm{H}_{G}^{*}(X-Y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}_{G}^{*}(Z-(Y\cap
Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}_{G}^{*-2c}(Y\cap
Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{G}^{*}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{G}^{*}(Z-(Y\cap
Z))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

###### Proof.

The groups $\mathrm{H}_{G}^{i}$ may be calculated by choosing a
$G$-representation $V$ and an open subset $U$ where $U\to U/G$ is a $G$-torsor
and $V-U$ has sufficiently high codimension. Then $\mathrm{H}^{i}_{G}(X)$ is
the cohomology of $(X\times U)/G$ and similarly for the other varieties. It
suffices therefore to prove that $(Y\times U)/G$ and $(Z\times U)/G$ intersect
transversely in $(X\times U)/G$, but this is implied by Lemma A.4. ∎

######  Lemma A.6.

Let $G$ be an algebraic group and $H$ a subgroup. Suppose $Z$ is a
$G$-invariant closed smooth subvariety of a smooth $G$-variety $X$ of constant
codimension $c$. Then there is a natural transformation of equivariant
Thom–Gysin sequences:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{G}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}_{G}^{*-2c}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{G}^{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\mathrm{H}_{G}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\cdots}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\mathrm{H}_{H}^{*-2c}(Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{H}_{H}^{*}(X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\textstyle{\mathrm{H}_{H}^{*}(X-Z)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\cdots}$

###### Proof.

Letting $U$ be as in the proof of Lemma A.5, it suffices to prove that

$\textstyle{(Z\times
U)/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X\times
U)/H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(Z\times
U)/G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(X\times
U)/G}$

is transversal. This can be verified after faithfully flat base change along
$(X\times U)\to(X\times U)/G$, whereupon it is isomorphic to

$\textstyle{Z\times
U\times(G/H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
U\times(G/H)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z\times
U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\times
U.}$

The maps are either projections or closed embeddings. This is obviously
transversal. ∎

######  Lemma A.7.

Let $X$ be a smooth variety equipped with a $G$-action. If $Z\to X$ is a
possibly-nonsmooth closed $G$-subvariety of codimension $d$ and if $U=X-Z$,
then the pullback map $\mathrm{H}^{i}_{G}(X)\to\mathrm{H}^{i}_{G}(U)$ is an
isomorphism when $i<2d-1$ and is injective when $i=2d-1$.

###### Proof.

Choose a $G$-representation $V$ and an open subset $U$ such that $U\to U/G$ is
a $G$-torsor and $V-U\hookrightarrow V$ has codimension greater than $d$. The
result now follows by applying Lemma A.1 to $(Z\times U)/G\to(X\times U)/G$. ∎

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# Design Considerations for Proposed Fermilab Integrable RCS

J. Eldred and A. Valishev  FNAL  Batavia  IL 60510  USA

###### Abstract

Integrable optics is an innovation in particle accelerator design that
provides strong nonlinear focusing while avoiding parametric resonances. One
promising application of integrable optics is to overcome the traditional
limits on accelerator intensity imposed by betatron tune-spread and collective
instabilities. The efficacy of high-intensity integrable accelerators will be
undergo comprehensive testing over the next several years at the Fermilab
Integrable Optics Test Accelerator (IOTA) and the University of Maryland
Electron Ring (UMER). We propose an integrable Rapid-Cycling Synchrotron
(iRCS) as a replacement for the Fermilab Booster to achieve multi-MW beam
power for the Fermilab high-energy neutrino program. We provide a overview of
the machine parameters and discuss an approach to lattice optimization.
Integrable optics requires arcs with integer-pi phase advance followed by
drifts with matched beta functions. We provide an example integrable lattice
with features of a modern RCS - long dispersion-free drifts, low momentum
compaction, superperiodicity, chromaticity correction, separate-function
magnets, and bounded beta functions.

## Introduction

Integrable optics is a development in particle accelerator technology that
enables strong nonlinear focusing without generating parametric resonances
[1]. A promising application of integrable optics is in high-intensity rings,
where it is necessary to avoid resonances associated with a large betatron
tune-spread while simultaneously suppressing collective instabilities with
Landau damping. The efficacy of accelerator design incorporating integrable
optics will undergo comprehensive experimental tests at the Fermilab
Integrable Optics Test Accelerator (IOTA) [2] and the University of Maryland
Electron Ring (UMER) [3] over the next several years. In this paper we discuss
a potential Fermilab integrable rapid-cycling synchotron (iRCS) as a high-
intensity replacement for the Fermilab Booster.

At Fermilab, a core research priority is to improve the proton beam power for
the flagship high-energy neutrino program [4]. In the current running
configuration, a 700 kW 120 Gev proton beam is delivered to a carbon-target
for the NuMI beamline that supports the NOvA, MINERvA, and MINOS neutrino
experiments. Next, the Proton Improvement Plan II (PIP-II) will replace the
400 MeV linac with a new 800 MeV linac that will increase the 120 GeV proton
power of the Fermilab complex to 1.2 MW [5].

The next flagship neutrino experiment at Fermilab will be the LBNF/DUNE [6].
The P5 Report referred to LBNF as “the highest priority project in its
lifetime” and set a benchmark for a 3$\sigma$ measurement of the CP-violating
phase over 75% the range of its possible values [7]. The P5 benchmark for the
CP-violating phase corresponds to a 900 kt$\cdot$MW$\cdot$year neutrino
exposure requirement [6, 4]. For a 1.2 MW proton power and a 50 kt LAr
detector, 15 years are required to meet that benchmark. For a 3.6 MW proton
power and a 36 kt LAr detector, 7 years are required.

In order to achieve a 120 GeV proton power significantly beyond the 1.2 MW
delivered by PIP-II, it will be necessary to replace the Fermilab Booster with
a modern RCS [4]. The Fermilab Booster is over 45 years old and faces
limitations from its magnets and its RF alike. There is no beampipe inside the
dipoles and the magnet laminations generate an impedance instability. The
impedance instability provides $\sim$200k deceleration during transition
crossing at current beam intensities [8]. The Booster dipoles are combined
function magnets which constrain tunability and amplify electron cloud
instabilities [9]. The Booster RF cavities underwent a refurbishment process
and cooling upgrade in order to achieve a 15-Hz Booster ramp rate [10] but the
ramp rate will not be able to exceed 20-Hz without replacing the Booster RF
entirely.

Figure 1 shows one of several siting options for an iRCS replacement for the
Fermilab Booster. Neither the RCS circumference nor the injection linac length
are constrained by the siting.

Figure 1: Site location for the proposed iRCS, relative to the PIP-II linac,
muon campus, and Main Injector [11].

## RCS Design Considerations

If the PIP-II era Booster were to fill the Main Injector without slip-
stacking, there would be 0.5 MW available at 120 GeV. Table 1 shows how
different parameters of a replacement RCS could modify that beam power. The
bolded parameters correspond to the particular scenario that relies on
integrable optics and a small increase in aperture to achieve 3.6 MW of beam
power.

Table 1: Multipliers on beam power relative to PIP-II Booster with boxcar stacking in the Main Injector. Bolded values shows an integrable RCS scenario that provides 3.6 MW of beam power. Booster-MI Beam Power | 0.5 MW
---|---
Laslett Tune-spread | $\Delta\nu/\Delta\nu^{(0)}$
-0.11 |  $\times$ 1.00
-0.27 |  $\times$ 2.45
-0.54 |  $\times$ 4.91
Injection Energy | $\beta\gamma^{2}/(\beta_{0}\gamma_{0}^{2})$
0.8 GeV |  $\times$ 1.00
1.2 GeV |  $\times$ 1.61
2.0 GeV |  $\times$ 3.21
95% Transverse Emittance | $\epsilon_{N}/\epsilon_{N}^{(0)}$
20 $\pi$-mm-mrad |  $\times$ 1.33
25 $\pi$-mm-mrad |  $\times$ 1.67
40 $\pi$-mm-mrad |  $\times$ 2.67
RCS ramp rate | $T_{MI}^{(0)}/T_{MI}$
12 Hz |  $\times$ 0.90
20 Hz |  $\times$ 1.00
40 Hz |  $\times$ 1.09

A high-intensity RCS is constrained by betatron resonance losses incurred by a
large Laslett betatron tune-spread. The Laslett tune-spread is given by

$\Delta\nu(z)\approx\frac{Nr_{0}}{2\pi\epsilon_{N}\beta\gamma^{2}}\left(\frac{\lambda(z)}{\langle\lambda\rangle_{z}}\right)F$
(1)

where $N$ is the number of particles, $r_{0}$ is the classical radius,
$\epsilon_{N}$ is the normalized transverse emittance, $\lambda(z)$ is the
local charge density at position $z$, and $F$ is a transverse form factor [4].
Phase-space painting from PIP-II Linac will substantially improve the
transverse and longitudinal beam uniformity. Eq. 1 can be rewritten to express
the beam intensity as a function of the other three parameters:

$N\sim\Delta\nu_{max}\times\epsilon_{N}\times\beta\gamma^{2}$ (2)

where $\nu_{max}$ is the maximum Laslett tune-spread that can be sustained
with minimal losses.

If integrable optics enable a significantly higher maximum Laslett tune-
spread, this would be a very cost-effective way to improve RCS performance.
The subsequent section discusses how integrability impacts the RCS lattice
optimization. Detailed simulations of space-charged dominated beams in
integrable lattices is an ongoing work [2, 12] and the ultimate limitation on
Laslett tune-spread is not fully determined.

For a fixed Laslett tune-spread, the intensity of an RCS can be improved by
increasing the injection energy. Table 1 shows how the $\beta\gamma^{2}$
parameter in Eq. 2 changes with an energy upgrade of the PIP-II Linac.
Increasing the injection energy also reduces the transverse emittance relative
to the normalized admittance, through adiabatic damping
$\epsilon=\epsilon_{N}/(\beta\gamma)$.

The Fermilab Booster has a 95% transverse admittance of 15 $\pi$-mm-mrad, but
the 95% admittance of the 5.7cm diameter beampipe is 20 $\pi$-mm-mrad without
the restriction from the dipole magnets [13, 4]. On the other hand, the 95%
transverse admittance of the Main Injector is 40 $\pi$-mm-mrad [14].

For a fixed lattice design, the transverse admittance increases quadratically
with increase aperture. Either the magnet current or the accelerator
circumference can be increased to compensate for the change in the magnet
aperture. Table 2 shows the transverse emittance as a function of injection
energy and aperture. Fermilab has designed RF cavities for an RCS with
apertures up to 8.255cm [13].

Table 2: 95% Transverse admittance (in $\pi$-mm-mrad) of RCS as a function of
aperture and injection energy. Asterisks indicates cases which exceed the Main
Injector admittance of 40 $\pi$-mm-mrad. 95% Transverse Admittance ($\pi$-mm-
mrad)
---
Injection | RCS Aperture
Energy | 5cm | 5.7cm | 6.35cm | 8.1cm
0.8 GeV | 15 | 20 | 25 | 40
1.2 GeV | 20 | 26 | 33 | 53*
2.0 GeV | 28 | 38 | 48* | 76*

Assuming conventional boxcar stacking, the impact of the RCS ramp rate on the
MI beam power can be calculated by:

$\displaystyle P_{MI}$ $\displaystyle=Nn_{b}E_{MI}/T_{MI}$ (3) $\displaystyle
T_{MI}$ $\displaystyle=T_{Ramp}+(n_{b}-1)T_{RCS}$ (4)

where $n_{b}$ is the number of batches. As long as the Main Injector ramp
remains long compared to the Main Injector fill time, the RCS ramp rate has
only a modest effect on the Main Injector beam power. However, the RCS ramp
rate will be an important parameter for any experiments which receive the RCS
beam while the Main Injector is ramping.

An alternate RCS design with an extraction energy of 21 GeV should also be
considered. Keeping the aperture and acceleration rate constant, the
extraction energy of an accelerator design can be scaled by increasing the
integrated dipole length, circumference, and number of RF cavities
proportionately.

For boxcar stacking, increasing the extraction energy has only a marginal
effect on Main Injector beam intensity. However it should be carefully
considered if a higher extraction energy would enable the RCS beam to be
stacked to a greater Main Injector intensity. For example, slip-stacking is an
accumulation technique currently used to double the intensity of the Main
Injector, but the feasibility of slip-stacking beyond the PIP-II era is still
under investigation [14, 15]. An RCS with an extraction energy of 21 GeV would
avoid transition crossing in the Main Injector and that would address one of
several challenges associated with high-intensity slip-stacking. An alternate
accumulation approach could be transverse stacking via nonlinear-resonant
injection [16].

##  1 IRCS EXAMPLE LATTICE

The iRCS should incorporate the innovations in RCS design that have been
developed after the Fermilab Booster [17]. Periodicity and bounded beta
functions increase the dynamic aperture. Transition crossing can be avoided by
designing the lattice with a low momentum compaction factor. Modern RCS design
also uses separate-function dipole magnets and long dispersion-free drifts. In
this section we show an example integrable lattice with these features.

An accelerator can achieve integrable optics with alternating sections of
T-inserts and nonlinear magnets [1]. The T-inserts are arc sections with
$\pi$-integer betatron phase-advance in the horizontal and vertical plane. The
lattice should be dispersion-free in the nonlinear section and the horizontal
and vertical beta functions should be matched. A special nonlinear elliptical
magnet is matched to the beta functions to provide the nonlinear focusing.

Figure 2 shows an example iRCS lattice and Table 3 shows the key parameters of
this lattice. The lattice is composed of 12 identical achromatic arcs and
dispersion-free drifts. Every other drift hosts a nonlinear insert, so the
lattice forms 6 periodic cells with a T-insert section and a nonlinear insert
section. The drifts in the center of each T-insert arc are used for injection,
extraction, and RF acceleration.

Figure 2: TWISS parameters for one of the six periodic cells. (top) Horizontal and vertical beta functions shown in black and red respectively. (middle) Location and length of magnetic lattice elements where dipoles are shown as short blue rectangles, quadrupoles shown as tall orange rectangles, and sextupoles shown as green rectangles. (bottom) Linear dispersion function. Table 3: Parameters of iRCS Lattice Parameter | Value
---|---
Circumference | 486 m
Periodicity | 6 (12)
Vertical Aperture | 5 cm
Maximum Energy | 8 GeV
Bend Radius | 15.6 m
Peak Dipole Field | 1.25 T
Peak Quadrupole Field | 25 T/m
Peak Sextupole Field | 180 T/m2
Max Beta Function | 35 m
Max Dispersion | 0.8 m
Insertion Length / Cell | 8.1 m
Total Insertion Length | 97 m
Single Dipole Length | 2.8 m
Number of Dipoles | 48
Number of Quadrupoles | 156
Number of Sextupoles | 48
Momentum Compaction | 2$\times 10^{-3}$
Extraction Phase-Slip Factor | -6$\times 10^{-3}$
Betatron Tunes | 19.7
Linear Chromaticities | -10
Second-order Chromaticities | 50

This example lattice is compatible with the 8-GeV lattice described in the
previous section. By increasing the circumference and number of cells, the
8-GeV lattice can easily be scaled up to 21-GeV lattice with a lower momentum
compaction factor.

The effect of linear chromaticity on integrable motion was examined in [18]
and the effect of nonlinear chromaticity was examined in [19]. Chromaticity is
compatible with integrability if the horizontal and vertical chromaticities
are matched. In the example iRCS lattice the chromaticities were matched and
reduced in both the first and second order. The chromaticity values shown in
Fig. 3 are the result of correction by the sextupole magnets (green in Fig.
2).

To preserve integrability, sextupole magnets should also be located so that
their effect cancels harmonically (separated by a $\pi$-odd phase-advance)
[18]. To maintain the flexibility of the early design, this constraint was not
imposed on the example lattice shown here. This constraint can be met by
requiring a $\pi$-odd phase-advance for the 12 linear-periodic cells or by
combining into 6 complex linear-periodic cells.

##  2 SUMMARY & FUTURE WORK

To achieve multi-MW beam power for the Fermilab high-energy program, an
integrable RCS replacement for the Fermilab Booster is an option that merits
careful scrutiny. In this paper we explore some of the preliminary design
concerns and provide an example integrable RCS lattice design. Experimental
and numerical work on the interaction between integrable optics and space-
charge dominated beams is still ongoing [2, 3, 12].

Upcoming efforts will study the space-charge dynamics of the lattice,
integrate sextupoles with harmonic cancellation, and investigate longitudinal
and transverse stacking schemes. The design of the high-power H- stripping
foil and injection chicane will also be developed.

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