11institutetext:  IBM Research – Zurich22institutetext: ETH Zürich
33institutetext: Max Planck Institute for Intelligent Systems
33email<EMAIL_ADDRESS>
# Terminating Differentiable Tree Experts

Jonathan Thomm 1122 Michael Hersche 11 Giacomo Camposampiero 1122 Aleksandar
Terzić 1122 Bernhard Schölkopf 2233 Abbas Rahimi 11

###### Abstract

We advance the recently proposed neuro-symbolic Differentiable Tree Machine,
which learns tree operations using a combination of transformers and Tensor
Product Representations. We investigate the architecture and propose two key
components. We first remove a series of different transformer layers that are
used in every step by introducing a mixture of experts. This results in a
Differentiable Tree Experts model with a constant number of parameters for any
arbitrary number of steps in the computation, compared to the previous method
in the Differentiable Tree Machine with a linear growth. Given this
flexibility in the number of steps, we additionally propose a new termination
algorithm to provide the model the power to choose how many steps to make
automatically. The resulting Terminating Differentiable Tree Experts model
sluggishly learns to predict the number of steps without an oracle. It can do
so while maintaining the learning capabilities of the model, converging to the
optimal amount of steps.

###### Keywords:

Mixture of Experts Differentiable Tree Machine Tensor Product Representations
Structure-to-structure Transformation Termination

##  1 Introduction

††footnotetext: 18th Conference on Neural-Symbolic Learning and Reasoning
(NeSy 2024).

Neuro-symbolic AI aims to combine the strengths of statistical AI, like
machine learning, with the capabilities of symbolic AI, to address the
weaknesses of each. Recent neuro-symbolic AI methods exhibit notable benefits
from a tight integration of low-level statistical perception and high-level
reasoning, e.g., in various tasks demanding out-of-distribution (OOD)
generalization [1, 2, 3, 4, 5, 6, 7]. However, compared to pure neural
approaches, neuro-symbolic AI suffers from the non-differentiability of
inherently discrete symbolic operations, unlike real numerical values, which
makes them incompatible with gradient-based learning methods. One solution can
be reinforcement learning-based learning approaches; however, they suffer from
ill-defined gradients [8]. Another solution is to use a fully symbolic search.
For instance, DreamCoder [9] builds an increasing library of functions from
input-output examples and uses wake, abstraction, and sleep phases to find
meaningful algorithm primitives for the library. Another example is the
Neural-Symbolic Stack Machine [10] which uses a neural model giving
instructions in a fixed language to be executed by a non-deep-learning part.
This approach, however, is not differentiable, and therefore, during training,
a correct execution trace is searched to obtain the multi-step training
target. While such solutions excel at solving surprisingly complex examples,
their base language might possess strong inductive biases, and it is unclear
how to connect such a system to noisy or continuous signal streams.

One viable option is to integrate Tensor Product Representations (TPR) [11]
inside neural networks. TPR is a general schema for mapping symbolic
structures to numerical vector representations that allow continuous
manipulations. Moreover, TPR can express a general formalization of
compositional structure which is defined by an instance of a structure
resulting from assigning a set of roles to particular fillers [12]: a role
characterizes a position in the structure, and its filler is the substructure
that occupies that position. In TPR, such compositional structure is
constructed by binding the role vectors with the filler vectors using an outer
product between the two vectors, which grows exponentially in dimension with
the number of bound vectors. To alleviate this explosion, there are closely
related lossy compressed representational schemes [13, 14] in which the
vectors are closed under binding operations: i.e., all roles, fillers
(substructures), and resulting compositional structures themselves can be
represented by fixed-dimensional distributed vectors.

The TPR has been integrated into various deep neural network architectures
[15, 16, 17, 18] (see [19] for an overview). Recently, TPR has been combined
with a transformer to construct a Differentiable Tree Machine (DTM) [20] which
learns sequences of operations on trees. The DTM represents trees using TPR
tensors, defines a set of operations analogous to a subset of the Lisp [21]
language on the TPR tree representations, and learns to execute the right Lisp
operations to transform trees within multiple steps. A transformer model
predicts the operations and arguments, and a TPR interpreter executes them in
a differentiable manner. The DTM has shown OOD generalization in several tree
transformation tasks, and provides a fairly general idea for learning of
sequences of discrete symbolic operations on trees by making transformations
and computations differentiable.

The DTM, on which our work is based on, faces several limitations. First, it
uses a different parameterization of the transformer layer for each
computation step. This means that as the number of steps of computation
increases, the total model size increases linearly. Second, due to the first
limitation, the DTM model only operates in a fixed number of steps. With that,
the DTM also needs an oracle termination mechanism, i.e., one needs to know
how many steps to compute for a given task. Further, the operations defined in
the DTM have strong inductive biases towards certain tree operations.
Therefore, the OOD performance shown in the original paper seems to exist
because the model is invariant to those specific OOD cases. In contrast, its
true OOD generalization is not as good in general as we demonstrate on a new
tree reversal task in Section 4.2.

In this work, we enhance the DTM architecture by making the following
contributions that address these limitations:

  * •

We introduce a Mixture of Experts [22, 23, 24] to the DTM architecture,
resulting in a Differentiable Tree Experts (DTE) model. DTE uses the same
parameters at every step and soft-chooses different weight combinations from a
weight pool. This allows for more general use-cases as the DTE can iterate
arbitrarily long without needing more parameters.

  * •

We experiment with halting and introduce a new halting mechanism that adapts
delicately enough to work with the slightly unstable training convergence of
the DTM. This avoids the need for hyperparameter tuning and theoretically
allows the model to learn how much computation is needed for a given task,
similar to what has been used in Universal Transformers [25].

  * •

Our DTE and Terminating DTE architectures exhibit similar ID and OOD
performances as the DTM. However, the number of parameters of the proposed
architectures scales constantly with the number of transformations required to
solve the problem, compared to the linear scaling that characterizes the DTM.
For example in Table 4, DTM has $47\,M$ parameters compared to $27\,M$ for our
DTE, with the difference getting larger with the number of steps. We also
observe that sparsifying expert selection, which reduces the computational
burden during training and inference, has no significant impact on
performance.

  * •

We ablate the DTM and DTE architectures on a novel tree reversal task. We
demonstrate that the OOD performance of the DTM architecture is attributable
to invariance rather than generalization (see Section 4.2).

##  2 Background

In this section, we briefly introduce Tensor Product Representations (TPRs)
and the architecture of DTM.

###  2.1 Tensor Product Representations

Tensor Product Representation (TPR) provides a general encoding of structured
symbolic objects in vector space. A TPR consists of roles and fillers [11].
While fillers describe the data, the roles define its context, and therefore,
the TPR allows for a compositional symbolic representation via distributed
vectors and tensors.

To represent a symbolic object, one computes the outer product ($\otimes$) of
the filler ($\mathbf{f}$) and the role ($\mathbf{r}$) vectors, resulting in a
matrix $\mathbf{M}=\mathbf{f}\otimes\mathbf{r}=\mathbf{f}\mathbf{r}^{T}$. A
set of $N$ symbolic objects is represented by the superposition of the role-
filler products:

$\displaystyle\mathbf{T}=\sum_{n=1}^{N}\mathbf{f}_{n}\otimes\mathbf{r}_{n}.$
(1)

If all role vectors are orthonormal, the individual fillers can be retrieved
from $\mathbf{T}$ using an associative recall, e.g.,

$\displaystyle\mathbf{f}_{i}=\mathbf{T}\mathbf{r}_{i}=\sum_{n=1}^{N}\mathbf{f}_{n}\mathbf{r}^{T}_{n}\mathbf{r}_{i}.$

The orthonormality constraint on the roles requires their dimensionality to be
$\geq N$.

Equation (1) allows us to encode tree structures as well [20]. Let us consider
an example of a binary tree of depth 4, illustrated in Fig. 1. The leaves of
the tree can be represented with TPR by
$\mathbf{T}=\mathbf{f}_{\text{some}}\otimes\mathbf{r}_{00}+\mathbf{f}_{\text{sad}}\otimes\mathbf{r}_{100}+\mathbf{f}_{\text{sheep}}\otimes\mathbf{r}_{1100}$.
Here, the subscript ($x$) of a role ($\mathbf{r}_{x}$) describes the path from
the root to the leaf, e.g., $x=100$ describes the sequence
right$\rightarrow$left$\rightarrow$left.

###  2.2 Differentiable Tree Machine

The Differentiable Tree Machine (DTM) [20] manipulates TPR-based tree
representations using three Lisp operations: car, cdr, and cons. Given a tree
($\mathbf{T}$), Lisp car extracts the subtree that is the left child of the
root by $\textsc{car}(\mathbf{T})=\mathbf{D}_{0}\mathbf{T}$. Here,
$\mathbf{D}_{0}=\mathbf{I}\otimes\sum_{x}\mathbf{r}_{x}\mathbf{r}_{0x}^{T}$ is
a linear operator that shifts all roles from the left subtree up to the root
by one level, and $\mathbf{I}$ corresponds to the identity matrix on the
filler space. Applying Lisp car to the example tree in Fig. 1 would yield
$\mathbf{T}_{0}=\textsc{car}(\mathbf{T})=\mathbf{f}_{\text{some}}\otimes\mathbf{r}_{0}$.
Lisp cdr extracts the right child by
$\textsc{cdr}(\mathbf{T})=\mathbf{D}_{1}\mathbf{T}$, where
$\mathbf{D}_{1}=\mathbf{I}\otimes\sum_{x}\mathbf{r}_{x}\mathbf{r}_{1x}^{T}$ is
the linear operator that shifts all roles from the right subtree up to the
root. Finally, the cons operation takes two trees ($\mathbf{T}_{0}$ and
$\mathbf{T}_{1}$) as arguments plus a new root node ($\mathbf{s}$) and
assembles a new tree by
$\textsc{cons}(\mathbf{T}_{0},\mathbf{T}_{1},\mathbf{s})=\mathbf{E}_{0}\mathbf{T}_{0}+\mathbf{E}_{1}\mathbf{T}_{1}+\mathbf{s}\otimes\mathbf{r}_{root}$.
The linear operators
$\mathbf{E}_{0}=\mathbf{I}\otimes\sum_{x}\mathbf{r}_{0x}\mathbf{r}_{x}^{T}$
and
$\mathbf{E}_{1}=\mathbf{I}\otimes\sum_{x}\mathbf{r}_{1x}\mathbf{r}_{x}^{T}$
shift all roles to the left and right subtrees down to the leaves,
respectively.

DTM generates a sequence of trees
($\mathbf{T}^{(0)},\mathbf{T}^{(1)},...,\mathbf{T}^{(L)}$), where the initial
tree ($\mathbf{T}^{(0)}$) is the source tree, and the final tree
($\mathbf{T}^{(L)}$) is the target tree (i.e., the result of the task). DTM
computes the tree at step $t$ as a convex combination of the results provided
by the three Lisp operations, which creates a TPR representation of a new tree
superposition:

$\displaystyle\mathbf{T}^{(t+1)}={w}^{(t)}_{\textsc{car}}\textsc{car}(\mathbf{T}^{(t)}_{\textsc{car}})+{w}^{(t)}_{\textsc{cdr}}\textsc{cdr}(\mathbf{T}^{(t)}_{\textsc{cdr}})+{w}^{(t)}_{\textsc{cons}}\textsc{cons}(\mathbf{T}^{(t)}_{\textsc{cons,0}},\mathbf{T}^{(t)}_{\textsc{cons,1}},\mathbf{s}^{(t)}).$

A transformer encoder layer (with the standard quadratic attention) predicts
the weights
(${w}^{(t)}_{\textsc{car}},{w}^{(t)}_{\textsc{cdr}},{w}^{(t)}_{\textsc{cons}}$)
for the three Lisp operations, and their arguments
($\mathbf{T}^{(t)}_{\textsc{car}},\mathbf{T}^{(t)}_{\textsc{cdr}},\mathbf{T}^{(t)}_{\textsc{cons,0}},\mathbf{T}^{(t)}_{\textsc{cons,1}},\mathbf{s}^{(t)}$).
Given a list of previously generated trees plus the input tree, each tree is
encoded as one token by encoding the tree TPR representation to a dense vector
using a deep learning encoder [26]. Each tree argument for the next Lisp
operation is computed as a weighted sum of all past trees, e.g.,

$\displaystyle\mathbf{T}_{\textsc{car}}^{(t)}=\sum_{i=0}^{t-1}a^{(i)}_{\textsc{car}}\mathbf{T}^{(i)}$

The weight of a tree ($a^{(i)}_{\textsc{car}}$) is predicted by the
transformer encoder layer at the position where the corresponding tree
($\mathbf{T}^{(i)}$) was put into the transformer encoder layer. The operation
weights and the new root weights for the cons operation are predicted at two
special classification tokens given as input to the transformer encoder layer.

The DTM runs for a predefined number of steps (between 12 and 28, configured
as a hyperparameter), each time generating a new tree superposition. Each step
however uses a different transformer encoder layer to predict the next action.
The output of the last step is taken as the model’s answer. See Fig. 1 for an
architecture diagram. In the experiments reported in this paper, we use a
newer version of the original DTM paper’s code which has improved training
stability and provided a lower parameter count [26].

Figure 1: The DTM architecture. In each step, a new tree superposition is
generated (in TPR) using a different transformer encoder layer for each step.
The instruction probabilities are predicted by the transformer encoder layer.
car, cdr, and cons are the three Lisp operations.

##  3 Terminating Differentiable Tree Experts

###  3.1 Differentiable Tree Experts

This section presents the main contribution of our paper: Differentiable Tree
Experts (DTE). Instead of learning a different transformer encoder in each
step for DTM, one could share the weights. However, according to our
experiments, using the same transformer encoder layer leads to a non-
converging DTM. We, therefore, propose integrating a Mixture of Experts in the
DTM architecture, which enables convergence again despite the weight sharing
between each step. This means that in every step, the same router in DTE
weights several experts (in our experiments, 16 experts) that then give
proposals for the operation and the arguments. Those predictions are weighted,
and then the DTE execution takes place.

In our router, a transformer encoder layer encodes all current trees. The
current step is encoded as a sinusoidal positional encoding [27]. From the
concatenation of the tree encoding and the step encoding, the router
probabilities are computed with a linear map. See Fig. 2 for an architecture
diagram of DTE.

This architecture modification scales much better when the number of steps
increases. While DTM needs an additional transformer encoder layer for every
step, DTE stays constant in size. Together with a termination algorithm, this
also allows for deciding the number of steps flexibly, e.g., allowing for a
different number of steps during inference (in our experiments between 12 and
28).

Figure 2: The architecture of the DTE. In each step, a new tree is generated
using the same model. Our transformer encoder layer is now a Mixture of
Experts (MoE) with the router itself being a combination of a transformer
encoder layer and a linear map. The router chooses the expert weights, which
then are used to weigh the outputs of each expert. In our sparse MoE
ablations, only the top 4 experts are activated.

###  3.2 Sluggish Termination

Several termination heuristics have been proposed in the literature [28, 29].
For this work, we found termination inspired by speculative execution to work
best for our Terminating DTE (TDTE). In general, the training convergence of
DTE is brittle, as also observed with the DTM. In particular, changing the
termination decision too often caused the model to not converge anymore. We,
therefore, use two termination predictors. One predictor follows the other as
soon as it is confident. This way, changes in the termination are only made if
a certain confidence is reached.

Let us denote
$i_{\text{damp}}:=\text{argmax}_{s}\left(p_{\text{damp}}(s)\right)$,
$i_{\text{expl}}:=\text{argmax}_{s}\left(p_{\text{expl}}(s)\right)$ the
predictions of the two predictors, and $p(i_{\text{damp}}),p(i_{\text{expl}})$
the probabilities of the predictors at those indices. The probabilities over
all steps sum up to 1 for each predictor. We define the loss label (i.e., the
target step) of the two termination predictors as:

Figure 3: Cases of the sluggish termination losses. The two arrows indicate
the labels of the termination predictors in each case. The cases are given by
whether the predictors are below or above the yellow confidence threshold. The
orange dots show where the main model loss is and the two (relatively small)
residual losses. The purple dot is the best local termination (e.g., best loss
with some step penalty). The green predictor is called the “explorer”, the
blue one the “damper”, as it will start to follow the explorer when the
explorer becomes confident and otherwise stays where it is.

$\displaystyle y_{\text{damp}}$
$\displaystyle=\begin{cases}i_{\text{expl}}&p(i_{\text{expl}})\geq 0.8\\\
i_{\text{damp}}&\text{otherwise}\end{cases}$ (2) $\displaystyle
y_{\text{expl}}$ $\displaystyle=\begin{cases}\underset{s\in
S}{\operatorname*{argmax}}\left(\text{loss}(s)*0.9^{\text{idx}(s)}\right)&\text{if
}p(i_{\text{damp}})\geq 0.8\land i_{\text{damp}}=i_{\text{expl}}\\\
i_{\text{expl}}&\text{otherwise}\end{cases}$ (3)

As shown in Equations (2) and (3), we use a confidence threshold of $0.8$
which was determined by a grid search based on the training convergence. $S$
denotes the local set of choices around the current prediction, i.e.
$S=\\{i_{\text{damp}}-4,i_{\text{damp}},i_{\text{damp}}+5\\}$ and
$\text{idx}(i_{\text{damp}}-4)=0$, $\text{idx}(i_{\text{damp}})=1$,
$\text{idx}(i_{\text{damp}}+5)=2$. The choices in $S$ are hyperparameters that
worked well in practice.

We learn one termination for the DTM on each task. Each predictor consists of
a series of constant parameters, one for each potential step, which is enough
for the datasets investigated here and in the datasets used in [20]. By
scaling the termination parameters (and initializing them to small values) by
a large factor, one can make sure that the gradient updates are high enough
for those single values. The method can also be applied to sample-wise
predictors from the main model, which remains to be explored in future work.

Fig. 3 visualizes Equations (2) and (3), i.e. which losses are applied to the
two termination predictors in which cases. The exploration predictor (blue)
will stay where it is until the damping predictor (green) is confident. When
this happens, the exploration predictor stays where it is, and the damping
predictor follows. As soon as both are at the same place and confident, the
exploration predictor starts exploring for better termination options again.

To compute which of the three considered termination steps is the best (see
Equation 3 top case), we calculate the loss at each of the three steps and
deduct a small multiplicative factor for later termination. This way the model
will choose to terminate earlier if iterating longer does not bring
significant improvements. We use cross-entropy loss to train the predictors,
which are single numbers for each step and predictor. Our main loss is on the
last choice, i.e., after the termination (see Fig. 3), to make sure the model
can learn to do intermediate work at the current termination and then decide
to terminate later.

The DTM and DTE architectures use the last tree as the final model output.
When the termination changes, a step previously computing the final answer now
computes an intermediate result. This could lower training performance;
therefore, having a separate read-out transformer layer and choosing the
termination over intermediate results only, could be an improvement for future
work.

##  4 Experiments

Figure 4:  Examples of the Car-Cdr-Seq, Passive$\leftrightarrow$Logical, and Active$\leftrightarrow$Logical dataset [20]. The model has to transform a source tree to the target tree. For the Passive$\leftrightarrow$Logical case we show the intermediate trees that the model could produce to get to the target tree. Moreover, we show an example of lexical generalization that uses unseen adjectives (in this case “funny”), as well as one for the structural generalization test set that adds additional adjectives. Dataset | DTM | DTE | TDTE
---|---|---|---
Car-Cdr-Seq |  |  |
-train | 0.99 ± 0.00 | 1.00 ± 0.00 | 0.99 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 0.99 ± 0.01 | 0.99 ± 0.01 | 0.98 ± 0.01
-test OOD structural | 0.96 ± 0.01 | 0.97 ± 0.01 | 0.94 ± 0.03
Active$\leftrightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 1.00 ± 0.00 | 0.94 ± 0.06
-test OOD structural | 1.00 ± 0.00 | 1.00 ± 0.00 | 0.96 ± 0.05
Passive$\leftrightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD structural | 1.00 ± 0.00 | 1.00 ± 0.00 | 0.99 ± 0.02
Active&Passive$\rightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD structural | 0.98 ± 0.02 | 0.99 ± 0.03 | 0.98 ± 0.02
Table 1: Performance of the reproduced DTM (our runs) and our new
architectures DTE and Terminating DTE (TDTE) on the same set of tasks used for
evaluating DTM [20].

We evaluate our Differentiable Tree Experts (DTE) and the Terminating DTE
(TDTE) on the same set of four tasks used for the evaluation of DTM [20]. Fig.
4 visualizes examples of these tasks. The first task, Car-Cdr-Seq, encodes
left-subtree and right-subtree Lisp operations in the root node of the input
tree. Those should be executed, and the resulting sub-tree is the answer. The
task Active$\leftrightarrow$Logical task contains sentence grammar trees in
either active or logical form; the task is to transform the tree into the
other grammatical form. The Passive$\leftrightarrow$Logical task is analogous,
having a sentence grammar in passive form instead of active. Finally, the
Active&Passive$\rightarrow$Logical task contains either active or passive
sentence grammar trees and the target is the logical form. All tasks come with
an ID test set and two OOD test sets. The lexical OOD set contains trees with
adjectives never seen on the leaves. The structural OOD test set contains
trees where additional adjectives are added.

While DTM and DTE use the same hidden dimension in the transformer encoder
layers (64), we observe that TDTE requires a larger one (256) in order to
obtain performance competitive with DTM and DTE. For the Mixture of Experts
router, unlike other works [22], we do not use a load balancing loss for the
router as it did not improve performance. In practice, the DTE and TDTE have
more operations during training because of the (parallelizable) mixture of
experts and the additional router. In more complex settings where the
termination is different for different samples, the DTE and TDTE can become
more efficient. The mixture of experts models become smaller in the number of
parameters than the DTE for high depth.

We used 16 experts for all of our experiments which turned out to work well.
This results in 17 transformer layer weights (one additional for the router)
compared to 16-28 layers of the DTM for the datasets in Table 1 and up to 56
in Section 4.2.

Table 1 shows the comparison results. We use the DTM as the strongest
baseline, as it outperformed vanilla Transformers, Tree Transformers, LSTMs,
and Tree2Tree LSTMs [20]. Similarly to the original DTM paper, runs that
reached validation accuracy less than $90\%$ were excluded because the
training is unstable for all models. 5 remaining runs were taken per data
point. As shown, both DTE and TDTE perform very similarly compared to DTM. At
the same time, our DTE and TDTE require a constant number of parameters with
respect to the number of steps, whereas DTM’s model size grows linearly with
the number of steps. Moreover, TDTE does not access the oracle knowledge of
the required number of steps, yet it performs on par with DTM.

###  4.1 Ablation: Sparse Mixture of Experts

Dataset | DTE | Sparse DTE | Sparse TDTE
---|---|---|---
Car-Cdr-Seq |  |  |
-train | 1.00 ± 0.00 | 0.99 ± 0.01 | 0.99 ± 0.01
-test ID | 1.00 ± 0.00 | 1.00 ± 0.01 | 1.00 ± 0.01
-test OOD lexical | 0.99 ± 0.01 | 0.99 ± 0.01 | 0.99 ± 0.01
-test OOD structural | 0.97 ± 0.01 | 0.96 ± 0.03 | 0.93 ± 0.03
Active$\leftrightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 0.94 ± 0.14 | 0.98 ± 0.05
-test OOD structural | 1.00 ± 0.00 | 0.98 ± 0.04 | 0.98 ± 0.04
Passive$\leftrightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 1.00 ± 0.00 | 0.90 ± 0.10
-test OOD structural | 1.00 ± 0.00 | 0.95 ± 0.09 | 0.99 ± 0.02
Active&Passive$\rightarrow$Logical |  |  |
-train | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test ID | 1.00 ± 0.00 | 1.00 ± 0.00 | 1.00 ± 0.00
-test OOD lexical | 1.00 ± 0.00 | 0.99 ± 0.01 | 0.96 ± 0.08
-test OOD structural | 0.99 ± 0.03 | 0.97 ± 0.07 | 0.97 ± 0.06
Table 2: Performance of the DTE and Terminating DTE with sparse Mixture of
Experts (with 4 out of 16 experts being active each time). Except for
outliers, the performance of the sparse models is very similar and within the
same range as the dense model.

One can further reduce the computational amount required for the DTE during
training and inference by introducing sparsity in the selection of experts. To
this end, we always select only the top four experts and normalize the
corresponding selection weights using the softmax function.

Table 2 shows the results. The performance of DTE is the same as in Table 1
and only repeated for the reader’s convenience. Again, we observe very similar
performance, with small OOD performance reductions for the Sparse TDTE and an
outlier in Test OOD Lexical for the Sparse DTE. This shows that sparse
experts, in principle, also work with this model and can provide up to four
times faster training and inference speed in the deep learning part.

###  4.2 New Task: Tree Reversal

Figure 5: Visualization of how the DTM and our (T)DTE can solve our novel tree reversal task. As shown, with the three operations cdr, car, and cons, reversing a tree requires several steps and more with growing tree size since every child of a branching node needs to be extracted to assemble the tree in reverse order afterward. Dataset | DTM | DTE | TDTE | SDTE | STDTE
---|---|---|---|---|---
Reverse |  |  |  |  |
-train | 0.25±0.18 | 0.37±0.34 | 0.02±0.04 |  0.80±0.19 | 0.16±0.23
-test ID | 0.23±0.17 | 0.37±0.34 | 0.02±0.04 |  0.79±0.20 | 0.14±0.22
-OOD lexical | 0.02±0.03 | 0.05±0.06 | 0.00±0.00 |  0.39±0.39 | 0.09±0.16
-OOD structural | 0.00±0.00 | 0.00±0.00 | 0.00±0.00 | 0.00±0.00 | 0.00±0.00
Table 3: Performance of the original DTM architecture and our DTE and TDTE architectures (both dense and sparse) on the tree reversal task. The training performance is highly dependent on the seed. Since these models are not invariant to this structural OOD, we observe 0% performance everywhere on this split. Dataset | DTM | DTE | TDTE
---|---|---|---
Reverse (56 steps) |  |  |
-train | 0.82 | 1.00 | 0.00
-test ID | 0.79 | 1.00 | 0.00
-test OOD lexical | 0.05 | 1.00 | 0.00
-test OOD structural | 0.00 | 0.00 | 0.00
Table 4: DTM and DTE architectures on the reversal task when given many steps.
The DTM and DTE architectures all use a hidden dimension size of 256 here
(instead of 64), because it performed better. Only one run is taken per model.

This section evaluates DTM and TDTE on a new tree reversal task. The model
gets a tree and has to reverse it exactly. This means that every inner node
that has two children has to be extracted and reversed. Because the trees
sometimes differ and subtrees to a higher depth have to be extracted, this
task is more challenging. Especially the structural OOD now requires more and
different operations. The input trees are the same as the input trees of the
Active$\leftrightarrow$Logical task. See Fig. 5 for a visualization with 28
steps.

As shown in Table 3, the models are able to learn tree reversals partially,
which is a good sign, since the model needs to choose different Lisp
operations for different samples. The structural OOD test set now needs other
Lisp instructions and here we see that the model does not generalize to them
at all.

We also train the model on many more steps (56 steps, see Table 4). This
should give the model enough room to execute enough Lisp operations to reverse
the whole tree. The ID accuracy increases significantly, especially for DTE,
which outperforms DTM while having a lower parameter count ($27\,M$ vs
$47\,M$). The good ID performance confirms that the model can learn more
complicated tree transformations, and its application to more complex datasets
looks promising in general. The TDTE does not learn.

Furthermore, it can be observed that the model generalizes to the lexical OOD
set, but not structurally. This can be explained because the model can
entirely ignore the precise adjectives, and as they are the tree leaves, they
are unlikely to be extracted by the Lisp operations. In other words, the
models are mostly invariant to OOD samples also for the reverse task. However,
this is not the case for the structural OOD (which adds an additional
adjective to the tree and therefore requires reversing two or more
adjectives), and we see that once the invariance drops, there is no
generalization at all anymore for all models in this experiment.

##  5 Discussion

DTM is a neuro-symbolic method for solving tree-to-tree transformation tasks,
effectively combining a neural controller (i.e., a transformer) with a
symbolic manipulator (TPR). The TPR engine performs symbolic manipulations in
a continuous vector space, which allows a convex combination of different
discrete operations (i.e., Car, Cdr, and Cons) and their operators (i.e.,
weighted superposition of past trees). This yields a fully differentiable
neuro-symbolic architecture that can be trained end-to-end, without requiring
reinforcement learning techniques. We further enhanced DTM by introducing a
mixture of experts and a novel automatic termination method, which reduces
both the number of parameters and the required knowledge about the number of
steps.

Both the DTM and TDTE still face some inherent limitations, which we elaborate
on in this section. Future work addressing the following limitations would
notably enhance the DTM/TDTE approach.

###  5.1 Limited Lisp operations

DTM and TDTE models focus on binary tree-to-tree transformation tasks that can
be solved with a limited subset of Lisp operations (Car, Cdr, and Cons). We
have seen that the tasks presented in [20] can be solved with a relatively low
number of sequential operations. By introducing the novel tree reversal task,
however, we could show that DTE can indeed learn to execute longer sequences
of the three operations.

Distributed representations are not restricted to the three Lisp operations.
For example, leveraging fractional power encoding (FPE) [13] would allow one
to perform arithmetic operations. In fact, FPE has been applied in
probabilistic abductive reasoning to solve Raven’s progressive tests [6].
Representing both logical and arithmetic rules with distributed
representations yielded a differentiable and fast symbolic engine, which can
even learn the underlying rules [30]. Using FPE to support arithmetic
operations in DTM or TDTE would further enrich the architectures and is an
interesting avenue for future work.

###  5.2 OOD generalization

Our novel tree reversal task reveals an important limitation of the DTM and
TDTE models in OOD generalization, showing that when the data does not fit the
strong inductive biases of Lisp operations, the models do not seem to
generalize well. The three Lisp operations have strong inductive biases to
allow certain tasks to be done in very few steps. However, other tasks, such
as tree reversal, require many steps. The fixation on the Lisp operations,
therefore, is a limitation for other tree-to-tree tasks than the tasks
evaluated in the previous sections.

Further, the tree-to-tree tasks evaluated in the original paper [20] require
mostly the same sequence of the three Lisp operations for all samples
including the OOD test sets. This allows DTM to generalize very well—the model
is invariant to the OOD variants tested, rather than generally having
excellent OOD capabilities. For Active$\leftrightarrow$Logical and
Passive$\leftrightarrow$Logical, the model only has to detect if the input is
a sentence tree in active or in logical form, which is possible by simply
looking at the children of the root node. These nodes are not really out of
distribution in the OOD splits, where the only trees changed are the ones
close to the leaves. See Fig. 4 for a visualized example. In
Active&Passive$\rightarrow$Logical the same holds and the task Car-CDR-Seq is
only about executing the steps encoded in the root node of the tree. Those
operations encoded in the root are ID in the OOD split samples (the trees are
different only close to the leaves but the model only needs to look at the
root).

For example, in the lexical generalization test, new adjective fillers are
used at the tree leaves. During execution, these values are never taken into
account during execution, as the TPR engine is invariant to the filler values.
In the structural generalization test set, additional adjectives are added to
the tree, while the operations that the model needs to learn are still the
same, i.e., the transformer can ignore the bigger tree and only look at the
inner nodes, which are still exactly equal. Such invariance is also the case
for the lexical generalization of the tree reversal task but not for the
structural OOD. This is because for the tree reversal, the fillers of the
leaves do not matter; the model does not need to reverse the leaves and only
needs to reverse the tree based on the inner nodes. However, when the
structure of the tree contains additional adjectives its shape changes, and
with it the inner nodes. Therefore, more different operations are necessary to
reverse the tree.

###  5.3 Training stability

Although the model combines deep learning and TPR successfully, the brittle
training convergence is still a limitation. Removing this issue would be
critical to allow the broader applicability of these hybrid models. The
dependence on the initialization suggests that the optimization landscape is
very non-convex or that the correct gradients are vanishing. Given the nature
of the model, which linearly superposes all operations, the latter seems
especially plausible. Investigating this problem and potentially finding
improved optimizers or in-model solutions would make the applications of deep
learning combined with TPR much more attractive to a broader audience. To
avoid vanishing gradients, one could also introduce a hybrid optimization
including elements to limit the number of superpositions and, therefore,
strengthen the remaining ones.

##  6 Conclusion

We have introduced Terminating Differentiable Tree Experts (TDTE) which
enhances the recently proposed DTM architecture. Our improvements allow the
model to scale constantly when the depth of computation increases. Based on
this, we are further able to introduce a new halting mechanism that changes
its decisions slowly and looks ahead multiple steps to be more precise and
have less impact on the model performance. This method makes it possible to
learn the right termination without having access to an oracle termination
information within the training data (which is usually not given).

## Acknowledgment

This work is supported by the Swiss National Science foundation (SNF), grant
200800. We thank Paul Soulos for providing the DTM code as well as inputs on
the work.

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#  Robustness of particle creation in the formation of a compact object

Kazumasa Okabayashi  okabayashi“at”ka.osaka-cu.ac.jp Department of Mathematics
and Physics, Osaka City University, Sumiyoshi, Osaka 558-8585, Japan  Tomohiro
Harada  harada“at”rikkyo.ac.jp Department of Physics, Rikkyo University,
Toshima, Tokyo 171-8501, Japan  Ken-ichi Nakao  knakao“at”sci.osaka-cu.ac.jp
Department of Mathematics and Physics, Osaka City University, Sumiyoshi, Osaka
558-8585, Japan  Nambu Yoichiro Institute of Theoretical and Experimental
Physics, Osaka City University, Sumiyoshi, Osaka City 558-8585, Japan

###### Abstract

Hawking predicted that the formation of a black hole by gravitational collapse
causes quantum particle creation and the spectrum of the particles is almost
thermal. This phenomenon is called the Hawking radiation. Recently, it has
been predicted that the particle creation may drastically change from the
Hawking radiation to a strong double burst if the gravitational collapse
suddenly stops just before the formation of the event horizon. In contrast
with the Hawking radiation, the burst may be so strong that it can be of
observational interest even for collapsing objects with astrophysical mass
scales. However, the burst phenomenon has been predicted through the studies
of idealized models in which a spherical hollow shell begins to collapse, but
stops shrinking and eventually settles down to a static “star”. Therefore, one
might guess that it could be particular to the hollow shell model. In this
paper, we study the particle creation due to the gravitational collapse of a
spherical object whose interior is filled with matter. In this model, we
obtain similar results to those in the case of the hollow shell model. This
implies that the double burst is a robust property of particle creation by the
sudden braking of gravitational collapse.

††preprint: OCU-PHYS-544, AP-GR-171, NITEP 112, RUP-21-11

##  I INTRODUCTION

The initial vacuum state could evolve to an excited state in curved spacetime.
For example, when an object collapses and a black hole forms, the initial
vacuum state changes and eventually settles down to a state in which particles
with a thermal spectrum are radiated steadily. This phenomenon is called
Hawking radiation Hawking (1974, 1975). Particle creation is a fundamental
phenomenon in quantum field theory in curved spacetime.

In the derivation of the Hawking radiation, the existence of a black hole was
originally assumed to justify the geometrical optic approximation. However,
the existence of the black hole is not a necessary condition for the creation
of particles with a thermal spectrum. In fact, particle creation with a
thermal distribution can be obtained when an adiabatic condition is satisfied
even if a black hole does not exist; i.e., one can obtain particle creation
with a thermal distribution in a horizonless spacetime Barceló  _et al._
(2011). This fact means that we cannot know whether an astrophysical object is
a black hole even when particle creation with thermal distribution is
observed.

A black hole is defined as a complement of the causal past of future null
infinity, and we cannot directly prove from observation that an object is
actually a black hole. Hence, all we can do is to show that an astrophysical
object is not a black hole. As astrophysical objects that cannot be
distinguished from black holes, various alternatives to black holes have been
proposed (see, e.g., Ref. Cardoso and Pani (2019) for a review).

When the collapsing hollow shell stops shrinking just before the formation of
an event horizon and becomes an ultra-compact object, it has been found that a
couple of bursts in the radiation power of particle creation occur Harada  _et
al._ (2019); Barceló  _et al._ (2019); Kokubu and Harada (2019). These bursts
might be so large that we can observe them although it is very difficult to
observe the Hawking radiation for astrophysical objects. Therefore, it is, in
principle, possible to obtain evidence that an object is not a black hole by
evaluating the radiation power. However, the bursts of particle creation might
be particular to the hollow shell model whose inside metric is the
Minkowskian. Thus, in this paper, we will consider the gravitational collapse
of a spherically symmetric object with a homogeneous density distribution, and
evaluate the radiation power when the object eventually becomes ultra compact.

The organization of this paper is as follows. In Sect. II, we briefly review
particle creation from a compact object with spherical symmetry. A description
of how an ultra-compact object forms after the gravitational collapse is given
in Sect. III. In Sect. IV, we evaluate the radiation power of the particle
creation for the present model and show that a similar result to the case with
the hollow shell model is obtained. In Sect. V, we explicitly show that the
distribution of mass of our model is totally different from that of the hollow
shell model. Finally, Sect. VI is devoted to concluding remarks. Throughout
this paper, we use the units of $c=G=\hbar=1$ in which the speed of light $c$,
Newton’s gravitational constant $G$, and the Dirac constant $\hbar$ are unity.

##  II Particle creation from a compact object with spherical symmetry

A spacetime is assumed to have spherical symmetry and the asymptotically flat
region. The line element is given as

$\displaystyle
ds^{2}=-N(t,r_{*})(dt^{2}-dr_{*}^{2})+R(t,r_{*})^{2}d\Omega^{2},$ (1)

where $d\Omega^{2}$ is the round metric. We introduce the null coordinates
defined as $u=t-r_{*}$, $v=t+r_{*}$. By this assumption, $N\rightarrow 1$ and
$R\rightarrow r_{*}$ for $v\rightarrow\infty$ or $u\rightarrow-\infty$. An
ingoing radial null $v=v_{\rm in}$ turns to an outgoing radial null $u=u_{\rm
out}$ after it crosses a point $R=0$. Since $r_{*}$ does not necessarily
vanish at a point $R=0$ in the curved spacetime, the relation between $v_{\rm
in}$ and $u_{\rm out}$ is non-trivial; the relation is denoted by
$v_{\text{in}}=G(u_{\text{out}})$.

We define the following quantity;

$\displaystyle\kappa(u_{\text{out}}):=-\frac{d}{du_{\text{out}}}\ln\frac{dv_{\text{in}}}{du_{\text{out}}}=-(\ln
G^{\prime})^{\prime}(u_{\text{out}}),$ (2)

where the prime denotes the derivative of $u_{\rm out}$. By the semiclassical
approximation, if the adiabatic condition $|\kappa^{\prime}(u_{\rm
out})|\ll\kappa^{2}(u_{\rm out})$ is satisfied, the radiation of the massless
scalar field with Planck distribution is observed at $u=u_{\rm out}$ in
asymptotically flat region Barceló  _et al._ (2011). The temperature of this
radiation is estimated as

$\displaystyle k_{B}T(u_{\rm out})=\frac{\kappa(u_{\rm out})}{2\pi},$ (3)

where $k_{B}$ is Boltzmann constant. The radiation power $P$ is the total
energy per unit time across a sphere. This represents the integration of the
expectation value of the energy-momentum tensor on the sphere. The energy-
momentum tensor of the scalar field is a quadratic operator and we need to
regularize it. By using the point-splitting regularization and the $s$-wave
approximation, the radiation power in the asymptotic region is evaluated as
Ford and Parker (1978); Birrell and Davies (1982)

$\displaystyle
P=\oint\bra{0}T_{~{}~{}t}^{r_{*}}\ket{0}R^{2}d\Omega\simeq\frac{1}{48\pi}(\kappa^{2}+2\delta\kappa^{\prime}),$
(4)

where $\delta$ respectively takes 1 and $0$ for the minimally and the
conformally coupled massless scalars and the vacuum $\ket{0}$ is defined at
early time.

For technical simplicity, we consider a spherically symmetric object with the
homogeneous density distribution. The geometry of its inside is described by
the Friedmann-Lemaître-Robertson-Walker(FLRW) metric:

$\displaystyle ds_{\text{in}}^{2}$
$\displaystyle=-d\tau^{2}+a^{2}(d\chi^{2}+\sin^{2}\chi d\Omega^{2}).$ (5)

The object is assumed to be composed of a perfect fluid with the equation of
state $p=w\rho$, where $\rho$ and $p$ are the energy density and the pressure,
respectively. Here, note that $w$ may depend on $\tau$. Then, the Einstein
equations lead to

$\displaystyle\dot{a}^{2}=\frac{8\pi}{3}\rho a^{2}-1,$ (6)
$\displaystyle\dot{\rho}+3H(1+w)\rho=0,$ (7)

where a dot means a derivative with respect to $\tau$. We introduce the null
coordinates as $V=\eta+\chi$ and $U=\eta-\chi$ where $d\eta=a^{-1}d\tau$.

The outside of the object is assumed to be vacuum and by Birkhoff’s theorem is
described by the Schwarzschild geometry:

$\displaystyle ds_{\text{ext}}^{2}$
$\displaystyle=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (8)

where $f(r)=1-2M/r$. We introduce the null coordinates as $v=t+r_{*}$ and
$u=t-r_{*}$ where $dr_{*}=f(r)^{-1}dr$.

Further, we assume that the surface of the object is comoving with the perfect
fluid. In this setting, the pressure gradient will normally diverge at the
surface of the object, and there will be a singular thin shell at the surface.
If a static compact object is formed after a gravitational collapse and the
spacetime does not have a horizon, then the conformal diagram of the collapse
to a compact object can be schematically depicted in Fig. 1.

Figure 1:  The conformal diagram of a collapse to a static compact object. The
surface of the object moves along a dashed curve whose parameter is $\lambda$,
and the matter of the object is represented by a gray region. An ingoing
radial null at $v=v_{\rm in}$ intersects the surface of the object at
$\lambda=\lambda_{\rm in}$. After the null crosses a point $R=0$, it finally
becomes to an outgoing radial null $u=u_{\rm out}$.

The surface of the object is parametrized by $\lambda$. The subscript s is
used to represent the coordinates of the surface (e.g. $r=r_{\rm
s}(\lambda)$). An ingoing radial null which intersects the surface of the
object at $\lambda=\lambda_{\rm in}$ is $v=v_{\rm s}(\lambda_{\rm in})$
outside the object whereas $V=V_{\rm s}(\lambda_{\rm in})$ inside the object.
At the center, the ingoing radial null $V=V_{\rm s}(\lambda_{\rm in})$ turns
to the outgoing radial null $U=U_{\rm s}(\lambda_{\rm out})$ which intersects
the surface of the object at $\lambda=\lambda_{\rm out}$. Since $V_{\rm
s}(\lambda_{\rm in})=U_{\rm s}(\lambda_{\rm out})$ should hold, we have
$\dot{V}_{\rm s}(\lambda_{\rm in})d\lambda_{\rm in}=\dot{U}_{\rm
s}(\lambda_{\rm out})d\lambda_{\rm out}$, or equivalently, $d\lambda_{\rm
in}/d\lambda_{\rm out}=\dot{U}_{\rm s}/\dot{V}_{\rm s}$, where the dot means
the derivative with respect to $\lambda$. This outgoing null is written as
$u=u_{\rm s}(\lambda_{\rm out})$ outside the object. Since $v_{\rm in}=v_{\rm
s}(\lambda_{\rm in})$ and $U_{\rm out}=U_{\rm s}(\lambda_{\rm out})$, we
obtain

$\displaystyle G^{\prime}(u_{\text{out}})$ $\displaystyle=\frac{dv_{\rm
in}}{du_{\rm out}}=\frac{\dot{v}_{\rm s}(\lambda_{\rm in})d\lambda_{\rm
in}}{\dot{u}_{\rm s}(\lambda_{\rm out})d\lambda_{\rm out}}$
$\displaystyle=\frac{\dot{v}_{\rm s}(\lambda_{\rm in})}{\dot{u}_{\rm
s}(\lambda_{\rm out})}\frac{\dot{U}_{\rm s}(\lambda_{\rm out})}{\dot{V}_{\rm
s}(\lambda_{\rm in})}$ $\displaystyle=\frac{A(\lambda_{\rm
out})}{B(\lambda_{\rm in})},$ (9)

where

$\displaystyle
A(\lambda)=\frac{\dot{U}_{\text{s}}(\lambda)}{\dot{u}_{\text{s}}(\lambda)}\quad{\rm
and}\quad
B(\lambda)=\frac{\dot{V}_{\text{s}}(\lambda)}{\dot{v}_{\text{s}}(\lambda)}.$
(10)

Then, $\kappa$ is written as

$\displaystyle\kappa(u_{\text{out}})$
$\displaystyle=C(\lambda_{\text{out}})-\frac{A(\lambda_{\text{out}})}{B(\lambda_{\text{in}})}D(\lambda_{\text{in}}),$
(11)

where we introduce

$\displaystyle C(\lambda)=-\frac{1}{\dot{u}_{\rm
s}}\frac{d\ln{A}(\lambda)}{d\lambda}\quad{\rm and}\quad
D(\lambda)=-\frac{1}{\dot{v}_{\rm s}}\frac{d\ln{B}(\lambda)}{d\lambda}.$

##  III Collapse to an ultra-compact object

We consider the case in which a static ultra-compact object whose radius is
slightly larger than its Schwarzschild radius is a final state of the
collapsing object. The dynamical evolution of the object is classified into
several phases specified as intervals with respect to the retarded time $u$
outside the object. Each phase is labeled by a number $i$. A radial null
entering the object in phase $j$ and escaping from it in phase $i$ is called
the null $(i,j)$.

We assume that the surface of the object moves along a timelike curve with
$\chi:=\chi_{0}={\rm constant}$. The time coordinate $\tau$ inside the object
is chosen as the parameter $\lambda$. Since $\tau$ is the proper time of the
surface, we have

$\displaystyle f\dot{t}_{\rm s}^{2}-\frac{\dot{r}_{\rm s}^{2}}{f}=1,$

and hence,

$\displaystyle\dot{t}_{\rm s}=\frac{1}{f}\sqrt{\dot{r}_{\rm s}^{2}+f}.$

By using this equation, we obtain

$\displaystyle\dot{u}_{\rm s}$ $\displaystyle=\dot{t}_{\rm
s}-\frac{\dot{r}_{\rm s}}{f}=\frac{1}{f}\left(\sqrt{f+\dot{r}_{\rm
s}^{2}}-\dot{r}_{\rm s}\right),$ $\displaystyle\dot{v}_{\rm s}$
$\displaystyle=\dot{t}_{\rm s}+\frac{\dot{r}_{\rm
s}}{f}=\frac{1}{f}\left(\sqrt{f+\dot{r}_{\rm s}^{2}}+\dot{r}_{\rm s}\right).$
(12)

We also have

$\displaystyle\dot{U}_{\rm s}=\dot{V}_{\rm s}=\frac{1}{a(\tau_{\rm s})},\quad
r_{\rm s}=a(\tau_{\rm s})\sin\chi_{0}.$ (13)

If the object is composed of the dust $w=0$, its surface shrinks along a
radial timelike geodesic:

$\displaystyle\dot{r}_{\rm s}^{2}=E^{2}-1+\frac{2M}{r_{\rm s}},\quad{\rm
and}\quad\ddot{r}_{\rm s}=-\frac{M}{r_{\rm s}^{2}},$ (14)

where $E$ is a conserved specific energy. However, as Ref. Harada  _et al._
(2019), we are interested in the case of the matter content with an exotic
equation of state which eventually causes the object to stop its gravitational
collapse and settle in a static state with a radius $r_{\rm s}=R_{\rm F}$.
Thus, we assume that the dynamics of the object is divided into the following
five phases.

  1. (i)

Phase 0: an early-collapse phase for $u<u_{0}$ $(r_{\rm s}>4M)$. In this
phase, the shell satisfies

$\displaystyle f(r_{\rm s})>\frac{1}{2},\quad\dot{r}_{\rm
s}^{2}<1,\quad|\ddot{r}_{\rm s}|<\frac{1}{2M}.$

  2. (ii)

Phase 1: a late-collapse phase for $u_{0}<u<u_{1}$ $(R_{\rm B}<r_{\rm s}<4M)$.
In this phase, the shell satisfies

$\displaystyle f(r_{\rm s})<\frac{1}{2},\quad f(r_{\rm s})<\dot{r}_{\rm
s}^{2}<1,$ $\displaystyle|\ddot{r}_{\rm s}|=O\left(\frac{1}{2M}\right).$

  3. (iii)

Phase 2: an early-braking phase for $u_{1}<u\leq u_{2}$ $(R_{2}\leq r_{\rm
s}<R_{\rm B})$. In this phase, the shell still satisfies

$\displaystyle f(r_{\rm s})\leq\dot{r}_{\rm s}^{2}<1.$

The conditions $\dot{r}_{\rm s}^{2}=f(r_{\rm s})$ and $r_{\rm s}=R_{\rm 2}$
hold at $u=u_{2}$.

  4. (iv)

Phase 3: a late-braking phase for $u_{2}<u<u_{3}$ $(R_{\rm F}<r_{\rm
s}<R_{2})$. In this phase, the shell satisfies

$\displaystyle\dot{r}_{\rm s}^{2}<f(r_{\rm s}).$

The collapse of the object stops, i.e., $\dot{r}_{\rm s}^{2}=0$ at $u=u_{3}$.
The condition $\dot{r}_{\rm s}^{2}\ll f(r_{\rm s})$ should hold in the domain
sufficiently close to $u=u_{3}$.

  5. (v)

Phase 4: a final static phase for $u>u_{3}$.

Note that Phase 0 and Phase 1 describe a collapsing dynamics which includes
not only the dust case but also the exotic matter cases.

We parametrize $R_{\rm F}$ and $R_{\rm B}$ by $\epsilon$ and $\beta$ as

$\displaystyle R_{\rm F}=2M(1+\epsilon^{2}),$

and

$\displaystyle R_{\rm B}-R_{\rm F}=2M\epsilon^{2\beta},$

where $\epsilon^{2}$ is small and $\beta$ is positive. This scenario is
schematically depicted in Fig. 2.

Figure 2:  The collapse to a compact object is depicted. The black solid curve
is the world line of the object’s surface. The surface is in the collapse
phase for $\tau<\tau_{1}$. For $\tau_{1}<\tau<\tau_{3}$, it brakes to become
static. Finally, it is static for $\tau>\tau_{3}$. The vertical axis and the
horizontal axis are the time coordinate $t$ and the radial coordinate $r_{*}$
outside the surface, respectively. These coordinates do not cover the region
inside the object. A blue line represents the ingoing null, and a red line
does the outgoing null to which the ingoing null becomes. Since these
coordinates do not cover the region inside the object, we use dashed lines to
specify the relation between the ingoing and the outgoing null curves.

##  IV Evaluation of particle creation

Following the scenario in Sect. III, we evaluate particle creation through the
gravitational collapse of the object. The functions $A$, $B$, $C$, and $D$
related to $\kappa$ are written as

$\displaystyle A$ $\displaystyle=\frac{\dot{U}_{\rm s}}{\dot{u}_{\rm
s}}=\frac{\sin\chi_{0}}{r_{\rm s}}\left(\sqrt{f+\dot{r}_{\rm
s}^{2}}+\dot{r}_{\rm s}\right),$ (15) $\displaystyle B$
$\displaystyle=\frac{\dot{V}_{\rm s}}{\dot{v}_{\rm
s}}=\frac{\sin\chi_{0}}{r_{\rm s}}\left(\sqrt{f+\dot{r}_{\rm
s}^{2}}-\dot{r}_{\rm s}\right),$ (16) $\displaystyle C:=-\frac{1}{\dot{u}_{\rm
s}}\frac{d}{d\tau}\ln A=-a\dot{A}$ $\displaystyle=\left(-\frac{\ddot{r}_{\rm
s}}{\sqrt{f+\dot{r}_{\rm s}^{2}}}+\frac{\dot{r}_{\rm s}}{r_{\rm
s}}\right)\left(\sqrt{f+\dot{r}_{\rm s}^{2}}+\dot{r}_{\rm
s}\right)-\frac{M\dot{r}_{\rm s}}{r_{\rm s}^{2}\sqrt{f+\dot{r}_{\rm s}^{2}}},$
$\displaystyle D:=-\frac{1}{\dot{v}_{\rm s}}\frac{d}{d\tau}\ln B=-a\dot{B}$
$\displaystyle=\left(\frac{\ddot{r}_{\rm s}}{\sqrt{f+\dot{r}_{\rm
s}^{2}}}+\frac{\dot{r}_{\rm s}}{r_{\rm s}}\right)\left(\sqrt{f+\dot{r}_{\rm
s}^{2}}-\dot{r}_{\rm s}\right)-\frac{M\dot{r}_{\rm s}}{r_{\rm
s}^{2}\sqrt{f+\dot{r}_{\rm s}^{2}}}.$

Hereafter, for notational simplicity, $r_{\rm s}(\tau_{\rm in})$ and $r_{\rm
s}(\tau_{\rm out})$ are denoted by $r_{\rm in}$ and $r_{\rm in}$,
respectively.

  1. (i)

Phase 0 ($u<u_{0}$): the null related to $\kappa$ is classified as $(0,0)$. In
this case, since it is just the beginning of the gravitational collapse, the
object is in an almost static state in this phase, and hence the radiation
power is so small that we do not discuss it in detail.

  2. (ii)

Phase 1 ($u_{0}<u<u_{1}$): the null related to $\kappa$ is classified as
$(1,0)$ or $(1,1)$.

We assume that, in this phase, the gravitational collapse proceeds until
$|f(r_{\rm s})|$ becomes much less than unity. In order to evaluate $\kappa$,
we need to specify the gravitational collapse model. However, $A(\tau_{\rm
out})$ is the order of $|f(r_{\rm s})|$ and $C(\tau_{\rm out})$ is the order
of $M^{-1}$ if $|f(r_{\rm s})|$ is much less than $\dot{r}_{\rm s}^{2}$. By
contrast, $B(\tau_{\rm in})$ and $D(\tau_{\rm in})$ are at most the order of
$M^{-1}$ for the null (1,0) and (1,1). Hence, we obtain

$\displaystyle\kappa=\frac{1}{4M}+O(|f(r_{\rm s})|).$ (19)

Then, the radiation power is evaluated as

$\displaystyle P\simeq\frac{1}{48\pi}\frac{1}{16M^{2}}.$

This is the same as the asymptotic value of the radiation power in the
formation of the Schwarzschild black hole. However, in our case, the
Schwarzschild black hole does not form as a final product. In this sense, we
call this type of radiation the transient Hawking radiation.

  3. (iii)

Phase 2 ($u_{1}<u\leq u_{2}$): the null related to $\kappa$ is classified as
$(2,0)$, $(2,1)$, or (2,2). In this phase, $|\dot{r}_{\rm s}|$ is not
necessarily small but at most the order of unity.

As in (ii), for the null (2,0) and (2,1), $B(\tau_{\rm in})$ and $D(\tau_{\rm
in})$ are at most the order of $M^{-1}$ whereas $A(\tau_{\rm out})$ is the
order of $|f(r_{\rm s})|$. By contrast, $\ddot{r}_{\rm out}$ can be too large
to ignore terms proportional to $f(r_{\rm s})\ddot{r}_{\rm s}|_{\tau=\tau_{\rm
out}}$ since the gravitational collapse should be stopped within a very short
period in the braking phase. From this, $\kappa$ is roughly evaluated as

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
C(\tau_{\text{out}})+O(|f(r_{\rm s})|)$ (20) $\displaystyle\simeq$
$\displaystyle\frac{1}{4M}-\left.\frac{f(r_{\rm s})\ddot{r}_{\rm
s}}{2\dot{r}_{\rm s}^{2}}\right|_{\tau_{\text{out}}},$

where $\dot{r}_{\rm s}^{2}\gg f(r_{\rm s})$ has been assumed, whence we obtain
at $u=u_{2}$

$\displaystyle\kappa$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4\sqrt{2}M}-\frac{2-\sqrt{2}}{2}\ddot{r}_{\rm out}.$
(21)

Since $|\kappa|$ could take a large value for $u_{1}<u\leq u_{2}$, the
radiation power $P$ can be very large in this phase. We call this radiation
the post-Hawking burst. Note that the first term in Eq.(20) is the same as
$\kappa$ of the transient Hawking radiation, and hence the radiation power
also includes the effect of the transient Hawking radiation. By contrast, the
first term in Eq.(21) is less than $\kappa$ of the transient Hawking
radiation. This is because the collapsing speed is so small that $\dot{r}_{\rm
s}^{2}=f(r_{\rm s})\ll 1$ holds at $u=u_{2}$.

For the null (2,2), $\kappa$ is roughly evaluated as

$\displaystyle\kappa$ $\displaystyle\simeq$
$\displaystyle\frac{1}{4M}-\left[\frac{\ddot{r}_{\rm out}}{\dot{r}_{\rm
out}}+\frac{\ddot{r}_{\rm in}}{\dot{r}_{\rm in}}\right]\left.\frac{f(r_{\rm
s})}{2\dot{r}_{\rm s}}\right|_{\tau_{\text{out}}},$ (22)

where $\dot{r}_{\rm s}^{2}\gg f(r_{\rm s})$ has been assumed at both
$\tau=\tau_{\rm in}$ and $\tau=\tau_{\rm out}$, whence we obtain at $u=u_{2}$

$\displaystyle\kappa\simeq\frac{1}{4\sqrt{2}M}$
$\displaystyle-\left[\frac{1}{\sqrt{2}}\frac{\ddot{r}_{\rm out}}{\dot{r}_{\rm
out}}+\frac{\ddot{r}_{\rm in}}{\dot{r}_{\rm in}}\right]\left.\frac{f(r_{\rm
s})}{(\sqrt{2}+1)\dot{r}_{\rm s}}\right|_{\tau_{\text{out}}},$ (23)

where $\dot{r}_{\rm s}^{2}\gg f(r_{\rm s})$ has been assumed only at
$\tau=\tau_{\rm in}$.

  4. (iv)

Phase 3 ($u_{2}<u<u_{3}$): the null related to $\kappa$ is classified as
$(3,j)$ $(j=0,\cdots,3)$. For $u_{2}<u<u_{3}$, $0<\dot{r}_{\rm s}^{2}<f(r_{\rm
s})$ holds. At the end of this phase, $\dot{r}_{\rm s}^{2}$ is much less than
$f(r_{\rm s})$.

For (3,0) and (3,1), $\kappa$ is evaluated as

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle
C(\tau_{\text{out}})+O(\sqrt{f(r_{\rm s})})\simeq-\ddot{r}_{\text{out}},$ (24)

where $\dot{r}_{\rm s}^{2}\ll f(r_{\rm s})$ has been assumed at
$\tau=\tau_{\rm out}$. For $u_{2}<u<u_{3}$, $|\kappa|$ could also take a large
value since the acceleration of the surface could be large to stop the
collapse as for Phase 2, $u_{1}<u\leq u_{2}$.

For the null (3,2), $\kappa$ is roughly evaluated as

$\displaystyle\kappa\simeq-\ddot{r}_{\text{out}}-\sqrt{f(r_{\rm
out})}\frac{\ddot{r}_{\rm in}}{|\dot{r}|_{\rm in}},$ (25)

where $\dot{r}_{\rm s}^{2}\gg f(r_{\rm s})$ and $\dot{r}_{\rm s}^{2}\ll
f(r_{\rm s})$ has been assumed at $\tau=\tau_{\rm in}$ and $\tau=\tau_{\rm
out}$, respectively. On the other hand,

$\displaystyle\kappa\simeq-\ddot{r}_{\text{out}}-\sqrt{\frac{f(r_{\rm
out})}{2f(r_{\rm in})}}\ddot{r}_{\rm in},$ (26)

where $\dot{r}_{\rm s}^{2}=f(r_{\rm s})$ and $\dot{r}_{\rm s}^{2}\ll f(r_{\rm
s})$ has been assumed at $\tau=\tau_{\rm in}$ and $\tau=\tau_{\rm out}$,
respectively.

For the null (3,3), $\kappa$ is roughly evaluated as

$\displaystyle\kappa\simeq-\ddot{r}_{\text{out}}-\sqrt{\frac{f(r_{\rm
out})}{f(r_{\rm in})}}\ddot{r}_{\rm in},$ (27)

where $\dot{r}_{\rm s}^{2}\ll f(r_{\rm s})$ has been assumed at both
$\tau=\tau_{\rm in}$ and $\tau=\tau_{\rm out}$.

  5. (v)

Phase 4 ($u_{3}<u$): the null related to $\kappa$ is classified as $(4,j)$
$(j=0,\cdots,4)$ and crosses the surface from the inside of the object to its
outside in this phase. Hence $C(\tau_{\rm out})$ vanishes.

For (4,4), $\kappa$ is exactly zero since the surface is in the static phase.

For (4,0) and (4,1), $A(\tau_{\rm out})$ is proportional to $|f(r_{\rm out})|$
whence $B(\tau_{\rm in})$ and $D(\tau_{\rm in})$ are at most the order of the
$M^{-1}$. Hence, $|\kappa|$ is much less than $M^{-1}$.

For (4,2), $\kappa$ is roughly evaluated as

$\displaystyle\kappa$ $\displaystyle=-A(\tau_{\rm out})\frac{D(\tau_{\rm
in})}{B(\tau_{\rm in})}\simeq-\epsilon\frac{\ddot{r}_{\rm in}}{|\dot{r}_{\rm
in}|},$ (28)

where $\dot{r}_{\rm s}^{2}\gg f(r_{\rm s})$ has been assumed at
$\tau=\tau_{\rm in}$. On the other hand, when $\dot{r}_{\rm s}^{2}=f(r_{\rm
s})$ holds at $\tau=\tau_{\rm in}$, we obtain

$\displaystyle\kappa$ $\displaystyle=-A(\tau_{\rm out})\frac{D(\tau_{\rm
in})}{B(\tau_{\rm in})}\simeq-\frac{\epsilon}{\sqrt{2}}\frac{\ddot{r}_{\rm
in}}{|\dot{r}_{\rm in}|}.$ (29)

For (4,3), $\kappa$ is roughly evaluated as

$\displaystyle\kappa$ $\displaystyle=-A(\tau_{\rm out})\frac{D(\tau_{\rm
in})}{B(\tau_{\rm in})}\simeq-\epsilon\frac{\ddot{r}_{\rm in}}{\sqrt{f(r_{\rm
in})}},$ (30)

where $\dot{r}_{\rm s}^{2}\ll f(r_{\rm s})$ has been assumed at
$\tau=\tau_{\rm in}$,

Since $\ddot{r}_{\rm in}$ can be very large, a burst of the particle creation
may arise in this phase. We call this the late-time burst.

To evaluate the post-Hawking burst and the late-time burst in detail, we will
consider two concrete braking dynamics in the following, which are called
models A and B. In model A, the shell slows down exponentially in the braking
phase. In model B, the shell decelerates by constant acceleration in the
braking phase. In both models A and B, we parametrize $|\dot{r}_{\rm
s}|_{\tau=\tau_{1}}=\alpha$ and $R_{\rm B}-R_{\rm F}=2M\epsilon^{2\beta}$
$(\beta>0)$, where $R_{\rm B}$ has been defined as the radius of the surface
at $\tau=\tau_{1}$ when the speed of collapse begins to decrease. In order to
compare the radiation energy of the post-Hawking burst and the late-time burst
with that for the late-collapse phase $(u_{0}<u<u_{1})$, i.e., the transient
Hawking radiation, we evaluate the radiation energy for the collapsing phase:

$\displaystyle E\simeq
P(u_{1}-u_{0})\simeq\frac{1}{48\pi}\frac{\ln\epsilon^{-1}}{2M}\min\\{\beta,1\\},$
(31)

where the time interval $u_{1}-u_{0}$ is estimated as
$4M\min\\{\beta,1\\}\ln{\epsilon^{-2}}$ in Appendix A.

###  IV.0.1 Model A: Exponentially slowed-down model

We assume that the dynamics of the surface is written as $r_{\rm s}-R_{\rm
F}\propto e^{-\sigma\tau}$ for $\tau_{1}<\tau<\tau^{\prime}_{3}$ except for a
short interval $\tau^{\prime}_{3}<\tau<\tau_{3}$. For the short interval
$\tau^{\prime}_{3}<\tau<\tau_{3}$, the surface of the object is assumed to
smoothly settle down to the final static state $R_{\rm F}$. In this model, we
obtain

$\displaystyle r_{\rm s}$
$\displaystyle=2M[1+\epsilon^{2}+\epsilon^{2\beta}e^{-\sigma(\tau-\tau_{1})}],$
(32) $\displaystyle\sigma$ $\displaystyle=\frac{|\dot{r}_{\rm
s}|_{\tau=\tau_{1}}}{R_{\rm B}-R_{\rm F}}=\frac{\alpha}{2M\epsilon^{2\beta}}.$
(33)

From $\dot{r}_{\rm s}^{2}=f(r_{\rm s})$ at $\tau=\tau_{2}$, we obtain a
quadratic equation for $X:=e^{-\sigma(\tau_{2}-\tau_{1})}$:

$\displaystyle\alpha^{2}X^{2}-\epsilon^{2\beta}X-\epsilon^{2}=0,$ (34)

where we used $\epsilon^{2}+\epsilon^{2\beta}e^{-\sigma(\tau_{2}-\tau_{1})}\ll
1$. The positive roots of this equation is given as

$\displaystyle
X=\frac{\epsilon^{2\beta}+\sqrt{\epsilon^{4\beta}+4\alpha^{2}\epsilon^{2}}}{2\alpha^{2}}.$
(35)

If $\epsilon$ is sufficiently small, $X$ is classified as

$\displaystyle X\simeq\begin{cases}\dfrac{\epsilon}{\alpha}&{\rm
for}~{}~{}\beta>1/2\\\
\dfrac{1+\sqrt{1+4\alpha^{2}}}{2\alpha^{2}}\epsilon&{\rm
for}~{}~{}\beta=1/2\\\ \dfrac{\epsilon^{2\beta}}{\alpha^{2}}&{\rm
for}~{}~{}0<\beta<1/2\end{cases}.$ (36)

In this model, we assume $\tau_{3}-\tau_{2}\simeq\sigma^{-1}$. The radial
coordinate of the surface and its derivatives at $\tau=\tau_{1}$ and
$\tau_{2}$ are summarized in Table.1.

$\tau$ |  $\tau_{1}$ |  $\tau_{2}$
---|---|---
$r_{\rm s}(\tau)-R_{\rm F}$ |  $2M\epsilon^{2\beta}$ |  $2M\epsilon^{2\beta}X$
$|\dot{r}_{\rm s}(\tau)|$ |  $\alpha$ |  $\alpha X$
$\ddot{r}_{\rm s}(\tau)$ |  $\dfrac{\alpha^{2}}{2M\epsilon^{2\beta}}$ |  $\dfrac{\alpha^{2}}{2M\epsilon^{2\beta}}X$

Table 1: The dynamics of the surface in the braking phase for the
exponentially slowed-down model.

In evaluating the radiation power of particle creation, it is sufficient to
take into account the null of types $(2,0)$, $(2,1)$, $(3,0)$, $(3,1)$ (see
Appendix B). In these cases, $C(\tau)$ is the dominant term in $\kappa$. In
particular, the following term proportional to $\ddot{r}_{\rm s}$ in $C(\tau)$
will be dominant:

$\displaystyle C_{\rm Acc}:=-\ddot{r}_{\rm s}\left(1+\frac{\dot{r}_{\rm
s}}{\sqrt{f+\dot{r}_{\rm s}^{2}}}\right).$ (37)

For the early stage in $u_{1}<u<u_{2}$, since $\dot{r}^{2}\gg f$ holds, the
quantity $C_{\rm Acc}$ is estimated as

$\displaystyle C_{\rm
Acc}\simeq-\frac{1+\epsilon^{2(1-\beta)}}{4\alpha^{2}M}.$ (38)

If $\beta>1$, we obtain $|\kappa|_{u=u_{1}}\gg M^{-1}$ for $0<\epsilon\ll 1$.
As has been mentioned, in order that $|\kappa|$ is larger than the value of
the transient Hawking radiation, $|\ddot{r}_{\rm s}|$ should be much larger
than $M^{-1}$. Hence, $\beta$ should be larger than $1/2$ so that
$|\ddot{r}_{\rm s}|_{\tau=\tau_{2}}\gg M^{-1}$ for $0<\epsilon\ll 1$. If
$\beta>1/2$, $\kappa$ at $u=u_{2}$ is estimated as

$\displaystyle\kappa\simeq-\frac{2-\sqrt{2}}{4M\epsilon^{2\beta-1}}\alpha.$
(39)

For $0<\epsilon\ll 1$, we have $|\kappa|_{u=u_{2}}\gg|\kappa|_{u=u_{1}}$. For
the late stage of $u_{2}<u<u_{3}$, $|\kappa|$ goes to zero. The radiation
energy for $u_{2}<u<u_{3}$ can be roughly evaluated as

$\displaystyle
E\simeq\frac{1}{2}P(u_{3}-u_{2})\simeq\frac{1}{48\pi}\frac{(2-\sqrt{2})^{2}\alpha}{16M\epsilon^{2\beta-1}},$
(40)

where the time interval $u_{3}-u_{2}$ is estimated as
$2M\alpha^{-1}\epsilon^{2\beta-1}$ is Appendix A. Comparing it with Eq.(31),
we find that the radiation energy of the post-Hawking burst is much larger
than that of the transient Hawking radiation.

By a similar argument for the late-time burst, it is sufficient in evaluating
$\kappa$ to consider the null of types $(4,j)$ $(j=0,\cdots,4)$. If $\beta$ is
larger than $1/2$, we find that $\kappa$ takes a nearly constant value for
$\tilde{u}_{1}<u<\tilde{u}_{2}$

$\displaystyle\kappa\simeq-\frac{\alpha}{2M\epsilon^{2\beta-1}},$

and goes to zero for $\tilde{u}_{2}<u<\tilde{u}_{3}$. The radiation energy for
$\tilde{u}_{1}<u<\tilde{u}_{2}$ is given by

$\displaystyle E\simeq
P(\tilde{u}_{2}-\tilde{u}_{1})\simeq\frac{1}{48\pi}\frac{\alpha\ln(\alpha\epsilon^{-1})}{2M\epsilon^{2\beta-1}},$
(41)

where the time interval $\tilde{u}_{2}-\tilde{u}_{1}$ is estimated as
$2M\alpha^{-1}\epsilon^{2\beta-1}\ln(\alpha\epsilon^{-1})$ in Appendix A.
Comparing it with Eq.(31), the radiation energy of the late-time burst is also
much larger than that of the transient Hawking radiation.

###  IV.0.2 Model B: Constant-deceleration model

We consider that the constant deceleration $\tilde{a}$ of the surface is given
by

$\displaystyle\ddot{r}_{\rm s}=\tilde{a}=\frac{\alpha^{2}}{2(R_{\rm B}-R_{\rm
F})},$ (42)

for $\tau_{1}<\tau<\tau_{3}$ so that $|\dot{r}_{\rm s}|_{\tau=\tau_{3}}$
vanishes. The deceleration $\tilde{a}$ is much larger than $1/4M$ for
$0<\epsilon\ll 1$. In this model, the radial coordinate of the surface is

$\displaystyle r_{\rm s}=2M[1+\epsilon^{2}+\epsilon^{2\beta}(Y-1)^{2}],$ (43)

where

$\displaystyle
Y:=\frac{\tau-\tau_{1}}{\tau_{3}-\tau_{1}}=\frac{\alpha}{4M\epsilon^{2\beta}}(\tau-\tau_{1}).$
(44)

In this model, the proper time of the surface, the radial coordinate of the
surface and its derivative at $\tau=\tau_{1}$, $\tau_{2}$, and $\tau_{3}$ are
summarized in Table.2.

$\tau$ |  $\tau_{1}$ |  $\tau_{2}$ |  $\tau_{3}$
---|---|---|---
$\tau_{3}-\tau$ |  $\alpha\tilde{a}^{-1}$ |  $\epsilon\tilde{a}^{-1}$ |  $0$
$r_{\rm s}(\tau)-R_{\rm F}$ |  $2M\epsilon^{2\beta}$ |  $2M\alpha^{-2}\epsilon^{2\beta+2}$ |  $0$
$\dot{r}_{\rm s}(\tau)$ |  $\alpha$ |  $\epsilon$ |  $0$

Table 2: The dynamics of the surface in the braking phase for the constant-
deceleration model.

From Appendix B, it is sufficient in evaluating $\kappa$ in the epoch of the
braking phase of this model to consider the null of types $(2,0)$, $(2,1)$,
$(3,0)$, $(3,1)$. In these cases, $C(\tau)$ is the dominant term in $\kappa$.
The quantity $C_{\rm Acc}$ is estimated as

$\displaystyle C_{\rm
Acc}=-\frac{\alpha^{2}}{4M\epsilon^{2\beta}}\left[1-\frac{\alpha(1-Y)}{\sqrt{\epsilon^{2}+(\epsilon^{2\beta}+\alpha^{2})(1-Y)^{2}}}\right].$
(45)

For the early stage in $u_{1}<u<u_{2}$, i.e., the stage with $\dot{r}^{2}\gg
f$, the quantity $C_{\rm Acc}$ is estimated as

$\displaystyle C_{\rm
Acc}\simeq-\frac{1+\epsilon^{2\beta-2}(1-Y)^{2}}{8M\epsilon^{2\beta-2}(1-Y)^{2}}.$
(46)

If $\beta>1$, we have $|\kappa|\gg M^{-1}$ for $0<\epsilon\ll 1$. On the other
hand, we find that $\kappa$ for $u_{2}<u<u_{3}$ take a nearly constant value:

$\displaystyle\kappa\simeq-\frac{\alpha^{2}}{4M\epsilon^{2\beta}}.$ (47)

For $0<\epsilon\ll 1$, we have
$|\kappa|_{u_{2}<u<u_{3}}\gg|\kappa|_{u=u_{1}}$. The radiation energy for
$u_{2}<u<u_{3}$ is given as

$\displaystyle E\simeq
P(u_{3}-u_{2})\simeq\frac{1}{48\pi}\frac{\alpha^{2}}{4M\epsilon^{2\beta}},$
(48)

where the time interval $u_{3}-u_{2}$ is estimated as
$4M\alpha^{-2}\epsilon^{2\beta}$ in Appendix A. Comparing this with Eq.(31),
we find that the radiation energy of the post-Hawking burst is much larger
than that for the transient Hawking radiation.

By a similar argument for the late-time burst, it is sufficient in evaluating
$\kappa$ to consider the null of types $(4,j)$ $(j=0,\cdots,4)$. If
$\beta>1/2$ and $0<\epsilon\ll 1$, $|\kappa|$ for $\tilde{u}_{1}\leq
u<\tilde{u}_{2}$ increases from $\alpha/(4M\epsilon^{2\beta-1})$ to
$\alpha^{2}/(4M\epsilon^{2\beta})$. For $\tilde{u}_{2}\leq u<\tilde{u}_{3}$,
$\kappa$ takes a nearly constant value:

$\displaystyle\kappa\simeq-\frac{\alpha^{2}}{4M\epsilon^{2\beta}}.$

For $0<\epsilon\ll 1$, we have
$|\kappa|_{\tilde{u}_{2}<u<\tilde{u}_{3}}\gg|\kappa|_{u=\tilde{u}_{1}}$. The
radiation energy for $\tilde{u}_{2}<u<\tilde{u}_{3}$ is given by

$\displaystyle E\simeq
P(\tilde{u}_{3}-\tilde{u}_{2})\simeq\frac{1}{48\pi}\frac{\alpha^{2}}{4M\epsilon^{2\beta}},$
(49)

where the time interval $\tilde{u}_{3}-\tilde{u}_{2}$ is estimated as
$4M\alpha^{-2}\epsilon^{2\beta}$ in Appendix A. Comparing this with Eq.(31),
we find that the radiation energy of the late-time burst is also much larger
than that for the transient Hawking radiation.

##  V The Misner-Sharp mass inside the shell and that of the shell

We obtain the radiation power of the transient Hawking radiation, post-Hawking
burst, and late-time burst caused by the gravitational collapse of a
homogeneous spherical object with a singular surface, i.e., an infinitesimally
thin shell. Their leading terms are similar to those in the hollow shell
model, except for the coefficients. This result seems to be non-trivial.
However, there is a possibility that the mass except for that of the shell is
so small compared to that of the shell that the contribution from the inside
matter to the radiation power is negligible, and hence the radiation power in
our model becomes similar to the hollow shell model. If this is true, our
result is not very new because it is physically same as the hollow shell model
in such a case. Hence, we will show that our model is different from the
hollow shell model by comparing the mass inside the shell and that of the
shell. Although, in General Relativity, it is a difficult problem to define a
mass, there is a quasi-local definition of a mass in a spherically symmetric
spacetime, which is called the Misner-Sherp mass Hayward (1996). We adopt this
for our present purpose.

Using the double null coordinates, the metric of a spherically symmetric
spacetime is given as

$\displaystyle ds^{2}=-\tilde{N}(u,v)dudv+\tilde{R}(u,v)^{2}d\Omega^{2},$ (50)

where $u$ and $v$ are respectively outgoing and ingoing null coordinates. The
expansion for outgoing and ingoing null congruence can be written as
$\theta_{+}=2\tilde{R}^{-1}\partial_{u}\tilde{R}$ and
$\theta_{-}=2\tilde{R}^{-1}\partial_{v}\tilde{R}$, respectively. Then, the
Misner-Sherp mass $\mathcal{M}$ is defined as

$\displaystyle\mathcal{M}=\frac{\tilde{R}^{3}}{2}\left(\tilde{N}^{-1}\theta_{+}\theta_{-}+\frac{1}{\tilde{R}^{2}}\right).$
(51)

The masses of the object seen from outside and inside the shell are denoted by
$\mathcal{M}_{\pm}$, respectively. Then, the mass of the shell is given as
$\mathcal{M}_{+}-\mathcal{M}_{-}$.

In our model, the metric outside the shell is the Schwarzschild metric and we
obtain

$\displaystyle\mathcal{M}_{+}=\frac{r_{\rm s}}{2}\left[1-f(r_{\rm
s})\right]=M.$ (52)

By contrast, the metric inside the shell is a closed FLRW metric and we obtain

$\displaystyle\mathcal{M}_{-}$
$\displaystyle=\frac{1}{2}a(\tau)\left[1+\dot{a}(\tau)^{2}\right]\sin^{3}\chi=\frac{4\pi}{3}\rho
r_{\rm s}^{3}.$ (53)

From Eqs. (52) and (53), the mass of the shell can be evaluated. To do this,
we have to specify the collapsing phase and the braking phase. As the braking
phase, we use the models A and B. By contrast, as the collapsing phase, we use
the Oppenheimer-Snyder collapse.

In the Oppenheimer-Snyder collapse, matter inside the object is assumed to be
dustOppenheimer and Snyder (1939). A solution of Eqs.(6), (7) is given by

$\displaystyle a(\eta)=\frac{1}{2}a_{\rm
max}(1+\cos\eta),\quad\tau(\eta)=\frac{1}{2}a_{\rm max}(\eta+\sin\eta),$

where we choose the origin of the conformal time $\eta=0$ such that the object
starts to collapse, and $a_{\rm max}$ is the scale factor at $\eta=0$. Let
$\eta_{1}$ be the conformal time such that the object starts to brake so that
$\tau(\eta_{1})=\tau_{1}$. At $\eta=\eta_{1}$, the radius and the velocity of
the shell are assumed to be

$\displaystyle r_{\rm
s}(\tau_{1})=2M+2M\epsilon^{2\beta},\quad\left.\frac{dr_{\rm
s}}{d\tau}\right|_{\tau=\tau_{1}}=-\alpha\quad(\alpha>0).$

By contrast, in the stage of the Oppenheimer-Snyder collapse, there is no
shell, $\mathcal{M}_{+}-\mathcal{M}_{-}=0$, and we have

$\displaystyle M=\frac{4\pi}{3}\rho r_{\rm s}^{3}=\frac{1}{2}a_{\rm
max}\sin^{3}\chi_{0}.$ (55)

After all, we obtain the following equations

$\displaystyle a_{\rm max}=\frac{2M}{\sin^{3}\chi_{0}},$
$\displaystyle\eta_{1}=2\arccos\left[\sin\chi_{0}\sqrt{\frac{r_{\rm
s}(\tau_{1})}{2M}}\right],$
$\displaystyle\alpha=\tan\frac{\eta_{1}}{2}\sin\chi_{0}.$

When the parameters $\chi_{0}$ and $\beta$ are fixed, the remaining
parameters, $a_{\rm max},\eta_{1}$, and $\alpha$, are also fixed. Thus, we
could continuously connect the Oppenheimer-Snyder collapse and the braking
phase.

In Figs. 3 and 4, the relation between the mass inside the shell and that of
the shell is given for models A and B, respectively. As seen in both figures,
the mass of the shell increases through the braking phase although the mass of
the shell is zero in the collapsing phase. However, the mass inside the shell
could be larger than the mass of the shell, and hence is not necessarily
negligible. Our result is non-trivial because it is physically different from
the hollow shell model.

Figure 3:  The relation between the mass inside the shell and that of the
shell in the exponentially slowed-down model. The blue and orange lines are
respectively the mass inside the shell and that of the shell. The dashed line
is the mass outside the shell which corresponds to $M$. In this figure, we
choose $M=1$, $\beta=2/3$, $\epsilon=10^{-2}$, $\chi_{0}=\pi/3$.  Figure 4:
The relation between the mass inside the shell and that of the shell in the
constant-deceleration model. The blue and orange lines are respectively the
mass inside the shell and that of the shell. The dashed line is the mass
outside the shell which corresponds to $M$. In this figure, we choose $M=1$,
$\beta=2/3$, $\epsilon=10^{-2}$, $\chi_{0}=\pi/3$.

##  VI Concluding remarks

In the current analysis, we treat the spherically symmetric object with the
homogeneous density distribution and evaluate the radiation power of the
particle creation.

Through the evaluation of the quantity $\kappa$, we show that the transient
Hawking radiation occurs for the collapsing phase, and that the post-Hawking
burst and the late-time burst occur for the braking phase. The transient
Hawking radiation is obtained in the hollow shell model studied in Ref.Harada
_et al._ (2019), and hence the transient Hawking radiation in the present
model is similar to the hollow shell model. The radiation power for the
braking phase depends on the detail of the dynamics of the surface, and hence
we consider the two scenarios (model A and B). For the two examples, we find
that these bursts could have the radiation power much larger than the
transient Hawking radiation if the acceleration is sufficiently large. The
radiation power of our model is different from that of the hollow shell model
in the numerical factor of the order of unity (e.g. the parametrization of
$|\dot{r}_{\rm s}|_{\tau=\tau_{1}}$ and the numerical factor in Eq.(39)). This
does not come from the physical difference between the models but is rather
because the present analysis is more accurate than the previous one. In spite
of this little difference, it is more important that the exponents of
$\epsilon$ for the post-Hawking burst and the late-time burst in the present
matter-filled model completely agree with those in the hollow shell model. In
this sense, these two bursts we obtained are also similar to the hollow shell
model. This result implies that the bursts of the particle accompanied by the
sudden braking of the gravitational collapse has a robust property. In
addition to the radiation power of particle creation, the radiation energy of
the transient Hawking radiation, the post-Hawking burst, and the late-time
burst also have the robust property since we find that the time intervals for
these phase are also similar to those in the hollow shell model.

We discuss why the similar result is obtained. In the collapsing phase, the
contribution from matter inside the surface becomes subdominant when the
radius of the surface is close to the Schwarzschild radius, and hence the
transient Hawking radiation is obtained. In the braking phase, the leading
term of the radiation is proportional to the acceleration of the surface since
the absolute value of the acceleration of the shell becomes sufficiently
large. The contribution from matter inside the surface does not affect the
exponent of $\epsilon$ in the leading term. Hence, the post-Hawking burst and
the late-time burst in the present model are also similar to the hollow shell
case.

By contrast, one might guess that our model is effectively the same as the
hollow shell model. To compare our model with the hollow shell model, we
evaluate Misner-Sherp mass inside the shell and that of the shell for the
timelike shell model. Since we find that Misner-Sherp mass inside the shell is
dominant in the total mass, our shell model is different from the hollow shell
model. Nevertheless, we obtain a similar radiation power from our model with
the hollow shell model. This implies that this type of particle creation does
not depend on the inside detail and has a robust property. At this point, this
robustness is just implied from our model and not proven in general. Hence,
this robustness should be examined in a more general situation.

## Acknowledgments

This work was supported in part by JSPS KAKENHI Grant Numbers JP21J15676(KO),
JP19K03876, JP19H01895, JP20H05853 (TH), and JP21K03557(KN).

##  Appendix A Time intervals

In order to estimate the total radiation energy, we evaluate a time interval
of each phase. From Eqs.(12) and (13), we obtain

$\displaystyle
u\simeq\left\\{\begin{array}[]{ll}\displaystyle{\frac{\tau}{\sqrt{f(r_{\rm
s})}}+\text{const.}}&(u<u_{0})\\\ \displaystyle{-4M\ln\left[\frac{r_{\rm
s}}{2M}-1\right]-2r_{\rm s}+\text{const.}}&(u_{0}<u<u_{2})\\\
\displaystyle{\displaystyle{\int\frac{d\tau}{\sqrt{f(r_{\rm
s})}}+\text{const.}}}&(u_{2}<u<u_{3})\\\
\displaystyle{\frac{\tau}{\epsilon}+\text{const.}}&(u_{3}<u)\end{array}\right.$
(60) (61)

From Eq.(61), we obtain the time interval $u_{1}-u_{0}$ as

$\displaystyle
u_{1}-u_{0}\simeq\left\\{\begin{array}[]{ll}\displaystyle{4M\beta\ln\epsilon^{-2}}&(0<\beta<1)\\\
\displaystyle{4M\ln\epsilon^{-2}}&(\beta\geq 1)\end{array}\right.,$ (64)

where $r_{\rm s}=4M$ at $u=u_{0}$ and $r_{\rm s}=R_{\rm B}$ at $u=u_{1}$. The
interval $\tilde{u}_{3}-u_{3}$ is given as

$\displaystyle\tilde{u}_{3}-u_{3}$
$\displaystyle=\frac{2M}{\epsilon\sin\chi_{0}}(\tilde{U}_{3}-U_{3})=\frac{4\chi_{0}M}{\epsilon\sin\chi_{0}},$
(65)

where we used $\tilde{U}_{3}-U_{3}=2\chi_{0}$. The other intervals depend on
the detail of the model in the braking phase. In what follows, we focus on
$\beta>1/2$ since the radiation much larger than the transient Hawking
radiation is obtained in this case.

###  A.0.1 Model A

From Eq.(36) for $\beta>1/2$, we obtain

$\displaystyle\tau_{2}-\tau_{1}=\frac{1}{\sigma}\ln\alpha\epsilon^{-1},$ (66)

and the time interval $u_{2}-u_{1}$ is evaluated through Eq.(61) as

$\displaystyle
u_{2}-u_{1}\simeq\left\\{\begin{array}[]{ll}4M(1-\beta)\ln\epsilon^{-2}&(1/2<\beta<1)\\\
4M&\left(\beta=1\right)\\\
4M\epsilon^{2\beta-2}&\left(\beta>1\right)\end{array}\right..$ (70)

To evaluate the time interval $u_{3}-u_{2}$, we have to determine the
$\tau_{3}-\tau_{2}$. If we assume $\tau_{3}-\tau_{2}=\sigma^{-1}$, the
interval $u_{3}-u_{2}$ is evaluated through Eq.(61) as

$\displaystyle u_{3}-u_{2}\simeq\frac{\tau_{3}-\tau_{2}}{\epsilon}\simeq
2M\alpha^{-1}\epsilon^{2\beta-1}.$ (71)

As for the time intervals $\tau>\tau_{3}$, the surface of the object is
static, and hence the time interval $\tilde{u}_{2}-\tilde{u}_{1}$ is evaluated
as

$\displaystyle\tilde{u}_{2}-\tilde{u}_{1}$
$\displaystyle=\frac{1}{\epsilon}(\tilde{\tau}_{2}-\tilde{\tau}_{1})\simeq\frac{1}{\epsilon}(\tau_{2}-\tau_{1})$
$\displaystyle\simeq 2M\alpha^{-1}\epsilon^{2\beta-1}\ln\alpha\epsilon^{-1},$
(72)

and the same calculation can be done for the time interval
$\tilde{u}_{3}-\tilde{u}_{2}$:

$\displaystyle\tilde{u}_{3}-\tilde{u}_{2}$
$\displaystyle=\frac{1}{\epsilon}(\tilde{\tau}_{3}-\tilde{\tau}_{2})\simeq\frac{1}{\epsilon}(\tau_{3}-\tau_{2})$
$\displaystyle\simeq 2M\alpha^{-1}\epsilon^{2\beta-1}.$ (73)

###  A.0.2 Model B

From Fig.2, we obtain

$\displaystyle\tau_{2}-\tau_{1}\simeq\alpha\tilde{a}^{-1},$ (74)

and hence the time interval $u_{2}-u_{1}$ is evaluated as

$\displaystyle
u_{2}-u_{1}\simeq\left\\{\begin{array}[]{ll}4M(1-\beta)\ln\epsilon^{-2}&(1/2<\beta<1)\\\
4M&\left(\beta=1\right)\\\
4M\epsilon^{2\beta-2}&\left(\beta>1\right)\end{array}\right..$ (78)

The interval $u_{3}-u_{2}$ is given as

$\displaystyle
u_{3}-u_{2}\simeq\frac{\tau_{3}-\tau_{2}}{\epsilon}=\tilde{a}^{-1}=4M\alpha^{-2}\epsilon^{2\beta}.$
(79)

As for the time intervals $\tau>\tau_{3}$, the surface of the object is
static, and hence the time interval $\tilde{u}_{2}-\tilde{u}_{1}$ is evaluated
as

$\displaystyle\tilde{u}_{2}-\tilde{u}_{1}$
$\displaystyle=\frac{1}{\epsilon}(\tilde{\tau}_{2}-\tilde{\tau}_{1})\simeq\frac{1}{\epsilon}(\tau_{2}-\tau_{1})$
$\displaystyle\simeq\alpha\epsilon^{-1}\tilde{a}^{-1}=4M\alpha^{-1}\epsilon^{2\beta-1},$
(80)

and the same calculation can be done for the time interval
$\tilde{u}_{3}-\tilde{u}_{2}$:

$\displaystyle\tilde{u}_{3}-\tilde{u}_{2}$
$\displaystyle=\frac{1}{\epsilon}(\tilde{\tau}_{3}-\tilde{\tau}_{2})\simeq\frac{1}{\epsilon}(\tau_{3}-\tau_{2})$
$\displaystyle\simeq\tilde{a}^{-1}=4M\alpha^{-2}\epsilon^{2\beta}.$ (81)

##  Appendix B The null classification of $\kappa$

In evaluating $\kappa$ for a phase, we have to know the type of the null
$(i,j)$ for the phase. For $u_{0}<u<u_{1}$, the null is classified as $(1,0)$
or $(1,1)$. For $u>u_{3}$, the null is classified as $(4,j)$ $(j=0,\cdots,4)$.
By contrast, the classification of the null for $u_{1}<u<u_{3}$ depends on the
detail of the dynamics of the surface. In general, the null for
$u_{1}<u<u_{3}$ might be classified as $(2,j)$ $(j=0,\cdots,2)$ or $(3,j)$
$(j=0,\cdots,3)$. However, we show that the null classified as $(2,2)$ or
$(3,3)$ does not exist in model A and B.

###  B.0.1 Model A

In the braking phase, the conformal time $d\eta:=d\tau/a$ is written as

$\displaystyle d\eta\simeq\frac{\sin\chi_{0}}{2M}d\tau.$ (82)

For the early-braking phase, we obtain the interval of the conformal time
$\eta_{2}-\eta_{1}$ as

$\displaystyle\eta_{2}-\eta_{1}=\frac{\sin\chi_{0}}{2M}(\tau_{2}-\tau_{1}).$
(83)

For $\beta>1/2$ and $\epsilon\ll 1$, we find from Eq.(36)
$X:=e^{-\sigma(\tau_{2}-\tau_{1})}=\epsilon/\alpha$, and hence the interval of
the conformal time $\eta_{2}-\eta_{1}$ is evaluated as

$\displaystyle\eta_{2}-\eta_{1}=\frac{\epsilon^{2\beta}}{\alpha}\sin\chi_{0}\ln\frac{\alpha}{\epsilon}\ll
1.$ (84)

This means that if any null crosses inwardly the surface for
$\tau_{1}<\tau<\tau_{2}$, the time interval of the conformal time
$\eta_{2}-\eta_{1}$ is too small for the null to cross outwardly the surface
for $\tau_{1}<\tau<\tau_{2}$. Hence, we do not have to consider the null
$(2,2)$ in evaluation of $\kappa$.

For the late-braking phase, by the similar argument, the interval of the
conformal time $\eta_{3}-\eta_{2}$ is given as

$\displaystyle\eta_{3}-\eta_{2}=\frac{\sin\chi_{0}}{2M}(\tau_{3}-\tau_{2}).$
(85)

In model A, we choose the interval of the comoving time
$\tau_{3}-\tau_{2}=\sigma^{-1}$, and hence the interval of the conformal time
$\eta_{3}-\eta_{2}$ is evaluated as

$\displaystyle\eta_{3}-\eta_{2}=\frac{\epsilon^{2\beta}}{\alpha}\sin\chi_{0}\ll
1.$ (86)

This also means that if any null crosses inwardly the surface for
$\tau_{2}<\tau<\tau_{3}$, the interval of the conformal time
$\eta_{3}-\eta_{2}$ is too small for the null to cross outwardly the surface
for $\tau_{2}<\tau<\tau_{3}$. Hence, we do not have to consider the null
$(3,3)$ in evaluation of $\kappa$.

###  B.0.2 Model B

For the early-braking phase, we obtain the interval of the conformal time
$\eta_{2}-\eta_{1}$ as

$\displaystyle\eta_{2}-\eta_{1}=\frac{\sin\chi_{0}}{2M}(\tau_{2}-\tau_{1}).$
(87)

From Table.2, we obtain the interval of the comoving time
$\tau_{2}-\tau_{1}=(\alpha-\epsilon)\tilde{a}^{-1}$, and hence the interval of
the conformal time $\eta_{2}-\eta_{1}$ is evaluated as

$\displaystyle\eta_{2}-\eta_{1}\simeq\frac{2\epsilon^{2\beta}}{\alpha^{2}}\sin\chi_{0}\ll
1.$ (88)

Hence, if any null crosses inwardly the surface for $\tau_{1}<\tau<\tau_{2}$,
the time interval of the conformal time $\eta_{2}-\eta_{1}$ is too small for
the null to cross outwardly the surface for $\tau_{1}<\tau<\tau_{2}$. Thus, we
do not have to consider the null $(2,2)$ in evaluation of $\kappa$.

For the late-braking phase, by the similar argument, the conformal time
$\eta_{3}-\eta_{2}$ is given as

$\displaystyle\eta_{3}-\eta_{2}=\frac{\sin\chi_{0}}{2M}(\tau_{3}-\tau_{2}).$
(89)

From Table.2, we obtain the interval of the comoving time
$\tau_{3}-\tau_{2}=\epsilon\tilde{a}^{-1}$, and hence the interval of the
conformal time $\eta_{3}-\eta_{2}$ is evaluated as

$\displaystyle\eta_{3}-\eta_{2}=\frac{\epsilon^{2\beta+1}}{\alpha}\sin\chi_{0}\ll
1.$ (90)

Hence, if any null crosses inwardly the surface for $\tau_{2}<\tau<\tau_{3}$,
the conformal time interval $\eta_{3}-\eta_{2}$ is too small for the null to
cross outwardly the surface for $\tau_{2}<\tau<\tau_{3}$. Thus, we do not have
to consider the null $(3,3)$ in evaluation of $\kappa$.

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