Laurin Wagner1, Mario Zusag1, Theresa Bloder1

# Careful Whisper - leveraging advances in automatic speech recognition for
robust and interpretable aphasia subtype classification

###### Abstract

This paper presents a fully automated approach for identifying speech
anomalies from voice recordings to aid in the assessment of speech
impairments. By combining Connectionist Temporal Classification (CTC) and
encoder-decoder-based automatic speech recognition models, we generate rich
acoustic and clean transcripts. We then apply several natural language
processing methods to extract features from these transcripts to produce
prototypes of healthy speech. Basic distance measures from these prototypes
serve as input features for standard machine learning classifiers, yielding
human-level accuracy for the distinction between recordings of people with
aphasia and a healthy control group. Furthermore, the most frequently
occurring aphasia types can be distinguished with $90\%$ accuracy. The
pipeline is directly applicable to other diseases and languages, showing
promise for robustly extracting diagnostic speech biomarkers.

Index Terms: speech recognition, aphasia classification, interpretable speech
biomarkers

##  1 Introduction

Aphasia is defined as an acquired language disorder following brain injury
[1], for instance, after stroke or traumatic head injury. Following a stroke,
the most frequently occurring aphasia types are global, anomic, Wernicke’s,
and Broca’s aphasia [2]. Kirshner and Wilson [3] summarized the
characteristics according to these syndromes. In Broca's aphasia, the speech
pattern is nonfluent, often referred to as “agrammatic” or “telegraphic”.
Patients with Broca’s aphasia often make phonemic errors that are inconsistent
from utterance to utterance. In contrast, the speech pattern of patients with
Wernicke’s aphasia is effortless, sometimes even overly fluent, containing
verbal paraphasias, neologisms, and jargon productions. Auditory comprehension
is impaired, which often renders patients unconscious to their nonsense speech
productions. Global aphasia might be considered a summation of the symptoms
observed in Broca’s and Wernicke’s aphasia, characterized by nonfluent speech
paired with poor comprehension skills. Finally, in anomic aphasia, the main
deficit concerns word access or retrieval from the mental lexicon which is
characterized by pauses and circumlocutions. The distinction between these
four aphasia types is, however, not always that clear cut and, several mixed
aphasia types exist. In clinical practice, therapeutic strategies to approach
any language-related deficits have to be aligned with the particular symptoms
to maximize rehabilitative potential [4]. The accurate classification of
aphasia is therefore highly relevant. However, the process of correctly
evaluating these subtleties of speech and language can be quite time-
consuming. Employing automated approaches on speech recordings with
interpretable insights would enable clinicians to initiate speech therapy more
quickly, thus expediting rehabilitation.

Several attempts have been made to classify different aphasia types and other
speech pathologies from manual transcripts [5, 6]. Fraser et al. [5] conducted
a study with the objective of discerning between individuals diagnosed with
semantic dementia, progressive nonfluent aphasia and healthy controls,
utilizing features extracted from manual transcripts for syntactic complexity
and measures of fluency. Fromm et al. [6] employed the CHAT format manual
transcriptions [7] to extract features on subtasks of AphasiaBank [8] data and
study k-means clusters of these features. Other works have developed
approaches to detect aphasia automatically from spontaneous speech [9] [10].
Notably, Chatzoudis et al. [11] use the XLSR-53 model from Conneau et al.
[12], which was pre-trained on multiple languages using the contrastive self-
supervised task and wav2vec2.0 architecture from Baevski et al. [13] to
extract language-invariant linguistic features for classifying aphasia. These
models learn high-quality audio representations using Transformer encoder
blocks [14] and are successively fine-tuned for downstream tasks. To fine-tune
these models, Conneau et al. [12] have added a classifier representing the
output vocabulary of the respective downstream task on top of the XLSR-53
model by training on labeled datasets using a Connectionist Temporal
Classification (CTC) loss [15].

Even though the underlying wav2vec2.0 architecture contains multiple
Transformer layers, which can encode contextual information, fine-tuned
XLSR-53 models without an additional language model tend to produce more
literal, acoustic outputs than the encoder-decoder Whisper model from Radford
et al. [16], therefore leaving filler words and recurring utterances as
artifacts. Whisper was trained in a supervised fashion on weakly-supervised
large-scale raw text labels without significant text standardization, such
that the decoder part of Whisper works as an audio-conditioned language model
and eliminates many filler words, recurring utterances and other artifacts. We
propose to combine the output of these two models and apply carefully selected
feature extraction methods on aligned and annotated transcripts to build
prototypes of healthy speech. These prototypes are used to normalize features
for inference in order to create interpretable insights for anomalous speech
recordings.

The contributions of our work are as follows:

  * •

With a fully automated approach, we achieve human-level accuracy on
distinguishing speech recordings of aphasic patients from a healthy control
group.

  * •

To the best of our knowledge, it is the first approach exploiting the
different decoding mechanisms of CTC and encoder-decoder based methods to
create rich annotations for automatic feature extraction.

  * •

With a fully automated approach, we achieve over 90% accuracy on classifying
speech between a healthy control group, anomic, Wernicke’s and Broca’s
aphasia.

  * •

We provide several unseen scores for improving the accuracy of automatically
assessing speech anomalies.

##  2 Proposed method

In this section, we describe the proposed pipeline for creating robust speech
characteristics for assessing speech impairments. The pipeline consists of a
prototype construction and classification phase. Both phases use feature
extraction methods in three successive steps with the difference that the
prototype phase uses extracted features from a population of healthy speech
samples to estimate healthy speech distributions, while the classification
phase uses distance measures from these distributions to generate classifiable
features. We have used the 89 hours of free speech samples from the Europarl
speech translation corpus [17] to construct the healthy prototypes.

The three successive steps consist of first constructing acoustic and clean
transcripts leveraging the speech recognition models XLSR-53 and Whisper on
speech chunks as described in Section 2.1. We have used the large version of
XLSR-53 with 24 Transformer encoder blocks and the large version of Whisper.
Second, we enrich the aligned transcripts with fully-automated annotations for
pauses, filler words, and part-of-speech (POS) tags. Third, we are computing
scores for coherence, fluency, syntax, lexical richness, and pronunciation as
described in Section 2.2. For classification, speech recordings are fed into
the alignment, annotation and score pipeline to compute the distance between
each score and the score prototypes. These distances are then used for
training simple machine learning classifiers and are used for the experiments
in Section 3. The steps for computing prototypes are portrayed in Figure 1.

Figure 1: Alignment, annotation and score procedure

###  2.1 Alignment and annotation

The whole pipeline operates directly on speech waves, which are sampled at
$16\text{\,}\mathrm{kHz}$. The speech wave is fed into Whisper to produce
syntactically correct transcripts containing punctuations, removing single
utterances, repetitions and other artifacts, and into XLSR-53 to produce the
more literal and standardized outputs. In the following, we will call the
decoded XLSR-53 output acoustic and the Whisper output clean transcripts. Due
to the configuration of wav2vec2.0, tokens are decoded with a
$20\text{\,}\mathrm{ms}$ stride resulting in precise timing information on
token level and by extension on word level [18]. Whisper provides timings on a
sentence level only.

We combine the strengths of these two transcripts by aligning their
standardized output via the minimization of the Levenshtein distance [19] as
portrayed in Figure 1. This enables us to identify unmatched words or
utterances and tokens as potential filler words or recurring utterances and
allows us to augment the clean transcripts with accurate timings for the
matched tokens from the acoustic transcripts in a fully-automated fashion.

The recordings were segmented into contiguous speech chunks, each containing a
minimum of 200 words, based on clean transcriptions. To ensure accurate syntax
scores, we confirmed that the final word in each chunk corresponded to the end
of a sentence, with Whisper providing punctuation marks. The aligned acoustic,
accurate timings, and clean text data were used in our annotation process,
illustrated in Figure 1. In this step, we are augmenting the aligned
transcripts with POS tags using RoBERTa from Liu et al. [20], detect pauses
from silence tokens with more than $300\text{\,}\mathrm{ms}$ of duration and
filler words from timings and unmatched utterances. These augmented
transcripts are then fed into our score pipeline, as described in Section 2.2.

###  2.2 Scores for language abilities

The goal of the proposed scores is to capture multiple aspects of language
abilities as shown in Table 1. We accumulate the distribution of each score
and approximate it with a normal distribution, since they fit the empiric
distributions well and serve as prototypes for healthy speech.

Table 1: Scores for identifying speech characteristics. Language ability | Feature
---|---
Fluency | Words and phonemes per second
|  | Percentage time spoken
|  | Productive time ratio
|  | Pause length and distance
|  | Pause per word
|  | Mean phoneme length nouns
Lexical richness | TTR, MATTR
|  | gzip ratio
|  | HD-D, MTLD
|  | Word information
Syntax | Noun, verb and adjective ratio
|  | Grammar acceptance
|  | Mean sentence length
Pronunciation | Word error rate acoustic
|  | Character error rate acoustic
Coherence | CTRLEval
|  | Word vector coherence
|  | GPT2 perplexity

####  2.2.1 Fluency

The words per second score is computed on clean transcripts, while percentage
time spoken uses the aggregated time of the acoustic transcripts divided by
the total time of the provided audio chunk. For measuring the productive time
ratio, we divide the strict spoken time (discounting pauses) of the acoustic
transcript by the strict spoken time of the clean transcript. We count every
successive set of silence tokens from acoustic transcripts that are longer
than $300\text{\,}\mathrm{ms}$ as a pause to compute pause length scores.
Sampling all pause lengths as a distribution gives insights into the frequency
and duration of speech breaks. We use different quantiles (10, 25, 50, 75, 95)
of this distribution as features. Pause distance by contrast computes the time
between successive pauses and again, samples the pause distances as a
distribution and uses quantiles (10, 25, 50, 75, 95) as features. The pause
per word score is defined as the ratio of pauses longer than
$300\text{\,}\mathrm{ms}$ and the total number of words from the clean
transcript. We also included two phoneme scores, for which we use the
automatic phonemizer from Bernard and Titieux [21] to transform the clean and
acoustic text into phonemes. For mean phoneme length nouns, we aggregate and
average the lengths of all words tagged as nouns from clean transcripts.
Finally, the phonemes per second score is computed on the phonemized version
of clean transcripts.

####  2.2.2 Lexical richness

All lexical richness scores are computed on clean transcripts. We have
computed type token ratio (TTR) and MATTR [22] with window sizes of 10, 25 and
50 words. Using the intuition that redundant text is easier to compress, we
have used the gzip algorithm, creating gzip ratio as a measurement for
redundancy, dividing the length of compressed by uncompressed byte string
representations. Further we use HD-D and MTLD as described in [23], which were
proven to capture lexical diversity well, even under varying text length. Word
information is defined as the entropy of the normalized token count
distribution of the lemmatized tokens [24].

####  2.2.3 Syntax

All syntax scores are computed on clean transcripts. Noun, verb and adjective
ratio are simply the ratio of the specified POS tags to the number of words in
the clean transcript. The grammar acceptance feature is defined as the softmax
probability of a RoBERTa-large model [20] fine-tuned for grammar acceptability
on the CoLA Corpus [25]. Finally, we use the mean sentence length as an
indicator of noticeable short or sprawling phrases.

####  2.2.4 Pronunciation

The word/character error rate acoustic scores are the word/character error
rates between the acoustic and clean transcripts, intuitively capturing speech
intelligibility proportional to the diverging outputs of Whisper and XLSR-53.

####  2.2.5 Coherence

All coherence scores are evaluated on Whisper transcripts. CTRLEval is a
reference-free metric to assess the coherence for controlled text generation
tasks as detailed by Ke et al. [26]. For word vector coherence, we use the
cosine similarity to assess the local coherence between averaged word vectors
of neighboring sentences. Word vectors are taken from spaCy's [27]
en_core_web_sm corpus. At last, with the GPT2 perplexity score, we evaluate
the perplexity from GPT2 by Radford et al. [28] with a sliding window size of
512 tokens using the standard GPT2 tokenizer.

##  3 Experiments and results

###  3.1 Experimental setup

To validate our approach, we are using speech samples extracted from the
AphasiaBank database [8] for English. AphasiaBank comprises a collection of
interviews between clinicians and subjects afflicted with aphasia, as well as
healthy control subjects. These interviews are transcribed in the CHAT
transcription format [7] and include timestamps for interviewer and patient
speech segments. Most interviews contain a label indicating which form of
aphasia the subject is suffering from. There are $272$ individuals from the
control group, $136$ with anomic, $80$ with Broca's, $60$ conduction, $29$
with Wernicke's, $9$ with transmotor, $3$ individuals with global aphasia and
$35$ individuals who do not exhibit symptoms of aphasia as determined by the
Western Aphasia Battery (WAB-R) [29]. For $348$ subjects the (WAB-R) Aphasia
Quotient (AQ) is included. We remove all speech uttered by the interviewer as
given by the timestamps from the CHAT protocols and concatenate the resulting
audio clips, which are input to our pipeline. For each speech chunk, we
compute the scores from Section 2.2 as outlined in Figure 1 and average
multiple score vectors from the same interview. The distance for each averaged
score $s_{i}$ to the healthy prototype of this score, estimated by
$X\sim\mathcal{N}(\mu_{s},\,\sigma_{s}^{2})$, are then used as features for
classification and computed by $s_{i}=\frac{\sigma_{s}}{|\mu_{s}-s_{i}|}$, if
$|s_{i}-\mu_{s}|>\sigma_{s}$ else it is set to 1.

To ensure comparability with the results from Chatzoudis et al. [11], we adopt
a Leave-One-Subject-Out (LOSO) cross-validation strategy for all
classification experiments. We use a standard linear Support Vector Classifier
SVC [30] with regularization parameter $C=0.1$ and max_iter $=50000$ for
classification. We do not use task-specific feature selection and train on all
of our features. All experiments were run on a virtual machine using an
NVIDIA-A100 GPU.

###  3.2 Results

####  3.2.1 Aphasia versus Control

To facilitate a direct comparison with Chatzoudis et al. [11], we have
included the classification results of our approach for aphasic (including all
subtypes) versus healthy control patients in Table 2. The outcome of this
analysis reveals that our fully automatic method exhibits superior performance
relative to the one of [11], even when comparing ourselves to features
extracted directly from the manually annotated CHAT transcriptions.
Furthermore, since we classify on averaged chunks of similar length, instead
of the full transcript, we are not benefiting from the discriminatory
information provided by total word output. These findings provide compelling
evidence of the robustness and efficacy of our fully-automatic approach for
real-world applications.

Table 2: Classification metrics for aphasia versus healthy control group using linguistic features on AphasiaBank. Method | Manual annotation | Accuracy | $\mathbf{F_{1}}$
---|---|---|---
SVC [11] | ✗ | $93.9$ | -
SVC [11] | ✓ | $97.4$ | -
SVC [31] | ✓ | - | $85.9$
Ours (SVC) | ✗ | $\mathbf{98.6}$ | $\mathbf{98.5}$

####  3.2.2 Aphasia subtype classification

To validate the effectiveness and robustness of the feature extraction
pipeline, we use a linear SVC to differentiate three types of aphasia and the
healthy control group as a multi-class classification problem. We have used
three of the four main aphasia subtypes, i.e. anomic, Wernicke’s, and Broca’s
aphasia. We exclude global aphasia since there are only 3 people with this
diagnosis in AphasiaBank. Further details on the classification performance
are summarized in Table 3.

Table 3: Classification metrics for a 4-class classification problem differentiating between aphasia subtypes and the control group of the AphasiaBank dataset. Class | Precision | Recall | $\mathbf{F_{1}}$
---|---|---|---
Control | $97.5$ | $98.5$ | $98.0$
Anomic | $83.1$ | $86.8$ | $84.9$
Broca | $80.0$ | $75.0$ | $77.5$
Wernicke | $91.7$ | $68.8$ | $78.6$
Weighted average | $90.6$ | $90.6$ | $90.6$

####  3.2.3 Classification results per score category

Each score category should contribute meaningful information to the multi-
class subtype classification problem from section 3.2.2. To this end, we
trained a linear SVC that only utilizes the subset of scores from the
respective score category to differentiate the three aphasia subtypes and the
healthy control group. The results for the weighted average precision, recall
and $F_{1}$ scores for the aphasia subtypes and control group per score
category are condensed in Table 4.

Table 4: Weighted average precision, recall and $F_{1}$ scores for a 4-class classification problem differentiating between aphasia subtypes and the control group of the AphasiaBank dataset using only features from a single score category. Score Category | Precision | Recall | $\mathbf{F_{1}}$
---|---|---|---
Fluency | $81.8$ | $82.0$ | $81.6$
Coherence | $68.9$ | $71.0$ | $69.1$
Lexical Richness | $76.5$ | $77.9$ | $76.5$
Syntax | $76.9$ | $78.3$ | $77.4$
Pronunciation | $61.7$ | $67.3$ | $63.9$

####  3.2.4 WAB-R AQ Regression

The automatic estimation of the aphasia quotient (AQ) from spontaneous speech
has numerous potential benefits, including progress monitoring without the
need for frequent repetition of the WAB-R assessment procedures. We follow the
evaluation procedure from Le et al. [32] and frame the WAB-R AQ prediction as
a regression problem, with our proposed score distances as features and the AQ
scores as the target labels for the $348$ individuals, for which the AQ was
included. The results are portrayed in Table 5.

Table 5: WAB-R AQ Prediction. MAE=Mean Absolute Error, PC = Pearsons's Correlation, SVR = Support Vector Regression Method | Manual annotation | PC | MAE
---|---|---|---
SVR [32] | ✗ | $0.776$ | $9.90$
SVR [32] | ✓ | $0.787$ | $9.54$
Ours (SVR) | ✗ | $\mathbf{0.815}$ | $\mathbf{8.19}$

##  4 Conclusions

The proposed approach showed high level of robustness for identifying aphasia
and aphasic subtypes from free speech recordings and presents a pipeline that
can be adapted to other diseases and languages with minimal effort. We used
the same pipeline to evaluate the proposed set of features for a dementia
classification challenge, beating the baseline by a large margin. In the near
future, we want to investigate the distance measures from healthy prototypes
to build accurate and interpretable speech biomarkers for multiple diseases.
Also, the optimal number of words per chunk for practical applications, the
transferability of features across multiple languages and the correlation
between multiple features need further investigation.

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# Thermal Casimir effect in the Einstein Universe with a spherical boundary

1 H. F. S. Mota<EMAIL_ADDRESS>2C. R. Muniz<EMAIL_ADDRESS>1 V.
B. Bezerra<EMAIL_ADDRESS>1Departamento de Física, Universidade
Federal da Paraíba,
Caixa Postal 5008, João Pessoa, Paraíba, Brazil.  2 Universidade Estadual do
Ceará, Faculdade de Educação, Ciências e Letras de Iguatu, Av. Dário Rabelo,
s/n, 63500-000, Iguatu, CE, Brazil.

###### Abstract

In the present paper we investigate thermal fluctuation corrections to the
vacuum energy at zero temperature of a conformally coupled massless scalar
field whose modes propagate in the Einstein universe with a spherical
boundary, characterized by both Dirichlet and Neumann boundary conditions.
Thus, we generalize the results found in literature in this scenario, which
has considered only the vacuum energy at zero temperature. To do this, we use
the generalized zeta function method plus Abel-Plana formula and calculate the
renormalized Casimir free energy as well as other thermodynamics quantities,
namely, internal energy and entropy. For each one of them we also investigate
the limits of high and low temperatures. At high temperatures we found that
the renormalized Casimir free energy presents classical contributions, along
with a logarithmic term. Also in this limit, the internal energy presents a
classical contribution and the entropy a logarithmic term in addition to a
classical contribution as well. Conversely, at low temperatures, it is shown
that both the renormalized Casimir free energy and internal energy are
dominated by the vacuum energy at zero temperature. It is also shown that the
entropy obeys the third law of thermodynamics.

##  I Introduction

The Einstein universe is a particular solution of the general relativity
equations, which originally describes a spatially closed universe with
constant positive curvature and uniform distribution of matter on the largest
scales. It is in fact the Friedman-Robertson-Walker (FRW) spacetime that
characterizes an universe under the assumption of the cosmological principle,
but with a constant scale factor Ford:1975su ; Ford:1976fn ; Bezerra:2011zz ;
Bezerra:2011nc ; Ozcan:2001cr ; Bayin:1993yq ; Kennedy:1980kc . Hence, the
Einstein universe is a model describing a closed static universe.

Such a cosmological model is certainly idealized, considering recent
observations related to the current universe geometry, which appears to be
flat in its overall scale Planck:2018vyg . However, more realistic
cosmological scenarios where closed static universes play a central role are
described by emergent models which consider the Einstein universe as the
suited geometry to characterize the stage preceding the inflationary epoch.
During this age, close to the Planck era, the non-negligible vacuum
fluctuations associated with quantum relativistic fields may provide robust
contributions to determine parameters for those models Ellis:2002we ;
Ellis:2003qz . Furthermore, modified theories that point towards the quantum
gravity has also been investigated taking into consideration the Einstein
universe geometry Boehmer:2009fey ; Boehmer:2007tr ; Li:2013xea ;
Bezerra:2017zqq ; Shabani:2016dhj .

The Casimir effect is a phenomenon associated with vacuum fluctuations of
quantum-relativistic fields, and was originally thought of as caused by
modifications, in Minkowski spacetime, of the zero-point oscillations of the
electromagnetic field due to the presence of two ideal parallel conducting
plates, at zero temperature Casimir1948dh . The present status of the
phenomenon is that it can also occur due to other fields (such as the scalar,
spinor, and Elko fields), with thermal influences, and also due to nontrivial
topologies associated with different geometries of the spacetime (for reviews,
see Milton:2001yy ; bordag2009advances ). As an iconic example, we cite
wormholes Khabibullin:2005ad ; Sorge:2019kuh ; Garattini:2019ivd ;
Santos:2020taq ; Santos:2021jjs ; Alencar:2021ejd , in which exotic negative
energies like the Casimir one would be even more necessary in order to
stabilize them.

The Casimir effect arising as a consequence of the nontrivial topology of the
spacetime has been considered, for instance, in Refs. Dowker:1978vy ;
DeWitt:1979dd ; Lima:2006rr . In the static Einstein universe (as well as in
the FRW one), the effect has been investigated in a variety of works
Ford:1975su ; Ford:1976fn ; Bezerra:2011zz ; Bezerra:2011nc ; Ozcan:2001cr ;
Bayin:1993yq ; Kennedy:1980kc ; Zhuk:1996xc ; Bezerra:2014pza ; Mota:2015ppk ;
Bezerra:2016qof ; Bezerra:2021qnw ; Herdeiro:2007eb , including thermal
correction analysis. The Casimir effect in the Einstein universe was also
investigated in the presence of a spherical boundary Bayin:1993yq ;
Kennedy:1980kc , as well as in a D-dimensional spacetime Herdeiro:2007eb ;
Ozcan:2001cr . In general, the role of the quantum vacuum energy in the
primordial universe is highlighted in such scenarios. Therefore, as the early
stages of the universe are known to have been extremely hot, an investigation
of thermal corrections to the vacuum energy at zero temperature in the
Einstein universe becomes of great interest. The Casimir effect is also
important to be explored in curved spacetimes, like the one intended here,
from the fundamental point of view, with the aim to investigate the role
played by the spacetime nontrivial topology. In this work, we investigate
thermal fluctuation corrections to the vacuum energy at zero temperature of a
conformally coupled massless scalar field whose modes propagate in the
Einstein universe with a spherical boundary. The latter is characterized by
both Dirichlet and Neumann boundary conditions. We, thus, generalized the
results found in Refs. Bayin:1993yq ; Kennedy:1980kc  where the authors
considered the vacuum energy at zero temperature by adopting the cutoff
method. We show that there exists remarkable differences between thermal
Casimir quantities obtained in these two cases, unlike the scenario of zero
temperature analysed in Refs. Bayin:1993yq ; Kennedy:1980kc , where the vacuum
energy is the same for both boundary conditions mentioned.

The structure of the paper is as follows: In Section II, we present the
Einstein universe spacetime with the spherical boundary and the zeta function
method to calculate the temperature corrections. In Section III, we obtain
finite temperature contributions that must be subtracted in order to define
the renormalized Casimir free energy. In Section IV, we calculate the
renormalized thermodynamic quantities and the corresponding asymptotic limits.
Finally, in Section V, we present our conclusions.

##  II Einstein Universe and generalized zeta function method to calculate
thermal corrections

In this section we present the spacetime we intend to work with, i.e., the
Einstein universe with a spherical boundary, as well as go through some
elements of the generalized zeta function method to obtain temperature
corrections to the vacuum energy at zero temperature. This Einstein universe
is different from the one considered in Refs. Bezerra:2011zz ; Bezerra:2011nc
in the sense that now it is viewed as a geometry inside a spherical boundary
as suggested by the authors in Ref. Bayin:1993yq  (see also Ozcan:2001cr ).
The topology of the spacetime is $S^{3}\times R^{1}$, but with $S^{3}$
equatorially bounded by $S^{2}$ Kennedy:1980kc . The spherical boundaries
considered in Refs. Bayin:1993yq ; Kennedy:1980kc  are the ones characterized
by Dirichlet and Neumann boundary conditions satisfied by a conformally
coupled massless scalar field, as we shall see below.

###  II.1 Einstein universe with a spherical boundary

The line element describing the Einstein universe inside a spherical boundary,
in accordance with the authors in Refs. Bayin:1993yq ; Ozcan:2001cr , is given
by

$\displaystyle
ds^{2}=c^{2}dt^{2}-a^{2}\left[d\chi^{2}+\sin^{2}\chi\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}\right)\right],$ (1)

where $-\infty<t<\infty$ and $0\leq\chi\leq\frac{\pi}{2}$,
$0\leq\theta\leq\pi$, $0\leq\varphi\leq 2\pi$ are dimensionless coordinates on
a three-space of constante curvature +1. The parameter $a$ is the constant
scale factor characterizing the static closed metric and $c$, of course, is
the speed of light. The finite spatial volume associated with the model
considered here can be calculated as

$\displaystyle V$ $\displaystyle=$ $\displaystyle\int\sqrt{|g^{(3)}|}d\chi
d\theta d\varphi$ (2) $\displaystyle=$ $\displaystyle
a^{3}\int_{0}^{\frac{\pi}{2}}\sin^{2}\chi d\chi\int_{0}^{\pi}\sin\theta
d\theta\int_{0}^{2\pi}d\varphi=\pi^{2}a^{3},$

where $g^{(3)}$ is the determinant of the spatial part of the metric related
to the line element (1). Note that, in the absence of the spherical boundary,
the spatial volume of the Einstein universe is given by $2\pi a^{3}$
Bezerra:2011zz ; Bezerra:2011nc ; Bezerra:2014pza ; Mota:2015ppk , that is,
twice the one in Eq. (2), as it should be.

We wish to consider a real massless quantum scalar field, $\phi$, conformally
coupled to the geometry codified in the metric present in the line element
(1). The massless scalar field, hence, obeys the equation

$\displaystyle\Box\phi+\xi R\phi=0,$ (3)

where $\Box$ is the D’Alembertian operator in curved spacetime MotaDispiration
, $R=6a^{-2}$ is the scalar curvature for the Einstein universe geometry and
$\xi$ is the curvature coupling. As has been said, a conformally invariant
theory is to be considered in the present work, in which case
$\xi=\frac{1}{6}$. The solution of the Klein-Gordon equation (3) for a
conformally coupled massless scalar field is given in terms of Gegenbauer
functions and spherical harmonics. This has been obtained previously, for
instance, in Refs. Ford:1975su ; Ford:1976fn ; Bayin:1993yq . We are in fact
interested in the eigenfrequencies associated with such a solution when
submitted to Dirichlet and Neumann boundary conditions at
$\chi=\frac{\pi}{2}$, characterizing spherical boundaries Bayin:1993yq ;
Kennedy:1980kc ; Ozcan:2001cr  . The eigenfrequencies in this case, along with
the degeneracy $g_{n}$, are

$\displaystyle\omega_{n}=\frac{nc}{a}\qquad\qquad g_{n}=\frac{1}{2}n(n\pm
1)\qquad\qquad n=1,2,3,...,$ (4)

where the minus sign stands for Dirichlet and the plus sign for Neumann
boundary conditions. The expression above will be useful to calculate
temperature corrections to the vacuum energy at zero temperature already
obtained in Refs. Bayin:1993yq ; Kennedy:1980kc ; Ozcan:2001cr .

###  II.2 Zeta function method and temperature corrections

Let us now briefly point out some necessary elements of the zeta function
method to implement temperature corrections. In this spirit, an operator
$\hat{A}_{4}$, in a four-dimensional Euclidian spacetime111In essence, a four-
dimensional Euclidian spacetime is reached by making $t=-i\tau$ in Eqs. (1)
and (3)., with eigenvalues $\lambda_{\sigma}$, has an associated generalized
zeta function written as Hawking1977 ; Elizalde1994book ; Aleixo:2021cfy

$\zeta_{4}(s)=\sum_{\sigma}\lambda^{-s}_{\sigma},$ (5)

where $\sigma$ stands for the spectrum of eigenvalues222The set of quantum
modes in our case. of $\hat{A}_{4}$, which may not always be discrete. Note
that the expression above converges, in four dimensions, for Re$(s)>2$ and it
is regular at $s=0$. It can also be analytically extended to a function of
$s$, with poles at $s=2$ and $s=1$ Elizalde1994book .

The analytic extension of the generalized zeta function defined in Eq. (5)
carrying temperature terms has already been known and is written as
Hawking1977 ; Elizalde1994book ; Aleixo:2021cfy

$\zeta_{4}(s)=\frac{\beta}{\sqrt{4\pi}\Gamma(s)}\left\\{\Gamma(s-1/2)\zeta_{3}(s-1/2)+2\sum_{n=1}^{\infty}\int^{\infty}_{0}\eta^{s-\frac{3}{2}}e^{-\frac{(n\beta)^{2}}{4\eta}}\text{Tr}\left[e^{-\eta\hat{A}_{3}}\right]d\eta\right\\},$
(6)

where $\hat{A}_{3}$ is a differential operator defined as the spatial part of
$\hat{A}_{4}$, $\zeta_{3}(s-1/2)$ is the generalized zeta function written in
terms of the eigenvalues of $\hat{A}_{3}$ and $\beta=\frac{1}{k_{B}T}$, with
$k_{B}$ being the Boltzmann constant and $T$ the temperature.

The partition function is, thus, given in terms of the zeta function (6) and
its derivative with respect to $s$, both calculated at $s=0$, as follows

$\displaystyle\ln
Z=\frac{1}{2}\zeta_{4}^{\prime}(0)+\frac{1}{2}\ln\left(\frac{\pi\mu^{2}}{4}\right)\zeta_{4}(0),$
(7)

where $\mu$ is a constant with dimension of mass. Hence, the free energy as
usual is defined as Aleixo:2021cfy

$\displaystyle F$ $\displaystyle=$ $\displaystyle-\frac{\ln Z}{\beta}$ (8)
$\displaystyle=$
$\displaystyle\frac{1}{2}\zeta_{3}(-1/2)-\frac{1}{\sqrt{4\pi}}\sum_{n=1}^{\infty}\int^{\infty}_{0}\eta^{-\frac{3}{2}}e^{-\frac{(n\beta)^{2}}{4\eta}}\text{Tr}\left[e^{-\eta\hat{A}_{3}}\right]d\eta.$

Furthermore, if one assumes that $\epsilon^{2}_{j}$ are the eigenvalues of the
three-dimensional spatial operator, $\hat{A}_{3}$, then, the trace above is
given by

$\displaystyle\text{Tr}\left[e^{-\eta\hat{A}_{3}}\right]=\sum_{j}e^{-\epsilon^{2}_{j}\eta},$
(9)

where $j$ stand for the quantum modes associated with the operator
$\hat{A}_{3}$. Hence, by performing the integration in $\eta$ present in Eq.
(8), and taking the sum in $n$ afterwards, we obtain

$\displaystyle F^{\text{ren}}$ $\displaystyle=$ $\displaystyle
E_{0}^{\text{ren}}+F_{T}$ (10) $\displaystyle=$
$\displaystyle\frac{1}{2}\zeta_{3}(-1/2)+\frac{1}{\beta}\sum_{j}\ln\left(1-e^{-\epsilon_{j}\beta}\right).$

Note that, in our case, $\epsilon_{j}=\hbar\omega_{j}$, where $\omega_{j}$
stands for the eigenfrequencies, (4), associated with the conformally coupled
massless scalar field subjected to spherical boundary conditions and $\hbar$
is the Planck’s constant. The operator $\hat{A}_{3}$ is the spatial part of
Eq. (3) plus the curvature coupling. Note also that the zeta function
$\zeta_{3}(s)$, through the parameter $s$, already regularizes and
consequently provides a renormalized vacuum energy at zero temperature given
by the first term on the r.h.s. of Eq. (10). The second term on the r.h.s.,
though, accounts for temperature corrections.

At high temperatures the correction $F_{T}$, obtained in a given restricted
volume $V$, may contain terms of quantum nature with the following structure:

$\displaystyle\alpha_{0}\frac{(k_{B}T)^{4}}{(\hbar
c)^{3}},\qquad\qquad\alpha_{1}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}},\qquad\qquad\alpha_{2}\frac{(k_{B}T)^{2}}{\hbar c},$ (11)

where $\alpha_{0},\alpha_{1},\alpha_{2}$ have, respectively, dimensions of
cm3, cm2 and cm and are given in terms of the heat kernel coefficients
Bezerra:2011zz ; Bezerra:2011nc ; Geyer:2008wb ; Dowker:1978md . It has been
argued that these terms should be subtracted from $F_{T}$ in order to provide
the dominant and correct classical limit at high temperatures, which is
independent of Planck’s constant. This is in accordance with investigations
conducted for material bodies, within the framework of Lifshitz theory
Lifshitz:1956zz ; Bezerra:2011zz ; Bezerra:2011nc ; Geyer:2008wb . Thereby,
$F_{T}$ must go through a finite renormalization process. This subtraction is
also motivated by the fact that the coefficients
$\alpha_{0},\alpha_{1},\alpha_{2}$ have the same geometric structure as in the
corresponding infinite terms to be subtracted from the vacuum energy at zero
temperature, as shown for instance in Ref. Geyer:2008wb , in the case of
retangular boxes. In the case of the Einstein universe considering the
conformally coupled massless scalar field investigated in Ref. Bezerra:2011zz
, it has been shown that the only contribution is the one for which
$\alpha_{0}\neq 0$, that is,

$\displaystyle\alpha_{0}=-\frac{\pi^{2}}{90}V,\qquad\qquad\alpha_{1}=0,\qquad\qquad\alpha_{2}=0.$
(12)

However, as we shall see in the next section, in the case of a conformally
coupled massless scalar field in the Einstein universe with spherical
boundaries, there also exists an additional contribution for which
$\alpha_{1}\neq 0$, besides the one in Eq. (12), associated with the black-
body radiation. In this case

$\displaystyle\alpha_{0}=-\frac{\pi^{2}}{90}V,\qquad\qquad\alpha_{1}=\mp\frac{\zeta(3)a^{2}}{2},\qquad\qquad\alpha_{2}=0,$
(13)

where we should emphasize that the up sign always stands for Neumann boundary
condition. Therefore, in our case, it should be subtracted from $F_{T}$ not
only the black-body radiation contribution but also the contribution
proportional to $(k_{B}T)^{3}$, with coefficient $\alpha_{1}\neq 0$ given
above.

The free energy in Eq. (10) allows us to calculate other thermodynamics
quantities such as the internal energy given by

$U=-T^{2}\frac{\partial}{\partial T}\left(\frac{F}{T}\right),$ (14)

as well as the entropy

$S=-\frac{\partial F}{\partial T}.$ (15)

Therefore, along with the free energy (10), the last two thermodynamics
quantities complement our analysis. In the next section we provide analytical
expressions, for each one of them, suitable to investigate the limits of high
and low temperatures.

##  III Finite renormalization terms and renormalized Casimir free energy

As it has been remarked in the previous section, the first term on the r.h.s
of the free energy (10) provides the renormalized vacuum energy at zero
temperature. Thus, by making use of the eigenfrequencies and their degeneracy
in Eq. (4) we are able to show that333We call attention to the fact that
$\zeta_{3}(s)$ is defined by (5), with $\lambda_{j}=\omega_{j}^{2}$ and $j=n$.

$\displaystyle E^{\text{ren}}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\zeta_{3}(-1/2)$ (16) $\displaystyle=$
$\displaystyle\frac{\hbar c}{480a}.$

This result has already been obtained previously by the authors in Refs.
Bayin:1993yq  by making use of the cutoff method. Here we have shown that the
generalized zeta function method also provides the same result, as it should
be. The result is the same for Dirichlet and Neumann boundary conditions. This
is because the term $n$ in the degeneracy gives a contribution that is
$\zeta(-2)=0$. Note that the vacuum energy given above is half of the one for
the Einstein universe without spherical boundary considered in Refs.
Bezerra:2011zz ; Bezerra:2011nc ; Bezerra:2014pza ; Mota:2015ppk .

As to temperature corrections, we need to investigate the second term on the
r.h.s of the free energy (10), taking into account the eigenfrequencies in
(4), that is,

$\displaystyle
F_{T}=\frac{1}{\beta}\sum_{n=1}^{\infty}g_{n}\ln\left(1-e^{-\hbar\omega_{n}\beta}\right).$
(17)

In order to perform the sum in $n$ above we can make use of the Abel-Plana
formula Saharian:2000xx

$\sum_{n=1}^{\infty}\Phi(n)-\int_{0}^{\infty}\Phi(t)dt=-\frac{1}{2}\Phi(0)+i\int_{0}^{\infty}\frac{\Phi(it)-\Phi(-it)}{\exp{(2\pi
t)-1}}dt.$ (18)

For our discussion we use Eq. (17) to make the following identification:

$\Phi(n)=\frac{k_{B}T}{2}\sum_{n=1}^{\infty}(n^{2}\pm
n)\ln\left(1-e^{-\frac{\hbar cn}{ak_{B}T}}\right).$ (19)

As a consequence, $\Phi(0)=0$, and the integral in the l.h.s of the Abel-Plana
formula (18) provides

$\displaystyle\int_{0}^{\infty}\Phi(t)=-\frac{\pi^{2}}{90}V\frac{(k_{B}T)^{4}}{(\hbar
c)^{3}}\mp\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar c)^{2}},$ (20)

where the spatial volume $V$ is given by Eq. (2). We can see that the first
term on the r.h.s is the black-body radiation contribution, also present for
the Einstein universe case without boundary as analyzed in Ref. Bezerra:2011zz
. Here, as a consequence of the spherical boundaries, a second contribution
comes about, given by the second term in the r.h.s. of Eq. (20). As discussed
in Ref. Bezerra:2011zz , and already mentioned in the previous section, these
two contributions must be subtracted from (17), allowing us to adopt a finite
renormalization process that provides the correct classical limit at high
temperatures.

Let us then define the renormalized temperature correction term by subtracting
the contributions in Eq. (20) from $F_{T}$ in Eq. (17), i.e.,

$\displaystyle F^{\text{ren}}_{T}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\Phi(n)-\int_{0}^{\infty}\Phi(t)dt$ (21)
$\displaystyle=$ $\displaystyle
i\int_{0}^{\infty}\frac{\Phi(it)-\Phi(-it)}{\exp{(2\pi t)-1}}dt.$

Now, by using the following series expansion for the logarithmic function,

$\displaystyle\ln(1-z)=-\sum_{n=1}^{\infty}\frac{z^{n}}{n},$ (22)

we are able to obtain, by considering Eq. (19), part of the integrand present
in the r.h.s of (21), in the form

$\displaystyle\Phi(it)-\Phi(-it)=$ $\displaystyle-$ $\displaystyle
ik_{B}Tt^{2}\sum_{n=1}^{\infty}\frac{1}{n}\sin\left(\frac{\hbar
cnt}{ak_{B}T}\right)$ (23) $\displaystyle\mp$ $\displaystyle
ik_{B}Tt\sum_{n=1}^{\infty}\frac{1}{n}\cos\left(\frac{\hbar
cnt}{ak_{B}T}\right).$

Hence, the renormalized temperature correction, (21), can be re-written as

$\displaystyle F^{\text{ren}}_{T}$ $\displaystyle=$ $\displaystyle
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\int_{0}^{\infty}\frac{t^{2}\sin\left(\frac{\hbar
cnt}{ak_{B}T}\right)}{\exp{(2\pi t)-1}}dt$ (24) $\displaystyle\pm$
$\displaystyle
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\int_{0}^{\infty}\frac{t\cos\left(\frac{\hbar
cnt}{ak_{B}T}\right)}{\exp{(2\pi t)-1}}dt.$

The result above, thus, allows us to obtain the renormalized Casimir free
energy of the conformally coupled massless scalar field in the Einstein
universe with spherical boundaries as given by

$\displaystyle F^{\text{ren}}_{\text{C}}=\frac{\hbar
c}{480a}+F^{\text{ren}}_{T}.$ (25)

This is in fact the expression in Eq. (10) when the contributions in Eq. (20)
are subtracted from $F_{T}$. Note that we could straightforwardly solve the
integrals in Eq. (24), which would lead to an expression adequately convenient
only to analyze the low-temperature limit of the renormalized Casimir free
energy defined above. Instead, we wish to obtain a result that is able to
provide us with appropriate expressions to consider the limits of both, high
and low temperatures, something that will be made in the next section.

##  IV Renormalized thermodynamic quantities and asymptotic limits

In this section we are interested in deriving analytic expressions for the
renormalized Casimir free energy that will allow us more conveniently to
investigate the limits of low and high temperatures. In order to do that let
us firstly consider the identity

$\displaystyle\frac{1}{e^{2\pi t}-1}=\sum_{\ell=1}^{\infty}e^{-2\pi\ell t}.$
(26)

This makes possible to write Eq. (24) alternatively as

$\displaystyle F^{\text{ren}}_{T}$ $\displaystyle=$ $\displaystyle
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\ell=1}^{\infty}\int_{0}^{\infty}t^{2}\sin\left(\frac{\hbar
cnt}{ak_{B}T}\right)e^{-2\pi\ell t}dt$ (27) $\displaystyle\pm$ $\displaystyle
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\ell=1}^{\infty}\int_{0}^{\infty}t\cos\left(\frac{\hbar
cnt}{ak_{B}T}\right)e^{-2\pi\ell t}dt.$

Consequently, the integrals above can be performed and the result is given in
terms of a double sum by

$\displaystyle F^{\text{ren}}_{T}$ $\displaystyle=$ $\displaystyle\frac{2\hbar
c}{a}\sum_{n=1}^{\infty}\sum_{\ell=1}^{\infty}\frac{3(2\pi\ell)^{2}-\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}}{\left[(2\pi\ell)^{2}+\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}\right]^{3}}\pm
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\ell=1}^{\infty}\frac{(2\pi\ell)^{2}-\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}}{\left[(2\pi\ell)^{2}+\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}\right]^{2}}.$ (28)

This is an exact and convergent result. As we shall see below, by performing
first the sum in $\ell$ we obtain a convenient expression to investigate the
low-temperature limit of (28). Conversely, by performing first the sum in $n$
we obtain a convenient expression to investigate the high-temperature limit.
Note that the first term on the r.h.s has already been analyzed444This term
comes from the $\frac{n^{2}}{2}$ part of the degeneracy in Eq. (4). Apart from
the factor 1/2, $n^{2}$ is exactly the degeneracy of the vacuum energy in the
Einstein universe considered in Ref. Bezerra:2011zz . in Ref. Bezerra:2011zz
and we do not intend to revisit this again. We will only quote the results
when necessary. We are in fact interested in investigating the effects of the
second term in the r.h.s. of Eq. (28), which from now on we denote as

$\Delta F_{T}^{\text{ren}}=\pm
k_{B}T\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\ell=1}^{\infty}\frac{(2\pi\ell)^{2}-\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}}{\left[(2\pi\ell)^{2}+\left(\frac{\hbar
cn}{ak_{B}T}\right)^{2}\right]^{2}}.$ (29)

This nonzero temperature contribution arises from the $\frac{n}{2}$ part of
the degeneracy in Eq. (4), which as we have seen does not contribute to the
vacuum energy at zero temperature but it is somewhat relevant for temperature
corrections.

Let us firstly perform the summation in $\ell$ now. For this purpose we may
re-write Eq. (29) as

$\Delta
F_{T}^{\text{ren}}=\pm\frac{k_{B}T}{(2\pi)^{2}}\sum_{n=1}^{\infty}\frac{1}{n}\sum_{\ell=1}^{\infty}\frac{\ell^{2}-\frac{n^{2}}{(2\pi\nu)^{2}}}{\left[\ell^{2}+\frac{n^{2}}{(2\pi\nu)^{2}}\right]^{2}},$
(30)

where we have defined the dimensionless parameter
$\nu\equiv\frac{ak_{B}T}{\hbar c}$. Consequently, the summation in $\ell$ can
be performed by using the following formula:

$\sum_{\ell=1}^{\infty}\frac{\ell^{2}-q^{2}}{\left(\ell^{2}+q^{2}\right)^{2}}=\frac{1}{2q^{2}}\left[1-q^{2}\pi^{2}\text{csch}^{2}(\pi
q)\right],$ (31)

where in our case $q=\frac{n}{2\pi\nu}$. Thus, using the result given by Eq.
(31) in Eq. (30), we find

$\displaystyle\Delta
F_{T}^{\text{ren}}=\pm\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\mp\frac{k_{B}T}{8}\sum_{n=1}^{\infty}\frac{1}{n}\text{csch}^{2}\left(\frac{\hbar
cn}{2ak_{B}T}\right),$ (32)

where $\zeta(s)$ is the Riemann zeta function, calculated at $s=3$
Elizalde1994book . The corresponding renormalized internal energy associated
with the temperature contribution in Eq. (32) is obtained by making use of Eq.
(14), i.e.,

$\displaystyle\Delta U^{\text{ren}}=\mp\zeta(3)a^{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\pm\frac{\hbar c}{8a}\sum_{n=1}^{\infty}\coth\left(\frac{\hbar
cn}{2ak_{B}T}\right)\text{csch}^{2}\left(\frac{\hbar cn}{2ak_{B}T}\right).$
(33)

In addition, from Eq. (15), we can also calculate the corresponding
renormalized entropy associated with (32). This is given by

$\displaystyle\Delta
S^{\text{ren}}=\mp\frac{3\zeta(3)k_{B}}{2}a^{2}\frac{(k_{B}T)^{2}}{(\hbar
c)^{2}}\pm\frac{k_{B}}{8}\sum_{n=1}^{\infty}\text{csch}^{2}\left(\frac{\hbar
cn}{2ak_{B}T}\right)\left[\frac{1}{n}+\frac{\hbar
c}{ak_{B}T}\coth\left(\frac{\hbar cn}{2ak_{B}T}\right)\right].$ (34)

Let us consider again Eq. (29) now written in the form

$\Delta F_{T}^{\text{ren}}=\pm
k_{B}T\nu^{2}\sum_{\ell=1}^{\infty}\sum_{n=1}^{\infty}\frac{1}{n}\frac{(2\pi\nu\ell)^{2}-n^{2}}{\left[(2\pi\nu\ell)^{2}+n^{2}\right]^{2}}.$
(35)

Thereby, the summation in $n$ is performed by using the formula

$\sum_{n=1}^{\infty}\frac{1}{n}\frac{p^{2}-n^{2}}{\left(p^{2}+n^{2}\right)^{2}}=\frac{1}{2p^{2}}\left\\{2\gamma_{\text{E}}+\psi\left(1-ip\right)+\psi\left(1+ip\right)+ip\left[\psi^{(1)}\left(1-ip\right)-\psi^{(1)}\left(1+ip\right)\right]\right\\},$
(36)

where $\gamma_{\text{E}}\simeq 0.577216$ is the Euler’s constant and
$\psi^{(k)}(z)$ are the polygamma functions, with $\psi^{0}(z)=\psi(z)$ being
the digamma function abramowitz ; gradshteyn2000table . Thus, since in our
case $p=2\pi\nu\ell$, Eqs. (35) and (36) provide

$\displaystyle\Delta F_{T}^{\text{ren}}=\pm\frac{\gamma_{\text{E}}}{24}k_{B}T$
$\displaystyle\pm$
$\displaystyle\frac{k_{B}T}{8\pi^{2}}\sum_{\ell=1}^{\infty}\frac{1}{\ell^{2}}\left\\{\psi\left(1-i2\pi\nu\ell\right)+\psi\left(1+i2\pi\nu\ell\right)\right.$
(37)
$\displaystyle\left.+i2\pi\nu\ell\left[\psi^{(1)}\left(1-i2\pi\nu\ell\right)-\psi^{(1)}\left(1+i2\pi\nu\ell\right)\right]\right\\}.$

We can note that there exists a classical contribution proportional to
$k_{B}T$ in the exact expression above, something that we would expect to
appear only in the high-temperature limit. Moreover, the corresponding
internal energy, from Eqs. (14) and (37) is found to be

$\displaystyle\Delta U^{\text{ren}}=\mp\frac{1}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}a^{2}\sum_{\ell=1}^{\infty}\left[\psi^{(2)}\left(1-i2\pi\nu\ell\right)+\psi^{(2)}\left(1+i2\pi\nu\ell\right)\right].$
(38)

Likewise, the associated entropy is obtained by considering Eqs. (15) and
(37), which gives us the following result

$\displaystyle\Delta S^{\text{ren}}=$ $\displaystyle\mp$
$\displaystyle\frac{k_{B}}{8\pi^{2}}\sum_{\ell=1}^{\infty}\frac{1}{\ell^{2}}\left\\{2\gamma_{\text{E}}+\psi\left(1-i2\pi\nu\ell\right)+\psi\left(1+i2\pi\nu\ell\right)\right.$
(39)
$\displaystyle\left.+i2\pi\nu\ell\left[\psi^{(1)}\left(1-i2\pi\nu\ell\right)-\psi^{(1)}\left(1+i2\pi\nu\ell\right)\right]\right.$
$\displaystyle\left.+(2\pi\nu\ell)^{2}\left[\psi^{(2)}\left(1-i2\pi\nu\ell\right)+\psi^{(2)}\left(1+i2\pi\nu\ell\right)\right]\right\\}.$

It is important to keep in mind that the expressions (32) and (37) are
equivalent, which is equally true for the corresponding expressions for the
internal energy and entropy derived above. Nevertheless, the expression (32)
is convenient to investigate the low-temperature limit whilst the expression
(37) to investigate the high-temperature limit. The results derived above,
associated with the temperature contribution (29), are to be considered along
with the results obtained in Ref. Bezerra:2011zz  for the first term on the
r.h.s. of (28) to calculate the renormalized Casimir free energy (25). Due to
its lengthy nature, we will not write the whole expression here. The analysis
of the asymptotic limits at high and low temperatures is to be considered in
the next (sub-) sections.

To end the discussion of the thermodynamics quantities let us mention another
one, namely, the pressure, calculated as minus the derivative of the
renormalized Casimir free energy with respect to the volume of the spacetime.
In analogy with what has bee done in Refs. Bezerra:2011zz ; Bezerra:2011nc  it
is straightforward to show in the case we are considering that an equation of
state is also satisfied, i.e.,

$P_{\text{C}}=\frac{\epsilon^{\text{ren}}}{3},$ (40)

where $\epsilon^{\text{ren}}=\frac{U_{\text{C}}^{\text{ren}}}{V}$ is the
renormalized internal energy density, with $V$ given by Eq. (2).

###  IV.1 Low-temperature limit

The first limiting case to be considered is the one of high temperature, that
is, $k_{B}T\ll\frac{\hbar c}{a}$. In this case, the most convenient
expressions to perform this analysis are given in Eqs. (32), (33) and (34).
Hence, the mentioned asymptotic limit for the free energy temperature
contribution in Eq. (32) is written as

$\Delta
F_{T}^{\text{ren}}\simeq\pm\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\mp\frac{k_{B}T}{2}e^{-\frac{\hbar c}{ak_{B}T}}.$ (41)

Therefore, the low-temperature limit for the renormalized Casimir free energy
takes into consideration the renormalized free energy in Eq. (28). In the
latter, the low-temperature limit for the first term in the r.h.s has been
considered in Ref. Bezerra:2011zz . Thus, the low-temperature limit for the
renormalized Casimir free energy (25) is written as

$\displaystyle F^{\text{ren}}_{\text{C}}$ $\displaystyle\simeq$
$\displaystyle\frac{\hbar
c}{480a}+\frac{\pi^{2}}{90}V\frac{(k_{B}T)^{4}}{(\hbar
c)^{3}}-\frac{k_{B}T}{2}e^{-\frac{\hbar c}{ak_{B}T}}$ (42) $\displaystyle\pm$
$\displaystyle\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\mp\frac{k_{B}T}{2}e^{-\frac{\hbar c}{ak_{B}T}}.$

Note, however, that the low-temperature limit of the first term in the r.h.s
of Eq. (28) has been considered in Ref. Bezerra:2011zz  without the 1/2 factor
coming from the degeneracy $g_{n}$ in Eq. (4). Thus, the first line in Eq.
(42) is just the result from Ref. Bezerra:2011zz  multiplied by 1/2. Another
important point to note is that, in the case of Dirichlet boundary condition,
the exponentials above cancel out, surviving only the terms proportionals to
$(k_{B}T)^{4}$ and $(k_{B}T)^{3}$ as corrections to the vacuum energy at zero
temperature, in the first term in the r.h.s. of Eq. (42). Conversely, in the
case of Neumann boundary condition, the exponentially suppressed terms sum up,
giving the same order exponential correction as in Ref. Bezerra:2011zz .

The discussion above follows in a similar way for the Casimir internal energy
obtained from Eqs. (14) and (28). Hence, by taking the low-temperature limit
of Eq. (33) and considering the result from Ref. Bezerra:2011zz , we obtain

$\displaystyle U^{\text{ren}}_{\text{C}}$ $\displaystyle\simeq$
$\displaystyle\frac{\hbar
c}{480a}-\frac{\pi^{2}}{30}V\frac{(k_{B}T)^{4}}{(\hbar c)^{3}}+\frac{\hbar
c}{2a}e^{-\frac{\hbar c}{ak_{B}T}}$ (43) $\displaystyle\mp$
$\displaystyle\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\pm\frac{\hbar c}{2a}e^{-\frac{\hbar c}{ak_{B}T}},$

which is also dominated by the vacuum energy at zero temperature, as we can
see.

Finally, to end our analysis of the low-temperature limit let us consider the
Casimir entropy. This can be reached by making use of Eqs. (15) and (28).
Thus, by taking into consideration the low-temperature limit of Eq. (34) and
considering the result from Ref. Bezerra:2011zz , we found

$\displaystyle S^{\text{ren}}_{\text{C}}$ $\displaystyle\simeq$
$\displaystyle-\frac{2\pi^{2}}{45}k_{B}\left(\frac{k_{B}T}{\hbar
c}\right)^{3}V+\frac{\hbar c}{2aT}e^{-\frac{\hbar c}{ak_{B}T}}$ (44)
$\displaystyle\mp$
$\displaystyle\frac{\zeta(3)a^{2}}{2}\frac{(k_{B}T)^{3}}{(\hbar
c)^{2}}\pm\frac{\hbar c}{2aT}e^{-\frac{\hbar c}{ak_{B}T}}.$

Again, for Dirichlet boundary condition the small exponential corrections
cancel out, while for Neumann boundary condition they sum up. We can notice
that in the limit $T\rightarrow 0$ the Casimir entropy goes to zero and the
third law of thermodynamics is satisfied, according to Nernst heat theorem
Landau ; Bezerra:2011zz .

###  IV.2 High-temperature limit

We now turn to the high-temperature limit case, i.e., $k_{B}T\gg\frac{\hbar
c}{a}$. The most convenient expressions for this investigation are the ones in
Eqs. (37), (38) and (39). Thereby, the asymptotic limit for high temperatures
of the renormaized free energy contribution in Eq. (37) is given by the
dominant terms

$\displaystyle\Delta F_{T}^{\text{ren}}\simeq\mp\frac{k_{B}T}{24}\left(1-12\ln
A\right)\pm\frac{k_{B}T}{24}\ln\left(\frac{k_{B}Ta}{\hbar
c}\right)\pm\frac{1}{5760}\left(\frac{\hbar c}{a}\right)^{2}\frac{1}{k_{B}T},$
(45)

where $A\simeq 1.28$ is the Glaisher’s constant and the largest (small)
correction to the linear and logarithmic terms has been written. Thus, the
renormalized Casimir free energy is obtained, from Eq. (25), by making use of
Eq. (28), the result of Ref. Bezerra:2011zz  for high temperatures and the
expression above. This gives

$\displaystyle F_{\text{C}}^{\text{ren}}$ $\displaystyle\simeq$
$\displaystyle\frac{k_{B}T}{8\pi^{2}}\zeta(3)+2\pi^{2}\left(\frac{ak_{B}T}{\hbar
c}\right)^{2}\frac{\hbar c}{a}e^{-4\pi^{2}\frac{ak_{B}T}{\hbar c}}$ (46)
$\displaystyle\pm$ $\displaystyle\frac{k_{B}T}{24}\left(12\ln
A-1\right)\pm\frac{k_{B}T}{24}\ln\left(\frac{k_{B}Ta}{\hbar
c}\right)\pm\frac{1}{5760}\left(\frac{\hbar c}{a}\right)^{2}\frac{1}{k_{B}T}.$

Note that the first term in the r.h.s of Eq. (25) has been canceled with the
same contribution, but with opposite sign, coming from the high-temperature
limit of the first term in the r.h.s. of (28) Bezerra:2011zz . We can also
notice that, compared with the results in Ref. Bezerra:2011zz  for the
Einstein universe without boundaries, the asymptotic expression above brings
one more classical term, as well as a logarithmic term similar to the
electromagnetic case analyzed in Ref. Bezerra:2011nc . Moreover, the largest
correction to the linear and logarithmic terms is the one proportional to
$\frac{1}{k_{B}T}$, not the exponential as it is the case in Ref.
Bezerra:2011zz .

The Casimir internal energy at high temperatures can be obtained by
considering the result from Ref. Bezerra:2011zz  and by using Eqs. (14), (28)
and (38). The complete expression is given by

$\displaystyle U^{\text{ren}}_{\text{C}}$ $\displaystyle\simeq$ $\displaystyle
8\pi^{4}\left(\frac{k_{B}Ta}{\hbar c}\right)^{4}\frac{\hbar
c}{a}e^{-4\pi^{2}\frac{ak_{B}T}{\hbar c}}$ (47) $\displaystyle\mp$
$\displaystyle\frac{k_{B}T}{24}\pm\frac{1}{2880}\left(\frac{\hbar
c}{a}\right)^{2}\frac{1}{k_{B}T}.$

Thereby, the expression above brings a classical contribution and a correction
that is proportional to $\frac{1}{k_{B}T}$, which is bigger than the
exponential correction in the first line. Therefore, the Casimir internal
energy is completely dominated by the second line of the expression (47). In
the Einstein universe analyzed in Ref. Bezerra:2011zz , the Casimir internal
energy is exponentially suppressed.

To end the analysis of the high-temperature limit, we consider the Casimir
entropy. The latter can be obtained by using the result of Ref. Bezerra:2011zz
, and Eqs. (15), (28) and (39). The Casimir entropy at high temperatures is,
then, written as

$\displaystyle S_{\text{C}}^{\text{ren}}$ $\displaystyle\simeq$
$\displaystyle-\frac{k_{B}}{8\pi^{2}}\zeta(3)+8\pi^{4}k_{B}\left(\frac{ak_{B}T}{\hbar
c}\right)^{2}e^{-4\pi^{2}\frac{ak_{B}T}{\hbar c}}$ (48) $\displaystyle\mp$
$\displaystyle\frac{k_{B}}{2}\ln
A\mp\frac{k_{B}}{24}\ln\left(\frac{k_{B}Ta}{\hbar
c}\right)\mp\frac{k_{B}}{5760}\left(\frac{\hbar c}{ak_{B}T}\right)^{2},$

where we can see two constant classical terms, We can also see that the
Casimir entropy is dominated by a logarithmic term of quantum origin,
differently from Ref. Bezerra:2011zz  where the entropy is dominated by a
constant classical term at high temperatures.

##  V Conclusions

In this work we have investigated temperature corrections to the vacuum energy
at zero temperature of a conformally coupled massless scalar field in the
Einstein universe with a spherical boundary. The field is subjected to obey
both Dirichlet and Neumann boundary conditions. Hence, we have generalized the
results found in Refs. Bayin:1993yq ; Kennedy:1980kc  where the authors
considered only the vacuum energy at zero temperature by adopting the cutoff
method. We have then calculated the renormaized Casimir free energy, internal
energy and entropy by adopting the generalized zeta function technique, as
well as Abel-Plana formula.

We have also studied the limits of low and high temperatures for all
thermodynamics quantities considered. In this context, we have shown that, at
low temperatures, both the Casimir free energy and internal energy are
dominated by the vacuum energy at zero temperature, and the entropy obeys the
third law of thermodynamics in accordance with the Nernst heat theorem.
Furthermore, at high temperatures we have also shown that the Casimir free
energy exhibit both classical term contributions and logarithmic contributions
with respect to the temperature. In this regard, the internal energy presents
also a classical contribution and the entropy presents a constant classical
contribution plus a logarithmic term with respect to the temperature. This is
in contrast with Ref. Bezerra:2011zz  where the authors have shown that, at
high temperatures, the Casimir free energy presents only a classical term, the
internal energy goes to zero and the entropy is dominated by a constant
classical contribution.

As a future perspective, we intend to extend this analysis to a
$D$-dimensional Einstein universe, with a hyper spherical boundary.

###### Acknowledgements.

The authors are partially supported by the National Council for Scientific and
Technological Development (CNPq) under grants No 311031/2020-0 (H.F.S.M.),
308268/2021-6 (C.R.M.) and 307211/2020-7 (V.B.B.).

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