11institutetext: Computer Vision Center and Computer Science Department,
Universitat Autònoma de Barcelona, Catalunya
11email<EMAIL_ADDRESS>
11email<EMAIL_ADDRESS>
#  Date Estimation in the Wild of Scanned Historical Photos: An Image
Retrieval Approach

Adrià Molina  0000-0003-0167-8756 Pau Riba  0000-0002-4710-0864 Lluis Gomez
0000-0003-1408-9803 Oriol Ramos-Terrades  0000-0002-3333-8812 Josep Lladós
0000-0002-4533-4739

###### Abstract

This paper presents a novel method for date estimation of historical
photographs from archival sources. The main contribution is to formulate the
date estimation as a retrieval task, where given a query, the retrieved images
are ranked in terms of the estimated date similarity. The closer are their
embedded representations the closer are their dates. Contrary to the
traditional models that design a neural network that learns a classifier or a
regressor, we propose a learning objective based on the nDCG ranking metric.
We have experimentally evaluated the performance of the method in two
different tasks: date estimation and date-sensitive image retrieval, using the
DEW public database, overcoming the baseline methods.

###### Keywords:

Date Estimation Historical Photographs Image Retrieval Ranking Loss Smooth-
nDCG.

##  1 Introduction

Historical archives and libraries contain a large variability of document
sources that reflect the memory of the past. The recognition of the scanned
images of these documents allows to reconstruct the history. A particular type
of archival data are historical photographs which are full of evidence that
tells us the story of that snapshot in time. One just needs to pay attention
to the subtle cues that are found in different objects that appear in the
scene: the clothes that people wear, their haircut styles, the overall
environment, the tools and machinery, the natural landscape, etc. All of these
visual features are important cues for estimating its creation date. Apart
from that, texture and color features might also be of great help to
accurately estimate of image creation date since photographic techniques have
evolved throughout history and have imprinted a specific date fingerprint on
them.

Date estimation of cultural heritage photographic assets is a complex task
that is usually performed by experts (e.g. archivists or genealogists) that
exploit their expert knowledge about all the features mentioned above to
provide precise date estimations for undated photographs. But their manual
labor is costly and time consuming, and automatic image date estimation models
are of great interest for dealing with large scale archive processing with
minimal human intervention.

Most approaches in date estimation for historical images try to directly
compute the estimation through classification or regression [5, 9, 12]. As
alternative of these classical approaches, in this paper we present a method
for date estimation of historical photographs in a retrieval scenario. Thus,
the date estimation of photographs is incorporated in the ranked results for a
given query image. This allows to predict the date of an image contextualized
regarding the other photographs of the collection. In the worst case, when the
exact date is not exactly estimated, the user can obtain a relative ordering
(one photograph is older than another one), which is useful in archival tasks
of annotating document sources. The proposed model for historical photograph
retrieval is based in a novel ranking loss function _smooth-nDCG_ based on the
Normalized Discounted Cumulative Gain ranking metric, which is able to train
our system according to a known relevance feedback; in our case the distance
in years between images.

The main idea in our approach relies on optimizing rankings such that the
closer is image’s date to the query’s date for a certain photograph the higher
will be ranked. When receiving an unknown image as query the method computes
the distances towards a support dataset, consisting of a collection of images
with known dates. The date is estimated assuming that the highest ranked
images are images from the same date.

In contrast to the literature reviewed in Section 2, our method allows not
only to predict but to rank images given a query. This means that considering
an image from a certain year our system is capable of retrieving a list of
images ordered by time proximity. This may be useful for many applications and
systems that rely on retrieving information from a large amount of data.

The rest of the paper is organized as follows: In Section 2 we present the
most relevant state of the art related to our work. The key components of the
proposed model are described in Section 3, where we describe the learning
objectives considered in the training algorithms, and in Section 4, where we
outline the architecture and training process of the model. Section 5 provides
the experimental evaluation and discussion. Finally, Section 6 draws the
conclusions.

##  2 Related Work

The problem of automatic estimation of the creation date of historical
photographs is receiving increased attention by the computer vision and
digital humanities research communities. The first works go back to Schindler
et al. [14, 13], where the objective was to automatically construct time-
varying 3D models of a city from a large collection of historical images of
the same scene over time. The process begins by performing feature detection
and matching on a set of input photographs, followed by structure from motion
(SFM) to recover 3D points and camera poses. The temporal ordering task was
formulated as a constraint satisfaction problem (CSP) based on the visibility
of structural elements (buildings) in each image.

More related with the task addressed in our work, Palermo et al. [10] explored
automatic date estimation of historical color images based on the evolution of
color imaging processes over time. Their work was formulated as a five-way
decade classification problem, training a set of one-vs-one linear support
vector machines (SVMs) and using hand-crafted color based features (e.g. color
co-occurrence histograms, conditional probability of saturation given hue, hue
and color histograms, etc.). Their models were validated on a small-scale
dataset consisting of 225 and 50 images per decade for training and testing,
respectively.

In a similar work, Fernando et al. [3] proposed two new hand-crafted color-
based features that were designed to leverage the discriminative properties of
specific photo acquisition devices: color derivatives and color angles. Using
linear SVMs they outperformed the results of [10] as well as a baseline of
deep convolutional activation features [2]. On the other hand, martin et al.
[8] also showed that the results of [10] could be slightly improved by
replacing the one-vs-one classification strategy by an ordinal classification
framework [4] in which relative errors between classes (decades) are
adequately taken into account.

More recently, Müller et al. [9] have brought up to date the task of image
date estimation in the wild by contributing a large-scale, publicly available
dataset and providing baseline results with a state-of-the-art deep learning
architecture for visual recognition [15]. Their dataset, Date Estimation in
the Wild (DEW), contains more than one million
Flickr111https://www.flickr.com/ images captured in the period from 1930 to
1999, and covering a broad range of domains, e.g., city scenes, family photos,
nature, and historical events.

Beyond the problem of unconstrained date estimation of photographs, which is
the focus of this paper, there are other related works that have explored the
automatic prediction of creation dates of certain objects [7, 16], or of some
specific types of photographs such as yearbook portraits [5, 12].

In this work, contrary to all previously published methods, we approach the
problem of date estimation from an image retrieval perspective. We follow the
work of Brown et al. [1] towards differentiable loss functions for information
retrieval. More precisely, our model learns to estimate the date of an image
by minimizing the the Normalized Discounted Cumulative Gain.

In all our experiments we use Müller et al.’s Date Estimation in the Wild [9]
dataset. We also share with [9] the use of state of the art convolutional
neural networks in contrast to classic machine learning and computer vision
approaches, such as [3].

##  3 Learning Objectives

As mentioned in Section 1 our retrieval system relies on a neural network [6]
trained to minimize a differentiable ranking function [1]. Traditionally,
information retrieval evaluation has been dominated by the mean Average
Precision (mAP) metric. In contrast, we will be using the Normalized
Discounted Cumulative Gain (nDCG). The problem with mAP is that it fails in
measuring ranking success when the ground truth relevance of the ranked items
is not binary.

In the task of date estimation the ground truth data is numerical and ordinal
(e.g. a set of years in the range $1930$ to $1999$), thus given a query the
metric should not punish equally for ranking as top result a 1-year difference
image than a 20-years difference one. This problem is solved in the nDCG
metric by using a relevance score that measures how relevant is a certain
sample to our query. This allows us to not only to deal with date estimation
retrieval but to explicitly declare what we consider a good ranking.

In this section we derive the formulation of the smooth-nDCG loss function
that we will use to optimize our date estimation model.

Mean Average Precision: First, let us define _Average Precision_ (AP) for a
given query $q$ as

$\operatorname{AP}_{q}=\frac{1}{|\mathcal{P}_{q}|}\sum_{n=1}^{|\Omega_{q}|}P@n\times
r(n),$ (1)

where $P@n$ is the precision at $n$ and $r(n)$ is a binary function on the
relevance of the n-th item in the returned ranked list, $\mathcal{P}_{q}$ is
the set of all relevant objects with regard the query $q$ and $\Omega_{q}$ is
the set of retrieved elements from the dataset. Then, the mAP is defined as:

$\operatorname{mAP}=\frac{1}{Q}\sum_{q=1}^{Q}\operatorname{AP}_{q},$ (2)

where $Q$ is the number of queries.

Normalized Discounted Cumulative Gain: In information retrieval, the
_normalized Discounted Cumulative Gain_ (nDCG) is used to measure the
performance on such scenarios where instead of a binary relevance function, we
have a graded relevance scale. The main idea is that highly relevant elements
appearing lower in the retrieval list should be penalized. In the opposite way
to mAP, elements can be relevant despite not being categorically correct with
respect to the query. The _Discounted Cumulative Gain_ (DCG) for a query $q$
is defined as

$\operatorname{DCG}_{q}=\sum_{n=1}^{|\Omega_{q}|}\frac{r(n)}{\log_{2}(n+1)},$
(3)

where $r(n)$ is a graded function on the relevance of the n-th item in the
returned ranked list and $\Omega_{q}$ is the set of retrieved elements as
defined above. In order to allow a fair comparison among different queries
that may have a different sum of relevance scores, a normalized version was
proposed and defined as

$\operatorname{nDCG}_{q}=\frac{\operatorname{DCG}_{q}}{\operatorname{IDCG}_{q}},$
(4)

where $\operatorname{IDCG}_{q}$ is the ideal discounted cumulative gain, i.e.
assuming a perfect ranking according to the relevance function. It is formally
defined as

$\operatorname{IDCG}_{q}=\sum_{n=1}^{|\Lambda_{q}|}\frac{r(n)}{\log_{2}(n+1)}$
(5)

where $\Lambda_{q}$ is the ordered set according the relevance function.

As we have exposed, we want to optimize our model at the retrieval list level,
but the two classical retrieval evaluation metrics defined above are not
differentiable. In the following we define two ‘smooth’ functions for ranking
optimization that are inspired in the $\operatorname{mAP}$ and
$\operatorname{nDCG}$ metrics respectively.

Ranking Function (R). Following the formulation introduced in [11], these two
information retrieval metrics can be reformulated by means of the following
ranking function,

$\mathcal{R}(i,\mathcal{C})=1+\sum_{j=1}^{|\mathcal{C}|}\mathbb{1}\\{(s_{i}-s_{j})<0\\},$
(6)

where $\mathcal{C}$ is any set (such as $\Omega_{q}$ or $\mathcal{P}_{q}$),
$\mathbb{1}\\{\cdot\\}$ is the Indicator function, and $s_{i}$ is the
similarity between the $i$-th element and the query according to $S$. In this
work we use the cosine similarity as $S$ but other similarities such as
inverse euclidean distance should work as well. Let us then define cosine
similarity as:

$S(v_{q},v_{i})=\frac{v_{q}\cdot v_{i}}{\|v_{q}\|\|v_{i}\|}.$ (7)

Still, because of the need of a indicator function $\mathbb{1}\\{\cdot\\}$,
with this formulation we are not able to optimize following the gradient based
optimization methods and we require of a differentiable indicator function.
Even though several approximations exists, in this work we followed the one
proposed by Quin et al. [11]. Thus, we make use of the sigmoid function

$\mathcal{G}(x;\tau)=\frac{1}{1+e^{\frac{-x}{\tau}}}.$ (8)

where $\tau$ is the temperature of the sigmoid function. As illustrated in
Figure 1 the smaller is $\tau$ the less smooth will be the function. For small
values the gradient is worse defined but the approximation is better since the
sigmoid becomes closer to the binary step function.

Figure 1: Shape of the smooth indicator function (top) and its derivative
(bottom) for different values of the temperature: 1 (left), 0.1 (middle),
0.001 (right).

Smooth-AP: Following [1] and replacing the term $P@n\times r(n)$ in Eq. 1 by
the Ranking Function introduced in Eq. 6, the $\operatorname{AP}$ equation
becomes

$\operatorname{AP}_{q}=\frac{1}{|\mathcal{P}_{q}|}\sum_{i\in\mathcal{P}_{q}}\frac{1+\sum_{j\in\mathcal{P}_{q},j\neq
i}\mathbb{1}\\{D_{ij}<0\\}}{1+\sum_{j\in\Omega_{q},j\neq
i}\mathbb{1}\\{D_{ij}<0\\}}.$ (9)

where we use $D_{ij}=s_{i}-s_{j}$ for a more compact notation. Finally,
replacing the indicator function by the sigmoid function (Eq. 8) we obtain a
smooth approximation of $\operatorname{AP}$

$\operatorname{AP}_{q}\approx\frac{1}{|\mathcal{P}_{q}|}\sum_{i\in\mathcal{P}_{q}}\frac{1+\sum_{j\in\mathcal{P}_{q},j\neq
i}\mathcal{G}(D_{ij};\tau)}{1+\sum_{j\in\Omega_{q},j\neq
i}\mathcal{G}(D_{ij};\tau)}.$ (10)

Averaging this approximation for all the queries in a given batch, we can
define our loss as

$\mathcal{L}_{AP}=1-\frac{1}{Q}\sum_{i=1}^{Q}AP_{q},$ (11)

where $Q$ is the number of queries.

Smooth-nDCG: Following the same idea as above, we replace the $n$-th position
in Eq. 3 by the ranking function, since it defines the position that the
$i$-th element of the retrieved set, and the DCG metric is expressed as

$\operatorname{DCG}_{q}=\sum_{i\in\Omega_{q}}\frac{r(i)}{\log_{2}\left(2+\sum_{j\in\Omega_{q},j\neq
i}\mathbb{1}\\{D_{ij}<0\\}\right)},$ (12)

where $r(i)$ is the same graded function used in Eq. 3 but evaluated at
element $i$. Therefore, the corresponding smooth approximation is

$\operatorname{DCG}_{q}\approx\sum_{i\in\Omega_{q}}\frac{r(i)}{\log_{2}\left(2+\sum_{j\in\Omega_{q},j\neq
i}\mathcal{G}(D_{ij};\tau)\right)}$ (13)

when replacing the indicator function by the sigmoid one.

The smooth-nDCG is then defined by replacing the original
$\operatorname{DCG}_{q}$ by its smooth approximation in Eq. 4 and the loss
$\mathcal{L}_{nDCG}$ is defined as

$\mathcal{L}_{nDCG}=1-\frac{1}{Q}\sum_{i=1}^{Q}nDCG_{q},$ (14)

where $Q$ is the number of queries.

##  4 Training process

The main contribution of this paper relies on using the smooth-nDCG loss
defined in the previous section for optimizing a model that learns to rank
elements with non-binary relevance score. Specifically, our model learns to
project images into an embedding space by minimizing the smooth-nDCG loss
function. In all our experiments we use a Resnet-101 [6] convolutional neural
network pretrained on the ImageNet dataset. As illustrated in Figure 2 the
idea is that in the learned embedding space the cosine similarity between
image projections is proportional to the actual distance in years. We will
discuss in Section 5 how teaching the system to rank images according to the
ground truth criteria leads the embedding space to follow a certain
organization (illustrated in Figure 7).

Figure 2: Objective function example, images with closer ground truth should
maximize the similarity in the output space.

In order to obtain such embedded representation of the photographs, we present
the training algorithm (see Algorithm 1) that mainly relies on using the
equations presented in the previous Section 3 in the output space for
optimizing the CNN so it minimizes the smooth-nDCG loss function (see Eq. 14)
and, consequently, maximizes the nDCG metric in our rankings, meaning that
images will be closer in the output space as they are closer in the ground
truth space.

Input: Input data $\\{\mathcal{X},\mathcal{Y}\\}$; CNN $f$; distance function
D; max training iterations $T$
Output:  Network parameters $w$

1:repeat

2: Process images to output embedding $h\leftarrow
f_{w}(\\{x_{i}\\}_{i=1}^{N_{batch}})$

3: Get Distance matrix from embeddings, all vs all rankings $M\leftarrow D(h)$

4: Calculate relevance from training set $Y$ Eq. 15

5: Using the relevance score, $\mathcal{L}\leftarrow$ Eq. 14

6: $w\leftarrow w-\Gamma(\nabla_{w}\mathcal{L})$

7:until Max training iterations $T$

Algorithm 1 Training algorithm for the proposed model. Figure 3: Proposed
baseline. Once ranked the images in the nDCG space we’ll be computing smooth-
nDCG in order to back-propagate the loss through the CNN.

As shown in Figure 3, once we process the images through the CNN, we compute
the rankings for each one of the batch images. We use cosine similarity as
distance function, however it could be replaced by any distance function model
over the vector space. Since nDCG requires a relevance score for each sample,
the design of the relevance function is an important part of the proposed
method. A possible relevance function could be just the inverse distance in
years as defined in Eq. 16.

Additionally, we decided to consider only the closest decade in the relevance
function (illustrated in Figure 4), so every image further than 10 years is
considered as relevant as any other. Eq. 15 is the relevance function used in
our model (see Eq. 14) but other relevance settings could be, like Eq. 17 in
order to exponentially punish more the images that should not be as far as
they are.

$\displaystyle r(n;\gamma)$ $\displaystyle=\max(0,\gamma-|y_{q}-y_{n}|)$ (15)
$\displaystyle r(n)$ $\displaystyle=\frac{1}{1+|y_{q}-y_{n}|}$ (16)
$\displaystyle r(n)$ $\displaystyle=e^{\frac{1}{1+|y_{q}-y_{n}|}}$ (17)

where $y_{q}$ and $y_{n}\in\mathcal{Y}$ are the dates of the query and the
$n$-th image in the training set, respectively; and $\gamma$, in Eq. 15, is an
hyper-parameter that has experimentally set to 10.

Figure 4: Different examples of relevance functions for ground truth data.

Once trained, minimizing the retrieval loss function, we can predict the year
for an unknown image through k-Nearest Neighbors (k-NN) to a support set.

In one hand, we can use a set of N train images as support set; each batch is
randomly selected for each prediction. Note that, since prediction relies on
the k-th most similar images, the bigger the support set the better should
perform the prediction. In Figure 5 we show the date estimation mean absolute
error of our method on the DEW training set depending on the k parameter for
the k-NN search. Hence, we would be using 10-Nearest Neighbors in all our
experiments for the prediction task.

Figure 5: Mean Absolute Error (MAE) on prediction by k-Nearest Neighbors with
respect to each k using a randomly selected support set of 128 images

Additionaly, we can compute the retrievals within the entire training set with
Approximate Nearest Neighbours222Using public Python library ANNOY for
Approximate Nearest Neighbours https://github.com/spotify/annoy, which allow
us to compute the retrievals efficiently. As it’s shown in Figure 6, the
possibility of using the whole train set makes the $k$ parameter more likely
to be huge. We don’t observe a notable divergence in the results until using a
an enormous $k$ parameter, which may indicate quite good result in terms of
retrieval since worst responses are at the bottom of the dataset.

Figure 6: Mean Absolute Error with different $k$ for k-NN using the entire
train set as support set. Note that the horizontal axis grows logarithmically.

##  5 Experiments

In this section we evaluate the performance of the proposed method in two
different tasks: date estimation and date-sensitive image retrieval. In all
our experiments we use the DEW [9] dataset, which is composed of more than one
million images captured in the period from $1930$ to $1999$. The test set is
balanced across years and contains $1120$ images, $16$ images per year. The
temperature parameter, $\tau$, in Eq. 8, is set to 0.01.

For the task of date estimation we use the Mean Absolute Error (MAE) as the
standard evaluation metric. Given a set of $N$ images and their corresponding
ground truth date annotations (years) $y=\\{y_{1},y_{2},\dots,y_{N}\\}$, the
$\operatorname{MAE}$ for a set of predictions
$y^{\prime}=\\{y_{1}^{\prime},y_{2}^{\prime},\dots,y_{N}^{\prime}\\}$ is
calculated as follows:

$\operatorname{MAE}=\frac{1}{N}\sum_{i}^{N}{|y_{i}-y_{i}^{\prime}|}$ (18)

For the image retrieval task we use the mAP and nDCG metrics defined in
Section 3. In Table 1 we present a comparison of our methods with the state of
the art in the task of date estimation and with a visual similarity baseline,
that ranks images using the euclidean distance between features extracted from
a ResNet CNN [6] pre-trained on ImageNet. As we will discuss in the
conclusions section, our approach is not directly comparable to Müller et
al.’s [9]. On one hand, we are not trying to directly improve the prediction
task, but a ranking one; this means that our model has to learn semantics from
the images such that it learns a continuous ordination. On the other hand, the
way we are estimating image’s date is by using a ground truth dataset or
support dataset. Nevertheless, we observe that our model isn’t too far with
respect to the state of the art prediction model. We consider this a good
result since estimating the exact year of a image is not the main purpose of
the smooth-nDCG loss function. Note that in Table 1 we mention a ”cleaned test
set”. This is because we observed the presence of certain placeholder images
indicating they are no longer available in Flickr. We computed the MAE for the
weights given by Müller et al.[9] 333This is an extended version of Müller et
al. [9] made publibly available by the same authors at https://github.com/TIB-
Visual-Analytics/DEW-Model for a proper comparison. The presence of this kind
of images may have a low impact while evaluating a classification or
regression, but they can have a higher impact on predicting with k-NN, since
all the retrieved images will be placeholders as well independently of their
associated years.

We present as well a result of predicting the year by weighted k-NN where the
predicted year will be the sum of the $k$ neighbours multiplied by their
similarities to the query.

Table 1: Mean absolute error comparison of our baseline model on the test set of the Date Estimation in the Wild (DEW) dataset. Baseline | MAE | mAP | nDCG
---|---|---|---
Müller et al. [9] GoogLeNet (regression) | 7.5 | - | -
Müller et al. [9] GoogLeNet (classification) | 7.3 | - | -
Müller et al. [9] ResNet50 (classification) (cleaned test set) | 7.12 | - | -
Visual similarity baseline | 12.4 | - | 0.69
Smooth-nDCG 256 images support set | 8.44 | 0.12 | 0.72
Smooth-nDCG 1M images support set | 7.48 | - | 0.75
Smooth-nDCG 1M images support set (weighted kNN) | 7.52 | - | 0.75

Since our model is trained for a retrieval task, we provide further
qualitative examples of good and bad rankings. This exemplifies how our loss
function is not mean to predict years, but finding good rankings according to
query’s year. Note we computed nDCG using as relevance the closeness in years
between images’ ground truth clipped at 10 years (see Eq. 15). Despite it may
seem nDCG is so high and mAP so low; there are many subtle bias in this
appearance. Since the ground truth can be interpreted as a continuous value
(such as time) there are not many matches in any retrieval of N random images
for a certain query regarding to query’s year compared with how many negative
samples are there. Then, mAP hardly punishes results in this situation. In the
opposite way, the same thing happens to nDCG. Since nDCG has this ‘continuous’
or ‘scale’ sense, any retrieval will have some good samples that satisfies the
relevance function (see Eq. 15), being the results way less punished if there
aren’t many exact responses in the retrieval. Briefly, we could say nDCG
brings us a more balanced positive/negative ratio than mAP for randomly
selected sets to rank (which is absolutely more realistic than forcing it to
be balanced). Additionally, nDCG allows us to approach retrievals that are not
based on obtaining a category but a wide number of them, or even retrieving
non-categorical information such as date estimation.

Additionally, as we observe in Figure 7, we compute the average cosine
similarity between clusters from different groups of years. This shows us how
our output space is distributed with respect to the ground truth data. As we
expected; clusters for closer years are the most similar between themselves.
This is because ranking is nothing but taking the closer points to our query;
by forcing our system to take those points that satisfies the relevance
function (see Eq. 15), points in the embedding space will have to be organized
such that distances are proportional to ground truth’s distances.

Figure 7: Clusters average similarity for each 5 year interval. We observe a
certain visible pattern in the diagonal; the closer the years the closer the
clusters.

Finally, in Figure 9, we compare the top-5 retrieved images for a given query
with our model and with the visual similarity baseline model, that ranks
images using the distance between features extracted from a ResNet CNN [6]
pre-trained on ImageNet. We appreciate that the visual similarity model
retrieves images that are visually similar to the query but belong to a
totally different date. On the contrary, our model performs better for the
same query in terms of retrieving similar dates although for some of the top
ranked images the objects layout is quite different to the query.
Additionally, we present the results with a mixed model that re-ranks images
with the mean ranking between both methods for each image in the test set; in
this way we obtain a ranking with visually similar images from the same date
as the query.

Figure 8: Qualitative results for our date estimation model. We show a set of
images from the DEW test set and indicate their ground truth year annotation
and distance to the query. Figure 9: Qualitative results for a retrieval
system that combines our date estimation model and a visual similarity model.
The top-left image is the query, in each row we show the top-5 retrieved
images using a visual similarity model (top), our model (middle), and the
combination of the two rankings (bottom).

##  6 Conclusions

In this paper we have proposed a method for estimating the date of historic
photographs. The main novelty of the presented work is that we do not
formulate the solution to the problem as a prediction task, but as an
information retrieval one. Hence metrics such as mean absolute error do not
have to be considered as important as retrieval metrics. Even so, we managed
to build an application-level model that can estimate the date from the output
embedding space with a decent error according to previous baselines [9]. Since
our prediction method is essentially a clustering or k-Nearest Neighbors
method we are using a known support set, so predictions relies on already
labeled data.

However, considering the output embedding space Figure 7 we conclude that
smooth-nDCG function works pretty well for ranking large-scale image data with
deeper ground truth structures; unlike smooth-AP [1], smooth-nDCG cares about
the whole sorted batch retrieved, not only how many good images are, but how
are the bad ones distributed along the ranking. This allowed us to retrieval
ordered information (such as years), or tree-structured information where
different categories can be closer or further depending on how the tree is
like.

In the case of date estimation in the wild [9] dataset, we found out some
problematic patterns that can lead the retrieval task to certain bias. As
there is not a clear balance between categories in the dataset, many classes
such as trains or vehicles may be clustered together regardless the actual
dates of the images Figure 10. However, it is not easy to say when a certain
category is most likely to be linked to a certain period of time. For example,
selfies are a category way more common nowadays than 20 years ago. Something
similar may happen with certain categories that could be biasing the retrieval
to a pure object classification task; however we found out many good clues,
Figures 7, 8, that indicate our embedding for ranking images is working
properly.

Figure 10: Possible biased retrieval due lack of balance in the training set.

Despite further research is needed; smooth-nDCG usage for large-scale image
retrieval is a promising novel approach with many practical applications.
Since Brown et al.[1] smoothed us the path to minimize the average precision,
we propose smooth-nDCG to model more complex retrieval problems where
different labels should not be punished equally according to a certain
criteria. As we commented in smooth-nDCG brings a new way of ranking images
beyond categorical labels from a neural network approach; then we would like
to consider this approach as an application example for larger information
retrieval problems with a more complex structure.

## Acknowledgment

This work has been partially supported by the Spanish projects
RTI2018-095645-B-C21, and FCT-19-15244, and the Catalan projects
2017-SGR-1783, the Culture Department of the Generalitat de Catalunya, and the
CERCA Program / Generalitat de Catalunya.

## References

  * [1] Brown, A., Xie, W., Kalogeiton, V., Zisserman, A.: Smooth-ap: Smoothing the path towards large-scale image retrieval. In: Proceedings of the European Conference on Computer Vision. pp. 677–694 (2020)
  * [2] Donahue, J., Jia, Y., Vinyals, O., Hoffman, J., Zhang, N., Tzeng, E., Darrell, T.: Decaf: A deep convolutional activation feature for generic visual recognition. In: Proceedings of the International Conference on Machine Learning. pp. 647–655 (2014)
  * [3] Fernando, B., Muselet, D., Khan, R., Tuytelaars, T.: Color features for dating historical color images. In: Proceedings of the International Conference on Image Processing. pp. 2589–2593 (2014)
  * [4] Frank, E., Hall, M.: A simple approach to ordinal classification. In: Proceedings of the European Conference on Machine Learning. pp. 145–156 (2001)
  * [5] Ginosar, S., Rakelly, K., Sachs, S., Yin, B., Efros, A.A.: A century of portraits: A visual historical record of american high school yearbooks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops. pp. 1–7 (2015)
  * [6] He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 770–778 (2016)
  * [7] Lee, S., Maisonneuve, N., Crandall, D., Efros, A.A., Sivic, J.: Linking past to present: Discovering style in two centuries of architecture. In: Proceedings of the International Conference on Computational Photography (2015)
  * [8] Martin, P., Doucet, A., Jurie, F.: Dating color images with ordinal classification. In: Proceedings of International Conference on Multimedia Retrieval. p. 447 (2014)
  * [9] Müller, E., Springstein, M., Ewerth, R.: “when was this picture taken?”–image date estimation in the wild. In: Proceedings of the European Conference on Computer Vision. pp. 619–625 (2017)
  * [10] Palermo, F., Hays, J., Efros, A.A.: Dating historical color images. In: Proceedings of the European Conference on Computer Vision. pp. 499–512 (2012)
  * [11] Qin, T., Liu, T.Y., Li, H.: A general approximation framework for direct optimization of information retrieval measures. Information retrieval 13(4), 375–397 (2010)
  * [12] Salem, T., Workman, S., Zhai, M., Jacobs, N.: Analyzing human appearance as a cue for dating images. In: Proceedings of the IEEE Winter Conference on Applications of Computer Vision. pp. 1–8 (2016)
  * [13] Schindler, G., Dellaert, F.: Probabilistic temporal inference on reconstructed 3d scenes. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 1410–1417 (2010)
  * [14] Schindler, G., Dellaert, F., Kang, S.B.: Inferring temporal order of images from 3d structure. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 1–7 (2007)
  * [15] Szegedy, C., Liu, W., Jia, Y., Sermanet, P., Reed, S., Anguelov, D., Erhan, D., Vanhoucke, V., Rabinovich, A.: Going deeper with convolutions. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015)
  * [16] Vittayakorn, S., Berg, A.C., Berg, T.L.: When was that made? In: Proceedings of the IEEE Winter Conference on Applications of Computer Vision. pp. 715–724 (2017)


# On the duality of Schwarzschild-de Sitter spacetime and moving mirror

Diego Fernández-Silvestre<EMAIL_ADDRESS>Departamento de Física Teórica
and IFIC, Universidad de Valencia-CSIC, C. Dr. Moliner 50, 46100 Burjassot,
Spain.  Joshua Foo<EMAIL_ADDRESS>Centre for Quantum Computation
& Communication Technology, School of Mathematics & Physics, University of
Queensland, St. Lucia, Queensland, 4072, Australia.  Michael R.R. Good
<EMAIL_ADDRESS>Department of Physics & Energetic Cosmos Laboratory,
Nazarbayev University, Kabanbay Batyr Ave 53, Nur-Sultan, 010000, Kazakhstan.

###### Abstract

The Schwarzschild-de Sitter (SdS) metric is the simplest spacetime solution in
general relativity with both a black hole event horizon and a cosmological
event horizon. Since the Schwarzschild metric is the most simple solution of
Einstein’s equations with spherical symmetry and the de Sitter metric is the
most simple solution of Einstein’s equations with a positive cosmological
constant, the combination in the SdS metric defines an appropriate background
geometry for semi-classical investigation of Hawking radiation with respect to
past and future horizons. Generally, the black hole temperature is larger than
that of the cosmological horizon, so there is heat flow from the smaller black
hole horizon to the larger cosmological horizon, despite questions concerning
the definition of the relative temperature of the black hole without a
measurement by an observer sitting in an asymptotically flat spacetime. Here
we investigate the accelerating boundary correspondence (ABC) of the radiation
in SdS spacetime without such a problem. We have solved for the boundary
dynamics, energy flux and asymptotic particle spectrum. The distribution of
particles is globally non-thermal while asymptotically the radiation reaches
equilibrium.

QFT in curved spacetime, black holes, cosmological horizons, moving mirrors

###### pacs:

04.62.+v, 03.67.Hk, 04.70.-s

##  I Introduction

Historically, black holes have been important systems for investigating and
understanding our universe. Theoretically, they have played a prominent role
in attempts at unifying quantum theory and gravity. Along astrophysical lines
of research, investigations into their formation and evolution have
contributed to our knowledge of the large-scale structure of galaxies, while
the detection of gravitational lensing and gravitational waves have
corroborated the predictions of general relativity. More largely, the study of
cosmological spacetimes is of great interest because we live in an accelerated
expanding universe Riess  _et al._ (1998); Perlmutter  _et al._ (1999). The
simplest expanding spacetime is de Sitter, which is especially appropriate to
describe the early phase of exponential expansion predicted by inflation Guth
(1981). Then, the study of cosmological black holes (e.g. asymptotically de
Sitter black holes) is well-motivated, for example, to investigate black holes
formed during the early phase of inflation or even the global structure of
black holes in the current phase of accelerating expansion.

To theoretically analyze black holes embedded in an expanding universe, the
simplest spacetime to start is Schwarzschild-de Sitter spacetime (SdS), which
is an exact black hole solution to the Einstein’s equations with a positive
cosmological constant. In the context of quantum field theory, the SdS
background is particularly convenient to study the effects of event horizons
on the propagation of quantum fields, since there are two different horizons
in a single spacetime metric: a black hole horizon and a cosmological horizon.

In the asymptotically flat Schwarzschild black hole case, Hawking discovered
the quantum particle creation effects that occur when a black hole forms from
the gravitational collapse of a star Hawking (1975) (see also Unruh (1976)).
The particles are produced at late times in a thermal spectrum (with a grey-
body factor) at a temperature that is proportional to the surface gravity at
the event horizon of the black hole. It was further shown in analogy with the
Schwarzschild black hole result, that the de Sitter cosmology also involves
thermal particle production with temperature proportional to the surface
gravity at the cosmological event horizon Gibbons and Hawking (1977).

In the asymptotically de Sitter Schwarzschild black hole case, the particle
production and thermodynamics were first investigated in Gibbons and Hawking
(1977). One can demonstrate, using the mechanism of Hawking (1975), that the
black hole horizon of SdS spacetime, and the cosmological horizon of SdS
spacetime, will create particles in a Planck distribution (with a grey-body
factor) at a temperature that is proportional to the surface gravity of the
corresponding horizon of SdS spacetime Bhattacharya (2018). The grey-body
effect caused by the effective potential is an important physical aspect of
the radiation. However, in the geometric optics approximation (i.e. no grey-
body effect), there is only particle production in a thermal spectrum for the
black hole at late times (after the collapse). Analogously, at early times
(before the collapse), there is only cosmological particle production in a
thermal spectrum. Our investigation, in part, is to compute and analyze the
interesting interaction behaviour between both early-time and late-time
thermal spectra.

The above mentioned results of SdS spacetime came from considering that the
particle detection is made by accelerating (but static) observers not at but
close to the (future) cosmological horizon Bhattacharya (2018). It is more
natural to study the quantum particle creation effects with respect to
inertial observers instead of non-inertial observers. However, particle
production in SdS spacetime and detection by inertial observers can be
understood from detection by non-inertial (who are static) observers through
amplitudes defined via the Bogoliubov coefficients Qiu and Traschen (2020). In
this sense, it is simpler to assume the perspective of accelerating observers
in the analysis of particle production effects in SdS spacetime. Regardless of
this question, there may be a gravitational analog model that has a well-
defined and intuitive asymptotically flat spacetime inertial observer ready to
detect the corresponding radiation. This analog system is the moving mirror
model.

The analogy of moving mirrors to black holes DeWitt (1975), being a
(1+1)-dimensional case of the dynamical Casimir effect Moore (1970), was
initially introduced by Fulling and Davies Fulling and Davies (1976); Davies
and Fulling (1977). They determined that an accelerated boundary (moving
mirror) with the appropriate trajectory in Minkowski spacetime creates
particles in a thermal spectrum (at late times), analogously to Hawking’s
particle creation by black holes result Hawking (1975). A standard approach to
the mirror model is to assume a trajectory that fulfills given physical
requirements and then calculate the corresponding radiation and resulting
spectrum Walker (1985); Carlitz and Willey (1987). A central theoretical
achievement of the mirror model has been the demonstration that these
accelerated point mirrors non-trivially disturb the quantum vacuum via a non-
zero Bogoliubov coefficients (and renormalized stress tensor), resulting in
particle production, energy flux, entanglement and other interesting physics
(see e.g. Bianchi and Smerlak (2014a, b); Good and Linder (2018); Cong  _et
al._ (2019); Romualdo  _et al._ (2019); Lee and Yeh (2019); Good  _et al._
(2020a); Foo  _et al._ (2020)). Since the stress tensor is technically
divergent in the model, a point-splitting regularization technique is used to
determine finite meaningful particle production results (see e.g. Stargen  _et
al._ (2016); Good and Abdikamalov (2020)).

Based on this 50 year old prescription, the particle production and energy
flux are generally thought to be a consequence of the mirror’s acceleration
and jerk, respectively Birrell and Davies (1984); Fabbri and Navarro-Salas
(2005). Moreover, the close relation with thermodynamics has been analyzed
(see e.g. Davies (1982); Helfer (2001)), as a product of the entanglement,
entropy, and relativistic quantum information issues associated with the
effect (see e.g. Gianfelici and Mancini (2017); Chen and Yeom (2017); Good and
Linder (2019)). Physical detection is an ongoing priority Chen and Mourou
(2017, 2020), especially since superconducting quantum interference devices
act as moving mirrors and Bose-Einstein condensates have demonstrated
experimental success for investigation of the dynamical Casimir effect Dodonov
(2020).

The main approach in this paper is the use of an accelerated boundary
correspondence in flat spacetime, which creates particles in a fundamentally
related way to the radiation in a curved spacetime. In flat spacetime the
particle production is caused by the dynamical acceleration of the boundary
(Davies-Fulling effect Fulling and Davies (1976); Davies and Fulling
(1977)).111In flat spacetime there also exists particle creation due to
acceleration without the presence of a boundary: an accelerating observer
detects particles in flat spacetime (Unruh effect Unruh (1976)). In curved
spacetime the particle production is caused by dynamical gravitational fields
in the formation of black holes (Hawking effect Hawking (1975)), but it can
also be caused by the acceleration of observers to remain static around black
holes (Unruh effect Unruh (1976)) or in an expanding universe (Gibbons-Hawking
effect Gibbons and Hawking (1977)). Accordingly, the moving mirror analogy is
a simple and useful way to study and understand the quantum effects produced
by gravity. This is the aim of solving for the accelerated boundary
correspondences (ABCs). The associated mirrors have perfect reflection, and
although these models are limited in scope (e.g. no backscattering, grey-body
effect, superradiance, etc.), the solutions successfully model the ‘pure’
quantum particle creation effects in either black hole spacetimes or even
cosmological spacetimes. This implies that only the s-waves of ‘pure’
gravitational particle production are accounted by this kind of mirrors Fabbri
and Navarro-Salas (2005).

In this context, the ABCs for the Schwarzschild black hole Good  _et al._
(2016), the Reissner-Nordström black hole Good and Ong (2020), the Kerr black
hole Good  _et al._ (2021a) (and the Kerr-Newman black hole Foo and Good
(2021)), and the Taub-NUT black hole Foo  _et al._ (2021) have been recently
solved. The range of ABC utility is quite broad, as particular asymptotic
dynamics characterize well-known non-thermal curved spacetime black hole end-
states, including:

  * •

extremals (uniform acceleration Liberati  _et al._ (2000); Rothman (2000);
Good (2020); Good  _et al._ (2021a); Foo and Good (2021)),

  * •

remnants (constant-velocity Good and Ong (2015); Good  _et al._ (2017); Good
(2017, 2018); Myrzakul and Good (2018); Good  _et al._ (2019)),

  * •

effervescents (zero-velocity Walker and Davies (1982); Good and Linder (2018,
2019); Good  _et al._ (2020a, 2013); Good and Linder (2017); Good  _et al._
(2020b)).

Even the de Sitter and anti-de Sitter cosmology have a respective ABC Good
_et al._ (2020c). Despite this progress, the amplification of quantum vacuum
fluctuations that result in the all-time particle spectrum (static observers)
for the formation of a black hole embedded in an expanding universe have not
been resolved, in the sense that the Bogoliubov coefficients are unknown. That
is to say, the SdS moving mirror has also not been explored, even though the
Schwarzschild mirror Good  _et al._ (2016) and de Sitter mirror Good  _et al._
(2020c) Bogoliubov coefficient solutions are known. In this paper we analyze
the particle creation by the SdS mirror directly, obtaining analytical
expressions for the Bogoliubov coefficients. The analog SdS mirror will
describe the results detected by static observers near the (future)
cosmological horizon of SdS spacetime. We combine our understanding of both
the Schwarzschild black hole and the de Sitter cosmology to derive a particle
spectrum which is consistent in both asymptotic regimes. This contribution is
a demonstration and touchstone of the established program where the simplicity
of analog gravitational systems in a flat spacetime with lower dimensions have
shed light on the more complicated curved spacetime higher dimensional
physics. This approach is expected to be tractable in part because in
(1+1)-dimensional SdS spacetime one can analytically solve the wave equation
Markovic and Unruh (1991); Anderson and Traschen (2020), where there is no
effective potential and so the effects of the causal structure on the initial
waveforms are accentuated. Despite significant simplifications that take place
in (1+1) dimensions, nontrivial quantum effects are present.

The results in the above mentioned papers suggest that the properties of the
radiation of SdS spacetime combination are qualitatively different from those
of the two constituents. This is one motivation to concentrate on the
radiation solution describing the asymptotically de Sitter Schwarzschild black
hole with respect to static observers. In agreement with De-You  _et al._
(2006), which used energy conservation and self-gravitation interaction, our
derived all-time SdS particle spectrum determined is explicitly shown to be
non-thermal (see also Choudhury and Padmanabhan (2007)), although this work
confirms two asymptotic Planck distributions (at their respective
temperatures).

The paper is organized as follows: in Sec. II we briefly introduce the SdS
spacetime, obtaining the horizons as well as the corresponding surface
gravities. We also solve for the Regge-Wheeler (tortoise) coordinate. In Sec.
III we deal with the SdS moving mirror by correctly interpreting and mapping
the SdS spacetime into the corresponding flat spacetime plus moving mirror
equation of motion. In Sec. IV we demonstrate that the energy flux of this
system is consistent with the known physics of the SdS spacetime black hole
and cosmological horizons. In Sec. V the particle production results are
shown. In Sec. VI we briefly investigate the corresponding spacetime
associated with the approximated asymptotic SdS moving mirror. In Sec. VII we
conclude. Throughout we use natural units: $G=c=\hbar=k_{B}=1$.

##  II Schwarzschild-de Sitter Spacetime

We consider the form of a general spherically symmetric static spacetime. In
spherical coordinates, the metric reads

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (1)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$. The SdS metric is
the simplest solution describing a black hole embedded in an expanding
universe with positive cosmological constant222The SdS metric is otherwise
known as the Kottler metric Kottler (1918).. It follows from Eq. (1) with

$f(r)\equiv f_{SdS}=1-\frac{2M}{r}-\frac{\Lambda r^{2}}{3},$ (2)

where $M$ is the black hole mass and $\Lambda>0$ is the cosmological constant
Stephani  _et al._ (2003). One can directly verify that the Schwarzschild case
is recovered for $\Lambda=0$ and the de Sitter case is recovered for $M=0$.
All along, we will use the parameter $L\equiv\sqrt{3/\Lambda}$, so that

$f(r)\equiv f_{SdS}=1-\frac{2M}{r}-\frac{r^{2}}{L^{2}}.$ (3)

By setting $f_{SdS}=0$, and also assuming that $3\sqrt{3}M/L<1$, the black
hole and cosmological horizons are, respectively, at $r=r_{B}$ and $r=r_{C}$
with

$r_{B}\equiv\frac{2L}{\sqrt{3}}\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{3\sqrt{3}M}{L}\right)+\frac{\pi}{3}\right],$
(4)

and

$r_{C}\equiv\frac{2L}{\sqrt{3}}\cos\left[\frac{1}{3}\cos^{-1}\left(\frac{3\sqrt{3}M}{L}\right)-\frac{\pi}{3}\right],$
(5)

where $r_{B}<r_{C}$. There also exists a negative, nonphysical solution given
by $r=r_{0}\equiv-(r_{B}+r_{C})$. This solution is unphysical since one cannot
extend the coordinate range beyond the curvature singularity at $r=0$. In
terms of these parameters we rewrite Eq. (3) as

$f_{SdS}=\frac{(r-r_{B})(r_{C}-r)(r-r_{0})}{L^{2}r},$ (6)

which imply the following relations:

$\begin{split}M&=\frac{(r_{B}+r_{C})r_{B}r_{C}}{2(r_{B}^{2}+r_{C}^{2}+r_{B}r_{C})},\\\
L^{2}&=r_{B}^{2}+r_{C}^{2}+r_{B}r_{C}.\end{split}$ (7)

The surface gravity associated with each horizon $r_{i}$ is defined from the
relation $\kappa_{i}=\frac{1}{2}\left|df_{SdS}/dr\right|_{r=r_{i}}$.
Explicitly, we get

$\begin{split}\kappa_{B}&=\frac{(r_{C}-r_{B})(r_{B}-r_{0})}{2L^{2}r_{B}},\\\
\kappa_{C}&=\frac{(r_{C}-r_{B})(r_{C}-r_{0})}{2L^{2}r_{C}},\\\
\kappa_{0}&=-\frac{\kappa_{B}\kappa_{C}}{\kappa_{C}-\kappa_{B}}.\end{split}$
(8)

Finally, we will be interested in writing the SdS metric in a double null
(light-cone) coordinate system $(u,v)$ and, for that, we will need the Regge-
Wheeler or tortoise coordinate $r^{*}$ defined by $r^{*}\equiv\int
f_{SdS}^{-1}\,dr$. The horizons $r_{i}$ and surface gravities $\kappa_{i}$
allow us to write $f_{SdS}^{-1}$ from Eq. (6) simply as

$f_{SdS}^{-1}=\frac{1}{2\kappa_{B}(r-r_{B})}+\frac{1}{2\kappa_{C}(r_{C}-r)}+\frac{1}{2\kappa_{0}(r-r_{0})},$
(9)

thence

$\begin{split}r^{*}&=\frac{1}{2\kappa_{B}}\ln\left(\frac{r-r_{B}}{r_{B}}\right)-\frac{1}{2\kappa_{C}}\ln\left(\frac{r_{C}-r}{r_{C}}\right)\\\
&+\frac{1}{2\kappa_{0}}\ln\left(-\frac{r-r_{0}}{r_{0}}\right).\end{split}$
(10)

The Regge-Wheeler or tortoise coordinate will prove to be more than
convenience; it is physically related with the trajectory of the moving mirror
in the analog gravitational system.

##  III Trajectory Dynamics

We are now ready to solve the accelerated boundary correspondence with SdS
spacetime. This is achieved by considering the matching of two metrics. This
matching condition appears when one considers the dynamical gravitational
model of a null shell collapsing in a (1+3)-dimensional spacetime. A
(1+3)-dimensional spacetime of this type allows us to deduce the corresponding
trajectory for the perfectly reflecting SdS mirror analog in (1+1)-dimensional
flat spacetime.

###  III.1 Geometric considerations of the (1+3)-dimensional gravitational
collapse of a null shell

The (1+3)-dimensional spacetime describing the dynamical gravitational
collapse of a null shell at $v=v_{0}$ takes advantage of matching two metrics:
the metric inside the shell ($v<v_{0}$) and the metric outside the shell
($v>v_{0}$) Fabbri and Navarro-Salas (2005). In this case, we apply the
matching along a null shell between an inside (“in”) Minkowski metric and an
outside (“out”) SdS metric.

The “in” metric is

$ds^{2}=-dt_{\mathrm{in}}^{2}+dr^{2}+r^{2}d\Omega^{2},$ (11)

which we rewrite in the double null coordinate system $(u_{\mathrm{in}},v)$
with $u_{\mathrm{in}}=t_{\mathrm{in}}-r$ and $v=t_{\mathrm{in}}+r$ as

$ds^{2}=-du_{\mathrm{in}}dv+r^{2}d\Omega^{2}.$ (12)

The “out” metric is

$ds^{2}=-f_{SdS}dt_{\mathrm{out}}^{2}+f_{SdS}^{-1}dr^{2}+r^{2}d\Omega^{2},$
(13)

with $f_{SdS}$ given by Eq. (3) or Eq. (6), which we rewrite in the double
null coordinate system $(u_{\mathrm{out}},v)$ with
$u_{\mathrm{out}}=t_{\mathrm{out}}-r^{*}$ and $v=t_{\mathrm{out}}+r^{*}$ with
$r^{*}$ given by Eq. (10) as

$ds^{2}=-f_{SdS}du_{\mathrm{out}}dv+r^{2}d\Omega^{2}.$ (14)

The matching condition between both “in” and “out” metrics333Note that the
“out” metric is the SdS metric written in static coordinates. As a
consequence, the analog SdS mirror trajectory in Minkowski spacetime will
represent static observers in SdS spacetime. demands the continuity of the
coordinates $r$ and $v$ along the null-shell $(v=v_{0})$444This is why we add
no subscripts for the coordinates $r$ and $v$. and reads simply as

$r(u_{\mathrm{in}},v_{0})=r(u_{\mathrm{out}},v_{0}).$ (15)

Accordingly, if we write

$r(u_{\mathrm{in}},v_{0})=\frac{1}{2}(v_{0}-u_{\mathrm{in}})$ (16)

and

$r^{*}(u_{\mathrm{out}},v_{0})=\frac{1}{2}(v_{0}-u_{\mathrm{out}}),$ (17)

we arrive at the coordinate relation
$u_{\mathrm{out}}=u_{\mathrm{out}}(u_{\mathrm{in}})$ through Eq. (10),

$\begin{split}u_{\mathrm{out}}(u_{\mathrm{in}})=v_{0}&-\frac{1}{\kappa_{B}}\ln\left(\frac{v_{0}-u_{\mathrm{in}}-2r_{B}}{2r_{B}}\right)\\\
&+\frac{1}{\kappa_{C}}\ln\left(\frac{2r_{C}-v_{0}+u_{\mathrm{in}}}{2r_{C}}\right)\\\
&-\frac{1}{\kappa_{0}}\ln\left(-\frac{v_{0}-u_{\mathrm{in}}-2r_{0}}{2r_{0}}\right).\end{split}$
(18)

From the above expression we check that if $u_{\mathrm{in}}=v_{0}-2r_{B}$ we
have that $u_{\mathrm{out}}\rightarrow+\infty$ whereas if
$u_{\mathrm{in}}=v_{0}-2r_{C}$ we have that
$u_{\mathrm{out}}\rightarrow-\infty$. Hence, we define $v_{B}\equiv
v_{0}-2r_{B}$ and $v_{C}\equiv v_{0}-2r_{C}$, denoting the null rays that
represent the horizons of SdS spacetime. The coordinate relation
$u_{\mathrm{out}}=u_{\mathrm{out}}(u_{\mathrm{in}})$ is, then,

$\begin{split}u_{\mathrm{out}}(u_{\mathrm{in}})=v_{0}&-\frac{1}{\kappa_{B}}\ln\left(\frac{v_{B}-u_{\mathrm{in}}}{2r_{B}}\right)\\\
&+\frac{1}{\kappa_{C}}\ln\left(\frac{u_{\mathrm{in}}-v_{C}}{2r_{C}}\right)\\\
&-\frac{1}{\kappa_{0}}\ln\left(-\frac{v_{0}-u_{\mathrm{in}}-2r_{0}}{2r_{0}}\right),\end{split}$
(19)

with $v_{C}<u_{\mathrm{in}}<v_{B}$. We can set (without loss of generality)
$v_{0}=0$ so $v_{B}=-2r_{B}$, $v_{C}=-2r_{C}$, and

$\begin{split}u_{\mathrm{out}}(u_{\mathrm{in}})=&-\frac{1}{\kappa_{B}}\ln\left(\frac{v_{B}-u_{\mathrm{in}}}{2r_{B}}\right)\\\
&+\frac{1}{\kappa_{C}}\ln\left(\frac{u_{\mathrm{in}}-v_{C}}{2r_{C}}\right)\\\
&-\frac{1}{\kappa_{0}}\ln\left(\frac{u_{\mathrm{in}}+2r_{0}}{2r_{0}}\right).\end{split}$
(20)

###  III.2 (1+1)-dimensional SdS mirror trajectory

A massless quantum scalar field $\phi(x)$ propagating in a (1+3)-dimensional
spacetime with spherical symmetry must satisfy the condition $\phi(t,r=0)=0$,
since the correspondent modes are spherical-type waves. This regularity
condition at $r=0$ allows determination of the exact form of the “out” modes
in the “in” (Minkowski) part of the spacetime where at $r=0$ we identify the
(Minkowski) coordinates $u_{\mathrm{in}}$ and $v$, i.e. $u_{\mathrm{in}}=v$.
On the other hand, a massless quantum scalar field $\phi(x)$ propagating in
(1+1)-dimensional Minkowski spacetime with a mirror must satisfy the boundary
condition $\phi(t,z(t))=0$, where $z(t)$ parameterizes the mirror perfectly
reflecting trajectory. Accordingly, we can think of the origin of coordinates
$r=0$ in the (1+3)-dimensional spacetime as a mirror in flat spacetime with a
trajectory that we can read straightforwardly as the the coordinate relation
$u_{\mathrm{out}}=u_{\mathrm{out}}(v)$. At this point, we move to the
(1+1)-dimensional acceleration model. The SdS moving mirror is completely
described by the following function of advanced time that parameterizes the
perfectly reflecting trajectory in flat spacetime:

$\begin{split}f(v)=&-\frac{1}{\kappa_{B}}\ln\left(\frac{v_{B}-v}{2r_{B}}\right)\\\
&+\frac{1}{\kappa_{C}}\ln\left(\frac{v-v_{C}}{2r_{C}}\right)\\\
&-\frac{1}{\kappa_{0}}\ln\left(\frac{v+2r_{0}}{2r_{0}}\right),\end{split}$
(21)

Finally, let us rewrite the SdS mirror trajectory, Eq. (21), as

$\begin{split}f(v)=&-\frac{1}{\kappa_{B}}\ln\left[\bar{\kappa}_{B}(v_{B}-v)\right]\\\
&+\frac{1}{\kappa_{C}}\ln\left[\bar{\kappa}_{C}(v-v_{C})\right]\\\
&-\frac{1}{\kappa_{0}}\ln\left[\bar{\kappa}_{0}(v+v_{B}+v_{C})\right],\end{split}$
(22)

where the constants $\bar{\kappa}_{B}\equiv(2r_{B})^{-1}$,
$\bar{\kappa}_{C}\equiv(2r_{C})^{-1}$ and
$\bar{\kappa}_{0}\equiv(2r_{0})^{-1}$ have been conveniently defined. In the
SdS mirror trajectory, Eq. (22), one should understand $\kappa_{B}$ (or
$\kappa_{C}$) as an acceleration parameter in the future (or in the past), and
$v_{B}$ (or $v_{C}$) as the null asymptote of the mirror trajectory in the
future (or in the past). In gravitational terms, $\kappa_{B}$ (or
$\kappa_{C}$) is the black hole (or cosmological) surface gravity and $v_{B}$
(or $v_{C}$) is the null ray representing the black hole (or cosmological)
horizon. The causal history of Eq. (22) is depicted in a Penrose diagram in
Fig. 1.

Figure 1: Penrose diagram showing the trajectory, Eq. (22), of the SdS moving
mirror. We see the two acceleration horizons (yellow and red lines), which are
analogs of the past cosmological horizon and the future black hole horizon,
respectively. The purple line is the SdS moving mirror. Here the gravitational
parameters are taken as $M=1/24$ and $L=2/3$ for clarity.

As a check, one can evaluate the Schwarzschild limit $(L\rightarrow\infty)$
and de Sitter limit $(M\rightarrow 0)$ of the accelerated boundary
correspondence with SdS spacetime, Eq. (22). In the limit $L\rightarrow\infty$
we get

$f(v)\simeq v-\frac{1}{\kappa}\ln\left[\kappa(v_{H}-v)\right],$ (23)

with $\kappa=(4M)^{-1}$ and $v_{H}=-4M$, recovering the result of the
accelerated boundary correspondence with the Schwarzschild black hole Good
_et al._ (2016). In the limit $M\rightarrow 0$ we get

$f(v)\simeq\frac{2}{\kappa}\tanh^{-1}\left(\frac{\kappa v}{2}\right),$ (24)

where $\kappa=L^{-1}$ and $v_{H}=\pm 2L$, recovering the result of the
accelerated boundary correspondence with de Sitter cosmology Good  _et al._
(2020c).

##  IV Energy Flux

After having analyzed the accelerated boundary correspondence with SdS
spacetime and before studying the particle creation by the SdS moving mirror,
we find it insightful to investigate the energy flux radiated by the perfectly
reflecting trajectory. An analysis of the energy-momentum tensor in the
(1+1)-dimensional quantum theory of a massless scalar field disturbed by the
boundary dynamics in flat spacetime yields an expression for the energy flux
Fulling and Davies (1976); Davies and Fulling (1977), which one can write as

$F(v)=\frac{1}{24\pi}\left(\frac{df(v)}{dv}\right)^{-2}\left\\{f(v),v\right\\}.$
(25)

The brackets define the Schwarzian derivative,

$\left\\{f(v),v\right\\}\equiv\frac{f^{\prime\prime\prime}(v)}{f^{\prime}(v)}-\frac{3}{2}\left(\frac{f^{\prime\prime}(v)}{f^{\prime}(v)}\right)^{2},$
(26)

where the prime denotes derivative with respect to $v$. The explicit
expression that follows from Eqs. (22) and (25) is lengthy but, from it, we
can get an idea of the constant energy flux and the distribution of particles
near the acceleration horizons (null asymptotes) of the mirror trajectory. A
plot of the energy flux is given in Fig. 2. On the one hand, at early times
$(v\rightarrow v_{C})$ and to leading order we have that

$F(v)=\frac{\kappa_{C}^{2}}{48\pi}+\mathcal{O}((v-v_{C})^{2}).$ (27)

On the other hand, at late times $(v\rightarrow v_{B})$ and to leading order
we have that

$F(v)=\frac{\kappa_{B}^{2}}{48\pi}+\mathcal{O}((v-v_{B})^{2}).$ (28)

These results show different constant energy flux at early and late times for
the SdS moving mirror. Furthermore, the above results, Eqs. (27) and (28)
suggest two Planck distributions of particles at different temperatures
$T_{C}$ and $T_{B}$ that are proportional to the acceleration parameter in the
past and future $\kappa_{C}=2\pi T_{C}$ and $\kappa_{B}=2\pi T_{B}$,
respectively. These two thermal spectra will be explicitly calculated in the
next section.

Figure 2: Energy flux, Eq. (25), as a function of advanced time $v$;
demonstrating the different constant energy flux at the horizon locations
(yellow and red lines). Here the gravitational parameters are taken as
$M=1/24$ and $L=2/3$ for consistency with Fig. 1. The key takeaway is that the
energy flux (purple line) increases monotonically from the smaller constant
value of the analog cosmological horizon in the past to the larger constant
value of the analog black hole horizon in the future. The black hole horizon
is hotter than the cosmological horizon. See Eqs. (27) and (28) for the early-
time and late-time energy flux asymptotic behavior.

##  V Particle Production

We are now in a position to address the particle creation by the SdS moving
mirror. The main goal is to obtain the solution to the Bogoliubov coefficient
$\beta_{\omega\omega^{\prime}}$ Birrell and Davies (1984),

$\beta_{\omega\omega^{\prime}}=\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{v_{C}}^{v_{B}}dv\,e^{-i\omega^{\prime}v-i\omega
f(v)},$ (29)

where $\omega$ $(\omega^{\prime})$ are the frequencies of the outgoing
(ingoing) modes. The calculation of the beta Bogoliubov coefficient directly
via Eq. (22) is intractable. As a first approximation, let us consider the
following form for the SdS mirror trajectory:

$f(v)\approx-\frac{1}{\kappa_{B}}\ln\left[\bar{\kappa}_{B}(v_{B}-v)\right]+\frac{1}{\kappa_{C}}\ln\left[\bar{\kappa}_{C}(v-v_{C})\right],$
(30)

instead of the full expression, Eq. (22). Note that we have ignored the third
logarithmic term. This approximation of the SdS mirror trajectory, Eq. (30),
finds its justification at early and late times, as it presents the same
asymptotic behaviour in the far past and far future as the exact expression.
Notice that, in the exact SdS mirror trajectory, Eq. (22), the first
logarithmic term dominates at late times $(v\rightarrow v_{B})$ and the second
logarithmic term dominates at early times $(v\rightarrow v_{C})$, which are
the two terms in the approximated SdS mirror trajectory, Eq. (30). The third
logarithmic term has significance not at early and late times but at the
intermediate times of the trajectory. However, in the regime
$3\sqrt{3}M/L\simeq 1$ (which we have considered in Figs. 3 and 4), one finds
that the magnitude of this third term is negligible in comparison to the first
and second terms, justifying our approximation. See Fig. 3 for a Penrose
diagram with the approximated trajectory overlaid on the exact trajectory. See
Fig. 4 for a comparison of the energy fluxes of the approximated trajectory
overlaid on the energy flux of the exact trajectory. As shown in Figs. 3 and
4, the asymptotic and exact SdS mirrors are qualitatively similar (both
dynamically and energetically), and hence one can have confidence that they
will likewise possess a similar particle production spectra as a result.

Figure 3: Penrose diagram showing the approximated and exact SdS moving mirror
trajectories, Eq. (30) (black dashed line) and Eq. (22) (purple line). Wee see
that asymptotic version of the SdS mirror trajectory also has the exact same
two horizons, which are analogs of the past cosmological horizon (yellow line)
and the future black hole horizon (red line). Here $M=1/24$ and $L=2/3$ as
before. At the intermediate times of acceleration the bifurcation of both
trajectories manifests. Figure 4: Energy fluxes $F_{a}(v)$ and $F(v)$ for the
approximated and exact SdS moving mirror trajectories, Eqs. (30) and (22),
obtained from Eq. (25) as a function of advanced time $v$; demonstrating the
different constant energy fluxes at the horizon locations (yellow and red
lines). Here $M=1/24$ and $L=2/3$ as before. The key takeaway is that the
energy fluxes, $F_{a}(v)$ and $F(v)$, are similar. Again, like in Fig. 2, the
energy flux increases monotonically from the smaller constant value of the
analog cosmological horizon in the past to the larger constant value of the
analog black hole horizon in the future, highlighting that the black hole
horizon is hotter than the cosmological horizon. Eqs. (27) and (28) are the
usual early-time and late-time energy flux asymptotic behavior. The same color
scheme is used in the Penrose diagram of Fig. 3.

This approximation is inspired by the method used in the first papers on
analog moving mirrors by Fulling and Davies Fulling and Davies (1976); Davies
and Fulling (1977). In them, they began by considering an asymptotic form of a
mirror trajectory at late times since the behaviour of the mirror trajectory
at early times was considered irrelevant to analyze the particle production in
the asymptotic late-time region. What Fulling and Davies did was to
approximate the calculation of the Bogoliubov coefficients by using the
asymptotic late-time form of the mirror trajectory at all times. Then, the
beta Bogoliubov coefficient derived in that way was a good approximation in
describing the particle production in the asymptotic late-time region. That
analog mirror asymptotic solution has the same form as the asymptotic
Schwarzschild mirror. In the case of the analog SdS moving mirror developed in
this paper, we have determined the exact form of the mirror trajectory, Eq.
(22), but since the computation of the Bogoliubov coefficients is intractable,
and the asymptotic early-time and late-time regions are relevant to study the
particle production, we approximate the mirror trajectory by using the
asymptotic form, Eq. (30), at all times; just as Fulling and Davies did. Now,
the beta Bogoliubov coefficient to be derived in this way will be a good
approximation in describing the particle production in the asymptotic early-
time and late-time regions. Importantly, the asymptotic SdS mirror will
certainly give approximate but physically relevant particle production results
in the intermediate-time region.

Thus, bearing in mind the regimes of validity of the approximation and the
subsequent discussion, we will derive the beta Bogoliubov coefficient from Eq.
(29) with $f(v)$ given by Eq. (30). Then, we define

$a\equiv\frac{i\omega^{\prime}}{\kappa^{\prime}}\quad,\quad
b\equiv\frac{i\omega}{\kappa_{B}},\quad,\quad
c\equiv\frac{i\omega}{\kappa_{C}}\quad.$ (31)

with $\kappa^{\prime}=(v_{B}-v_{C})^{-1}$. The final result is

$\begin{split}\beta_{\omega\omega^{\prime}}=&\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\frac{(\bar{\kappa}_{B})^{b}(\bar{\kappa}_{C})^{-c}}{(\kappa^{\prime})^{1+b-c}}e^{-i\omega^{\prime}v_{B}}\\\
&\cdot\frac{\Gamma(1+b)\Gamma(1-c)}{\Gamma(2+b-c)}{}_{1}F_{1}(1+b;2+b-c;a),\end{split}$
(32)

where $\Gamma(x)$ is the Gamma function and ${}_{1}F_{1}(p;q;z)$ is the
confluent hypergeometric function of the first kind. The explicit derivation
of the above expression is developed in Appendix A. Thereby, using the
regularized confluent hypergeometric function
${}_{1}\widetilde{F}_{1}(p;q;z)\equiv{}_{1}F_{1}(p;q;z)/\Gamma(q)$, we obtain

$\begin{split}\beta_{\omega\omega^{\prime}}=&\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\frac{(\bar{\kappa}_{B})^{b}(\bar{\kappa}_{C})^{-c}}{(\kappa^{\prime})^{1+b-c}}e^{-i\omega^{\prime}v_{B}}\\\
&\cdot\Gamma(1+b)\Gamma(1-c){}_{1}\widetilde{F}_{1}(1+b;2+b-c;a).\end{split}$
(33)

The particle spectrum per mode $\omega$ and per mode $\omega^{\prime}$ is the
modulus square of the beta Bogoliubov coefficient in Eq. (33),

$N_{\omega\omega^{\prime}}\equiv\left|\beta_{\omega\omega^{\prime}}\right|^{2}.$
(34)

Therefore, we write $N_{\omega\omega^{\prime}}$ as

$N_{\omega\omega^{\prime}}=\frac{a^{2}bc}{\omega\omega^{\prime}}\frac{e^{i\pi(b+c)}}{\left(e^{2i\pi
b}-1\right)\left(e^{2i\pi
c}-1\right)}\left|{}_{1}\widetilde{F}_{1}\right|^{2},$ (35)

Notice the not-so-inconspicuous hint of two Planck distributions in the
denominator. In the high frequency regime, i.e. $\omega^{\prime}\gg\omega$, we
expect to see a late-time thermal spectrum, as is the case in Hawking
radiation Hawking (1975), see e.g. the analog Schwarzschild black hole mirror
Good  _et al._ (2016). Importantly, in this situation we also expect to see a
early-time thermal spectrum, corresponding the de Sitter cosmological horizon.
Expanding Eq. (35) for large $\omega^{\prime}$, gives to leading order,

$N_{\omega\omega^{\prime}}=N_{B}+N_{C}+N_{BC},$ (36)

where

$N_{B}\equiv\frac{1}{2\pi\kappa_{B}\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa_{B}}-1},$
(37)

which is the expected late-time thermal spectrum for the black hole horizon.
This result agrees with particle creation by the asymptotic version of the SdS
moving mirror in a Planck distribution at a temperature $T_{B}$ that is
proportional to acceleration parameter of the boundary trajectory in the
future, $\kappa_{B}=2\pi T_{B}$. What about the expected early-time thermal
spectrum for the cosmological horizon? This term also makes an appearance:

$N_{C}\equiv\frac{1}{2\pi\kappa_{C}\omega^{\prime}}\frac{1}{e^{2\pi\omega/\kappa_{C}}-1}.$
(38)

This result agrees with particle creation by the asymptotic version of the SdS
moving mirror in a Planck distribution at a temperature $T_{C}$ that is
proportional to acceleration parameter of the boundary trajectory in the past,
$\kappa_{C}=2\pi T_{C}$. With the above arguments, we confirm that the
(Schwarzschild black hole) late-time thermal spectrum in Eq. (37) and the (de
Sitter cosmology) early-time thermal spectrum in Eq. (38) are present in SdS
moving mirror system. Furthermore, an additional complicated cross term is
obtained, indicating the approximate form of the interaction between both the
black hole horizon and the cosmological horizon of SdS spacetime. The presence
of this term is reminiscent of the treatment of radiation resistances for
resistors in ‘parallel’ (see e.g. Gray  _et al._ (2016)). This term is

$N_{BC}=\bar{N}_{BC}+\bar{N}_{BC}^{*},$ (39)

where

$\begin{split}\bar{N}_{BC}\equiv&-\frac{1}{4\pi^{2}\omega\omega^{\prime}}\left(\frac{\omega^{\prime}}{\kappa^{\prime}}\right)^{-\frac{i\omega}{\kappa}}e^{-\frac{\pi\omega}{2\kappa}-\frac{i\omega^{\prime}}{\kappa^{\prime}}}\\\
&\cdot\Gamma\left(1+\frac{i\omega}{\kappa_{B}}\right)\Gamma\left(1+\frac{i\omega}{\kappa_{C}}\right),\end{split}$
(40)

where we have defined $\kappa^{-1}=\kappa_{B}^{-1}+\kappa_{C}^{-1}$, and
$\kappa^{\prime}=(v_{B}-v_{C})^{-1}$ as in Eq. (31). This result indicates the
particle creation by the asymptotic version of the SdS moving mirror at
intermediate times. It demonstrates a non-thermal particle distribution
between the early-time and late-time thermal equilibria. Note that this result
implies (for SdS) that the regime $\omega^{\prime}\gg\omega$ captures both the
late time and the early time radiation (this limit is most commonly associated
with late-time radiation in the context of Schwarzschild Hawking (1975)). Eq.
(40) has a counterpart in the SdS moving mirror system, and it should take a
different but similar form since it shows the effect of the acceleration
horizons and the interaction behaviour between them resulting in the non-
thermal spectrum. The radiation will almost certainly be influenced by the
intermediate-time trajectory dynamics. Eq. (40) is one of the more interesting
results of our analysis and a key novelty of the SdS-type radiation where two
Planck distributions ‘interact’ to give an overall non-thermal spectrum.

##  VI Dual-Temperature ABC

Once having obtained the early-time and late-time particle production results
for the SdS moving mirror through the use of the trajectory Eq. (30), an open
question remains: What is the approximate form of SdS spacetime that gives the
asymptotic SdS mirror trajectory? A brief calculation gives an approximate
metric of the form of Eq. (1), with

$f(r)\approx\frac{A(r-r_{B})(r_{C}-r)}{r+B}.$ (41)

Here the horizons locations $r=r_{B}$ and $r=r_{C}$ are obtained from the SdS
metric, Eq. (3), with $A>0$ and $B>0$ being two constants determined by
imposing that the corresponding surface gravities obtained from the SdS
metric, Eq. (3), are the same as those obtained from the approximate metric,
Eq. (41).

Additionally, we can be more precise about the all-time particle evolution in
the corresponding exact spacetime. This is a tractable problem which can be
answered by solving for the exact spacetime analog for Eq. (30). The
accelerated boundary correspondence (ABC) metric has a simplified spherically
symmetric form of Eq. (1), with

$f(r)\equiv 1-\frac{2M}{r}-\frac{r}{L},$ (42)

where $M>0$ and $L>0$ are two constants. We keep this notation for the
constants to contrast with the SdS curved spacetime system, Eq. (3). Despite
the resemblance between Eqs. (3) and (42), the constants $M$ and $L$ of Eq.
(42) do not have an immediately clear physical meaning, unlike the SdS metric.
The corresponding horizons are, respectively, at $r=r_{B}$ and $r=r_{C}$ with

$r_{B}=\frac{1}{2}\left[L-\sqrt{L(L-8M)}\right],$ (43)

and

$r_{C}=\frac{1}{2}\left[L+\sqrt{L(L-8M)}\right],$ (44)

where $r_{B}\leq r_{C}$. In this case there exists no negative, nonphysical
solution. This is the key point to eventually get an exact mirror with
trajectory of the form of Eq. (30). The surface gravities associated with each
horizon are, respectively,

$\kappa_{B}=\frac{r_{C}-r_{B}}{2Lr_{B}},$ (45)

and

$\kappa_{C}=\frac{r_{C}-r_{B}}{2Lr_{C}}.$ (46)

If we follow the same steps as in Secs. II and III we arrived at exactly the
mirror trajectory of Eq. (30) with analog horizons $r_{i}$ and surface
gravities $\kappa_{i}$ obtained from the ABC metric. In this case, the
particle production results in Sec. V are completely valid at all times. Note
that the SdS mirror trajectory, Eq. (22), has analog horizons $r_{i}$ and
surface gravities $\kappa_{i}$ obtained from the SdS metric, where the
particle production results in Sec. V are completely valid at early and late
times.

We conclude by noting that the model in Good  _et al._ (2021b) likewise
presents an accelerated mirror trajectory with two different temperatures
(Planck distributions) at asymptotically early and late times. This previously
studied mirror has early-time and late-time thermal particle spectra, just
like the SdS system. However, in Good  _et al._ (2021b) there is no horizon at
early times, whereas the SdS moving mirror has a cosmological horizon in this
region.

##  VII Final Remarks

In this work, we have analyzed the horizon radiation of SdS spacetime with
respect to static observers by using the moving mirror model. In particular,
there a few salient steps forward made:

  * •

We have derived the accelerated boundary correspondence, Eq. (21) or Eq. (22),
to SdS spacetime and determined the energy flux of the SdS moving mirror, Eq.
(25).

  * •

We have derived the particle spectrum, Eq. (35), of the asymptotic SdS moving
mirror, Eq. (30).

  * •

We have determined the corresponding curvature analog to the all-time particle
spectrum of the asymptotic SdS moving mirror, Eq. (30).

The temperature and thermodynamics of the SdS spacetime has been the object of
study because a static observer in this universe will have to interact with
both the black hole horizon and the cosmological horizon and their quantum
particle creation effects. The open question of what actually constitutes a
thermodynamic equilibrium in this scenario can be ignored with a small
cosmological constant but becomes important with a large cosmological
constant. While many previous studies have defined effective temperatures for
the SdS spacetime and numerous proposals for temperature have been introduced
(see e.g. Pappas and Kanti (2017); Robson  _et al._ (2019); Singha (2021) and
references therein), our work conclusively delivers on the non-thermal
distribution identified with the interacting spectra. The all-time non-thermal
spectrum contains definitive early-time and late-time regimes of energy flux
in thermal equilibrium indicative of the particle production in different
Planck distributions. It highlights the limited utility of associating an
interactive temperature to describe the SdS spacetime as a whole (in agreement
with Choudhury and Padmanabhan (2007)). While the temperature of the black
hole horizon defined in terms of the corresponding surface gravity fails to
take into account the lack of an asymptotically flat limit, the well-defined
and expected Planck distributed particle radiation spectrum is nevertheless
present in the analog moving mirror system, which is correspondingly
asymptotically flat.

### Future Work

While we have addressed the study of the analog SdS moving mirror and
concretely tackled the problem of particle production using standard tools of
quantum field theory in curved spacetime, i.e. the formalism of the Bogoliubov
transformations Birrell and Davies (1984), there are future extensions. A
physically appropriate approximation, i.e. the asymptotic SdS moving mirror,
has been introduced and so we have managed to obtain the beta Bogoliubov
coefficient, but the next step is to try to arrive at particular,
intermediate-time, physical results without using any approximation. With this
aim, alternative approaches that avoid the calculation of the Bogoliubov
coefficients can be considered, and so we point out the formalism introduced
in Fabbri  _et al._ (2004) which takes advantage of the conformal symmetry
present in (1+1)-dimensional systems (like the mirror model). This formalism
has been applied in black hole and cosmological scenarios Agullo  _et al._
(2007, 2008, 2009, 2010) and it has also been applied recently to analyze the
particle creation by wormholes Gurrea-Ysasi and Olmo (2020). It would be
insightful to try to use it in gravitational analog mirror models like the SdS
ABC.

Alternatively, the extension to general cosmological black hole spacetimes
would be interesting as well. In particular, the generalization to Kerr-de
Sitter spacetime Bhattacharya (2018); Gregory  _et al._ (2021) could provide
insight into the radiation of a rapidly spinning black hole and its
evaporation process in an expanding universe. Our approach would confirm the
cosmological constant quenching effect on the amplification of quantum vacuum
fluctuations, among other things.

The analogy between black holes and moving mirrors has encouraged recent
fruitful investigations (see e.g. Akal  _et al._ (2021a); Reyes (2021)). In
this year (2021) alone, we have seen special attention given to the moving
mirror model with treatments ranging from various fields and topics like
entanglement capacity Kawabata  _et al._ (2021), harvesting Liu  _et al._
(2021), holography Akal  _et al._ (2021b), qubits Agustí  _et al._ (2021),
relativistic semi-transparency Lin and Chen (2021), Page curves Ageev (2021)
and complexity Sato (2021). We expect the model to continue to be enriched by
diverse and novel ideas connecting the nature of quantum vacuum radiation to
the resulting effects of acceleration and gravity.

###### Acknowledgements.

Thanks in memory to the late Thanu Padmanabhan. We are grateful to Alessandro
Fabbri for his comments on an early version of this paper. DFS is appreciative
for continuous discussions and support, and also thanks Gonzalo J. Olmo for
his observations on possible future work. MG thanks Aizhan Myrzakul for
discussions. Funding from the state-targeted program “Center of Excellence for
Fundamental and Applied Physics” (BR05236454) by the Ministry of Education and
Science of the Republic of Kazakhstan is acknowledged, as well as the
FY2021-SGP-1-STMM Faculty Development Competitive Research Grant No.
021220FD3951 at Nazarbayev University. JF acknowledges support from the
Australian Research Council Centre of Excellence for Quantum Computation and
Communication Technology (Project No. CE170100012).

##  Appendix A Bogoliubov coefficient for the approximated asymptotic SdS
moving mirror

In this Appendix, we determine the expression of the beta Bogoliubov
coefficient in Eq. (32). We begin by considering Eq. (29) and the asymptotic
SdS mirror trajectory, Eq. (30), to end up with

$\begin{split}\beta_{\omega\omega^{\prime}}=&\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\int_{v_{C}}^{v_{B}}dv\\\
\cdot&\left[\bar{\kappa}_{B}(v_{B}-v)\right]^{\frac{i\omega}{\kappa_{B}}}\left[\bar{\kappa}_{C}(v-v_{C})\right]^{-\frac{i\omega}{\kappa_{C}}}e^{-i\omega^{\prime}v}.\end{split}$
(47)

A change of variable $x=v_{B}-v$ allows us to arrive at

$\begin{split}\beta_{\omega\omega^{\prime}}=&-\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}e^{-i\omega^{\prime}v_{B}}\int_{v_{B}-v_{C}}^{0}dx\\\
\cdot&\left[\bar{\kappa}_{B}(x)\right]^{\frac{i\omega}{\kappa_{B}}}\left[\bar{\kappa}_{C}(v_{B}-v_{C}-x)\right]^{-\frac{i\omega}{\kappa_{C}}}e^{i\omega^{\prime}x}.\end{split}$
(48)

Another change of variable $x^{\prime}=\kappa^{\prime}x$ with
$\kappa^{\prime}=(v_{B}-v_{C})^{-1}$ leads us, after some steps, to

$\begin{split}\beta_{\omega\omega^{\prime}}=&\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\frac{(\bar{\kappa}_{B})^{b}(\bar{\kappa}_{C})^{-c}}{(\kappa^{\prime})^{1+b-c}}e^{-i\omega^{\prime}v_{B}}\\\
\cdot&\int_{0}^{1}dx^{\prime}\,(x^{\prime})^{b}(1-x^{\prime})^{-c}e^{ax^{\prime}},\end{split}$
(49)

where we have defined

$a\equiv\frac{i\omega^{\prime}}{\kappa^{\prime}}\quad,\quad
b\equiv\frac{i\omega}{\kappa_{B}},\quad,\quad
c\equiv\frac{i\omega}{\kappa_{C}}\quad.$ (50)

Finally, the integral in Eq. (49) is analytically solved in terms of the Gamma
function $\Gamma(x)$ and the confluent hypergeometric function of the first
kind ${}_{1}F_{1}(p;q;z)$ as

$\begin{split}&\int_{0}^{1}dx^{\prime}\,(x^{\prime})^{b}(1-x^{\prime})^{-c}e^{ax^{\prime}}=\\\
&\frac{\Gamma(1+b)\Gamma(1-c)}{\Gamma(2+b-c)}{}_{1}F_{1}(1+b;2+b-c;a),\end{split}$
(51)

and we get

$\begin{split}\beta_{\omega\omega^{\prime}}=&\frac{1}{2\pi}\sqrt{\frac{\omega^{\prime}}{\omega}}\frac{(\bar{\kappa}_{B})^{b}(\bar{\kappa}_{C})^{-c}}{(\kappa^{\prime})^{1+b-c}}e^{-i\omega^{\prime}v_{B}}\\\
&\cdot\frac{\Gamma(1+b)\Gamma(1-c)}{\Gamma(2+b-c)}{}_{1}F_{1}(1+b;2+b-c;a),\end{split}$
(52)

which is the beta Bogoliubov coefficient in Eq. (32).

## References

  * Riess  _et al._ (1998) A. G. Riess _et al._ (Supernova Search Team), Astron. J. 116, 1009 (1998), arXiv:astro-ph/9805201 .
  * Perlmutter  _et al._ (1999) S. Perlmutter _et al._ (Supernova Cosmology Project), Astrophys. J. 517, 565 (1999), arXiv:astro-ph/9812133 .
  * Guth (1981) A. H. Guth, Phys. Rev. D 23, 347 (1981).
  * Hawking (1975) S. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * Unruh (1976) W. G. Unruh, Phys. Rev. D 14, 870 (1976).
  * Gibbons and Hawking (1977) G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977).
  * Bhattacharya (2018) S. Bhattacharya, Phys. Rev. D 98, 125013 (2018), arXiv:1810.13260 [gr-qc] .
  * Qiu and Traschen (2020) Y. Qiu and J. Traschen, Class. Quant. Grav. 37, 135012 (2020), arXiv:1908.02737 [hep-th] .
  * DeWitt (1975) B. S. DeWitt, Phys. Rept. 19, 295 (1975).
  * Moore (1970) G. T. Moore, Journal of Mathematical Physics 11, 2679 (1970).
  * Fulling and Davies (1976) S. A. Fulling and P. C. W. Davies, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 348, 393 (1976).
  * Davies and Fulling (1977) P. Davies and S. Fulling, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences A356, 237 (1977).
  * Walker (1985) W. R. Walker, Phys. Rev. D 31, 767 (1985).
  * Carlitz and Willey (1987) R. D. Carlitz and R. S. Willey, Phys. Rev. D 36, 2327 (1987).
  * Bianchi and Smerlak (2014a) E. Bianchi and M. Smerlak, Phys. Rev. D 90, 041904 (2014a), arXiv:1404.0602 [gr-qc] .
  * Bianchi and Smerlak (2014b) E. Bianchi and M. Smerlak, Gen. Rel. Grav. 46, 1809 (2014b), arXiv:1405.5235 [gr-qc] .
  * Good and Linder (2018) M. R. Good and E. V. Linder, Phys. Rev. D 97, 065006 (2018), arXiv:1711.09922 [gr-qc] .
  * Cong  _et al._ (2019) W. Cong, E. Tjoa, and R. B. Mann, JHEP 06, 021 (2019), arXiv:1810.07359 [quant-ph] .
  * Romualdo  _et al._ (2019) I. Romualdo, L. Hackl, and N. Yokomizo, Phys. Rev. D 100, 065022 (2019), arXiv:1908.00835 [quant-ph] .
  * Lee and Yeh (2019) D.-S. Lee and C.-P. Yeh, JHEP 06, 068 (2019), arXiv:1904.06831 [hep-th] .
  * Good  _et al._ (2020a) M. R. Good, E. V. Linder, and F. Wilczek, Phys. Rev. D 101, 025012 (2020a), arXiv:1909.01129 [gr-qc] .
  * Foo  _et al._ (2020) J. Foo, S. Onoe, M. Zych, and T. C. Ralph, New J. Phys. 22, 083075 (2020), arXiv:2004.07094 [quant-ph] .
  * Stargen  _et al._ (2016) D. J. Stargen, D. A. Kothawala, and L. Sriramkumar, Phys. Rev. D 94, 025040 (2016), arXiv:1602.02526 [hep-th] .
  * Good and Abdikamalov (2020) M. Good and E. Abdikamalov, Universe 6, 131 (2020), arXiv:2008.08776 [gr-qc] .
  * Birrell and Davies (1984) N. Birrell and P. Davies,  _Quantum Fields in Curved Space_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, 1984).
  * Fabbri and Navarro-Salas (2005) A. Fabbri and J. Navarro-Salas,  _Modeling Black Hole Evaporation_ (Imperial College Press, London, 2005).
  * Davies (1982) P. C. W. Davies, Phys. Lett. B 113, 215 (1982).
  * Helfer (2001) A. D. Helfer, Phys. Rev. D 63, 025016 (2001), arXiv:hep-th/0001004 .
  * Gianfelici and Mancini (2017) G. Gianfelici and S. Mancini, Sci. Rep. 7, 15747 (2017), arXiv:1704.04635 [quant-ph] .
  * Chen and Yeom (2017) P. Chen and D.-h. Yeom, Phys. Rev. D 96, 025016 (2017), arXiv:1704.08613 [hep-th] .
  * Good and Linder (2019) M. R. Good and E. V. Linder, Phys. Rev. D 99, 025009 (2019), arXiv:1807.08632 [gr-qc] .
  * Chen and Mourou (2017) P. Chen and G. Mourou, Phys. Rev. Lett. 118, 045001 (2017), arXiv:1512.04064 [gr-qc] .
  * Chen and Mourou (2020) P. Chen and G. Mourou, Phys. Plasmas 27, 123106 (2020), arXiv:2004.10615 [physics.plasm-ph] .
  * Dodonov (2020) V. Dodonov, MDPI Physics 2, 67 (2020).
  * Good  _et al._ (2016) M. R. R. Good, P. R. Anderson, and C. R. Evans, Phys. Rev. D 94, 065010 (2016), arXiv:1605.06635 [gr-qc] .
  * Good and Ong (2020) M. R. R. Good and Y. C. Ong, Eur. Phys. J. C 80, 1169 (2020), arXiv:2004.03916 [gr-qc] .
  * Good  _et al._ (2021a) M. R. R. Good, J. Foo, and E. V. Linder, Class. Quant. Grav. 38, 085011 (2021a), arXiv:2006.01349 [gr-qc] .
  * Foo and Good (2021) J. Foo and M. R. R. Good, JCAP 01, 019 (2021), arXiv:2006.09681 [gr-qc] .
  * Foo  _et al._ (2021) J. Foo, M. R. R. Good, and R. B. Mann, Universe 7, 350 (2021), arXiv:2012.02348 [gr-qc] .
  * Liberati  _et al._ (2000) S. Liberati, T. Rothman, and S. Sonego, Phys. Rev. D 62, 024005 (2000), arXiv:gr-qc/0002019 .
  * Rothman (2000) T. Rothman, Phys. Lett. A 273, 303 (2000), arXiv:gr-qc/0006036 .
  * Good (2020) M. R. R. Good, Phys. Rev. D 101, 104050 (2020), arXiv:2003.07016 [gr-qc] .
  * Good and Ong (2015) M. R. R. Good and Y. C. Ong, JHEP 07, 145 (2015), arXiv:1506.08072 [gr-qc] .
  * Good  _et al._ (2017) M. R. R. Good, K. Yelshibekov, and Y. C. Ong, JHEP 03, 013 (2017), arXiv:1611.00809 [gr-qc] .
  * Good (2017) M. R. R. Good,  _Reflecting at the Speed of Light_ (World Scientific, Singapore, 2017).
  * Good (2018) M. R. Good, Universe 4, 122 (2018).
  * Myrzakul and Good (2018) A. Myrzakul and M. R. Good, in  _15th Marcel Grossmann Meeting on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories_ (2018) arXiv:1807.10627 [gr-qc] .
  * Good  _et al._ (2019) M. R. Good, Y. C. Ong, A. Myrzakul, and K. Yelshibekov, Gen. Rel. Grav. 51, 92 (2019), arXiv:1801.08020 [gr-qc] .
  * Walker and Davies (1982) W. R. Walker and P. C. W. Davies, Journal of Physics A: Mathematical and General 15, L477 (1982).
  * Good  _et al._ (2013) M. R. R. Good, P. R. Anderson, and C. R. Evans, Phys. Rev. D 88, 025023 (2013), arXiv:1303.6756 [gr-qc] .
  * Good and Linder (2017) M. R. R. Good and E. V. Linder, Phys. Rev. D 96, 125010 (2017), arXiv:1707.03670 [gr-qc] .
  * Good  _et al._ (2020b) M. R. R. Good, E. V. Linder, and F. Wilczek, Mod. Phys. Lett. A 35, 2040006 (2020b), arXiv:2108.11188 [quant-ph] .
  * Good  _et al._ (2020c) M. R. R. Good, A. Zhakenuly, and E. V. Linder, Phys. Rev. D 102, 045020 (2020c), arXiv:2005.03850 [gr-qc] .
  * Markovic and Unruh (1991) D. Markovic and W. G. Unruh, Phys. Rev. D 43, 332 (1991).
  * Anderson and Traschen (2020) P. Anderson and J. Traschen, (2020), arXiv:2012.08494 [hep-th] .
  * De-You  _et al._ (2006) C. De-You, J. Qing-Quan, L. Hui-Ling, and Y. Shu-Zheng, Chinese Physics B 15, 1425-1429 (2006).
  * Choudhury and Padmanabhan (2007) T. R. Choudhury and T. Padmanabhan, Gen. Rel. Grav. 39, 1789 (2007), arXiv:gr-qc/0404091 .
  * Kottler (1918) F. Kottler, Annalen der Physik 361, 401 (1918).
  * Stephani  _et al._ (2003) H. Stephani, D. Krammer, M. MacCallum, C. Hoenselaers, and E. Herlt,  _Exact Solutions of Einstein’s Field Equations_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, UK, 2003).
  * Gray  _et al._ (2016) F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, Class. Quant. Grav. 33, 115003 (2016), arXiv:1506.03975 [gr-qc] .
  * Good  _et al._ (2021b) M. R. R. Good, A. Mitra, and V. Zarikas, Astronomy Reports (2021b), arXiv:2102.00158 [gr-qc] .
  * Pappas and Kanti (2017) T. Pappas and P. Kanti, Phys. Lett. B 775, 140 (2017), arXiv:1707.04900 [hep-th] .
  * Robson  _et al._ (2019) C. W. Robson, L. D. M. Villari, and F. Biancalana, (2019), arXiv:1902.02547 [gr-qc] .
  * Singha (2021) C. Singha, (2021), arXiv:2108.11704 [gr-qc] .
  * Fabbri  _et al._ (2004) A. Fabbri, J. Navarro-Salas, and G. J. Olmo, Phys. Rev. D 70, 064022 (2004), arXiv:hep-th/0403021 .
  * Agullo  _et al._ (2007) I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Phys. Rev. D 76, 044018 (2007), arXiv:hep-th/0611355 .
  * Agullo  _et al._ (2008) I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Phys. Rev. D 77, 124032 (2008), arXiv:0804.0513 [hep-th] .
  * Agullo  _et al._ (2009) I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Phys. Rev. Lett. 103, 061301 (2009), arXiv:0901.0439 [astro-ph.CO] .
  * Agullo  _et al._ (2010) I. Agullo, J. Navarro-Salas, G. J. Olmo, and L. Parker, Phys. Rev. Lett. 105, 211305 (2010), arXiv:1006.4404 [hep-th] .
  * Gurrea-Ysasi and Olmo (2020) G. Gurrea-Ysasi and G. J. Olmo, Int. J. Mod. Phys. D 29, 2041009 (2020), arXiv:2004.06073 [hep-th] .
  * Gregory  _et al._ (2021) R. Gregory, I. G. Moss, N. Oshita, and S. Patrick, Class. Quant. Grav. 38, 185005 (2021), arXiv:2103.09862 [gr-qc] .
  * Akal  _et al._ (2021a) I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Phys. Rev. Lett. 126, 061604 (2021a), arXiv:2011.12005 [hep-th] .
  * Reyes (2021) I. A. Reyes, Phys. Rev. Lett. 127, 051602 (2021).
  * Kawabata  _et al._ (2021) K. Kawabata, T. Nishioka, Y. Okuyama, and K. Watanabe, JHEP 05, 062 (2021), arXiv:2102.02425 [hep-th] .
  * Liu  _et al._ (2021) Z. Liu, J. Zhang, and H. Yu, JHEP 20, 2021 (2021), arXiv:2101.00114 [quant-ph] .
  * Akal  _et al._ (2021b) I. Akal, Y. Kusuki, N. Shiba, T. Takayanagi, and Z. Wei, Class. Quant. Grav. 38, 224001 (2021b), arXiv:2106.11179 [hep-th] .
  * Agustí  _et al._ (2021) A. Agustí, L. García-Álvarez, E. Solano, and C. Sabín, Phys. Rev. A 103, 062201 (2021), arXiv:2011.02822 [quant-ph] .
  * Lin and Chen (2021) K.-N. Lin and P. Chen, (2021), arXiv:2107.09033 [gr-qc] .
  * Ageev (2021) D. S. Ageev, (2021), arXiv:2107.09083 [hep-th] .
  * Sato (2021) Y. Sato, (2021), arXiv:2108.04637 [hep-th] .