11institutetext:  Faculty of Computing, Blekinge Institute of Technology, 371
79 Karlskrona, Sweden
11email: {mengqiao.zhao<EMAIL_ADDRESS>22institutetext: Pontifical
Catholic University of Parana (PPGIa/PUCPR), Curitiba, Brazil
R. Imaculada Conceição, 1155, Curitiba, PR, Brazil - 80215-901
22email<EMAIL_ADDRESS>
# End-to-End Approach for Recognition of Historical Digit Strings

Mengqiao Zhao 11 Andre G. Hochuli 22 Abbas Cheddad 11

###### Abstract

The plethora of digitalised historical document datasets released in recent
years has rekindled interest in advancing the field of handwriting pattern
recognition. In the same vein, a recently published data set, known as ARDIS,
presents handwritten digits manually cropped from 15.000 scanned documents of
Swedish church books and exhibiting various handwriting styles. To this end,
we propose an end-to-end segmentation- free deep learning approach to handle
this challenging ancient handwriting style of dates present in the ARDIS
dataset (4-digits long strings). We show that with slight modifications in the
VGG-16 deep model, the framework can achieve a recognition rate of 93.2%,
resulting in a feasible solution free of heuristic methods, segmentation, and
fusion methods. Moreover, the proposed approach outperforms the well-known
CRNN method (a model widely applied in handwriting recognition tasks).[Cite
as: Mengqiao Zhao, Andre G. Hochuli and Abbas Cheddad, End-to-End Approach for
Recognition of Historical Digit Strings, to appear in the 16th International
Conference on Document Analysis and Recognition (ICDAR 2021), LNCS, Springer,
Lausanne, Switzerland.]

###### Keywords:

Handwriting Digit String Recognition Segmentation-Free Historical Document
Processing

##  1 Introduction

Due to the rapid growth of document storage in modern society, handwriting
recognition has become an important research branch in the field of machine
learning and pattern recognition. In the context of handwriting digit strings
recognition (HDSR), there are several application scenarios such as postcode
recognition, bank checks, document indexing[3] and word spotting [1, 4, 14].

Most state-of-the-art approaches focus on the segmentation of connected
components, followed by training a model to classify each component. However,
the performance collapses when two or more digits are touching. Besides,
variant approaches based on multiple segmentation algorithms have been
proposed. For such algorithms, to generate a potential segmentation cut, a
heuristic analysis based on background and foreground information, contour,
shape, or a combination of these is required to be implemented[26]. The over-
segmentation strategy is frequently used to optimize the segmentation of
touching components by over segmenting the string. Although over-segmentation
increases the probability of obtaining a plausible segmentation cut, it also
increases the computational cost since the hypothesis space expands
exponentially with the increase of the segmentation cuts. However, the use of
heuristic to deal with touching digits has shown that performance degrades by
the presence of noise, fragments, and the lack of context in digit strings,
such as a lexicon and the string length.

To better address these issues and to take advantage of deep learning models,
segmentation-free methods came to the surface, offering unique capabilities to
the research community in the domain [19, 2, 17, 23]. Along this line, the
proposed approaches rely on implicit segmentation through a deep model or a
set of them. Recently, Hochuli et al.[18] evaluated object recognition models
into the HSDR context. For that, a digit string is considered a sequence of
objects. The advantage is that these models can encode the background, the
shape, and the neighbourhood of digits efficiently. Nonetheless, the
annotation of digit bounding boxes (ground-truth) is a drawback when synthetic
data are not available. Finally, sequence-to-sequence approaches were proposed
resulting in feasible end-to-end models [30] for word strings’ recognition.
The primary strategy is to split the input string into fragments to feed a
recurrent model (RNN), and then, a transcription layer determines the
resulting string. As stated in [23, 18], due to the lack of context, this
approach did not achieve outstanding results in HDSR; however, it produces a
good trade-off between data annotation, training complexity and accuracy.

An essential aspect of the strategies mentioned above is that most of them
have been proposed for modern handwritten digit strings recognition. There is
still a lack of approaches in the context of historical (ancient) document
recognition.The challenges are different from modern ones, such as paper
texture deterioration, noise, ancient handwriting style, ink failure, bleed-
through, and the lack of data [20]. Remarkably, the performance of the modern
approaches applied to historical document context is a matter of discussion.

Recently, Kusetogullari et al.[20] released to the public the ARDIS dataset,
composed of historical handwriting digit strings extracted from the Swedish
church record. Their comprehensive analysis reveals that a model trained with
modern isolated digits (MNIST, USPS, etc.) fails by a fair margin to correctly
encode the isolated digit from ARDIS due to its unique characteristics.
However, a comprehensive analysis using the historical digit strings of ARDIS
dataset is missing. In light of this, we propose to assess two state-of-the-
art approaches by adapting slight modifications into their architecture to
better fit the ARDIS digit strings characteristics. Moreover, we introduce
data augmentation techniques to represent the classes more efficiently. This
work’s resulting analysis eventually proposes a baseline for the ARDIS strings
dataset and pin-points, which efforts are needed to implement feasible
solutions for historical digit strings recognition. Moreover, it highlights
the research gaps for further investigations.

The rest of this document is organized as follows: The related work is
presented in Section 2. The assessed approaches are described in Section 4,
and then Section 5 provides a discussion around the experiments. Section 6
presents the conclusion and future directions.

##  2 Related Work

We surveyed the state-of-art and divided the related work into two main
categories: (a) Segmentation-based and (b) Segmentation-free approaches. The
related work is discussed in the following paragraphs.

Segmentation-based approaches: By segmenting the connected components as much
as possible, we attain the concept of over-segmentation, which is the most
commonly used strategy. It maximizes the probability of generating the optimal
segmentation point. However, as mentioned earlier, it also increases the
hypothesis space, resulting in a higher computational cost to classify all the
candidates compared with a strategy based on single segmentation.

An implicit filter is proposed by Vellasques et al. [32] to reduce the
computational cost of over-segmentation,in which a Support Vector Machine
(SVM) classifier is used to determine whether the cut produces reliable
candidates. The proposed filter succeeded in eliminating up to 83% of
unnecessary segmentation cuts in their experimental results.

In Roy et al. [27], a segmentation-based approach is devised to segment out
destination address block for postal applications; a review of postal and
check processing applications is warranted in [16].

Besides the strategies mentioned above, several segmentation algorithms were
proposed in the last decade. Ribas et al. [26] assessed most of them
considering their performance, computational cost, touching types, and
complexity. This characterization aimed to identify the limitations of the
algorithms based on a given pair of touching digits. Moreover, the work
reveals that most of the heuristic segmentation strategies are biased towards
the characteristic of the dataset’s characteristic under scrutiny; thus, a
suitable method that works for all touching types is impractical.

Segmentation-free approaches: To the best of our knowledge, the first attempt
along this line was proposed by Matan et al. [22]. A convolutional neural
network (CNN) based model is displaced from left to right over the input. The
proposed approach is termed SDNN (Space Displacement Neural Network), which
reported 66% of correct classification on 3000 images of ZIP Codes. LeCun et
al. [21], stated that SDNN is an attractive technique but has not yielded
better results than heuristic segmentation methods.

Years later, Choi and Oh [7] presented a modular neural network composed of
100 sub-networks trained to recognize 100 classes of touching digits (00..99).
The recognition rate of 1374 pairs of digits extracted from the NIST database
reaches 95.3%. A similar concept was presented by Ciresan [10], in which
100-class CNN was trained with 200,000 images reporting a recognition rate of
94.65%.

An image-based sequence recognition was proposed by Shi et al.[29]. The end-
to-end framework, named Convolutional Recurrent Neural Network (CRNN),
naturally handles sequences in arbitrary lengths without character
segmentation or horizontal scale normalization. The approach achieved
outstanding performance on recognising the scene text (text in the wild) and
music scores.

To make handwriting digit recognition less dependent on segmentation
algorithms, Hochuli et al.[19] proposed a segmentation-free framework based on
a dynamic selection of classifiers. The authors postulate that a set of
convolutional neural networks trained to (a) predict the size of touching
components and (b) specific-task models to recognize up to three touching
digits performs better than if the digits were segmented. However, this
algorithm’s generalisation to other datasets needs further verification since
there is a lack of diversity of the used datasets in the experimental
protocol.

Cheng et al. [5] proposed a strategy based on the improved VGG-16 model to
overcome the lack of texture features in handwriting digit recognition. The
model was examined on the extended MNIST dataset, eventually achieved a high
accuracy of 99.97%, which indicates that this VGG-based model has a robust
feature extraction ability than traditional classifiers and can meet the
requirements of handwriting digits classification and recognition. Besides, a
VGG-like model with multiple sub classifiers was built to recognize CAPTCHAs.
Although the CAPTCHA images for the test are featured by a lot of noise and
touching digits, the model accuracy reached 98.26% without any pre-
segmentation.

End-to-end approaches are frequently proposed in recent years, including those
tackling writer identification [8] and document analysis and recognition [24,
25]. Recently, approaches based on object recognition models have been
exploited with the HDSR task [17, 18]. Considering that a string is a sequence
of objects, these models can efficiently encode the background, shape, and
neighbourhood of digits, providing an end-to-end solution for the problem.
Additionally, they reduce the restrictions imposed on the number of touching
components or string length. However, the annotation of each digit bounding
box (ground-truth) is a bottleneck when synthetic data are not applicable.

##  3 ARDIS dataset

The Arkiv Digital Sweden (ARDIS) Dataset comprises historical handwriting
digit strings extracted from Swedish church document images written by
different priests from 1895 to 1970. The dataset is fully annotated [20],
including the digits bounding-boxes [17]. The sub-datasets of ARDIS are
exemplified in Figure 1.

Figure 1: Examples from the different representations of ARDIS subsets [20].

The dataset (I) is composed of 4-digit strings that represent the year of a
record. Most of the samples were cropped with the size of 175 x 95 pixels from
the document image and stored in its pristine RGB colour space. The dataset
(I) contains 75 classes mapping to different years. However, due to
insufficient samples, classes later than 1920 were not considered in this
work. The class distribution used in this study is shown in Figure 2.

Figure 2: Class distribution of ARDIS Strings[20]. Classes later than 1920
were not considered in this work due to an insufficient number of samples.

The historical digit strings pose several challenges to classification,
including variations in terms of variability of handwriting styles, touching
digits, ink failure, and noisy handwriting. Figure 3 demonstrates some of the
aforementioned challenges. Moreover, as observed in Figure 2, the classes are
not equally distributed.

Figure 3: Challenging samples in ARDIS Dataset (I) representing noisy
handwritten digits (i)(ii), ink failure (iii) and touching digits (iv).

The dataset (II) comprises cropped digits from the original digit strings,
containing artefacts and fragments of the neighbour digits. In dataset (III),
the isolated digits were manually cleaned. For completeness, a uniform
distribution of each digit’s occurrences was ensured in the dataset (III),
resulting in 7600 de-noised digit images in RGB colour space. For dataset
(IV), the isolated digits are normalised and binarised.

###  3.1 Synthetic Data

Recently, data augmentation techniques in handwriting digit were proposed to
generate a synthetic training set [30, 2, 19]. In order to improve the data
representation of ARDIS strings (Dataset I), we propose the creation of
synthetic data by permuting and concatenating several single digits from
dataset III. Figure 4 depicts two synthetic samples. Although both digit
strings belong to the label “1987”, the representation (e.g., style) is
remarkably different.

Figure 4: Synthetic strings creation from isolated digits of dataset III.

We combine synthetic data and real data from up to 1000 samples for each class
during the training phase to balance the distribution of classes. The number
of real and synthetic data are summarized in Table 1. Considering the limited
amount of data in the ARDIS data set, it is relatively reasonable (closer to
standard practice) to retain about 43% of the real data for testing
(2651/(3474+2651)) and use the remaining 57% of the real data set for
training. It is worth mentioning that the synthetic data was used only for
training, which represents 88.8% of the training samples. In total, 31000 and
2651 images are used for training and testing, respectively. All the data used
in this work are already publicly available through the ARDIS
Website111https://ardisdataset.github.io/ARDIS/ by the authors of [20].

Table 1: Training protocol using synthetic and real data

Label |  Training Samples | Test Samples
---|---|---
Synthetic | Real
1890 | 1000 | 0 | 15
1891 | 1000 | 0 | 15
1892 | 1000 | 0 | 15
1893 | 1000 | 0 | 15
1894 | 1000 | 0 | 15
1895 | 875 | 125 | 100
1896 | 846 | 154 | 100
1897 | 846 | 154 | 100
1898 | 825 | 175 | 100
1899 | 833 | 167 | 100
1900 | 794 | 206 | 100
1901 | 761 | 239 | 100
1902 | 765 | 235 | 100
1903 | 764 | 236 | 100
1904 | 781 | 219 | 100
1905 | 809 | 191 | 100
1906 | 830 | 170 | 100
1907 | 824 | 176 | 100
1908 | 844 | 156 | 100
1909 | 863 | 137 | 100
1910 | 862 | 138 | 100
1911 | 883 | 117 | 100
1912 | 896 | 104 | 100
1913 | 910 | 90 | 100
1914 | 909 | 91 | 100
1915 | 925 | 75 | 100
1916 | 955 | 45 | 100
1917 | 962 | 38 | 100
1918 | 983 | 17 | 100
1919 | 981 | 19 | 100
1920 | 1000 | 0 | 76
Total | 27526 | 3474 | 2651

##  4 Approaches for Historical Handwriting Digit String Recognition

As stated by [26] and [19], the segmentation problem has been overcome by
segmentation-free approaches in the recent advances in deep learning models.
In light of this, we propose to evaluate two segmentation-free approaches on
the ARDIS dataset to tackle the task of historical handwriting digit string
recognition. The first approach (Section 4.1) is based on the well-known
VGG-16 model. The second one (Section 4.2) is based on a sequence-to-sequence
model.

Motivations on the choice of models: Given the VGG-16 model’s decent
performance in other character recognition tasks, we consider it the baseline
of this experiment to evaluate against other alternative models. The VGG
network model [31] was proposed by the Visual Geometry Group (VGG) at Oxford
University. When first created, the focus of this network was to classify
materials by their textural appearance and not by their colour. Due to the
excellent generalisation performance of VGG-Net, its pre-trained model on the
ImageNet dataset is widely used for feature extraction problems [9, 13] such
as: object candidate frame (object proposal) generation [15], fine-grained
object localization, image retrieval [34], image co-localization [35], etc. On
the other hand, our new approach is based on modifying the concept of CRNN
[29]. The CRNN is mainly used for end-to-end recognition of indefinite length
text sequence. It does not require pre-segmentation on long continuous text.

###  4.1 Specific-Task Classifiers

In this new approach, we adapted the well-known VGG-16 model to the context of
the ARDIS dataset, which is composed of 4-digit strings. Instead of using a
dynamic selection of classifiers [19], we proposed to parallelize the
classification task by adding four dense layers (classifiers) on the bottom of
the architecture. The final architecture is depicted in Figure 5. With this
simple modification, we produce an end-to-end pipeline avoiding both the
heuristic segmentation and fusion methods.

Figure 5: Structure of the 4 Classifiers CNN framework

The rationale here is that each specific-task classifier ($C_{1}$, $C_{2}$,
$C_{3}$, and $C_{4}$) should determine the ten classes (0..9) for each digit
of the 4-digit string. The prediction of input digit string is defined as
follows:

Let $\mathcal{C}(x)=\max\limits_{0\leq i\leq 9}p^{i}(x)$ be the probability
produced by the digit classifier (10-classes). Then, an input digit $x$ is
assigned to the class $\omega_{j}$ (j=0…9) according to Equation 1.

$P(\omega_{j}|x)=\max(\mathcal{C})$ (1)

Considering that the input image $I$ contains $n=4$ digits, the most probable
interpretation of the written amount $M$ is given by Equation 2.

$P(M|I)=\prod_{i=1}^{n}P_{i}(\omega_{j}|x_{i})$ (2)

###  4.2 CRNN

A Convolutional Recurrent Neural Networks (CRNN) [33, 30, 12] is a sequence-
to-sequence model that can be trained from end to end. The pipeline for such a
network is depicted in Figure 6a. First, convolutional layers extract features
from an input image, and then a sequence of feature vectors is extracted from
feature maps.

Figure 6: CRNN architecture proposed by [30]: (a) the pipeline from
convolutional layers to transcription layer and (b) the receptive field for
each feature vector.

Since each region of the feature map is associated with a receptive field in
the input image, each vector in the sequence is a descriptor of this image
field, as illustrated in Figure 6b. Next, this sequence is fed to the
recurrent layers, which are composed of a bidirectional Long-Short Term Memory
(LSTM) [28] network, producing a per-frame prediction from left to right of
the image. Finally, the transcription layer determines the correct sequence of
classes to the input image by removing the repeated adjacent labels and the
blanks, represented by the character ‘-’. This solution is well suited when
the past and future context of a sequence contributes to recognising the whole
input. With the aid of contextual information, such as a lexicon, this
approach achieves high text recognition performance. The application of this
solution to handwriting digits is a matter of discussion since we have fewer
classes (0..9) as compared to words, but there is no lexicon to mitigate
possible confusion.

To address the context of historical digit string recognition, we propose a
modification of the Recurrent Layers Architecture. Due to the lack of data, we
replaced the LSTM with a Gated Recurrent Unit (GRU) [6]. Since the latter has
fewer parameters, besides reducing training time, the vanishing gradient’s
impact is minimised. Moreover, we combined two identical GRU Layers to process
the feature maps from left to right and vice-versa, and then, the output of
both GRUs are merged. It is worth mentioning that the feature maps fed to the
GRU are reshaped to vectors to provide a sequence of information. Further,
another two identical GRUs repeat the process; however, their outputs are
concatenated. Finally, a fully connected layer determines the class
probabilities, and the connectionist temporal classification (CTC) layer
determines the final prediction. The architecture and characterisation of our
modified CRNN approach are depicted in Figure 7.

Figure 7: CRNN architecture for HDSR.

##  5 Experiments

In this section, we assess all reported models in the context of HDSR. We also
added to the comparison the native VGG16 model (i.e., without our
modification). All metrics used to measure the performance are described in
Section 5.1. The training protocol is presented in Section 5.2. Finally, the
results are discussed in Section 5.3.

###  5.1 Evaluation Metrics

To better assess the performance of the proposed approaches, besides the well-
known accuracy and F1-score, we propose using the Normalized Levenshtein
Distance (NLD) and the Average Normalized Levenshtein Distance (ANLD).

The Normalized Levenshtein Distance (NLD) [11] describes how close a predicted
string is from the ground truth by eliminating the influence of string length
on performance measurement. The NLD can be defined as follow:

$NLD(a_{T},a_{R})=\frac{LD(a_{T},a_{R})}{\left|a_{T}\right|}$ (3)

where $\left|a_{T}\right|$ presents the length of the string $a_{T}$ and LD
represents the Levenshtein distance, which refers to the minimum number of
editing operations of each character (insert, delete, and substitute) to
convert from one string to another. A zero value indicates a correct
prediction.

Complementary to this, the Average Normalized Levenshtein Distance (ANLD) is a
soft metric to evaluate the model performance:

$ANLD=\frac{\sum_{i=1}^{T}NLD\left(a_{T}^{i},a_{R}^{i}\right)}{T}$ (4)

where T indicates the number of evaluated strings. Lower values represent a
good performance.

###  5.2 Training

The models were fine-tuned using the data described in Table 1, which
comprises real and synthetic data that sums up to 1000 training samples for
each one of 31 classes (number of years in the selected period). All the
classifiers were trained up to 100 epochs, and the over-fitting was prevented
through early-stopping when no convergence occurs after ten epochs. The
training parameters are summarized in Table 2.

Table 2: Training Parameters Frameworks | Batch | Loss | Reg. | Opt. | Learning | Batch
---|---|---|---|---|---|---
Size | Function | Rate | Norm
Specific-Task | 32 | Cat. Crossentropy | Dropout (0.25) | Adam | $10^{-3}$ | TRUE
CRNN | 128 | CTC | Dropout (0.25) | Adadelta | $10^{-3}$ | FALSE
VGG-16 | 32 | Cat. Crossentropy | L2 ($5*10^{-4}$) | Adam | $10^{-5}$ | FALSE

###  5.3 Results and Discussion

The performance of the evaluated models in this work is reported in Table 3.
As described in Section 4.1, we modified the last layer of VGG-16 architecture
by adding four classifiers instead of one. As stated in Table 3, this
modification achieved the best performance since it can encode the string by
implicit segmenting the digits with the specific-task classifiers.
Comprehensive analysis reveals that the whole image’s information is difficult
to encode in several cases by only one classifier (VGG-16) due to the
handwriting variability. For example, let us assume that we have the following
strings “1890” and “1819”. From a computational perspective, the global
representation poses challenges in discriminating the digit ‘0’ and ‘9’ that
mislead the classifier. However, for the specific-task approach, the
information can be implicitly segmented according to each classifier’s domain
space. Regarding this, the models $C_{1}$ and $C_{2}$ exhibit a reduced
complexity when compared to the models $C_{3}$ and $C_{4}$ since the former
two classifiers need to discriminate fewer classes ([1],[8,9]).

Table 3: Comparisons among the three frameworks in terms of different metrics Frameworks | Accuracy | F1-score | ANLD
---|---|---|---
VGG-16 | 36.2 | 38.5 | 28.7
CRNN | 85.0 | 83.5 | 5.8
Specific-Task | 93.2 | 90.8 | 2.3

Regarding the CRNN, we believe that the model suffers due to a lack of
context. Contrary to the word recognition, the CTC layer missed the prediction
since there is no lexicon to mitigate some confusions, such as fragment
recognition and repeated labels. The issue is quite similar to the over-
segmentation.

##  6 Conclusion and Future Work

In this work, we explored the recognition of historical digit strings. Based
on this context, an image-based dataset containing 31 classes representing
handwriting years ranging from 1890 to 1920 is utilised.

To this end, we proposed to evaluate three models implementing end-to-end
solutions. The use of synthetic data was employed to overcome the lack of
data.

The proposed approach that combines four specific-task classifiers achieved
outstanding results. This promising performance can be explained by the
implicit segmentation of the input string made by the domain space of each
specific-task classifier. On the other hand, the approach based on a single
classifier suffers due to the handwriting variability represented in a global
perspective. Regarding the CRNN approach, it suffers from the lack of lexicon,
as also stated in [18].

For future endeavours, once we have context information about the first and
second digits, a reduced number of classifiers for the specific-task approach
could be examined. Also, we will investigate a dynamic approach on VGG-16 that
can implicitly handle different length of strings.

####  Acknowledgments. This project is supported by the research project
“DocPRESERV: Preserving and Processing Historical Document Images with
Artificial Intelligence”, STINT, the Swedish Foundation for International
Cooperation in Research and Higher Education (Grant: AF2020-8892).

## References

  * [1] Almazan, J., Gordo, A., Fornés, A., Valveny, E.: Word spotting and recognition with embedded attributes. IEEE Transactions on Pattern Analysis and Machine Intelligence 36(12), 2552–2566 (2014). https://doi.org/10.1109/TPAMI.2014.2339814
  * [2] Aly, S., Mohamed, A.: Unknown-length handwritten numeral string recognition using cascade of pca-svmnet classifiers. IEEE Access 7, 52024–52034 (2019). https://doi.org/10.1109/ACCESS.2019.2911851
  * [3] Cecotti, H.: Active graph based semi-supervised learning using image matching: application to handwritten digit recognition. Pattern Recognition Letters 73, 76–82 (2016)
  * [4] Cheddad, A.: Towards query by text example for pattern spotting in historical documents. In: 2016 7th International Conference on Computer Science and Information Technology (CSIT). pp. 1–6 (2016). https://doi.org/10.1109/CSIT.2016.7549479
  * [5] Cheng, S., Shang, G., Zhang, L.: Handwritten digit recognition based on improved vgg16 network. In: Tenth International Conference on Graphics and Image Processing (ICGIP 2018). vol. 11069, p. 110693B. International Society for Optics and Photonics (2019)
  * [6] Cho, K., van Merrienboer, B., Gulcehre, C., Bougares, F., Schwenk, H., Bengio, Y.: Learning phrase representations using rnn encoder-decoder for statistical machine translation. In: Conference on Empirical Methods in Natural Language Processing (EMNLP 2014) (2014)
  * [7] Choi, S.M., Oh, I.S.: A segmentation-free recognition of handwritten touching numeral pairs using modular neural network. International journal of pattern recognition and artificial intelligence 15(06), 949–966 (2001)
  * [8] Cilia, N., De Stefano, C., Fontanella, F., Marrocco, C., Molinara, M., Scotto Di Freca, A.: An end-to-end deep learning system for medieval writer identification. Pattern Recognition Letters 129, 137–143 (2020). https://doi.org/https://doi.org/10.1016/j.patrec.2019.11.025, https://www.sciencedirect.com/science/article/pii/S0167865519303460
  * [9] Cimpoi, M., Maji, S., Vedaldi, A.: Deep filter banks for texture recognition and segmentation. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR). pp. 3828–3836 (2015). https://doi.org/10.1109/CVPR.2015.7299007
  * [10] Ciresan, D.: Avoiding segmentation in multi-digit numeral string recognition by combining single and two-digit classifiers trained without negative examples. In: 2008 10th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing. pp. 225–230. IEEE (2008)
  * [11] Diem, M., Fiel, S., Kleber, F., Sablatnig, R., Saavedra, J.M., Contreras, D., Barrios, J.M., Oliveira, L.S.: Icfhr 2014 competition on handwritten digit string recognition in challenging datasets (hdsrc 2014). In: 2014 14th International Conference on Frontiers in Handwriting Recognition. pp. 779–784. IEEE (2014)
  * [12] Dutta, K., Krishnan, P., Mathew, M., Jawahar, C.V.: Improving cnn-rnn hybrid networks for handwriting recognition. In: 2018 16th International Conference on Frontiers in Handwriting Recognition (ICFHR). pp. 80–85 (Aug 2018). https://doi.org/10.1109/ICFHR-2018.2018.00023
  * [13] Gao, B.B., Wei, X.S., Wu, J., Lin, W.: Deep spatial pyramid: The devil is once again in the details. arXiv preprint arXiv:1504.05277 (2015)
  * [14] Gao, Y., Mishchenko, Y., Shah, A., Matsoukas, S., Vitaladevuni, S.: Towards data-efficient modeling for wake word spotting. In: ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). pp. 7479–7483 (2020). https://doi.org/10.1109/ICASSP40776.2020.9053313
  * [15] Ghodrati, A., Diba, A., Pedersoli, M., Tuytelaars, T., Gool, L.V.: Deepproposals: Hunting objects and actions by cascading deep convolutional layers. International Journal of Computer Vision 124(2), 115–131 (2017). https://doi.org/10.1007/s11263-017-1006-x
  * [16] Gilloux, M.: Document analysis in postal applications and check processing. Handbook of Document Image Processing and Recognition p. 705–747 (2014). https://doi.org/10.1007/978-0-85729-859-1_22
  * [17] Hochuli, A.G., Britto, A.S., Barddal, J.P., Sabourin, R., Oliveira, L.E.S.: An end-to-end approach for recognition of modern and historical handwritten numeral strings. In: 2020 International Joint Conference on Neural Networks (IJCNN). pp. 1–8 (2020). https://doi.org/10.1109/IJCNN48605.2020.9207468
  * [18] Hochuli, A.G., Britto Jr, A.S., Saji, D.A., Saavedra, J.M., Sabourin, R., Oliveira, L.S.: A comprehensive comparison of end-to-end approaches for handwritten digit string recognition. Expert Systems with Applications 165, 114196 (2021). https://doi.org/https://doi.org/10.1016/j.eswa.2020.114196, http://www.sciencedirect.com/science/article/pii/S0957417420309271
  * [19] Hochuli, A.G., Oliveira, L.S., Britto Jr, A., Sabourin, R.: Handwritten digit segmentation: Is it still necessary? Pattern Recognition 78, 1–11 (2018)
  * [20] Kusetogullari, H., Yavariabdi, A., Cheddad, A., Grahn, H., Hall, J.: ARDIS: A swedish historical handwritten digit dataset. Neural Computing and Applications pp. 1–14 (2019)
  * [21] LeCun, Y., Bottou, L., Bengio, Y., Haffner, P.: Gradient-based learning applied to document recognition. Proceedings of the IEEE 86(11), 2278–2324 (1998)
  * [22] Matan, O., Burges, C.J., LeCun, Y., Denker, J.S.: Multi-digit recognition using a space displacement neural network. In: Advances in neural information processing systems. pp. 488–495 (1992)
  * [23] Neto, A.F.D.S., Bezerra, B.L.D., Lima, E.B., Toselli, A.H.: Hdsr-flor: A robust end-to-end system to solve the handwritten digit string recognition problem in real complex scenarios. IEEE Access 8, 208543–208553 (2020)
  * [24] Neudecker, C., Baierer, K., Federbusch, M., Boenig, M., Würzner, K.M., Hartmann, V., Herrmann, E.: Ocr-d: An end-to-end open source ocr framework for historical printed documents. In: Proceedings of the 3rd International Conference on Digital Access to Textual Cultural Heritage. p. 53–58. DATeCH2019, Association for Computing Machinery, New York, NY, USA (2019). https://doi.org/10.1145/3322905.3322917, https://doi.org/10.1145/3322905.3322917
  * [25] Palm, R.B., Laws, F., Winther, O.: Attend, copy, parse end-to-end information extraction from documents. In: 2019 International Conference on Document Analysis and Recognition (ICDAR). pp. 329–336 (2019). https://doi.org/10.1109/ICDAR.2019.00060
  * [26] Ribas, F.C., Oliveira, L., Britto, A., Sabourin, R.: Handwritten digit segmentation: a comparative study. International Journal on Document Analysis and Recognition (IJDAR) 16(2), 127–137 (2013)
  * [27] Roy, K., Vajda, S., Pal, U., Chaudhuri, B.B., Belaid, A.: A system for indian postal automation. In: Eighth International Conference on Document Analysis and Recognition (ICDAR’05). pp. 1060–1064 Vol. 2 (2005). https://doi.org/10.1109/ICDAR.2005.259
  * [28] Schuster, M., Paliwal, K.K.: Bidirectional recurrent neural networks. IEEE Transactions on Signal Processing 45(11), 2673–2681 (Nov 1997). https://doi.org/10.1109/78.650093
  * [29] Shi, B., Bai, X., Yao, C.: An end-to-end trainable neural network for image-based sequence recognition and its application to scene text recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence 39(11), 2298–2304 (nov 2017). https://doi.org/10.1109/TPAMI.2016.2646371
  * [30] Shi, B., Bai, X., Yao, C.: An end-to-end trainable neural network for image-based sequence recognition and its application to scene text recognition. IEEE transactions on pattern analysis and machine intelligence 39(11), 2298–2304 (2016)
  * [31] Simonyan, K., Zisserman, A.: Very deep convolutional networks for large-scale image recognition. In: International Conference on Learning Representations (2015)
  * [32] Vellasques, E., Oliveira, L.S., Britto Jr, A., Koerich, A.L., Sabourin, R.: Filtering segmentation cuts for digit string recognition. Pattern Recognition 41(10), 3044–3053 (2008)
  * [33] Voigtlaender, P., Doetsch, P., Ney, H.: Handwriting recognition with large multidimensional long short-term memory recurrent neural networks. In: 2016 15th International Conference on Frontiers in Handwriting Recognition (ICFHR). pp. 228–233 (Oct 2016). https://doi.org/10.1109/ICFHR.2016.0052
  * [34] Wei, X.S., Luo, J.H., Wu, J., Zhou, Z.H.: Selective convolutional descriptor aggregation for fine-grained image retrieval. Trans. Img. Proc. 26(6), 2868–2881 (Jun 2017). https://doi.org/10.1109/TIP.2017.2688133, https://doi.org/10.1109/TIP.2017.2688133
  * [35] Wen, W., Wu, C., Wang, Y., Chen, Y., Li, H.: Learning structured sparsity in deep neural networks. In: Proceedings of the 30th International Conference on Neural Information Processing Systems. p. 2082–2090. NIPS’16, Curran Associates Inc., Red Hook, NY, USA (2016)


# Scalar field dark matter with time-varying equation of state

Gaspard Poulot  Elsa M. Teixeira  Carsten van de Bruck  and Nelson J. Nunes

###### Abstract

We propose a new model of scalar field dark matter interacting with dark
energy. Adopting a fluid description of the dark matter field in the regime of
rapid oscillations, we find that the equation of state for dark matter is non-
zero and even becomes increasingly negative at late times during dark energy
domination. Furthermore, the speed of sound of dark matter is non-vanishing at
all length scales, and a non-adiabatic pressure contribution arises. The
results indicate that there are still unexplored possible interactions within
the dark sector that lead to novel background effects and can impact structure
formation processes.

##  1 Introduction

The widely accepted standard cosmological model, known as $\Lambda$CDM,
standing for “a cosmological constant ($\Lambda$) plus cold dark matter
(CDM)”, is in impressive agreement with various observational findings. This
framework explains the accelerated expansion of the Universe at late times,
structure formation in the Universe and the properties of the cosmic microwave
background radiation (CMB) (see, e.g. [1] for a recent overview). However,
despite its success in explaining phenomena at large scales, the $\Lambda$CDM
model still faces significant observational challenges when applied to
clustering on smaller scales and consistently reconstructing the Universe’s
expansion history. In particular, the so-called Hubble tension is still a
persistent challenge, together with other statistically less significant
tensions. We refer to [2] for an overview of these tensions and possible
solutions. Moreover, from a theoretical perspective, the nature of the dark
sector components, from which the model derives its name, remains a puzzle.

There is overwhelming observational evidence for dark matter (DM) from
different cosmological and astrophysical probes, including galaxy rotation
curves [3], the peculiar motion of clusters [4, 5], the Bullet cluster system
[6], observations from the CMB and baryonic acoustic oscillations [7, 8]. CDM
constitutes approximately $26\%$ of the Universe’s current energy density;
therefore, illuminating its exact nature remains one of the most significant
problems in cosmology and particle physics. A key factor is that dark matter
must interact predominantly through gravity with the particles of the Standard
Model of particle physics to assist structure formation in the Universe. The
lack of direct detection of weakly-interacting massive particles (WIMPs) [9,
10, 11, 12, 13, 14, 15, 16], the simplest and most popular CDM candidates,
motivates the search for alternative proposals rooted in theories of high-
energy particle physics. One such proposal of relevance for this work is the
concept of fuzzy dark matter [17, 18, 19, 20, 21], which consists of modelling
dark matter as an ultralight scalar field or axion-like particle with a mass
around $10^{-22}$ eV. The ultra-light nature of these particles translates
into a large de Broglie wavelength, effectively suppressing structure
formation on small scales, while retaining the successes of the CDM paradigm
on larger scales [22, 23, 24, 25, 26]. Ultralight scalars often emerge in
particle physics and string theory compactifications, where axion-like
particles arise as Kaluza-Klein zero modes of anti-symmetric tensor fields
[27, 19, 20]. Arguments based on measurements of CMB anisotropies constrain
the mass of these ultralight fields to be $m_{\phi}\gtrsim 10^{-24}$ eV [28,
29, 30, 21, 31], while observations of the Lyman$-\alpha$ forest extend this
lower bound to about $m_{\phi}\sim 10^{-21}$ eV [32, 33, 34, 35, 36], assuming
these particles account for over $30\%$ of the total dark matter content.
Albeit less widely agreed upon, studies on the kinematics of ultra-faint dwarf
galaxies further bring the lower limit to around $m_{\phi}\sim 10^{-19}$ eV
[37, 38, 39]. Other probes include galaxy clustering [40, 25], weak lensing
measurements [18, 41] and 21 cm observations [42, 43, 44]. Furthermore, recent
pulsar timing array reports of a stochastic gravitational wave background
impose significant constraints on ultralight axions [45, 46]. It has been
shown that the travel time of pulsar radio beams is influenced by the
gravitational potential induced by such ultralight DM particles [47, 48, 49,
50]. The analysis of the second data release from the European Pulsar Timing
Array [51] peaks at $m_{\phi}\sim 10^{-23}$ eV and rules out particles with
masses ranging from $10^{-24}$ eV to $10^{-23.3}$ eV. It should be noted that
the analyses mentioned above often rely on the assumption that the ultralight
DM makes up the whole of DM and interacts solely through gravitational means.

The other dark ingredient of the standard model of cosmology is the
cosmological constant $\Lambda$, the simplest realisation of dark energy in
the form of a background energy component which accounts for the current
accelerated expansion of the Universe. Given that the value of the
cosmological constant has to be very small, alternative candidates for DE have
been proposed, including slowly evolving scalar fields [52, 53, 54, 55, 56],
three-form fields [57] and other more exotic proposals (see [58, 59, 60] for
reviews on dark energy models). If DE is a dynamical degree of freedom,
couplings to other matter forms are generally expected unless a particular
symmetry forbids such interactions [61]. Constraints coming from the Cassini
probe [62] suggest that the coupling of a slowly rolling DE scalar field to
the standard model fields has to be much weaker than gravity, as the field’s
minute mass results in a very large interaction range. On the other hand,
these constraints can be relaxed for couplings to DM only, derived based on
model-dependent cosmological observations. Some interacting dark energy (IDE)
models may address shortcomings of the $\Lambda$CDM model such as the Hubble
tension [2, 63, 64, 65, 66, 67], providing a guiding direction and framework
for further study. Of particular interest are a class of models in which the
CDM-DE coupling is directly proportional to the energy density of dark energy,
as studied for example in Refs. [68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78,
79, 80].

In this paper, we propose a model of scalar field DM, which interacts with a
quintessence-like DE field. A conformal coupling between the DM and the DE
field describes this interaction. In the literature, this interaction is
mediated by the slowly evolving scalar degree of freedom identified with DE
[81, 82]. In this work, however, we propose an alternative formulation with
the twist that the coupling depends instead on the (fast-oscillating) DM
scalar degree of freedom. We note here that even if the DE sector is
considered here to be a slowly evolving scalar field, we expect many of our
results to still hold if DE is a cosmological constant (but conformally
coupled to DM) or a three-form field [57]. Nevertheless, our formalism is not
circumscribed to the conformal coupling. In Appendix B we present an
alternative formulation of a theory with a DM-DE coupling in which DM has a
very similar phenomenology.

This article is organised as follows. In section 2, we motivate and describe
the proposed framework. In this section, we also collect the field equations
and useful relations, which will be relevant to the subsequent calculations.
In section 3, we derive an effective fluid-field description, a valid
approximation from the onset of the DM field quick oscillations around the
minimum of an effective potential. The treatment of the associated fluid-field
cosmological perturbations is discussed in section 4. We summarise our
findings in section 5.

##  2 Model

Our setup is inspired by that of field theories of dark energy, such as
coupled quintessence [55, 83, 81], but with the role of the fields for DM and
DE swapped. That is, we treat $\phi$ in the action below as a dark matter
scalar field, and the dark energy sector is coupled conformally to the DM
sector, resulting in the following effective action for this model:

$\displaystyle{\cal S}=\int d^{4}x\sqrt{-g}\left(\frac{M_{\rm Pl}^{2}}{2}{\cal
R}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\phi)\right)+{\cal
S}_{\rm SM}+{\cal S}_{\rm DE}\,,$ (2.1)

where ${\cal R}$ is the Ricci-scalar and $M_{\rm Pl}\equiv 1/\sqrt{8\pi G}$ is
the reduced Planck mass. In the action above, ${\cal S}_{\rm SM}$ denotes the
Lagrangian containing the standard model fields propagating in geodesics of
the metric $g_{\mu\nu}$. In what follows, we assume that the DE sector can be
described by a slowly evolving scalar field $\chi$, described by the action

${\cal S}_{\rm DE}=\int
d^{4}x\sqrt{-{\tilde{g}}}\left(\frac{1}{2}\tilde{g}^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-V(\chi)\right)~{},$
(2.2)

where the metric ${\tilde{g}}$ is related to the metric $g$ via a conformal
transformation of the form ${\tilde{g}}_{\mu\nu}=C(\phi)g_{\mu\nu}$. Rewriting
${\cal S}_{\rm DE}$ in terms of the metric $g_{\mu\nu}$ results in

${\cal S}_{\rm DE}=\int
d^{4}x\sqrt{-g}\left(\frac{C(\phi)}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-C^{2}(\phi)V(\chi)\right)\,.$
(2.3)

In contrast to models such as coupled quintessence, in which $C$ depends on
the dark energy field, the function $C$ in this framework is dependent on dark
matter properties111In Appendix A we show the corresponding equations for the
case of a conformal coupling depending also on the kinetic energy of the DM
field. However, this is a different model and its discussion is out of the
scope of this work..

The equation of motion for the DE field $\chi$ is derived from the
corresponding variation of the action and reads

$\nabla^{\mu}\nabla_{\mu}\chi-C\frac{dV}{d\chi}=-\frac{\nabla_{\alpha}C}{C}\nabla^{\alpha}\chi\,.$
(2.4)

The equation for the DM field $\phi$ is derived in an analogous manner and
yields

$\nabla^{\mu}\nabla_{\mu}\phi-\frac{\partial U}{\partial\phi}=Q\,,$ (2.5)

where the coupling $Q$ was defined as

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\left(\frac{\partial
L_{\chi}}{\partial(\nabla_{\mu}\phi)}\right)-\frac{\partial
L_{\chi}}{\partial\phi}$ (2.6) $\displaystyle=$
$\displaystyle\frac{C_{,\phi}}{2C}\left[Cg^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+4C^{2}U\right]\,.$

The Einstein equations result from variation of the action with the
gravitational metric $g_{\mu\nu}$, resulting in

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=\kappa^{2}\left(T_{\mu\nu}^{(\phi)}+T_{\mu\nu}^{(\chi)}+T_{\mu\nu}^{(\rm
SM)}\right)\,,$ (2.7)

where $R_{\mu\nu}$ is the Ricci tensor, $\kappa\equiv 1/M_{\rm Pl}$ and
$T_{\mu\nu}^{(i)}$ are the energy-momentum tensors for each $i$-th fluid,
which for the dark sector scalar fields $\phi$ and $\chi$ read

$\displaystyle T_{\mu\nu}^{(\phi)}=\nabla_{\mu}\phi\nabla_{\nu}\phi-
g_{\mu\nu}\left(\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\phi\nabla_{\beta}\phi+V(\phi)\right)\,,$
(2.8) $\displaystyle T_{\mu\nu}^{(\chi)}$ $\displaystyle=$ $\displaystyle
C\nabla_{\mu}\chi\nabla_{\nu}\chi-
g_{\mu\nu}\left(\frac{C}{2}g^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+C^{2}U(\chi)\right)\,,$
(2.9)

respectively, and obey the following conservation equations:

$\nabla^{\mu}T_{\mu\nu}^{(\phi)}=Q\nabla_{\nu}\phi\,,\quad\nabla^{\mu}T_{\mu\nu}^{(\chi)}=-Q\nabla_{\nu}\phi\,,$
(2.10)

which are physically equivalent to the Klein-Gordon equations.

For the background cosmology we consider the case of a flat Friedmann-
Lemaître-Robertson-Walker Universe, for which the field equations read

$\ddot{\phi}+3H\dot{\phi}+U_{,\phi}=\frac{1}{2}C_{,\phi}\dot{\chi}^{2}-2CC_{,\phi}V\,,$
(2.11)
$\ddot{\chi}+\left(3H+\frac{C_{,\phi}}{C}\dot{\phi}\right)\dot{\chi}+CV_{,\chi}=0\,,$
(2.12)

where $H$ is the Hubble expansion rate and over-dots indicate derivatives
according to cosmic time $t$. The energy densities $\rho_{\phi}$ and
$\rho_{\chi}$ and the corresponding pressures $p_{\phi}$ and $p_{\chi}$ are
given by

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+U(\phi),\ \ \
p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-U(\phi)\,,$ (2.13)
$\rho_{\chi}=\frac{1}{2}C\dot{\chi}^{2}+C^{2}V(\chi),\ \ \
p_{\chi}=\frac{1}{2}C\dot{\chi}^{2}-C^{2}V(\chi)\,.$ (2.14)

The Friedmann equations, which govern the expansion of the Universe and which
can be derived from Einstein’s equations, Eq. (2.7), read

$H^{2}=\frac{\kappa^{2}}{3}\left[\frac{1}{2}\dot{\phi}^{2}+U(\phi)+\frac{1}{2}C(\phi)\dot{\chi}^{2}+C(\phi)^{2}V(\chi)\right]\,,$
(2.15)
$\dot{H}=-\frac{\kappa^{2}}{3}\left[\dot{\phi}^{2}+C(\phi)\dot{\chi}^{2}\right]\,.$
(2.16)

Finally, the Klein-Gordon equations for $\phi$ and $\chi$, Eqs. (5.1) and
(5.1), become

$\dot{\rho}_{\phi}+3H\left(\rho_{\phi}+p_{\phi}\right)=-\frac{C_{,\phi}}{2C}\left(\rho_{\chi}-3p_{\chi}\right)\dot{\phi}\,,$
(2.17)
$\dot{\rho}_{\chi}+3H\left(\rho_{\chi}+p_{\chi}\right)=\frac{C_{,\phi}}{2C}\left(\rho_{\chi}-3p_{\chi}\right)\dot{\phi}\,.$
(2.18)

Since we wish for $\phi$ to behave as dark matter and $\chi$ as dark energy,
in this analysis we focus on the following self-interacting scalar field
potentials:

$V(\chi)=V_{0}e^{-\kappa\lambda\chi}\,,$ (2.19)
$U(\phi)=\frac{1}{2}m^{2}\phi^{2}\,.$ (2.20)

The function $Q$ defined in Eq. (2.6) can be written as

$Q=\frac{C_{,\phi}}{2C}\left(\rho_{\chi}-3p_{\chi}\right)\,,$ (2.21)

for a field dependent conformal function $C$. Unlike standard interacting dark
energy-dark matter models, we emphasise that the coupling in the present model
depends on the DE energy density and pressure, rather than the DM density.

This concludes the derivation of the relevant equations for this theory, which
will be used in the next section to derive a fluid-field description for the
system.

##  3 Towards a fluid-field description

We now seek to solve the equations derived in the previous section to study
the evolution of both the DM and DE species. The main difference and challenge
of our model compared to interacting quintessence models is that the coupling
described here by $Q(t)$ depends on $\phi$ and $\dot{\phi}$, which must be
rapidly oscillating for $\phi$ to behave like DM. Therefore, we have to
perform an analysis based on the time-averaged character of the field over a
period of oscillation to extract its average evolution and make the equations
more manageable. For the particular conditions under study, the Klein-Gordon
equation for $\phi$ reads

$\ddot{\phi}+3H\dot{\phi}+m^{2}\phi=-Q(t)\,.$ (3.1)

We now consider a conformal coupling function linearly dependent on $\phi$,
i.e. $C=1+2\kappa\beta\phi$, where $\beta$ is a constant and $\kappa=1/M_{\rm
Pl}$. This allows us to decompose the source term in Eq. (3.1) as

$Q(t)=Q_{0}(t)+Q_{1}(t)\phi\,,$ (3.2)

with

$\displaystyle Q_{0}(t)\equiv-\kappa\beta(\dot{\chi}^{2}-4V)\,,\hskip
56.9055ptQ_{1}(t)\equiv 8\kappa^{2}\beta^{2}V\,,$ (3.3)

such that the equation of motion for $\phi$ can be recast into

$\ddot{\phi}+3H\dot{\phi}+m^{2}_{\rm eff}(t)\phi=-Q_{0}(t)\,,$ (3.4)

where we have defined the effective mass

$\displaystyle m^{2}_{\rm eff}(t)\equiv m^{2}+Q_{1}(t)\,.$ (3.5)

Before solving these equations, we want to stress that the following analysis
hinges on the form of Eq. (3.4) and any theory which predicts this form for
the equation of motion for the DM field, will result in a similar DM
phenomenology, as long as the functions $Q_{0}$ and $Q_{1}$ are slowly varying
(and it’s not even a necessary condition that these quantities must depend on
the dark energy scalar field). What will potentially change is the dynamics of
the DE field. We give another example of such a theory in Appendix B.

We will be interested in the case in which $m>H$ and the scalar field is
under-damped, allowing for the oscillations to begin before the DM dominated
epoch begins. To solve Eq. (3.4) we employ an ansatz for $\phi$:

$\displaystyle\phi(t)=\phi_{\rm osc}(t)+A(t)\,,$ (3.6)

where $\phi_{\rm osc}$ is the solution to the homogeneous version of the
Klein-Gordon equation:

$\ddot{\phi}_{\rm osc}+3H\dot{\phi}_{\rm osc}+m^{2}_{\rm eff}(t)\phi_{\rm
osc}=0\,,$ (3.7)

while $A(t)$ solves

$\ddot{A}+3H\dot{A}+m^{2}_{\rm eff}(t)A=-Q_{0}(t)\,.$ (3.8)

On the basis that $A$ is sourced by a slowly-evolving function, $Q_{0}$, we
make the assumption that $A(t)$ itself is slowly varying such that
$\ddot{A},3H\dot{A}\ll m^{2}_{\rm eff}A$ and thus, we can neglect the first
two terms of the above equation. This results in an approximated analytical
solution for $A$:

$\displaystyle A(t)\approx-\frac{Q_{0}(t)}{m^{2}_{\rm eff}(t)}\,.$ (3.9)

Making use of the Wentzel-Kramer-Brillouin (WKB) approximation, we find the
following solution for $\phi_{\rm osc}$:

$\phi_{\rm osc}(t)=\left(\frac{a_{0}}{a}\right)^{3/2}\left(\frac{m_{0}}{m_{\rm
eff}}\right)^{1/2}\left(\phi_{+}\sin{(m_{\rm eff}t)}+\phi_{-}\cos{(m_{\rm
eff}t)}\right)\,,$ (3.10)

where $m_{0}$ is the effective mass taken at $t=t_{0}$, and $\phi_{+}$ and
$\phi_{-}$ are constants. We can then average $\phi$ over one oscillation
period to get

$\langle\phi\rangle=A\,,$ (3.11)

and for its velocity

$\langle\dot{\phi}\rangle=\dot{A}\,.$ (3.12)

Likewise, the variance of these quantities becomes,

$\langle\phi^{2}\rangle=\frac{1}{2}(\phi_{+}^{2}+\phi_{-}^{2})\left(\frac{m_{0}}{m_{\rm
eff}}\right)\left(\frac{a_{0}}{a}\right)^{3}+A^{2}\,,$ (3.13)
$\langle\dot{\phi}^{2}\rangle=\frac{1}{2}m_{\rm
eff}^{2}(\phi_{+}^{2}+\phi_{-}^{2})\left(\frac{m_{0}}{m_{\rm
eff}}\right)\left(\frac{a_{0}}{a}\right)^{3}\,,$ (3.14)

where in the last line we have ignored an $\dot{A}^{2}$ term, since under our
assumptions $\dot{A}\ll m_{\rm eff}A$.

The above expressions are crucial to the final results shown in this section.
For standard uncoupled scalar field DM, both the field value and its
derivative average out to zero. These equations encode the fact that, in our
case, the field does not oscillate around zero, but rather, its averaged value
is shifted by $A$ due to the interaction. Therefore, the average energy
density and pressure of the field are

$\langle\rho_{\phi}\rangle=\frac{1}{4}(\phi_{+}^{2}+\phi_{-}^{2})\left(\frac{m_{0}}{m_{\rm
eff}}\right)\left(\frac{a_{0}}{a}\right)^{3}(m_{\rm
eff}^{2}+m^{2})+\frac{m^{2}A^{2}}{2}\,,$ (3.15)

and

$\langle
p_{\phi}\rangle=\frac{1}{4}(\phi_{+}^{2}+\phi_{-}^{2})\left(\frac{m_{0}}{m_{\rm
eff}}\right)\left(\frac{a_{0}}{a}\right)^{3}(m_{\rm
eff}^{2}-m^{2})-\frac{m^{2}A^{2}}{2}\,,$ (3.16)

respectively. Here we see clearly that the interaction between DM and DE
results in a change in the DM pressure, causing its departure from zero. The
pressure can also be expressed as

$\langle
p_{\phi}\rangle=\left(\langle\rho_{\phi}\rangle-\frac{m^{2}A^{2}}{2}\right)\frac{m^{2}_{\rm
eff}-m^{2}}{m^{2}_{\rm eff}+m^{2}}-\frac{m^{2}A^{2}}{2}\,,$ (3.17)

in which case the equation of state becomes

$w_{\phi}\equiv\frac{\langle
p_{\phi}\rangle}{\langle\rho_{\phi}\rangle}=\left(1-\frac{m^{2}A^{2}}{2\langle\rho_{\phi}\rangle}\right)\frac{m^{2}_{\rm
eff}-m^{2}}{m^{2}_{\rm
eff}+m^{2}}-\frac{m^{2}A^{2}}{2\langle\rho_{\phi}\rangle}\,.$ (3.18)

Provided the kinetic energy of the dark energy field is small comparatively to
its potential, $Q_{1}$ is positive and $m_{\rm eff}>m$. This means that the
equation of state of dark matter starts by being positive in its early stages.
Around matter-dark energy equality, the equation of state begins to decrease
and, in fact, becomes increasingly negative, as illustrated in Fig. 1. The
very late-time occurrence of this transition is ascribed to the dependence of
the coupling on the dark energy density, implying that the coupling only has a
significant impact on $w_{\phi}$ when DE begins to dominate. Indeed, from Eq.
(3.18), we see that the equation of state parameter for dark matter becomes
negative when

$\langle\rho_{\phi}\rangle<\frac{1+Z}{Z}\,\frac{m^{2}A^{2}}{2}$ (3.19)

where $Z=(m^{2}_{\rm eff}-m^{2})/(m^{2}_{\rm eff}+m^{2})\approx Q_{1}/2m^{2}$.
As $A\approx-Q_{0}/m^{2}$, it turns out that $w_{\phi}<0$ when
$\langle\rho_{\phi}\rangle$ drops below

$\langle\rho_{\phi}\rangle<\frac{Q_{0}^{2}}{Q_{1}}\approx
V\approx\rho_{\chi}\,.$ (3.20)

Consequently, the DM pressure becomes appreciably negative only at very low
redshift.

Differentiating the averaged density, Eq. (3.15), we obtain

$\frac{d\langle{\rho_{\phi}}\rangle}{dt}+3H\left(\langle\rho_{\phi}\rangle+\langle
p_{\phi}\rangle\right)=\frac{3H}{4}(\phi_{+}^{2}+\phi_{-}^{2})\left(\frac{m_{0}}{m_{\rm
eff}}\right)\left(\frac{a_{0}}{a}\right)^{3}(m_{\rm
eff}^{2}-m^{2})+m^{2}A\dot{A}\,,$ (3.21)

or equivalently

$\frac{d}{dt}\langle{\rho_{\phi}}\rangle+3H\left(\langle\rho_{\phi}\rangle-\frac{m^{2}A^{2}}{2}\right)=m^{2}A\dot{A}\,.$
(3.22)

Had we done the time-averaging of Eq. (2.17) directly, we would conclude that,
in fact

$\displaystyle\frac{d\langle{\rho_{\phi}}\rangle}{dt}=\langle{\dot{\rho}_{\phi}}\rangle\,.$
(3.23)

We now define an effective equation of state parameter for DM as

$w_{\rm
eff}=-\frac{1}{\langle{\rho_{\phi}}\rangle}\left(\frac{m^{2}A^{2}}{2}+\frac{m^{2}A\dot{A}}{3H}\right)\,,$
(3.24)

such that Eq. (3.22) becomes

$\langle{\dot{\rho}_{\phi}}\rangle+3H\langle{\rho_{\phi}}\rangle(1+w_{\rm
eff})=0\,.$ (3.25)

The qualitative behaviour of $w_{\rm eff}$ is shown in Fig. 1, derived from a
modified version of the publicly available
CLASS222https://github.com/lesgourg/class_public [84] code to account for the
dynamics of the present model. From this, we see that the DM component
effectively evolves like DM during matter domination and then transitions into
a negative pressure fluid behaviour during DE domination.

Finally, and for completeness, the equation of motion for the dark energy
field, obtained by substituting Eqs. (3.11) and (3.12) into Eq. (2.12), yields

$\ddot{\chi}+\left(3H+\frac{2\kappa\beta\dot{A}}{1+2\beta\kappa
A}\right)\dot{\chi}+V_{,\chi}\left(1+2\beta\kappa A\right)=0.$ (3.26)

Before discussing cosmological perturbations in this setup, we emphasise once
more that this framework is indeed more general than the setup presented. In
particular, the slowly varying functions $Q_{0}$ and $Q_{1}$ might arise from
interactions with non-scalar field dark energy sources. For example, if the DE
field’s dynamics is negligible, our system reduces to DM coupled conformally
to a cosmological constant. Another possibility motivated by string theory
would be to have DM interacting with a 3-form DE field [57], which could also
reduce to this same phenomenology.

Figure 1: Evolution of the equation of state of dark matter $w_{\phi}$ (red
curves) and the effective dark matter equation of state $w_{\rm eff}$ (blue
curves), defined in Eq. (3.18) and Eq. (3.24) respectively. The model
parameters used in this illustrative example are $m=10^{-17}$ eV and
$\lambda=-0.2$, according to Eqs. (2.19) and (2.20). Filled lines label the
curves with $\beta=5\times 10^{14}$, while dashed lines are used for
$\beta=3\times 10^{14}$.

##  4 Perturbations

We now turn to the study of linear perturbations for our model. This will lead
to the derivation of the sound speed, which is crucial to solving the fluid-
scalar field equations of motion.

We follow the approach developed in Ref. [85] to compute the sound speed,
considering only scalar perturbations. We adopt the following conventions for
the metric scalar perturbations in a general gauge:

$\displaystyle\delta g_{00}$ $\displaystyle=-2\Phi\,,$ (4.1)
$\displaystyle\delta g_{i0}$ $\displaystyle=a\nabla_{i}B\,,$ (4.2)
$\displaystyle\delta g_{ij}$
$\displaystyle=-2a^{2}(\delta_{ij}\Psi-\nabla_{i}\nabla_{j}E)\,,$ (4.3)

where $\Phi$, $\Psi$, $B$ and $E$ are the four scalar degrees of freedom which
can be expanded and decomposed in independently evolving Fourier modes $k$.
This results in the following equations of motion for the perturbations of the
scalar fields $\delta\chi$ and $\delta\phi$:

$\displaystyle\delta\ddot{\chi}$ $\displaystyle+$
$\displaystyle\left(3H+\frac{C_{,\phi}}{C}\dot{\phi}\right)\delta\dot{\chi}+\left(\frac{k^{2}}{a^{2}}+C(\phi)V_{,\chi\chi}\right)\delta\chi=\dot{\chi}\left(\dot{\Phi}+3\dot{\Psi}-\frac{k}{a}B+\dot{E}\right)$
(4.4) $\displaystyle-$ $\displaystyle 2C(\phi)V_{,\chi}\Phi-
C_{,\phi}V_{,\chi}\delta\phi-\frac{C_{,\phi}}{C}\dot{\chi}\delta\dot{\phi}-\left(\frac{C_{,\phi}}{C}\right)_{,\phi}\dot{\phi}\dot{\chi}\delta\phi$

and

$\delta\ddot{\phi}+3H\delta\dot{\phi}+\frac{k^{2}}{a^{2}}\delta\phi+m^{2}\delta\phi=\dot{\phi}\left(\dot{\Phi}+3\dot{\Psi}-\frac{k}{a}B+\dot{E}\right)+2(\ddot{\phi}+3H\dot{\phi})\Phi-\delta
Q\,,$ (4.5)

Where we have defined $\delta Q$ as the perturbation of the coupling function.
As for the background evolution, we can expand it as

$\delta Q=\delta Q_{0}+\phi\delta Q_{1}+\delta\phi Q_{1}\,,$ (4.6)

such that Eq. (4.5) becomes

$\delta\ddot{\phi}+3H\delta\dot{\phi}+\frac{k^{2}}{a^{2}}\delta\phi+m^{2}_{\rm
eff}\delta\phi=\dot{\phi}\left(\dot{\Phi}+3\dot{\Psi}-\frac{k}{a}B+\dot{E}\right)+2(\ddot{\phi}+3H\dot{\phi})\Phi-\delta
Q_{0}-\phi\delta Q_{1}\,.$ (4.7)

Note that the $\delta Q$ term encloses a term proportional to $\Phi$, arising
due to the covariant derivatives in the definition of $Q$ in Eq. (2.6). The
exact form of $\delta Q_{0}$ and $\delta Q_{1}$ for the particular setup given
by Eqs. (2.19)-(2.21) is the following:

$\displaystyle\delta Q_{0}$
$\displaystyle=-\kappa\beta\left(2\dot{\chi}\delta\dot{\chi}-4V_{,\chi}\delta\chi-2\dot{\chi}^{2}\Phi\right)\,,$
(4.8) $\displaystyle\delta Q_{1}$
$\displaystyle=8\kappa^{2}\beta^{2}V_{,\chi}\delta\chi\,.$ (4.9)

The perturbed perfect fluid quantities are defined in terms of the time-
averaged oscillating DM scalar field and its perturbation:

$\delta\rho=\langle\dot{\phi}\delta\dot{\phi}-\dot{\phi}^{2}\Phi+m^{2}\phi\delta\phi\rangle\,,$
(4.10) $\delta
p=\langle\dot{\phi}\delta\dot{\phi}-\dot{\phi}^{2}\Phi-m^{2}\phi\delta\phi\rangle\,,$
(4.11) $\frac{a}{k}(\rho+p)(v-B)=\langle\dot{\phi}\delta\phi\rangle\,,$ (4.12)

where it should be understood that the fluid variables are themselves averaged
quantities. Bringing together Eqs. (4.7)-(4.12), we arrive at the following
expressions for the perturbed continuity equations:

$\delta\dot{\rho}+3H(\delta\rho+\delta
p)=(\rho+p)\left(3\dot{\Psi}+\dot{E}-\frac{k}{a}v\right)-\langle\delta
Q\dot{\phi}\rangle-\langle Q\delta\dot{\phi}\rangle\,,$ (4.13)
$\frac{1}{a^{4}(\rho+p)}\frac{d}{dt}[a^{4}(\rho+p)(v-B)]=\frac{k}{a}\Phi+\frac{k}{a(\rho+p)}[\delta
p-\langle Q\delta\phi\rangle]\,.$ (4.14)

For convenience and to simplify the calculations, we consider the axion-
comoving gauge, which amounts to setting $B=v$. Thus, in this context, Eq.
(4.14) reduces to

$\Phi=-\frac{\delta p}{\rho+p}+\frac{\langle Q\delta\phi\rangle}{\rho+p}\,.$
(4.15)

Analogous to background fluid-field treatment, we specify the following ansatz
for the perturbation of the $\phi$-field:

$\delta\phi(k,t)=\delta\phi_{+}(k,t)\sin(m_{\rm
eff}t)+\delta\phi_{-}(k,t)\cos(m_{\rm eff}t)+\delta{\mathcal{A}}(t)\,.$ (4.16)

It is important to note that $\delta{\mathcal{A}}(t)$ is not just the
perturbation of $A(t)$ defined in Eq. (3.9), as will be shown explicitly
below.

For the rest of the calculation, we will assume a quasi-static approximation
following [85], and will also be discarding derivatives of $A$ and
$\delta{\mathcal{A}}$ on similar grounds. In comoving gauge we have
$\langle\dot{\phi}\delta\phi\rangle=0$ from Eq. (4.12), which implies

$\delta\phi_{+}\phi_{-}=\delta\phi_{-}\phi_{+}\,.$ (4.17)

Now we can solve Eq. (4.7) to leading order in $H/m$, since the field is
oscillating if $m\gg H$. This implies assuming that metric perturbations vary
only on cosmological time scales $t\sim H^{-1}\gg m^{-1}$ and then grouping
the terms in powers of $H/m$,

$\delta\ddot{\phi}+\frac{k^{2}}{a^{2}}\delta\phi+m_{\rm
eff}^{2}\delta\phi=-2(m^{2}_{\rm eff}\phi+Q_{0})\Phi-\delta Q_{0}-\phi\delta
Q_{1}\,.$ (4.18)

Once again, splitting the equation and matching the non-oscillating terms, we
get:

$\delta\ddot{{\mathcal{A}}}+\frac{k^{2}}{a^{2}}\delta{\mathcal{A}}+m^{2}_{\rm
eff}\delta{\mathcal{A}}=-\delta Q_{0}-\delta Q_{1}A\,,$ (4.19)

which, in the quasi-static approximation, yields

$\delta{\mathcal{A}}=-\frac{\delta Q_{0}+\delta
Q_{1}A}{\frac{k^{2}}{a^{2}}+m^{2}_{\rm eff}}=-\frac{\delta
Q_{0}-\frac{Q_{0}}{m_{\rm eff}^{2}}\delta
Q_{1}}{\frac{k^{2}}{a^{2}}+m^{2}_{\rm eff}}\,.$ (4.20)

Substituting the ansatz in Eq. (4.16) into Eq. (4.18) we obtain:

$\Phi=-\frac{1}{2}\left(\frac{a}{a_{0}}\right)^{3/2}\left(\frac{m_{0}}{m_{\rm
eff}}\right)^{-1/2}\frac{\delta\phi_{+}}{\phi_{+}}\frac{k^{2}}{m^{2}_{\rm
eff}a^{2}}-\frac{1}{2m^{2}_{\rm eff}}\delta Q_{1}\,,$ (4.21)

up to leading order in $H/m$. Replacing this expression into the perturbed
fluid equations for $\delta p$ and $\delta\rho$, Eqs. (4.10) and (4.11) we
arrive at

$\displaystyle\delta p$ $\displaystyle=$
$\displaystyle\frac{a^{-\frac{3}{2}}}{2}\left(\frac{m_{0}}{m_{\rm
eff}}\right)^{\frac{1}{2}}(\phi_{+}^{2}+\phi_{-}^{2})\frac{\delta\phi_{+}}{\phi_{+}}m^{2}_{\rm
eff}\left[\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}_{\rm
eff}}+\frac{Q_{1}}{m^{2}_{\rm eff}}\right]$ (4.22) $\displaystyle-$
$\displaystyle
m^{2}A\delta{\mathcal{A}}+\frac{a^{-3}}{4}\left(\frac{m_{0}}{m_{\rm
eff}}\right)(\phi_{+}^{2}+\phi_{-}^{2})\delta Q_{1}\,,$
$\displaystyle\delta\rho$ $\displaystyle=$
$\displaystyle\frac{a^{-\frac{3}{2}}}{2}\left(\frac{m_{0}}{m_{\rm
eff}}\right)^{\frac{1}{2}}(\phi_{+}^{2}+\phi_{-}^{2})\frac{\delta\phi_{+}}{\phi_{+}}m^{2}_{\rm
eff}\left[\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}_{\rm
eff}}+2-\frac{Q_{1}}{m^{2}_{\rm eff}}\right]$ (4.23) $\displaystyle+$
$\displaystyle
m^{2}A\delta{\mathcal{A}}+\frac{a^{-3}}{4}\left(\frac{m_{0}}{m_{\rm
eff}}\right)(\phi_{+}^{2}+\phi_{-}^{2})\delta Q_{1}\,.$

From here, and using Eq. (4.21), we can compute the pressure perturbation,
given by

$\displaystyle\delta
p=c^{2}_{a}\delta\rho-m^{2}A\delta{\mathcal{A}}(1+c^{2}_{a})+\frac{1}{2}\frac{\rho+p}{m^{2}_{\rm
eff}}\delta Q_{1}(1-c^{2}_{a})\,.$ (4.24)

where we have defined the effective sound-speed

$c^{2}_{a}=\frac{\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}_{\rm
eff}}+\frac{Q_{1}}{m^{2}_{\rm eff}}}{\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}_{\rm
eff}}+2-\frac{Q_{1}}{m^{2}_{\rm
eff}}}=\frac{\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}}+\frac{Q_{1}}{m^{2}}}{\frac{1}{2}\frac{k^{2}}{a^{2}m^{2}}+2+\frac{Q_{1}}{m^{2}}}\,,$
(4.25)

and the second equality follows from the definition of $m^{2}_{\rm eff}$ in
Eq. (3.5). It is important to note that this effective sound speed is akin to
that of the non-interacting axion, presented in Refs. [85, 86]. Accordingly,
and as expected, Eq. (4.25) reduces to that case in the limit where the
coupling $\beta$, and therefore $Q_{1}$, is zero. The effect of this coupling
can be appreciated in Figs. 2 and 3. At early times, the $k^{2}/a^{2}m^{2}$
term dominates, and we get a similar behaviour to that of the non-interacting
case. On the other hand, at late times, when $k^{2}/a^{2}m^{2}\ll Q_{1}$, the
sound-speed becomes practically constant, according to
$c^{2}_{a}\approx\frac{Q_{1}}{2m^{2}}$. As expected, in the absence of the
coupling, $c^{2}_{a}$ falls to zero at small redshifts.

Furthermore, according to Eq. (4.24), in contrast to the non-interacting case,
the pressure perturbation is not precisely proportional to the density
perturbation. This can be interpreted as a non-adiabatic contribution to the
pressure perturbation caused by the interaction with the dark energy scalar
field [87, 88, 81, 82].

Figure 2: Evolution of the effective sound-speed in terms of redshift $z$, as
defined in Eq. (4.25), for a different set of $k$ values
$\\{0.001,0.01,0.1,1,10\\}$ (in Mpc-1), for the following set of parameters:
$m=10^{-17}\rm{eV}$, $\beta=5\times 10^{14}$ and $\lambda=-0.2$. Figure 3:
Evolution of the effective sound-speed for the coupled model, defined in Eq.
(4.25), compared with a non-coupled standard axion, with the same parameters
used as in Fig. 2 and $k$ = 10 Mpc-1.

##  5 Summary and Conclusions

In this study we have introduced a novel model of the dark sector, consisting
of scalar field dark matter interacting with quintessence dark energy. Our
approach involves a conformal coupling between DM and DE via a conformal
coupling, with a unique twist: the conformal factor depends on the (fast
oscillating) DM field rather than the slowly evolving DE field, as is
typically the case. As a result, we have found an effective theory of
interacting dark matter and dark energy in which the coupling term is linked
to the energy density of DE rather than DM. Our framework not only offers a
theoretical basis for many extensively studied models of interacting dark
energy found in the existing literature, e.g. in Refs. [73, 69, 89, 65, 79,
90, 91, 80], but also introduces significant differences. Namely, within our
setup, the coupling between DM and DE serves to offset the oscillations of the
scalar field, resulting in a non-zero average for its equation of state
parameter. Upon averaging over these rapid oscillations, we have derived a
fluid-field description of the system, revealing a non-zero average value for
the field determined by the coupling. Consequently, the effective DM fluid
exhibits a non-zero physical pressure, starkly contrasting with standard IDE
models. This physical pressure transitions from slightly positive at early
times to negative after DM-DE equality. The main observational effect most
likely comes from the change in the DM equation of state at very late times,
at a redshift smaller than the redshift of the DE-DM equality. Thus, our work
carries implications in the context of searches for non-standard CDM physics
and provides theoretical support for such models (see [92, 93, 94]). Notably,
the non-standard equation of state of DM impacts the analysis and
interpretation of cosmological observational data, such as Pantheon+ [95] or
the recently published DESI data [96].

At the linear perturbation level, we have computed the pressure perturbation
and sound speed of the averaged DM fluid. Our findings reveal terms
proportional to the density perturbation $\delta\rho$ plus non-adiabatic
pressure terms depending on the perturbation of the DE scalar field. Notably,
due to the coupling, the adiabatic term deviates from the standard axion
scalar field scenario. This results in an effective sound speed that remains
non-zero across all scales, unlike the uncoupled case where it vanishes at
small $k$. Consequently, we anticipate a slight power suppression in the
matter power spectrum at all scales within the interacting scenario considered
here. Furthermore, the influence of non-adiabatic pressure contributions on
observables in this context warrants further investigation. A comprehensive
study of these points requires a full implementation of the theory in publicly
available Boltzmann codes such as CLASS or
CAMB333https://github.com/cmbant/CAMB [97], a task left for future work.

## Acknowledgments

We are grateful to Eleonora Di Valentino, William Giarè and Vivian Poulin for
useful discussions and encouragement. EMT is supported by funding from the
European Research Council (ERC) under the European Union’s HORIZON-ERC-2022
(grant agreement no. 101076865). EMT was also supported by the grant
SFRH/BD/143231/2019 from Fundação para a Ciência e a Tecnologia (FCT) during
part of this work. CvdB is supported (in part) by the Lancaster–Sheffield
Consortium for Fundamental Physics under STFC grant: ST/X000621/1. NJN
acknowledges support from FCT via the following projects: UIDB/04434/2020 &
UIDP/04434/2020 and PTDC/FIS-AST/0054/2021. This article is based upon work
from COST Action CA21136 Addressing observational tensions in cosmology with
systematics and fundamental physics (Cosmoverse) supported by COST (European
Cooperation in Science and Technology).

## Appendix A

In the main body of the paper, we considered a conformal coupling function
$C$, which depends only on the DM field value, i.e. we considered the case of
$C(\phi)$. It is possible to consider a more general scenario, in which the
$C$ depends both on the field and its kinetic term
$X=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi$. For
completeness, in this appendix, we provide the equations for that case, that
is, $C\equiv C(\phi,X)$. The equation of motion for the dark energy field
$\chi$ can be derived from the action in Eq. (2.1) and reads

$\nabla^{\mu}\nabla_{\mu}\chi-C\frac{dV}{d\chi}=-\frac{\nabla_{\alpha}C}{C}\nabla^{\alpha}\chi\,.$
(5.1)

In the same way, the equation for the DM field $\phi$ is

$\nabla^{\mu}\nabla_{\mu}\phi-\frac{\partial U}{\partial\phi}=Q\,,$ (5.2)

where

$\displaystyle Q$ $\displaystyle=$
$\displaystyle\nabla_{\mu}\left(\frac{\partial
L_{\chi}}{\partial(\nabla_{\mu}\phi)}\right)-\frac{\partial
L_{\chi}}{\partial\phi}$ (5.3) $\displaystyle=$
$\displaystyle\nabla_{\mu}\left[\nabla^{\mu}\phi\left(Cg^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+4C^{2}U\right)\frac{C_{,X}}{C}\right]+\frac{C_{,\phi}}{2C}\left[Cg^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+4C^{2}U\right]\,.$

The energy momentum tensor for $\phi$ and $\chi$ are given by

$\displaystyle T_{\mu\nu}^{(\phi)}=\nabla_{\mu}\phi\nabla_{\nu}\phi-
g_{\mu\nu}\left(\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\phi\nabla_{\beta}\phi+V(\phi)\right)\,,$
(5.4) $\displaystyle T_{\mu\nu}^{(\chi)}$ $\displaystyle=$ $\displaystyle
C\nabla_{\mu}\chi\nabla_{\nu}\chi-
g_{\mu\nu}\left(\frac{C}{2}g^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+C^{2}U(\chi)\right)$
(5.5) $\displaystyle-$
$\displaystyle\frac{C_{,X}}{2C}\nabla_{\mu}\phi\nabla_{\nu}\phi\left[Cg^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi+4C^{2}U\right]\,.$

We have the following conservation equations:

$\nabla^{\mu}T_{\mu\nu}^{(\phi)}=Q\nabla_{\nu}\phi;~{}~{}~{}\nabla^{\mu}T_{\mu\nu}^{(\chi)}=-Q\nabla_{\nu}\phi\,.$
(5.6)

Note that

$T^{\mu}_{(\chi)\mu}=-\left[C\nabla^{\mu}\chi\nabla_{\mu}\chi+4C^{2}U(\chi)\right]\left(1+\frac{C_{,X}}{2C}\nabla^{\mu}\phi\nabla_{\mu}\phi\right)\,,$
(5.7)

and so the two terms in Eq. (5.3) are proportional to $T^{\mu}_{(\chi)\mu}$.

In full generality, these equations are difficult to analyse and to recast
into a fluid-field description. Nevertheless, the sub-case of $C\equiv C(X)$
might be worth investigating. We leave this for future work.

## Appendix B

In the main body of the paper we have dealt with an interaction mediated by a
conformal coupling which depends on the fast oscillating DM field. In the
following we briefly present another example in which DE and DM couple, but
not via a conformal coupling. The phenomenology turns out to be very similar
to the model presented in Sections 3 and 4.

The action we consider is

$\displaystyle{\cal S}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left(\frac{M_{\rm Pl}^{2}}{2}{\cal
R}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-U(\phi)-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-V(\chi)\right)$
$\displaystyle-$ $\displaystyle\int d^{4}x\sqrt{-g}\left(\phi
Q_{0}\left(\chi,X^{(\chi)}\right)+\frac{1}{2}\phi^{2}Q_{1}\left(\chi,X^{(\chi)}\right)\right)+{\cal
S}_{\rm SM}\,,$

where $Q_{0}$ and $Q_{1}$ are coupling functions, depending of the slowly
varying DE field $\chi$ and its kinetic term
$X^{(\chi)}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi$. Let
us focus here on the case in which $Q_{0}$ and $Q_{1}$ depend on $\chi$ only.
Then, in a flat FRW universe, the equation of motion for $\phi$ is given by
(using $U=\frac{1}{2}m^{2}\phi^{2}$)

$\ddot{\phi}+3H\dot{\phi}+m^{2}\phi=-Q(t)\,.$ (5.9)

where $Q(t)=Q_{0}(\chi)+Q_{1}(\chi)\phi$. This equation is of the same form as
Eq. (3.1). Assuming that $\chi$ evolves slowly as it represents DE, an
analysis similar to that performed in Section 3 would apply for this theory,
with the displacement $A$ in (3.9) now given by

$\displaystyle A(t)\approx-\frac{Q_{0}(\chi)}{m^{2}_{\rm eff}}\,,$ (5.10)

and with an effective mass that is still given by Eq. (3.5).

The equation for the DE field reads

$\ddot{\chi}+3H\dot{\chi}+V_{,\chi}+\phi
Q_{0,\chi}+\frac{1}{2}\phi^{2}Q_{1,\chi}=0\,.$ (5.11)

We can now employ the time-averaged behaviour of the DM field. The end result
is that the dynamics of the DE field is described by

$\ddot{\chi}+3H\dot{\chi}+V_{,\chi}-\frac{Q_{0}}{m_{\rm
eff}^{2}}Q_{0,\chi}+\frac{2\langle\rho_{\phi}\rangle+\frac{Q_{0}^{2}}{m_{\rm
eff}^{2}}}{2\left(m_{\rm eff}^{2}+m^{2}\right)}Q_{1,\chi}=0\,,$ (5.12)

where we have used Eqs. (3.13) and (3.15).

At the linear perturbations level, the equation for the DM field is given by
Eq. (4.5) and so the expression for the sound-speed is unchanged.

## References

  * [1] P.J.E. Peebles, _Cosmology’s Century: An Inside History of Our Modern Understanding of the Universe_ , Princeton University Press (4, 2022).
  * [2] E. Abdalla et al., _Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies_ , _JHEAp_ 34 (2022) 49 [2203.06142].
  * [3] V.C. Rubin, J. Ford, W. K. and N. Thonnard, _Rotational properties of 21 SC galaxies with a large range of luminosities and radii, from NGC 4605 (R=4kpc) to UGC 2885 (R=122kpc)._ , _Astrophys. J._ 238 (1980) 471.
  * [4] F. Zwicky, _Die Rotverschiebung von extragalaktischen Nebeln_ , _Helvetica Physica Acta_ 6 (1933) 110.
  * [5] F. Zwicky, _On the Masses of Nebulae and of Clusters of Nebulae_ , _Astrophys. J._ 86 (1937) 217.
  * [6] D. Clowe, M. Bradac, A.H. Gonzalez, M. Markevitch, S.W. Randall, C. Jones et al., _A direct empirical proof of the existence of dark matter_ , _Astrophys. J. Lett._ 648 (2006) L109 [astro-ph/0608407].
  * [7] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [8] G. Hinshaw, D. Larson, E. Komatsu, D.N. Spergel, C.L. Bennett, J. Dunkley et al., _Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results_ , _Astrophys. J. Sup._ 208 (2013) 19 [1212.5226].
  * [9] A.S. Göttel, A. Ejlli, K. Karan, S.M. Vermeulen, L. Aiello, V. Raymond et al., _Searching for scalar field dark matter with LIGO_ , 2401.18076.
  * [10] XENON collaboration, _Dark Matter Search Results from a One Ton-Year Exposure of XENON1T_ , _Phys. Rev. Lett._ 121 (2018) 111302 [1805.12562].
  * [11] LUX collaboration, _Results from a search for dark matter in the complete LUX exposure_ , _Phys. Rev. Lett._ 118 (2017) 021303 [1608.07648].
  * [12] F. Kahlhoefer, _Review of LHC Dark Matter Searches_ , _Int. J. Mod. Phys. A_ 32 (2017) 1730006 [1702.02430].
  * [13] G. Bertone and T. Tait, M. P., _A new era in the search for dark matter_ , _Nature_ 562 (2018) 51 [1810.01668].
  * [14] P.W. Graham, I.G. Irastorza, S.K. Lamoreaux, A. Lindner and K.A. van Bibber, _Experimental Searches for the Axion and Axion-Like Particles_ , _Ann. Rev. Nucl. Part. Sci._ 65 (2015) 485 [1602.00039].
  * [15] K. Choi, S.H. Im and C. Sub Shin, _Recent Progress in the Physics of Axions and Axion-Like Particles_ , _Ann. Rev. Nucl. Part. Sci._ 71 (2021) 225 [2012.05029].
  * [16] A. Amruth et al., _Einstein rings modulated by wavelike dark matter from anomalies in gravitationally lensed images_ , _Nature Astron._ 7 (2023) 736 [2304.09895].
  * [17] W. Hu, R. Barkana and A. Gruzinov, _Cold and fuzzy dark matter_ , _Phys. Rev. Lett._ 85 (2000) 1158 [astro-ph/0003365].
  * [18] M. Dentler, D.J.E. Marsh, R. Hložek, A. Laguë, K.K. Rogers and D. Grin, _Fuzzy dark matter and the Dark Energy Survey Year 1 data_ , _Mon. Not. Roy. Astron. Soc._ 515 (2022) 5646 [2111.01199].
  * [19] P. Svrcek and E. Witten, _Axions In String Theory_ , _JHEP_ 06 (2006) 051 [hep-th/0605206].
  * [20] A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, _String Axiverse_ , _Phys. Rev. D_ 81 (2010) 123530 [0905.4720].
  * [21] R. Hlozek, D. Grin, D.J.E. Marsh and P.G. Ferreira, _A search for ultralight axions using precision cosmological data_ , _Phys. Rev. D_ 91 (2015) 103512 [1410.2896].
  * [22] J.S. Bullock and M. Boylan-Kolchin, _Small-Scale Challenges to the $\Lambda$CDM Paradigm_, _Ann. Rev. Astron. Astrophys._ 55 (2017) 343 [1707.04256].
  * [23] D.H. Weinberg, J.S. Bullock, F. Governato, R. Kuzio de Naray and A.H.G. Peter, _Cold dark matter: controversies on small scales_ , _Proc. Nat. Acad. Sci._ 112 (2015) 12249 [1306.0913].
  * [24] L. Hui, J.P. Ostriker, S. Tremaine and E. Witten, _Ultralight scalars as cosmological dark matter_ , _Phys. Rev. D_ 95 (2017) 043541 [1610.08297].
  * [25] K.K. Rogers, R. Hložek, A. Laguë, M.M. Ivanov, O.H.E. Philcox, G. Cabass et al., _Ultra-light axions and the S 8 tension: joint constraints from the cosmic microwave background and galaxy clustering_, _JCAP_ 06 (2023) 023 [2301.08361].
  * [26] A. Laguë, J.R. Bond, R. Hložek, D.J.E. Marsh and L. Söding, _Evolving ultralight scalars into non-linearity with Lagrangian perturbation theory_ , _Mon. Not. Roy. Astron. Soc._ 504 (2021) 2391 [2004.08482].
  * [27] M.B. Green, J.H. Schwarz and E. Witten, _SUPERSTRING THEORY. VOL. 1: INTRODUCTION_ , Cambridge Monographs on Mathematical Physics (7, 1988).
  * [28] L. Amendola and R. Barbieri, _Dark matter from an ultra-light pseudo-Goldsone-boson_ , _Phys. Lett. B_ 642 (2006) 192 [hep-ph/0509257].
  * [29] R. Hložek, D.J.E. Marsh, D. Grin, R. Allison, J. Dunkley and E. Calabrese, _Future CMB tests of dark matter: Ultralight axions and massive neutrinos_ , _Phys. Rev. D_ 95 (2017) 123511 [1607.08208].
  * [30] R. Hlozek, D.J.E. Marsh and D. Grin, _Using the Full Power of the Cosmic Microwave Background to Probe Axion Dark Matter_ , _Mon. Not. Roy. Astron. Soc._ 476 (2018) 3063 [1708.05681].
  * [31] G.S. Farren, D. Grin, A.H. Jaffe, R. Hložek and D.J.E. Marsh, _Ultralight axions and the kinetic Sunyaev-Zel’dovich effect_ , _Phys. Rev. D_ 105 (2022) 063513 [2109.13268].
  * [32] V. Iršič, M. Viel, M.G. Haehnelt, J.S. Bolton and G.D. Becker, _First constraints on fuzzy dark matter from Lyman- $\alpha$ forest data and hydrodynamical simulations_, _Phys. Rev. Lett._ 119 (2017) 031302 [1703.04683].
  * [33] E. Armengaud, N. Palanque-Delabrouille, C. Yèche, D.J.E. Marsh and J. Baur, _Constraining the mass of light bosonic dark matter using SDSS Lyman- $\alpha$ forest_, _Mon. Not. Roy. Astron. Soc._ 471 (2017) 4606 [1703.09126].
  * [34] T. Kobayashi, R. Murgia, A. De Simone, V. Iršič and M. Viel, _Lyman- $\alpha$ constraints on ultralight scalar dark matter: Implications for the early and late universe_, _Phys. Rev. D_ 96 (2017) 123514 [1708.00015].
  * [35] M. Nori, R. Murgia, V. Iršič, M. Baldi and M. Viel, _Lyman $\alpha$ forest and non-linear structure characterization in Fuzzy Dark Matter cosmologies_, _Mon. Not. Roy. Astron. Soc._ 482 (2019) 3227 [1809.09619].
  * [36] K.K. Rogers and H.V. Peiris, _Strong Bound on Canonical Ultralight Axion Dark Matter from the Lyman-Alpha Forest_ , _Phys. Rev. Lett._ 126 (2021) 071302 [2007.12705].
  * [37] K. Hayashi, E.G.M. Ferreira and H.Y.J. Chan, _Narrowing the Mass Range of Fuzzy Dark Matter with Ultrafaint Dwarfs_ , _Astrophys. J. Lett._ 912 (2021) L3 [2102.05300].
  * [38] N. Dalal and A. Kravtsov, _Excluding fuzzy dark matter with sizes and stellar kinematics of ultrafaint dwarf galaxies_ , _Phys. Rev. D_ 106 (2022) 063517 [2203.05750].
  * [39] I.S. Goldstein, S.M. Koushiappas and M.G. Walker, _Viability of ultralight bosonic dark matter in dwarf galaxies_ , _Phys. Rev. D_ 106 (2022) 063010 [2206.05244].
  * [40] A. Laguë, J.R. Bond, R. Hložek, K.K. Rogers, D.J.E. Marsh and D. Grin, _Constraining ultralight axions with galaxy surveys_ , _JCAP_ 01 (2022) 049 [2104.07802].
  * [41] A. Kunkel, T. Chiueh and B.M. Schäfer, _A weak lensing perspective on non-linear structure formation with fuzzy dark matter_ , _Mon. Not. Roy. Astron. Soc._ 527 (2023) 10538 [2211.01523].
  * [42] S.C. Hotinli, D.J.E. Marsh and M. Kamionkowski, _Probing ultralight axions with the 21-cm signal during cosmic dawn_ , _Phys. Rev. D_ 106 (2022) 043529 [2112.06943].
  * [43] J.B. Bauer, D.J.E. Marsh, R. Hložek, H. Padmanabhan and A. Laguë, _Intensity Mapping as a Probe of Axion Dark Matter_ , _Mon. Not. Roy. Astron. Soc._ 500 (2020) 3162 [2003.09655].
  * [44] J. Flitter and E.D. Kovetz, _Closing the window on fuzzy dark matter with the 21-cm signal_ , _Phys. Rev. D_ 106 (2022) 063504 [2207.05083].
  * [45] EPTA collaboration, _The second data release from the European Pulsar Timing Array: V. Implications for massive black holes, dark matter and the early Universe_ , 2306.16227.
  * [46] NANOGrav collaboration, _The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background_ , _Astrophys. J. Lett._ 951 (2023) L8 [2306.16213].
  * [47] A. Khmelnitsky and V. Rubakov, _Pulsar timing signal from ultralight scalar dark matter_ , _JCAP_ 02 (2014) 019 [1309.5888].
  * [48] N.K. Porayko and K.A. Postnov, _Constraints on ultralight scalar dark matter from pulsar timing_ , _Phys. Rev. D_ 90 (2014) 062008 [1408.4670].
  * [49] N.K. Porayko et al., _Parkes Pulsar Timing Array constraints on ultralight scalar-field dark matter_ , _Phys. Rev. D_ 98 (2018) 102002 [1810.03227].
  * [50] Z.-Q. Xia, T.-P. Tang, X. Huang, Q. Yuan and Y.-Z. Fan, _Constraining ultralight dark matter using the Fermi-LAT pulsar timing array_ , _Phys. Rev. D_ 107 (2023) L121302 [2303.17545].
  * [51] European Pulsar Timing Array collaboration, _Second Data Release from the European Pulsar Timing Array: Challenging the Ultralight Dark Matter Paradigm_ , _Phys. Rev. Lett._ 131 (2023) 171001 [2306.16228].
  * [52] C. Wetterich, _Cosmology and the Fate of Dilatation Symmetry_ , _Nucl. Phys. B_ 302 (1988) 668 [1711.03844].
  * [53] P.J.E. Peebles and B. Ratra, _Cosmology with a Time Variable Cosmological Constant_ , _Astrophys. J. Lett._ 325 (1988) L17.
  * [54] B. Ratra and P.J.E. Peebles, _Cosmological Consequences of a Rolling Homogeneous Scalar Field_ , _Phys. Rev. D_ 37 (1988) 3406.
  * [55] C. Wetterich, _The Cosmon model for an asymptotically vanishing time dependent cosmological ’constant’_ , _Astron. Astrophys._ 301 (1995) 321 [hep-th/9408025].
  * [56] R.R. Caldwell, R. Dave and P.J. Steinhardt, _Cosmological imprint of an energy component with general equation of state_ , _Phys. Rev. Lett._ 80 (1998) 1582 [astro-ph/9708069].
  * [57] T.S. Koivisto and N.J. Nunes, _Coupled three-form dark energy_ , _Physical Review D_ 88 (2013) .
  * [58] L. Amendola and S. Tsujikawa, _Dark Energy: Theory and Observations_ , Cambridge University Press (1, 2015).
  * [59] E.J. Copeland, M. Sami and S. Tsujikawa, _Dynamics of dark energy_ , _Int. J. Mod. Phys. D_ 15 (2006) 1753 [hep-th/0603057].
  * [60] M. Li, X.-D. Li, S. Wang and Y. Wang, _Dark Energy_ , _Commun. Theor. Phys._ 56 (2011) 525 [1103.5870].
  * [61] S.M. Carroll, _Quintessence and the rest of the world_ , _Phys. Rev. Lett._ 81 (1998) 3067 [astro-ph/9806099].
  * [62] B. Bertotti, L. Iess and P. Tortora, _A test of general relativity using radio links with the Cassini spacecraft_ , _Nature_ 425 (2003) 374.
  * [63] N. Schöneberg, G. Franco Abellán, A. Pérez Sánchez, S.J. Witte, V. Poulin and J. Lesgourgues, _The H0 Olympics: A fair ranking of proposed models_ , _Phys. Rept._ 984 (2022) 1 [2107.10291].
  * [64] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al., _In the realm of the Hubble tension—a review of solutions_ , _Class. Quant. Grav._ 38 (2021) 153001 [2103.01183].
  * [65] E. Di Valentino, A. Melchiorri and O. Mena, _Can interacting dark energy solve the $H_{0}$ tension?_, _Phys. Rev. D_ 96 (2017) 043503 [1704.08342].
  * [66] E. Di Valentino, A. Melchiorri, O. Mena and S. Vagnozzi, _Interacting dark energy in the early 2020s: A promising solution to the $H_{0}$ and cosmic shear tensions_, _Phys. Dark Univ._ 30 (2020) 100666 [1908.04281].
  * [67] W. Yang, S. Pan, E. Di Valentino, O. Mena and A. Melchiorri, _2021-H0 odyssey: closed, phantom and interacting dark energy cosmologies_ , _JCAP_ 10 (2021) 008 [2101.03129].
  * [68] L.L. Honorez, B.A. Reid, O. Mena, L. Verde and R. Jimenez, _Coupled dark matter-dark energy in light of near universe observations_ , _Journal of Cosmology and Astroparticle Physics_ 2010 (2010) 029–029.
  * [69] M.B. Gavela, L. Lopez Honorez, O. Mena and S. Rigolin, _Dark Coupling and Gauge Invariance_ , _JCAP_ 11 (2010) 044 [1005.0295].
  * [70] J.-H. He, B. Wang and E. Abdalla, _Stability of the curvature perturbation in dark sectors’ mutual interacting models_ , _Phys. Lett. B_ 671 (2009) 139 [0807.3471].
  * [71] J. Valiviita, E. Majerotto and R. Maartens, _Instability in interacting dark energy and dark matter fluids_ , _JCAP_ 07 (2008) 020 [0804.0232].
  * [72] W. Yang, S. Pan and D.F. Mota, _Novel approach toward the large-scale stable interacting dark-energy models and their astronomical bounds_ , _Phys. Rev. D_ 96 (2017) 123508 [1709.00006].
  * [73] M.B. Gavela, D. Hernandez, L. Lopez Honorez, O. Mena and S. Rigolin, _Dark coupling_ , _JCAP_ 07 (2009) 034 [0901.1611].
  * [74] W. Yang, S. Pan, O. Mena and E. Di Valentino, _On the dynamics of a dark sector coupling_ , _JHEAp_ 40 (2023) 19 [2209.14816].
  * [75] W. Yang, O. Mena, S. Pan and E. Di Valentino, _Dark sectors with dynamical coupling_ , _Phys. Rev. D_ 100 (2019) 083509 [1906.11697].
  * [76] W. Yang, E. Di Valentino, O. Mena, S. Pan and R.C. Nunes, _All-inclusive interacting dark sector cosmologies_ , _Phys. Rev. D_ 101 (2020) 083509 [2001.10852].
  * [77] R.C. Nunes, S. Vagnozzi, S. Kumar, E. Di Valentino and O. Mena, _New tests of dark sector interactions from the full-shape galaxy power spectrum_ , _Phys. Rev. D_ 105 (2022) 123506 [2203.08093].
  * [78] C. van de Bruck, G. Poulot and E.M. Teixeira, _Scalar field dark matter and dark energy: a hybrid model for the dark sector_ , _JCAP_ 07 (2023) 019 [2211.13653].
  * [79] E. Di Valentino, A. Melchiorri, O. Mena and S. Vagnozzi, _Nonminimal dark sector physics and cosmological tensions_ , _Phys. Rev. D_ 101 (2020) 063502 [1910.09853].
  * [80] W. Giarè, Y. Zhai, S. Pan, E. Di Valentino, R.C. Nunes and C. van de Bruck, _Tightening the reins on non-minimal dark sector physics: Interacting Dark Energy with dynamical and non-dynamical equation of state_ , 2404.02110.
  * [81] M. Carrillo González and M. Trodden, _Field theories and fluids for an interacting dark sector_ , _Physical Review D_ 97 (2018) .
  * [82] J.P. Johnson and S. Shankaranarayanan, _Cosmological perturbations in the interacting dark sector: Mapping fields and fluids_ , _Phys. Rev. D_ 103 (2021) 023510 [2006.04618].
  * [83] L. Amendola, _Coupled quintessence_ , _Phys. Rev. D_ 62 (2000) 043511 [astro-ph/9908023].
  * [84] D. Blas, J. Lesgourgues and T. Tram, _The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes_ , _JCAP_ 07 (2011) 034 [1104.2933].
  * [85] J.-c. Hwang and H. Noh, _Axion as a cold dark matter candidate_ , _Physics Letters B_ 680 (2009) 1–3.
  * [86] V. Poulin, T.L. Smith, D. Grin, T. Karwal and M. Kamionkowski, _Cosmological implications of ultralight axionlike fields_ , _Physical Review D_ 98 (2018) .
  * [87] A.J. Christopherson and K.A. Malik, _The non-adiabatic pressure in general scalar field systems_ , _Physics Letters B_ 675 (2009) 159–163.
  * [88] N. Koshelev, _Non-adiabatic perturbations in multi-component perfect fluids_ , _Journal of Cosmology and Astroparticle Physics_ 2011 (2011) 021–021.
  * [89] G. Caldera-Cabral, R. Maartens and B.M. Schaefer, _The Growth of Structure in Interacting Dark Energy Models_ , _JCAP_ 07 (2009) 027 [0905.0492].
  * [90] Y. Zhao, Y. Liu, S. Liao, J. Zhang, X. Liu and W. Du, _Constraining interacting dark energy models with the halo concentration–mass relation_ , _Mon. Not. Roy. Astron. Soc._ 523 (2023) 5962 [2212.02050].
  * [91] Y. Zhai, W. Giarè, C. van de Bruck, E. Di Valentino, O. Mena and R.C. Nunes, _A consistent view of interacting dark energy from multiple CMB probes_ , _JCAP_ 07 (2023) 032 [2303.08201].
  * [92] S. Ilić, M. Kopp, C. Skordis and D.B. Thomas, _Dark matter properties through cosmic history_ , _Phys. Rev. D_ 104 (2021) 043520 [2004.09572].
  * [93] S. Pan, W. Yang, E. Di Valentino, D.F. Mota and J. Silk, _IWDM: the fate of an interacting non-cold dark matter — vacuum scenario_ , _JCAP_ 07 (2023) 064 [2211.11047].
  * [94] M. Khurshudyan and E. Elizalde, _Constraints on prospective deviations from the cold dark matter model using a gaussian process_ , 2024.
  * [95] D. Brout et al., _The Pantheon+ Analysis: Cosmological Constraints_ , _Astrophys. J._ 938 (2022) 110 [2202.04077].
  * [96] DESI collaboration, _DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations_ , 2404.03002.
  * [97] A. Lewis, A. Challinor and A. Lasenby, _Efficient computation of CMB anisotropies in closed FRW models_ , _Astrophys. J._ 538 (2000) 473 [astro-ph/9911177].