11institutetext: Department of Oncology, University of Cambridge, UK
22institutetext: Cancer Research UK, Cambridge Institute, University of
Cambridge, UK33institutetext: Early Cancer Institute, University of Cambridge,
UK 44institutetext: School of Computation, Information & Technology, Technical
University of Munich 55institutetext: Institute of Machine Learning in
Biomedical Imaging, Helmholtz Munich, Germany 66institutetext: Department of
Information and Electrical Engineering, University of Salerno, Italy
77institutetext: School of Biomedical Engineering & Imaging Sciences, King’s
College London, UK 88institutetext: Research Center for Medical Image Analysis
and AI, Danube University, Austria 99institutetext: Department of Radiologic
Sciences, Università Cattolica del Sacro Cuore, Italy 1010institutetext:
Department of Radiology, Boston Children’s Hospital, Harvard Medical School
1010email<EMAIL_ADDRESS>
# A Self-Supervised Image Registration Approach for Measuring Local Response
Patterns in Metastatic Ovarian Cancer

Inês P. Machado*(🖂) 112233 Anna Reithmeir 4455 Fryderyk Kogl 4455 Leonardo
Rundo 66 Gabriel Funingana 112233 Marika Reinius 112233 Gift Mungmeeprued
112233 Zeyu Gao 112233 Cathal McCague 1122 Eric Kerfoot 77 Ramona Woitek 88
Evis Sala 99 Yangming Ou 1010 James Brenton 1122 Julia Schnabel 445577 Mireia
Crispin 112233

###### Abstract

High-grade serous ovarian carcinoma (HGSOC) is characterised by significant
spatial and temporal heterogeneity, typically manifesting at an advanced
metastatic stage. A major challenge in treating advanced HGSOC is effectively
monitoring localised change in tumour burden across multiple sites during
neoadjuvant chemotherapy (NACT) and predicting long-term pathological response
and overall patient survival. In this work, we propose a self-supervised
deformable image registration algorithm that utilises a general-purpose image
encoder for image feature extraction to co-register contrast-enhanced
computerised tomography scan images acquired before and after neoadjuvant
chemotherapy. This approach addresses challenges posed by highly complex
tumour deformations and longitudinal lesion matching during treatment.
Localised tumour changes are calculated using the Jacobian determinant maps of
the registration deformation at multiple disease sites and their macroscopic
areas, including hypo-dense (i.e., cystic/necrotic), hyper-dense (i.e.,
calcified), and intermediate density (i.e., soft tissue) portions. A series of
experiments is conducted to understand the role of a general-purpose image
encoder and its application in quantifying change in tumour burden during
neoadjuvant chemotherapy in HGSOC. This work is the first to demonstrate the
feasibility of a self-supervised image registration approach in quantifying
NACT-induced localised tumour changes across the whole disease burden of
patients with complex multi-site HGSOC, which could be used as a potential
marker for ovarian cancer patient’s long-term pathological response and
survival.

###### Keywords:

Foundation Models Deformable Image Registration Medical Imaging Cancer
Research Treatment Response

##  1 Introduction

Despite notable advances in the treatment of ovarian cancer, it continues to
be one of the leading causes of gynaecological cancer death among women [1].
High-grade serous ovarian carcinoma (HGSOC) is the most common type and is a
highly heterogeneous disease that typically presents with advanced, multi-site
metastatic disease. Neoadjuvant chemotherapy (NACT) followed by delayed
primary surgery is becoming the most frequent treatment strategy for advanced
HGSOC [2]. HGSOC often exhibits lesion-wise heterogeneous response to NACT
with some lesions disappearing or shrinking, some remaining stable, some
growing, and new lesions appearing despite ongoing treatment. The response of
individual lesions to treatment has been shown to drive progression and
provides predictive information for the effectiveness of delayed primary
surgery and survival [3]. Consequently, treatment response assessment
following NACT is critical to properly plan subsequent stages of the patient’s
clinical pathway.

In current clinical practice, tumour response assessment is usually based on
two-dimensional measurements to approximate change in tumour size on serial
contrast-enhanced computerised tomography (CE-CT) scan images. Although
Response Evaluation Criteria in Solid Tumours (RECIST 1.1) is the current
clinical guideline to assess size change of solid tumours after therapeutic
treatment, it is based on measurements of lesion diameters, and therefore is
not able to capture the three-dimensional complexity of tumours. Additionally,
radiologists have to choose which target lesions to track and measure, which
adds subjectivity to the clinical evaluation process.

Future clinical trials for ovarian cancer require stronger, more sensitive
response metrics that can provide more robust discrimination between different
response patterns. Here, a first challenge is how to effectively monitor
volumetric tumour changes across multiple disease sites during the course of
NACT and, more importantly, how to effectively reveal heterogeneous changes in
various sub-regions within tumours rather than tumours as a whole, since the
heterogeneity may be more indicative of long-term pathological response. A
second challenge is how to match multiple lesions between longitudinal scans.
Lesions may be numerous and densely packed within a single organ or tissue.
For example, metastatic ovarian cancer patients with a high disease burden can
exhibit dozens of disease sites with hundreds of small connected components
[4]. Additionally, lesions may grow, shrink, split, or merge over time. Thus,
matching lesions between longitudinal images is critical to ensure that
heterogeneous response patterns can be identified, and imaging data can be
best utilised to inform treatment decisions. Figure 1 illustrates pre- and
post-treatment CE-CT scans that highlight the metastatic nature of HGSOC and
the lesion-wise heterogeneity in response to chemotherapy, with some lesions
disappearing or shrinking, some remaining stable, some growing, and new
lesions appearing.

Registration of longitudinal CE-CT images acquired during the course of
treatment has received increasing interest as a useful tool to enable
localised lesion tracking during therapy. Despite its importance, longitudinal
registration of ovarian cancer CE-CT scans has been less studied, and the
registration performance is less understood compared to other organs (e.g.,
lung [5, 6], liver [7, 8], brain [9], prostate [10], cervix [11] and head and
neck [12]).

Figure 1: Overview of the treatment schedule and characteristics of an
advanced-stage high-grade serous carcinoma. Initial staging and diagnosis were
based on CE-CT imaging and biopsy. Surgery was performed after two cycles of
NACT. According to RECIST 1.1 criteria, the patient was classified as a
partial responder. Pre- and post-treatment CE-CT scans revealed the metastatic
nature of the disease, identifying 16 sites of disease burden with varying
numbers of connected components (CC) and highlighting heterogeneity in lesion
response.

With the emergence of deep learning, there has been a notable shift in image
registration techniques. These approaches directly predict a displacement
field by leveraging self-learned hierarchical features extracted from both the
input moving and reference images. In particular, the introduction of
foundation models in computer vision marks a significant paradigm shift in the
approach to complex visual tasks. Self-supervised vision foundation models
demonstrate this evolution by providing pre-trained models that have learned
rich feature representations from large-scale unlabeled datasets. This is
particularly advantageous in scenarios like HGSOC, where little data is
available and no publicly available imaging datasets exist. In such cases,
self-supervised learning becomes instrumental in overcoming data scarcity
challenges.
This work proposes a deformable image registration framework based on a
general-purpose image encoder to assess tumour changes over time as a measure
of treatment response in metastatic HGSOC. The assessment of registration
performance covers various anatomical structures in the pelvic, abdominal, and
chest regions. Specifically, it examines registration accuracy among disease
sites and patient subgroups with different treatment responses. Volumetric
disease change is quantified for the first time for each lesion individually,
and for different sub-regions within tumours. Our ultimate goal is to use
registration-quantified tumour changes in HGSOC to more accurately measure
treatment response and predict long-term pathological response and patient
survival after treatment.

##  2 Methods

Registration framework. The source and target images in the registration
problem are denoted by $I_{A}:\Omega\rightarrow\mathbb{R}$ and
$I_{B}:\Omega\rightarrow\mathbb{R}$, respectively. The transformation map is
denoted by $\Phi_{AB}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ with the
intention that $I_{A}\circ\Phi_{AB}\sim I_{B}$. We adopt the GradICON [13]
registration network which uses a two-step registration process. In the first
step, images are passed through a three-level multiresolution registration
network. At each level, a U-Net accepts the warped image from the previous
level and the target image. Additionally, the input images are downsampled to
$\frac{1}{4}$ and $\frac{1}{2}$ for each level, respectively. In the second
step, images are passed through a single U-Net, which takes the warped image
from the first step and the target image at full resolution. All four U-Nets
share the same architecture as illustrated in Figure 2.

Volumetric feature encoder and training. We adopt the publicly-available
uniGradICON [14] feature encoder, a foundation model trained and evaluated on
12 public medical image datasets from the Learn2Reg Grand Challenge111
https://learn2reg.grand-challenge.org/, covering various anatomical regions,
imaging modalities, and deformation patterns. Given an ordered image pair
$(I_{A},I_{B})$, the registration network outputs the transformation map
$\Phi_{AB}$ which maps $I_{A}$ to the space of $I_{B}$. By swapping the input
pair $(I_{B},I_{A})$, we obtain the estimated inverse map $\Phi_{BA}$. The
similarity loss $L_{\text{sim}}$ is computed between the warped image
$I_{A}\circ\Phi_{AB}$ and the target image $I_{B}$, and vice versa following
the formulation:

$L=L_{\text{sim}}(I_{A}\circ\Phi_{AB},I_{B})+L_{\text{sim}}(I_{B}\circ\Phi_{BA},I_{A})+\lambda\left\|\nabla(\Phi_{AB}\circ\Phi_{BA})-I\right\|^{2}_{F}$

We use localised normalised cross correlation (1 - LNCC) as the similarity
measure following the implementation described in [13]. The third term is the
gradient inverse consistency regularizer, which penalizes differences between
the Jacobian of the composition of $\Phi_{AB}$ with $\Phi_{BA}$ and the
identity matrix $I$; $\|\cdot\|_{F}^{2}$ is the Frobenius norm, $\lambda>0$.
The first and second stages of the registration network were trained for 800
and 200 epochs, respectively, with a learning rate of $5\times 10^{-5}$.

Quantification of volumetric tumour progression. The obtained deformation
vector fields were further used to calculate the Jacobian maps. The Jacobian
map was computed as the determinant of the gradient of the deformation vector
field (DVF). The Jacobian determinants indicate the volumetric change ratio at
each voxel, where a value greater than 1 indicates voxel expansion, a value
less than 1 indicates shrinkage, and a value of exactly 1 indicates volume
preservation.

Intra- and inter-tumoural heterogeneity. HGSOC is characterised by high levels
of macroscopic heterogeneity, with frequent hyper-dense (calcified), hypo-
dense (cystic/necrotic), or intermediate dense (soft tissue) tumour regions,
each of which can have different prognostic significance and influence therapy
response. To understand how different tumour regions respond to treatment, an
automated sub-segmentation of these regions was performed using a tissue-
specific sub-segmentation framework based on an unsupervised fuzzy clustering
technique previously described in [15]. Jacobian maps were computed as the
determinant of the gradient of the deformation vector field to quantify local
changes for each tissue-specific sub-segmentation.

Figure 2: Image registration follows a multi-step, multi-resolution approach
trained with a two-stage process (A) via the Two Step (C) and the Downsample
(D) operators. The basic component $\Psi$ of the registration network is
represented by a U-Net instance, taking as input two images and returning a
displacement field (B). The downsample operator predicts the warp between two
high-resolution images using a network that operates on low-resolution images,
and the two-step operator predicts the warp between two images in two steps,
first capturing the coarse transform via $\Phi$ and then the residual
transform via $\Psi$.

##  3 Experiments

Patient cohort. A total of 99 patients with a confirmed histopathological
diagnosis of HGSOC, who underwent neoadjuvant chemotherapy before delayed
primary surgery, were included in this study. All patients were treated at
Cambridge University Hospitals NHS Foundation Trust and were recruited into a
clinical observational study approved by the local research ethics committee
(REC reference number: 08/H0306/61). Clinical CE-CT scans covering the abdomen
and pelvis were acquired. Figure 3 (a) shows the median (min - max) age of the
patient cohort in years, the total number of NACT cycles administered, and the
FIGO stage (I to IV), which describes how far the cancer has spread. The
radiological response is measured based on the RECIST 1.1 criteria and is
categorised into four groups: complete response (CR), partial response (PR),
stable disease (SD), or progressive disease (PD). Chemotherapy Response Scores
(CRS) describe the histopathologic response to NACT in the omentum (fatty
tissue layer in the abdominal cavity) and are categorized into I to III.

Image segmentation and labelling. On CE-CT axial images reconstructed with a
slice thickness of typically 5 mm, all cancer lesions were segmented semi-
automatically by a board-certified radiologist with ten years of experience in
clinical imaging, using Microsoft Radiomics (project InnerEye; Microsoft,
Redmond, WA, USA). The volumes of interest were annotated for their anatomic
location with the following disease sites and volume percentages: ovaries and
pelvis (66.4%), omentum (24.4%), right upper quadrant (5.1%), left upper
quadrant (1.1%), epigastrium (0.1%), mesentery (0.3%), right paracolic gutter
(0.8%), left paracolic gutter (0.3%), infrarenal abdominal lymph nodes (0.8%),
suprarenal abdominal lymph nodes (0.2%), inguinal lymph nodes (0.2%), and
supradiaphragmatic lymph nodes (0.1%). Cystic and solid tumour parts were
included in these segmentations. Intra-tumoural sub-regions were automatically
segmented as described in [15]. Tumour in the omentum and pelvic/ovarian
locations accounts for the majority of the disease burden at baseline and are
the most frequent tumour locations. Figure 3 (b) and (c) show the sites of
primary and metastatic disease in HGSOC and the distribution of tumour
volumes, volume change (follow-up/ baseline) and number of connected
components by disease site, as well as the total, omental and pelvic/ovarian
disease volume change stratified by RECIST 1.1 response status. Additionally,
segmentations of 10 anatomical structures, including left and right kidneys,
liver, spleen, lungs, stomach, pancreas, colon, vertebra and sacrum in the CE-
CT images were automatically computed for each scan using the publicly-
available segmentation tool Total Segmentator v2 [16] and used for
registration evaluation. Baseline and follow-up CE-CT scans were evaluated
according to RECIST 1.1 for treatment response assessment.

Figure 3:  (A) Main characteristics of the patient cohort. (B) Sites of
primary and metastatic HGSOC disease. (C) Distribution of tumour volumes by
disease site.

Comparative approaches and evaluation criteria. The proposed foundation model
was evaluated against two state-of-the-art techniques: NiftyReg [17], a
conventional registration technique, and VoxelMorph [18], the latest benchmark
for deformable image registration. To provide a comprehensive evaluation of
the registration performance, we consider a number of complementary metrics
that assess the accuracy, plausibility, and speed of the proposed approach.
The agreement between registration-propagated contours and ground-truth
contours was assessed with the Dice Similarity Coefficient (DSC) for tumour
and non-tumour regions. The spatial Jacobian and deformation vector field
images were used to assess the plausibility of the resulting transformations.
The plausibility (smoothness) of a displacement field is captured using the
standard deviation of the logarithm of the Jacobian determinant (SDlogJ) of
the displacement field [19]. Visual inspection of the deformed images was also
performed to detect any large, noticeable issues with the transformation. In
addition, we measure the test-time registration runtime (RT). The registration
method obtaining the highest accuracy was used to quantify localised
volumetric tumour changes in HGSOC patients undergoing NACT.

##  4 Results

Registration performance. Table 1 (A) and (B) show the Dice coefficient and
the standard deviation of the logarithm of the Jacobian determinant of
different registration methods across different disease sites (A), anatomical
structures (B), and patient subgroups with different treatment responses.

Quantification of tumour deformation. Table 1 (C) shows the average median
Jacobian for all disease sites, and separately for omental and pelvic ovarian
disease, as these comprise 90.8% of the total tumour volume. Results are
stratified by macroscopic areas, including hypo-dense (i.e., cystic/necrotic),
hyper-dense (i.e., calcified), and intermediate dense (i.e., soft tissue)
portions and categorised by patient subgroups based on their treatment
responses. Figure 4 shows the baseline image, the follow-up image, the
deformation vector field and Jacobian map in axial view for HGSOC patients
classified as a responder (top row) and a non-responder (bottom row) according
to RECIST 1.1. In the top row, pelvic ovarian disease shrinkage is visible in
the follow-up image. Tumour shrinkage is illustrated by the DVF overlaid on
top of the baseline image (red contour). Converging vectors generated a sink
point in the middle of the tumour region where median Jacobians were smaller
than 1 (dark blue) indicating shrinkage. Quantitatively, the majority of
voxels (97.53%) were shrinking with average Jacobian of 0.46±0.15, indicating
overall a 54% shrinkage in the tumour volume. The bottom row shows a non-
responder tumour with its DVF and Jacobian map. In the follow-up image, a
large expansion appeared on the left distal side of the pelvic ovarian lesion.
DVF overlaid on the baseline image illustrates that vectors are diverging
toward the right distal side of tumour (red contour) where median Jacobians
were larger than 1 (dark red), indicating expansion. In this case, majority of
voxels (98.26%) were expanding with Jacobian greater than 1. The mean Jacobian
was 1.34±0.18, indicating overall a 34% expansion in the tumour volume. Figure
5 shows examples of tissue-specific sub-segmentations. The whole tumour,
hyper-dense and hypo-dense region contours are displayed as light blue, yellow
and magenta solid lines, respectively.

##  5 Discussion

We benchmarked state-of-the-art medical image registration algorithms,
including traditional methods (NiftyReg), deep learning-based methods
(VoxelMorph), and foundation models (uniGradICON), for the first time in the
context of longitudinal CE-CT registration for HGSOC. Table 1 shows that the
proposed foundation model outperforms the traditional and DL-based methods in
terms of registration accuracy in both tumour and non-tumour regions and
speed. We also show that registration accuracy varies across different regions
(disease sites vs. anatomical structures) and among patient subgroups with
different responses to chemotherapy. Omental disease poses a more challenging
registration task compared to the pelvic ovarian disease, as omental lesions
are usually numerous and densely packed within a single organ.

Figure 4: Baseline, follow-up, DVF, and Jacobian maps in axial view for HGSOC
patients classified as a responder (top row) and a non-responder (bottom row)
according to RECIST 1.1. Tumour segmentations are overlaid on top of both the
DVFs and the Jacobian maps, showing longitudinal volume changes. Red and blue
colours in Jacobian maps represent expansion and shrinkage respectively,
whereas white colour shows volume preservation.

Location/Registration Method | NiftyReg [14min] | VoxelMorph [8s] | GradICON [8s]
---|---|---|---
(A) DSC±SD (SDlogJ) in Tumour Regions
POD | Responder | 0.575±0.089 (0.061) | 0.645±0.042 (0.052) | 0.796±0.261 (0.025)
Non-responder | 0.599±0.071 (0.059) | 0.751±0.212 (0.035) | 0.802±0.145 (0.018)
OD | Responder | 0.501±0.073 (0.036) | 0.602±0.056 (0.061) | 0.701±0.383 (0.071)
Non-responder | 0.572±0.076 (0.083) | 0.700±0.021 (0.041) | 0.725±0.178 (0.020)
(B) DSC±SD (SDlogJ) in Healthy Anatomical Structures
Kidneys | 0.824±0.048 | 0.815±0.067 | 0.913±0.052
Liver | 0.801±0.059 | 0.807±0.031 | 0.937±0.076
Sacrum | 0.889±0.051 | 0.845±0.078 | 0.901±0.033
Vertebra | 0.811±0.086 | 0.978±0.062 | 0.912±0.220
Colon | 0.510±0.052 | 0.525±0.043 | 0.720±0.068
Stomach | 0.522±0.076 | 0.560±0.035 | 0.779±0.040
Spleen | 0.645±0.083 | 0.712±0.028 | 0.800±0.031
Lungs | 0.661±0.084 | 0.724±0.084 | 0.889±0.010
Pancreas | 0.609±0.035 | 0.739±0.071 | 0.891±0.091
(C) Median Jacobians stratified by Tumour Sub-Regions
POD | Responder | 0.80±0.10 | Cyst/Necrotic | 0.918±0.55
Calcified | 1.129±0.36
Soft Tissue | 0.804±0.39
Non-responder | 1.05±0.15 | Cyst/Necrotic | 1.028±0.29
Calcified | 1.215±0.53
Soft Tissue | 1.106±0.31
OD | Responder | 0.64±0.14 | Cyst/Necrotic | 0.962±0.46
Calcified | 0.606±0.21
Soft Tissue | 0.640±0.43
Non-responder | 1.02±0.15 | Cyst/Necrotic | 1.004±0.39
Calcified | 1.182±0.41
Soft Tissue | 1.116±0.39

Table 1: Mean and standard deviation of Dice coefficient and the standard
deviation of the logarithm of the Jacobian determinant as DSC±SD(SDlogJ), and
running time of registration methods across disease sites (A), anatomical
structures (B), and patient subgroups. Initial DSC is 0.376±0.45 for tumour
regions and 0.546±0.71 for non-tumour regions. (C) Median Jacobian
determinants stratified by disease sites - omental disease (OD) and pelvic
ovarian disease (POD) - and their macroscopic areas, including hypo-dense
(i.e., cystic/necrotic), hyper-dense (i.e., calcified), and intermediate dense
(i.e., soft tissue) portions. Figure 5: (A) 3D and axial CT reconstructions
showing segmented ROIs: hyper-dense, hypo-dense and soft tissue components
relative to the whole omental tumour. (B) Total percentage of each tissue type
for two disease sites at baseline and follow-up for the cohort.

These lesions split or merge over time more frequently than those in the
pelvic region, where the number of connected components between pre- and post-
treatment scans is generally more stable. Additionally, patients with complete
responses exhibit greater tumour changes, posing a more significant
registration challenge, as shown by the lowest DSC in Table 1 compared to non-
responders.

Due to the higher accuracy of the proposed foundation model compared to other
registration methods, we further evaluated its ability to quantify
longitudinal tumour changes induced by NACT. The average median Jacobian for
all disease sites was 0.82±0.11 and 1.06±0.15 for responders and non-
responders, respectively. These indicated that on average responders had 18%
median volume shrinkage while non-responders had 6% median volume expansion.
For pelvic ovarian disease, the average median Jacobian was 0.80±0.10 for
responders and 1.05±0.15 for non-responders, respectively. This indicates
that, on average, responders experienced a 20% median pelvic tumor volume
shrinkage, while non-responders experienced a 5% median pelvic tumor volume
expansion. For omental disease, the average median Jacobian was 0.64±0.14 for
responders and 1.02±0.15 for non-responders, indicating that, on average,
responders had a 36% median volume shrinkage, while non-responders had a 2%
median volume expansion. Omental disease showed a significantly better
response than pelvic ovarian disease. We also quantify, for the first time,
localised volumetric deformation in different sub-regions (soft tissue,
calcified, and necrotic regions) within tumors. We show that cysts are more
prevalent in the POD compared to omental disease, and that calcified regions
are typically more prominent in follow-up scans, representing areas of
calcified fibrosis or calcified deposits resulting from treatment or disease
progression. Table 1 (C) shows that soft tissue shrinks in responders and
expands in non-responders, while cyst regions preserve their volume and
calcified areas generally increase during NACT.

Upon further evaluation of the performance of registering longitudinal CE-CT
images to quantify localised tumour change in HGSOC, our future work includes
(a) quantitative evaluation of registration accuracies on longitudinal images
with more than one follow-up time point; (b) evaluation of the performance of
registration in larger datasets and across a broader range of registration
algorithms; and (c) the use of registration-quantified changes and
registration-revealed heterogeneity of changes to predict long-term
pathological responses to treatment. In conclusion, this work quantifies, for
the first time, localised heterogeneous tumour changes across different
disease sites in ovarian cancer and demonstrates the ability of registration
to characterise spatially heterogeneous effects induced by treatment, which
may ultimately be used as more sensitive markers for evaluating treatment
response and predicting patient outcomes.

###  5.0.1 Acknowledgements

We acknowledge funding and support from Cancer Research UK (A22905) and the
Cancer Research UK Cambridge Centre [CTRQQR-2021-100012 and A25177], The Mark
Foundation for Cancer Research [RG95043], GE HealthCare, and the CRUK National
Cancer Imaging Translational Accelerator (NCITA) [A27066]. Additional support
was provided by the National Institute for Health Research (NIHR) Cambridge
Biomedical Research Centre [NIHR203312 and BRC-1215-20014] and the EPSRC
Tier-2 capital grant [EP/T022221/1]. This work was further supported by the
backing of the Federal Ministry of Education and Research (BMBF, Grant Nos.
01ZZ2315B and 01KX2021), the Bavarian Cancer Research Center (BZKF, Lighthouse
AI and Bioinformatics), and the German Cancer Consortium (DKTK, Joint Imaging
Platform). Further support was received through the St. Baldrick’s Career
Development Grant, NIH R61NS126792, NIH R21TR004265, and NIH R21NS21735.
Additionally, we recognize the contribution of the Helmholtz Information and
Data Science Academy. The authors would like to acknowledge the support of the
Scientific Computing team from the Cancer Research UK Cambridge Institute.

###  5.0.2

The authors have no competing interests to declare that are relevant to the
content of this article.

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# Testing $f(T)$ cosmologies with HII Hubble diagram and CMB distance priors

Rodrigo Sandoval-Orozco<EMAIL_ADDRESS>Instituto
de Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito
Exterior C.U., A.P. 70-543, México D.F. 04510, México.  Celia Escamilla-Rivera
0000-0002-8929-250X<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U.,
A.P. 70-543, México D.F. 04510, México.  Rebecca Briffa
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy,
University of Malta, Msida, Malta  Department of Physics, University of Malta,
Msida, Malta  Jackson Levi Said  0000-0002-7835-4365<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University of Malta, Msida, Malta
Department of Physics, University of Malta, Msida, Malta

###### Abstract

In this work, we present independent determinations of cosmological parameters
and new constraints on $f(T)$ cosmologies, employing two new catalogs related
to HII galaxy Hubble and CMB distance priors, along with the local standard
measurements, SNIa, $H(z)$ measurements, growth rate data (RSD), and BAO
baselines. We found that the marginalised best-fit C.L. $H_{0}$ and
$\sigma_{8}$ parameters within these cosmologies can relax the current
cosmological tensions using HIIG data. This produces a larger range of
admissible values for the current Hubble constant, and when all baselines are
considered, the uncertainty bands for $H_{0}$ and the matter density parameter
reduce significantly.

##  I Introduction

Several cosmological measurements indicating a cosmic accelerated expansion
have suggested that standard cosmology itself cannot explain all the
observational evidence given by the wide range of surveys and catalogs. Within
these collections, observational data from Supernovae Type Ia (SNIa)
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns ; Pan-
STARRS1:2017jku , Baryon Acoustic Oscillations (BAO) Addison:2013haa ;
Aubourg:2014yra ; Cuesta:2014asa ; Cuceu:2019for , and the Cosmic Microwave
Background Radiation (CMBR) Planck:2018vyg  provided a piece of strong
evidence that suggests that modifications, in gravitational or matter sectors,
needs to be made. Therefore, the easiest modification is to consider a
positive cosmological constant $\Lambda$, into the gravitational field
equations described by General Relativity (GR)
SupernovaCosmologyProject:1998vns ; Weinberg:1988cp ; Copeland:2006wr ;
Frieman:2008sn , this is the so-called $\Lambda$-Cold Dark Matter
($\Lambda$CDM) concordance model, which is well-constrained using several
observational datasets. However, while this scenario can give us a good
description of the current cosmic accelerated expansion, the $\Lambda$CDM
model has been facing both theoretical and observational issues Verde:2019ivm
; Riess:2019qba ; DiValentino:2020zio ; Riess:2021jrx ; DiValentino:2020vvd .
From the theoretical side, so far, we have not reached a clear explanation of
the nature of the cosmological constant $\Lambda$, and how the dark energy can
be explained by it or whether there are dynamical features to this parameter.
Furthermore, the explanation of dark matter is still an open problem within
high-energy physics landscapes. On the other hand, on the observational side,
there have been careful studies that suggest a revision of the $\Lambda$CDM
cosmology since it presents cosmological tensions and discrepancies associated
with measured parameter values such as the Hubble constant Knox:2019rjx ;
Jedamzik:2020zmd ; DiValentino:2021izs ; Abdalla:2022yfr ;
Kamionkowski:2022pkx ; Escudero:2022rbq ; Vagnozzi:2023nrq ; Khalife:2023qbu .

Following this line of thought, one main question arises: Can extended
theories of gravity be the description of a fundamental theory of gravity that
allows us to address theoretical and observational issues with viable
solutions in the observed discrepancies? According to extensive literature, we
currently have a global description of cosmological scenarios which can be
obtained from the following classifications: (i) A standard cosmology with an
effective dark energy fluid. (ii) A modified gravity theory in which extra
terms due to gravitational effects are related to dark energy effects. (iii)
An extension of the teleparallel gravity (TG), where a dark energy fluid can
be introduced as an effective fluid Bahamonde:2021gfp  and references therein,
or where the gravitational effects can be related directly to a cosmic late-
time acceleration under the non-fluid approach Aguilar:2024cga . These
classifications have been successful in giving us an insight into the
cosmological tensions, including statistical constraint analysis using SNIa
catalogs, Baryon Acoustic Observations (BAO) and $H(z)$ measurements
Briffa:2023ern , Quasars (QSO) datasets using ultraviolet, X-ray and optical
plane techniques Sandoval-Orozco:2023pit , and CMB data Nunes:2018evm ;
Kumar:2022nvf ; Nunes:2018xbm . However, under the third classification, it
has been reported that some deviations can be observed when we consider
perturbations of the tetrad field Aguilar:2024cga .

As an extension of the role of constraining cosmological models with late-time
observational catalogs above mentioned, lensed quasars are interesting objects
since they allow us to measure time delays from gravitational effects. Some of
these measurements are available through the H0LiCOW Collaboration
Bonvin:2016crt  and due to their brightness and variability, we can compute
$H_{0}$ as for standard cosmological scenarios Kumar:2014vvy ; Sereno:2013ona
, for cosmological models derived from modified gravity DAgostino:2020dhv ;
Vagnozzi:2023nrq ; Dainotti:2021pqg , and extended cosmologies derived from TG
Sandoval-Orozco:2023pit , which includes X-ray and optical plane techniques to
constraint $H_{0}$.

Another important measurement in the local universe includes the
implementation of HII galaxy Hubble catalogs (Gonzalez-Moran:2021drc, ), which
have been previously done with observational expansion rates derived using
Gaussian Processes (GP) methods in order to optimise values of the model
parameters from $f(R)$ models Sultana:2022qzn . However, in these analyses
when we include spatial curvature it complicates the statistical fitting and
leads to high correlations between the free parameters of the models. HII
galaxy measurements provide a good source of luminosity distance to probe the
cosmic acceleration. One of the interesting methods is the luminosity-velocity
dispersion relation $L-\sigma$, which is satisfied by elliptical galaxies,
globular clusters and bulges of spiral galaxies. Using this method for the HII
observations beyond the current reach of SNIa, we can derive a value for the
matter density parameter $\Omega_{m}$ Wu:2019mjm . Moreover, while the
constraints obtained for this parameter are in good agreement with Planck 2018
data, this result suggests that the dynamical properties of these objects can
have a significant impact on the likelihood distributions, especially the ones
that constrain the properties of dark energy. Further efforts to extend the
HII baseline have been investigated in Ref.Wang:2016pag , where it is possible
to use a combination of model-dependent and model-independent methods, like GP
reconstructions, which give higher values of $H_{0}$ in comparison to R16
results Riess:2019cxk . Further analysis includes constraints on the spatial
flatness parameter $\Omega_{k}$, which avoids biases introduced by a prior
model. This analysis was performed using measurements of the Hubble parameter
$H(z)$, cosmic chronometers, and its extensions with GP reconstructions of the
HII galaxy Hubble diagram Yennapureddy:2017vvb ; Ruan:2019icc . From
astrophysical methodologies, in Gonzalez-Moran:2021drc  was considered a
correlation between the distance estimator and Balmer line luminosity of HII
galaxies. This allows us to obtain, in an independent way, constraints on the
cosmological parameters derived from standard and modified cosmologies e.g.
$H_{0}$, which are in good agreement with those of similar analyses using
SNIa.

At early times, also there have been interesting results regarding extended
teleparallel cosmologies, in particular for $f(T)$ cosmologies Nunes:2018evm ;
Kumar:2022nvf ; Nunes:2018xbm ; dosSantos:2021owt ; Hashim:2021pkq . However,
the scenarios treated so far, while well-constrained with CMB data, consider
the extra torsion-like terms like an effective dark energy fluid in the
Boltzmann equations. This later method brings relevant changes in the
constraint analysis on cosmological tensions at early times, where the
possibility to introduce a non-fluid-like approach as a direct consequence of
$f(T)$ extensions has been discussed Aguilar:2024cga . As part of this
observational analysis, it is possible to work with a physical and sufficient
compression of the full CMB catalog, the so-called CMB distance priors (or
shift parameters), which proves to be useful to include CMB data when
obtaining cosmological C.L., especially when we are probing dark energy
dynamics. The importance of using this kind of catalog is that the CMB shift
parameters are the least model-dependent set of parameters that can be
extracted from the original CMB catalog, e.g. from Planck 2018 Wang:2006ts .
In this discussion, there have been studies using this CMB distance priors to
understand dark energy dynamics, the primordial power spectrum, and neutrino
properties Zhai:2019nad , and more recently, using this baseline with multiple
tracers method in beyond-standard cosmological probes Moresco:2022phi .

As part of revising and finding new constraints on teleparallel cosmologies,
in particular $f(T)$ cosmologies, in this work, we will employ two new
catalogs related to HII galaxy Hubble and CMB distance priors, along with the
local standard measurements, e.g. SNIa, $H(z)$ measurements, growth rate data
(RSD), and BAO baselines. The primordial goal is to find better cosmological
constraints for $H_{0}$ and $\sigma_{8}$, that allow $f(T)$ cosmologies to
relax (or solve) the current cosmological tensions.

This paper is divided as follows: In Sec. II we summarise the TG background
theory and the most promising $f(T)$ cosmologies available in the literature
to relax the $H_{0}$ tension. All of these models are described through their
normalised $E(z)$ Friedmann evolution equation. Furthermore, we are going to
consider the standard $\Lambda$CDM model in addition to the three $f(T)$
cosmologies described to proceed with comparisons between them. In Sec. III we
present the statistical methodology employed for the baseline datasets
mentioned. We divided this discussion into baseline (low-z) local observations
and a high-z Hubble diagram. Our results on new constraints are developed in
Sec. IV. Finally, discussions are presented in Sec. V.

##  II Teleparallel cosmology background

Regarding possible extensions to GR, one of the approaches that has been
gaining field in cosmological analyses is the one that can be constructed
through a metric-affine gravity, which related the connection with other
possible geometries BeltranJimenez:2019esp . It is along this line of thought
that TG offers a curvature-free connection Bahamonde:2021gfp ; Krssak:2018ywd
and produces an interesting formulation of gravity. A novel scenario insight
into this approach involves a teleparallel equivalent of general relativity
(TEGR), which is dynamically equivalent to GR at the level of field equations,
and thus phenomenological predictions, however, it can differ at IR scales
Mylova:2022ljr .

As discussed in the extensive literature, TG theories are sourced by the
interchange of the curvature-based Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu}$111Over-circles
notation are used to denote objects determined using the standard Levi-Civita
connection. with the teleparallel connection given by
$\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu}$ Hayashi:1979qx ; Aldrovandi:2013wha
; Bahamonde:2021gfp . Under this scheme, any curvature-less term of the
teleparallel connection means that curvature-based geometries will vanish when
we compute with this object, therefore the new objects are needed to construct
a set of gravitational theories Krssak:2018ywd ; Aldrovandi:2013wha .
Following this construction, TG can be directly written with the tetrad
$e^{A}_{\phantom{A}\mu}$ (and its inverses $E_{A}^{\phantom{A}\mu}$), and
additionally a spin connection $\omega^{A}_{\phantom{A}B\mu}$. This tetrad
$e^{A}_{\phantom{A}\mu}$ is constructed to the metric using the following
expressions

$\displaystyle
g_{\mu\nu}=e^{A}_{\phantom{A}\mu}e^{B}_{\phantom{B}\nu}\eta_{AB}\,,$
$\displaystyle\eta_{AB}=E_{A}^{\phantom{A}\mu}E_{B}^{\phantom{B}\nu}g_{\mu\nu}\,.$
(1)

We will use Latin indices to denote coordinates on the tangent space. The
Greek indices denote coordinates on the general manifold. Notice that the
tetrad is not the only non-inertial variable in the description of gravity in
the standard GR. Regarding the metric, the tetrad preserves the orthogonality
conditions given by:
$e^{A}_{\phantom{A}\mu}E_{B}^{\phantom{B}\mu}=\delta^{A}_{B}\,$ and
$e^{A}_{\phantom{A}\mu}E_{A}^{\phantom{A}\nu}=\delta^{\nu}_{\mu}\,.$ Notice
that the spin connection $\omega^{A}_{\phantom{A}B\mu}$ is a flat object which
could be responsible for adding the local Lorentz transformation invariance
into the equations of evolution. We can join both, the tetrad and spin
connection, to define the teleparallel connection as Weitzenbock1923 ;
Krssak:2018ywd

$\Gamma^{\sigma}_{\phantom{\sigma}\nu\mu}\equiv
E_{A}^{\phantom{A}\sigma}\left(\partial_{\mu}e^{A}_{\phantom{A}\nu}+\omega^{A}_{\phantom{A}B\mu}e^{B}_{\phantom{B}\nu}\right)\,,$
(2)

We notice that the spin connection and the tetrad represent the local degrees
of freedom and the gravitational terms of the system, and as a consequence,
fulfil the diffeomorphism and local Lorentz invariance.

Furthermore, the torsion tensor can be written from the teleparallel
connection given above through the expression Hayashi:1979qx
$T^{\sigma}_{\phantom{\sigma}\mu\nu}\equiv
2\Gamma^{\sigma}_{\phantom{\sigma}[\nu\mu]}\,,$ where the subindex in square
brackets describes the antisymmetric operator. We can compute the torsion
scalar by considering the contraction of the torsion tensor as
Aldrovandi:2013wha ; Bahamonde:2021gfp

$T\equiv\frac{1}{4}T^{\alpha}_{\phantom{\alpha}\mu\nu}T_{\alpha}^{\phantom{\alpha}\mu\nu}+\frac{1}{2}T^{\alpha}_{\phantom{\alpha}\mu\nu}T^{\nu\mu}_{\phantom{\nu\mu}\alpha}-T^{\alpha}_{\phantom{\alpha}\mu\alpha}T^{\beta\mu}_{\phantom{\beta\mu}\beta}\,.$
(3)

An equivalent action, so-called TEGR, is denoted by a linear Lagrangian form
of the torsion scalar $R=\accentset{\circ}{R}+T-B=0\,,$ Bahamonde:2015zma ;
Farrugia:2016qqe , where $R\equiv 0$ denotes curvature-less.
$\accentset{\circ}{R}\neq 0$ due to the Levi-Civita connection, and the
boundary term $B$ is a total divergence term. Notice that the Einstein-Hilbert
action is dynamically equivalent to the description of a linear torsion
scalar, therefore it guarantees alike equations of motion for both actions.
Modification of TEGR can be written directly with an arbitrary generalisation
of the TEGR Lagrangian to the so-called $f(T)$ gravity. In such case, we can
parametrise this function of the torsion as $f(T)=-T+\mathcal{F}(T)$, and the
action can be written as Ferraro:2006jd ; Ferraro:2008ey ; Bengochea:2008gz ;
Linder:2010py ; Chen:2010va ; RezaeiAkbarieh:2018ijw

$\mathcal{S}_{\mathcal{F}(T)}=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\;e\left[-T+\mathcal{F}(T)\right]+\int\mathrm{d}^{4}x\;e\mathcal{L}_{\text{m}}\,,$
(4)

with $\kappa^{2}=8\pi G$, $\mathcal{L}_{\text{m}}$ is the Lagrangian
associated with matter effects, and
$e=\det\left(e^{a}_{\phantom{a}\mu}\right)=\sqrt{-g}$ is the tetrad
determinant. It is important to mention that when $\mathcal{F}(T)\rightarrow
0$, the standard $\Lambda$CDM model is recovered when this functional of the
torsion tends to a constant $\Lambda$ value. Calculating the standard
variation of the above action, we obtain the field equations

$S_{\rho}^{\phantom{\rho}\mu\nu}=K^{\mu\nu}_{\phantom{\mu\nu}\rho}-\delta_{\rho}^{\mu}T_{\sigma}{}^{\sigma\nu}+\delta_{\rho}^{\nu}T_{\sigma}{}^{\sigma\mu}=-S_{\rho}{}^{\nu\mu}\,.$
(5)

where

$K^{\rho}_{\phantom{\rho}\mu\nu}:=\Gamma^{\rho}{}_{\mu\nu}-\accentset{\circ}{\Gamma}^{\rho}{}_{\mu\nu}=\frac{1}{2}\left(T_{\mu}{}^{\rho}{}_{\nu}+T_{\nu}{}^{\rho}{}_{\mu}-T^{\rho}{}_{\mu\nu}\right)\,,$
(6)

is the contortion tensor denoting the difference between the Levi-Civi and
teleparallel connections. Each tetrad and spin connection field equations are
given by $W_{(\mu\nu)}=\kappa^{2}\Theta_{\mu\nu}\,,$ and $\quad
W_{[\mu\nu]}=0\,.$ When we consider a specific metric, a unique tetrad-spin
connection pair is allowed and it is in agreement with a vanishing spin
connection, this is the so-called Weitzenböck gauge Krssak:2018ywd ;
Bahamonde:2021gfp .

For our study, we consider a flat homogeneous and isotropic cosmology denoted
by the following tetrad Krssak:2015oua ; Tamanini:2012hg

$e^{A}_{\phantom{A}\mu}=\text{diag}\left(1,\,a(t),\,a(t),\,a(t)\right)\,,$ (7)

where $a(t)$ is the scale factor in physical time $t$. This tetrad satisfies
the Weitzenböck gauge condition Hohmann:2019nat . Using Eq. (1), we can write
the flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric as
misner1973gravitation

$\mathrm{d}s^{2}=\mathrm{d}t^{2}-a^{2}(t)\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right)\,,$
(8)

where the Hubble parameter is $H=\dot{a}/a$ with over-dots referring to
derivatives with respect to physical time. Under this description, we notice
that $T=-6H^{2}$ and $B=-6\left(3H^{2}+\dot{H}\right)$. Therefore, the $f(T)$
gravity Friedmann equations are given by the following expressions
Bahamonde:2021gfp

$\displaystyle H^{2}+\frac{T}{3}\mathcal{F}_{T}-\frac{\mathcal{F}}{6}$
$\displaystyle=\frac{\kappa^{2}}{3}\rho\,,$ (9)
$\displaystyle\dot{H}\left(1-\mathcal{F}_{T}-2T\mathcal{F}_{TT}\right)$
$\displaystyle=-\frac{\kappa^{2}}{2}\left(\rho+p\right)\,,$ (10)

where $\mathcal{F}_{T}=\partial\mathcal{F}/\partial T$ and
$\mathcal{F}_{TT}=\partial^{2}\mathcal{F}/\partial T^{2}$ and we indicate the
energy density and pressure of the total matter sector by $\rho$ and $p$,
respectively. These will be the set of equations to analyse through a
selection of viable $f(T)$ cosmologies.

###  II.1 $f(T)$ scenarios: models and their cosmologies

In this section, we describe the models to be constrained through the
observational baselines. Notice that these models have been studied
extensively in the literature Bengochea:2008gz ; Xu:2018npu ; Briffa:2023ozo ;
Briffa:2021nxg ; Linder:2010py ; Nesseris:2013jea ; Bahamonde:2021gfp , and
references therein. However, a few of them have been successful to specific
baselines. We will discuss each of them along this argument.

  * •

Power Law Model – This model was first studied in Bengochea:2008gz  due to its
ability to reproduce the late-time cosmic acceleration behaviour in the $f(T)$
scheme. We can describe it through

$f_{1}(T)=p_{0}\left(-T\right)^{p_{1}}\,,$ (11)

where $p_{1}$ is a constant and $p_{0}$ can be computed through $p_{1}$ at
current times by using the
$p_{0}=(6H_{0}^{2})^{1-p_{1}}(1-\Omega_{m}-\Omega_{r})/(1-2p_{1})\,$. We can
write the Friedmann equation for this model as

$E^{2}(z)=\Omega_{m}\left(1+z\right)^{3}+\Omega_{r}\left(1+z\right)^{4}+\left(1-\Omega_{m}-\Omega_{r}\right)E^{2p_{1}}(z)\,,$
(12)

which recovers $\Lambda$CDM model for $p_{1}=0$. For $p_{1}=1$, the extra
component in the Friedmann equation gives a re-scaled gravitational constant
term in the density parameters, i.e. the GR limit. Also, we can obtain an
upper bound such that $p_{1}<1$ for an accelerating Universe.

In (Xu:2018npu, ) using BAO, SNIa, $H(z)$ and scalar perturbations for the
CMB, the authors found that $H_{0}=69.4\pm 0.8$ km s-1 Mpc-1,
$\Omega_{m}=0.298\pm 0.07$ and $p_{1}=-0.10^{+0.09}_{-0.07}$, with a Gaussian
prior on $H_{0}$ in concordance with a local high value. The result is a hint
of a slight deviation from the $\Lambda$CDM model for the $p_{1}$ parameter.
In (Briffa:2023ozo, ) the $f_{1}(T)$ model results in $H_{0}=69.90\pm 0.58$ km
s-1 Mpc-1, $\Omega_{m}=0.289^{+0.016}_{-0.018}$ and
$p_{1}=0.014^{+0.091}_{-0.125}$, that indicate a value of $p_{1}$ close to
zero consistent with $\Lambda$CDM. Those results are obtained using $H(z)$,
SNIa, BAO and Redshift Space Distortion (RSD) data. Using $H(z)$, SNIa and
BAO, in (Briffa:2021nxg, ) $H_{0}=67.1\pm 1.6$ km s-1 Mpc-1,
$\Omega_{m}=0.294\pm 0.015$ and $p_{1}=0.06\pm 0.13$ in agreement with
$p_{1}=0$. In (Sandoval-Orozco:2023pit, ) the combination between $H(z)$, BAO,
SNIa and Quasars results in $H_{0}=67.8^{+1.0}_{-0.9}$ km s-1 Mpc-1,
$\Omega_{m}=0.321\pm 0.012$ and $p_{1}=-0.08^{+0.14}_{-0.15}$ that agrees with
$p_{1}=0$.

  * •

Linder Model. – This model was proposed to produce late-time accelerated
expansion through Linder:2010py

$f_{2}(T)=p_{0}T_{0}\left(1-\text{Exp}\left[-p_{2}\sqrt{T/T_{0}}\right]\right)\,,$
(13)

where $p_{2}$ is a constant and $T_{0}=T|_{t=t_{0}}=-6H_{0}^{2}$. Similar to
the previous model, $p_{0}$ is a constant which can be computed from the
Friedman equation at present time through
$p_{0}=(1-\Omega_{m}-\Omega_{r})/[(1+p_{2})e^{-p_{2}}-1]\,.$ The corresponding
Friedmann equation for this model can be written as

$E^{2}\left(z\right)=\Omega_{m}\left(1+z\right)^{3}+\Omega_{r}\left(1+z\right)^{4}+\frac{1-\Omega_{m}-\Omega_{r}}{(p_{2}+1)e^{-p_{2}}-1}\left[\left(1+p_{2}E(z)\right)\text{Exp}\left[-p_{2}E(z)\right]-1\right]\,,$
(14)

which reduces to $\Lambda$CDM as $p_{2}\rightarrow+\infty$. To obtain
numerical stability, we consider the values for $1/p_{2}$, so
$1/p_{2}\rightarrow 0^{+}$.

In (Xu:2018npu, ) using the CMB, BAO, SNIa and $H(z)$ the Linder model results
in $H_{0}=69.6\pm 0.9$ km s-1 Mpc-1, $\Omega_{m}=0.296\pm 0.07$ and
$1/p_{2}=0.13^{+0.09}_{-0.11}$ that is not consistent with $1/p_{2}\to 0$
which returns the $\Lambda$CDM model, therefore reporting an interesting
deviation. Again, a Gaussian prior on $H_{0}$ is used, consistent with local
values. In (Briffa:2023ozo, ) the authors found that using $H(z)$, SNIa, BAO
and RSD $H_{0}=69.38^{+0.67}_{-0.68}$ km s-1 Mpc-1, $\Omega_{m}=0.282\pm
0.011$ and $1/p_{2}=0.275^{+0.083}_{-0.096}$ that seems to be consistent with
the cosmological standard model. Using $H(z)$, SNIa, BAO and Quasars, the
authors in (Sandoval-Orozco:2023pit, ) report that $H_{0}=68.0\pm 0.7$ km s-1
Mpc-1, $\Omega_{m}=0.300^{0.0008}_{-0.00001}$ and
$1/p_{2}=0.048^{+0.178}_{-0.046}$ which is in agreement with $1/p_{2}\to 0$
that is a confirmation of the standard cosmological model. Meanwhile in
(Briffa:2021nxg, ) $H_{0}=66.9^{+1.5}_{-1.6}$ km s-1 Mpc-1,
$\Omega_{m}=0.294\pm 0.016$ and $1/p_{2}=0.22^{+0.12}_{-0.15}$ using $H(z)$,
SNIa and BAO, resulting in a deviation from the $\Lambda$CDM represented in
the $1/p_{2}$ value.

  * •

Variant Linder Model – $f_{3}(T)$ Model. A variant version of the latter model
can be described by Nesseris:2013jea

$f_{3}(T)=p_{0}T_{0}\left(1-\text{Exp}\left[-p_{3}T/T_{0}\right]\right)\,,$
(15)

where $p_{3}$ is constant and $p_{0}$ in this case can be computed as
$p_{0}=(1-\Omega_{m}-\Omega_{r})/[(1+2p_{3})e^{-p_{3}}-1]$. The Friedmann
equation for this model can be written as

$E^{2}\left(z\right)=\Omega_{m}\left(1+z\right)^{3}+\Omega_{r}\left(1+z\right)^{4}+\frac{1-\Omega_{m}-\Omega_{r}}{(1+2p_{3})e^{-p_{3}}-1}\left[\left(1+2p_{3}E^{2}(z)\right)\text{Exp}\left[-p_{3}E^{2}(z)\right]-1\right]\,,$
(16)

which goes to $\Lambda$CDM as $p_{3}\rightarrow+\infty$ similar to $f_{2}$CDM.

In (Xu:2018npu, ) using BAO, CMB, SNIa and $H(z)$ the Variant Linder model
results in $H_{0}=69.5\pm 0.8$ km s-1 Mpc-1, $\Omega_{m}=0.297\pm 0.07$ and
$1/p_{3}=0.41\pm 0.31$ that is not consistent with $1/p_{3}\to 0$ and
therefore a confirmation of $\Lambda$CDM is not present in the analyses. The
results are presented using a Gaussian prior on $H_{0}$. Using BAO, SNIa,
$H(z)$ and RSD in (Briffa:2023ozo, ) is found that
$H_{0}=69.54^{+0.64}_{-0.66}$ km s-1 Mpc-1, $\Omega_{m}=0.282\pm 0.012$ and
$1/p_{3}=0.197^{+0.038}_{-0.092}$ which shows a slight deviation from the
expected $1/p_{3}\to 0$. Previously, in (Briffa:2021nxg, ) the combination of
$H(z)$, SNIa and BAO reported $H_{0}=67.35^{+0.94}_{-0.97}$ km s-1 Mpc-1,
$\Omega_{m}=0.292^{+0.012}_{-0.014}$ and $1/p_{3}=0.043^{+0.101}_{-0.026}$
that again is a slight deviation from the $\Lambda$CDM base model. In
(Sandoval-Orozco:2023pit, ) is presented that $H_{0}=67.2^{+0.8}_{-0.7}$ km
s-1 Mpc-1, $\Omega_{m}=0.317^{+0.013}_{-0.012}$ and
$1/p_{3}=0.071^{0.097}_{-0.070}$ using SNIa, $H(z)$, BAO and a Quasar Sample.
This result is in the range of the expected $1/p_{3}\to 0$.

##  III Observational data treatment

In this section, we will evaluate each $f(T)$ cosmological model by employing
the constraining parameters method through Markov chain Monte Carlo (MCMC)
analysis, utilizing the publicly available code 222emcee.readthedocs.io for
our cosmology. Our analysis will include a baseline dataset for reference,
comprising $H(z)$ measurements, Supernovae Type Ia (SNIa) data, BAO
measurements, and two sets of quasar data utilizing ultraviolet, X-ray, and
optical plane techniques. Expanding our analysis, we will incorporate
additional observations: HII galaxy Hubble, Growth rate data and CMB distance
priors. These observations will provide further insights into our cosmological
models.

###  III.1 Baseline (low-z) local observations

  * •

$H(z)$ measurements. We considered the observational measurements of the
Hubble parameter through the Cosmic Chronometers (CC) in which the system of
galaxies evolves in a cluster without interacting so small differences in
redshift can be measured along with age estimates. The galactic spectra are
used to obtain $\mathrm{d}t/\mathrm{d}z$ using redshift and the age difference
(Moresco:2016mzx, ). The resulting sample contains 31 data points up to $z\sim
2$.

The corresponding $\chi^{2}_{\mathrm{CC}}$ is given by:

$\chi^{2}_{\mathrm{CC}}=\Delta H(z_{i},\Theta)^{T}C^{-1}_{H(z)}\Delta
H(z_{i},\Theta),$ (17)

where $\Delta H(z_{i},\Theta)=H(z_{i},\Theta)-H_{obs}(z_{i})$ and
$C^{-1}_{\mathrm{CC}}$ is the covariance matrix generated given in
(Moresco:2020fbm, ).

  * •

Pantheon+ SNIa dataset. We use the 1701 data points provided by the Pantheon+
sample Scolnic:2021amr  that measures the apparent distance for 1550 distinct
Supernovae Ia (SNIa) events in a redshift range $0.01<z<2.3$. The dataset for
the Pantheon+ sample provides SNIa magnitudes corrected for the stretch and
colour effects along with the maximum brightness, the mass of the host galaxy,
and sky position bias, so to obtain a cosmological useful quantity we need to
calculate the distance modulus $\mu=m-M$. The Pantheon+ dataset also includes
SH0ES measurements of Cepheid distances to determine $H_{0}$ and $M$ at the
same time without the necessity of including a prior in the parameters
Sandoval-Orozco:2023pit . Hereafter, this data set will be referred to as PN+.

Thus, the $\chi^{2}_{\text{SN}}$ for the Pantheon+ sample is given by

$\chi^{2}_{\text{PN${}^{+}$}}=\Delta\mu(z_{i},\Theta)^{T}C^{-1}_{\text{PN${}^{+}$}}\Delta\mu(z_{i},\Theta)+\ln\quantity(\frac{S}{2\pi})-\frac{k^{2}(\Theta)}{S},$
(18)

where $C^{-1}_{\text{PN${}^{+}$}}$ is the total covariance matrix for the
data, $S$ is the sum of all components of the inverse of the matrix and
$k(\Theta)=\Delta\mu(z_{i},\Theta)^{T}C^{-1}_{\text{PN${}^{+}$}}$. To use
SH0ES Cepheids host distances, the SNIa, and the residuals will be calculated
as (Brout:2022vxf, ):

$\Delta\mu(z_{i},\Theta)=\left\\{\begin{matrix}\mu_{\text{obs}}(z_{i})-\mu_{i}^{\text{Cepheids}}&i\in\text{Cepheid
hosts,}\\\ \mu_{\text{obs}}(z_{i})-\mu(z_{i},\Theta)&\text{ in other
case.}\end{matrix}\right.$ (19)

For this cases, the distance modulus $\mu(z)$ can be calculated as:

$\mu(z_{i},\Theta)=5\log\quantity[D_{L}(z_{i},\Theta)]+M,$ (20)

and where $D_{L}(z_{i},\Theta)$ is the luminosity distance given as:

$D_{L}(z_{i},\Theta)=c(1+z_{i})\int_{0}^{z_{i}}\frac{dz^{\prime}}{H(z^{\prime},\Theta)},$
(21)

where $c$ is the speed of light and $H(z_{i},\Theta)$ is the Hubble parameter.

  * •

Baryon acoustic oscillation (BAO) data. We will consider a joint baryon
acoustic oscillation (BAO) data set composed of several independent data
points from different sources. This BAO data set includes measurements from
the SDSS Main Galaxy Sample at $z_{\mathrm{eff}}=0.15$ Ross:2014qpa , the six-
degree Field Galaxy Survey at $z_{\mathrm{eff}}=0.106$ 2011MNRAS.416.3017B ,
and the BOSS DR11 quasar Lyman-alpha measurement at $z_{\mathrm{eff}}=2.4$
Bourboux:2017cbm . We will also use the angular diameter distances and $H(z)$
measurements of the SDSS-IV eBOSS DR14 quasar survey at
$z_{\mathrm{eff}}=\\{0.98,1.23,1.52,1.94\\}$ Zhao:2018gvb , along with the
SDSS-III BOSS DR12 consensus BAO measurements of the Hubble parameter and the
corresponding comoving angular diameter distances at
$z_{\mathrm{eff}}=\\{0.38,0.51,0.61\\}$ Alam:2016hwk . For these two BAO data
sets, we consider the full covariance matrix in our MCMC analyses as there is
an overlap in the redshift slices.

To use the aforementioned datasets we need to compute additional distances as
the angular diameter distance:

$D_{V}(z)=\left[(1+z)^{2}D_{A}(z)^{2}\frac{cz}{H(z)}\right]^{1/3},$ (22)

where $D_{A}(z)=(1+z)^{-2}D_{L}(z)$ is the angular diameter distance. An
additional quantity is necessary to perform the BAO analyses as it appears
$r_{s,\mathrm{fid}}$ and $r_{s}(z_{d})=r_{d}$. The value of the first one will
be fixed according to the mentioned references as the different results
correspond to a specific value. The $r_{d}$ value will be a free parameter in
the MCMC analysis.

The corresponding $\chi^{2}$ for the BAO data is calculated using

$\chi^{2}_{\text{BAO}}(\Theta)=\Delta
G(z_{i},\Theta)^{T}C_{\text{BAO}}^{-1}\Delta G(z_{i},\Theta)$ (23)

where $\Delta G(z_{i},\Theta)=G(z_{i},\Theta)-G_{\text{obs}}(z_{i})$ using
$G(z_{i},\Theta)$ the corresponding vector for each of the observations used.
$C_{\text{BAO}}$ is the covariance matrix of all the considered BAO
observations.

###  III.2 High-z Hubble diagram

  * •

HIIG and GHIIR Hubble Diagram (High-z HIIG)

The HII Galaxies or Giant Extragalactic HII regions GHIIR (that will be
referred to as HII from now on) are known to satisfy a relation between the
Luminosity in the Balmer Line of Hydrogen $L\mathrm{H}\beta$ and the velocity
dispersion $\sigma$ (Gonzalez-Moran:2021drc, ). This relation is used as a
cosmological distance indicator. HII regions are compact mass systems whose
luminosity is composed almost entirely from a young burst of stellar formation
with strong narrow emission lines that can be studied in detail
(Chavez:2014ria, ). As we mentioned in the Introduction, the $L-\sigma$
relation can be written

$\log{L}=\alpha\log{\sigma}+\beta,$ (24)

that will be used to calculate the observed distance modulus, which can be
expressed from the complete set of observables as (Gonzalez-Moran:2019uij, )

$\mu_{i,\mathrm{obs}}=2.5(\alpha+\beta\log{\sigma}-\log{f}-40.08),$ (25)

where $\sigma$ is the mentioned velocity dispersion of the object. $\alpha$
and $\beta$ are the intercept and slope of the $L-\sigma$ relation and
$\log{f}$ is the measured flux around the continuum. The data points are
provided with the flux, the Luminosity $L$ and the dispersion $\sigma$.

The corresponding $\chi^{2}$ for the HII sample is calculated using:

$\chi^{2}_{\mathrm{HII}}=-\frac{1}{2}\sum_{i}\frac{\left[\mu_{i,\mathrm{obs}}-\mu(z_{i},\Theta)\right]^{2}}{\varepsilon^{2}},$
(26)

where $\mu(z_{i},\Theta)$ is the theoretical distance modulus. $\varepsilon$
is the error associated with the measurements that include the statistical
uncertainties for every mentioned parameter

$\varepsilon^{2}=6.25(\varepsilon^{2}_{\log{f}}+\beta^{2}\varepsilon^{2}_{\log{\sigma}}+\varepsilon^{2}_{\beta}\log{\sigma^{2}}+\varepsilon_{\alpha}^{2}),$
(27)

Although $\alpha$ and $\beta$ are free nuisance parameters, the variation
between the numerical values of the parameters for the different models is too
small to be considered in this analysis. We will employ $\alpha=33.268\pm
0.083$ and $\beta=5.022\pm 0.058$ consistent with the values reported in
(Gonzalez-Moran:2019uij, ; Gonzalez-Moran:2021drc, ; Yang:2024epu, ;
Cao:2023eja, )

A sample of 181 points is used in this analysis from (Gonzalez-Moran:2021drc,
). Due to the bright nature of the star formation burst, HII composes a
promising dataset as they reach $z\sim 4$ and could help to complete the
Hubble diagram at higher redshifts where SNIa is lacking data points.

  * •

Growth rate data (RSD). The growth rate catalog considered in this work is
reported in Alestas:2022gcg . The catalog consists of measurements of the
growth of cosmic structure, so-called Redshift Space Distortion (RSD). In
standard scenarios, the peculiar velocity of galaxies causes cosmic high-
density regions larger in the line-of-sight direction at small $z$, given by
consequence maps of galaxies where distances are measured from spectroscopic
redshifts show anisotropic deviations from the true galaxy distribution. From
these results, this baseline can be important to test modified or extended
theories of gravity. To derive the growth rate $f(z)$, we can use RSD
cosmological probes to constrain cosmological models Lambiase:2018ows ;
Gonzalez:2016lur ; Gupta:2011kw . However, instead of deriving the growth rate
directly, Large Scale Surveys (LSS) report the density-weighted growth rate,
$f\sigma_{8}\equiv f(z)\sigma_{8}(z)$. This RSD baseline can be used to
constrain cosmological parameters, in particular $\sigma_{8,0}$, e.g. the
$\chi^{2}_{\mathrm{min}}$, which is given by

$\chi^{2}_{\mathrm{RSD}}=\Delta Q(z_{i},\Theta)^{T}C_{\text{RSD}}^{-1}\Delta
Q(z_{i},\Theta)$ (28)

where
$Q(z_{i},\Theta)=(f\sigma_{8}(z_{i},\Theta)_{\mathrm{theo}}-f\sigma{{}_{8}{}_{\mathrm{obs}}}(z_{i}))$
and $C_{\text{RSD}}^{-1}$ is the inverse covariance matrix assumed to be a
diagonal matrix. The total covariance matrix is given by

$C_{\text{RSD}}=\begin{pmatrix}\sigma_{1}^{2}&0&0&\dots\\\
0&C_{\text{WiggleZ}}&0&\dots\\\ 0&0&\dots&\sigma_{N}^{2}\\\ \end{pmatrix}\,.$
(29)

Notice that $f\sigma_{8}(z_{i},\Theta)_{\mathrm{obs}}$ is derived from the
compiled observational baseline described above.

  * •

CMB (Planck 2018) distance priors. The additional sample used for the high
redshift universe will be the CMB distance priors (referred to only as CMB in
the text). This is a dataset created to provide information on the CMB power
spectrum without the calculation of the complete set of perturbations
Zhai:2019nad ; Chen:2018dbv .

The presented quantities for this dataset are the shift parameter $R$ that
measures the peak spacing of the temperature in the power spectrum, the
acoustic scale $l_{\mathrm{A}}$ that measures the temperature in the
transverse direction and the combination $\Omega_{b}h^{2}$ Chen:2018dbv .
Those quantities are known as the distance priors and are going to be
calculated as:

$l_{\mathrm{A}}=(1+z_{*})\frac{\pi D_{A}(z_{*})}{r_{s}(z_{*})},$ (30)

where $z_{d}$ is the redshift at photon the decoupling epoch. This redshift
value will be fixed to $z=1089$ in concordance with (Planck:2018vyg, ).
Additionally:

$R=\frac{(1+z_{*})D_{A}(z_{*})\sqrt{\Omega_{m}H_{0}^{2}}}{c},$ (31)

with $c$ the light speed. Similar to the analyses performed in BAO,
$r_{s}(z_{*})=r_{*}$ will be used as an additional free parameter. We will use
the distance priors as $R=1.7502\pm 0.0046,l_{\mathrm{A}}=301.471\pm
0.090,\Omega_{b}h^{2}=0.02236\pm 0.00015$ with the correlation matrix $c_{ij}$
given as

$c_{ij}=\begin{pmatrix}1&0.46&-0.66\\\ 0.46&1&-0.33\\\
-0.66&-0.34&1\end{pmatrix},$ (32)

that need to be transformed to a covariance matrix $C_{ij}$ using
$C_{ij}=c_{ij}\sigma_{i}\sigma_{j}$ with the associated errors for every
measurement. So, the $\chi^{2}$ used for the CMB distance priors is

$\chi^{2}_{\mathrm{CMB}}=\Delta V(\Theta)^{T}C_{\text{CMB}}^{-1}\Delta
V(\Theta),$ (33)

where
$V(\Theta)=\left[R(\Theta),l_{\mathrm{A}}(\Theta),\Omega_{b}h^{2}\right]$ and
$V=(R,l_{\mathrm{A}},\Omega_{b}h^{2})$ with $\Delta V=V(\Theta)-V$. The
normalized Hubble parameter also appears as $h=H_{0}/100$ km/s/Mpc.

##  IV Constraint analyses results

This section presents the results for the different models under consideration
in this analysis. We compare these results with our baseline $\Lambda$CDM
model analysis, which we present first. We consider several combinations of
data sets that are chosen to expose important nuances between the behaviours
of the models against observational constraints.

  1. (a)

Constraints for the $\Lambda$CDM model. As a baseline, we consider
$\Lambda$CDM cosmology with posterior outputs shown in Fig. 1 with constraints
shown in Table 1. Immediately, the constraints appear to be strong in that
they have robust uncertainties. Also, as is somewhat expected, the PN+ data
set appears to push the value of $H_{0}$ up while the CMB shift data gives a
comparatively lower value of the Hubble constant. The differences are smaller
when for the matter density parameter $\Omega_{m}$ and essentially nonexistent
for the $\sigma_{8}$ parameter. One also observes that the absolute magnitude
$M$ remains largely consistent throughout the different data sets. While
offering a lower number of data points with less certainty, the HIIG data set
appears to be largely consistent with the PN+ when considering the value of
the Hubble constant. On the other hand, the matter density parameter suffers a
significant shift upward in value albeit with a lower level of certainty. This
is likely a result of the higher redshift ratio of the data points where the
projected Hubble constant value is consistent with PN+ but where more
substantial information is obtained on the matter density parameter.

Dataset |  $H_{0}$ [km/s/Mpc] | $\Omega_{m}$ | $M$ |  $r_{d}$ [Mpc] |  $r_{\ast}$ [Mpc] | $\sigma_{8}$ |  $S_{8}$ (derived)
---|---|---|---|---|---|---|---
CC+PN++HII | $71.72\pm 1.6$ | $0.325^{+0.032}_{-0.030}$ | $-19.292^{+0.047}_{-0.046}$ | – | – | – | –
CC+PN++RSD | $73.0^{+1.8}_{-2.1}$ | $0.293^{+0.026}_{-0.025}$ | $-19.261^{+0.053}_{-0.063}$ | – | – | $0.821\pm 0.060$ | $0.811^{+0.077}_{-0.075}$
CC+PN++HIIG+RSD | $72.3^{+1.6}_{-1.6}$ | $0.290^{+0.023}_{-0.023}$ | $-19.287^{+0.047}_{-0.048}$ | – | – | $0.817^{+0.057}_{-0.058}$ | $0.803^{+0.076}_{-0.074}$
CMB+BAO+RSD | $67.90^{+0.46}_{-0.45}$ | $0.290^{+0.010}_{-0.010}$ | – | $150.1\pm 3.1$ | $148.3\pm 2.3$ | $0.818^{+0.053}_{-0.054}$ | $0.805^{+0.057}_{-0.058}$
All | $68.17\pm 0.45$ | $0.293^{+0.011}_{-0.010}$ | $-19.410^{+0.017}_{-0.016}$ | $149.5^{+3.1}_{-3.0}$ | $146.8^{+2.1}_{-2.2}$ | $0.822^{+0.053}_{-0.056}$ | $0.811^{+0.057}_{-0.060}$

Table 1: Constraints for the $\Lambda$CDM model that include the parameters
$H_{0}$, $\Omega_{m,0}$ and $\sigma_{8}$. The $S_{8}$ parameter (derived) and
the nuisance parameter $M$ were computed for each baseline. The full baseline
(All) includes CC+PN++HIIG+RSD+BAO+CMB.

  2. (b)

Constraints for the Power Law Model – $f_{1}(T)$ model. The power-law model,
or $f_{1}$CDM, is first considered with the constraint table shown in Table 2
while the posterior contours are shown in Fig. 1. As in the $\Lambda$CDM case,
the same division arises between data set combinations involving the PN+
sample and those that do not. As in the $\Lambda$CDM scenario, the Hubble
constant is increased for the PN+ sample while the HIIG data set appears to be
more consistent with PN+. On the other hand, the HIIG sample allows for a
lower value of the matter density parameter as compared with the other data
sets. Moreover, there is a clear anti-correlation with the model parameter
$p_{1}$. Indeed, the $p_{1}$ parameter appears to be counter-balancing the
$\Omega_{m}$ parameter. The constraints on $p_{1}$ are generally consistent
with previous studies Briffa:2021nxg ; Briffa:2023ozo  with a reasonable
consistency with its $\Lambda$CDM limit. Another important point to take away
from these constraints is how consistent they are at the level of
$\sigma_{8}$, and thus with large-scale structure constraints.

Dataset |  $H_{0}$ [km/s/Mpc] | $\Omega_{m}$ | $p_{1}$ | $M$ |  $r_{d}$[Mpc] |  $r_{\ast}$ [Mpc] | $\sigma_{8}$ |  $S_{8}$ (derived)
---|---|---|---|---|---|---|---|---
CC+PN++HII | $71.8\pm 1.6$ | $0.267^{+0.073}_{-0.062}$ | $0.31^{+0.25}_{-0.34}$ | $-19.282^{+0.047}_{-0.048}$ | – | – | – | –
CC+PN++RSD | $72.9^{+2.0}_{-2.1}$ | $0.284^{+0.033}_{-0.032}$ | $0.06^{+0.14}_{-0.16}$ | $-19.266^{+0.056}_{-0.059}$ | – | – | $0.824^{+0.061}_{-0.059}$ | $0.802^{+0.085}_{-0.080}$
CC+PN++HIIG+RSD | $72.3^{+1.6}_{-1.6}$ | $0.280^{+0.033}_{-0.032}$ | $0.06^{+0.14}_{-0.16}$ | $-19.286^{+0.047}_{-0.047}$ | – | – | $0.819^{+0.059}_{-0.058}$ | $0.791^{+0.084}_{-0.079}$
CMB+BAO+RSD | $67.82^{+0.49}_{-0.48}$ | $0.294^{+0.012}_{-0.012}$ | $0.029^{+0.082}_{-0.083}$ | – | $150.0^{+3.1}_{-3.0}$ | $148.2^{+2.3}_{-2.2}$ | $0.826^{+0.055}_{-0.055}$ | $0.817^{+0.061}_{-0.059}$
All | $68.05^{+0.47}_{-0.51}$ | $0.297\pm 0.011$ | $0.059^{+0.069}_{-0.071}$ | $-19.408^{+0.017}_{-0.019}$ | $149.3^{+3.0}_{-3.0}$ | $146.7^{+2.1}_{-2.3}$ | $0.834^{+0.057}_{-0.057}$ | $0.830^{+0.062}_{-0.061}$

Table 2: Constraints for the $f_{1}(T)$ model that include the parameters
$H_{0}$, $\Omega_{m,0}$, $p_{1}$ and $\sigma_{8}$. The $S_{8}$ parameter
(derived) and the nuisance parameter $M$ were computed for each baseline. The
full baseline (All) includes CC+PN++HIIG+RSD+BAO+CMB.

  3. (c)

Constraints for the Linder Model – $f_{2}(T)$ model. The posterior constraints
for the $f_{2}$CDM model are shown in Table 3 while their contour fits are in
Fig. 1. A lot of the nuances that the $f_{1}$CDM model exhibits are already
prevalent in this scenario. On the other hand, the HIIG sample produces a much
wider range of admissible values for the Hubble constant, while the value of
the matter density parameter is noticeably reduced, while still being
consistent with the expected value. Saying that the model parameter $p_{2}$
remains consistently within the positive range as in previous works
Briffa:2021nxg ; Briffa:2023ozo  which also acts as a consistent check on the
rest of the analysis. In the perturbative sector, the fits for the
$\sigma_{8}$ parameter are similarly tight and well within the expected range
from the plethora of survey releases.

Dataset |  $H_{0}$ [km/s/Mpc] | $\Omega_{m}$ | $1/p_{2}$ | $M$ |  $r_{d}$[Mpc] |  $r_{s}$[Mpc] | $\sigma_{8}$ |  $S_{8}$ (derived)
---|---|---|---|---|---|---|---|---
CC+PN++HIIG | $71.7^{+1.8}_{-1.7}$ | $0.233^{+0.082}_{-0.071}$ | $0.63^{+0.39}_{-0.39}$ | $-19.280^{+0.052}_{-0.049}$ | – | – | – | –
CC+PN+ \+ RSD | $72.6^{+1.9}_{-1.9}$ | $0.219^{+0.050}_{-0.048}$ | $0.65^{+0.28}_{-0.26}$ | $-19.258^{+0.056}_{-0.056}$ | – | – | $0.796^{+0.059}_{-0.057}$ | $0.680^{+0.10}_{-0.094}$
CC+PN++HIIG+RSD | $72.0^{+1.6}_{-1.6}$ | $0.213^{+0.050}_{-0.048}$ | $0.66^{+0.28}_{-0.28}$ | $-19.276^{+0.047}_{-0.050}$ | – | – | $0.798^{+0.065}_{-0.074}$ | $0.67^{+0.10}_{-0.11}$
CMB + BAO + RSD | $67.73^{+0.52}_{-0.51}$ | $0.294^{+0.012}_{-0.012}$ | $0.20^{+0.12}_{-0.20}$ | – | $149.8^{+3.2}_{-3.1}$ | $148.2^{+2.2}_{-2.2}$ | $0.822^{+0.055}_{-0.056}$ | $0.814^{+0.061}_{-0.062}$
All | $67.96^{+0.52}_{-0.62}$ | $0.295^{+0.013}_{-0.012}$ | $0.263^{+0.078}_{-0.076}$ | $-19.405^{+0.019}_{-0.023}$ | $148.9^{+3.0}_{-2.9}$ | $147.2^{+2.3}_{-2.3}$ | $0.827^{+0.055}_{-0.060}$ | $0.819^{+0.060}_{-0.065}$

Table 3: Constraints for the $f_{2}(T)$ model that include the parameters
$H_{0}$, $\Omega_{m,0}$ and $\sigma_{8}$. The $S_{8}$ parameter (derived) and
the nuisance parameter $M$ were computed for each baseline. The full baseline
(All) includes CC+PN++HIIG+RSD+BAO+CMB.

  4. (d)

Constraints for the Variant Linder Model – $f_{3}(T)$ model. The posteriors
and constraint table of this model are shown in Fig. 1 and Table 4. This is
different to the Linder model in that it has a quadratic Hubble dependency in
the exponents. In this, the model parameter, which appears in the exponent
retains a preference for positive values as in the regular Linder model,
albeit with smaller but more confident values. This point is particularly true
for the instances where RSD data is used, while the HIIG sample shows a
relatively strong preference for a nonzero value for this parameter. On the
other hand, and as with the other two models, the CMB, BAO and RSD data push
the Hubble constant to lower values while the PN+ data sample prefers higher
values of $H_{0}$. As for the matter density parameter, this can tolerate a
lower value which may be more compatible with most independent literature
values.

Dataset |  $H_{0}$ [km/s/Mpc] | $\Omega_{m}$ | $1/p_{3}$ | $M$ |  $r_{d}$[Mpc] |  $r_{*}$[Mpc] | $\sigma_{8}$ |  $S_{8}$ (derived)
---|---|---|---|---|---|---|---|---
CC+PN++HII | $71.5^{+2.0}_{-2.0}$ | $0.278^{+0.055}_{-0.057}$ | $0.24^{+0.11}_{-0.13}$ | $-19.278^{+0.054}_{-0.049}$ | – | – | – | –
CC+PN++RSD | $72.0^{+2.1}_{-2.5}$ | $0.261^{+0.032}_{-0.046}$ | $0.264^{+0.12}_{-0.077}$ | $-19.251^{+0.091}_{-0.083}$ | – | – | $0.780^{+0.062}_{-0.057}$ | $0.728^{+0.093}_{-0.092}$
CC+PN++HIIG+RSD | $71.6^{+1.7}_{-1.6}$ | $0.262^{+0.031}_{-0.036}$ | $0.254^{+0.045}_{-0.043}$ | $-19.274^{+0.049}_{-0.048}$ | – | – | $0.779^{+0.064}_{-0.062}$ | $0.728^{+0.092}_{-0.097}$
CMB + BAO + RSD | $67.72^{+0.52}_{-0.52}$ | $0.294\pm 0.012$ | $0.156^{+0.082}_{-0.156}$ | – | $149.6\pm 3.2$ | $148.2\pm 2.2$ | $0.819^{+0.055}_{-0.053}$ | $0.810^{+0.060}_{-0.058}$
All | $67.86^{+0.24}_{-0.26}$ | $0.295\pm 0.005$ | $0.210\pm 0.017$ | $-19.406\pm 0.011$ | $150.0\pm 2.5$ | $147.3\pm 1.3$ | $0.817\pm 0.027$ | $0.809\pm 0.028$

Table 4: Constraints for the $f_{3}(T)$ model that include the parameters
$H_{0}$, $\Omega_{m,0}$ and $\sigma_{8}$. The $S_{8}$ parameter (derived) and
the nuisance parameter $M$ were computed for each baseline. The full baseline
(All) includes CC+PN++HIIG+RSD+BAO+CMB.

Figure 1: Confidence contours (C.L) and posterior distributions for: Top left:
the $\Lambda$CDM model parameters, including $H_{0}$, $\Omega_{m,0}$, and
$\sigma_{8}$. In cases where the RSD data is incorporated (dark blue, green,
grey, light blue contours), the $\sigma_{8,0}$ parameter is also displayed.
Top right: for the $f_{1}(T)$ model parameters, including $H_{0}$,
$\Omega_{m,0}$, $p_{1}$, and $\sigma_{8}$. In cases where the RSD data is
incorporated (dark blue, green, grey, light blue contours), the $\sigma_{8,0}$
parameter is also displayed. Bottom left: for the $f_{2}(T)$ model parameters,
including $H_{0}$, $\Omega_{m,0}$, $1/p_{2}$, and $\sigma_{8}$. In cases where
the RSD data is incorporated (dark blue, green, grey, light blue contours),
the $\sigma_{8,0}$ parameter is also displayed. Bottom right: for the
$f_{3}(T)$ model parameters, including $H_{0}$, $\Omega_{m,0}$ and $1/p_{3}$.
In cases where the RSD data is incorporated (dark blue, green, grey, light
blue contours), the $\sigma_{8,0}$ parameter is also displayed.

##  V Conclusions

In this study, we have presented the behaviour of parameters across various
combinations of datasets. We assessed three prominent $f(T)$ gravity models,
evaluating their performance against five distinct combinations of
observational datasets. Our combinations included CC+PN+, coupled with either
HIIG or RSD datasets, as well as a combined set incorporating all four
datasets. Additionally, we explored a dataset combination integrating CMB
distance priors and BAO data with RSD data. Finally, we amalgamated all
datasets for a holistic analysis.

The inclusion of the RSD dataset proved particularly insightful, as it offers
sensitivity to the growth of structure formation, thereby aiding in
constraining the $\sigma_{8,0}$ of the models under examination. CMB data was
conveniently incorporated to enrich cosmological constraints, and their
amalgamation with other cosmic measurements—such as PN+, BAO, and Hubble
parameter data—further enhanced our ability to constrain cosmological
parameters and alleviate degeneracies between them. Moreover, the utilization
of HIIG galaxies, providing luminosity distance measurements, offered further
valuable constraints on parameters.

The three primary models under examination are the Power Law model, denoted as
$f_{1}(T)$, the Linder Model, referred to as $f_{2}(T)$, and a variant of the
Linder model, labelled as $f_{3}(T)$. For reference, we also provide the
$\Lambda$CDM constraints for each dataset combination considered. The results
of the constrained parameters are summarized in Fig. 2. Notably, when all
datasets are included, the uncertainty bands for $H_{0}$ and the matter
density parameter reduce drastically, with the values generally decreasing as
well. Additionally, we include values for the derived parameter $S_{8}$, which
remains relatively consistent across various datasets and the three different
model types.

Our study provides valuable insights into the behaviour of these $f(T)$
models, particularly in understanding their response to different combinations
of observational baseline at higher redshift. It sheds light on their
cosmological viability in TG. Moving forward, the intention is to use these
observations and extend the analysis to encompass the full comprehensive CMB
power spectra, along with other early Universe datasets. This broader approach
will allow for a more thorough assessment of the competitiveness of these
models and their ability to accurately describe the Universe’s evolution.

Figure 2: Whisker plot containing the results from all the combinations of
baselines and the different $f(T)$ models. The bands in pink and blue colour
denote Planck 2018 and SH0ES $H_{0}$ values respectively.

###### Acknowledgements.

RS is supported by the CONACyT National Grant. CE-R is supported by the
CONACyT Network Project No. 376127 and acknowledges the Royal Astronomical
Society as Fellow FRAS 10147. This research has been carried out using
computational facilities procured through the Cosmostatistics National Group
ICN UNAM project. 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). JLS would also like to acknowledge funding from “The
Malta Council for Science and Technology” as part of the REP-2023-019
(CosmoLearn) Project.

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